7142-21 T IRIA Stote-of- he~Art Report BAND-MODEL METHODS FOR COMPUTIN ATMOSPHERIC SLANT-PA TH MOLECULAR ABSORPTION DAVID ANDING February 1967 This document-is dsubjen t to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of the Office o! Naval Research (Code 421), Washington, D. C. 20360....*r Aisd O t st A -rbto *Mk g'e't T( tresrIT e OF SCit NCt t ANO TECH(OLOy' I H! U N Vf ~A $ A b t o F M t Mk C t G A N Ann Arbof, MichEgan

-I-WILLOW RUN LA BORATORIES —--- ACKNOWLEISC MENTS The author wishes tc acknowledge the assistance of Harvey M. Rose, who provided first-draft versions of sections 4, 5, and an and who contributed largeiy to the many computational and plotting labors encountered in the writing o! this report. I am particularly indebted to George Oppel for his technical advice and suggestions, and for providing the necessary information for section 2.3. The author is also indebted to William L. Wolfe of the Radiation Center of Honeywell, Lnc., and Thomas Lmperis of the Willow Run Laboratories for their advice and comments.

... -- WILLOW RUN LABORATORIES PREFACE The IRIA (Infrared Information and Analysis) Center, a part of the Willow Run Laboratories of The University of Michigan's Institute of Science and Technology, is responsible for collecting, analyzing, and disseminating to authorized recipients all information concerning military infrared research and development. To this end, IRIA prepares annotated bibliographies, request subject bibliographles, stateof-the-art reports, The Proceedings of IRIS, and other special publications. IRIA also sponsors sympositums anr maintains a staff of specialists in infrared technology to assist and advise visitors. The consulting services offered by IRIA are available to qualfed requesters without charge. For its first ten years IRIA was supported by a trl-service contract, NOnr 1224(!2), and reports were Issued under Project 2389. Effective March 1965, IRIA is being supported by a new trt-Pervtce contract, NOnr 1224(52), and reports wili oe issued under Project 7142. The (ontract is administered by the Office of Naval Research, Physics Branch; a steering committee consisting of representatives of the three military services assists in the technical direction of the work. The quarterly IRIA Annotated Bibliographies and The Proceedins of IRIS are sent' ) all those on t': IRIS distribution list. In addition to receiving these publications, those on the IRIS distribution list are notified of all other IRIA publicaticois. To be added to the list send a request via appropriate security channels to Mr. T. B. Dowd, Office of Naval eresearch, 495 Summer St., Boston, Massachusetts 02210.

-- - ----- WILLOW RUN LABORATORIES ABSTRACT The general transmisslvity equation for computing slant-path molecular absorption spectra is developed and two methods for evaluating this equation, the direct integration and that which assumes a model of the band structure, are discussed. Five band models are discussed and twelve methods for computing molecular absorption based on these band models Pre presented. Spectra comruted by band-mtodel methods are compared with spectra calculated by direct integration of the general transmissivtty equation and with open-air field measurements of absorption spectra. Conclusions concerning the capability of Land-model methods for predicting slant-path absorption spectra are stated and recommendations for future research are outlined. A summary of open-air field measurements of absorption spectra and laboratory measurements of absorption spectra for homogec.eous paths is presented and-a computer program for comiuting the equivalent sea-level path. the Curtis-Godson equivalent pressure, and the absorber concentration for atmospheric slant paths for any model atmosphere is given in appendix I.

-....- WILLOW RUN LABORATORIES CONTENTS Acknowledgments............................... Preface..........................ll Abstract................. v. List of Figures................ x........... x List of Tables................................ xii 1. Introduction,............................. 1 2. General Theory of Atmospheric Absorption............. 4 2.1. General Fxuation for Slant-Path Absorption 6 2.2. Band Models for Computing Spectral Absorption 10 2.2.1. Introduction to the Band-Model Concept 10 22.2. A Single Lorentz Line 11 2.2.3. Elsasser Model 14 2,2.4. Statistical Band Model 20 2,2.5, Random-Elsasser Band Model 26 2,2.6. Quasi-Random Model 27 2.2.7. King Model 30 2.2.8. Band-Model Limitations 33 2.2.9. Temperature and Frequency Dependence of Band-Model Parameters 34 2.3. Determination of Frequency-Dependent Parameters 38 2.4. Conversion of Slant- ath to Equivalent-Palh Parameters 44 2.4.1. Weak-Line Nonove'2pping Condition 45 2.4.2. Strong-Line Equivalent Pressure P 46 2.4,3. Equivalent Sea-Level Absorber Concentration V* 48 2.4.4. General Comments on W, P, and W 49 2.4.5. A Generalized Calculation of W, P, and W 50 2.5. Summary 52 3. Methods for Computing Mo. cular Absorption Spectra Based upon Band Models.............................. 54 3.1. Introduction 54 3.2. Method of W. M. Elsasser 54 3.2.1. Transmissivity Function: Carbon Dioxide 55 3.2.2. Transmissivity Function: C one 59 3.2.3. Transmissivity Function: Water Vapor 64 3.3. Method of T. L. Atsnuler 72 3,3.1. Transmissivity Function: Water Vapor 72 3.3.2. Transmissivity Function: -Carbon Dioxide 76 3.3.3. Transmissivity Function: Ozone and Nitrous Oxide 82 3.3.4. Procedure for Calculating Absorption 82 3.4. Method of A. Zachor 87 3.4.1. Ozone 88 3.4.2. Carbon Dioxide 88 3.5, Method of J. Howard,, D. Burch, and D. Williams 97 3.6. Method of G. Lindquist 98 3.7. Method of G. E. Oppel 104 3.7.1. C'arbon Dioxide 104 3.7.2. Water Vapor 108 3.7.3. Minor Constituents (N20, CH4, and CO) 119 vii

-- WILLOW RUN LABORATORIES 3.8. Method of W. R. Bradford 119 3.9. Method of A, E. S. Green, C. S. Lindenmeyer, and M. Griggs 125 3.10. Method of R. 0. Carpenter 127 3.11. Method of V. R. Stull, P. J. Wyatt, and G. N. Plass 137 3.12. Method of A. Thomson and M. Downing 142 3.13. Method of T. Elder and J. Strong 143 3.14. Summary. 146 4. Distribution of Atmospheric Molecular Absorbing Gases........ i54 4.1. Introduction i 5 4.2. C.rbon Dioxide 15. 4^,. Nitrous Oxide 4.4. Carbon Monoxide and Methane l. 4.5. Ozone 159 4.6. Water Vapor 164 5. Summary of Open-Air Field Meas..rements of Absorption Spectra..... 182 6. Summary of Laboratory Measurements of Homogeneous-Path Absorption Spectra.......,................ 185 7. Comparison of Atmospheric Molecular Absorption Spectra......... 187 7.1. Comparison of Computed Spectra with Spectra Measured by Taylor and Yates 187 7.2. Comparison of Computed Spectra with the Rigorous Calculation for the 2.7-p H20 Band 191 7.3. Comparison of Computed Spectra with Slant-Path Field Measurements for the 2.7-l2 Band 204 7.4. Comparison of Computed Spectra with Slant-Path Field Measurements fo.: the 4.3-, Band 222 7.5 Comparison of Computed Spectra with the Rigorous Calculation for the 15-, Band 229 7.6 Ge ieral Comparison of Computed Spectra 234 6. Summary and Conclusions........................... 244 Appendix I: Computer Program for Calculating W, W*, and P........ 249 Appendix I: Computer Program for Calculating Elsasser Band-Model Transmissivity Function................... 265 References.................................. 270 viii

.-. WILLOW RUN LABORATORIES ----- FIGURES 1. The Near-Infrared Spectra of Solar Irradiation and of CO, CH4, N20, 03, CO2, and H2O....................... 5 2. Absorption vs. Frequency for a Single Line............ 13 3. Absorption by a Single Elsasser Band.................. 15 4. Absorption As a Function of i,3 = SW/d for the Elsasser Model..... 17 5. Absorption As a Function of (3,- 2raSW/d2 for the Elsasser Model. 19 6. Absorption Divided by,3 As a Function of 4 = SW/2ra for the Elsasser Model........1.................. 19 7. Absorption As a Function of P3 = SW/d for the Statistical Model.... 24 8. Absorption As a Function of 2 = 2, QSW/d2 for the Statistical Model. 24 9. Absorption Divided by fP As a Function of, SW/,ra for the Statistical Model..................... 25 10. The Effect of Temperature on the Distribution of Line Strength.. 37 11. Weak-Line Approxirmation to the Elsasser Model............. 42 12. Strong-Line Approximation to the Elsasser Model............ 42 13. Nonoverlapping-Line Approximation to the Elsasser Model....... 42 14. Working Graph for Estimating S/d.................... 42 15. Working Graph for Estimating 2ra /d2.......... 42 16. Working Graph for Estimating 2rai/d and S/2ra.......... 42 17. Transmissivity Curve for Carbon Dioxide................ 56 18. Generalized Absorption Coefficient for 15-4 Carbon Dioxide Band... 56 19. Actual Absorption of 15-p Carbon Dioxide Band............. 60 20. Computed and Measur2d Band Areas for 15-m Carbon Dioxide Band. 61 21. Transmissivity Curve for Ozone..................... 62 22. Generalized Absorption Coefficient fcr 9.6-p and 9-p Ozone IBads.. 62 23. Computed and Observed Band Areas for 9.6-g Ozone Band....... 63 24. Transmissivity Curves for Water Vapor................. 66 25. Comparison of Cdntours for 6.3-u Water Vapor Band.......... 67 26. Generalized Absorption Coefficient for 6.3-4 Water Vapor Band.... 67 27. Computed and Measured Band Areas for 6.3-4 Water Vapor Band.... 68 28. Generalized Absorption Coefficient for Rotational Water Vapor Band. 70 29. Generalized Absogpticn Coefficient of Rotational Water Vapor, u i; d Win.diow7........................... 71 30. Transmission vs. Wavelength for Water Vapor...... 77 31. Transmissivity Curve for Carbon Dioxide......... 79

...- WILLOW RUN LABORATORIES 32. TrausmI.sston vs. Wavelength for Carbon Dioxide............ 80 33. Transmission vs. Wavelength for Ozone................. 33 34. Transmissivity Curves Showing the Divergence ef Absorption from That Predicted by the Elsasser, Goody, and Experimental Band Models. 85 35.,tegionn of Linear Absorpt4on for Horizontal Atmospheric Paths at Various Altitudes............................ 86 36. Absorption by a Single Eltasser Band.................. 89 37. Fractional Absorption......................... 95 38. Absorption vs. Wavenumber for 2.7-.t'Water Vapor Band........ 105 3U. Transmission vs. Wavenumber for 2.7-ji Carbon Dioxide Band..... 126 40. Procedure for Determininn W and qr................... 128 41. Valvu of W at 740 mm -ind 3003K for Atmospheric -Absorbing.e Species.................................. 129 42. Transmission vs. Wavenumber for 4.3-4 Carbon Dioxide Band at 2800K............................ 136 43. Transmission to Top of Atmosphere from Indicated Altitudes As a Function of Wavenumber for 3S16-cmr 1 Ci rton Dioxide Ea-id..... 144 44. Absolute Infrared Transmission of the Atmosphere........... 145 45 Wir.ow Transmission.......................... 147 46. Representative Ozone Concentration Profile................ 160 47. Seasonal Variation of Ozone Concentraticn at Flagstaff..... 161 48. Latitudinal Variation of Ozone Concentration.............. 161 49. Average Distribution of Total Ozone over the Northern Hemisphere in the Spring............................ 162 50. Average Distribution of Total Ozone over'*e Northern Hemisphere in the Fall....................... 163 51. Shifting of Profile HI-1 To Contain Total Ozone of 0.42 atn cm.... i65 52. Variation of Atmospheric Water Vapor with Altitude........... 167 53. Water Vapor Mixing-Ratlo Pro;iles................... 170 54. Summer and Winter Mixing Ratios of Barrett......,. 171 55. Zachor's Summer and Winter Mixing-Ratio Approximations..... 172 56. IRMA Approximate Water Vapor Profiles................ 174 5/. Average Water Vapor Profile over Florida................. 175 58. Gutnick's Average Water Vapor Profile................... 176 59. "mnparison of Gutnick's Average Profile to Zachor's Approximation of Barrett's Data................... 178 60. Mixing-Ratio Profiles Bounding the Data Showi, in Figure 58...... 179 61. Cor^parison of Spectra for Horizontal Path 19 km Long at 15.5-km Altitude............................. 181 X

------- WILLOW RUN LABORA TORIES - 62. Comparison of Measured Spectra of Taylor and Yates with Computed Spectra of Al:'ihuler; Elsassr; Stull, Wyatt. and Plass; and Carpente-. 183 63_ Comparison of Measured Spectra. of Taylor and Yates with Cmnpuated Spectra ao Bradford: Green and Griggs, Oppel; and Zachor....... 189 64. Comparison of Calculated (Rigorous) Spectra of Gates vith Computed Spectra of Altshuler................... 192 65. Comparison of Calculated (Rigorous) Spectra of Gates with Computed Spectra of Howard, Burch,.nd Wi;liams.......... 193 66. Comparison of Calculated (Rigorous) Spectra of Gatea with Computed Spectra of Lindquist.............,.......... 194 67. Comparison of Calculated (Rigorous) Spectr.A of Gates with Computed Spectra of Greecr and Griggs.................... 195 68. Comparison of Calculated (Rigorous) Spectra of Gates with Computed. Spectra of Stull, Wyatt, and Plass..................... 196 69. Comparison of Calculated (Rigorous) Spectra of Gates with Computed Spectra of Oppel............................. 197 70. Comparison uf CaLculated (Rigorous) Spectra of Gates with Computed Spectra of Altshuler........................ 193 71. Comparison of Calculated (Rigorous) Spectra of " ies with Computed Spectra of Howard, Burch, and Williams................. 99 72 Comparison of Calculated (Rigoro'is) Spectra of Gates with Computed Spectra of i ndquist.......................... 200 73. Comparison of Calculated (Rigorous) Spectra of Gates with Computed Spectra of Green and'Griggs........................ 201 74. Compariso of Calculated (Rigorous) Spectra of Gates with Computed Spectra of Iull, Wyatt, arnd Piass.................... 20C2 75. Conra. ison of Calculated (Rigcrous) Spectra of Gates with Computed Spectra of Oppel.................... 203 76. Comparison of CARDE Solar Scectra with Cor;Futed Spectra of Altshuler.................. 207 77. Comparison of CARDE Solar Spectra. with Computed Spectra of Zachor................................ 208 73. Comparison of CARDE Solr Spectra with Computed Spectra of Green and Griggs....................... 209 79. Comparisor. of CARDE Solar Spectra with Computed Spectra of Stuli, Wyatt. and Plass......................... 210 80. Comparison of CARDE Solar Spectra with Computed Spectra of Oppel................. i.................... i 81. Comparison of CARDE Sola-'Spc -tta with Computed Spectra of Alt'shuier................................ 212 82. Comparison of CARDE Solar Spectra wtti Computed Spectra of Zachor....................... 213 63. Comparls.o of CARDE Solar Spectra with Cor'puted Spectra of Green'nd Griggs.......................... 214 xi

-.WILLOW RUN LABORATORIES 84. Comparison of CARDE Solar Spectra with Computed Spectra c. Stull. Wyatt, and Plass......................... 215 85. Comparison of CARDE Solar Spectra with Computed Spectra of O pe.................................. 216 86. Comparison of CARDE Solar Spectra with Computed Spectra of Altshuler......................... 217 87. Comparison of CARDE Solar Spectra with Computed Spectra of Zachor...................... &Ao 88. Comparison of CARDE Solar Spectra with Computed Spectra of Green and Griggs............................ 219 89. Comparison of CARDE Solar Spectra with Computed Spectra of Stull, Wyatt, and Plass.................... 220 90. Comparison of CARDE Solar Spectra with Computed Spectra of Oppel................................. 221 91. Comparison of Solar Spectra of Kyle with Computed Spectra of Bradford; Plass; Carpenter; Oppel; Altshuler; and Gieen and Griggs.. 224 92. Comparison of Solar Spectra of Kyle with Computed Spectra of Bradford; Plass; Carpenter; Oppel; Green and Griggs; and Altshuler.. 227 93. Comparison of Solir Spectra of Kyle with Computed Spectra of Plass; Carpenter; Bradford; Oppel; Green and Griggs; and Altshuler.. 228 94. Comparison of Calculated (Rigorous) Spectra of Drayson with Computed Spectra of Elsasser, Altshvler. and Plass.......... 230 95. Comparison of Calculated (itigorous) Spectra of Drayson with Con puted Spectra of Elsasser, Altshuler, and Plass.......... 231 96. Comparison of Calculated (Rigorous) Spectra of D ayson with Ccmputed Spectra of Altshuler and Elsasser...............; 232 97. Comparison of Calculated (Rigorous) Spectra of Drayson with Computed Spectra of Altshuler and Elsasser............... 233 98. Comparison of Computed Spectra for High-Altitude RervLmaissance Path....................,...... 236 99. Comparison of Computed Spectra for Low-Altitude Reconnaissance Path................... 237 100. Compariso: of Computed Spectra for Re-Entry Vehicle Tracking Path (Ground-Based)........................... 238 101. Comparison of Computed Spectra for Re-Entry Vehicle Tracking Path (Air-borne Platform)........................ 239 102. Comparison of Computed Spectra for Air-to-Air Path......... 940 103. Parameters Defining Geometry of Slant Path.............. 251 101. Schematic Diagram of Atmosptheric Refracted Path........... 252 xii

- WILLOW RUN LABORATORIES TABLES 1. Regions of Validity of Various Approximations for Band Absorption. 17 2. Summary of Closed-Form Expressions for Spectral Band Absorpton. 36 3. Transmissivity Function and Absorption Coefficients for Carbon Dioxide.................. 57 4. Transmissivity Function and Absorption Coefficients for Ozone.... 65 5. Transmissivity Function and Absorption Coefficients for Water Vapor................. 73 6. Summary of Laboratory Data and Path Parameters........... 75 7. Empirical Constants for Ozone...................... 90 8. Empirical Constants for Carbon Dioxide................. 92 9.. Wo vs.. Wavelength for Water Vapor.................. 99 10. Absorption Coefficients for Water Vapor............... 101 11. Absorption Constants for Carben Dioxide for the 2.7-p Band... 106 12. Absorption Constants for Carbon Dioxide for the 4.3-/ Band...... 107 13. Absorption Coefficlents for Water Vapor................. 109 14. Absorption Coifficients for Nitrous Oxide for the 4.3-ti l3and...... 120 15. Absorption Coefficients for Mthane............ 121 16. Absorption Coefficients for Carbo,. Monoxide............ 123 17. Band Parameters for Carbon Dioxide for the 4.3-A Band........ 124 18. Values of'............,................. 130 19. Latoratory Data Usea To Determine Values of W and?...... 134 20. Absorption Coefficients for Carbon Dioxide.............. 136 21. Hon.ogeneous Paths for Which Transmission Data Are TaLbuated.. 138 22. Mixing Ratio for "Wet" Stratosphere Mode".......... 140 23. Window Transm!ss.on........................... 49 24. Empirical Constants from Figure 5........... 150 25. Summary of Band-Model Methods for Computing Atmospheric Absorption.................................. 151 26. Concentration of Carbon Dioxide............ 15. 5 27. Atmospheric Carion Dioxide for 1857-1906 and 1907-1956, Various Categories..................... 156 28. Concentration of Nitroui Oxide....................... 158 29 Concen'"ation of Carbon Monoxide................. 158 30. Instruments, Investigators, and Or-galizatlons Responsible for NonEystemattc St.-atospheric Humidit; Measurements.......... 168 31. Summary zf Open-Air Field Measurlments of Absorption Spectra... 83

-- - -' WILLOW RUN LABORATORIES 32. Summary of Laboratory Me.surements of Homogeneous-Path Absorption Spectra.,,................... 186 33. Values of Average Transmission and Total- Band Absorption for the Spectra Presented in Figures 64-69................ 205 34. Values of Average Transmission and Total-Band Absorption for the Spectra Presented in Figures 70-75.......... 205 35. Values of Average Transmission and Total-Band Absorption for the Spectra Presented in Figures 76-80......... 223 36. Values of Average Trarsmission and Total-Band Absorption for the Spectra Presented in Figures 81-85................. 223 37. Values of Average Trzasmission and Total-Band Absorption for the Spectra Presented in Figures 86-90................. 223 38. Values of Average Transmission and Total-Band Absorption for the Spectra Presented in Figure 91.................. 226 39. Values of Aver.ge T ansmission and Total- Band Absorption for the Spectra Presente& in Figure 92................... 226 40. Values o: Average Transmission and Total-Band Absorption for the Spectra Presented in Figure 93................... 226 xiv

------------ - WILLOW RUN LABORATORIES BAND-MODEL METHODS FOR COMPUTING ATMOSPHERIC SLANT-PATH MOLECULAR ABSORPTION INTRODUCTION The literature devoted to the development.nd presentation of methods for computing atmospheric slant-path molecular abso.:ption based on assuming models of the band structure I1. extensive. This state-of-the-art report summarizes the literature, discusses the advantages and limitations o ouch methods, and demonstrates the degree of correctness with which absorption iswctra computed by band- model methods can predict the true absorption spectra for an atmospheric slant path. In recent years there has been considerable interest in predicting the amount of atmospheric attenuation of electromagnetic radiation for a wavelength range extending from the visible to the far infrared. Knowledge of the amount of radiation which will be transmitted by the atmosphere between p heat source and a detector is required for the design of infrared equipmlent. Other concerns are the amount of sky-background raliation a detector will see and problems of radiation transfer in the atmosphere. These di.'erse problems require values of attenuation spectra ranging fronm mere approximate values rf attenuation for broad spectral intervals to fairly accurate, high-resolution attenuation spectra. In order to fulfill sucF diverse requirements, zxtensive research has been done i, the past decade with the objective, first. of spec.;ytng and defining the problemn of atmospheric attenuation, and second, of developtng methods for predicting the a:tenuation tor arbitrary atmospheric slant paths for assumed model atmospheres. In general, there are five attenuating mnechanisms which may be associated with the atmosphere: (1) AsAorption by the ozone in the Hartley continuum (2) Rayleigh scattering by molecular nitrogen and oxygen (3) Rarticultte or Mie scattering (4) R.:sonance atomic absorption (5) Molecular band absorption In the spectral range from 0.2 to 0.32 it. photo-absorption associated with the decomposition of ozone plays a dominant role in attenuation. In this range,'i;enuation is very sensitive to wavelength and the czone density. Unfortunately, the o07'ne der.sity is a complex function of altitude, the functional relationship being variable with tia-e and geographic location.

.... -.. WILLOW RUN LABORATORIES Rayleigh scattering by molecular nitrogen and oxygen must be considered in the ultraviolet and visible regions. From the standpoint of both the distribution of the scattering matter and the wavelength dependence of the absorption coefficients, this is the simplest of the attenuating mechanisms to be considered here. The presence of haze or trace distributions of particulate matter in the atmosphere causes MiL scattering-a phenomenon of particular importance to ground-based observations. This scattering problem becomes less acute as reasonable altitudes are achieved since the particulate matter is generally confined to lower altitudes. It is a rather complex problem, however, because the wavelength dependence of the absorption coefficient is a function of particle size. The spatial distribution of particulate matter is fairly complex and still relatively uniknown. The most complicated and difficult attenuation problem arises from molecular band absorption. The molecules in the atmosphere which have vibration-rotation resonance frequencies in the infrared, and hence give rise to absorption, are water vapor (H20), carbon dioxide (CO), ozone (O3); nitrous oxide (N20), methane (CH,), and carbon monoxide (CO). H2, C02, and 03 cause the greatest amount of absorption because they huve strong absorption bands and exist in the atmosphere at relatively high concentration. The molecular species CO. N20, and CH4 demonstrate significant amounts of absorption only wnen the line of sight passes through a large number of air masses. To completely specify the amount of attenuation of some source of radiation as it traverses an atmospheric slant path, it would be necessary to consider each of the five mechanisms. However, each attenuation mechanism constitutes i complex problem warranting separate treatment. In this report we consider only the problem of molecular absorption, and place particular emphasis on band-model methods of computing the absorption caused by an atmospheric slant path. The problem of calculating the molecular absorption by any one of the absorbing species mentioned above is extremely complex because of the many variables that must be known before the computation can be performed. In general, it is necessary to specify the meteorological conditions that exist at each point along the path, including pressure, temperature, and the concentration of each abtorber. Also, for a given wavelength interval, the location, intensity, and shape of each absorption line must be specified as well as the functional relationships between these parameters and meteor ological conditions. If all of the above quantities are known, then the absorption spectra for any desired resolution can be determinr, e by su;. —-, it wi tion to absorption of each spectral line throughout a given wavelength interval. This method is most easily applied to homogeneous paths but can be extended to slant paths by performing a direct pressure integration along the path. 2

-. —. —-------— WILLOW RUN LABORATORIES - The method of summing the contribution,d efch spectral line is the most rigorous way of determining atmospheric absorption since it involves a direct integration cf the general transmiesivtty function which specifies the absorption of an atmospheric slant path. Although rigorous, this method has not received ge'.ral application for all absorbing species and all spectral regions since the necessary line parameters are known fairly accurately for only a small portion of the spectrum for the major absorbing species and are almost unknown for the minor constituents. Because knowledge of the band parameters is limited and because the direct integration involves such extensive computational labor, almost all of the currently available methods for computing absorption use approximations which reduce the general transmissivity function to a form which expresses the transmission averaged over some interval, in terms of elementary functions. The simplified equation is then resed in conjunction with laboratory homogeneouspath data to predict absorption for other homogeneous paths. This last restriction necessitates the reduction of a slant path to an equivalent homogeneous path. The standard approach used in performing the simplification of the general transmissivity function is to use a model of the band structure, that is, to assume that the line positions and strengths are distributed in a way that can be presented by a mathematical model. It should be pointed out that both the rigorous calculations and the band-model methods require laboratory data, but with this difference: the rigorous calculations dpend upon highresolution spectra to obtain the necessary line parameters prior to performing the direct integration. Band model methods involve empirically fitting the integrated transmissivity function to laboratory spectra to specify equivalent line parameters. The rigorous calculations would appear to yield results that are more exact since the line positions are not simulated. However, the model calculations are more accurate at the present time for some molecules in some spectral regions. Tue reason for this is related to the Inaccuracies in the shape, intensity, and half-width of the spectral lines. The great advantage of using a direct integration is the ability to extrapolate over extreme ranges of pressure, temperature, and absorber concentration and the fact that much higher resolution is obtainable than from band- nodel calculations. In section 2 of this report, the general transmissivity function for determining atmospheric slant-path absorption is derived and the simplification of this function through the use of band models is performed. The equations for determining an equivalent homogeneous path for an Atn-mose rtc slant pats are also deiiveud and niiethods of empir'icaily lti.ing lband-.mwif,. aosorption equations to homogeneous-path absorption spectra are presented. 3

----- WILLOW RUN LABORATORIES ---- 2 GENERAL THEORY OF ATMOSPHERIC ABSORPTION The atoms of any molecule not at absolute zero are constantly oscillating about their positions of equilibrium. The amplitudes of these oscillations are extremely minute and their frequencies are htih. Since these frequencies are of the same orders of magnitude as are infrared radiations, some direct relationship might be expected between motions of the atoms within molecules and their effects on infrared radiation incident upon the atoms. Actually, those molecular vibrations which are accompanied by a change of dipole moment, so-called "infraredactive" vibrations, absorb by resonance all or part of the incident radiation, provided the frequencies of the latter coincide exactly with those of the intramolecular vibrations. Thus, if a sample of molecules of a single absorbing species is irradiated in succession by a series of monochromatic bands of infrared, and the percentage of radiation transmitted is plotted as a function of either wavelength or frequency, the resulting graph will show regions of absorption centered at each of the resonant frequencies. The percentage transmission is generally different for each resonant frequency, depending upon the energy of the molecular transition. These regions of absorption are known as spectral lines. If the sample of molecules being irradiated is at a pressure so low that there is a minimal amount of molecular interaction and at a temperature so low that there is little relative molecular motion, then each of the spectral lines would be extremely narrow and in the limit would have an infinitesimal width. For atmost,.i conditions the pressures are such that there is a significant molecular interaction giving l: 4o spectral lines which absorb over a range of frequencies. Although asymmetrical lines have been observed for some molecules under some conditions, for the most part the absorption of a spectral line is a maximum at the resonant frequency and decieases to zero asymptotically at smaller and greater frequencies in a symmetrical manner. Because the spectral lines of atmospheric gases are generally clustered in bands of frequencies, there are certain broad regions where no absorption exists and oTher regions where the absorption is almost continuous. For example, water vapor has two absorption bands near 2.7 / containing approximately 4000 spectral lines. Figure. 1 shows the near-infrared solar spectrum of atmospheric air and laboratory absorption spectra for each of the infrared-active atmospheric gases. In order of presentation, carbon mono.dide has one fairly weak band at approximately 4.8,i. Methane has two regions of absorption at 3.2 and 7.8 A. The second methane band is almost completely obscured by nitrous oxide which also absorbs at 7.8 tL.'The strongest of the twc nitrous oxide bands is centered at approximately 4.7. Ozone has two bands, one at 9.6 l and. the other at 14 ji. The two remaining gases, carbon dioxide and water vapcr, are the 4

—.. WILLOW RUN LABORATORIES - -- 0 o _ 0 -9N20 0 A - Vco ^ l * *1. L a o. 100 1 1 Solar Spectrum 8000 5000 3000 2000 1400 1000 800 - I. I. I j I.... I... 1 2 3 4 5 5 7 8 9 10 11 12 13 14 15 WAVELENGTH (4' FIGURE 1. THE NEAR-INFRARED SPECTRA OF SOLAR IRRADIATION AND OF CO, CH4, N20, 03, CO AND 20 41~~~~~~~~~~~~~~^ 2 3 *: — 2 2j

--- - WILLOW RUN LABORATORIES most significant contributors to atmospheric absorption. Carbon dioxide has three strong bands at 2.7, 4.3, and 15 t. Water vapor has a greater number of ibsorption bands than any other absorbing species with bands at 1.14, 1.38, 1.88, 2.7, 3.2, and 6.3 I. The absorption of infrared radiation by atmospheric gases is then characterized by discrete bands consisting of a large number of overlapping spectral lines of various strengths which are distributed throughout the band. The degree of overlapping depends upon the line half-width, and the distribution of lines throughout an absorption band depends upen the absorbing molecule.'or example, the line positions for water vapor are distributed unevenly, in contrast to carbon dioxide, which displays a relatively regular spectral-line distribution. 2.1. GENERAL EQUATION FOR SLANT-PATH ABSORPTION The general equation which specifies the molecular absorption of an atmospheric slant path composed of a single absorbing gas may be derived by considering the Lambert-Beer law. This law is rigorous when the absorption coefficient does not vary over the spectral bandwidth wvder consideration. It states that the decrease in spectral intensity dl(x, v) which is caused by abscl ptici. in a (thin) differentia! section of path dx, is proportional to dx, to the intensity I(x, v) of the radiation incident on the front face, and to the concentration p(x) of the absorbing molecule. The proportionality constant is designated k(x, v). Mathematically, OI(x, t) -I(, k(x,, v)p(x)dx Integration gives the transmission T over a path from x = 0 to x =X. T(X,; v! exp J k(xx, r)p(x)dx Uo j To determine the transmission over a finite frequency interval the average is taken: ix-,= { exp Jk(x. vp(x) dx dv Rewriting in terms of absorption, we have. the average absorption for the frequency interval AV given by ifA f - r -l 6

. — - WILLOW RUN LABORATORIES -- where pA) is the density of the absorbing gas x is the distance along the path k(x, v) is the absorption coeffic'.nt The absorption and its functional form with respect to the path therefore depend upon the form of the absorption coefficient within the interval A v. Let us assume that the interval A v contains a single spectral line and let us specify the various forms of k(x, v), where each form depends upon the environmental conditions of the absorbing gas. If only one molecule of gas were present, the form of k(x, v) would simply be a constant having a value only at v0, the center;:equency of the spectral line. In general, there are many gas molecules present and the fa:t that gas nolecules collide gives rise to the Lorentz pressure-broadened line which is considered fundamental for the whole theory of atmospheric transmission. The form of k(x, v) as given by the Lorentz theory is S aL k(x, v) = (2) (vj -v) +aI. In this equation, vU is the frequency at the line center and S is the line strength which depends upon temperature according to S= S (TO/T)3/2 exp (./) where E" is the vibration-rotation energy of the initial state. Tne ha.L-widkh of the line (aL) depends upon pressure and temperature in the following manner: /p \(TON where P is the effective broadening pressure. It is different from the total pressure because the absorbing gas molecules are more effective in broadening than are the foreign gas molecules. Burch et al. [1] have shown for CO2 that P equals the total pressure plus 0.3 times the partial pressure of CO2. T is the absolute temperature, and n depends upon the nature of the broadening gal and line center frequency. For nitrogen-water vapor collisions, Benedict and Kaplan [2] show n = 0.62 to be a good representative value. Here S< and aLo refer to the -strength and half-width, respectively, calculated at a reference temperature Tg. A study by Winters et al. [3] of an isolated CO2 spectral line showed that the intensity in the wings of the line decreased more rapidly than predicted by the Lorentz expression. On the

.-.L. WILLOW RUN LABORATORIES ---- basis of this study Benedict proposed the following empirical expression for the spectral line shape of CO2:. a L Aa exp ajv- vl S L, v | > d (z) +a where d is the average spacing between spectral lines a - 0.0675 b= 0.7 d = 2.5 cm 1 The value of A is chosen to make the two forms continuous at | - o= d 2.5 cm. There is a second cause of line broadening known as the Doppler effect which is related to the relative motions o! the molecules. The pure Doppler broadened line [4] is given by 2 k(x, L)- kle'y (3) where 1/2 Y — ( n 2)1/2 Da and aD, the Doppler half-width at hatf maximum, ia given bv -7 T1/2, a 3.58 x 10 -) v where M = molecular weight. For lower altitudes where the atmospheric pressure is high, the Lorentz half-width, aL, is dominant. As the atmospheric pressure decreases, the Lorentz half-width decreasces and the influence of Doppler broadening becomes more marked. For the 15- CO2 band, the Doppler and Lorentz half-width., are equal at about 10 mb [5] and at lower pressures the Doppler 8

.WILLOW RUN LABO A1 OR ES... half-width is greater. Thus, over a wide range of ntmospheric pressures it is necessary to consider the mix?d Doppler-Lorentz line shape to be completely accurate. The absorption, coefficient for the mixed broadening is given by reference 6; k(x, v) = — dt (4) 1St 2 0 2 )u u (1-t) where k0, Y are as before and u = -(in 2)1/2 Four different forms for k(x, v) have been stated, each form.ccuratelv representing the shape of a spectral line if the appropriate conditions are satisfied. Therefcre, to completely specify the absorption as given by equation I it is necessary to subs, tute the appropriate form cf k(x, v). Let us v.ssume that the soectral interval over which enuatitn 1 is defined cont.ins many spectral lines. Then we have AA,J | -^ exp LJt- k(x, v)p(x)dx (5) Equation 5 is the general equation for specifying the absorption of a given species over an atmnospheriic slant. path, the ringe of pressures encompassed by the pa.th defnin.nc the specific;ox m for the absorption coefficient k(x,,); To better understand the param:ters t:ia must be specified before equation 5 "an be evaluatec let us assume that the slant path is such that the.orentz lin3 shape is valid. Then equation 5 becom s - X J. S..I aip( )4 Aay= { L exp dx dv (6) The summation is over the total number of lires in the spectral interval anc oc is the strength of the nth line, a is the half -width of the nth line. and un is the center frequency o; toe nih line. n "'in T:e problem of determining slaat-path transmission sp.ctra is then to evaluate equation 6 for each frequency within the spectral inter:'al of interest. To perform this evaluation, the following four parameters must be specified: 9

-- -- WILLOW RUN LABORATORIES -- (1) The shape of each spectral line (2) The location of each spectral line (3) The intensity and half-width of each line and their variations with temperature and pressure (4) The density of the absorber at each point in the path In theory, if all of these parameters were known exactly for all absorption bands and for all absorbing species, the infinity-resolved transmittance could be determined simply by summing the contribution to absorption of each spectral line at each wavelength. Unfortunately, although the line positions are documented, the line parameters such as intensity and half-width are known fairly accurately for only the 2.7-1. H 0 band and the 15-p CO2 band. Drayson [5] has evaluated equation 6 for the 15-, CO2 band for 200 paths ranging from sea level to the limit of the atmosphere for Zenith angles extending from the horizontal to the vertical. Some representative spectra are presented in section 7 of this report. Gates et al. [7] have also evaluated equation 6 for the 2.7-M H2O band for homogeneous paths. Samples of their spectra are also presented. The method for computing absorption by the direct integration shown in equation 6 is classified as the rigorous method since it sums the contribution of each spectral line* after first determining its position and intensity. This is in contrast to the method which assumes that the positions and intensities obey some mathematical model. The rigorous method requires a great amount of computational labor and the process must be completely repeated for each slant path. The advantages of this method are its ability to extrapolate over extreme conditions and that any resolution is obtainable since Av in equation 6 may be made as small as desired. 2,2. BAND MODELS FOR COMPUTING SPECTRAL ABSORPTiON 2.2.1. INTRODUCTION TO THE BAND-MODEL CONCEPT. Many applications do not require a highly accurate determination of the absorption spectra, but only a first-order approximation to the true spectra, and in many cases the spectra may have relatively low resolution. Therefore, almost all of the available methods for computing atmospheric absorption use approximations which reduce equation 6 to a form which expresses tee transmision, averaged ~All spectral lines are not neceasmrily included, but only those having an intensity greater than a certain minimum value. 10

—. --- WILLOW RUN LABORATORIES over some interval, in terms of elementary functions. The simplified equation is then used with laboratory homogeneous-path data to predict absorption for other homogeneous paths. The classical approach used in performing the simplification of equation 6 is that of using a model of the band structure. That is, it is assumed that the line positions and strengths are distributed in a way that can be represented by a simple mathematical model. The most commonly used band models are listed here.* (1) The Elsasser or regular model [8-101 assumes spectral lines of equal intensity, equal spacing, and identical half-widths. The transmission function is averaged over an interval equal to the spacing between the line centers. (2) The statistical or random model, originally developed for water vapor assumes that the positions and strengths of the lines are given by a probability function. The statistical model was suggested by Teiles and worked out by Mayer [11] and (independently) Goody [12]. (3) The random-Elsasser moal [10] is a generalization oI the Elsasser model and the statistical model. It assumes a random superposition of any number of Elsasser bands of different intensities, spacings, and half-widths. Therefore, as the number of bands ranges from one of infinity, the band model extends, respectively, from the Elsasser model to the purely statistical model. This generalization, therefore, yields an infinite set of absorption curves between those of the Elsasser and statistical models. (4) The best available model is the quasi-random model [13]. It is capable of fairly accurate representation of the band provided the averaging interval is made sufficiently small. However, of the five models, it requires the greatest amount of computation. (5) The King model [14] consists of an infinite array of lines, either random or regular, which is modulated by a band envelope whose intensity falls off exponentially from the band center. 2.2.2. A SINGLE LORENTZ LINE. Let us consider the absorption caused by a single spectral line of a homogeneous path of a single absorbing gas. Let us assume that the shape of this line is represented by the Lorentz equation. For these conditions the absorption is given by * Other band models have been developed and are discussed in detail by R. M. Goody (Atmospheric Radiation, Oxford University Press, 1964). The models listed here are those that have received general^appiication. 11

- ---- sWILLOW RUN LABORATORIES Aav I| - exp -I S X dx dv (7) Ll /Jo( -V0 +a a For a homogeneous path S, a and p are constant; equation 7 further reduces to AAP= exp~ 2- 2-fdv (8) AL (v- + j2 X where W pdx = pX and is defined as tile optical path length. A plot of absorption versus frequency is shown in figure 2 for different path lengths, or for different values of W. The absorption caused by this line for a path of length Xl would be considered a weak line since the absorption is small even at the line center. For a path of length X3 the center of the line is completely absorbed away and any further increase in path length would only change the absorption in the wings of the line. Absorption by paths of length equal to or greater than X3 would, therefore, be considered strong-line absorption. In equation 8, if it is assumed that the interval ALv is such that the entire line is included, then the limits of integration can be taken from -oo to o0 without introducing a significant error. When these limits are used, equation 8 can be solved exactly for the total absorption. Ladenburg and Reiche [15] have solved the integral to obtain AAv= 27Tr1o'V[I0(4) + I1(%)] (9) where 4 = SW/2r a and I and I1 are Bessel functions of imaginary argument. Examining equation 9 under conditions of weak-line and strong-line absorption, we have for weak-line absorption iP << 1; equation 9 reduces to AAv = 2,r = SW (10) and absorption is linear with the optical path length W. Under conditions of strong-line absorption V/ is large and equation 9 reduces to AAv - 2 ISW (11) which is known as the suuare-root approximation. The above derivations are for a single spectral line but are also valid for absorption when many spectral lines are present but do not 12

—.. WILLOW RUN LABORATORIESoI -v ^ ^2>o FREQUENCY FIGURE 2. ABSORPTION VS. FREQUENCY FOR A S&.GLE LINE 13

WILLOW RUN LABORATORIES overlap. Therefore, for the nonoverlapping approximation the absorption is simply given by equation 9, 10, or 11, depending upon the value of'. 2.2.3. ELSASSER MODEL, The Elsrsser model of an absorption band is formed by allowing a single Lorentz line to repeat itself periodically throughout the interval Av. This gives rise to a series of lines that are equally spaced and tha tve a constant intensity and halfwidth throughout the interval. This arrangement of sp'.al lines was first proposed by Elsasser [8] and his derivation is presented here. The absorption coefficient for a periodic band is given by ^'k( r l - L 2 2 (12) 7n=-co ('- nd) + 2 n —oo It is possible to express equation 12 in terms of an analytical function owing to the fact that if such a function has only single poles, it is uniquely defined by these poles. Therefore, equation 12 is equivalent to S sinh [ k(v) cosh cos s (13) where 3 = 2na/d s = 277 /d If the averaging interval Av is taken. as cne period of the band (A. = 2'), then equation 6 becomes A.- { 1 - exp [-Wk(, s)]} ds (14); 2 This integral has been evaluated by Elsasser.[16] to give the general expression for absorption by an Elsasser band, namely Y A = sinh fI I(Y) exp (-Y cosh f) dY (15) where 3 = 2ra/d Y =4/sinh,. = SW/27ra d. mean spacing between s pectral lines A plot of this function for various values of f is given in figure 3. Because the function in its present form is difficult to evaluate: considerable effort has been expended comparing 14

* WILLOW RUN LABORATORIES ---- - I X, Iti r "' tifl nM-1 N l lt'.t-i'<. tt t ^l -ti L c/ H ifi!i!~i:!::,'!ii l:::.;i ^lil' O 4^!......... ~t- itr a" " -tiir t_,.,.,,., I,.;, J \. ot t> ~> f ~? *-'.,*''t.- * - o I -......, i~ -....iid os.av, -... tii ii-i: i~~~~~~i~~i1

.-. - WILLOW RUN LABORATORIES approximate formulae and evaluating the integral. Kaplan [17] has expanded the integral into a series which is convergent only for values of 0 less than 1.76. D. Lundholm [18] derived an algor.thm for the Elsasser integral which is convergent for all values of { and,. The algorit'\m,id a -omputer program, written by Oppel, for the evaluation of the Elsasser integral bas~k....'nis algorithm Are presented in appendix I. However, it is frequently desirable to w?3! with approximations to the function which are valid for certain conditions. 2.2.3.1. Weak-Line Approximation. In figure 4 the absorption given by equation 15 is plotted as - iuCcio: of the product 04 = SW/d for four values of f3. It is noted for f: 1 that the absorption curves become superimposed for all values of 4. Since the parameter 0t measures the ratio of line width to the distance of neighboring line, 0 < 1 implies that the spectral lines are strongly overlapped and spectral line structure is not observable. This condition corresponds to large pressures which would be realistic for atmospheric paths at low altitudes. For 3 2 1, equation 15 can be approximated by A- I- e's (16) Further, equation 16 is a good approximation to equation 15 whenever the absorption is small at the line centers (small'/) regardless of the value of,. Therefore, this approximation is referred to as the weak-line approximation, and, as will be shown later, is independent of the position of the spectral lines within the band. Table 1 summarizes the regions of 3 and 4 for which the weak-line approximation is valid with an error of less than 10%. This approximation is particularly useful in extrapolating the absorption to small values of 4 and to large values of pressure. Note that the weak-line approximation reduces to the linear approximation when the absorption is small even if the lines overlap. 2.2.3.2. Strong-Line Approximation, Of increasing interest are the long atmospheric paths at high altitudes which give rise to large values of W and small values of pressure. Under these conditions the absorption at the line centers is usually complete (large 4), the halfwidths are narrow, and the lines do not overlap strongly (small 0). For large 4 and small 3, equation 15 may be approximated by A = erf (2) 2(17) where erf (t) - et dt ~~~~16 ~VJ 16

-- WILLOW RUN LABORATORIES ----- 1.0 Elsasser 3 10. Model 4 60'^ 0. o 01 0,1 o.0oo 0.001 0.001 0.01 0.1 1.0 10 100,B = SW/d FIGURE 4. ABSORPTION AS A FUNCTION OF 13 = SW/d FOR THE ELSASSER MODEL. The weak-line approximation is the uppermost curve. TABLE I. REGIONS OF VALIDITY OF VARIOUS APPROXIMATIONS FOR BAND ABSORPTION* Statistical Statistical Model; Model; All Exponential Elsasser Lines Equally Line Intensity Approximation 3 = 2ra/d Model Intense Distribution Strong-line 0.001 4 > 1.63 4 > 1.63'0 > 2.4 approximation: 0.0! 4 > 1.63 4/ > 1.63 4"0 > 2.4 0.1 g > 1.63' > 1.63'0 > 2.3 Equations 17 and 25- 1',, 1.35 4 > 1.1 I'o > 1.4 10 4 > 0.24 4 > 0.24 0 > 0.27 100 > 0.024' > 0.024 4t' >0.24 Weak-line 0.001'< <0.20 V <0.20 40 <0.10 approximation: 0.01 4V <0.20 4 < 0.20 0 <0.10 0.1' < 0.20 < 0.20 < 00,Equation 16: 1 < co 4' <0.23 < 011 10 4'<oo 4'<oo 10' <oo 4 <oo'0 <o 130 3p < 00p < co ak l; < C Nonoverlapping- 0.001' < 600,000 4 < 63,000 line 0.01 4' < 6000' < 630 0 < 800 approximation: 0.1 <' <60' < 6.3'0 <8 1' <0.7' <0.22 10 < 023 Equations 26 and 33: 10' < 0.02' < 0.020 I% < 0,020 o00 4 < 0.002 4 < o0.oo020 < 0.0020 *When /, = SW/2ra satisfies the given inequalities, the indicated approximation for the absorption is valid with an error of less than 10%. For the exponential line intensity distribution;'Q = SoW/2na, where P(S) - SO-1 exp (-S/So). 17

WILLOW RUN LABORATORIESwhich is known as the strong-line approximation to the Elsasser band model. Figure 5 is a plot of equation 15 with absorption as a function of f2 /. For F3 = 0.01, equations 15 and 17 are superimposed for3 24i > 0.0003. Clearly, given small 3 and large 4/, equation 17 is a particularly good approximation for representing the absorption when 3 < 1. If (3 < 1, then equation 17 is valid whenever U.1 1 A < 1. This includes most values of absorption'that are usually of interest, This case differs from the square-root approximation in that it is not necessary that the lines do not overlap. For overlapping spectral lines (larger /), the values of P for which the approximation is valid are simply restricted to large values of Q/. The specific regions of validity are given in table 1. 2,2.3.3. Nonoverlapping Approximation. The third approximation to the Elsasser band model is known as the nonoverlapping approximation.. The regions of validity for the strongand weak-line approximations depend only upon the value of absorption at the frequeacy of the line centers and do not depend upon the degree of overlapping of the spectral lines. On the other hand, the only requirement for the validity of the nonoverlapping-line approximation is that the spectral lines do not overlap appreciably. It is valid regardless of the value of the absorption at the line centers. This approximation is particularly useful for extrapolating the absorption to small values of W and small values of pressure which correspond to short paths at high altitudes. Under these conditions equation 15 reduces to A = 34/e -'[Io() + I (W)] (18) which is exactly the same expression as that obtained for the absorption by a single spectral line. This is an expected result, for if the lines do not overlap there will be only one line that contributes to the absorption at a given frequency. In figure 6, A/f is given as a function-of 4. The uppermost curve is the nonoverlapping approximation. For, << 1, the curve has a slope of 1 (i.e., a region where the weak-line approximation is valid) and for q/ >>. 1 the curve has a slope of one-half (i.e., a region where the strong-line approximation is valid). The region of validity for.various values of (3 and 4/ are iven in table 1. The general expression for absorption by an Elsasser band given by equation 15 and the strong-line approximation (eq. 17) are useful for determining absorption by CO2 since the bands consist of fairly regularly spaced lines. However, the bands of H20 and 0, have a highly ir* regular fine structure. nd cannot be wel described by equation 15. To develop an analytical expression for the transmissivity function for H 0 and 03 we must ernploy statistical methods. 18

WILLOW RUN LABORATORIES Elsaseer Model - Constant Pressure — Constant Amount of Absorbing Gas 0.1. i 0.01 - 0.0001 0.001 0.01 0.1 1.0 10 3 4" 2iraSW/d2 FIGURE 5. ABSORPTION AS A FUNCTION OF 82P = 2orSW/d2 FOR THE ELSASSER MODEL. Curves are shown for constant pressure (3 constant) and for constant amount of the absorbing gas (/3 = constant). The strong-line approximation is the uppermost curve. 10 Elsasser Model = 0.01 0.1 0.01 0.001 0.01 0.1 1 10 100 W= SW/2ira FIGURE 6. ABSORPTION DIVIDED BY, AS A FUNCTION OF, = SW/2-ra FOR THE ELSASSER MODEL. The nonoverlapping-line approximation is the uppermost curve. 19

---- WILLOW RUN LABORATORIES —-- 2.2.4. STATISTICAL BAND MODEL. Let Av be a spectral interval in which there are n lines of r.ean distance d: Av= nd (19) Let P(Si) be the probability that the ith line will have an intensity Si and let P be nornlaiized; then IP(S)dS = (20) *0 We assume that any line has equal probability of being anywhere in the interval Av. The mean absorption clearly does not depend upon v provided we are far enough away from the edges of the interval. We shall, therefore, determine it for v= 0, the center of the interval. If we let the center of the ith line be at o = vi, then the absorption coefficient becomes ki(Si, vi) -k(v = v, o = 0) The mean tra-smissivity is found by averaging over all positions and all intensities of the lines; thus T= a @ dv,... P dvn e P( )e kdS.. P()e dS )n. A ^ J.1'' n But since all integrals are alike, LT d IP(S)e kW ydS| L d v P(s)(1 - ekW) n Since v = nd, when r.becomes large the last expression approaches an exponential; therefore,, T= exp [- JP (S)(A V)d (21) where [A1 Av] =- ( e kW)dv A1 is the'absorptivity of a single line taken over the interval Av. 20

W —- WILLOW RUN LABORATORIES - 2.2.4.1. Equal Intensity Lines. Equation 21 can be evaluated for two special cases. Firt, when all the lines have equal intensities equation 21 reduces to -A lA/d = e - exp -BPe (I (Q) + I+ )] Rewriting in terms of absorption, we have A = 1 -:'p -PeP'I(11 ) + I1()]1 (22) If each of the lines absorbs weakly so that 41 is small, then equation 16 reduces to A 1 - exp (-34) (23) This is the weak-line approximation to the statistical model with all lines equally intense. If the lines absorb strongly, then equation 22 reduces to A = 1 - oxp (-2\'rSaW/d) (24) In terms of P and 4, A 1 - exp -L ) | (25) This is the strong-line approximation to the statistical model hi which all lines are equally intense. The nonoverlapping approximation to equation 22 is obtaiined from the fir:t t-"m in the expansion of the exponential, so that A = te e[I0(I) + I1()] (26) This is the same expression as that obtained for the nonoverlapping approximation to the Elsasser model, equation 18. 2.2.4.2. Line Strength by Poisson Distribution. Next let us consider the case where the lines are of different strength and the distribution for the probability of their strength is a simple Poisson distribution, namely, 1 -'/0 P(S) = -e (27) 21

. —----.. - WJLLOW RUN LABORATORIES- __ By letting k a KS li equation 21, we obtain - ( exp 7 KWS ) (28) We now introduce the Lorentz line shape (v a 2 This vanishes fast enough for large Av that we caa extend the integration in equation 28 from -o to oo, giving T-=exp., _ - (29) Ld aa (WS0 )/n ReTriting in terms of absorption and i and',. we have A 1 exp - 0/(Il + ZO-)12 (30) where O 2 = S0W/2ra. This is the formu!a first developed by Goody and is therefore referred to as the Goody band model. The weak-line approximation to equation 30 is obtained when ~0 < 1. Under these conditions equation 30 b;ccmes A - exp (-;3 ) (31) The strong-line approximAtion to the statistical model with an exponential distribution of line strengths is obtained when 0 >> 1. Under these conditlons equation 30 becomes A = I - exp (i2^l,/) (32) The nonoverlapping approximation is obtained from equation 30 when the exponent is small and is. thereiore given by the firfst two terms of the expansion, or A 3i,0/(l + 20)1'/2 (33) 2.2.4.3. Strong-Line, Weak-Lne, and NonoverllappinE Approximations. The three approximaLionll tu the twp statistical models will be.discu3sed concurrently because they are so 22

- - ---- WILLOW RUN LABORATORIES closely related. First, we shall consider the weak-line approximation. Recall for this case that absorption was given by A = I - exp (-Pb), which is exactly the expression obtained for the weak-line approximation to the Elsasser band mcdel. The saw.a expression results when an exponential distribution of line intensities is assumed with 4,::eplaced by 40. This confirms our earlier statement that under weak-line absorption, absorptton is independent of the arrangement of the spectral lines within the band. Absorption versus Pip is plotted for equation 22 in figure 7. The solid curves give the absorption for the statistical model for the case in which all lines are equally intense, The dashed curves give the absorption for the statistical model with an exponential distribution of line strengths. The uppermost solid and dashed curves represent the weak-line approximations for those intensity distributions. The regions of validity are given in table 1. For the case in which all lines are equally intense, the weak-line approximation is always valid within 10% when 4/ < 0.2. It is valid for the exponential intensity distribution when,c < 0.1. If 1% accuracy is required, these values of 4, should be divided by 10. The strong-line approximation to the statistical model for all lines of equal intensity and for an exponential distribution of line strengths are given respectively by equations 25 and 32. The absorption for these models as a function of i 4, is shown in figure 8. The strong-line approximation is the uppermost curve in the figure. The absorption can even become greater than the limiting values given by this curve. It is evident that the distribution of line intensities in a band only slightly influences the shape of the absorption curve. As for the strongline approximation to the Elsasser model, the strong-line approximation to the statistical model for either distribution of line strengths is always valid when l s 1 and 0.1 _< A I. The n,,gplete regions of validity are given in table 1. The last approximation to be discussed for the statistical model is the nonoverlapping approximation. For all lines of equal intensity the absorption is given by the expression used for the nonoverlapping approximation to the Elsasser model, namely, A lV/e[I0() + 1 ()] For an exponential distribution of line strengths the absorption is given by equation 33. Therefore, the intensity distribution, but not the regular or random spacing of the spectral lines, influences the absorption curve in this approximation. In figure 9, A/F is plotted as a function of 4, for the statistical model Note that if this figure is compared with the corresponding one for the Elsasser model the nonuverlapping approximation to the Elrasser model has a considerably larger region of validity. This is because the spectral lines begin to overlap at considerably larger path lengths for the Elsasser model than for the statistical model (cf. table 1). 23

---.WILLOW RUN LABORATORIESStatistical Model - All Lines Equal. Tntense, Constant PressuL, *Sta~ti~stic~al hoel --— Exponential In:tensity Distribution, Statistical Model Con:tant Pressure -- All Lines E-quaA- ll I^ntense — All Lines Equally Intense, Equally Intense Coabstant Amount — Exponential Intensity of Asor of Absorblnr Gas Distribution 1.0 " —.., ~/ 0.1 | / /// / -1 /o/ ~0.0l o., 0n.1/.^^.00^...^.o /, / / 0.001 ___________________ o J00001 0.001 0.01 0.1 1.0 10 X - SW/d 3 4 = 2-7aSW/d FIGURE 7. ABSORPTION AS A FUNCTION OF FIGURE 8. ABSORPTION AS A FUNCTION OF, = SW/d FOR THE STATISTICAL MODEL. 324' = 2raSW/d2 FOR THE STATISTICAL-'1MODEL. The absorption for a model in which the spec- Curves are show?~ for constant pressure ( = contral lines are all of equal intensity is comp;.red stant) and for constant amount of the abserbing with that for a model in which the spectral lines gas (23'= constant). The absorption is shown when have an exponential intensity-distribution func- all thespectral lines have equal intensity and when tion. The weak-line approximation is the upper- there is an exponential intensity-distribution fun.most curve. tlon. The strong-line approximation is the uppermost curve. 24

— WILLOW RUN LABORATORIES Statistical Mode! -- All Lines Equally Intense - Exponential Intenoitv Distribution S10 - ^ 1.0,~ -- 0.1 // 0.0 1................................. 0.001. 0. 0.1. 1 1 10' 00.= SW/2n7 FIfURE 9. ABS0UP'TION DIVIDED BY a AS A FUNCTION OF W - -W/Wj2Tra FOR THE S''IT3STICAL MODEL. The absurption for a model in which the Ypectral li,~es are all of equal intensJty is compared w.th hat lor a model in which ihe.scpctral lines have'an eyponential intensity-disiribution, function. The nono, erlapping-line approxi.nationr is tle uppermoit ca.;ve

---- - WILLOW RUN LABORATORIES - Three important approximations to the band models of Elsasser and Goody have been dis cussed above. These three approximations provide a reliable means for the extrapolation of laboratory absorption data to values of the pressure and path length that cannot easily be reproduced in the laboratory. For example, the absorption for large values of pressure can be obtained from the strong- and weak-line approximations, depending upon whether W is relatively large or small. For extrapolation to small values of pressure all three approximations may be used in their respective regions of validity; however, the nonoverlapping-line approximation is valid over the largest range of values of W. For extrapolation to large values of W, either the strong- or weak-line approximation may be used, but the former approximation is valid over a much vider range of pressure than the latter. For extrapolation to small values of W, all three approximations may be used in their respective regions of validity; however the nonoverlapping-line approximation is valid over a wider range of pressure. in general, atmospheric slant paths that are of interest to the systems engineer contain rel.tively large amounts of absorber and the range of pressures are such that the strong-line approximation to any of the models is applicable. As will be seen in section 3, almost all researchers used only the strong-line approximations to the various band models to develop equations for predicting absorption over a specified frequency band. 2.2.5. RANDOM-ELSASSER BAND MODEL. At small values of'p the same absorption is predicted by the statistical and Elsasser models, and is determined by the total strength of all the absorbing lins. However, as' increases the results calculated from these two models begin to diverge; the Elsasser theory always gives more absorption than does i;,^ statistical model for a given value of P. The reason for this is that there is always more overlapping of spectral lines with the statistical model than with the regular arrangement of lines in the Elsasser model; thus, for a given path length and pressure, the total line strength is not used as effectively for absorption in the statistical model. An actual band has its spectral lines arranged neither completely at random nor at regular intervals. The actual pattern is formed by the superposition of many systems of lines. Therefore, for some gases and some spectral regions the absorption can be represented more accurately by the random Elsasser band model than by either the statistical or Elsasser model alone. The random Elsasser band model is a natural generalization of the original models which assumes that the absorption can be represented by the random superposition of Elsasser bands.. The individual bands may have different line spacings, half-widths, a'id intensities. As the number of superposed Elsasser bands becomes large, the predicted absorption approaches that of the usual statistical model. 26

.-.... - WILLOW RUN LABORATORIES —The absorption for N randomly superposed Elsasser bands with equal intensities, halfwidths, and line spacings is given by A 1 - e- rf 2W ) ~A &,uctAv-,~t1Ul U uIL equation and tie more general equation for different intensities, halfwidths, and line spacings is discussed in detail by Plass [10]. The more general result is not presented here since the result given above is the only function which has received application. 2.2.6. QUASI-RANDOM MODEL. The fourth band model that has been developed [13, 19] - the quasi-random band model-reportedly presents a more realistic model of the absorption bands of water vapor and carbon dioxide. It does not require that the lines be uniformly nor randomly spaced, but can represent any type of spacing which may occur. It also accurately simulates the intensity distribution of the spectral lines including as many of the weaker lines as actually contribute to the absorption. Furthermore, it provides a means of accurately calculating the effect of wing absorption from spectral lines whoLe centers are outside the given frequency interval. The quasi-randor model allows for more accurate prediction of absorption than do the other three models but sacrifices simplicity in the process. When this model is used the general transrnissivity function cannot be expressed in terms of elementary functions. Moreover, to determine accurate absorption spectra by this method, a priori knowledge of the band structure is required,-as it is in the rigorous method. The quasi-random model is characterized by the following features: (1) The frequency interval Av for which the transmission is desired is divided into a set of subintervals of width D. Within the small interval D the lines are assumed to be arranged at random. The transmission over av >> D is then calculated by averaging the results from the smaller intervals. (2) The interval Av is divided into subintervals D in an arbitrary manner. The transmission is first calculated for one mesh, which defines a certain set of frequency intervals. The calculation is repeated for another mesh which is shifted in frequency by D/2 relative to the first set. In principle, the mesh can be shifted n times by an amount D/n. The final calculation is then the average of the results for each of these meshes.

- -- WILLOW RUN LABORATORIES (3) The transmi3sion for each subinterval D is calculated from expressions which are valid for a finite and possibly small number of spectral lines in the interval. The expression is not used which is valid in the limit as the number of lines becomes large. (4) The spectral lines in each frequency interval are divided into intensity subgroups which are fine enough to simulate the actual intensity distribution. In the calculation, the average intensity of all the lines in each subgroup is used with the actual number of lines. (5) The contribution from the wings of the spectral lines whose centers are outside a given interval is included. The wing effect is treated in the same detailed manner used for evaluating the contribution from the lines within an interval. Let us consider one interval Dk of a given mesh. If there are nk lines in the interval with their line centers at the frequencies vi (i = 1, 2,..., n) then the transmission at the frequency L is affected by these nk lines so that n- n — =-k 7 I exp [-Wk(v, )] d34) D Dk where W is the amount of absorber and k(v, vi) may be expressed as k(v, vi) = Sib(v, vi), b(v, v.) is called the linershape factor (Lorentz, Doppler, etc.), and S. is the intensity of the ith line. The transmission is calculated at some frequency v which for convenience is usually taken at the center of the interval Dk. Since this transmission corresponds to the average of all permutations of the positions of the. spectral lines within the interval, it is also assumed to represent the most prouable transmission for the interval. To be completely rigorous the transmission of each spectral line should be calculated from its intensity and position and then substituted into equation 34. This procedure would require a computation time analogous to that required for the rigorous method and hence would defeat the purpose of employing a band model. Thus, a method is used which simulates the actual intensity distribution. The procedure adopted for intensity simulation is to divide the lines in each frequency interval into subgroups by intensity decades. When the line intensities are calculated, the results are grouped according to these intensity decades. The average intensity S of the lines in each dcade iS tihsi;n ClIcuilted. Wyatt, Stull,.and Plass [19] have found that only the first five intensity decades for each frequency interval influence the value of transmission. Thus only the data for the five strongest 28

- --.WILLOW RUN LABORATORIES - intensity decades need be retained for each frequency interval. It should be emnhusized that the numerical range of the intensity decades retained is quite different for a fi equency Interval which contains strong lines than for one which has weak lines. The criterion is applied separately to each frequency interval. Thus, the transmission can be calculated by the following equation from the average value of the intensity Si together with the number of lines n in each intensity decade: 5 -SiWb(v, v) T (v) i-7 D Pd nv i. (35) The number of lines nk in tLt frequency interval is given by 5 Iik-( ni (36) i=l Equation 35 represents the transmission at a frequ: icy v for the interval Dk as affected by nk spectral lines whose centers are within the interval Dk. The integral in equation 35 represents the transmission over the finite interval Dk of a single line representative of the average intensity of the decade. The transmission at a frequency v is also influenced by the wings of the spectral lines whose centers lie in interval. outside the interval Dk, denoted by Di. The wing transmission, Tn., from each of these intervals Di is calculated by assuming a random distribution of the lines within the interval D.. Thus, by the random hypothesis,thetransmission at the frequency v is the product of all transmissions calculated for the lines with centers in intervals, outside Dk, taken with the transmission for the interval Dk. Therefore 00 T() 77 Tn.() (37) where the subscript ni indicates that there are ni lines in the frequency interval Di. The transmission for the lines whose centers are in the interval D are given by equation 35. The lines in distant intervals from the particular frequency v have a negligible effect on the transmission, so that in practice the product (eq. 37) usually needs to be evaluated over only a few terms. It is clear that the quasi-random model can reproduce the actual line structure fairly accurately if the chosen interval Dk is small enough. However, the smaller the interval the 29

- WILLOW/ RUN- LABORATORIES... greater the amount of computation required. The tradeoff between the two has not been discussed by the authors, but such an investigation would indeed be of interest since any particular application has a tolerable error which might be used. to establish the frequency interval and hence the degree of computation. 2.2.7. KiNG MODEL. All previous models generate expressions which can be used to calculate absorption averaged over spectral intervals in the order of 5 to 10 wavenmmbers. To calculate total-band absorption using any one of these models one would necessarily calculate the narrowband absorption at each wavelength throughout an absorption band and then evaluate the integral A dA, where A is the absorption averaged over some A- which is very small compared to the total band King [14] sought an expression which would yield the total band absorption directly by performing the frequency integration prior to tIn empirical fitting procedure. He first assumed the band structure was described by the statistical or Elsasser nmodels. Since each of these models assumes an infinite spectrum and absorption bands are finite, the frequency integration cannot be performed without some modification of the model. Every absorption band contains many spectral lines of varying intensities. ToTse rtw the center of the bind are more intense than those near the wings of the band. This distribution of line strengths gives rise to a band envelope that King assumed to be approximately exponential. That is, the line strengths decrease exponentially from the band center. irrom this assumption, King used an exponential envelope to modulate the spectral-line arrays as given by the statistical or Elsasser model. Either modulated array was then integrated to yield ah expression for the total-hand absorption. Recall that the narrowband transmittance as given by the statistical model is T = exp J. 7 S/ai. t [{ (s)w/ d ]1 where S, d, and a are all taken within the band element. In the opaque line-center approximation (strong-line approximation), SW/ra >> 1. Therefore, the banad-elerr.ent transmittance becomes r 2 exp -(naSW/d ) | exp [-(2k W)] (36) where we have delined the harmonic mean absorption coefficient k by. g v Jexpp(52) i~dW=n 30dW exp dW =d (37)'3Cn

-- -- WILLOW RUN LABORATORIES Let us assume that these elements are modulated by a band profile whose shape is given by kn =(k) exp(-I i/) - (38) max leading to dk /d= -kn/k e (3s) where v is the band scale-width and is a measure of the steepness of the band profile.'e The total band transmittance is the average of the narrowband transmittance over the finite bandwidth Av. So T I /2 F,xp E(2knW)/ dv (40) AP/2 By changing the dummy variable of integration from v to the harmonic mean-absorption coefficient k, one can erpress the transmittance as a Laplace-like transform. Thus k,2 min r 1/2 d exp (- 2k W) 1 dkn (41) max Substituting equation 39 into equation 41, we have k, u t.n min dk T /= 2 exp[( 2kw) l2] k n2 nmax 2 E -2in E12k (42) ( I mi Iin maxT) ] where E1 is the first exponential integral and is given by. oo -tx 1.For the strong line approximation, kn < 1 and k W >> 1. By expanding the exponential nmt nmin max integral, one obtains an expression for the.total-band absorption. Thus 31

*-_..-.- WWI LLOW RU N LABORATOR I E S,2k A =1- t = 2( (7 ) Fn +-' a+2 kW] 2kn min und AAv= A dv=2 W + ln2k +.2a (43) max where a = 0.5772156 is the Euler-Mascheroni constant. For the weak-line approximation, k W << 1 and k < 1, and equation 42 can be approximated by nmin - max maxn and AAv= |A dv= 4ve2k W (44) ~~J vmax Expressions 43 and 44 then represern the total-band absorption for the strong- and we ak-line approximations, respectively, If one assumes that the ordering of the line array in the band elements Is statistical. If we assume that the lines in the subset are spaced regjlarly, we have, from the Elsasser model in the opaque line-center approximation, 1/2 = erf (raSW/d ) erf (knW/2)1/ Using a procedure parallel to that used in the development of equation 42, this equation leads directly to the following expression for transmittance: -v tnax k(Wif/2-dk T " e r \2 nerf n min 32 2( T)I F - 2 32 i~~~~~~~~~~~~~ W k W

-WILLOW RUN LABORA'TORIES where an error-function integral is defined by analogy to the exponential integral ly to t Fl(x) = erf t x The expansion of Fi(x) fer small argument is given by Fl(X) - - In x + 2x/a1/ - 2x3/971/2 where 6 - 0.981. Similarly, the strong-fit expression is obtained by analogy.:;h equation 43: A^ v 2v (n W + 1/2 In k + 26 (45) nmax / The weak fit is obtained using the expansion for F (x) and a procedure simila.r to that used in equation 44. 1/2 -/2 nax A. v = 8ve ) Four expressions have been derived which express tota? integrated- bana absorption under conditions of stronr. and weak absorption in terms of three parameters v, W, and kn e nmax These parameters can be determined by empirically fitting the respective functions to laboratory homogeneous-path data. Once these parameters are specified these expression's can be used to predict total-band absorption for other homogeneous paths. It is emphasized that one should reduce the laboratory data to standard temperature and pressure (STP) before one empirically evaluates the band parameter. The absorber concentration at STP is denoted by W* and is given by W*' p [ T n and is given by W = W (, where P = effective broadening pressure and n is a function of the absorbing molecule. Then the respective equations can be used to calculate absorption for atmospheric slant paths by first reducing the absorber concentration i.L the path to an eq.;ivalent sea-level concentration W*. The procedures for evaluating W* are discur.ed later. 2.2.8. BAND-MODEL LIMITATIONS. In the p"eeedingr pira<,- ph,'uye mnortpl R of tho hoIn structure were discussed, four of whic,. ed to the derivation of clos.d-form expressions which can be used to calculate absorption spectra. Although the use of these models greatly re?. ces 33

......... WILLOW RUN LABORATORIES the amount of computation required by the method of summing the contribution of each spectral line, their use is fubject to several limitations: (1) By their very nature the models are such that they can only simulate the actual line intensities and distribution. Since the different absorbers have different line structures, a model that may be applicable to one gas may be inapplicable to another. Also, the model may be reasonably accurate for one spectral interval but very inaccurate for another. (2) The solutions lose their simplicity when a mixed-line shape is used rather than the pure pressure-broadened Lorentz line shape. Therefore, models are applicable only for a range of pressure for which the Lorentz shape is valid. (3) The spectral resolution for most models is limited. For the quasi-random model this is not the case if the averaging interval is small enough, but under these conditions the simplicity of the model is sacrificed. (4) A band model can be used only to predict absorption for homogeneous paths. Therefore, atmospheric slant paths must be reduced to equivalent sea-level paths or to equivalent homogeneous paths, both of.which involve the use of the Curtis-Godson approximation. This approximation is valid only for constant-temperature paths under conditions of strong-line absorption. (5) The use of band models is further limited by the accuracy of the laboratory data upon which the final solution is based. These limitations may sound rather severe, but for the range of atmospheric paths for which the systens engineer wishes to determine absorption spectra the restrictions imposed by the band models are sati'sfied. It has been shown by Plass (201 that for altitudes below 50 km (^ category encompassing most paths bf interest) under conditions of strong- or weak-line absorption, the absorption can be represented by the Lorentz equation. Also, the slant paths of interest are usually lon.g enough that strong line conditions are present and therefore the Curtis-Godson.approximation is usually applicable.' It is emphasized that before a band-model method is used to compute the absorption spectra for an atmospheric slant path, it should be verified that the approximations inherent in the band-model methods are satisfied. This is done by, first, approximating the values of 3 and i and then comparing the results with the information in table 1. 2.2.9. TEMPERATURE AN -FDFREQUENCY DEPFNDECEr OF -ASND-MODEL PARAMETERS. A summary of the band models that yield closed-form expressions for spectral-band 34

WILLOW RUN LABORATORIES - absorption is presented in table 2. All of these expressions are functions of two parameters, P and 4, which are functions of temperature, pressure, absorber concentration, and frequency. Recall that 2ra 22 a P (T.n o d =d P VT and SW SW Po \' -2r 2ra0 J'T) including the dependence of line half-width on pressure and temperature. Also, the line strength is a function of temperature and frequency and is given by 3/2 )S = * exp- E z-. where SO is the line strength at standard temperature and E" it; the ground-level energy of a given spectral line. Because E" varies from line to line, the variation of line strength with temperature is different for each spectral line. Figure 10 clearly demonstrates this temperature dependence. As the temperature decreases, the line strengths near, the center of the band increase and the line strengths in the wings of the band decrease; the area under the curve decreases slightly. If two homogenecua paths having different temperatures were compared, the integrated absorption for the lower, temperature path would be less than the integrated absorption for the higher temFeratare path. Drayson et al. [5] have shown that for two identical slant paths having temperature profiles that differ by a constant amount of 100K the transmission varies as much as 10% in the wings of the band and somewhat less nearer the center of the band. They also concluded that the effect of temperature on line half-width has a secondary effect on trar.sm.ission. Since the variation of line strength with temperature is different for each spectral line, it would be impossible to include the effect of temperature on line strength and still:etdin the band-model expressions in closed form. For this reason, the band-modt I expressions in their present form can be used only to predict absorption for homogeneous,pai. that are at stindard temperature, Further, since Drayson et al. i5] have shown that the eflect of temperature on 35 L,E:i

..- WILLOW RUN LABORATORIES TABLE 2. SUMMARY OF CLOSED-FORM EXPRESSIONS FOR SPECTRA BAND ABSORPTION Band Model Approximation.. Equation. 1. Single Lorentz line None A 2s 4 I ) + ) 2. Single Lorentz line Linear A - SW/AV 3. Single Lorentz line Square root A = 2SoW/A7^ 4. Elsasser band None A = sinh J I0(Y) exp (-Y cos h3)dY 5. Elsasser band Weak A"- 1 - e4/ 6. Elsasser band Strong A = erf (1/2ji20)112 7. Statistical band (Poisson) None A I - exp [-3 /(L 2, 2" j 8. Statistical band None A = 1- exp-{L e (') 9. Statistical band (Poisson). Weak A = - e " 10. Statistical band (Poisson Strong A = 1 - exp jo) ]J and equal) 11. Statistical band (equal) Strong A I - expL[ t2 ], j 12. Random Elsasser band Strong A - - erf(2 3 6) 36

R — -WILLOW RUiS LABORATORIES —... T 1 00oo~K +~ 0 -5 v ^i1') +5 0 -5.v T 1= R.OK _"R~ P ~.10 +5 0 -5 -10 -15 v FIGURE 10. THE EFFECT OF TEMPERATURE ON THE DISTRHBUTION OF LINE STRENGTH 37

- - -.. WILLOW RUN LABORATORIES.. —lne half-width has a secondary effect on absorption, this dependence is neglected al'o. Therefore, if we let T = T0, the expressions for P and * become 20d P r d and S o S W (48) 200 P 2aa P(48) where ao is the half-width at standard temtwrature, per unit pressure. The expressions listed in table 2 are of the following general fcmn: (1) A = A(, /) when no approximation to the model is assumed 2 (2) A A A2i gP).for the strong-line approximation (3) A = A3, /) for the weak-line approximation Substituting equations 47 and 48 into 1, 2, and 3, we have the first expression as a function of two frequency-dcpendent parameters, 2ra./d and S/2na, and two path parameters, W and P. The second expression,.ves absorption as a function of one frequency-dependent parameter, 2raoS/d and two path parameters, W and P. The last is a function of S/d and W, being independent of press-ue. The next p:..olem in completely specifying the absorption expressions is that of evaluating the frequency-dependent parameters by empirically fitting the respective equations to laboratory homogeneous absorption spectra. The empirical procedure is the most involved when no approximations to the model are assumed since, in this case, two parameters must be evaluated, rather than only one for both the strong- and weak-line approximations. After the frequencydependent parameters have been specified and the values of W and P have been determined for a given slant path, it becomes a simple matter to generate absorption spectra. 2.3. DETERMINATION OF FREQUENCY-DEPENDENT PARAMETERS BY EMPIRICAL PROCEDURES The determination of the frequency-dependent parameters is undoubtedly the most critical aspect of developing a function for use in computing homogeneous-path spectral absorption. The theory of band models is reasonably well defined and the resulting functions should be capable of specifying spectral absorption to a good approximation if, first, the conditions inherent in the band-model development are satisfied, and second, rigorous empirical proce38

`,s-*4-w~z-o ~n - 7:v -... -I ~,.**<'':'... -r - Wl I L WILLOW RUN LABORATORIES dures are applied to high-quality laboratory homogeneous-path data to specify the absorption conrstants. It has been stated that the first conditional restrictions are satisfied by most atmospheric shnt paths of interest. Therefore, it seems hopeful that usable transmissivity furctions could be developed if the second procedural restrictions are satisfied. Oppel [21] has postulated and employed certain rules for selecting laboratory data and performing an empirical fit. His approach is sound and is analogous to that suggested by Plass [22]. Hence, the methods of Oppel are those which are recommended for the determination of the spectral absorption constants and are stated below. It has been found by most researchers that the quality and the reduction of the laboratory data have constituted the most critical step in determining the absorption constants. In order to insure satisfactory results from an empirical fitting procedure the following rules should be observed. (1) The data. should be consistent from one run to another; that is, all runs for a particular spectral interval should have the same resolution and spectral calibration. The absolute spectral calibration is not nearly as important as the relative spectral position from run to run, because the absorption constants can be determined for relative frequency units. Also, the spectrum can later be shifted if necessary. (2) Only data with absorption between 5% and 95% should be used. Usually the data for very high or very low absorption values are questionable because of the uncertairty of the 0% and 100% levels. and relatively small differences may have an inordinate.y great effect on the determination of the unknown parameters. (3) It is desirable to have spectra obtained over a range of paths and pressures which is wide enough that the function can be properly fitted to the data. Ideally, for each frequency, one would like to have data which would give absorptions between 5% and 95% and which include the following conditions:. (a) Pressure and absorption path are sufficiently small that there is negligible overlapping of the spectral lines. (b) Pressure is high and absorption path is small so that there is heavy overlapping of the spectral lines. (c) Pressure is low and absorption path is large so that the spectral lines are opaque at their centers and the transmission is achieved only in the wings of the lines.

' - * -';':/' -WILLOW RU'N LABORATORIES... (4) Measurements for which pressure self-broadening is predominant, that is, where the absorbing-gas pressure is an appreciable part of the total pressure, should be discarded. This is necessary since self-broadened spectral lines need not be evaluated under real atmospheric conditions. (5) In general, better results are obtained if spectral frequencies are selected at which absorption achieves a local maximum, minimum, or inflection point. Thus, the absorption constants should not necessarily be selected at uniforn intervals but for spectral positions that are more easily measured. It has been frequently observed that published spectra are hand plotted and that their author will tend to be more exact on the maxima, minima, a!:'l inflec- tion points than he will be on any other part of the curve. Atmospheric paths that are of greatest interest to the infrared researcher are long paths through an atmosphere of relatively low pressure. Such paths give rise to large values of V/A and small values of equivalent pressure P. The dominant spectral lines, those which account for 95% of the absorption, are opaque at their centers but do not display a significant amount of overlapping. Therefore, for such paths 3 is small (/3 < 1) and V* is large (V >> 1) and the strong-line conditions are valid. It has been stated that strong-line absorption must be present for an atmospheric path to be reduced to an equivalent homogeneous path. For these reasons most researchers assume that absorption is represented by the strong-line approximation, A = A(i3 4), and empirically fit this fmnction to laboratory data. When such a procedure is used, if the data used do not satisfy strong-line absorption conditions erroneous constants will be determined.I' After selecting the laboratpry homogeneous-path spectra according to the above rules and the appropriate model for represerting absorption for a given absorption band, one has the proper foundation for performing an empirical fit which will yield a valid closed-form expression for predicting absorption for other homogeneous paths. Basically, the absorption coefficients are determined by minimizing the sums of the squares of the errors between measured and calculated values of absorption. Before the least- squares iteration can begin, an enitial estimate of the two parameters 2,rad/d and S/27ra must be made. The method of determination and the accuracy of these initial parameters does not affect the final accuracy except that poorly chosen estimates will considerably increase the iteration time. Te method suggested is based on the fact that the three approximations to the Eisasser integral, described in section 2 always show greater absorption than that given by the complete Elsasser integral and approach the complete expression in the limit. Thit method can be described as follows: 40 I:::

WILLOW RUN LABORATOR ES- F (1) Three master graphs are drawn on log-log paper. The first graph (fig. 11), is a plot of Beers' law, given by SW A = 1 - exp (-#3); 3/ =, | The second master graph (fig. 12) gives the strong-line approximation to the Elsasser integral: A=erfj[(^22q) f:3 4= — 2 WP The third master graph (fig. 13) gives the nonoverlapping-line approximation: A -,[I40(p) + i1()] exp (-). 2ra': d PI * - 2-rl P | (2) Three working graphs are plotted for each spectral frequency for which absorption constants are to be determined. (a) In the first working graph (fig. 14) fractional absorption A is plotted as a function of the absorption path W (atm cm). (b) The second graph (fig. 15) gives fractional absorption versus absorption path times effective broadening pressure WP (atm cm x mm Hg). (c) The third working graph (fig. 16) is a plot of the ratio of fractional absorption to effective broadening pressure, A/P (mm Hig ) versus absorption path/effective broadening pressure W/P (atm cm/mm Hg).. (3) The six graphs are now paired off in the following order: figure 11 is superposed over figure 14, figure 12 over figure 15, and figure 13 over figure 16. With a good spread of data, it now becomes apparent that the several approximations are limiting functions and all points on the graphs will fall below the limiting curve. Figure 11 is adjusted in the W direction until all plotted points of figure 14 fall below ancd to the right of the curve draw n on figure 11. Thus, the curve of figure 11 forms the liit ing continuum for the points of figure 14. The relative positions yield the approximate value (S/d) = (fi4/W) Similarly, figure 12 is superposed over figure 15 and adjusted along the axis'r..il the limiting continuum is found, yielding the approximate value

Z 1~~~~ 2 FIGURE 11. WEAK-LINE APPROXIMATION TO FIGURE 12. STRONG-LINE APPROXIMATION TO THE ELSASSER MODEL. A = 1 - exp (-34);.i V/21 -4 = SW/d. THE ELSASSER MODEL. A - erfn(l V/); I~ E 101 10 10.- 0.1 l l 1 00 >- 0 -3 l02 0 -1 1 4-' W (a tm cm) FIGURE 13. NONOVERLNE APP INLINE TO FIGURE 1. WORKNGINE APH FPROXIMA ESTION TO HMODEL. A/ = 4 [I(,) + ^A I - exp -; 4 = (S/2lra)(W/PY. 001,L_ ^. r.., 00' S/d1 w.:.....,*~ 1 — 3 2'd1 4 10" 10a 10 1 10V 10-3 10- 10 S WPe (atm m x mm Hg) W/P (atm cm/mm Hg) - FIGURE 15. WORKING GRAPH FOR FSTI- FIGURE 16. WOtKING CRAPH FOR ESTI - MATING 2na6S/d2. For these data 2avcbS/d2 MATING 4na&O/d and S/2ffac. ror these data 4 4.5. 2flfcf/d s 0.3 and S/2~ac $ 500. 2. 0.1 I 0 100 3 -',.. W (ztm cm) FIGURE 13. NoNOVERLAPPING-LINE FIGURE 14. WORKING GRAPH FOR ESTIAPPROXL-AATION TO TSHE ELSASSER MATING S/d. For tliese~data S/d ~ 15. MODEIL. A/p [1O() +!l(~exp -i; (i4i wp e (atm cm x mm Hg) W/pe (at~n cm/mm Hg)4 42

- WILLOW RUN LABORATORIES O.-. 2 / =(WP) The third pair (figs. 13 and 16) can be adjusted in both -directions to give approximations - - to two parameters: (2ra\ A/Pe \T~ = (A-. and. The three plots indicated above are not all necessary if sufficient data are available since only two constants are necessary to evaluate the Elsasser integral, however, experience has shown that sufficient spread of data is usually not available to adequately construct allof the graphs. For this reason, the best graphs should be chosen. In the event that thz..onoverlapping approximation gives a good continuum, then it may be used alone for both approximations to the unknown parameters. At this. point it is assumed that the initial estimates of the parameters have been made and the laboratory data have been properly chosen. Beginning with the first estimates, the absorption is calculated for each value of absorber concentration and each value of equivalent pressure and compared with the measured value. The parameters are adjusted until the sums of the squared residuals between computed and measured absorption values arc minimized. - A computer program has been written by Baumeister and Marquardt and distributed as a SHARE program [23]. This program, when coupled with the Elsasser-model alogorithm described in section 2, will optimize the parameters for the Elsasser function. The procedure described above is that which should be used when fitting the complete Elsasser integral to laboratory data. It is emphasized that if the strong-line or weak-line approximation to the. Elsasser model is used to represent atmospheric absorption, then the laboratory data must be such that these conditions are satisfied. This can be determined by using the initial estimates of 21rab/d and S/da% and the values of W and P to calculate ]t and V/. These results can then -<'9omrparmed with the values in table 2. A completely analogous procedure should be used when fitting the Goody model to a set of laboratory data. That is, all three.equations should be fitted to the data to obtain a set of 43

:El.~,., -! ~?; -:..:.....,. - -; -- - WILLOW RUN LABORATORIES - - - frequency-dependent parameters. Parameters obtained in this manner will give valid results whether the absorption is strong, weak, or of some intermediate value. The method described above for performing an empirical fit of a band model to laboratory data is the suggested procedure. However, it has been used by only two of the many researchers that have developed functions for calculating atmospheric absorption. Of the various methods that have been employed, some are similar to it and others are very different; in section 3 each of the empirical functions developed by the various researchers is presented and the em- pirical procedures employed by each of these researchers are discussed.- Empirically fitting a band-model expression to homogeneous data generates coefficients that represent the best value for all available data. The absorption constants then do not actually represent the strengths, widths, and spacings of the spectral lines but can be interpreted as a set of lines giving equivalent absorptioln. It can be expected that calculations should be satisfactory for conditions bounded by the laboratory data. However, for extreme cases the results may be rather poor. It is difficult to theoretically analyze the accuracy with which a band model can be used to predict absorption, so the approach used here is simply to compare the results with field measurements. Such comparisons are presented and discussed in section 7.;.4. CONVERSION OF SLANT-PATH TO EQUIVALENT-PATH PARAMETERS - Since the band-.iodel expressions can be applied only to homogeneous paths at standard temperature, any real atm3spheric slant path must be reduced to an equivalent homogeneous path at standard tempe-rature or an equivalent sea-level path which is a homogeneous path at standard temperature and pressure. In the discussions that follow, the methods for determining the equivalent paths are derived for various types of absorption and the assumptions and approximations used are discussed. The absorption spectra for any homogeneous path, assuming a constant temperature, is defined by the absorber concentration W of each gas *ad the equivalent broadening pressure P which determines the half-width of the spectral lines. Therefore, to determi: - an equivalent homogeneous path for an atmospheric slant path, it is necessary to determine a value of W and P for each absorbing species such that the slant pati will absorb exactly as does the homoge- neous path. Although this cannot be achiev ed exactly, since the intensities and half-widths are continually changing from point so point along a slant path, under certain conditions i:ie conce::.n lenrt path is a reasonable apprcximation. 44;a,

I. Recall that the general equation for absorption over an atmospheric slant path is given by The absorption of a homogeneou~s path is given by h f ft L ( n (5d) 2.4.1. WEAK-LINE NONOVERBLAPPING CONDITION. Let us consider the case where every spectral line in the interval Av is a weak absorber and the spectr. lines do not overlap strongly. This will be true in general for short paths at high altitudes. Under these cunditions the pressures are lo-v and the half-widths are narrow. If tenc absorption is small at the lI'.ne center for each spectral line within the interval Qv, the slant-p.th absorptior: may be approxi-.;ated by taking the first two terms in the expansin of the exponential. Thub euation 49 reduces to SN Sp N 1h (hI) A A^A^ If ^ \ —iP dv (5A1 e LV"^ 2 + a Further, for each v0 within Av the entire spectral line is within Av, sc the order of inteeration may be interchanged, giving Sence Av encompasses the entire spectr2l line,;he limits may be taken as -to to oo with no loss in generality. Now..... de to I 2 d \ - - + a.

__-__WILLOW RUN LABORATORIES Io euatton 523 becomes A3Av |( t)dx | or N X ASA, t (X;pdx (54) I! it IS further assunme that the line streitht St a.e tpdpendent o tShe ptth, the. n. N rK A IvN' St pdx (:S5) t~ ~ ~ ~ ~ ~~~~~~~~~~ t I, I:! 1t we prform han aa..* e rvelb r ient tor a. horntere. u -ithn.e ot, W * * A:. W \, - (5tj A A(LA's 1' s w AD 46, Ntxti *, f. S~ i, -'',.', *. * -'" "'rX'-'. "I 12w r IS i te'G! d >t tk s 11W atxortrAne i nadepeeni o1 pressure l td v ts cthw*te tetparatf$ed y'Y....,.,I.. StN LINE **d.QWA E i Rt $ E Pi. flecalt t.at th Aso. t.,rtpic.' )n A ~ At'4, W) (vreakfl*n p a roximnato~i j:I: 46: I;

- --.. WILLOW RUN LABORATORIES -_... To develop an expression for the equivalent pressure for a slant path, let us consider'he case In which all of the lines within Av are strong-line absorbers. This ctaracteristic implies that absorption is complete to wavenumbers that arv in excess rf t*h live half-width. Therefore, a further increase in path length simply increases the absorption in the wings of a sfectral line and has noaffect on the absorption at the line center. For the Lorents line shape this means that ( - >^ > a for all lines. Using this approximauon, we may write equation 49 as aSN P A,,, I - xp t } S ax dv ( For a homogeneous path, we have af' h W..,- hexp._ —. h (59) LI t [ _J3 L If it is assumed a hbefore that St does n00 vary along the stant path, then the slant path wtll aibsorb approximately as a h'mogeneous pah with -x t~ h %, ] tpSc dy ({0) Recall that h, O POT (61) Al) P o, pA nT "0 As wras mentioned previously, th-e value o pressure used In equation 62 should he the effective broadening pressure. However, the partal pressure of any absorbing gas is very small compart-d to atmospheric pressuree, so the error involved in assuming PS to be total atmospheric pressure is assumed to be negligible, Substitut'ng these expresstins into equation 60, we obtain 47

....WILLOW RUN LABORATORIES WhP1. P\ PS Suwtitutlng In the slatt-path integral expression for Wh, we have IX n T'Ihe assumptions imposed thus far have been strong-line absorption and line dtrength indepe.,dent of path. U we further assume that the path s Isothermal and equal to Th, which in effect completely neglects the effect of temperature on half-width, then the pressure of the nomogeneous path. such that the homogeneuus and slant paths absorb the same, is g:iven by P ~PS d -a --- (65) h X i ~sdx Equation 65 is then the expression for obtaining the eqaivalent path p ressure. This pressure will tb denoted by P. This concept was first considered by Curtis and Godson (24, 25] and is therefore referred to as the Curtis-Godson approximation. The equivalent -path parameters are given by equations 5'7 \nd 65. Equation 57 is valid under any conditions of atsorption. assuming only that the line strengths,.re inoependent of path. Equation 65 is valid only for those conditions of strong-tine absorption in which the effect of temperature on the spectral-line half-width and line strength are neglected. 2 4.3. EQUIVALENT SEA-LEVEL ABSORBER CONCENTRATION W~. Some authors have emprically determined their frequency-dependent parameters from laboratory data that are at j standard pressure as well as standard temperature.. Therefore, their newthcds can only be used to predict absorption for taher homogeneLus puths that are at standard concitions. To apply i such methods to atmosphleric slaat paths, the paths would have to be reduced to equiv;alent honogeneous paths at STP. Such paths have been defined as equivalent sea-level paths and are denoted by W. P p dx S S~~~~~~~~~~~~~~

—.WILLOW RUN LABORATORIES Returning to equation 60, a stlnt path will absorb approximately as a homogett ovs path witf %hWh 0 u pS (d} As was stated previously, thia expression is valtd only for strong-line atsorption with S independent of path, For this case the homogeneous path is at STP so o and equation 66 becomes *hi i and, finally, the equivalent sea-lev l absorber concentration is given by " WhJ PG I Ps dx It is Jieiul to note that under conditions of strong-line absorption (neglecting the effect of temperature on half-width and line strength) the equivalent sea-level co-entration is equal to the product of the weak-line noroverlapping absorber concentration and the equivalknt-path pressure normalized to..tindard pressure. Or w w Zp (68) 24.4.. GENERAl. COMMIENTS ON W AN W, tD W. ree expressions have been derived which will be used throughout the rerrainder of the report. First is the weak-line nonovertapping. bsorber concentration W, which is simply the grams of absorber in the slant path. Second is t e Curtis-Godson equivalent pressure P. The concept of an equivalent slant-path pressure is va id only, under conditions of strong-line absorption and isothermal slant paths. Also, this con~cep requires that the effect of temperature on half-width be neglected. The last quantity which was derived is the "equivalent sea-level path," W*. This equivalent concentration is effectively the grams of absorblr in the slant path adjusted in accordance with the dependence of half-width on temperature and pressure. This concept is valid only under conditions of strong-line absorption and for isothermal slant paths. 49

------— WILLOW RUN LABORATORIES — _. — To develop closed-form bndr m..el expressions for absorpt' n it was nece!laury to ne'V..~t the effect of temperature on line strenrth an: l'ne half-width. Such approximations were also required to determine the equs tlent-path parameters. AMthough this assumption d)es, of coure, introduce errors in,~te final result. Drayson has shown that swu:h errors are not large. Aso, there is an aided fact which further reduces the error introduced by this assumrpton. Consider the Integrated absorption for an entire spectral band, It has been stated previously that as the temperature decreased the integrated absorption decreases. However, the line half-wdths be- come larger for smaller temperatures and thus give ride to increased absorption. Therefore, rather than being additive the errors tend to cancel each other. j The nr.tegral erpressions for W and W' given by equatIons 57 and 6'; yield, respectively, the grams per square centimeter and edfective grams per square centimeter.',r a lant path. The accepted practice is o normalize these quantities by dividtng by the t:erilsty of tht absorbing gas at standard t'mnprattu.re anci pressure. For the? gases CO. 03 N,0, CO. an- CH4 W and 2* 3 an 4 C W* wilt have units of centimeters slince for each of these gases PST is expressed in grams t per cubic centimeter. Actually, W represents the lengrth oW a column o piure ga. 1 cm in in crosx. ectionT ^.t ST? which conrxns the same number of grams as the sla.tt path. For W* the se-ngth ti adjusted in accordance with the varia' ion of half-width with pressure and temperature, Accordirn to this deflnititx. the units of W and W* are commonly referred to as atmospheric centimeters.'Trve r.its ol W and W are dclfned somnewhat differently for water vapor. For a given slant prtth the amount of water va,-)r.s retu.coi, to an equivalent amount of liquid water at STP or 3 precipitabie centimeters Oi water. she normalizing (factor is, therefore, the density of liquid water rather than water vapor. There ss no justification for using different units for the two cases: it has simply been the acceped practice since transmission work began. 2.4.5. A GENERALIZED CALCULATION OF W. P AND W*. To clarify the exact technique for evaluating W. We, and P for each absorbing species for a slant path, consider a path that begins at the earth's surface and extends outward to the limits of the atmosphere. As a result of the refracting characteristics of the atmosphere. the line of sight will be slightly curved rather than straight, and the amount of curvature will increase as the path becomes more hoi-t} zontal. The first step in determining the reduced optical path is to determl;e the actual refractive path thro'uh the atmosphere. A computer program has been written in FORTRAN which performs this computation. The analytical procedures and computer program are discussed tn I~ lrFe

---- -- --- WILOW RUN LABORATORIEES detail in appendix I. After the refracted pith is determineS, the values of pressure and temperature and the mixing ratio of each absorbing species can be expressed s f.uctions of distance along the puth. These are represented by P(x). T(x), and M(x). respectively. For the computation of W we have from equation 67: *w',,......i_ h P(x) (( w'J wo Ela~l —r-\ ~aLJ ~di (69) MP4 a uP P, T(x). RT(x) HT) where P0' surface pressure TO surface temperature M w molecular weight of absorbing gaas gas.M i molecular weight of air R S gas constant M(x) mixing ratio in grams of gas per gram of air X path length in centimeters Since the mixing ratio (or gases is usually given in moles of gas per mole of air, the mixing ratio must Jte converted to grams of gas per gram of air by multiplying by the ratio of their o;-l^.. w,:ghts. The equation for determining W= for a slant path becomes 0g~r Tx ) Of n.,,(7) W - ( ",:(x),iO where SM(x) is the mixing ratio in moles of gas per mole of air. The equation for computing the equivalent slant-path precipitable centimeters of water differs slightly from equation 70 sinco;e miring ratio for water Is given in grams of H20 per kilogram of air, and the density of H12 at S7P is used as the normalizing factor. For this case it is easily showr that the precipitable centimeters of H20 in the slant path are giv2n by 2 I where M(x) is the mixing ratio iri g/kg. In summary, equation 70 is used to compute atmospheric centimeters for gases for which the mixing ratio is given in moles of gas per lii-A of.*i. Eui.ji *1 i6 ujev ciii i.51 {*

--..-. —-—. WILLOW RUN LABORATORIES --- preciptiable centimeters of water vapor if the mixing ratio is given in grams of H. 0 per kilogram of air. To rompute the reseective values of W, simply let the exponents of the pressure and temperature terms in equations 70 and 71 be unity. To determine P for a slant path, fir6.ompute the appropriate valae of W*, letting n * 0. Next, compute the value of W as stated abo\;.. Then P will be given by Pwl lnQPO P w In cojunction with the r:,fractive-path computer program, programs for computing W W, and P have been written in FORTRAN for the IBM 7090. These programs are also included and discussd in appendix I. The use of a computer program to evaluate the quantities W, W*, and P for each absorb,,.. gas for an atmospiheric slant path is a rigorous yet practical method for determining the equiv:lent-path parameters. Many recearchers (26-28] have proposed met,ods for evaluating equ. L tions 70 and 71 without the aid of a computer by approximating the mixing ratio, pressure, and temperature proliles with analytical functions and thus allow the integrals to be evalated by direct interpretation. Such methods yield adequate results for a fixed set of atmospheric con- cltions, tnt are not capable of handling vartable meteorological conditions. This can be a ratler severe limitation in the case of water vapor and oxone. Therefore, it is recommended that thtse methods not be used unless only iirst-order approximations to the path parameters are required. The cdmputer pi,gram is simple to apply and perforns an exact numerical integration. Therefore, it is flt that there is no need to introduce approximations to a problem that already i has a relatively simple solution, 2.5. SUMMARY In this section four basic methods for computing atmospheric-slant-path molecular a bsorption have been discussed. The first involved a direct-pressure integration over the slant path,'summing the contribution of each spectral line. including the effect of temperature and'pressure on line strength and half-width. This method requires complete a priori knowledge of the band amrameters, including line strength, line half-width, and line location. The absorption spectra are computed by direct integration of the general transmissivity function. The second method employed the use of the quasi-random model of the band structure.. This model simulates the band structure in a prescribed way such that the absorption spectr2 * 52

._............ WILLOW RUN LABORATORIES - for any homogeneous path may be. determined. This method requires some a priori knowledge of the band paramete s;as does the -'-orous method), but once the absorrlton spectra are determined for one homogeneous path, a set of coefficients is generated so that the summing procedure need not be repeated to determine the absorption for a different h.omogeneous path. Homogeneous laboratory spectra are not a prerequisite for the use of this metaod but are ususily employed to normalize the absorption coefficients. The third method assumt-. models of the band structure which allowed for the derivat;,n of closed-form analytical expres,:;ons that could be used in conjunction witi: laboratory homo. geneous-path data to predict spectral absorption for other homogeneous paths. The models assumed were the Elsasser, statistical, and random-Elsasser models. The fourth method assumed models of the band structure and then modulated these line arrays with an exponential envelope. The resulting band model was integrated directly to obtain expressions for the total inte,'rated band absorption, A di. These expressions, in conjunctionr with laboratory homogeneous-path data, could then be used to predict irtegrated-band ab-" sorption values for atmospheric slant paths. In order to use the closed-form expressions derived by methods three and four to predict absorption for slant paths, the slant paths need first be reduced to equivalent homogeneous paths. Some of the expressions require that the equivalent path need only be homogeneous and express absorption in terms of W and P. Others require that the path be reduced to standard conditions and therefore express the absorption in terms of W*. The reiscn for developing intth.ds for computing absorption based on band models was to generate a method that required a minimal amount of computation yet would give results that we-:e at least a first-order approximation to the true results. Later in the report absorption spectra as.computed by the various methods are compared both with one another and with field measurements; the resulis of these comparisons are analyzed and discussed. 53 N.. "'"i' _*9*^ a*^j - Tmi!'Vgi>M _i w niii,.*t'""..|'."'

~-= WILLOW RUN LABORATORIES f N V3 ^ L t MErHODS FOR COMPUTING MO.ECULAR ABSORPTION SPECTRA IASED UPON BAND MODELS 3.1. INTRODUCTION In section 2, the general theor7'or computing atmospheric slant-path molecular absorption was formulated. The most useful results developed from the theory were the closed-form bandmodel functions. These functions express absor ptior as a function of one or two frequency* dependent parameters which are determined by empirically fitting the band-model expressions to homogeneous-path. laboratory data. Because many researchers have made such empirical fits, no two methods for computing absorption are exactly the same. This inconsistency results from the fact that oiffe'ent researchers selected different land models (or ditfe:ent forms of the same model), had different collections of laboratory data: and employed differing procedures to achieve the desired empirical fit. In this section titose methods which are considered most representative of the state of the art are represented. Each method will be discussed separately, and the band models, laborztry data, and empirical proc..dures us-ed by esch author will be stated. V 3.2.- NMETHOD C. W. M.':LSASSER Elsasser [9] developed transmissfvlty functions for the absorption bands of carbon dioxide, ozone, an. water vapor using a method which is a simple extension of the band-model procedures discussed in section 2.2. The strong-line approximation to the Elsasser and Goody models can be expressed.s T T (k(X)'.W*)' where k(XW is defL'^e as a generalized absorption coefficient and the functional frm of T is given by either the Elsasser or the Goody model. The empirical procedure outlined in section 2.3 was to assume:he appropriate form for T, which depended upon the absorbing gas under consid- } eration. The laboratory homogeneouts-r.n data were usvd to evaluate the frequency-depnendent parameter or parameters for each wavelength throughout the band. Elsasser did not assume 5 that the funct.nal form of T *aas given by either model, but used the laboratory data to define i both the functional (fo m of t and the absorption coefficient kAX). The procedure he used is dc- scribed here: 54: ell

WILOW RUN LABORATORIES — The homogeneous-path laboratory data we;e reduced to equivalent W* val;ee using the expreseson h PO The absorption band was'hen divided Into subintervals approximately 10 \iavenu;nber, wide a'd for each subinterval the mean trarnsmission for th, interval was plotted as a function of W*. Elsass r found that each of the curveft obtained for the various int.trv;ls had approximately the same functional form. Therefore, a st:and'-rd transmnissl ity curve (tbh-* unctional form of 7) was adoptea which expressed the transmission as a function of the product W',k).X)''.erefore, Elsasser proceeded on the inherent assumpt!on that an absorpt'~n band may be represented by a strong-line band-mo'el expression, but determined the functionral form of;.i;s expression empirically rather tnar. anr'ytically. The empiri-alll derived t,'aismissivity function unde:goes a shift alorg thp W*,l.te as one passes from one interval in the spectrum to an adjaceni interval. Elsaser plotted this shitl.s a function of wavelength and MittCd i curve to the data using a least-squares proctuire. This curve defines the generalized absorption coeflicient, k(A). Elsa.s.cr's results are preserted i,} the f''rm of tables which represent the twe functions, 7(k(x)) and k(X). Since the concept of stror.g-iine aLsorption was i:tilized 1i the develr^ment of these finctions, the method should be appl!,d o0ly to slant paths for which thLs condition is satisfied. 3.2.1. TRANSMhISSvTfY FUNCTION,: CA'RBON D'IOXI)E. Elsasser d.evelop.d trarsmiss;vity functions for only the 15-,t absorption band of CO2. The laboratory data up.n wih'ch the emplrical procedjres were:ased e.re taken from the work of Cloud [29]. Elsasser later considered the work of Hfoward, Burch, and Williams [30], Kaplan and Eggers [31], and Yamamroto and Sasarnort [321 and pioposed modifications io the or!grna? transmissivity funt.ions to obtain functions that are in agreneent with all three sets of lauoratory atta.:rhe lransmissivity function obtained is shown in figure 17, -nd the generalized?Vssozptlun coefficient derived frorir the shift in the 7 curves is shown in figurc 18. For convenience in determining atmospheric-transmission values, botr the transmissivity curve and the teneralizcd absorption coefficient curve are tabulated. The resu':s are,iven in table 3. Using the trantniisstivity curve shown In figure 17 and the ausorption coefficient shown in fi u re 1R, F!aer r^!.,''*-t ctcd th; atuoi;-iu;ii;tl.-i b contours ior vaious values of W- which 55

_-.. WILLOW RUN LABORATORIES... soo J 40 Accuracy of Data ie I 20 - --. - —. -4 -3 -2 -1 0 +1 LOG WI FIGURE 17. TRANSMISSIVITY CURVE FOR CARBON D!OXIDE 800 750 700 650 600 560 - 1 -2 -4' 800 750 700 65C 00 560 WAVENUMBER (em l) FIGURE 18. GENERALIZED ABSORPTION COEFFICIENT FOR 15-iL CARBON DIOXIDE DAND 56 <~* ^rnrl

-— WILLOW RUN LABORATORIES-. TABLE 3. THANSMISSIVWTY FUNCTION ANI AkBORPTION COEFFIClZNTS FOR CARBON DIOXIDE x v _..._k_,,ls.J_^ (4) (cm 1) (%) 11.77 8a0 1.59 E-7 100,00 8.31 E-6 11.95 837.5 5.01 E-7 99.97 7.94 E-6 12.13 825 3.16 E-6 99.91 1.00 E-5 12.35 810 1.48 E-5 99.82 1.26 E-5 12.50 800 4.07 E-5 i9.'0 1.59 E-5 12.6f 790 1.17 E-4 99.55 1.99 E5 12.82 780 3. 02 E-4 99.37 2 51 E-5 12.99 770 8 32 E-4 99.16 S.S6 E-5 13.16' 60 2.14 F-3 98.92 3.98 E-5 13.33 750 5.37 E-S 98.65 5,01 E-5 13.51 740 1.32 E-2 93.35 8.31 E-5 13.70 730 3.16 E-2 98.02 7.94 E-5 13.89 720 8.32 E-2 91.66 1.00 E-4 14.08 710 1.91 E-1 97.27 1.26 E-4 14.29 700 4.57 E-l 96S85 1.59 E-4 14 49 690 1.00 96.39 1.99 E-4 14.71 680 2.09 95.90 2.51 E-4 14.93 670 2.69 95,38 3..16 E-4 15.15 660 2.40 94.81 3.98 E-4 15.38 650 t.45 94.22 5.01 E-4 15.63 640 6.17 E- 3).J58 6.31 E-4 15.87 630 2.45 E-1 92.9. 7.94 E-4 16.13 620 8.91 F-2 92.18 1.00 E-3 16.39 610 3.16 E-2 91.-:1 1.26 F-3 16.67 600 8.17 E-3 90.59.59 3 E3 16.95 590 2.34 E-3 89.72 1.99 E-3 17.24 580 5.' l -4 88.79 2.51 E-3 17.54 7 570 8.9'1 E-5 87.80 3.IS E-3 17.86 560 1.0.7 E-5 8 -74 3.98 E-3 1a.2 550 1.26 E-6 85.63 5 01 E-3 - ~' 84.40 8.31 E-3 8JO.1.0?794 E-1, 1.70 l.00 E-2 80.19 1.26 E-2 78.56 1.59 E-S 76.80 1.99 E-2 74.91 2.51 E-2 " 2.88 3.16 E-2 70.71 3.98 E-2 68.39 5.01 E-2 65.92 6C31 E-2 63.30 7.94 E-,a 60.53 1.00 E- I 57 62 I.2 ^ F54.58 1.59 E-1 57

----- WILLOW RUN LA BOR ATORI ES -----— I ( TABLE 3, TRASMLSSIVrITY FUNCTION AND ABSORPTION COEFFICIENTS FOR CARBON DIOXDE (Continued) u) icml) (%) 51.42 1.99 E-} 48.15 245i E-1 44.78 3.16 I- 1 41.33 s.98E* l1 37.82 5.01 E34.28 6.31 E-1 30.75 7,94 E-1 i7.27. 1.00 23.89 1.26 20.66 1.59 17.63 1.99 14.84 3.51 12.30 3.16 10.02 3,98 8.01 5.01 6.27 6.31 4.80 7.94 3.60 1.00 1.47 1.99 1.13 2.51 0.89 3.16 0(7. 3.98 0.?8 5.01?,42 6.31 (I28 7.94 0.14 1.00 0.00 1.26 I 58

-- WILLOW RUN LABORATORIES -- are shown in fIrure 19. In order to compare the results using Cloud's data with the results of Howard, Yamamoto, and Kaplan and Eggers, Elsasser planimetered the curves shown in figure 19 to obtain values of the band area for each value of Wk'. This procedure evaluates the integral;Adp for each vale of W*. The band area as a function of W was then plotted and compared with the results obtained by other authors. The results are show n in tige 20. It is noted that Elsasser proposed a method of adjusting his absorption coefficients so that the curve of figure 20 would be in agreement twth other existing data. This method involved various steps. First, a curve (labeled "adopted") was drawn which most closely approximated the composite set of data given in figure 20. The difference between this curve and the curve marked "trial" along the Wm scale was used to relabel the curves of figure 19. For example, for W 1, the band area is approximately 85 cm. From figure 20, using the'adopted" curve, the new value of W is approxinvtely 0.4. After allthe curves shown in fie 1 are relabeled. they may be used in conjunction with the transmissivity curve of figre 17 to find new values for the atsorption coefficients. Since the method outlined above for modifying the absorption coefficients was not applied to Elsasser's original data, the data presented in this section are the result of Elsasser's original work on Cloud's data.. The amount of tompu'atlon labor required to perform the suggested modif.cation wocid be comparable to that required to perform the initial empirical fit, anid such a modification appears to be impractical. ft s1 suggested that the present absortion coefficients be used and the results obtatned be ad;usted according to figure 2C. 3.2.2. T'IANSMISSTVrTY FUNCTION: OZONE. Because the procedure used to ideelop the transmissivity function and absorption coefficients for ozone is completely analogous in all details to the one used for carbon dioxide, the:escriflXon in this section will be brief. Tht transmisstvlty function and absorption coefficients for ozone were first developed from the laboratory data of Summeifield [331. The results of the empirical fit are shown in figurfs 21 and 22. The transmissivity curve differs clearly from the corresponding curve for CO2 (fig. 17) rnt only In shape but also in that there is a much larger scatter of thl observed points. Using the transmissivity curve and the absorption coefticlents, Elsasser computed band spectra for various values of We (as was done for CO)2. These curves were planimetered and plotted as W' versus band area and comparedwith the experimental results of Walshaw [34]. The results are shown in figure 23. Note that Elsasser's curve is below Walshaw's curve for every. value of band area. Elsasser proposed that this a result of the nam. re of the pressure correction used in reducing the laboratory 59

WILLOW RUN LABORATORIES - WAVELENGTH () 12 13 14 15 16 1'7 18 X00 o.f9.r1t 850 8 750 70a h1P650 60 550 WAVENlbMBER(cm1) i FIGURE 19. ACTUAL ABSORPTION OF 15-, CARBON DtOXIDE BANPD: Core t responds to trial curve in figure 20.. 60'' te~~~I 60 0 <00 04

WIL1LCW RUN LABORATORIES ~~3.~~~~' ", —"?\t. *,' W*, 19,000 1_ Adoptd _W ~.^' Area 304 - Yamamnoto o \ \>: -, u O. *', / &'r, al Cw.ve -1 |-'j / Cloud's Data // - /;'. -2 /o Howard'~ Measurements [30 t/ I/ "* ~Cloud's Measurements f291!/'. Kaplan & Eggers (P w 950 mm Hg) [31] -3 25 50 75 100 125 150 175 200 225 250 BAND ARA (cm 1) FIGURE 20. COMPUTED AND MEASURED BAND AREAS FOR 15-u CARBON DIOXIDE BAND 61

-WILLOW RUN LABORATORIES -.- - R I 100 60 - U- 20-.......- p 0.6, 1.3, 2.6, 5.8 cm "' p- 72cm\........ I -4 -3 2 -1 LOG W* FIGURE 21. TRANSMISIVTY CURVE FOR OZONE WAVELENGTH (p) 8.89 9.09 9.30 9.52 9.76 10.0 10.25 0i Band Center i Center -2 | 1125.1100 1075 1050. 1025 000 975 WAVENUMBER (c;" ) FIGURE 22. GENERALIZED ABSORPTION COEFFICIENT FOR 9.6-4i AND 9-j OZONE BANDS

0.5/ 0-' ---- --— =....'0. 60 03.7 0 / *0- 5',20 30 0.20.22 0.J / I 0. / 10 cmS*2-1.5 60.303 f -/2. ~ P < 10 cn,' ~ P > 10 cm -2.55 0. in O Io0 20 30 40 50 60 70 80 90 BAND AREA (cmrn1) FIGURE 23. COMPUTED AND OEgSt;ir:D BAND AREAS FOR 9.6-u OZONE BAND 63 S~~~~~~~~~~~~~~~~~~~~~~~~~~ -~':

- ---- WILLOW RUN LABORATORIES --. data. Recall that W* = Wh-p ) where Pe is the effective broadening pressure. Elsasser WhP T /T e assumed a linear pressure correction for CO2 and 03 when reducing the W values to equivalent WI values. He felt that this assumption was valid for CO2, although it gave poor results for 0. 2'3 Elsasser further suggested that had he used a pressure correction of approximately 0.b, his original data would have been in close agreement with Walshaw's data. However, sinc e d'e dd not make any corrections to his original data, it is recommended that these ortglnil data b-. used for trancmisslon calculations; the user should remember, however, that the flas. Alues will overrr-ic~ absorption by approximately 10%. i As for CO2, the transmissivity function and the generalized absorption coefficients for 03 are tabulated. The results are given in table 4. 3.2.3. TRANSMISSIVITY FUNCTION: WATER VAPOR. Elsasser considered only the far- f infrared spectrum of H20; that is, he considered the 6.3-{A vibration-rotation band and the pure- { ly rotational band at the long-wave end of the spectrum. The transmissivity curves developed by Howard et al. [301 were used for both ilh bt.3-ts band and the rotational water band. Their two curves, representing a total pressure of 740 mm Hg and 125 mm Hg are plotted in figure 24, I together with the curves derived by Yamamoto [35] and D)aw [36]. All c, rves were shifted along the abscissa so they would coincide at the point of 50' absorption. F.sa:;Lur adopted the curve of Howard et al. which corresponds to a pressure of 125 mm Hg. This is the pressure to which the laboratory data were reduced before the empirical fit was made. it seems that the curve j for 740 mm Hg would be mole zealistic for absorption calculations since that is the pressure for which the "equivalent sea-level path" is defined. However, since Elsasser desired the greatest accuracy for high-altitude slant paths, he chose 125 mm Hg. The differences between t a.iy two of the curves is small enough that the selection of any one of the transmissivity curves would not produce an appreciable difference in the final absorption calculation. Elsasser adopted a curie for the generalized absorption coefficient for the 6.3-/i band using a trial-and-error procedure based on the work of Paw [36?, Yamamoto [35], and Howard [30]. Daw and Yamamoto separately constructed curves of the generalized absorption coefficient for the 6.3-z band; their results are shown-in figure 25. A plot of band area vers:'s W* for Howard's data is shown in figure 2.6. Elsasser assumed that the shape of Daw's c'urve was cor- rect but that the curve was high; he shifted Daw's curve (the top of which is sketched in fig. 25) downward by an amount which would produce a satisfactory fit to the band areas shown in figure 26. The final curves for the generalized absorption coefficient are shown in figure 27. 64.,.~~~~~~~~~~~~~~~~~~~~~

WILLOW RUN LABORATORIES _ -- TABLE 4. TRANSMISSVITY FUNCTION AND AESORPTION COEFFICIENTS FOR OZONE J- v, k r(kW*) k*W.() (-" - (%) 8.85 1130 3.16 E-3 100.0 5.01 E-5 8.93 1120 5.01 E-2 99.2 6.31 E-5 9.01 1110 1.00 E-1 98.4 7.94 E-5 9.09 1100 6.31 E-2 97.6 1.00 E-4 9.17 1090 5.01 E-2 96.8 1.26 E-4 9.26 1080 5.62 E-2 96.0 1.59 E-4 9.35 1070 2.00 E-1 95.2 1.99 E-4 9.43 1060 2.24 94.4 2.51 E-4 9.51 1052 3.16 93.6 3.16 E-4 9.57 1045 7.94 E-1 92.7 3.98 E-4 9.64 1037 3.16 91.7 5.01 E,.4 9.71 1030 2.00 90.6 6.31 E-4 9.80 1020 1.59 89.4 7.94 E-4 9.90 0100 8.91 E-1 88.1 1.00 F-3 10.00 1000 3.98 E-1 86.6 1.26 E-3 10.10 990 1.59 E-1 84.9 1.59 E-3 10.20 980 3.93 E-2 83.0 1.99 E-3 10.31 970 6.31 E-3 80.9 2.51 E-3 10.42 960 6.31 E-4 78.6 3.16 E-3 10.47 955 5.01 E-5 76.1 3.98 E-3 73.4 5.01 E-3 12.00 70.5 6.31 E-3 12.20 820 6.31 E-5 67.3 7,94 E-3 12.50 800 3.98 E-4 63.8 1.00 E-2 12.82 780 1.59 E-3 60.0 1.26 E-2 13.16 760 3.98 E-3 55.9 1.59 E-2 13.51 740 7.94 E-2 51.5 1.99 E-2 13.79 725 1.00 E-2 46.8 2.51 E.2 14.08 710 5.01 E-3 41.8 3.16 E-2 14.49 690 7.94 E-3 36.6 3.98 E-2 14.93 670 5.01 E-3 31.2 5.01 E-2 15.38 650 1.99 E-3 25.7 6.31 E-2 15.87 630 5.01 E-4 20.2 7.94 E-2 16.39 610 2.51 E-5 15.1 1.00 ~-1 10.8 1.26 E- 1 7.4 1.59 E-1 4.8 1.99 E-1 2.9 2.51 E-1 1.5 3.16 E-l1. 0.5 3.98 E-1 0.0 5.91 E-1 65 <111' *~~~~~~

.\. - WIV/LLOW RUN LABORATORIES - -. I;_~ _.>YYamamoto [351 [361. p * 12.5 cm (aoe J 3oard e a Howard et al. p 125 cm Ho rd et [36], p = 75 cm (adoted [3 6 60 I ~_ j I._, Yamamoto 5J> -3' -2' -1 0 +1' LOG W*'FIGURE 24. TRANSMISSIVITY CURVES FOR WATER VAP.'OR 66

Pi ^ —----- WILLOW RUN LABORATORIES.- A I t-^-lrl —-|At4optedV ~~2`~~~~~~ 2... 0 1 /Yamamoto (35J. <! /(Lok 4 0.04) - ~/0.,.' 2000 1800 1 400 400 1200 WAVENMhBER (cm 1) FIGURE 25. COMPARISON OF CONTrojRS FOR 6.3- WATER VAPOR BAND * -2f m i, _ ^c Computed BAND AREA (cm-) FIGURE 26. COMPUTED AND MPASURED BAND AREAS FOR 6.3, WATER VAPOR BAND. Numbers Indicate approximate pressure in centimeters of mercury. Where there is sto number. < 3 cm Hg. 67

E - WILLOW RUN LABORATOORIES -H Hi 1 -2-3 / \ 2300 2100.1900 1700 1500 1300 1100 900 WAVENUMBER (cm ) FIGURE 27. GENERALIZED ABSORPTION COEFFICIENT FOR 6.3-M WATER VAPOR BAND 68

- --— WILLOW RUN LABORATORIES ---- F As for CO2 and 03, band areas for a series of reduced optical paths were calculated and plotted. The results are represented by the "computed curve" shown in figure 26. Elsasser obtained the heavy curve by making a best fit to Howard's data [30], disregarding the values for P 2 3t) mm Hg. Note that the two curves compare closely for band areas greater than 200 cmin This value of band area corresponds to values at absorption less thi.n 20. For absorptionr values less than 20% Elsasser's data will yield values of absorption that are lower than Howard's by approximately 10ot. For the rotational band Elsasser used the same transmissivity curve as for the 6.3-/l band and used Yamamoto's absorption coefficients [35] for the rotational band. The results are shown in figure 28. Yamamoto determined these coefficients by summing the contributions from all of the spectral lines. The line intensities were computed only from the measurements of the electrostatic dipole movement of the water molecule. Yamamoto's results are compared with the experimental results of Palmer [37] and Bell [38] in finre 29. Elsasser considered the comparison sufficiently close to justify the use of Yamamoto's data without modification. The region between the two bands is known as the window; absorption there is weak. It is known that there are a number of weak lines within this interval and also that a certain amount of the total absorption is caused by the wings of the very strong lines concentrated at the peak of the rotational band and the 6.3-j/ band. Recall that the absorption coefficient for the Lorentz line shape is given by SIa ii- = -r — --— ~ 2 We see that at large distances from the line centers the term oa in the denominator may be 2 \ neglected compared to (v v) 2. The net effect of the wings of all the distant lines is then a continuous, absorption with k(. where the summation extends over all distant lines. This quantity for the window has been determinec by Roach and Goody [39] from atmospheric observations and the results of tieir work used by Elsasser.. To determine the total absorption for the window.region Elsasser proposes the following technique. As seen in figure 29, the extrapolated generalized absorption coefficients 69

F"'^ *''''.::.......:... - r... WILLOW RUN LABORATORIES —---- c2 It, -'"'' M R -4 70 1000 800 600 400 200 0 WAVENUMBER (cm 1) FIGURE 28. GENERALIZED ABSORPTION I COEFFICIENT FOR ROTATIONAL WATER VAPOR BAND I it 70' i

WILLOW RUN LABORATORIES. I 1' Bell31'3J" r Palmer 37j. 2 L Varlabe P i (Shifted 0 for Inspection of X r-Curves at 0.50) o Ymamoto, 2 _ta 4 __I Contintuumm -3 6.3 \ yland \1 /RotatioBl Band -5 j 1200 1000 800 600 400 200 0 WAVENUMBER (cm ) FIGURE 29. GENERALIZED ABSORPTION -COEFFICIENT OF ROTATIONAL WATE? VAPOR BAND AND WINDOW i 71's*pL(.~?l

v —— WILLOW RUN LABORATORIES - -. —--- for the two rriln bands are shown together with the absorption coefficients of the continuum. For the spectral region from 660 cm1 to 1220 cm' the absorption is first computed from the extrapolated values and then by the values given by the continuum. The total absorption throughout the r-egon is then given by?total Tcontinuum Tband An equation such as the one above is rigorously correct if two continuous transmission spectra are superimposed. Therefore, Elsasser assumed it should be a reasonable approximation If one of the components were a continuum. As for the previous gases, the transmissivity function and the absorption coefficients are tabulated and the results are given in table 5. Note that for the spectral region from 8.33 to 14.7, the two absorption coefficients kand and kcotnuum respectively, are tabulated. 3.3. METHOD OF T'' L. ALTSHULER The method Altshuler developed or d d computing slant-path absorption [26] is extremely compact and simple to use. The abtorption functions and the generalized absorption coefficients I are presented in the form of curves which allow for fast computation of spectral absorption for four different atmospheric gases, namely HZO, CO2, 03, and N20. Altshuler's method is basically applicable to only those patlh which give rise tc strong-line absorption. However, he also proposed a method for computing absorption for wak-line nonoverlapping conditions, } whsch is based upon the strong-line absorption functions. The laboratory data used in. performing the empirical fits and the method of empirical fit I will be discussed for each of the absorbing species and the exact procedure to be followed when t computing transmission will be otlined. Table 6 is a listing of the laboratory data and the val- ues of the many parameters used in obtaining the empirical fits for ach gas. 3.3.1. TRANSMISStI[TY FUNCTION: WATER VAPOR. The strong-line approximation to the statistical model with an exponential distilbution of line strengths v'as used as the transmissivity function fcr H 0. This function, expressed in terms of W*, is given by I woSW 1/2 A I exp -\d- (72) Note ia table 6 this equation was fitted to the data of Howard, Burch, and Williams [30] for the wavele igth region from 1.0 to 9.1 /, the data of Yates and Taylor [42] for the wavelength ~72

--- - WILLOW RUN LABORATORIES — -- TABLE 5. TRANSMISSIVITY FUNCTION AND ABSOPPTION COEFFICIENTS FOR WATER VAPOR.(.).. ald k Continuum T(k.W* Y.. (,) (cm- (%) 4.39. 2260 3.89 E-5 100.00 1.99 E-4 4.46 2240 2.04 E-4 99.90 2.51 E-4 4.55 2200 9.33 E-4 99.67 3.16 E-4 4.62. 2160.80 E-3.99.32 3.98 E-4 4.72. 2120 1.32 E-2 98.87 5.01 E-4 4.81 2080 3.98 E-2 98.33 6.31 E-4 4.90 2040 1.10 E-1 97.72 7.94 E-4 5.00 2000 2.57 E-1 97.05 1.00 E-3 5.10 1960 6.17 E-. 96.33 1.26 E-3 5.21 1920 1.38 95.56 1.59 E-3 5.32 1880 3.16 94.75 1.99 E-3 5.43 1840 8.13 93.89 2.51 E-3 5.56 1800 240 E 1 92.98 3.16 E-3 5.68 1760 7.24 E 1 92.00 3.98 E-3 5.81 1720 1.48 E 2 90.94 5.01 E-3 5.95 1680 1.95 E 2 89.78 6.31 E-3 6.10 1640 1.20 E 2 88.51 7.94 E-3 6.25 1600 3.63 E 1 87.11 1.00 E-2 6.29 1590 3.09 E 1 85.56 1.26 E-2 6.41 1560 1.02 E 2 83.84 1.59 E-2 6.58 1520 2.89 E 2 81.94 1.99 E-2 6.76 1480 1.12 E 2 79.84 2.51 E-2 6.94 1440 4.37 E 1 77.53 3.16 E-2 7.14 1400 1.86 E 1 75.00 3.98 E-2 7.35 1360 7.08 72.25 5.01 E-2 7.58 1320 2.24 69.28 6.31 E-2 7.81 1280 5.50 E-1 66.10 7.94 E-2 8.06 1240 1.17 E-1 62.72 1.00 E- 1 8.33 1200 2.95 E-2 1.18 E-2 59.15 1.26 E-1 8.62 1160 8.51 E-3 9.55.E-3 55.41 1.59 E-I 8.93 1120 2.40 E-3 8.13 E-3 51.52 1.99 E-1 9.26 1080 7.08 E-4 7.08 E-3 47.50 2.51 E-1 9.62 1040 2.04 E-4 6.76 E-3 43.38 3.16 E-1 10.00 1000 5.89 E-5 6.61 E-3 39.19 3.98 E- 10.42 960 1.78 E-5 6.61 E-3 34.97 5.01 ~-I 10.87 920 1.58 E-5 6.76 E-3 30.76 6.31 E-1 11.36 880 6.31 E- 5 7.24 E-3 26.61 7.94 E-l 11.90 840 2.57 E-4 7.76 E-3 22.58 1.00 12.50 80 -1.00 E-3 8.71 E-3 18.74 1.26 13.16 760 3.63 E-2 1.05'E-2 15.18 1.59 13.89 720 1.20 E-2 1.32 E-2 11.98 1.99 14.70 680 3.80 E-2 1.66 E-2 9.20 2.51 15.63 640 1.07 E-1.7. 73 ~ W_ 4 betev. Xf-wd.N f r

--—..... —-- WILLOW RUN LABORATORIES TABLE 5. TRANSMISSIVITY FUNCTION AND ABSORPTION COEFFICIENTS FOR WATER VAPOR (Continued) -.. (v) kBd k Continuum ki.* i (A) (cm-1) (%) 16.67 600 2.82 E-1 5.00 3.98 17.86 560 7.08 E-l 3.57 5.01 19.23 520 1.70 2.50 6.31 20.83 480 3.99 1.68 7.94 22.72 440 8.91 1.03. 1.00 E-1 25.00 400 2.00 E 1 0.49 1.26 E-1 27.78 360.4.47E 1 0.0 1.59 E-1 31.25 320 9.12 E 1 35.71 280 1.70 E 2 41.67 240 2.82 E 2 50.00 200 4.07 E 2 62.50 160 4.90 E 2 83.33 120 4.68 E 2 125.00 80 2.95 E 2 250.00 40 1.10 E 2 74 I' I 74

-- - - WILLOW RUN LABORATORIES —-- --- - TABLE 6. SUMMARY OF LABORATORY DATA AND PATH PARAMETERS Gas or Vapor v P.._ P,. T W n W' Ret. (atm-cm) (atm-cm) CO2.3-1.7 6000-7000 760 50.0 777 295 S100 0.50 8200 30 1.8-2.1 4600-5400 760 50.0 777 295. 830 0.7 8700 30 2.5-3.1 3500-3700 737 1. 738 295 22 0.86 21 30 2.5-3.1 3500-3700 755 10.0 758 295 1619 0.86 1614 30 4.0-5.4 1800-2500 1.0 10 1.3 295 18 0.80 0.11 30 4.0-5.4 1800-2500 734 9.8 737 295 1043 0.80 1018 30 9.0-11 9101100 760 608 942 295 5600 0.8 6580 40 9.0-11 910-1100 6080 6080 7900 295 8000 0.27 15,100 41 12-19 550-850 750 0.25 750 295 3 0.88 2.9 30 12-19 550-850 745 4.0 746 295. 173 0.88 i69 30 (pr cm) (pr cm) HO.1.0-1.05 95-10,000 1.05-1.2 8200-9500 750 295 3 0.52 2.9 30 1.05-2.2 4600-9500 740 295 50 0.60 49 30 1.05-2.2 4600-9500 740 225 0.05 0.60 0.049 30 1.2-2.2 4600-6250 740 295 0.71 0.60 0.70 30 2.3-3.65 2100-4400 740 16.3 821 295 1.68 0.62 1,75 30 2.3-3.65 2800-4400 740 16.3 821 295 0.140 0.62 0.147 30 2.3-3.65 2800-4400 740 2 750 295 0.017 0.62 0.017 30 3.65-4.5 2200-2800 4.5-9.1 1100-2200 746 295 1.03 0.6, 1.02 30 4.5-9.1 1100-2200 746 295 0.047 0.60 0.046 30 4.5-9.1 1100-2200 123 2.5 136 295 0.021 0.70 0.006 30 9.1.13 1100 770 760 291 1.18 1.0 4.18 42 S.1-13 1100-770 760 285 6,7 1.0 6.7 42(run 61) 9.1-13 1100-770 760 298 0.57 1.0 0.57 42(run 60) 13-16 625-770 16-20 500-625 7/60 298 0.57 1.0 0.57 42(run 60) 20-30 333-500 606 4.0 626 290 0.074 1.0 0.061 43 20-30 333-500 600 1.6 608 290 0.0325 1.0 0.026 43 20-30 333-500 590 0.2 591 290 0.0050 1.0 0.0039 43 30-40 250-333 603 0.15 604 283 0.00318 1.0 0.00255 43 40-2500 4-250 760 293 0.0104 1.0 0.0103 44 Gas (atm cm) (atmn cm) 03 4.4-5.1 1960-2270 230 230 370'95 9.08 0.3 7.38 45. 46 8,0-9,3 1075-1250 160 160 257 295 6.32 0.3 4.58 45 9.3-10.2 980-1075 0,31 0.13 47 12.0-16.5 606-833 160 160 257 295 6.32 0.3 4,58 46 (atm cm) (atmn cm) N 0 3.8-4.1 2240-2630 0.062 48, 49 4.24-4.36 2290-2360 400 400 448 300 12.6 0.8 8.1 50 4.4-4.7 2130-2270 750 295 5.8 0.7 5.7 51 4.4-4.7 2130-2270 759 295 0.1001 0.7 0.0100 51 7,6-8.1 1235-1315 100 100 112 286 0.79 0.8 0.17 52 8.5-9.1 1100-1175 240 240 269 288 1.9 0.8 n.3 52 16.0-18.0 155-625 160 160 179 288 S. 0.8 1.73 52 P total pressure p 5 partial gas pressure Pt equivalent pressure T = temperature W - actual quantity of gas in path - V* = equivalent sea-level path n = exponent for P and T correction 75

-... WILLOW RUN LABORATORIES —region from 9.1 to 20 s, that of Palmer [431 for the region from 20 to 40 jt, and the Yaroslavsky and Stanevich [44] data for the region from 40 to 2500 A. The empirical procedure used to evaluate the absorption coefficient was as follows. The laboratory data cited in table 6 were reduced to equivalent W* values from the equation We F W, AP@)X'' The values of n are given in table 6. The spectral Interval from 1.0 to 2000 M was civided into equal wavelength increments on a logarithmic scale and for each value of W* a mean absorption coefficient was evaluated for each wavelength increment, assuming the transmission for each wavelength interval obeys equation 72. The absorption coefficients for each wavelength increment were plotted as a function of wavelength and a curve was fitted to the data u'sing a leastsquares fitting procedure. The resulting curve was a plot of the generalized absorption coe-ffi2 clent a0Sr/d as a function of wavelength. Tils curve was then used to calculate values of transmission versus wavelength for certain values of W*. The curves for water vapor are given in figure 30. Note that each curve is a plot of transmission versus wavelength for a single value of W*. To determine the absorption for a different value of W*, simply shift the curve by an amount indicated by the index on the right. The transmission data for all of the gases are presented in the same format, and examples of computation are given in section 3.3.4. 3.3.2. TRANSMISSIVITY FUNCTION: CARBON DIOXIDE. The transmisslvity function for this gas was developed by a purely empirical technique and is, therefore, given by a curve of transmissiop versus the product of the generalized absorpion coefficient and the equivalent sealevel path W* or T =?(kW*). This curve is shown in figure 31. Since the procedure used in oh- taining this curve is exactly that v'hich was used by Elsasser, the reader is referred to sec- iion 3.2 where the specific deiails of this form of empirical fit'.ing are discussed. In general, when performing a purely empirical fit to a set cl absorptio, data one finds that one function will be determined for one absorption hand and another f, r a different absorption band. However, l Altshul;r fitted the same curve (fig. Z1) for the entire interval from 1.3 to 19 ft to the data of i Howard et al. [30] and King [14]. Therefore; one function represents the CO2 data for the entire interval. Using this transmissti ity function he determined the absorption coefficients exactly i as those for H 0 and the results are plotted in the same format. The curves are represented by figure 32. The units of W' for CO2 are atmospheric centimeters, but the units on the curves 2 76 I.

— W!LL OW RUN LABORATORIES -........ WAVELENGTH (I) WAVELENGTH (i) 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.8 2.0 2.2 2.4 99.9 -'- ~ ~"-V'-'~' -'- - 99.9'TTT!1 1 1 -1j -H~j. —L-. —.-. — Index Inde*. l ~ o x LOG- -- W AVELENGTH (p).) L Ot0 WAVELENGT., (n) { -...0.01 -Z:' lii,/,'!l ii^^ ij^tj!i' - 9: - 01 99: -q' - - w4-. I * - -A *' i tii'4 0. t 0.00 0.04 0.08 0.52 0.6 0.60 0.0 0.24 0.28 0.72 0.76 0.80 LOG. WAVELENGTH (I) LOG0 WAVELENGTE (t).1 - A. 100 0, r_*'+ 1 +- 1t *0 14 d (c)(b) 2.6 3.0 3.4 3.8 4.0 5.0 6. 06.32 99.,9; rnt |^^^T^TfT*^TT^* 899.9 8P,: I FIGURE 30. TRANSMISSION VS. WAVELENGTH GFC WATER VAPOR 6^ ~ ^U-w'~i^ * W_ 0.H Pr cm77 _ g 90| I' Irf-~~"i~~u~i~e~~t- 90 10 d * ^^^^i^^ ^.^^: ^ 100 $ ^H-}-i-:00 - 00~ 0.40 0.44 0.48 0.52 0.560.60 0.60 0.64 0.68 0.72 0.76 0.80 LOG WAVELENGTH (1i)LOG,. WAVELENGTH (T)! FIGURE 30. TRASMISSION VS. WAVELENGTH FCAt WATER VAPOR j 2,6 3,0..3,s 3.8 4,0 5.0 6),0 8.32~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

;'' ti t j afi i rlr~l;': w:-'' - -' - ",-.**,-,.... "..- _... -WILLOW RUN LABORATORIES WAVELENGTH () WAVELENGTH (g) 6.32 7.0 8.0 9.0 10.0 10 12 14 16 18202224 99.9 ^ ^ e9.9,i, 1,.'\ vf 4Index T de1x * r::^:::-^^ l; ^^ 0.01 0.01~:-fi,pf: EjO ll g99 0".i^ e _': 3: 99 0.10 W-.-^ pO WtO.I pr cm -1 1f^O00 44 4 i-^tt^ -100 -boo 1I 0 IFE ^!! 10- i.~ i 0.80 0.840.880.92 0.96 1.00 1.0 1.1 1.2 1.3 1.4 LOG!0 WAVELENGTH (M) LOG10 WAVELENGTH (), i 1 (e). 1 (f) t WAVELENGTH (i) WAVELEN'TH (T) 25 30 35 40 45505560 40 70 2004001000 3000'Ti+t4 tT-7t-,e i t, —. —!i4- Index i IO tl' -il = 10t00 t- 100 Lt"-,.-,;.,.: — -w-, - _@ __ f' 1 0 1E1 1 O99 0.1 AVELEGTH 99() LO:G - WAVELENG —:TH(, 0.1 ( 2Q ~ ~ t m i rrf-i rtti-i. I;c i. - i -i -.1 --..^^ ^ ^Jiooo "i i 1. j4 10 (g)'(h) FIGURE 30. TRANSMSSON VS. WAVELENGTH FOR WAT VAPOR (Continued) LOG0 WAVELEISGTH (.O~)WAVELENGTH I. OG10 WAVELENGTH (~) f (g) (h). FIGURE 30. TRANSMISSTION VS. WAVELENGTH FOR.WATERq VAPOR (Continued) 78

- W ILOW R UN LABORATORIES - 100 -------- Experimental 0o -Curve 1 / 80 70 - z 60 -j, SO- 0 8% 040 30 20 c Pressure - 10. 735 mm H 1 v L 1860-2540 cm -3 -2 -1 0 1 2 FIGURE 31. TRANSMISSIVITY CURVE FOR CARBON DIOXIDE ~~~\79t. 79

- -- WILLOW RUN LABORATORIES WAVELENGTH (n) WAVEI.'NGTH (p1) 1.3 1.4 1.5 1.6 L 7 1.7 1.8 1. 9 2.02.1 2. 2.3 " jd Index 91 Index -_d. 100 knm 4d 100 k (, i L 2 iWA V E GT ( *. t,...:j,...-.... -,:+ - i:........... +..~..!t i - -^-r-^-~ Irm+-..-.-, " - I-' I-, — if-!-"-I'-I,,-i —t — t-'+ — Ii — i(-~~-~-, 4- 4.....+0.12 0.16.20 0. 24 0.24 0.28 0.32 0.36 LOG10 WAVELENGTH () LOG10 WAVELENGTH (.), (a) (b) ) I 0 lu WAIELENGTH. WAVELENGTH OARN (I) d 4 - fokm:. F.-~, ~-de-;-;..... i], U- n^. 01 0- 1 t o^ -. 2':4 02 0-:-28! 03 0 36^ 90 -0 i ~ a +*" -+-j1 h l it_,10 t o2- -.-..'. -. -- -i —:'. i.... ) 4 2.4 2- -. 6- - 2;,.8 3.- 0 4.l - 4 —- t'i- 1 -' 3 20 4t 3 _ d...a O -m:-d 0 —.- k k —-— t mta 7 - t ~'-+-~t i P~* —-.~ —~-=^-t-~3- - - ~t.~.: /........ -t- ~-n —+t i;:.;' _:...' I f'.:; I..., l-.- -' ** i t — t- i-+ t, [-0;i -T —'o'-3 - 1-t' i.:. i, a 0.38 0.42.46 0,50 060.4 068 0.0.4 072 LOG10 WAVELENGTH (,) LOGC- WAVELENGTH (,) 4 0tt. 0!_'.Wri~~~~.r (c) /. (d) ) 80 0.38 0. 42 -46 0.50 0.60 0.64 0.68 0.72.80~~~~0r-+~,~J

WAVELENGTH (g) WAVELENGTH (M) 9.0 10.0 11.0:2 13 14 15 ^ ==ab==^ ==0.l ^:;^r. 0.1,-,I_]_~ ~!!~~ ~- [ndex I90 Wu= I 9?'] 0.94 0.98 1.02 1.06 1.06 1.10 1.14 1.18 L ) LO0 WAVELENGTH (M) LOG10 WAVELENGTH (g) (e) (f) WAVELENGTH (.) 1.4 1.5 1.6 1.7 1.8 1.9 99 -- --- 4.X; lO~t11.:2 3Index 41 _ g g -f^-4m — -4-. 99rT TA^ S 4E 1-ArI 10 10c10,. 1I^^. ~X 1kn I 1.14 1.18 1.22 1.26 LOG 10 WAVELENGTH (i) (g) FAGURE 32. TBANSMISSION VS. WAVELENGTH FOR CARBON DIOXIDE (Continued) 81 h:-=- --.^= — ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ct Jo J0

.... ---.WILLOW RUN LABORATORIES - ------ are kilometers of dry air. Therefore, to use the curves, convert atmospheric centimeters to kilometers of dry air by recalling that 1 km of dry air equals approximately 32 atm cm of CO2. 3.3.3. TRANSMISSIVITY FUNCTION: OZONE AND NITROUS OXIDE. T:e strong-line approximation to the Elsasser model which was used as the transmisslvity function for both 03 and N20 is given by (sa 0 = 1 - erf W) d / This equation was fitted to thedata in table 6 by the same procedure used for H0. The resulting absorption curves are given in figure 33. The curves for N20 use'he same units as do'those for CO, and can be converted to atmospheric centimeter, by recalling that 1 km of dry air contains 0.028 atm cm of N20. Because the curves for 03 are given directly in atmospheric centimeters, no conversion is necessary. 3.3.4. PROCEDURE FOR CALCULATING ABSORPTION. Figures 30, 32, and 33 represent all of Altshuler's absorption data. The computation of transmission becomes a trivial matter once the amount of each absorbing species in the slant path is determined. The technique for reducing a slant path to an equivalent sea-level path WI has been previously discussed and the result;: are given by equation 67 (sec. 2.4.3). Altshuler, however, suggests that W* be computed by an equation which is a slight modification of-equation 67: > S.:X wT = ) pdx. (73) 1 0" where n = 1.0 for CO2 - I n 1.0 for H f n =0.3 for 0 n = 1.0 for NO For.C02, H20, and N20 the two equations are identical since n = 1. However, for 03 Altshuler suggests a pressure correction on half-width, that is, 0.3 rather than linear. Altshuler determined this value empirically from Walshaw's [34] laboratory data. There is some controversy concerning the correct pressure correction which should be used when reducing laboratory data'to equivalent W* values. Theory indicates that half-width varies linearly with the effe ctive 82 82{

WILLOW RUN LABORATORIES- ifilii( 11 o i. E- ~ oS ^ o 8 ~ i}:.IF 8i[: 8'"Pi: ~' ~ s -o ~~ ~ -*'....- ^ - I:- t ii liji.i.j C',,i,.., j:t::: l tij:-:::- | | t *^1, - il 3!51. is^t ~': ~[>-t}eX 4iX 0 U = 0. rt_, * i: W w s l A n w 0 a^ ~w o*S o ~ s g 1 ( %) Nozsstvms vu a:. c (SOS o~ ISbO r o O O M O.'6 d...,. +....'.it tx._0_ -..^, -::..-._....s.o C > cn r- ---- -. O" I,: no I * <N^-.-^^+^..^^^T~iii-^n: —(,s * o *m *I''<-+X*.Ss,3Zw,>;4 5t-6-w'- t s............................... -.............. ~:..._ *5'" ibf^^:,, k.. o o T- il, -..ti::-:r~.::i 4-l'o t 2; co t _l:~.'. _,''~'-,:". 0'** ~ iii:X. -4 3. i. 0 0 O' *-'*^wwM~^A^**1^;^*^'-'^'^^^/''-**''-^^^''""^""'" " "' ". q?O 1

~'i: * -,.. — WILL'OW RUN LABORATORIES - boadening pressure P; this does not seem to be supported by laboratory data. A possible 8olutton, particularly for 03, would be to perform all empirical fits with W and P as parameters and not use the equivalent sea-level path concept. Under these conditions a pressure correction would not be required. The technique described in appendix I may be used to evaluate equation 73. The output of.the program will yield values of W* in precipitablecentimeters for H20 and in atmospheric centimeters for 03 which may be used directly on the curves. However, values of W* forCO2 and N20 must be converted to an equivalent length of di ir, d. For CO2,d = W*/32 and for N2O, d W* /0.028. After the appropriate values of W* r d have been computed the following procedure is used to compute transmission: (1) Wi,.h the use of dividers, mark off on the "index" the amount of absorbing gas present in the slant path, i.e., W* or d. Use the fine scale between 1 and 10 on the index for Interpolating between the major divisions on the index. (2) Locate the major division on the index corresponding to the values of W* or d identifying the curve on the figure. (3) The transmission is obtained by displacing the transrission curve by the same amount and direction that the entry value differs from the curve. (4) Read the transmission r for each gas at the same wavelength and calculate T(X)= T0 (h) jC (X) T (X) N7 (X) L JL2 J L 3 2 J (5) Repeat (1)-(4) for each wavelength through the interval of interest. Atshuler's method for computing transmission is valid only when there is enough absorb- g ing gas that strong-line conditions are present. Recall that when the quantities of gases and vapors are small and the pressures are low, absorption is such that the weak-line nonover- lapping approximation is valid and absorption varies linearly with W*. This occurs for values of absorption less than approximately 10%, regardless of which absorption model applies, the Elsasser model, the statistical model, or the experimental model. Examining figure 34 one may note that all band models overpredict absorption for small quantities of gas. It is neces-' sary, therefore, to correct these values of absorption to bring them in to agreement with the actual absorption. Figure 35 shows the linear absorption region for various gases and vapors. Absorption is plotted versus the altitude of a horizontal path. This figure, nsdetr t the ct th.at at h4,I;- t.. altitudes the maximum absorption for weak-line conditions becomes less and less owing to the Al 84 I..- -* -. i

WILLOW RUN LABORATORIES- - -_iE~~~~~~~~~~~~~~~~~~~~~~~~~~ r~~~~~~~~~~~~~~~~~~~I Elsasser Model Goody Modl --- 1.0 - Experimental Model - - Actual Curve --.. A 0. 05 O~bi 1- -.. -' -?- i4^ ~ 0.01 0~0 /^/'.0I/. II- _ 0.001 / 0.0001 0.001. 0.01 0.1 1.0 10.. l.kW* j p FIGURE 34. TRANSMISSIVITY CURVES SHOWING THE DIVERGENCE OF ABSORPTION FROM THAT PREDICTED BY THE ELSASSER, GOODY, AND EXPERIMENTAL BAND MODELS. For gases at sea level. 85 1~~~~~~~ r~~~~~~~~~~PX~~~~~~~~~~ll~~~~~~~~~~y kt~~~~~~~~~~~~~~~r-wcr-'~~~~~~~~~~~~~~~~~~~~~~~~.~~~~~~~ h*RLRB-rhCSDIP~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ni~~~~~~~~~~nur ~~~~~~~~~~~~~~~~~~~ —*r ~~ ~~ ~~~ ~~ ~~ ~~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~

- WILLOW RUN LABORATORIES - 1.0 1 -:- | r 1.0 0 3 0 Hf N30 0 10' ] - 10 10-5 L _ 0 10 20 30 40 50 60 70 ALTITUDE (km) FIGURE 35. REGIONS OF LINEAR ABSORPTION FOR HOI ZONTAL ATMOSPHERIC PATHS AT VARIOUS ALTITUDES Areas below the curves are linear absorption regions, i.e., where maximrum error in A is less than 10%. 86

-.. -WILLOW RUN LABORATORIES- i decrease in spectral-line width with decreasing pressure. In order to apply this figure to an actual calculation, simply determine the absorption for a given path and if it lies below the curve of figure 35 the use of the weak-line approximation is valid. If the path is not horizontal, the mean attitude of the path may be used to a good approximation. In order to compute transmission for weak-line conditions, first compute the amount of absorber that will yield a value of absorption equal to 5%. Then, using the values of W* for each gas, determine the absorption from the following equations (taken from figure 34 by inspection). W* W*H20 A 0.012 -- 2 (74) H20 W- 02 H 0 (9~52 W* C2 ACO = 0.0016 -(95 (75) 2 C W*O A =0.07 3(95_ _ (76) 3 93 W0 (95%) (?6) W* N20 N 0.22 (7; W*H WU* W*, and W* are the reduced equivalent paths for each gas and 2 2 3. 2 H 0 (95%) and W 03 (95%) are the equivalent paths that will yield a value of transmission equal to 95%. dCo (95%) and dN2 (95%) are the dry-air path lengths that give a transmission 2 2 of 95%. The values of absorption obtained from these equations are compared with the curves of figure 35 to test the validity of calculation, 3.4. METHOD OF A, ZACHOR Zachor [27j used four'differeit functions based on the band models of Elsasser apd the homogeneous laboratory data of Howard, Bursh, and AWillams [301 atif Watshaw [34] to develop transmissivity functions which can be used to calculate absorption versus wavelength for the spectral interval from 1.03 to 10.80 for CO2 and 0.. Zachor also Included In reference 27 a method for computing absorption caused by HO0 for the same spectral interval. Howpevr the method presented was taken directly from the results of Howard et al. (see sec. 3.5, 87

--- WILLOW RUN LABORATORIES..... ref. 30). Therefore, this section presents only the results of Zachor's original contribution: methods for computing absorption caused by CO2 and 03. 3.4.1. OZONE. Zachor made an empirical fit to the laboratory data of Walshaw[34] for the 9.6i ozone bandusing a modification of the Elsasser band mood:. Recall that the absorption as given by the Elsasser model is rY A = sinh I0(Y) exp (-Y cosh )) dY 2~0 ~ O P P where - d P- 0 0 S W Y = d sinh 3 According to this model, the mean half-width a of a spectral line, is u linear function of the effective broadening pressure. Zachor found that this linear relationship was not supported by Walshaw's data and that t3 was more accurately described by, P\C where c is a parameter that depends upon wavelength. Therefore, Zachor used this modifiation of the Elsasser model to describe absorption'byO3. The Elsasser model is plotted for different values of 13 in figure 36. The values of 13 and Y for which the weak- and strong-line approximations may be used are noted on the figure. The constants g%, S/d, and c are listed in table 7. 3.4.2. CARBON DIOXIDE. Zachor analyzed the laboratory data of Howard et al. [30] and found that a single model could not be used to accurately ccscribe the absorption by CO2 for the entire spectral region from 1.03 to 10.80 A. Therefore, Zach;.r used four different models, each model being used to describe the abso'ption over only a portion of the total spectral interval. The four models used are listed below. Model 1: The exact Elsasser band model SY A sinh ft J0(Y) exp (-Y cosh ) dY (78) Y S W dsinh 0) 2na0 0 d4 88

WILLOW RUN LABOr.ATORIES..:',: ~..".,..-..'*~H -i-'.q..'.i.~ —..'/;^.' i - u.,...4 - tt..*. ~\'': -, oi i ftit^ i -t** \i^t-Hf+ * * x wt - +*;;4*-N1X-\i\. a - T t C!:;: 1 -+;*+:,2,:;x.;;:.:-:t:,::'i' *-+f;.~ - i:f:; 0 -, —wO-*i - o - - d o *o c1 o o 89 Tn T-sF~t-f-t.~7r>+4 -:t;t: X:..t~:;^t':t::;:w:'~~::M:.r4 t'lxi4ll w i t + b.. 1 o~~N *Stt*++1, }b+XftN'|NMEt.;I;::::;+,:Xf~tt:;t::X \::.:;:;:x:......................................Nt+x*:;:;hw~-l~st"...................;.:X.:.tS:'':t::';t':t:''lt.::+l.''-t';;:-$;;'.X:'::':xHI:;N'4to:x:\xt8~ t1';t) Nfl;j++t \?*;t\t.\;\\\8\t;tl H. -4- *+tb fl +; +*-.9 ~ ~~t N * y\\l1<5:.~~~~~~~~~~~~~~~~~~~~~~~~~C t tt +=tifffl~tNO W HOStS V!i...............tt XM\I......... * *te.*+ --.* ^*+ ++-.t* -+ f- --- — * — * *\ W* -*- Slw 89b

-WI WILLOW RUN LABORATORIES TABLE 7. EMPIRICAL CONSTANTS FOR OZONE X S/d 0 c 9.335 0.555 0.678 0.602 9.398 0.476 0.712 0.477 9.463 10.0 1.70 0.654 9.494 14.3 1.48 0.648 9.526 9.43 2.07 0.699 9.590 4.58 1.36 0.602 9.652 7.15 2.03 0.602 9.713 8.34 1.49 0.648 9.773 8.00 1.67 0.699 9.834 7.14 1.86 0.756 9.893 5.20 1.74 0.725 9.953 3.22 1.82 0.748 10.07 0.944 2.27 0.745 10.19 0.318 0.554 0.301 90

...., WILLOW RUN LABORATORIES Model 2: The strong-line approxlimton to the Elsesser band model A = erf (KWaPc/) (79) Model 3 The strong-line approximation tl tvo overlapping Elsasser bands with equal line spacing and intensity 2 A = 1(1 - erf (Kw C/2)]J (80) Model 4: The strong-line approximation to the statistical band model A = 1 - exp (-KWaPc/2) (81) Each of these models Is a modification of one of the original models discussed in section2.2. If the values a = 1/2 and c:- 1 were used for every wavelength the models would be Identcal. Although there is no theoretical reason for employing such modifications, Zachor justified their use by the fact that the modified models allowed for better empirical fits to the laboratory data. The wavelength-dependent parameters K, a, and c and the model to be used to compute absorption for each wavelength are given in table 8. Let us consider the actual procedure used by Zachor to fit the empirical relations for 03 and CO2 to the absorption data. A number of discrete wavelengths were picked in each of the 2 absorption bands studied. These wzre picked to accurately descrIte the Howard. Burch, and Williams spectral data; that is, given measured absorption at each of tnese wavelengths, the band spectra could be duplicated almost exactly by simply sketching a curve through these measured values. For each of the chosen -vkvelengths, values of absorption corresponding to different (effective) pressures and concentrations were read from the spectra and plotted in the form A vs. log W for different constant values of P. Figure 37 shows plots of the data for the wavelengths 4.85, 2.76, and2.70 A. The plotted data were compared to the master plot of equations 78-81 by placing the plots of the equations, one over the other, on a light table; the plot of the data was shifted to the right or ieft until the curves best matching thp c:ata points were found. These curves were then traced directly onto the data plot as i.vdicated by the solid c.rves of figure 37. This procedure established the form of the empirical cquation, and led into the problem of determining the values of the three constants. Figure 37a demonstrates that tne absorption data at X = 4.85 g is accurately described by equation 78. The shape of the absorption curve given by equation 78 is taken directiy from the labeled curves of the master plot after they are traced through the data -;oints. The value of is determined by comparing the OB'sforanytwo of the curves; that is, Pl/P2 = f1//2)c. As would be e,.)ected, the values of c obtained from different pa;'s vwere usually not consistent on Wue lrsit attempt. The value of c for.whicn all the curves simultaneously fit the data points with 91

WILLOW RUN LABORATORIES- TABLE 8. EMPIRICAL CONSTANTS FOR CAREON DIOXIDE X Moel.k a c Scd /0 c 1.37 2 1.29 x 10 6 0.500 0.500 1,38 2 7.8x 10'6 0.500 0.500 1.39 2 1,71 x 10 5 0.500 0.500 1.40 2 1.42 x 10(4 0.500 0.2S4 1.41 2 2.02 10'4 0.500 0.295 1.42 4 6.38 x 104 0.399 0.358 1.44 4 1.02 x 103 0.399 0.292 1.46 2 2.30 x 104 0.500 0.28C 1.48 2 2.40 x 10 0.500 0.500 1.50 2 2.54 x 10 0.500 0.500 1.52 2 3.62 x 10 7 0.500 0.500 1.54 2 1.24 x 105 0.500 0.500 1.56 2 4.07 x 10'5 0.500 0.491 158 2 2.00 x 104 3.500 0.366 1.60 4 3.62 x 104 0.446 0.399 1.63 2 1.61 x 10 4 0.500 0.336 1.65 2 3.04x 10-' 0.500 0.500 1.67 2 1.1 x 1 0- 0.500 0.500 1.68 2 3.27 x 106 0500 0.500 1.09 I,iil 1.88.. 1.90 2 2.92 x 10 0.500 0.500 1.92 2 8.66 x 105 0.500 0.542 1.94 2 1.33 x 10 3 0.500 0.352 1.95 2 1.89x 103 0.46 0.35 1.96 2 1.29 x 10 3 0.500 0.387 1.97 2 9.9 x 10'4 0,528 0.356 1.98 2 7.92 x 10'4 0.500 0.383 1.99 2 2.32 x 10'4 0.676 0.402 2.00 4 1.83 x "-4 0.717 0.572 2.01 4 5.35 x 10 3 0.446 0.399 2.02 4 2.64 x 10'2 0.358 0.177 92

- --- WILLOW RUN LABORATORIES -_ TABLE 8, EMPIRICAL CONSTANTS FOR CARBON DIOXIDE (Continued) x Model k a c/2 S/d c 2.03 4 7.44 x 10'3 0.447 0.213 2.04 2 1.99 x 103 0.500 0.276 2.05 2 1.35 x 103 0.500 0.363 2.06 2 1.48 x 10-3 0.500 0.377 2.07 2 3.64 x 10'3 0.415 0.303 2.08 4 3.6 x 1003 0.399 0.292 2.10 2 5.0 x 105 0.500 0.207 2.12 T 31 2.62 2.64 4 1.04 x 10'3 0.446 0.424 2.66 4 1.38 x 10 0.375 0.316 2,68 4 2.31 x 10"2 0.448 0.378 2.70 4 3.29 x 10"2 0.446 0.435 2.72 4 2.36 x 102 0.414 0.455 2.74 4 1.07 x 10-2 0.500 0.493 2.76 3 5.87 x 103 0.500 0.485 2.78 4 1.82 x 10-2 0.500 0.490 2.80 4 2.89 x 10-2 0.399 0.360.2.82 4 3.07 x 10-2'0.292 0.344 2.84 4 3.25 x 10 2 0.216 0.257 2.86 4 2.25 x 1'2 6.189 0.217 2.88 4 8.87 X t03 0.292 0.141 2.90 3.96 4.00 4.05 2 2.11 10 6 0.500 1.17 4.10 1 2 x 103 0.212 1.81 4415 1 2.5 x 10 2 0.103 1.00 4.20 2 1.6 x 10'2 0.500 0.363 4.2'5 - 4.30 -- 93

-. — WILLOW RUN LABORATORIES.. TABLE 8. EMPIRICAL CONSTANTS FOR CARBON DIOXIDE (Continued) x Model k a c/2 S/d c (A) 4.35 4 1.12 x 10-' 0.446 0.372 4.40 4 4.81 x 10'2 0.500 0.367 4.45 4 1.77 x 10'2 0.500 0.380 4.50 1 7.5 x 10'3 0.409 0.675 4.55 2 1.48x 10'3 0.500 0.193 4.60 2 1.12 x 104 0.500 0.500 4.65 2 8.94 x 10-5 0.500 0.500 4.70 2 1.74 x 104 0.500 0.500 4.75 1 1.3 x 10'3 0.103 1.00 4.80 1 2.41 x 10'3 0.207 1.00 4.85 1 4.17 X 103 0.205 0.742 4.90 1 1.89 x 10 3 0.102 0.440 -5 4.95 1 7.40 x 105 0.203 0.433 5.00 - 5.05 T 1 5.10 1 1.08 X 10'4 0.021 1.00 5.15 2 9.9 x 10'5 0.500 0.500 5.2' 1 5.56 x 10'4 0.314 1.30 5.25 1 2.25 x 10'4 1.19 1.00 5.30 r I 94

WILlOW RUN LABORATORIES 1.0 _________ 0 p. )-.g / 0.0801X- / 0.Q (a) 4.85i 0.8 0.00417 --''061O~ 0' — l A si1 -i expf -y c oh7W1)J2yp.48dy5 0. 0.205 0.1 1 10 100 1,00G 10,000 W (atm cm) (a) 4.85 i FIGURE 37. FRACTIONAL ABSORPTION ----— 0.6 -- / / orl/ — g/' -- - * I o,.,, -------— 950.4 <LC A =1-(1-erf (0.00587WP/2-0'485)]2 1 10 100 000. 10,000'100,000 W (atm cm) (b) 2.76 6 FIGURE 37. FRACTIONAL ABSORPTION 95

.. WILLOW RUN tABORATORIES - E minimum scatter can eventually be determined by retracing the curves several times. From the determined values of c and of any one of the F's the value oi P. can be calculated from the relation 0 = 0O(P/P0)C. Since the abscissa of the master plot is 3 = (S/d)W, as in figure 36, the value of S/d is determined as soon as the curves are traced through the data points by dividing the value of (S/d)W on the abscissa of the master plot by the corresponding value of " oncentration W on data plot. Figures 37a and 37b show, except for the uppermost curve in figure 37b, that the absorption at 2.76 and 2.70 MI is described by equations 80 and 81, respectively. The procedure in determining the appropriate values of K, k, and c in either of these equations is much the same as the procedure used to make the empirical fit of equation 78 to the 4.85-M data. The value of k was determined by tracing the curves of the master plot through the data points. The value of c/2 could have been found in the same manner, by plotting A vs. log P = for constant W and using the master plot again; the usual process, however, was to equate the values of kP /2, which give the same absorption for two different pressures. If A(W1, P1) A(W2, P2), then (Wi/W 2) = (P2/P1)c/2, vhich gives c/2 when k is known. By substituting the known values of k and c and'2 1 the absorption corresponding to any P and W into the absorption equation one can determine the value of K. Zachor's method for computing absorption is unique in that he is the only author that incorporated a pressure correction on half-width that is nonlinear and a function of frequency. Such a pressure correction implies that a change in pressure affects the average line half-width differently, depending upon the frequency interval. Although such effects are not supported by theory, they were observed by Zachor upon. examining the laboratory data and were, therefore, incorporated into the fitting procedure. It seems to the author that thO observed anomalies should have been attributed fo experimental error rather than to a new physical phenomenon and that a linear correction should be used (at least one that was constant with frequency) rather than the one employed. This would result in an empirical fit that had a tendency to average out the errors rather than emphasizing them. Zachor's method for computing slant-path transmission is quite straight-forward. First, the values of W and P are determined for each absorbing gas. To compute absorption caused by 03, calculate'Y and: for each wavelength, using the constants given in table 7. The values of 3. Y and 3 can then be used tQ determine the spectral absorption from figure 36 or equation 78. The strong-line and weak-line'approximations to the exact Elsasser model should not be used unless the values of Y and 3 ale satisfied by the corditions,noted on figure 36. 96

-- --- WILLOW RUN LABORATORIES -__. To calculate absorption for CO2, one of four models is used: table 8 gives the appropliate model to use for a given wavelength. These models in conjunction with the constants k, a, c/2, S/d, 0 and c, also giyen in table 8, are then used to calculate slant-path absorption spectra. 3.5. METHOD OF J. HOWARD, D. BURCH; AND D. WILLIAMS The major contribution of Howard, Burch, and Williams to the field of atmospheric absorption has been the measurement of low- and high-resolution absorption spectra for homogeneous paths under laboratory conditions. The major results of their efforts are published in two reports [1, 30J which include a wealth of data. The earlier of these reports contained spectra measured at relatively low resolution compared to the resolution obtained from their more recent measurements. WATER VAPOR. In an effort to contribute further to the field of atmospheric transminsion, Howara et al. made an empirical fit of the Goody band model to their low-resolution absorption spectra [301. Let us consider the empirical procedures which were employed. Recall that the Goody model is givenby A 1 - exp --- - (82) Equation 82 expresses the absorption averaged over a narrow spectral interval in terms of the mean line intensity S for that interval as well as the mean line half-width and the absorber concentration. It was assumed by Howard et al. that the average intensity for a subinterval varied across an absorption band according to the relation S = k/W0, where k is a constant and W0 is the value of absorber concentration, at some pressure which yields a value of absorption equal to 50%. Substituting this expression into equation 82 and simplifying, we reduce that equation to d W A = I - exp 4 j (83) k 0W~i QL %j Before equation 83 can be used to predict absorption the various parameters must be evaluated. W0 is the only frequency-dependent parameter and has been evaluated as follows: a number of homogeneous absorption spectra were selected covering a broad range of absorber concentrations, each concentration reduced to an equivalent value at some common pressure. The absorption band was divided into subintervals and for each subinterval the absorption was plotted as a function of absorber concentration. A curve was fitted to these points and the value 97

--.WILLOW RUN LABORATORIES -- TOI- - i of absorber corresponding to an absorption of 50% defines W0 for that subinterval. This process w-~1:was continued until W0 was determined for the entire band. The values of W0 are given in table 9. The next step in performing the empirical fit was to determine the values of k/d and k/a0. rhis was done simply by plotting A vs. -(t)0 for all of the homogeneous laboratory spectra and adjusting the two constants until equation 83 gave a best fit to the data. The resulting expression for absorption as determined by Howard et al. is given by -1.97 0 A' 1 - exp/ (84) r+ 6.57 ( Equation 84 in conjunction with the values of W0 presented In table 9 can be used to compute H20 absorption for any path for which values of W and P are known. 3.6. METHOD OF G. LINDQUIST Lindquist [53] made an empirical fit c.f the Goody band-model absorption equation to the laboratory data of Howard, et ai. todevelop a transmtssivity function for H20 for the wavelength region from 1.06 to 9.2. The form of the absorption equation used by Lindquist is -C3W* A - 1 - exp (1+ (85) where W* is tie equivalent sea-level path and C3 and C4 are empirically derived constants. The procedure used to evaluate the constants was as follows: The Howard spectral data were reduced to equivalent W* values by the equation 0.62 P T0'62 W* =WwEquation 85 was then fitted to the reduced data, determining C3 and C4 such that the error be3 4 tween measured and calculated values of absorption was minimized in a least-squares sense. The resulting values of C3 and C4 are given in table 10. 98,,.. —--- - -::il I ____ ________ ^^^^^^^^^^^^r~P.';I1Ipifritfff^-"^ BP~" R.. D - -~ _. i ",' * - I*< **" ~ >-"- - - kV

WILLOW RUN LABORATORIES -- TABLE 9. w0 Vs. WAVELENGTH FOR WATER VAPOR A Wo Wo0 A o ( R -_ 1.03 3.6 x 103 1.38 6.2 x 10-2 1.97 2.15 2.72 4.75 x 10'3 1.04 1.62 x 103 1.39 7.5 x 10'2 1.98 3.45 2.74 8.00 x 10"3 1.05 7.2 x 102 1.40 9.5 x 102 1.99 5.55 2.76 9. x 10-3 1.06 3.0 x 102 1.41 1.3 x 101 2.00 8. 2.78 1.1 x 102 1.07 1.14 x 102 1.42 1.85 x 10-1 2.01 1.37 x 10 2.80 1.29 x 10-2 1.08 3.75 x 10 1.44 4.3 10'1 2.02 2.1 x 10 2.82 1.65 x 10'2 1.09 1.45 x 10 1.46 1.08 2.03 3.3 x 10 2.84 2.45 x 10 2 1.10 6.8 1.48 2.55 2.04 4.95 x 10 2.86 4.00 x 10-2 1.11 3.15 1.50 5.9 2.05.6 x 10 2.88 7.00x 10-" 2 1.12 1.91 1.52 1.59 x 10 2.06 1.1 x 10 2.O0 1.23 x 101 1.13 1.42 1.54 5.1 x 10 2.07 1.6 ( 102 2.92 2.14 x lU 1.14 1.62 1.56 1.86 x l02 2.08 2.35 x 102 2.94 3.00x 10-1 1.15 2.65 1.58 7.7 x 102 2.10 4.8 x 102 2.96 3.47 X 101 1.16 4.8 1.60 3.5 x 103 2.12 9.2 x 102 2.98 3.75 x 10-1 1.17 8.4 1.63 1.6 x 13 2.14 1.68 x 103 3.00 3.83 x -1 1.18 1.5 x 10 1.65 6.0 x 102 2.16 2.8 x 103 3.02 3.65 x 101 1.19 2.8 x 10 1.67 2.15 x 102 2.18 2.1 x 103 3.04 3.22 x 101.20 5.8 x 10 1.68 1.25 x 102 2.20 1.17 x 103 3.06 2.92 x 10-' 1.21 1.45 x 102 1.69 7.2 x 10 2.24 2.6 x 10 3.08 2.93 x 10 1 1.22 5.6 x!0 1.70 4.1 x 10 2.28 6.4 x 10 3.10 3.65 x 10 1.23 1.7 x 103 1.72 1.3 x 10 2.32 1.75 x 10 3.12 6.2 x 10'1 2 1.24 5.3 x 10 1.74 4.2 2.36 5.5 3.14 1.2 1.25 1.85 x102.1.76 1.3 2.40 1.95 3.16 1.7 1.26 6.6 x 10 1.78 4.25 x 10l 2.44 7 x 10 3.18 1.0 -1 -1 1.27 2.5 x 10 1.80 1.52 x 101 2.48 2 x 10-3 3.20 3.4 x 10 1.28 1.0 x 10 1.82 5.6 x 102 2.52 5 x 10-2 3.22 2.25 x 0-1 1.29 4.45. 1.84 2.6 x 10-2 2.54 2.25 x 102 3.24 2.27 x 10 1.30 2.08 1.86 2.35 x 102 2.56 1.12 x 102 3.26 2.8 x 10 1.31 1.0 1.88 3.95 x 10-2 2.58 8.5 x 10'3.28 3.5 x 101 1.32 5.05 x 10 1.90 8.7x102 2.60 7.1 x 03 3.30 4.5 x 10' 1.33 2.65 x I01 1.92 2.2 x 10-1 2.62 5.6 x 10 3 3.32 5.8x 101 1.34 1.5 x 10I 1.94 5.5 x 10 2.64 3.85 x 103 3.34 7.6 x 101 1.35 9.2 x 10 2 1.95 8.6 x 101 2.66 2.6 x 103 3.36 1.0 1.36 0.5 x 102 1.96 1.37 2.68 2.2 x 10' 3.40 1.80 1.37 5.7 x 10-2 -3 1.37 5.7 x 102 2.70 2.37 x 10 3.44 3.35 99

. - WILLOW RUN LABORATORIES -.... - TABLE 9. w0 Vs. WAVELENGTH FOR WATER VAPOR (Continued) X WO A Wo A W il (- ()3.48 6.40 3.52 1.3X10 5-10 8.9 x 90 6.90 2.75 x10 3.56 2.3 x 10 5.15 6.4 x 10 6.95 3.5 x 103.60 4.4 x 10 5.20 4.7 x 10'1 7.00 4;3 x 102 3.64 8.5 x 10 5.25 3.4 x 10 7.10 6.8 x 103 26 5.30 7.20 1.1k 0-2 3.68 1.59 xo 53020 1.1 x 102 3.72 3.0 2 5.s35 1.78 x 101 7.40 2.75 x lo3.72 3.0 x 10 3.76 5.7 xo2 5.40.25 x 101 7.60 7.3 x 10"2 3- 76 5.7 x 08 2 78 3.80 1.06 x 10s 5.45 8.8 x 10 7.80 1.9 x-1 383 5.bO 6.3 x 102 8.00 4.7 x 101 3.84 1.95 x 10 2 3 53.8 3.0 x 01 820 1.0 103.88 3.68 x 10 3.92 69 x03 5.70 1.4 x 102 8.40 2.0 J.92 6.9 ) 10?.96 5.80 7.2 x:03 8.60 3.85 -3 4.00 5.85 5.1 x 103 1.80 6.9 4.05 5.90 3.5 x 10 9.00 1.21 x 10 -3 4.10 5.95 2.65 x 10 9.20 2.17 x 10 4.10 -3 4.'.5 T7 ir1 6.00 2.1 x 10 9.40 3.95 x 10 4.20 6.05 1.97 x 103 960 7.4 x 10 4.25 6.10 2.15 x 10 9.80 1.43 x 10 4.30 6.15 2.95 x 10 10.00 3.00 x 10 4.35 6.20 4.15 x 103 10.20 6.6 x 102 43 6.25 5.8 x 10-3 10.40 1.65 x 103 4.40 4.45 x 103 4.45 1.7 x 3 6.30 7.7 103 10.60 4.65 x 103 4.50 8.0 x 02 6.35 8.05 x10' 10.80 T 51 4.50 8.0 x 103 2 6.40 7,1 x 10 4.55 3.6 x 10-3 4.60 1.85X 02 645 4.8 x 10 4.65 8. 10 6.o50 3.1 x 10 4.70 4.5 x 10 655 2.05 x 10; 4.75 2.3 x 10 60 15 x 0' 4.80 1.2 x 10 6.65 1.35 x 10 4.85 6.8 6.70 1.37 x 10 6.75 1.52 x 103 4.90 4.15 4.95 2.7 6.80 1.8 x0-3 5.00 6.85 2.2 x 103 5.05 1.26 100

-- - WILLOW RUN LABORATORIES _ I O o CM *~ X 0 c ^ ~ t o c: co o> oo oM c P c i e, ~ o 0 ap n ~ o n _ oI c o t, - c IOVc 3 0 0o e- ^ V4w e V t Lo n Lo n t 0 ~, C4 c::CO ") O O ocO c c C;.,o *N Mt CO 0,sa oo t-c. o:o -o COOt-M t -CO Lt 0000 W0 0000000 "t-t-0 t. COLO' OOCO -04 O t000000 O^ o.^, C> m O 4O vr o4 to O vs o ax " to v Fo m v +t-t Cfi C<lA'^ ^'^ ~q M "0 V CC _ C Q a0 C; t~ t- 0; C~ rO OS <7 C; C 0> > 0 o Q P 4 C Q tn C _M CO ) "4.-4 r4 0 dc de p d rt I co u tv % L cntto t-o coo U 1 r c:.-,-,.,' - -., "............................... C; _0: C)r,:, _ -4. V:4.,: <::,-4 _,4 _ Cr _t_ k1 O Q oO - O 0o' 0n -.o cI o. oo 0o 0-'o o 0 O Co oo c <140 tr cn O0 0'- O LO; ~( t-.t.t- C O Q 0 00 0 )) 0n C0M0 <01 O OOO v o ~ o o ^ H ^ Cx o o o oo Q O ca Q o C oC o O a O a b, X n" ~ v 101

WILLOW RUIN LABORATORIES o ~ ot oQ < < oi _< C1 ~ o> < o ~t - _q t- ~l~ o o t_ V c o o o to m to t l U o r 0 CI c C t, s O o c o> oo4 4 c 0 0 N~ o o n 0 1" y> ls ri e: ft o&^^" w a 00 "^ l cN C^. (O C ^oco retc V00 ~0QrS44rc~ P'N~4 C0(V-C~ N~4000 00000 00 -= t- Cm N N co Nr. t- o In v4 a CO Sn. s to in c)C CV, a 03 a C* bb a a m m m m C4 t-.4 L) In oo a g,< o0 to co UX 04 CV 0 ) Oi cq C.) 5 <0t o UO) c O t 0 )Cr1', t.-0 CC tc -.4 CO CD Co..,tO..-4 A 0 0,41 OC C UU^ D I 0' u c to. 0 < ".., Co0 C4 to (C O t o 0 c;O Q oo oo oo oo. in 0 0 3 ~ o..t oN1C O COONO-O s N to v C)N to C 0 o0 O o o CD o -c ) co t. It' W. to, 3 0000OC^'~y<000 0010 OCSIN NOO O LtoOO co C", >0 U'.%0o CO Nt1 "4 N"4 V-4 o.o o N.' ^. i"r 0'.4 - 0> a CD D i C ) atoo s1C CO tDC'to C O COn O tuO O'' S- c Mr 000 C - 4 n MO q V4 n 00 C4 t- - X'O co UD in 0 v CV) c- Lto N4tow ) o vt0 w ZQ l Lo V" 0 o t' ^ coV) W C3 co to CD 0 t' inKeq V4 V o~,I,,,,, ~ ~ ~'r * 4'..,'. ~ ~ * "e,, ~ * ~ ~ t'. co * ~ " " 1o0 2 cDct - ao O " o a _ co cq "O C) 0 o co cttoco ttocor^ 0 0P0 0 " " M VO O C O U t rt o 0N E-' c 1 0o 0 0 4 0 0 NN N C,'4 N tNN'N NNC4 NN 4C CNNNNN;4 CN4 N N O02 { U t - Cl<) XO;3 _CMCCO CO CO t O O ~l C0 _ _~t ddddd Crj m a a cu d <o~~?o~- Eo o ci r ctco~^< t v co la o a Co C cs O d O C 0 i0n 0 n< r< o C; t<< c tO O CSi* 00 OW 06 0 c a o;> aO Ce S c c Si c c cc ci N c; CQ cS CS ~s CV CS C Cc i N N N N N N N A N N C; Nb NC 102

WILLOW RUN LABORATORIES ~0'1 " ( M M to= m 0 mOD t-* tC- t- U 1 a << atR co 4 W t c co C 4 M " c a cA m b - co U" em t-I C, U3 C t- CQ N COC C IC C CO C 0 V.40 It O CO CO a m C w co 0 o a o v CW c i0 IV~ C 4 C4O 4 Ov-4- vH C.0OO' 0m ^ O S M00C r(WU <4a _r o oo o o oob r4 0 C c C C i O 0) t- a O 0 Pe cotsC 5? ett Ct4 _oc o: L a o c 9-4 _ 4 s-4 L a P4: CO ~ IO C0U)0> 000,t0 t-0t ( O 0 > C0 t 00, - LOCt0 D O - t M V U. " a ) v W La 0 ow C 00 0 t co Cs a >0 cC /*<1 u. ~0 - r r 4cd V tD CLoD CO LO U) <o OD vc t,i4 0 CS 61 0~C0 l l 0000 000000 t 00 000 C 4 00 E 0 0 ^{^ l^~^tD<Pt^ tCO U'4- 4 LO J0U v m 0 C3 00 N V0 - co t C o 0 c 00c m 00'I 00, (0 rt-.- co (0 0co0 VIIn UM t- ~r v m U c m C) C,_ l - m c v M m c 0 V 4' r. C;O,0.- O tO hO1 0 I Ct C CO n 0 ai C c, ti 0I, nC 03 r^~=, ~ ~ ~ ~ ~ ~ ~ -4,. Nr 0' CO 0, _ " _"O CO''' L' g r V3 O <t a X O a r 0> D Ln - v t;D t IC aZ O^ ta~< 3 a a le O v J CV C- C3 4)'o *t; Ln t~ C- 0S-O3 0U ~N o, a yX * C'', ~,i 4, 103

-....,- WILLOW RUN LAfSORATORIES-____ To predict H20 absorption by the method of L.indquist, the amount of H20 In an atmospheric slant path is reduced to an equivalent eea level concentration, W*. This value and the constants C3 and C4 are substituted in equation 85. 3.7. METHOD OF G. E. OPPEL Oppel [18. 21, 54j has applied the absorption band models of Elsasser and Goody to the laboratory data of Burch, Williams, Gryvnak, Singleton, and France [1] and the calculated homogeneous-path data of Stull, Wyatt, and Plass [19] to develop transmissivity functions for H20, CO2, N20, CH4, and CO for the wavelength range from 1.0 to 5.0 4. These functions express absorption as a function of two frequency-dependent parameters, 2nff/d and S/2rao, and two slant-path parameters, W and P. Absorption spectra computed by Oppel's method differ from spectra obtained by any'other published method which is based upon a closed-form band model in that the spectra have a relatively high resolution (approximately 0.008 o). The higher resolution resulted from the laboratory data that were used and the method of performing the empirical fit. A sample of the laboratory data of Burch et al. is shown in figure 38. Two curves are prese'ited for the 2.7-4 H20 band for two values of absorber concentration, each for a different equivalent broadening pressure. Note that the resolution is such that spectral line structure is observable. Oppel preszrved this structure by evaluating the frequency-dependent parameters at each maximum, minimuin, and inflection point of the absorption spectra. The procedure employed in;erforming this evaluation is exactly that which is described in section 2.3. This method is in contrast to that employed by other authors who either invoke a smoothing process that removes most of the spectral detail in the absorption ba.;d or use laboratory data that are of relatively low resolution. 3.7.1. CARBON DIOXIDE. Oppel fitted thb. Elsasser model to the data of Surch, et al. [] 2 for the 2.7- and 4.3-4 bands of CO, to otiain values of 2naoS/d and S/d which are presented in tables 11 and 12, respectively. The exact Elsasser equation and the strong-line and weak-line approximations are, respectively:.Y A = sinh I0(y) exp (-y cosh 3) dy (86) -'0 (2nrS) -1/2 A = erf 4 WP (87) 2 2 104

-WILLOW RUN LABORATORIES 20 20 0 i~m i ( 0 60I- I F Win t A P n mmM pr cm i 113 80 OAGiL t 862 100LL' 3000 3200 3600 4000 WAVENUMBER (cm'i) FIGURE 38. ABSORPTION VS. WAVENUMBER FOR 2.7-p WATER VAPOR BAND 105

-WILLOW RUN LABORATORIES TABLE H1. ABSORPTION CONSTANTS FOR CARBON DIOXIDE FOR THE 2.7?- BAND 2woS/d2 /d 2.6434 0.15 E-8 0.33 E-3 2.6504 0.8 E-7 0.1C E-" 2.6674 0.5 E-4 0.37 2.6724 0.25 E-3 0.50 2.6738 0.315 E-3 0.50 2.6802 0.55 E-3 0.55 2.6831 0.54 E-3 0.88 2.6882 0.28 E-3 0.86 2.6911 0.195 E-3 9.54 2.6940 0.26 E-3 0.38 2.6969 0.325 E-3 0.52 2.7027. 0.345 E-3 0.70 2.7086 0.28 E- 3 0.74 2.7137 0.135 E-3 0.62 2.7211 0.39 E-4 0.44 2.7322 0,10 E-4 0.15 2.7397 0.15 E-4 0.025 2.7473 0.75 E-4 0.076 2.7548 0.26 E-3 0.36 2.7579 0.30 E-3 0.55 2.7609 0.255 E-3 0.58 2.7663 0.13 E-3 0.53 2.7685 0.81 E-4 0.33 2.7739 0.24 E-3 0.26 2.7778 0.28 E-3 0.42 2.7801 0.27 E-3 0.4'" 2.7855 0.165 E-3 0.44 2.7917 0.7 E-4 0.29 2.7941 0.56 E-4 0.16 2.8027 0.35 E-4 0.13 2.8129 0.84 E-5 0.07 2.8153 0.58 E-5 0.037 2.8177 0.39 E-5 0.027 2.8241 0.2 E-5 0.017 2.8281 0.165 E-5 0.005 2.8345 0.155 E-5 0.003 2.8433 0.197 E-5 0.0039 2.8514 0.165 E-5 0.0033 2.&855 0.13 E-5 0.0024 2.8620 0.45 E-6 0.001 2.8686 0.97 E-7 0.3 E-4 2.8752 0.24 E-7 2.8777 0.1 E-'l 2.8810 0.7 E-9 106

WILLOW RUN LABORATORIES — TABLE 12. ABSORPTION CONSTANTS FOR CARBON DIOXIDE FOR THE 4.3-j BAND X l 22%S/d2 S/d 4.149 0.0 0.0 4.158 2.6 E-10 0.00026 4.167 2.95 E-8 0.0059 4.175 1.92 E- 5 0.24 4,184 1.39 E4 0.87 4.193 5.13 E-4 2.05 4.202 5,14 E-3 15.1 4.211 1.2 E 2 26.15 4.219 2.04 E-2 34.6 4.228 3.34 E-2 41.6 4.237 3.48 E-2 41.5 4.246 3.42 E-2 37.2 4.255 1.2 E-2 12.25 4.264 2.34 E-2 25.5 4.274 2,81 E-2 32.7 4.283 2.53 E-2 31.6..,292 2.07 E-2 26.5 4.301 2.33 E-2 28.4 4.310 1.07 E-2 12.16 4.319 1.34 E-2 13.9 4.329 8.46 E-3 7.7 4.338 4.58 E-3 3.53 4.348 3.7 E-3 2. 2 4.357 1.74 E-3 1.;5 4.367 1.34 E-3 0.974 4.376 4,31 E-4 0.322 4.386 5.07 E-4 0.39 4.396 1.03 E-3 0.825 4.405 1.47 E-3 1.224 4.415 9.87 E-4 0.866 4.425 5.67 E-4 0.522 4.435 4.25 E-4 0.34 4.444 1.23 E-4 0.131 4.454 7.1 E-5 0.0825 4.464 4.14 E-5 0.053 4.474 2.34 E-5 0.0304 4.484 9.92 E-6 0.0167 4.494 3.18 E-6 0.00618 4.505 9.6 E-7 0.0024 4.515 0.0 0.0 12V

-.WILLOW RUN LABORATORIES — - - A a 1 - exp (-S/dW) -where Y = - sinh ( 2so: s P 2-p $ W 2ire' P To compute absorption spectra by Oppel's method for a slant path, simply determine W and P by the appropriate equations given in section 2.4 or by the computer program given in appendix I. Substitute these values into the equations. If either of the two approximations to the Elsasser function is used, the values of f and vp should be compared with table 1 to confirm the valid application of the.^odel selected. 3.7.2. WATER VAPOR. The statistical model with an exponential distribution of line strength was used to describe absc:-tion by H20 for the spectral region from 1.0 to 5.0 A. This function was fitted to the data of Howard et al. for the three absorption bands centered at 1.14, 1.4, and 1.88 i, and the data of Burch et al. for the 2.7- and 3.1-u bands. High-resolution absorption spectra for H20 are not available for the 4 to 5 i region. However, Stull et al. have calculated homogeneous absorption spectra for this spectral region using the qutast-random model. These calculated absorption spectra were used by Oppel to determine the statistical band-model constants. The empirical procedure used to evaluate the frequency-dependent parameters S/2rc.nd S/e for the entire region from 1.0 to 5.0 p uwas analogous in every detail to that used for CO2. The results of the empirical fit are presented in table 13. Pbsorption for the exact statistical model is given by A = 1 - exp [-38/(1. 2)1/2] (89) and the strong-:lne and %eak-line approximations are, respectively, A = - exp [-(132) ] (90) and A 1 - exp (-gyp) (91) where ( and 4 are defined as for CO2. in general, neither of the approximations is used to compute absorption since the exact statistical- model is given in a convenient closed form. t the exact function is used, it is not necessary to check the limits of P and 4p since the function is defined for all values. 108

----.WILLOW RUN LABORATORIES... TABLE 13. ABSORPTION COEFFIC!ENTS FOR WATER VAPOR s/2r S/2S/d /do S/2d% S/d 1.07875 0.0 0.0 1.13379 4.7 E 5 1.9 E 1 1.07991 5. 7. E-3 1.13507 6.5 E 5 1.9 E1 1.08108 1. E 1 8.5 E-3 1.13636 2.2 E 6 2.4 E 1 1.08225 1.2 E 1 1.1 E-2 1.13766 d.2 E6 6.8 E 1 1.08342 1.8 E1 1.3 E-2 1.13895 2.5 E7 1.5 E2 1.08460 2.6 E 1 1.5 -2 1.14025 7.5 E 7 2.4 E2 1.08578 4.5 E 1 1.7 E-2 1.14155 2.3 E8 3.5 E 2 1.08696 7.5 E1 4. E-2 1.14286 4.3 E 8 4.8 E 2 1.08814 1.5 E 2 5.6 E-2 1.14416 8.5 E 8 6. E 2 1.08932 2.9E 2 7. E-2 1.14548 1.4 E 9 8.4 E 2 1.09051 4.2 E2 8.5 E-2 1.14679 2.1 E9 1.1 E 1.09170 6.8 E2 1.1 E-1 1.14811 2.5 E9 1.2 E3 1.09290 1.1 E3 1.3 E-1 1.14943 2.4 E9 1.1 E3 1.0940,t 1.7 E3 1.5 E-1 1.15075 2. E 9 9. E 2 1.09529 2.3 S 3 1.7 E-1 1.15207 1.2E 9 6. E 2 1.09649.2.7 E3 2. E-1 1.15340 4.5 E8 3.8 E 2 1.09769 2.9 E3 2.3 E-1 1.15473 1.3 E8 2,5 E2 1.09890 2.8 E3 2.5 E-1 1.15607 4. E 7 1.8 E 2 1.10011 2.6 E3 2.6 E-l 1.15741 1.5 E7 1. E 2 1.10132 2.4 E3 2.7 E-1 1.15875 4. E 6 6. E 1 1.10254 2.2 E3 2.9 E-1 1.16009 1.5 E6 2.9 E I 1.10375 2.15 E 3 3.1 E-1 1.16144 4.4 E 5 1.2 E 1 1.10497 2.2 E3 3.5 E 1 1.16279 9.5 E4 6.5 1.10619 2.3 E3 3.8 E-1 1.16414 4.5 E4 4.5 1.10742 2.4 E3 4.3 E-1 1.16550 2.8 E4 3. 1.10865 2.7 E3 4.8 E-1 1.16686 1.9 E 4 2. 1.10988 2.9 E3 5.5 E-1 1.16822 1.2 E4 1.4 1.11111 3.3 E3 6.2 E-1 1.16959 8.8 E3 1.1 1.11235 3.9 E 3 7.4 E-1 1.17096 6.6 E3 8.7 E-1 1.11359 4.7 E 3 8.5 E-1 1.17233 5. E3 6.8 E-1 1.11483 5.7 E3 9.6 E-1 1.17371 3.7 E3 5.1 E-1 1.1607 7.2 E 3 1.1 1.17509 2.8 3.8 E-1 1.11732 9.5 E3 1.4 1.17647 2.1 3.5 E-1 1.11857 1.3 E4 1.1 1.17786 1.5 3.3 E-1 1.11982 1.8 E4 2.1 1.17925 i.2 3.1 E-1 1.12108 2.5 E 4 2.7 1.18064 9.5 2.8 E-1 1.12233 3.9 E 4 3.4 1.18203 7.7 2.6 E-1 1.12360 5.1 E 4 5.2 1.18343 6.5 2.4 E-1 1.12486 66'E 4 8.5 1.18483 5.2 2.2 E-1 1.12613 8.8E 44 1 E1 1.18624 4.3 2. E-1 1.12740 1.2 E5 2.1 E 1 1.18765 3.3 1.8 E-1 1.12867 1.6 E5 2.9 E 1 1.18906 2.7 1.6 E-1 1.12994 2.1 E5 2.8 E 1 1.19048 2.1 1.5 E-1 1.13122 2.7 E 5 2.5 E 1 1.19190 1.6 1.3 E-1 1.13250 3.5 5 2.1 E I a."32.21.2 109

WILLOW RUN LABORATORIES TABLE 13. ABSORPTION COEFFICIENTS FOR WATER VAPOR (Continued) At0 AS/d S/20a S/d w i1.19474 9.7 1.1 E-1 1.30378 2.0 E 1 1.8 E-1 1.19617 7.5 9.0 E-2 1.30548 2.4 E 1 2.1 E-1 1.19760 5.8 7.5 E-2 1.30719 2.9 E 1 2.3 E-1 1.19760 4.5 61 E-2 1.30890 3.5 E 1 2.5 E-1 1.19904 3.5 5.2 E-2 1.31062 4.3 E1 2.8 E-1 1.20048 2.7 4.4 E-2 1.31234 5.4 E 1 3.1 E-1 1.20192 2.2 4. E-2 1.31406 6.6 E 1 3.5 E1.20337 1.8 3.6 E-2 1.31579 8. E 1 3.8 F1.20627 1.4 3.4 E-2 1.31752 9.5 E 1 4.2 E-1 1.20773 1.15 3.2 E-2 1.31926 1.1 E 2 4.7 E -l 1.20919 9.5 3.0 E-2 1.32100 1.3 E 2 5.2 F-1 1.21065 8, 2.7 E-2 1.32275 1.6 E2 5.8 E-! 1.22122 6.8 2.4 E-2 1.32450 1.9 E 2 6.4 E-1 1.21359 5.3 1.7 E-2 1.32626 2.3 E 2 7.2 E-1 1.21507 4.6 1.2 E-2 1.32802 2.7 E2 8.2 E-1 1.21654 3:9 8.0 E-2 1.32979 3.3 E 2 9.3 E-1 1.21803 3.3 4.5 E-3 1.33156 3.9 E 2 1.05 1.21951 2.8 2.6 E-3 1.33333 4.7 E 2 1.2 1.22100 2.3 1.6 E-3 1.33511 5.5 E 2 1.4 1.22249 2.0 1. E-3 1.33690 6.6 E 2 1.7 1.22399 1.0 E-3 1.0 E-5 1.33869 8.3 E2 1.9 1.22549 1.0 E-2 1.0 E-5 1.34028 1.1 E3 2.2 1.22699 1.0 E-3 1.0 E-5 1.34288 1.5 E3 2.8 1.22850 1.0 E-3 1.0 E-5 1.34409 1.9 E3 3.3 1.26904 1.1!.8 E-2 1.3 1590 2.6 E 3 4.1 1.27065 1.15 2. E-2 1.34771 3.5 E 3 5.1 1.27226 1.2 2.3 E-2 1.34953 4.6 E 3 6.3 1.27389 1,3 2.6 E-2 1.35135 7. E 3 8.3 1.27551 1.4 2.9 E-2 1.35318 1. E 4 1. E 1.27714 1.5 3.2 E-2 1.35501 1.4 E 4 1.2 E 1 1.27877 1:6 3.5 E-2 1.35685 2.2 E 4 1.5 E 1 1.28041 1.8 4. E-2 1.35870 3.8 E 4 2. E 1 1.28205 2.1 4.5 E-2 1.36054 5.3 E 4 2.5 E1 1.3370 2.5 5. E-2 1.36240 8.2 E 4 3.1 E 1.28535 3. 5.6 E-2 1.36426 1.4 E 5 4.1 E 1 1.28700 3.6 6.2 E-2 1.36612 2.8 E 5 5.4 E 1 1.28866 -4.4 7. E-2 1.36799 4.3 E 5 6.6 E 1 1.29032 5.2 8. E-2 1.36986 6.8 E 5 8.3 E 1 1.29199 6.4 9. E-2 1.37174 1.1 E6 1.0 E 2 1.29366 7.4 1.,.E-1 1.37363 1.9 E 6 1.2 E 2 1.29534 8.8 1.1 E-1 1.37552 2.9 E 6 1.6 E 2 1.29702 1. E 1 1.2 E-I 1.37741 4.5 E 6 2.5 E 2 1.29.870 1.2 E 1 1.4 E-i 1.37931 7. E 6 2.9 E 2 1.30039 1.4 E1 1.5 E-1 1.38122 1.1 E7 4.3 E 2 1.30208 1.7 E 1 1.7 E-1 1.38313 1.6 E7 6.0 E 2 110

WILLOW RUN LABORATORIES TABLE 13. ABSORPTION COEFFICIENTS FOR WATER VAPOR (Continued) S/2a d S/2% S/d 1.38504 2.1 E 7 7.4 E 2 1.47710 9.9 E3 3.5 1.38696 2.6 E 7 8.7 E 2 1.47929 8.5 E 3 3.1 1.38889 2.8 E 7 9.5 E 2 1,48148 7.4 E 3 2.8 1.39082 2.8 E 7 9.1 E 2 1.48368 6.5 E 3 2.4 1.39276 2.6 E7 8.3 E 2 1.48588 5.6 E3 2.1 1.39470 2.4 E 7 7.2 E 2 1.48810 4.9 E 3.9 1.39665 2.2 E 7 6.3 E 2 1.49031 4.3 E 3 1.6 1.39860 1.9 E 7 5.3 E 2 1.49254 3.6 E 3 1.4 1.40056 1.7 E 7 4.2 E 2 1.49477 3.1 E 3 1.2 1.40252 1.4 E 7 3.4 E 2 1.49701 2.6 E 3 1. 1.40449 1.25 E 7 2.8 E 2 1.49925 2.2 E3 8.8 E-1 1.40547 1.05 E 7 2.4 E 2 1.50150 1.9 E 3 7.9 E-1 1.40845 9. E 6 2.1 E 2 1.50376 1.6 E 3 6.9 E-1 1.41044 7.4 E 6 1.9 E 2 1.50602 1.3 E 3 6. E-1 1.41243 6.3 E 6 1.7 E 2 1.50830 1.1 E3 5.3 E-1 1.41443 5.3 E 6 1.6 E 2 1.51057 9.5 E 2 4.6 E-1 1.41643 4.5 E 6 1.5 E 2 1.51286 7.9 E 2 4. E-1 1.41844 3.6 E 6 1.4 E 2 1.51515 7.5 E 2 3. E-1 1.42045 3.1 E 6 1.35 E 2 1.51745 7. E 2 2.2 E-1 1.42248 2.6 E 6 1.3 E 2 1.51976 6.6 E 2 1.6 E-1 1.42450 2.2 E 6 1.25 E 2 1.52207 6.2 E 2 1.1 2-1 1.42653 1.8 E 6 1.2 E 2 1.52439.9 E 2 8.1 E-2 1.42857 1.5 E 6 1. E2 1.52672 5.2 E2 6.2 E-2 1.43062 1.3 E 6 9.4 E 1 1.52905 4.6 E 2 4.8 E-2 1.43266 1.05 E 6 8.6 E 1 1.;3139 4.1 E 2 3.8 E-2 1.43472 8.6 E 5 7.8 E 1 1.53374 3.55 E 2 3. E-2 1.43678 7.2 E 5 7. E 1 1.53610 3.3 E 2 2.4 E-2 1.43885 6. E 5 6.2 E 1 1.53846 1.3 E 2 2. E-2 1.44092 4.6 E 5 5.2 E 1 1.54083 5. E 1 1.6 E-2 1.44300 3.9 E 5 4.4 E 1 1.54321 2. E 1 1.4 E-2 1.44509 3.2 E5 3.5 -E 1 1.54560 8. 1.1 E-2 1.44718 2.5 E 5 2.8 E 1 1.54799 1.15 E 1 2.17 E-2 1.44928 1.8 E 5 2.4 E 1 1.55039 1.13 E 1 2.13 E-2 1.45138 1.2 E 5 2. E 1.55280 1.01 E 1 1.67 E-2 1.45349 8.7 F 4 1.7 E 1 1.55521 8.19 1.62 E-2 1.45560 6.2 E 4 1.4 E 1 1.55763 7.2 1.65 E-2 1.45773 4.3 E 4 1.2 E 1 1.56006 7.95 1.77 E-2 1.45985 3.6 E 4 9. 1.56250 9.18 1.85 E-2 1.46199 3.1 E 4 8.8 1.56495 9.2 2.13 E-2 1.46413 2.7 E 4 7.8 1.56740 8.89 2.45 E-2 1.46628 2.2 E 4 6.9 1.56986 1.35 E 1 3.46 E-2 1.46843 1.85 E 4 6.1 1.57233 2.35 E 1 3.18 E-2 1.47059 1.59 E 4 5.3 1.57480 1.12 E 1 1.62 E-2 1.47275 1.36 E 4 4.6 1.57729 1.55 E 1 1.98 E-2 1.47493 1.17 E 4 4. 1.57978 5.99 1.57 E-2 111

-- -- WILLOW RUN LABORATORIES -- TABLE 13. APORPTION COEFFICIENTS FOR WATER VAPOR (Continued) S/2 S/d x s/2ao S/d 1.58228 9.99 2.72 E-2 1.70358 1.2 E 3 1.9 E-2 1.58479 8.92 2.57 E-2 1.70648 7.8 E 3 2.2 E-2 1.58730 5.25 1.51 E-2 1.70940 8.8 E 3 2.5 E-2 1.58983 5.37 1.42 E-2 1.71233 9.2 E 3 3. E-2 1.59236 7.66 1.66 E-2 1.71527 9.3 E 3 3.5 E-2 1.59490 7.1 1.49 E-2 1.71821 9.2 E 3 4.1 E-2 1.59744 5.75 1.11 E-2 1.72117 9. E 3 4.7 E-2 1.60000 1.35 E 1 2.25 E-2 1.72414 7.2 E 3 5.4 E-2 1.60256 1.33 E 1 2.35 E-2 1.72712 5.8 E 3 6.3 E-2 1.60514 6.29 1.08 E-2 1.73010 4.8 E 3 7.2 E-2 1.60772 4.66 7.5 E-1 1.73310 4. E 3 8.2 E-2 1.61031 G.57 1.08 E-2 1.73611 3.2 E3 9.5 E-2 1.61290 6.47 9.75 E-1 1.73913 2.6 E 3 1.1 E-1 1.61551 5.28 7.64 E-1 1.74216 2.4 E 3 1.3 E-1 1.61812 4.24 6.18 E-1 1.74520 2. E 3 1.5 E-1 1.62075 4.41 6.44 E-1 1,74825 1.78 E 3 1.7 E-1 1.62338 1.72 E 1 1.88 E-2 1.751 1 1.5 E3 1.9 E-1 1.62602 2.25 E 1 1.92 E-2 1.75439 1.5 E 3 2.2 E-1 1.62866 1.56 E 1 1.2 E-2 1.75747 1.5 E 3 2.5 E-1 1.63132 5.45 5.08 E-1 1.76056 1.5 E 3 2.9 E-1 1.63391 4.41 3.95 E-1 1.76367 1.46 E 3 3.4 E-1 1.63666 3.97 3.89 E-1 1.76678 1.42 E 3 3.7 E-1 1.63934 4.44 4.31 E-1 1.76991 1.36 E 3 4.8 E-1 1.64204 1.84 E 1 1.43 E-2 1.77305 1.35 E 3 5. E-] 1.64474 2.42 E 1 1.41 E-2 1.77620 1.32 E 3 5.8 E-1 1.64745 3.11 2.35 E-1 1.77936 1.3 E3 6.7 E-1 1.65017 3.67 2.98 E-1 1.78253 1.3 E3 8. E-1 1.65289 1.29 E 1 1.07 E2 1.78571 1.35 F 3 9.3 E-1 1.63553 1.82 E 1 1.3 E-2 1.78891 1.45 E 3 1.1 1.65837 2.9 E 1 2.4 E-2 1.79211 1.6 E3 1.2 1.66113 3.02 E 1 2.13 E-2 1.79533 1.7 E3 1.4 1.66389 1.11 E 1 6.88 E-1 1.79856 2. 3 1.7 1.66667. 2.53 E-1 1.80180 2.5 E3 2.1 1.66945 4.19 3.13 E-1 1.80505 3.1 E 3 2.6 1.67224.6i 3.79 E- 1 1.80832 3.8 E 3 3.2 1.67504 7.79 5.48 E-1 1.81159 4.9 E 3 4.2 1.67785 1.24 E 1 8.36 E-1 1,81488 6.1 E 3 5.6 1.68067 1.29 E 1 8.48 E-1 1.81818 9. E 3 7. 1.68350 1.06 E 1 7.87 E-1 1.82149 1.04 E 4 9. 1.68634 8.05 6.44 E-1 1.82482 2.1 E 4 1.2 E 1 1,68919 8.08 6.31 E-1 1.82815 3.3 E 4 1.6 E I 1.69205 6.74 5.57 E-1 1.83150 4.3 E 4 2.3 E 1 1.69492 1. 1.2 E- 1.83486 6.4 E4 3.2 E 1 1.69779 4. E 1.4 E-2 1.83824 8.6 E 4 4.7 E 1 1.70068 2.2 E2 1.6 E-2 84162 1.2 I 5,.6 E 112

-WILLOW RUN LABORATORIES- - - TABLE 13. ABSORPTION COEFFICIENTS FOR WATER VAPOR (Continued) x 1 S/2na S/d 0 S/2a Sd __ ~ os~o0 Ss/d (I).i —_ 1.84502 1.8 E 5 1.6 E 2 2.01207 5. E 2 3.9 S-1 1.84843 2.5 E 5 2. E 2 2.01613 4.2 E 2 3.3 E-: 1.85185 3.3 E 5 2.5 E2 2.02020 3. E 2 2.8 E-1 1.85529 4.5 E 5 2. E 2 2.02429 2.15 E 2 2.3 E-1 1.85874 6.5 E 5 1.7 E 2 2.C2840 1.6 E 2 2.0 E-1 1.86220 8. E 5 1.3 E 2 2.03252 1.15 E 2 1.7 E-1 1.86567 9.2 E5 1.1 E 22.03666- 8.5 E I 1.4 E-1 1.86916 1. E6 1.2 E2 2.04082 6.8 E 1 1.2 E-1 1.87266 1.08 E 6 1.4 E 2 2.04499 5.3 E 1 9.8 E-2 1.87617 1.15 E 6 1.8 E 2 2.04918 3.8 E 1 8. E-2 1.87970 1.16 E:6 2.2 E 2 2.05339 2.7 E 1 6.8 E-2 1.88324 1.16E 6 2.3 E 2 2.05761 1.8 E 1 5.8 E-2 1.88679 1.12 E 6 2. E 2 2.06186 1.25 E 1 5.9 E-2 1.89036 1.05 E6 1.7 E2 2.06612 86 6.1 E-2 1.89394 9.2 E 5 1.. E 2 2.07039 6. 6.5 E-2 1.89753 7.5 5 8. E 1 2.07469 4. 7 E-2 1.90114 4.9 E 5 7.5 E 1 2.07900 2.8 7.2 E-2 1.90476 3.65 E 5 7.9 E 1 2.083333 1.9 6.6 E-2 1.908-0 3.15 E 5 1.1 E2 2.08768 1.3 5.2 E-2 1.91205 2.7 E5 1.3 E 2 2.09205 9. E-1 3.7 E-2 1.91571 2.55 E 5 1.5 E 2 2.09644 6.5 E-1 2.6 E-2 1'91939 2.35 E5 1.65 E 2 2.10084 4.5 E-1 1.7 E-2 1.92308 2.24 E 5 1.75 E 2 2.10526 3.2 E-1 1.08 E-2 1.92678 2.2 E 5 1.75 E 2 2.10970 1,0 E-3 1.0 E-5 1.93050 2.1 E 5 1.8 E 2 2.11416 1.0 E-3 1.0 E-5 1.93424 2. E 5 1.9 E 2 2.11864 1.0 E-3 1.0 E-5 1.93798 1.83 E 5 1.9 E 2 2.12314 1.0 E-3 1.0 E-5 1.94175 1.7 E 5 1.6 E 2 2.12766 5.7 E 1 3.43 E-2 1.9-553 1.45 E 5 1.1 E 2 2.13220 1.07 E 1 1.03 E-2 1.94932 1.24 E 5 8. E 1 2.13675 1.47 E 1 1.23 E-2 1.95313 1.02 E5 5.8 E 1 2.14133 4.49 E 2 1.24 E-1 1.95695 8.6 E 4 4. E 1 2.14592 5.54 E 2 1.59 E-1 1.96078 6.6 E 4 2.8 E 1 2.15054 1.09 E 2 4.33 E-2 1.96464 4.5 E 4 2.1 E 1 2.15517 1,48 E 1 1.24 E-2 1.96850 2.8 E 4 1.2 E 1 2.15963 2.04 E 1 1.58 E-2 1.97239 1.8 E 4 7.5 2.16450 1.36 E 1 1.14 E-2 1.97628 1.07 E 4 5.2 2.16920 4.32 E 4 9.7 E-1 1.98020 6.7 E 3 3.5 2,17391 2.91 E 2 7.42 E-2 1.98413 4. E 3 2.4 2.17865 6.34 6. E-1 1.98807 2.6 6 3 1.6 2.18341 6.55 6.59 E-1 1.99203.1.6 E 3 1.2 2.18818 1.36 E 2 6.2 E-2 1.99.601 1.1 E 3 9.5 E-1 2.19298 1.31 E 2 6.96 E-2 2.00000 9,2 E 2 7.5 E-1 2.19780 1.27 E 2 9.07 E-2 2.00401 7.6 E 2 6. E-i 2.20264 1.76 E 2 1.64 E-1 2.00803 3.2 E2.. 4.6 E-1 2.20751 2.62 E2 1.3 E-I 113

u —-—,.....W!LLOW RUN LABORATORIESTAi3UL 13. ABSORPTION COFFFICIENTS FOR WATER VAPOR (Continued) 2_ S/d A20 S/d 2.21239 1.67 E 2 1.66 E-1 2.45700 4.39 E 2 6.46 E-1 2.21729 1.51 E 2 1.26 E-1 2.46305 4.85 E 2 7.49 E-1 2.22222 2.64 E 2 1.74 E-1 2.4697 5.11 E 4 4.38 2.22711 3.78 E 2 2.48 E-1 2.4740 2.32 E 4 2.53 2.23214 2.'6 E 2 1.98 E-1 2.4802 2.11 E 4 6.65 2.23714 3.06 E 2 2.6 E - 12.4832 3.92 E 4 5.28 2.24215 6. E I 4.14 E-I 2.4876 3.14 E 4 8.63 2.24719 5.05 E 2 3.97 E-1 2.4938 2.15 E 4 1. E 1 2.25225 4.15 E 2 3.11 E-1. 2.5013.6'i E 4 9.80 2.25734 4.43 E 2 3.18 E-1 2.5050 5.09 E 4 1.03 E 1 2.26244 2.55 E 2 2.05 E-l1 2.5088 3.63 E 4 1.59 E: 2.26757 1.52 E 2 1.38 E-i1 2.5113 3.07 E 4 1.39 E 1 2.27273 1.61 E 2 1.45 E-1 2.5189 2.48 E 4 2.6 E I 2.27790 1.62 E 2 1.37 E-1 2.5240 1.73 E 5 3.45 E 1 2.28311 1.32 E 2 1.15 E-I 2.5284 9.29 E 4 5.16 E 1 2.28833 1.06 E 2 9.35 E-2 2.5361 1.15 E 5 9.2 E 1 2.29358 8.66 E 1 7.76 E-2 2.5394 1.03 E 5 7.22 E 1 2.29885 8.62 E 1 7.1 E-2 2.5445 1.52 E 5 1.44 E 2 2.30415 9.43 E 1 6.82 E-2 2.5497 5.65 E 5 9.06 E 1 2.30947 6.94 E 1 6.75 E-2 2.5562 2,7 E 5 1.12 E 2 2.31481 6.5 E 1 5.83'r 2 2.5628 6.69 E 6 1.41 E 3 2.32019 6.45 E 1 5.55 E-2 2.5661 S.5 E 5 4,57 E 2 2.32558 6.29 E 1 5.41 E-2 2.5707 1.3] E 5 2.59 E 2 2.33100 6.79 E 1 5.65 E-2 2.5773 1.15 E 5 2.32 E 2 2.33645 6.8 E 1 5.66 E-2 2.5840 1.3 E 6 8.13 E 2 2.34192 8.21 E 1 7.93 E-2 2.5893 7.88 E 5 5.98 E 2 2.34742 7.88 E 1 7.98 E-2 2.5az0 1.31 E 6 1.02 E 3 2.35294 8.24 E 1 8.39 E-2 2.5974 1.31 E 5 2.88 E 2 2.35849 8.64 E 1 8.82 E-2 2.6021 2.02 E 5 4.95 E 2 2.36407 8.67 E 1 9.31 E-2 2.6089 i.27 E 5 2.94 E 2 2.36967 9.27 E 1 9.89 E-2 2.5144!.36 E 5 4.07 E 2 2.37530 9.91 E 1 1.04 E-1 2.6219 1.74 E 5 2.51 E 2 2.38095 9.92 E 1 1.09 E-1 2.6288 6.22 E 4 2.38 E 2 2.38663 1.03 E 2 1.15 E- 2.6337 4.9 E 4 9.65 2 1 2.39234 1.05 E 2 1.21 E-1 2;6399 5.67 E 4 1.26 E 2 2.39808 1.05. 2 1.25 E-1 2.6434 7.03 E 4 1.13 E 2 2.40385 5.26 E 1 9.58 E-2 2.6504 7.88 E 4 1.93 E 2 2.40964 9.79 E1 1.37 E-1 2.6674 1.53 E 5 6.18 E 2 2.41546 1.03 E 2 1.5 E-1 2.6724 1.19 E 5 4.89 E 2 2.42131 9.81 E 1 1.62 E-1 2..6738 1,85 E 5 6.09 E 2 2.42718 1.06 E 2 1.86 E-1 2.6802 8.04 E 4 3.05 E 2 2.43309 1.32 E 2 2.16 E-1 2.6831 8.64 E 4 3.09 E 2 2.43902 1.69 F 2 2,49 E- 2.6882 7.07 E 4 2.15 F 2 2.44499 1.76 E 2 2.91 E-1 2.6911 7.98 E 4 2.59 E 2 2.45098 2.49 E 2 3.97 E-1 2.6940 8.88 E 4. 3.07 E2 114

---- WILLOW RUN LABORATORIES - - -- TABLE 13. ABSORPTION COEFFICIENTS FOR WATER VAPOR (Continued) X fS/2ao S/d A (/_a 8/d 7W[ST 2.6969 4.34 r 4 1.47 E 2 ".95r/ 5.43 E 4 1.21 E i 2.7027 4.83 E 4 1.31 E 2.9551 1.72 E 4,16 E 1 2.7086 3.05 F 4 2.49 E 2 2.9612 2.47 E 3 2.32 2,7137 1.2 E 5 2.13 E 2 2.9718 1.24 E 4 1.57 E I.2.7211 3.09 E 5 5.79 E 2 2.9762 2.13 E 4 1.34 E 1 2.7322 7.18 E 4 1.24 E 2 2.9895 6.27 E 3 2.05 2.7397 2.02 E 5 4.95 E 2 2.9940 1.63 E 4 8.18 2.7473 6.15 E 4 139 E 2 3.0030 3. E, 6.85 2.7548 6.89 E 4 2.78 E 2 3.0075.37 E 4 8.01 2.7579 1.1 E 5 2.72 E 2 3.0139 1.5 E 4.74 2.'609 1.52 E 5 2.66 E 2 3.0211 9.92 E 3 112 E I 2.7663 6.69 E 4 2.94 E 2 3.0248 5.22 E 4 7.;2 2.7685 6.64 E 4 2.5 E 2 3.0303 1.49 E 4 8.14 2.7739 6.46 E 4 1.71 E 2 3.0358 6.65 E 3 9.81 2.7778 7.2 E 4 1.9 E 2 3.0395 2.45 E 4 8.43 2.7801 7.63 E 4 1.8 E 2 3.0441 2.39 E 4- 1.1 E 1 2.7855 8.46 E 4 2.3 E 2 3.0488 1.77 E 4 1.67 E.1 2.7917 6.27 E 4 1.13 E 2 3.0534 3.21 ~' 4 7.6 2,7941 4.97 E 4 8.74 E 1 3.0600 1.94 i 4 8.28 2.8027 1.32 E 5 2.55 E 2 3.0C75 1.19. 4 9.68 2.8129 1.1 E 5 1. E 2 3.072i 4.48 F 7.19 2.8153 9.17 E 4 1.05 E 2 3.0779 2.19 E 4 1.28 E 1 2.8177 9.23 E 4 1.36 E 2 3.0386 1.05 E 4 7.03 2.8241 1.3 F5 8.17 E 3.08oS 7.01 E 3 8.86 2.8281 9.38 E 4 6.12 E 1 3,n94' 3.13 E 4 8-3 2.8345 6. E 4 1.07 E 2 3.0989 4. 5 E3 6,25 2.8433 6.24 E 4 3.74 E 1 3.1075 1.66 E 4 9.83 2.8514 5.74 E 4 6. E 1 3.1162 1.15 E 5 6.43 2.8555 6.88 E 4 8.03 E 1 3.1211 1.44 E 4 9.78 2.8620 3.49 E 4 3.87., 1 3.'55 9.55 E 3 8.12 2.8686 5.2 E 4 2.83 E 2 3.1299 ".48 ~ 3 2.2 2.8752 1.8 E4 1.5 E 1 3.1348 1.47 E 3 1.42 2.8777 1.6 E 4 1.49 E 13.1397 9.38 E 3 7.67 2.8810 1.78 E 4 1.5 E 1 3.1447 7.25 E 3 7. 2.8860 7.93 E 3 1.54 E 1 3.1496 3.54 P 3 4.3 2.8944 8.93 E3 C.91 3.1546 8.14' 2 1.24'"2.9028 5.7 E 4 3.89 E 1 3.1596 2.3 E2 4.57 E-I 2.9104 1.13 E 4 8.03 3.166 1.26 2 2.7 E-1 2 9146 1.07 E 4 8.73 3.1696 1.0 E 2 2.29 E~2.9163 1.55 E 4 -8.16 3.1746 3.3 2 i.26 E-1 2.9240 2.48 E 4 2.6 i;1 3.1797 1.31 E 3.30 2.9308 4.92 E 3 4.14 3.1847 1.28 E 3 1.'B 2.9326 9.71 E 3 $.81 3.1898 8.01.E 3 7.34 2.9412 6.61 E 3 7.173.1^,.3 i i 5,, 2.9481 3.02 E 4.1.56 E 1 3.2 7.69 E 3 1.1 E 1 il,

-— WILLOW RUN LABORATORIES TABLE 13. ADBSORPTION COEFFICIENTS FOR WATER VAPOR (Continurm: AX S/2?% S/d S/2O_ 3.2051 8.07 E 3 1. E 1 3.4542 9.16 E 1 1.72 L- 3.2103 1.1 E 4 1.29 E 1 3.4602 5.26 E 1 1.13 E-1 3.2154 1. E 4 1.11 E 1 3.4662 8.36 E I 1.11 E-I 3.2206 1,07 E 4 1.29 E 1 3.4722 1.6 E 2 2.28 E-1 3.2258 7.96 E 3 1.45 E 1 3.4783 5.21 E 1 1.19 E-1 3.2310 6.42 E 3 1.11 E 1 3.4843 3.61 E 1 5.31 E-2 3.2362 5.26 E 2 8.78 E-1 3.4904 4.46 L 1 8.09 E-2 3.2415 8.41 E 3 4.87 3.4965 7.35 E 1 1.13 E-1 3.2468 1.4 E 4 1.02 E 1 3.5026 7.05 E X 9.14 E-2 3.2520 5.07 E 2 6.47 E-1 3.5088 3.83 E 1 8.,84 E-2 3.2573 9.14 E3 4.35 3.5149 3.52 E 1.62 E-2 3.`:26 4.61 E4 1.5 E 1 3.5211 4.77 E 1 1.29 E-1 3.2680 4.67 3 3.33 3.5273 5.23 E 1 8.3 E-2 3 2735 4.15 E 2 6.4 E-1 3.5336 5.4 E 1 8.39 E-2 3.27I7 1.07 E 3 1.21 3.398 5.27 E 1 8.38 E-2 3.2840 5.82 E 2 6.47 E- 3.5461 3.78 E 1 5.25 E-2 3.2895 3.61 E 3 1.94 3.554 6.22 E 1 1.49 E3.2949 1.59 E 4 1.48 E I3.4.33 E 1 7.63 E-2 3.30') 8.88 E 3 8.47.5651 2.E1 4.39 E-2 3.3C8 3 4.64 9 Z 5.16 ".5714 5.31 E 1 1.55 E-1 3.31i3 4.t:8 ~3 3.83 3.5778 7.1 E 1 1.3 E-I 3.3167 3.8. E 3 2.01 3.5842 7.83 E 1 1.14 E-1 3.2 23 1,s56E 3 1.37 3.5907 1.27 E 2 1.9 E-1 3.3,73 fi.1:1 1.71 E-1 3.5971 3.7 E 1 3.28 E-2 3.3333. 1 1.09 E-1 3.6036 4.6 E 1 4.26 E-2 3.3389 5,05 E 2 4.56 E-1 3.6101. 4.67 E 1 7.68 E-2 3.3445 2:.78 E3 1.87 3,6166 2.62 E 1 3.83 E-2 3.3500 1.67 E 2 2.88 E-1 3.6232 4.36 1 3.47 E-2 3.3557 5.05 E 2 5.69 E-I 3.6298 5.41 E 1 5.92 E-2 3.3613 9.43 E3 5.3 3.6364 4.06 E 1 4.99 E-2 3.367 1.15 E 3 1.12 3.6430 2.83 E 1 3.82 E-2 3.3727 1.09 E3 1.13 3.6496 3.14 E 1 5.97 E-2 2.3784 1,31 E 2 2.02 E-1 3.6563 3.65 E 1 4.7 E-2 3.3841 1.22 E 3 8.97 E-I 3.6630 6.29 E 1 5 57 E-2 3.3898 2.22 E3 1.79 3.6697 2.02 E2 3.78 E-1 3.3956 2.15 E 3 1.73 3.6765.07 E 2 3..92 E-I 3.4014 1.54 E2 2.48 E-i 3.6832 4.18 E 1.42 E-1 3.4072 1.03 E 2 1.68 E-1 3.69 2.24 E 1 8.36 E-2 3.4130 1.05 E 3 8.31 E-I 3.6969 1.73 E 1 5.33 E-? 3.4188 8.84 E 1 7.17 E-1 3.7037 2.59 E 3.54 ~-2 *3.4247 2.8 E 2 3..32 E-I 3.710C 4.03 E 1 5.58 E-2 3.4305 1.33 E2 2.03 E-I 3.7175 3.11 E 1 3.53 E-2 3.4364 1.98 E 2 3.06 E-1 3,7244 3.74 E 1 2.32 E-2 3.4423 5.51 E 2 5.29 E-I 3.7314 5.99 E 1 6.8 E-2 3.4483 1.59E 2 2.09 E-I 3.7383 4.26 E 1 5.1.E-2 116

-- --- WILLOW RUN LABORATORIES TABLE 13. ABSORPTION COEFFICIT'NTS FOR WATER VAPOR (Corlinued) S//2ra_^/ S/2n S/dS/d 3.7453 1.23 E 2 5.89 E-2 4.089 3.91 E 1 2.2 E-2 3.7523 5.76 E 1 6.22 E-2 1.098 3.88 E 1 2.61 E-. 3.7594.92 E 1 5.88 E-2 4.107 7.32 E 1 4.97 E-.: 3.7665 3 39 E t 2.32 E-2 4.115 1.66 E 2 9.7 F'-. 3.7736 1.39 E 1 4.42 E-2 4.124 3.87 E 1 2.49 >-' 3.7807 55.5 E 1 7.12 E-2 4.132 4.16 E 1'.34 E2.' 3.7879 5.,2 ~ 1 2.71 E-2 4.141 4.77 E 1 3.19:-2 3.7951 2 65 E 1 2.29 E-2 4.149 3.87 E 1 2.74 E-2 3.8023 t.74 E 1 5.94 E-2 4.158 3.77 E 1 2.38 E-2 3.8095 4.13 E 1 4.12 E-2 4.167 4.81 E 1 2.56 2-2 3.8168 2.76 E 1 2.19 E-2 4.175 5.82 E 1 3.55 E-, 3.8441 3.41 E I 3.48 E-2 4.184 6.11 E 1 4.99 E-2 3.8314 3.55 E 1 4.78 E-2 4.193 3.91 E 1 2.75 E-2 3.8388 2.97 E 1 2.03 E-2 4.202 3.57 E 1 2.C E-2 3.8462 2.98 E 1 2.07 E-2 4.211 4.05 E 2.53 E-2 3.8536 3. E 1 3.79 E-2 4.219 6.97 E 5.36 E-2 3.861 2.92 E 1 2.9 E-2 4.228 2.07 E 2 1.17 S-1 3.8685 3 E 1 1.94 E-2 4.237 3.7 E 1 3.48 E-2 3.876 3.28 E 1 2.38 E-2 4.' 46 3.86 E 1 3.36 E-2 3.8835 2.62 E 1 3.18 E-2 4.255 3.36 E 1 2.98 E-2 3.8911 3.35 E 1 2.4 E-2 4.264 3.49 E 1 3.19 E-2 3.o986 4.07 E 1.94 E-2 4.274 4.37 E 1 4.5 E-2 3.9063 2.83 E 1 2.22 E-2 4.283 6.33 E 1 5.22 E-2 3.9139 2.45 E 1 2.27 E-2 4.292 3.74 E 1 4.22 E-2 3.9216 3.51 E 1 2.7 E-2 4.301 4.03 E 1 4.02 E-z 3.9293 5.5 E 1 1.99 E-2 4.310 4.42 E 1 4.07 E-2 3.937 2.'7 E 1 2.12 E-2 4.319 4.34 E-1 4.32 E-2 3.9448 3.18 1 2.01 E-2 4.329 3.21 E 1 3.46 E-2 3.9526 5.29 E 1 2,23 E-2 4.338 4.65 E. 5.46 E-2 3.9604 3.4 E 1 1.96 E-2 4.348 6.02 E 1 5.6'E-2 3.9683 3.42 E 1 2.12 E-2 4.357 6.17 E 1 6.86 E-2 3.9761 3.87 E 1.94 E-2 4.307 3.55 E 1 4.5 E-2 3.9841 5.75 E 1 2.34,E-2 4.376 4.6 E 1 5.59 E-2 3.9920 3.87 E 1 2.18 E-2 4.386 7.55 E 1 8.2 E-2 4. 3.93 E 1 2.44 E-2 4.396 5.73 E 1 6.84 S-2 4.008 4.2 E1 2.11 E-2 4.405 6.28 E I 7.62 E-2 4.016 4.12 E1 2. E-2 4.415 4.62 E 1.26 E-2 4.024 4.16 E 1 2.09 E-2 4.425 6.99 E 1 7.4c- E-2 4.032 6.2 E 1 3.4 - 2 4.435 8.3 E I 9.9, E-2 4.04 4.68 E 1 2.57 E-2 4.444 9.09 E 1.1e E-1 4.049 419 E 1 2.15 E-? 4.454 5.28 E' 6.32 E-2 4.057 4.09 E 2.26 E-2 4.464 7 7 E7E 9.36 E-2 4.065 4. E 1 2.2 E-2 4.471 2.'. f 2 2.49 E-1 4,073 7.81 E 4.97 E-2 4.484 2.3t4 E 2 2.87 E-1 4.082 5.28 E 3.28 E-2 4.494 1.25 A 2 1.49 E-1 117

- WILLOW RUN LABORATORIES —- - TABLE 13. ABSORPTION COEFFICIENT, F'Ot WATER VAPOR'Conttnued) s S /2, S/d; S/2ra Sd 4,305 4.81 E 1 5.97 E-2 4.773 1."3 E 3 2.02 4.515 2.0, 2 2.14 E-1 4.785 6.75 E 3 7.08 4.525 4- 9 E 2 7.31 E-1 4.796 6;42 E 2. 4.535 5.96 E 2 6.87 E-1 4.808 5,93 E 2 ".69 L1, 4.545 1.15 E 2 1.48 E-1 4.819 1.3' E 3 1.33 4.556 1.1 E 2 1.16 E-1 4.631 7.02 E 2 1.16 4.566 3.44 E 2 2.9 E-1 4.,43 4.39 E 4 2.62 E 1 4.576 6.5 E-2 7.31 E-1 4.854 1.35 E 4 7.99 4.587 2. E 2 1.86 E-1 4.866 6.61 E 1 2.22 E-1 4.598!.15 E 2 1.16 E-1 4.87T 2.97 E 2 4.7 E-1 4.608 2.35 E 2 2.57 E 1 4,U689 2.07 E 4 1.02 E 1 4.618 8.74 E 2 7.78 ~-1 4.902 1.6 E 4 1.31 E I 4.629 1.37 E 3 1.36 4.914 6.27 E 2 1.22 4.640 8.19 E 2 1.1 4.928 6.22 E 2 8.91 E-1 4.651 5.04 E 2 6.96 E-1 4.938 2.78 E 3 3.38 4 S'2 7.28 E 2 1.09 4.350 4.64 E 4 2.29 E 1 4.073 2.17 E 3 2.45 4.963 4.11 E 4 2.5 E 1 4.684 8.68 E 3 4.63 4.975 3.02 E 3 4.34 4.695 1.57 E 2 2.44 E-1 4.933 9.81 E 2 1.32 4.706 8.72 E 2, 988 E-1 5, 2.02 E 3 2.55 4.717 4. 1S E2 5.92:E-1 4.728 2.63.F 3 2.17 4.73'9 8.0 1: 2 1.17 4.'51 1.6 E. 2 4.86 E4.762 5.17 E 2 6.38 E-E 118

--- --- WILLOW R4UN LABORATORIES 3.7.3. MINOR CONSTITUENTS (N20, CH4, and CO). Oppel made empirical fits to the data of Burch [1, to develop transmissivity fusnctions for N20, CH4, and CO. The pro;edureas nvolved in performing the empirical fits and calculating absorption arc, the;ame as used for H20 and CO2. The resulting frequency-dependent parameters are given in tales 14, 15, and 16. For N20, absorption Is given by the Elsasser model (eqs. 86-88) and the parameters are given for the spectral region from 4.367 to 4.739 M. For CH4 and CO, absorption Is defined by the statistical model (eqs. 89-91) and the parameters are defined from j.0248 to 4.158; for CH4 and from 4.405 to 5.0 t for CO. r et foregoing methods for computing atmospheric slant-path aLsurption all'wt for the de'erz.'anation of spectra of extremely high resolution compared to the other methods discussed in thi' section. This primarily caused by the fact that Oppel's empirical flte were.ased on the most recent laboratory data and the empirical procedures were such that a smoothing process was not used and therefore the Spectral character, observable in the laboratory data, was retained. 3.8. METHOD Oi W. R..,PADFORD Bradford [55] made ar, empirical fit of the Elsasscr band-model equations to the laboratory data of Bradford, McCormick, and Selby [56] for the 4.3-g CO2 absorption band. Since the mrethod employed by Bradford to determine the frequency-dependent parameters is analogous to that described itn section 2.3 it will not be discussed in this section. The band-model equations and frequency-dependent parameters are exactly those used by Oppel; hence, absorption ic given by ~Y A sinh f I0(Y)(-Y cosh LdY (92) or the strong- or weak-line approxiri i;.ns, respectively, A = erf lt WP (93) A = 1 - exp (-S/dW) (94) The frequency-dependent parameters 2rao/d and S/27ra0 are given in table 17 for the wavelength range from 4.184 to 4.454 / with entries every 5 cm. The resolution is on the order of 20 cm. It is emphasized that the parameters 2:;a0/d and S/2ao. listed in the table are not 119

-----. WILLOW RUN LABORATORIES TABLE 14. ABSORPTION COEFFICIENTS FOR NITROUS OXIDE FOR THE 4.3-_i BAND XA S/d 2ro6S/d 4.367 0,0 0.0 4.3"'; 0.0495.2.4 8 E-6 3b6 0.123 8,62 E-6 4.396 0.396 475 E- 5 4.405 0.726 4.415 1.18 2 0 E-4 4.425 1.98 3.96 E-4 4.435 26.5 1.11 E-2 4.444 47.0 3.1 E-2 4.454 41.4 3.68 E-2 4.464 30.0 3.3 E-2 4.474 26.9 3.63 E-2 4.484 19.4 2.96 E-2 4.494 15.2 2.49 E-2 4.505 20.1 2.86 E-2 4.515 20.9 2.7 E-2 4.525 28.3 1.87 E-2 4.535 30.7 3.44 E-2 4.545 13.1 1.97 E-2 4,556 7.95 1.34 E-2 4.566 4.96 9.72 E-3 4.576 3.51 9.34 E-3 4.587 1.29 4.53 E 3 4.598 0.825 4.13 E-3 4.608 0.341 2.25 E-3 4.618 0.245 1.7 E-3 4.629 0.175 1.15 E-3 4.640 9.94 E-2 5.62 E-4 4.651 5.15 -2 2.41 E-4 4.662 3,48 E-2 1.38 E-4 4.673 ~.54 E-2 8.38 E-5 4.684 1.8 -2 4.21 E-5 4.695 8.6 E-3 2.42 E-5 4.706 3.3 E-3 2.48 E-5 4.717 2.2 E-3 1.08 E-5 -'.728 5.7 E-3 1.48 E.5 4.739 0.0 0.0 120

: - W I L.CW RUN LABORATORIES "'"<'"' C " C",~ C", f CN C' " C t 1-. - I CM C% i, CM CM, CM CN N N CN C C N CMN M CM CM Cs CMa I I I I I 0 t a I a I a a a a a a, t t I I I I a I a a a * a i t'-,n O > t 0 tin U 0 W (0 o U,, - t OOC v >..0 o - o I C C o )n C >c Ct P4 V U CPn 4b U c v c^ c~l;P~a~ od c~ ~c: ed ~j ui 9;.Tj F; ej c;~ c; c~c; n~c~c~rr Q6 q; m- V:u oira i f lj o 4i rcoi., C,-, C O CI~nO f,- u a In t0 Lo j O O F a I 0 1M 8 CCO Co 00 ut m t - v.n w t co "- 11f01) L co (IV) k- m m V t; I ir ~-e e s c' -4 c' C ) i O er 0 )e o CZ t4 L'c V ); CV; 0 W;L.- - P4oNc't *oP4 cco r P4 - coP4 Pcs4~ cY - N cs n m c-o - c c'. 00 - C tI t co PN a cO Z I^ Cs c o r o a c s c14 c ( Co t c s co a o o c t c(v i mt ac c L n r c E K co d( /) I,S C: CI -t- C'l )' " W.' C4 tO t, C, 0,' cS' C, a i. a, t- a a a0' Kr v Ztr v 1V v 1 V VS t) Ut v tr I v uLO ),n Lu" V u Lr" n of If -' U X t-4 it, CL, tu)),toco d -i X C C?; C n i ts.............. o ~ o )., 0 | I I t t ( I I I I ) I l ( I ) ( I |'?tM W W 41 W W W W o3 1W W im W W W W W W Wcu W W Wd W:7a' ) w m c c-' - u'I) uC' u'o~n t- j L C in eL n c Cn u'~, C'c.c t'I C) e- re- t -.0 co s C — &n " " o L- > inin C4 M 0C OO *- " - 0 co ococOc U) r 4 u L iw w,, tJi L:sl CJ fi kl w w W W;r1 W W W M w iJ w x 4 i W w w W sZ w_ n o Co m m a r - to LO uo V e O0' a) cn o a - t- a O o c C co Oa Cs to a4 - n O'W a a -e tO <JCO Q COU *O4 (O " CL- C CO.,, cO..".. - M. a. >.,...4 C..'.'.-... ^'j' e c" C4 Cn cn C o n (AD s C - co iO a C>o ct t Cs t- c 4cc t o LO V; > w t- U- co c; c C - 3c$N ictt c 0 s C 4^ Cq c-o —-P ccm' )' c O C c c! coeoc o eC C" m m VO Z.,CV. o) * n o r_ ) m VL fnL n m V C I m n > U4o 4, W o V m r N m V V m mrw re c co co - co; Cr o coc d 9' e7 S) c co ccc co mo m o cO o C4 c o Cq co c C Cq C co; o c c- 4 c cO 1 U" *II i t f I "t <* ~~~~~ - -."<l'^'^ NI( ^.~~~~~~~~~~~~~ ~~~~~1 " ~- CN (0 O ai( o" Oaf) ) 0)COC") 0 if)C4 0) 1Ct-W-C (0 ( b'000)'. u0 co c.'- c' CO 0"Ca) M oCot4 V4 C) t- at tt o Ctot o to t C" cn er <r t *o *t C 2 6 *! L 4 u,*~ t) * *0P I c'')' *c * " * * *f *U ) (0 C >' *0*0 0 C0D - 0 a _'.4 P 4 P4 P 4 PO p4 P 4 _O P4 P'.4 v N - CO OVn- C - P - C O C',C")'C")l, CO c")C")C') C' )C'C') CO C' CC, C C CO v) c - CO C) q ) )0 C C' C O C m C O)C C ) C t ) C0 c;j, c...............*.*.r.....~c...................cc~.................................. X C X X X C C X CXCNCA S N C XCA A n:nX nCNCA X C XC e XX XX

V- - - WILLOW RUN LABORATORIES... cm In cn cn cn cV) )C4) ce) V) m m ( * (V m V c4) MV V c4... Ia a4 a.4, e t,,,,.,,.,a, a 4,?1 CO c"? o <0 0 t r- n ~) t O C, Ct C us e. I m\ o a — 4' ) tV o C( IV o at to q co U) U.) a) _.. O C C C r _ o C o G Ct0 CV O O n C t to t- O * el _s C i cn m vt v V t I i:P t I t V 1 V m VI V4 ) c, C C C CO c - cCC) v) m c C' ",' "c C! c" " C, e. v Co ) _-' -'' ~""',O cn c U O C cO3 vcO co, t' - OC t- tc C t- o C") O c o t-; i, r I *. r. e a, to tn 1w tn U) U13 L if ui 4 C4O V v Cs CX M C4 C u^ *r>^ irt3i^ ~n in ir~tn ^ ^ ^ ^ ^^t 0 qw t co w c cVc) t Y f-o t tU a a a a a O a a a a; * a a. Za a aa I I I S I I I I ~, t: t ~ 4V 0: n <, a,,e <i V4-,-4:4.. ":'. V;' 4 "4I. c n U cor IV I) co esw ec - c - C ai U) ra O co t. 12. rr;;1 X u,* o Pc U CZ kn uo U 3 tn Lm i cn uc ( s L M <o&~lr>^- c3 ex, PI t*-;o(~> <*^^~tr'<?txr~L co r- co I vc oo t- co c1....................,... h.< _e _ tC c o 6m c o c co co tc w o. c (,,) LO U, co v c C b in r r Vn in C n. n' 4 n E- rnoo m ap z o: co co oco oo > r- w oo?> > in V ^- V a PI M v t tt v ~) v cv z cm c- c >. s m es cs N CO c cV (23 QC^u N a v o t:9 c t > a w i3Cr^ e v v Ca e O) L t th - co am C c as ) tt YZ O C t- r V 00 at a n q Cto A( 1 CS C4 C1 tr 1 w t t e t M a5 o M v CM a n V n U 0 a a c'< WD W - CO E- tD, C;> CO tV C - C- t- C- t- C-,r^ ^~^~^ a cs ^ ^ 75 oco cs o,c^r c: n cd cc^ o Ca 4 o C. C c lc I ) cn.;t C) t n C" 7 t C. ) m I vl' I i i 4 ii 8;Cj Cj a; c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-s,.e ew; c,<iss;;;t 4-;i,'-t gt r i~ C

WILLOW RUN LABORATORIES --'ABLE 16. ABSORPTION COEFFICIENT FOR CAR3ON4 MONOXIDE X S/2r S/d X S/2 S'd - ---- 4.405 1. E 1 1. E-2 4.717 1.01 E 4 2.14 4.415.2. E 1 2. E-2 4.728 1.07 E 4 2.53' 4.425 4. E 1 4. ~-2 4.739 1,0; E 4 2.4 4.435 6. 1 6. E-2 4.751 I. E 4 2.2 4.444 8. E 1 9. E-2 4.762 0.35 E 3 2, 4.454 1 E2 1.2 E-1 4.773 8. E3 1.66 4.464 1.2 E 2 1.6 E-1 4.785 6.5 E3 1.37 4.474 1.4 E 2 2.2 E-1 4.796 5.1 E3 1.12 4.484 2. 2 2.7 E-1 4.808 4. E 3 3.8 E-1 4.494 3.6 E2 3.3 E-i 4.819 2.9 E3 7.1 E..1 4.505 8. E 2 4. E-1 4.331 2.3 E 3 5.3 E-1 4.515 2.1 E3 5.5 E-I 4.843 1.6 E 3 3.7 E-1 4.525 4.8 E3 7.3 E-1 4.854 1.3 E 3.7 E-1 4.535 6.4 E3 1.1 4.866 8.4 E2 2.2 E-1 4.545 6.4 E3 1.5 4.878 5.7 E 2 1.6 E-1 4.556 9.5 E3 2.1 4.889 3.3 E 2 1.1 E4.566 1.05 E 4 2.63 4.902 2. E 2 7. E-2 4.576 1.15 E 4 3. 4.914!.62 E 2 5. E-2 4.587 1.26( E 4 3.26 4.926 1.49 E 2 4.5 E-2 4.598 1.36 E 4 3.4 4.938 1.38 E 2 4. E-2 4.6)8 1.4 E 4 3.45 4.950 1.Y E 2 3.5 E-2 4.618 1.3 E 4 3.36 4.963 1.23 F 2 3. E-2 4.629 1.07 E 4 3.03 4.975 1.17 E 2 3. E-2 4.640 8.3 E3 2.26 4.986 1.12 E 2 2.5 E-2 4.651 5.8 E3 1.53 5. 1.09 E 2 2. E-2 4.662 4.3 E 3 9.7 E-1 3.673 3.6 E 3 1.25 4.684 4.8 E 3 1.52 4.695 6.3,~ 3 i.83 4.706 8.30 E 3 2.14 123

WI-LtOW RUN LABORATORIESS TABLE 17. BAND PARAMETERS FOI C.REON DIOXIDE FOR THE 4.3-i BAND A v/(cm1) 2,ao/d S/2%a 2 2S/d2 4.184 2390 1.380 0.072 1,804 E — 4.193 2385 0.560 1.750 7.221 E-4 4.202 2380 0.580 8."00 3.851 E-3 4.211 2375 0.610 24.0 1.175 E-2 4.219 2370 0.880 31.0 3.159 E-. 4.228 2365 1.140 29.5 5.045 E-2 4.237 2360 1.180 28.5 5.^22 E-2 4.246 2355 1.220 21.0 4.113 E-2 4,255 2350 1.280 10.5 2.263 F 2 4,26, 2345 1.200 19.3 3.657 E-2 4.274 2340 1.080 24.0 3,683 E-2 4.283 "335 1.030 24.0 3.350E-2 4.292 2330 0.92e 22.5 2.506 E-2 4.301 2325 1.110 15.3 2.436 E-2 4.310 2320 0,960 12.7 1.540 E-2 4.319. 2315 1.00C 7.30 9.605 E-3 4.329 2310 1.0C 5.00 6.579 E-3 4.338 2305 1.! 2.Cb 4.220;-3 4.348 2300 1.02C 1.9 2.710 E-3 4.357 2295 1.100 1. 1.879 E-3 4.367 2290 0.980 0.70 8.346 E-4 4.376 2285 0.990 0.38 4.900 E-4 4.38u 2280 0.630 0.73 3.812 E-4 4.396 2275 0.420 1.45 3.36C E-4 4.405 22 0 0.380 1.55 2.945 E-4 4.415 2265 0.330 1.70 2.436 E-4 4.425 2260 0.365 1.05 1.840 E-4 4.435 2255 3.520 0.48 1.710 E-4 4.444 2250 0.700 0.265 1.710 E-4 4.454 2245 0.770 0.112 8.73. E-5 124

----- WILLOW RUN IABORATORIES -. values p^r unit pressure; hence, the prime (') has been deleted' froi the half-width term, a, To obtain the necessary parameters for equation 92 simply divide by sea-level pressure P0. For the spectral region from 4.184 to 4.454 g, the methods of Oppel and B3rdford are basically the same. The same equations were used and the parameters are defined for exactly the same wavelengths. The values are slightly different for the two authors, however, which undoubtedly resulted from Lne fact that different collections of laboratory data were used in performing the empirical fits. 3.3. METHOD OF A, E. S. GREEN, C. S. LINDENMEYER, AND M. GRIGGS A. E. S. Green, C. S. Lindennieyer, a;,d M. Griggs have pursued tie problem of developing methods for calculating atmospheric molecular absorption. The results of their efforts are published in three reports [57-59]. Basically their work consisted of enrpi icall. fitting the strc-ng-lir.e approxi.nation tc the.tatistical liodel to laboratory homcgeteous absorption spectra inr order to obtain transmissivity functions which define the spectral bsorpzion for the atmospheric gases, H20, CO02, 0 N2O, CO. and CH4. The transmissivity functions Pre given as functions of the equivalent sea-level path, W*, and two wavelength-dependent parameters W and n. Specifically, e T e xP- (95) where W is the quantity of absoroer.t 740 rm I ^ and 3000K cox responding to a transmission of T = e 1. This furction defines the transmission for all six absorbing species. The method employed by Green et ai. [57] tc determine the frequency-dependent parameters We and T7 was as follows. Consider a sample of laboratory data taken from t;ie work of Burc'shown in figure 39. Each curve represents the absorption as a function of wavelength for a homogeneous path at 300~K for.aome absorber concentrPtion W and some equ: valent pressux. P. Each value of W is reduced to an absorber concentration at standard pressure W* by t;e expression W EWp W* =-We P0 For cach wavelength throughout the ntetrvai the curve-s define several values ol transmission T1' T2,... Tn,'each corresponding to a value of W*. These transmiasson d-ta were plotted using graph paper in rnich the ordinate was Mn (-!n T) and the absctssa was f.. VW. The re125

WILLOW RUN LABORATORIES 100 r-i-& — r —-., cm 0~mm; 60 -1.38 J 1 -'- jL p < 1 -.38 228 346.08 360 3700 380 ^ 1770 3400 3500 3600 3700 3800 WAVENUMBER (cm 1) FIGURE 39. TRANSMISSION VS. WAVENUMBER FOR 2.7- 1CARBON DIOXIDE BAND 126

..... WILLOW RUN LABORATORIES.. suiting data points define a straight line whose slope is r? and abscissa intercept'a We. Five such curves are shown in figure 40. The case illustrated is for the 2.7-/ CO2 band. Each line defines W and rI for one wavelength. This process was repeated at approximately every 0.O:l, e throughout the absorption band to obtain a plot of We versus wavelength. This procedure was repeated tor each absorption band and each absorbing gas for the wavelength region from 1.0 to 10.0 j. The values of We and r at 740 mm Ir Ki nd 300~K are given in figure 41. The values of We for the same pressure and temperature are also given.n table 18. Table 19 is a listing of the laboratory data which were used ior the various gases and absorption bands to determine the values of W and 17. To apply the. mnethocd of Green et al. [57], one need simply determine, for a given slant path, the value of W* for eatl?hbsorbing gas as outlined in section 2.4.3. The values of transmission can then be computed for each gas using equation 95 and the values of W and r. The flaul value e of transmission for;. given wavelength would then be the product of the transmission of each of the absorbing gases. Noting figure 41, one may observe that the curves of W versus Uwa'elength are presented on a very compressed scale, limitirg tLe accuracy with which one ca.i.ictermine the value of wavelength which correspon is to a gi- i value of W. The data riven in table 18 were compiled, e from these curves; hence the above ir.Lcuracies are inherent in these dafa also. In an effort to increase the acevracy of the original work of Green et al. [57], J. A. Rowe [601] the Aerospace Corporation: teplotted the original data of We versu; wavelength on a highly expanded scale. Some adjustments in the W values were made based on other sources e of absorption spectra than those gtv.en in table 19. The exact sources of data used by Rowe and the specific reasons for modifying the results of the initial empirical fit made by Green et al. are not known at the present time. Rowe's modifications of Greens original work will be given in an Aerospace report which is to be published in the near future. 3.10. METHOD OF R. 0. CARPENTER R. O. Carpenter L61] developed a method for;alculating the emission from CO2 at elevated temperature for the 4.3-p absorption band. His method is based on the Elsasser band model, but the f equency-dependent parameters were calculated from theory rather than being determined from laboratory data using empirical procedures. Even though Carpenter's method was developed for predicting high-temperature C.-; emission, it is directly applic.'., to atmospheric slant-path problems. 127

: WILLOW RUN LABORATORIES i. — "...... -..-.. _ X o-5 ~ *X() -Io410 1 2.68 l 1o-s 2 2.70 -,' {2.~76. D o. 3 0.01 12.78 4 2.80 5 2.7 0.10 -0.10 lO -6 -5 - -3 - -1 e 2 3 3.50t I__ In (-in T) 0 / 6.50'',: O. 50 0.90 0.95 3 0.4 0.6 0.9:5 0.8 1.0 0,999......... - 0.001 0.01 0.1 1 10 100 fn iV* FIGURE 40. PROCEDURE FOR DETEIIMINING We AND Av. The case illustrated is for the 2.7-p carbon dioxide band. 128

---------- WILLOW RUN LABORATORIES —107 *- 0 o - H20(n 0.5) 106 Co2 C02(n - 0.47) 105 (2 * 0.60) co o5- ~ -co 2 10 Co 2(?7 0.6C 2 0~2, ~ ~ ~ ~S 1,o,' l t I, 1 j!,o01:= ii3 2,, 10 CH ) CH (P= 0.46) 10 to2 -1 i I 0.47).u I 10, i i 03 o t o1.0 2.0 3.0.4.0 5.0 6.0 7.0 80 9.0 10.0 1WAVELEN.4GTH (rj 4) 129 lo - 1.29

-- -.WILLOW,t LABORATORIES TA','; d. VALUES OF We H 2 co2 _ 2 H20 H CO2 TO^W)~~~~~~~H) 1.03 7,600 1.92 0.1 1010 1.05 1,500 1.94 0.4 20,000 1.075 130 1.95 0.86 3,500 1.1 14 1.96 1.9 1,700 1.13 3 1.98 7,2 10,000 1.15 5.6 2.00 18 1,100 1.8 30 2.95 35 450..2 12C 2.03 69 1,500..23 3,500 2.04 100 10,000 1.25 390 2.06 210 1,700 1.28 21 2.10 1,000 29,000 1.30 4.4 2.15 4,800 1.55 0.17 2.167 6,500 1.37 0.12 2.20 2,500 1.38 0.14 2.30 65 1.40 0.23 106 2.40 4.15 1.42 0.4 1.5 x 105 2.50 0.1' 1.43 0.56 1.1 X 105 2.59 0.00 1.44 0.8 1.6 x 105 2.63 0.02 1.46 1.95 4.7 x 105 2.66 0.0037 1,600 1.475 4.0 106 2.675 0.0015 3.2 1.50 14.0 2.68 0.0016 2.5 1.55 190 2.69 0.0022 5.4 1.555 106 2.70 0.003 2.8 1.58 1,650 1.2 x 10s 2.72 0.0058 40 1.595 5,000 9.6 x 104 2.735 0.0095 220 1.C1 10,000 1.5 x 105 2.74 0.01 120 1.65 1,250 106 2.76 0.005 3.6 1.70 360 2.77 0.0032 12 1.75 10 2.775 0.003 7 1.80 0.65 2.80 0.027 25 1.82 0.14 2.84 0.052 95C 1.835 0.044 2.90 0.26 1.86 0.08 2.93 0.52 1.87 0.03 2.96 0.74 1.882 0.15 3.00 0.80 1.91 0.056 q.06 0.62 3.10 0.78 130

-- WILLOW RUN LABORATORIES TAE LE 18. VALUES OF We (Continued) xH20 co2 CO N20 CI4 ~3 3.16 3.6 22 3.2 0.6 9 3.22 0.46 8 3.26 0.48 10 3.32 1.4 1.7 3.37 2.7 10 3.38 3.1 8 3.42 5.5 10 3.47 11 20 3.5 18 ~3.7 5000 3.975 30,000 4.1 10,000 4.2 0.6 4.235 0.026 4.255 0.054 4.275 0.036 4.3 27,000 0.12 4.35 1,500 1.6 4.4 9,,i0 200 1 4.4' 3,900 600 0.035 4.495 3,050 1,300 0.0425 4.52 2,200 1,900 20 0.037 4.60 1,650 3,300 35 0.M0 4.65 540 10,400 8 4.75 200 2,500 4.3 4.82 94 740 13.4 4.85 62 1,200 20 4.92 27 1,700 5.00 11 5.13 3 14,000 5.17 2 4,500 5.23 0.92 1,300 5.45 0.058 5.5 0.084 5.725 0.0088 5.8 0.0115 5.875 0.0033 6.025 0.01 6.1 0.0034 6.3 0.085 6.375 0.003 131

~-....' WILL W KU N LADUKA I tJKItC --- - TABLE 18. VALUES OF W (Continued) e HO co 2 CO N20 CH4 03 (O) 6.425 0.005 6.45 0.002 6.525 0X, )46 6.575 0.0025 6.60 0.0033 6.625 0.0025 6.75 0.03 6.775 0.0058 6.8-. 0.02 6.85 0.005 6.93 0.08 6.975 0.023 7.025 0.042 7.125 0.01 7.15 0.08 7.18 0.021 7.2 0.08 83 7.25 0.046 55 7.3 0.08 22 7.35 0.047 8 7.4 0.3 3 7.42 0.115 2.5. 7.45 0.044 4.6 7.50 0.076 4.6 10.6 7.5 0.325 0.42 26 7.60 0.080 0.3 47 7.63 0.23 0.21 0.3 7.675 1.0 0.13 0.55 7.775 1.0 0.4 5.4 7.8 0.4 0.31 10 7.85 0.98 0.19 3.8 7.875 0.58 0.17 4.3 7.95. 0.25 0.58 8 8.00 0.5 2.7 12.5 8.025 1.0 6.3 16 8.15 1.7 48 8.175 1.85 57 8.3 2.8 8.6 7.6 8.9 19 132

.... WILLOW RUN LABORATORIES. TABLE 18. VALUES OF W (Continsed) H20 c CO N2O CH4 ~3 9.2 48 9.34 74 1.5 9.4 88 0.062 9.49 115 0.011 9.55 140 0,022 9.61 170 0.035 9.65 200 0.0125 9.7 240 0.015 9 9 470 0.042 10.0 660 0.098 10.22 1,600 2.7 10.5 5,500 133

-_- - —.WILLOW RUN LABORATORIEC --- - - -- TABLE 19. LABORATORY DATA USED TO DETERMINE VALUES OF W AND rj Molecule Band Reference._ N20 4.4-4.6 Burch et al. [1] 0.54 N20 7.5-8.0 Burch et al. 0.48 CH4 3.15-3.45 Burch etal. 0.56 CH4 7.25-8.15 Burch et al. 0.46 CO 4.5-4.85 Blurch et al. 0.47 CO2 4.1-4.A Burch et al. 0.6 CO2 2.,S-2.84 Burch et al. 0.56 CO2 1.93-2.09 Howard et al. [30] 0.5 CO2 1.55-1.65 Howard et al. 0.5 CO2 1.4-1.,7 Howard et al. 0.5 H20:.03-1.24 Howard a1 al. 0.5 H20 1.24-1.61 Howard et al. 0.5 HO 1.61-2.16 Howard et L' and Burch et al. 0.53 H20 2.16-3.90 Howard et al, 0.54 H,20 4.40-10.0 Howard et al. and Burch et al. 0 5 03 9.3-10.2 Walshaw [34] 0.> 134

... ——. WILLOW RUN LABORATORIES Recall that the transmission averaged over one period of a regular Eluasser band under conditions of strong-line absorption is given by T = I - erf (, )2(96) The most common procedure followed at this point was to empirically fit this function to labora2 tory data to evaluate the frequency-dependent parameter,raS/d. Carpenter chose to evaluate this quantity from theoretical considerations for homogeneous paths at 280~K and 140 mm Hg. Therefore, the W in equation 96 becomes W* and the half-width becomes a0. For homogeneous paths at standard conditions the.,alf-width, line spacing, arid integrated a.,ins'.ty do not vary from point to point along the path. Further, for a regular band the halfwidth and line spacing are independent of frequency. Basically, Carpenter found it necessary only to determine the variation of the integrated intensity from period to period throughout the band. This was accomplished by tusing both theory and experimental result&. With all parameters specified, Carpenter calculated the transmission for several homogeneous paths. His results are shown in figure 42. Each curve represents the transmission for a given absorber 2 concentration where 1 gm/cm equals 509 atm cm of C0?. Since these curv.:s were calculated from equation 96 after first determining the spectral coefficient ra0S/ 2, each of the curves should define the same generalized absorption coefficient which could be used to calculate the absorption for any value of W. This is'not the case, however, and the reaso'ns for this inconsistency between curves is not known. For example, at 2273 cm we have T - 0.13 for W* 5.09 T2 0.54 for W* = 0.509 T, = 0.89 for W* = 0.0509 Applying these values to equation 96, we have (raS/d ) 00.229, (aS/d) 0.216, and trca0S/d ) = 0.190, which is a spread in the data of approximately 20%. Because of the inconsistency in the data, no single. curve should be us:d to specify the absorption coefficients. In order that the results obtained by Carpenter's method could be compared with other authors, a set of coefficients was determined for 21 discrete wavelengths by maling a best-fit to the data in a least-squares sense. The results of this effort are given in table 20. These coefficients.1/2 in conjunction with the.functior T = 1 - erf {(C(nr)W*l]/2} define a method for computing absorption for the 4.3-pt CO2 band. 135

. —- - WILLOW RUN LABORATORIES —- -- 100r 90- 5100 2 202510 o - \\ \ \ f\ I51 o05 60 F 50 * =W 0.051 2150 2200 2"50 2300 2350 2400 2450 2500 WAVENUJMBER (cm-1) FIGURL 42. TRANSMISSION V'S. WAVENUMBER FOR 4.3-g CARBON DIOXiDE 4.16-7 0.159 E-2 BAND AT 280~K 4.237 F 0 5C6 T.__..-___cl2_ __ 4.246 015.9 0 4.2554 0.6'318 4.202 0.527 4.214 14.153 4.283 18.813 4.301 13.487 4.320 5.239 4.348 2.266 4.367 0.648 4.386 0.102 4.405 0.159 4.425 0.906 X-1 4.444 0.175 E-1 4.464 0.123 E-1 4.484 0.498 E-2 4.505 0,235 E-2 136

-W —- WILLOW RUN IABORATORIES3.11. METHOD OF V. n. STUILL, P. J. WYATT. ANO G. N. PLA; q Stall, Wyatt, and Plass used the quasi-random mAoel to cac,'ilate the spectral transmission of H20 and CO..Spectral transmission of these two gaseb was cicul.ted for a large numaber of homogeneous paths and the results are presented in the forni of cxtensive tables. The total work of these.authors is published in two volumes (19, b2] which include a detailed description of the quasi-randm n,odel, the metho:.. used in calculating the rpectral transmission based on this model, and tabulated transmilsitn data. Basically, the method employei is t;,,: hich is described in section 2.2.6. Equations 31 and 35 were used assuming only five inte.-r'iy decades for carh subilterval, Dk. Also, calfulations were made for both the a.rentz line shape and the Benedicl line shape. To remove the artificial effects introduced by dividing the frequency interval into subintervals of width Dk, the interval war sln'xivided twice, each subdivision being called a mesh. The final transmission spectrum was then the average of the results obtained from the two meshes. In using the quasi-random model, the only parameter that was calculated from theory was that of line intensities, which were subsequently grouped into intensity decades; these decades in turn defined a vl^'.e for the average decade line intensity, S;i Wyatt, Stull, and Plass further defined Si by normalizing their results against the laboratcry data of Howard, Burch. aind Williams [1, 30]. The normalization was carried out in the following mainer. The values of Si were calculated for each decade and all intervals. Transmission data were then calculated based on these values of Si. Calculated integrated absorptions were compared with the laboratory data for large frequency i.tervals (>400 cm ) and the valves of S. were adjusted accordingiy. This process was repeated until a set of Si values was obtained that yielded a best fit of the calculated data with the laboratory data in a least-squares sense. The entire process was carried out for two mesh sizes (2C cm1 and 5 c:m ) and a two-dimensional generalization of the least-squares method was used to obtals a final set of Si val,:es. It is obvious from the above discussion that the initial selection of the Si values is immaterial since the empirical process will always converge. Table 21 summarizes the various homogeneous paths for which transmission data are tabulated. It is noted that for each gas there are six entries in the table. The first Ia the wavelength interval over which the computations were perfornmcd. The second gives the range of WV values, the third gives the pressure range, and the fourth gives the temperature range. The last two give, respectively, the frequency spacing between entries and the resolution. Since there are 15 different W values, 7 pressures, and 3 temperatures, each line in this ta"'e, for either gas, encompasses transmission spectra for 3!5 paths. 137

-------- WILLOW RUN LABORATORIES cc o I NC I 9 % 0 o' C4( C O r4g " - O N -, to -. QI U) X ( < 3 O O O O O < h'- 0000 0 0 00 4 l d - o's Q CO O C C C o o 000 C a 00 (s M co u V- V4 *, -4,,-I,OO 0 0 0 C O O 3: v-8 v 1 l s QOg ~ ~ ~ ~ 0( 000._ CW O O' o c: O:; & rJ a X; c P,~ c:; 13 N N N 2 ~ gN S ) 0 000 4 0.0 4 138

--- - -~- WILLOW RUN LABORATORIES Such an extensive set of data rot only gives transmission spet ra for numerous homogeneous paths but can easily be used to determine the absorption for any homogeneous path. If the values of W, P, ad T for a given path are all bounded by the values shown in table 21, then the transmission can be determined by simple interpolation procedures. Also, these data can be used in place of measured spectra to specify the band parameters for the Elsaoser or statistical model. This was done by Oppel who fitted the E;sasser model to these data for the spectral region from 4.0 to 5.0 u. This procedure allows for the calculation of transmission spectra for slant paths that are not bounded by the range of values in the tables. The interpolation procedure to be used is that suggested by the authors-simply a logarithmic interpolation between nonzero values of the absorption. The absorption is expressed in the form A = 1 - = ef(PTW) (97) where f is usually a complicated function of pressure P, temperature T, and abs cber concentration W. At points where one or both of the tabular values of transmission r is equal to unity, linear interpolation on 7 provides more than sufficient accuracy Otherwise it is far more accurate to interpolate I(P, T, W). Since only three significant figures are given in the tables, in general, a lirst-order interpolation provides sufficient accuracy. From equation 97 one has i(P, T, W) = in A When the absortiarce at constant P is plotted against W on a log-log graph, the resulting curve has only a small curvature and displays long linear sections. Similar remarks apply to an absorptance curve at constant W plotted against P. For these reasons, the most accurate interpolation is obtained by taking equal logarithmic intervals for P and W so that af af af f= aFx + AT + ay where x = n P and y = in W. Thus, f(P + AP, T + AT + W + AW) f(P, T, W)+ f and T(P + AP, T + AT, W + oW) = 1 - A(P + AP, T + AT, W + AW) 1- exp [-(f Af)] The following example illustrates the method described above. Given: P 1 atm, T = 3000K at W = 2.0, T = 0,648, A = 0.35? at W = 5.0, T = 0.425, A = 0.575 P = 0.5 atm, T = 300~K at W =2.0, T = 0.715. A = 0.285 at W = 5.0, T = 0.545, A = 0.455 139

.. WILLOW RUN LABORATORIES Find: r(P, W) at W a 3.2, P = 0.75, T = 300~K Procedure: A(1, 2.0) = 0.352; f 1.0441 A(I, 5.0) = 0.575; f = 0.5534 A(0.5, 2.0) = 0.285; f = 1.2553 A(0.5, 5.0) e 0.455; f = 0.7875 In (2.0) a 0.6931; in (1.0) = 0.0000 In (3.2) = 1.1631; Rn (0.75) =-a0.2877 In (5.0) a 1.6094; In (0.5) a -0.6931 Hence: Af at i ()( n W) + a(fh( P n P) 0.5534 - 1.441 1.2553 - 1.0441 1.6094 - 0.693(1.1631 - 0.693) + 93 (-0.2877 -0.2517 + 0.0877 = -0.1640 Therefore: f(0.75, 3.2) a f(1, 2.0) + Af = 1.0441 - 0.1640 = 0.8801 T(0.75, 3.2) = 1 - e 0'8801 = 0.585 In order to calculate transmission for atmospheric slant paths, the equivalent path paraineters P and W* are first determined by the methods used in section 2.3. The equivalent temperature may be assumed to be the average temperature without in.oducing a significant error. Once the equivalent hornogenBeous-path parameters are determined, the spectral transmission can be calculated according to the interpolation procedure. Plass [63] used this method to calculate the spectral transmission over slant paths from initial altitudes of 15, 25, 30, and 50 km to the outer limit of the atmosphere. Results are presented in the form of tables for paths from the vertical to the horizontal in 50 steps, Values are given for CO2 from 500 to 10,000 cm 1 and for H20 from 1000 to 10,000 cm1 for both a "dry" stratosphere and a "wet" stratosphere. Plass assumed that CO2 was uniformly inxed throughout the atmosphere at 0.033% by volume. H20 distributions were based on the. work of Gutnick [64, 65]. For the "dry" stratosphere H20 was assumed to be constant at 0.045 gm/km air. For the "wet" stratosphere the values used are given in table 22. The use of the quastlriiinro model to describe the spectral-line structure of atmospheric gases is most rigorous and undoubtedly results in a more accurate representation of the line structure than does any other band model. However, it is not possible to express the transmissivity functions generated through the use of this model In a closed form. It is therefore a dfiflcult and laborious task to apply them to arbitrary atmospheric slant paths. For this reason 140

________ IWILLOW RUN LABORATORIES ---------- TABLE 22. MIXING RATIO FOR'"ET" STRATOSPHERE MODEL Altitude Mixin Ratio, (km) (gm H20 per kg air) 15 0.0090 16 0.0095 17 0.0105 18 0.012 19 0.015 20 0.018 21 0.022 22 0.027 23 0.033 24 0.040 25 0.048 26 0.058 27 0.C69 28 0.087 29 0.11 30 0.13 31 0.16 32 0.18 33 and above 0.20 141

------- WILLOW RUN LABORATORIES -- it is felt that the n-ost useful aspect of the work of Wyatt, Stull, and L ass is not the methods they employed, but rather thetr extensive resultP.* 3.12. METHOD OF A. THOMSON AND M. DOWNING Thomson ai4d Downing were involved in a program that required knowledge of the transmission of the atmosphere from 2.0 to 5.0 p for atmospheric paths tthat extended from the surface of the earth to the limit of the atmosphere. To determine such transmission spectra they used the transmissivity functions developed by Howard, Lurch, and Willir..s for H20 for the 2.7-/ and 3.2-p absorption bands and the results of Carpenter for the 4.3-, CO2 bani. Their only original contribution consisted of determining a method for cr.lculating absorption ctased by CO2 for the 2.7?-M band. ThLs work was based on the results of Thomson [66!. Their complete work is presented in reference 67. Since the works of Howard ar.d Carpenter have t;en previously discussed, only the results of their work related to CO2 absorption at 2.7 A will be discussed here. The transmission for the 2.7-M CO2 band was calculated directly from the strong-line approximation to the Elsasser band model. Since Thomson and Downing developed their method for sea-level paths only, tne transmission is given by A = trf.- 0 j (S8) Thomson et al. derived an approximate expression for the mean or "smeared" integrated intensity of a linear molecule which is given as Tfv'- I -(I- )2/)2 S(v) SoT,-" )2 e where SO is the total-band intensity and 4v = 2/BkT7hc, in which k = Boltzmann's constant, h = Planck's constant, and c = velocity of light. B is the rotational constant and has the value of 0.39 cm-1 [68]. The 2.7-4 CO2 band consists of two, adjacent bands; as reported by Fhomson and Dowring,, -1 -2 -1 one is centered at 3716 cm and has a value for S at. 298~K of 42.3 cm -atm; the other cm-1and has value for 0 -2 -l band is centered at 3609 cm and has value for SO at 2980K of 28.6 cm -atm. The values *Unfort-nately, the tables could not be included in th'3 document because of their considerable length. However, copies are obtainable from the Defcnse )ocumentaton Center in Washin,,ton. (Note the AD numbers on ref. 19, 62, and 63.) 142

- I WILLOW RUN LABORATORIES —for a0 and d were determined to be 0.064 cm1 and 1.56 cm, respectively. With all of the parameters 7pecified, the transmission is easily calculated for any equiivalent path W*. The results of a sample calculation are shown in figure 43, which is the transmission to the top of the atmosphere for vertical paths extending from six different altitudes for the band centered at 3716 cm1. It is interesting to note the peak in transmission at the band center, anoroximately 3610 cm. Even for l',rge quantities of absorber these peaks reach 100% transmission. If the absorption of CO2 were measured at infinite resolution such peaks would indeed be observable; however foL a resoluttk. consistent with the Elsasser model (i.e., the averaging interval is equal to d = 1.56 cm' for this calculation), such peaks would be smoothed and a value of 100% transmission would not be observable. Therefore, it appears that the results obtained by this method are not in agreement with the initial assumption of a regular Llsasscr band model. The peaks are very narrow, however, apd from the standpoint of total-band absorption their method yields results that are comparable with other methods and experimental field measurements. 3.13. METHOD OF T. ELDER AND J. STRONG Elder ai.d Strong [69] derived a useful, simple method for predicting the average total absorption-the average being taken over broad spectral intervals-caused by H20 for homogeneous paths. The spectral regions of interest to Elder and Strong were the so-called infrared "windows." In figure 44 a low-resolution curve is presented of transmission versus wavenumber for th.! spectral region from, 0.7 to 14.0C. It is noted that there are eight regions of low absorption. Each of the'ow-absorption regions is called an atmospheric window and is spectrally defined in the figure.' The boundaries are indicated'by the dotted lines in the centers oi the absorption bands. Note that the windov, transmissions are not unity as they would be if the optical path were evacuated. This is primarily a result of continuoJs attenuation. Elder and Strong developed.empirical functions, based primarily on open-air field measurements, which specify the average window transmission resulting from molecular absorption for each of the window regions shown in figure 44. They assumed that the average window transmission for each of the eight windows could be represented to a first approximation by the expression T = -k log W* + to (99) where k uird i0 are empiri;al constants and W* is the equivalent sea-level concentration of HO0. Thus, when measured valu.s of T and W* are plotted for any window on semi-log graph paper, the experimental points should fall on a straight line with slope -k and with an intercept t0o 143

-.. WILLOW RUN LABORATORIES E —------ 100r 3 -1 10.1 -2 10-3!_____1 3520 3550 3600 3640 WAVENUMBER (cml ) FIGURE 43. TLRANSMISSION TO TOP OF ATMOSPHERE FROM INDICATED ALTITUDES AS A FUNCTION OF WAVENUMBER FOR 3716-CM'- CARBON DIOXIDE BAND. Altitudes: curve 1, 15,000 ft; curve 2, 20,000 ft; curve 3, 30,000 ft; curve 4, 50,000 ft; curve 5, 75,000 ft; curve 6, 100,000 ft. 144

WILLOW RUN LABORATORIES -.N * s-t -- c * 100 1 I- g$LVUJ i I t, 70 I I I 401tI I I ~ gL 80 0 I i 4 ^ 50 1I I,, I l l I I I 1ig I z z i 10 I I i 14,000 12,000 10,000 8000 6000 4000 2000 0 FREQUENCY (wavenumbers) FIGURE 44. ABSOLUTE INFRARED TRANSMISSION OF THE ATMOSPHERE. Adapted from reference 97. 145

WILLOW RUN LABORATORIES- - - - Figure 45 contains graphs of T vs. log W* for windows I to VIL Window VI was not treated since this procedure required adjacent windows to determine the envelope spectrum which was used to determine the amount of continuous attenuation. The data used in the curves are summarized in table 23. The circled dots on figures 45d, 45e, 45f, and 45g were not obtained from observed spectra, as were the other points, but were calculated indirectly from observations reported in references 84-87. The empirical constants associated with the solid straight lines on the graphs are given in table 24. The average straiget lines of these seven graphs show the validity of equation 99, while the pointz show the agreement of the observed data. In figures 45d, 45e, and 45f, a dashed curve is drawn. This curve represents the transmission as predicted for no-haze conditions from the published tables of Yates [88] (also see ref. 89). It can be seen that in the region of the Howard/Ohio State data, the fit with the straight line is good, but the extrapolations to long paths do not always agree with other observations, nor with the straight line. The wcrk of Elder and Strong is represented completely by equation 99 and the coefficients given in table 23. This empirically derived expression orovides a very useful and simple method for computing values of window transmission but should be used with some reservations. First, this work is based on data that were measured earlier than 1950. Therefore, the results are somewhat dated and no attempt has been m.ae to verify the empirical constants with more recent measurements. However, their method is the only one available for predicting values of window transmission without first computing spectral absorption. Secondly, their method should not be applied to paths containing concentrations greater than those shown in figure 45a-45f. In general, this value is approximately 2 atm cm H^O and therefore limits the method to relatively short paths. 3.14. SUMMARY In this section methods for computing absorption for homogeneous paths based on assumed mc!els of the band structure were presented. A summary of this information is given in table 25. Afr each researcher or research group, the gases treated, band models assumed, wavelength range over which the method is applicable, a. 4 the experimental data used to determine the frequency-dependent parameters are stated. All methods presented yield values of spectral absorption with the exception of that of Elder and Strong. Their method yields values of average absorption for only the window regions. Of the methods which can be used to compute sa tL: iril bsupltiun, the method oi Sull, Wyatt, and Plass is the only one which does not give the transmissivity function in closed form. This is because they used the quasi-random model. 146

-WILLOW RUN LABORATORIES - -iS' F z 9 FS S S - W C SGCS C S C gfi 100., ^,1 jt Ir, I, 100 —L,'1 — 1|... I I 80- ""t -I 80- Region I' 0.7-0.92 1u.60 Region H I ~ ~ 60- Regionf l " - 0.92-1.1 {. ~ 40- 40 1 2 5 10 20 50 100 200 400 1 2 10 20 50 100 200 4 2 1 2 5 10 20 50 100 200 WATER PATH (r m) WATER PATH (pm) [Lg cale WAr mm) [Log Scale] (a) Window I (b) Window I C'-' W C S GFS SC S C o 100+ — I'.. H!.....: I It 80 _ X X 860 - Region IIt X, 401 3~ _ 1.1-1.4 / 2 40 ----. r _ -- -_I..1. 12 5 10 20 50 100 200 WATER PATH (pr mm) [Log ScaleJ (c) Window III FH HH FW EC SGCSCS C 6 100( 80 _ S 9o 70 - z a 60 - Region IV " - 1.4-1.9 - 50 - Table 4 Peferences I 40 o Ref. 84......,0.01 0.050.10.2 0.51.0 2 5 10 20 50100 500 WATER PATH (pr mm) [Log Scale] (d) Window IV FiGURE 45. WINDOW TRANSMISSION. Letters are id.ntified with sources in table 23. 147

.-.... WILLOW RUN LABORATORIES' - FHHH F WCCC SGC SCS C, 106 u II! i' 1,I | I l1 I'l I 900 90-'0 70 Region V i Z60 1.9-2.7,1 <C, _ x Table 24 References ^x, x 50- Refs. 84 and 86 ^ -- 0- (No CO.)' 1_40 1 1 1..1 0.01 0.05 0.1 0.2 0.5 1.0 2 5 10 20 50100 500 WATER PATH (pr mm) [Log Scale] (e) Window V FH HH F WOE O 1lt! 1 i I 1 1 100 ~ goo 80 3 70 - Region VI x U 60- 2.7-4.3/ % _- _x Table 24 References,, 50- oRef 86 ^ -~ -- (No CO,) E 40 - * I I I I I I 0.01 0.050.1 0.2 0.51.0 2 5 10 20 50100 500 WATER PATH (pr mm) [Log S:aleJ (f) Window VI FH HH F 0 80 1 70 _ _O -_:_ 60- - 5o -' 5' Region VII 40- 4.3-5.9 r 3: - x Table 24 References 8 30 - a Ref. 85 ~ - (No CO2) x 20_ _-SN J 0.01 0.05.. 0f.2 0.5 1.0 2 5 10 20 50 100 WATER PAmTH (pr mem) [g Sc..l (g) Window VII FIGURE 45. WINDOW TRANSMISSION. Lettrs are identified with sources in table 23. (Continued) 148

- -...WILLOW RUN LABORATORIES- - TABLE 23. WINDOW TRANSMISSION (percent) Water Source Code* Path I H 1 rV V VI VII (pr mm) Fowle [70.75] F 0.08 - -89.9 83.5 77.1 77.5 Howard [76] H 0.09 --.- - 93.0 87.0 87.5 69.7 Howard [78] H 0.19 -- - - 92.0 82.0 82.2 66.2 Howard [76] H 0.26 - - - 88.0 78.4 81.5 61.2 Fowle [70-75] F 0.82 - - 79.6 68.9 69.0 52.4 Hettner [77] W 1.5 -- 98.5 88.6 77;0 68.5 76.2 -- Elliott etal. [78-79] E 3.05 -- -- - 63.5 67.5 Elliott et al. [78-79i E 3.9 - -- 75.0 65.4 65.5 Fischer [80] C 5.0 96.6 91.6 83.3 71.1 Strong [81] S 11.0 91.0 90.0 79.0 68.0 57.0. Gebbie [82-83] G 17 -- 82.6 68.0 63.4 55.8 55.6 25.0 Fischer [80] C 20 -- 88.8 78.4 63,2 54.8 Fowle [70-75] F 20 86.0 89.0 62.0 Strcng [47] S 24 85.0 89.0 58.0 Strorg [81] S 3G 82.0 83.0 67.0 61.0 55.0 Fischer [80] C 50 - 77.5 70.0 55.5 46.4 Strong [81] S 75 79 9 73.5 58.0 47.0 54.0 Strong [81] S 75 75.u 75.0 59.0 59.0 54.0 Fischer [80] C 200 -- 62.8 60.1 49.4. 40.6 *Reference designation on figure 45.

-......... WILLOW RUN trABORATORIES — TABLE 24. EMPIRICAL CONSTANTS FROM FIGURE 45 T = -k log W + to T is w4ldow transmission In percent W ai water path precipitable millimeters T = 100% if Window Region k t W is less than (U() (mm) 1 0.70 to 0.92 15.1 106.3 0.26 i 0.92 to 1.1 16.5 106.3 0.24 m 1.1 to 1.4 17.1 96.3 0.058 V/ 1.4 to 1.9 13.1 81.0 0.036 V 1.9 to 2,7 13.1 72.5 0.008 VI 2.7 to 4.3 12.5 72.3 0.006 VIH 4.3 to 5.9 21.2 51.2 0.005 VIn 5.9 to 14 (not treated) Example: for window VI and 0.03 precipitable millimeters of water, T = -12.5 (log 0.03) + 72.3 = -12.5 (8.477 - 10) + 72.3 = -12.5 (-1.523) + 72.3 = +19.0 + 72.3 15913%. 150

WILLOW RUN LABORATORIES E. ""' t -s j ~ 8t. o e z O eoG o ** oO o o o 5* - - " g 3 3~0 m 2'5 w s-y3:~-"C C!.V C". *C;; o 1 a K ^ W Q -^ OOC;,^,to4 4 4 V V) - ^ s M ak g _ O _ I'4 s0 c ~ 2 0 0 n ~ N ^. It~ If) t Cr i. 0 0 0 C 0 - V 0 6 o0 ot0 4 * M U3 e 0 o goq o e Q I; % -s. S *~. s,E -a a S S 6 g g. I 0 e4, eq 0 e qw~ 0 coa~ 1 oeto v) El'oww0 CO 4:-.J 0u s14 C CIv 0.4 H' o o fl CV 0 - 0 4.. 0 co 0 Q Cd cC 151

— WILLOW RUN LABORATORIES -... <St " " d' " * * **- * S t 3 -1.~ j20 & c kC "4 g. - *-.3,,. c g3 0 r r ^ o o * O. X t 0t:1 i - c o5 o u ~r ^~ t c. o o ed T?,S o oo o tf 0t ^ ~ ^ & s: g:a'"4'~I z M o O O N c 0 0 OS:x:? ~ 0~C O 0:: m::: t i I O _ z _ i_ 0 02 10 ( C3 > ^ ca O | 3 S?Sm C b M 00 Mc ~ 152

- -- - - WILLOW RUN LABORATORIES -.. — Since there are many different methods available for computing absorption for a given gas and for a given spectral interval (for the 4.3-0 C02 band there are seven methods available), the infrared researcher must select the best method to use for a given application considering the trade-offs between accuracy, resolution, and amount of computational labor. The resolution obtainable from any given method and the amount of computational labor involved have been made quite clear in the previous discussions. However, the accuracy with which a given method can predict atmospheric absorption is yet to be established. There are basically two approaches one can use to determine the accuracy with which a method based on a band model can predict spectral absorption. The first of these is to compute spectral absorption by the direct integration of the general transmissivity function as outlined in section 2.1 for broad ranges of absorber concentration and pressure. These results are then compared with the results obtained by each of the methods listed in table 25. Such comparisons would be indicative but not necessarily conclusive for several reasons. The rigorous calculation requires a priori knowledge of such band parameters as line strength, line half-width, and line location which are determined from both theory and experimental measurement. Also, the number of spectral lines that must be taken into account increases as the quantity of absorber increases. Therefore, even the direct integration calculation includes assumptions and approximations which, in the final Analysis, should be verified with experimental field measurements. Unfortunately, even if the rigorous calculation could yield results that approximate the true absorption spectra for an atmospheric slant path better than any ether technique, the state of the art is such that it could not be applied to al! spectral regions. This results from the fact that the band parameters are fairly accurately known for only a small portion of the spectrum. For the spectral regions for which the rigorous calculation has been performed, comparisons are made with band-model methods. These are presented and discussed in section 5. The second approach is to simply compare the results obtained from the band-model methods with experimental field measurements. It is felt that such comparisons yield the most conclusive results if the slant-path field measurements are made under such conditions that the quantity of absorber in the path can accurate;,f be determined. Comparisons of this type are also presented and discussed i,.section 5. It is emphasized that The motivation behind the development of the banr-model methods was to generate methods for computing slant-path absorption which yield good first-order approximation to the true absorption spectra, while at the same time requiring a minimal amount of computation. As will be shown in section 5, many of the methods have achieved this goal for a large portion of the infrapedspectrum. 153

- WILLOW RUN LABORATORIES -___ ~ DISTRIBUTION OF ATMOSPHERIC MOLECULAR ABSORBING GASES 4.1. INTRODUCTION In section 2 it was stated that the atmosphere contains six different gases that attenuate infrared radiation by a process known as molecular absorption. Those gases are C02, H20, 03, N20, CH4, and CO. In order to determine the amount of radiation absorbed by a particular absorbing species as the radiation traverses an atmospheric slant path it is necessary to reduce the slant-path absorber concentration to an equivalent concentration for a homogeneous path. To p, -form this reduction it is first necessary to determine the pressure, temperature, and density of each absorbing gas at each point along the slant path. Hence it is necessary to have a priori knowledge of the distribution of pressure, temperature, and de.sity as a function of altitude. The distribution of pressure and temperature is not a controversial subject and for purposes of atmospheric absorption calculations the values published in the U. S. Standard Atmosphere 1962 are adequate The distribution of density for the various gases, however, has been subject to considerable research. This section discusses the present state of knowledge pertaining to these density distributions. Recall that the dens ity of a given absorber is given by P up M gas air gas where Pair is the density of air and Mg is the mixing ratio of the gas expressed as percent air' gas by volume (equivalent to mole fraction). The mixing ratios of CO2, 03, CH4, CO, and N20 are normally expressed in these units. However, the mixing ratio for H20 is normally expressed as percent by weight, grams of H20 per kilogram of air being the standard unit. 4.2. CARBON DIOXIDE Carbon dioxide is u strong absorber of infrared radiation in three rather broad spectral regions centered at approximately 2.7, 4.3, and 15 M. CO2 data are of specific importance to two main groups of scientists; meteorologists and systems engineers. The former, who are concerned with radiative transfer, have special interest in the 15-g CO2 band and require highly accurate mixing ratio data. The latter find all three C02 bands of importance, and unlike the meteorologist the systems engineer does not need painstakingly accurate data. The vast majority of the work done in this area is pointed toward satisfying systems requirements, which is'also the goal of this report. The following study is presented in this light. 154

-,- - WILLOW RUN LABORATORIES All researchers are 1-. general agreement that CO2 is uniformly mixed throughout the atmosphere for altitudes up to 50 km. Since data are extremely sparse for greater altitudes, specific conclusions carnot be made. However, air density is minute for altitudes greater than 50 km, and, for purposes of atmospheric absorption, it may be considered negligible. The available information reporting values of CO2 content have been reviewed and the resuits are presented in table 26. Note that all authors are In close agreement. The average deviation is less than 2% and the percent difference between the maximum and minimum value is only 5%. These values are averages of a great many measurements made at many different locations. The most extensive measurement program was that of Keeling [91] which consisted of continuous measurements from 1957 to 1961 at Hawaii, Antartica, and California for altitudes to 5 km. His results indicated that the CO2 concentration shows monthly variations of approximately 2 ppm, having a maximum value in April and a minimum in December. He also indicated that the average yearly increase in CO2 is approximately 1.0 ppm. In addition, Callendar [96) and Bray [951 separately have shown evidence that the CO2 concentration has increased in the last 50 years. Callendar reported an average yearly increase of approximately 0.7 ppm, attributing the increase to industrial activity and the clearing, draining, and burning of vegetation. Presented in table 27 are some comparisons Bray made to demonstrate the CO2 increase over the last 100 years. It is emphasized that the data in table 27 are averages of a large number of measurements and that various isolated measurements show large deviations from the accepted average value. For example, concentrations less than 150 ppm have been recorded in polar areas while data taken near Africa have shown concentrations as high as 700 ppm. Also, most observations indicate that the CO2 content is higher near industrial or urban areas than in rural areas. In 2 general, the measurements indicate that the CO2 content is higher at night than during the day, Although little work has been performed at high altitudes, Glueckauff [93] has reported values between 50 and 70 ppm less than the norm for stratospheric balloon flights over England. On the basis of the data presented in table 26 and the fact that Bray's value is based u'xon an extensive literature survey, one could reasonably choose a value of 320 ppm for atmosphericabsorption calcilations. This value was used for all calculations presented in this report. 4.3. NITROUS OXIDE Nitrous oxide was first recognized as a permanent constituent of the atmosphere by Adel [97) in 1939, and since that time a number of other authors have measured the N20 concentration. Data of Goldberg and Muller [98] and Goody and Walshaw [93] support the assumption that the;55

.... WILLOW RUN LA.ORATORIESTABLE 26. CONCENTRATION OF CARBON DIOXIDE Source CO2 Content ppm) Handbook of Geophysics [90] 314 Keelt.g [91] 314 Callendar [92] 320 Glueckauff [93] 330 Fo:,selious [94] 321-329 Bray [95] 320 Average 321 * 5 TABLE 27. ATMOSPHERIC CARBON DIOXIDE FOR 1857-1906 AND 1907-1956, VARIOCUS CATEGORIES Category Meaa Mean JUnwelghted Values (1857- 1906) (190"- 1956) Yearly, all values 321 350 Yearly, non-urban 312 332 Summer, non-urban 305 314 April, non-urban 311 318 August, non-urban 295 315 Weighted Values Yearly, all values* 313 337 Yearly, non-urbanr 309 326 Summer, non-urban* 299 315 April, non-urban 316 317 August, non-urban 299 317 Yearly, non-urban** 293 319 Summer, non-urban** 293 315 *Weighted by opinions in literature and by maximumminimum variability in relation to mean. *Weighted by maximum-minimum variability only. 156

-. -WILIOW RUN LABORATORIESfractional concentration is uniform with altitude,'The former authors indicate tiat N20 is uniformly mixed to an altitude of 15 km, while Goody and Walshaw show uniform distributior to 10 km and suggedt a good probability of uniformity to 40 km. Presented in table 28 is a summary of the available data on the N20 concentration, a value which is not well agreed upon. One possible reason for the diversity was reported by Groth and Schierholz [105]. Using laboratory data, they suggest that aerobic bacterial decomposition of nitrogen comnounds is a major source of N20, which might imply that its concentration is dependent upon geographic location and seasonal variation. Bates and Witherspoon [106] also suggest that these photochemical reactions are the major sources of N20 in the atmosphere. For all calculations performed in this report, a value of 0.28 ppm for N20 concentration was used. The choice was based on the fact that this value was predominant in the most recent measurements. 4.4. CARBON MONOXIDE AND METHANE Carbon monoxide and methane are the two remaining gases that are, except for localized conditions, uniformly mixed as a function of altitude. These gases are elatively weak absorbers and exist in the atmosphere at concentrations which are relatively low compared to that of CO2. The abundance of CO has been investigated to a much greater extent than that of CH4. A summary of the CO data is presented in table 29. Note that there is considerable variation in the data, with Shaw [110] giving the lowest value of 0.075 ppm and Bowman [104] reporting the highest value of 1.1 ppm. The remaining data have an aver,?e value of approximately 0.12 ppm. The reason for these differences is undoubtedly a result of the different methods of measurement. For example, Shav's values were determined from solar spectra observed from a. ground level station. Therefore, the CO concentration represents a mean value for a slant path extending to the limit of the Ptmosphere. The possibility of an increa..d concentration caused by the industrial activity would have little effect on such a path. Bowman's data, in contrast, were determined from air samples taken at ground level stations in or near the city, and are therefore biased data. The remaining data were determined from solar spectra or from air samples far removed from industry or urban areas. If we neglect Bowman's data it appears that a good representative average mixing ratio for CO is 0.12 ppm. This value was used for all calculativ.;s in this report. The available data on CH4 is very sparse. Bowman [104] reports a value of 2.4 ppm based on an experimental procedure the same as that used for CO. Coldberg [115] reports a value of 157

.-........ - WILLOW RUN LABORATORIES —-—.. TABLE 28. CONCENTRATION OF NITROUS OXIDE Author Date Location Content (ppm) Adel [97] 1941 Arizona 0.38 Shaw, Sutherland, and WormeU [49] 1948 England 1.25 McMath and Goldberg [100] 1949 Michigan. 0.5 Slobod and Krogh [101] 1950 Texas 0.5 Birkeland, Burch, and Shaw [102] 1957 Chesapeake Bay 0.43 Birkeland [103] 1957 Ohio 0.28 Bowman [104] 1959 Ohio 0.28 U. S. Standard Atmosphere 1962 0.5 TABLE 29. CONCENTRATION OF CARBON MONO'.IDE Location Observers Mean Concentration..'.Range (ppm) (ppm) Columbus, Ohio Migeotte [107] 0.125* Columbus, Ohio Bowman [104] 1.1 0.4-2.2 Columbus, Ohio Shaw and Howard [108-110] 0.125** 0.075-0.25 Columbus, Ohio Snaw [lll] 0.075** 0.05-0.125 Otiawa, Canada Locke and Herzberg [112] 0.1625 0.09-0.18 Juni'fraujoch, Benesch, Migeotte, and Switzerland Neven [113] 0.075** 0.025-0.11 Flagstaff, Adel [114] Arizona 0.125 *Concentration determined from published spectra. *Concentration reduted to sea level assuming uniform vertical distribution. 158

WILLOW RUN LABORATORIES —--- 1.1 ppm. Since the available datn'.re so meager, it is fortunate that CH4 is a minor contributor to slant-path absorption. 4.5. OZONE Ozone is formed primarily in the mesosphere by photochemical dissociation of oxygen as a result of ultraviolet radiation from the sun. ts distribution is variable with altitude, showing maximum concentration between 15 and 30 km. Presented in figure 46 is a typical vertical profile of ozone distribution [116]. The units of 03 concentration are atmospheric centimeters (at STP) per kilometer of air. To convert these units to parts per million, two steps are required. First, convert the values on the a;bcissa to partial pressure of 03 in micromilllbars. his conversion is given by P03 10oT- x PO x atm cm/km, where T is the temperature at the given altitude. The 03 concentration in parts per million for any given altitude is then given by the ratio of the 03 partial pressure to air pressure at that altitude, or ppm = PO /Pair, 3 where P is in micromillibars ard Pair is in millibars. 03 air It is noted that the 03 concentration is approximately 0.004 atm cm/km at sea level, which wo:id cause negligible absorption even for very long paths. It increases to a maximum of approximately 0.02 atm cm/km at 23 km and then gradually decreases with increasing altitude. It is emphasized that, since this is only a typical profile, the concentration for a specific season or a given geographical location can vary markedly from that shown; The 03 mixing-ratio profile appears to have a significant seasonal and latitudinal variation, as is seen in figures 47 and 48 [90 ]. Note that the shape of the profile changes under differ -nt conditions. In figure 47 there appears to be a yearly cyclic change in the distribution. Starting in the colder months, the peak concentration increases to a maximum in August. There also appears to be a tendency for the peak to rise in altitude as it grows in magnitude. Figure 48 seems to indicate that the profile "flattens out" with increasing latitude at the same time at which the peak concentration becomes less. These changes in tae profile, however, do not be. come a critical concern unless a specific application calls for a long, near-horizontal path between 15 and 30 km. In thit case, use of different profiles could give significant differences and it is recommended that under these ci. cumstances a profile should be used that had been.obtained under very similar temporal, seasonal, and latitudinal conditions as the path in question. Figures 49 and 50 show the average distribution of total 03 (for a vertical path through the atmosphere) over the northern hemisphere for the maximum and minimum seasons [117]. Total 03 in equatorial regions averages about 0.24 atm cm and varies only slightly with season. The amount increases to the north.reachir, annual averages in excess of 0.4 atm cm. North of the 159

V ILtULVW K1UN LA'UK R-A I )K I l: 50... I ~- - I 45 Total Ozone 0.35 atm cm _ 40 - 35 30 215 5 10 _ K 4 8 12 16 20 24 OZONE CONCENTRATION (10-3 atm cm/km) FIGURE 46. REPRESENTATIVE OZONE CONCENTRATION PROFILE. So'.id curve developed from ozonesonde network data, dashed curve from chemical-equilibrium theory [116]. 160

---— WILLOW RUN LABORATORIES - -- February 50 ------ _ — May 45 ~. —- August 40. -.......... November 5 35 ^.... 30 -_ -..,15 i I 5. 2 4 6 8 10 12 14 16 18 OZONE CONCENTRATION (103 atm cm/km) FIGURE 47. SEASONAL VARIATION OF OZONE CONCENTRATION A' FLAGSTAFF [90] — 10 14' Kodaikanal - 78~')' Longyearbyen -- 18~ 31' Pooni 69~ i0' Tronms 60. - r- - - 4-, o --- 55r 35C 12' 47~ 0' 50 Flagstaff Arosa 45 Ars 40\ 3C' 25'' 20... 15 Latitde Latitude 10)/' Latitude Latitude o 2 4 6 8 10 12 0 2 4 6 6 0 2 4 6 8 10 0 2 4 6 8 10 12 OZONE CONCENTRATION (103 atm cm/km) FIGURE 48. LATITUDINAL VARIATION O.' OZONE CONCENTRATION [90] 161

------- -WILLOW RUN LAB-ORATORIES -- ooe FIGURE 49. AVERAGE DISTRIBUTION OF TOTAL OZONE OVER THE NORTHERN HEMISPHERE IN THE SPRING 162

W-IWILLOW RUN LABORATORIES - 240 FIGURE 50. AVERAGE DISTRIBUTION OF TOTAL OZONE OVER THE NORTHERN HEMISPHERE IN THE FALL 163

- WILLOW RUN LABORATORIES - tropical zone, the variation with season is approximately sinusoidal with a distinct maximum in early spring and a distinct minimum in the fall. The general features of the 0 distribution conform roughly to the features of the mean circulatory pattern of the lower atmosphere. Larger amounts of 03 are associated with regions of low pressure and waxm temperature above the tropopause. It can be seen that the typical profile in figure 46 can be adapted for different latitudes by simply shifting the curve right or left so that the area under the curve (with respect to the altitude axis) equals the total vertical 03 content. To illustrate the procedure, let us assume that one wanted to correct the profile in figure 46 to one which contains 0.42 atm cm of 03. The curve must now be shifted to the right to include an additional area of 0.07 atm cm. The new curve is presented in figure 51. This method of choosing an 03 profile will yield results which are reasonably accurate with respect to all the variable conditions. If a more precise 03 profile is required it is desirable to consult an extensive measurement program so that a curve more closely aligned with the particular application could be obtained. One such publication of 03 profiles is the result of an experimental program initiated by the Air Force Cambridge Research Laboratories [118]. A network of eleven ozonesonde stations was established in North America, ganging in location from the Canal Zone in the south to Thule, Greenland, in the north, which produced hundreds of high-resolution observations of vertical 03 distributions. The reports from this group include 03 data for both the winter and summer months. Also included are average 03 distribution for overlapping bimonthly periods. 3 4.6. WUATER VAPOR Water vapor is the most variable constituent in the atmosphere ar.a compared to other atmospheric gases is probably the most difficult to measure at low temperatures and concentrations. Because of these difficulties and the inaccuracies of the early measurements, the only data that are considered in this review are those from a few of the most recent measurements. Since air cannot remain in a supersaturate state, the maximum moisture content is temperature dependent. Combining this effect with the effect of evaporation, the previous environment or past history of a given.air mass affects the H20 concentration of that air mass. However, no tenable theory of atmospheric circulation has been able to explain how the history of the air mass determines its H20 con.ent. As a consequence of the temperature and evaporation effects, the H20 concertration has a great temporal and spatial variation; and the task of determining a correct H20 profile for a given season, geographic location, and for a given day therefore 164

WILLOW RUN LABORATORIES - 50 -- 45 Total Ozone 0.42 atm'cm 40 35 30 4 8 12 16 20 24 OZONE CONCENTRATION f- \~-3 (10 atm cm/km) 165 165

WILLOW RUN LABORATORIE$S.. becomes arduous indeed. The most desirable method would be to take on-the-spot measurements, thereby determining the "exact" profile for that location. This procedure has been done by some researchers, notably Kowall of Lockheed [119], but for the most part this type of measurement would be impractical if not impossible. If the on-the-spot profile is not feasible, a reasonable HO profile t) use is that which has been taken under approximately the same conditions and for a location consistent with a particular application. Another possibility would be to use an "average" profile (one averaged for all rarameters of interest) and to rely upon the fact that the mean error is tolerable. Available humidity data can be classified conveniently into two altitude ranges: sea level to about 7 km and from that point to 31 km. In this study the average elevation of annual midlatitude tropopause has been taken as 11 km, in accordance with the standard atmosphere; e'.Pvations above 11 km will be referred to as stratospheric and those below XI km will be referred to as tropspheric. tp to 7 km where the conventional radiosolde humidity element often functions, a veritable wealth of data exists. Summaries of radiosonde ascents are published as a part of the official publication of various weather services. A reasonably good woI-K.-wide sample of data is also published by the U. S. Weather Bureau as part of the CLIMAT reports (Monthly Climatic Data for the World). Low-level humidity data are so plentiiul that world-wide maps have appeared for the mid-seasonal months giving the amount of precipitable water from the surface to approximately 5 km [1201. It is important to note, however, that these maps give average H20 content; they by ro means specify the profiles with accuracy on any given day. Even the most casual observer is aware of the dayv-!-day fluctuations in near-grouni-level humidity, but even hourby-hour changes occur whilc affect the profile considerably. Shown in figure 52 are low-altitude measurements made by Kowall [119] which indicate deviations of as much as an order of magnitude of H20 content. Needless to say, no H20 profile other than An instantaneous on-the-spot one will yield exact corpliance with the conditions of a measurement. However, as is seen in figure 52, there are profiles available which give a reasonaoly close approximation to the average condition and which can be used with reasonable aecuracy when direct measurement is not feasible. Moistut'e data are'scarce above 7 km compared to those at the!ower levels. With two excepticn.s, all stratospheric moisture measurements have been taken norystematf-ally by individuals and/or organizations interested in the field. These ascents were made s r'radically,henever the time, location, funds, etc., were available. Loss than 50 such stratosyheric ascents have succeeded, and only;bo:v. 10 uAve reached or exceeded 31 km. Table 30 lis;s the 166

WILLOW RUN LABORATORIES - 30, so - 9 25 / 4 _/`/ / 0167 x 2- / --- /.. — Average of 5 Soundings [119] 1100, 1410, 1758, 2015, and 2140 hours / - ~.... A Mid-Latitude'~/ ~~~' Averaged Profile Total Deviation of 5 Soundings 4 / 5 1.0 0.5 0.1 0.05 MIXING RATIO (g/kg) FIGURE 52: VARIATION OF ATMOSPHERIC WATER VAPOR WITH ALTITUDE. 26 October 1964. 167

---- -— W — WILLOW RUN LABORATORIES -- - - TABLE 30. INSTRUMENTS, INVESTIGATORS, AND ORGANIZATLONS RESPONSIBLE FOR NONSYSTEMATIC STRATOSPHERIC HUMIDITY MEASUREMENTS Instrument Reference _ Orgarizationi Automatic frostpoint Barrett, Herndon, and Carter, University of Chicago hygrometer 1950 [121] Vapor trap Barclay et al., 1960 [122] United Kingdom Brown et al., 1961 [123] Atomic Energy Authority (UKAEA) Automatic frostpoint Brown and Pybus, 1960 [124] USA Ballistic Research hygrometer Laboratories (BRL) Automatic frostpoint Mastenbrook and Dinger, USN Research hygrometer 1960, 1961 [125, 126] Laboratory (NRL) and (CBA) Infrared spectrometer Murcray, Murcray, Williams, University of Denver and automatic frost- 1961 [127] point hygrometer Infrared spectrometer Murcray, Murcray, Williams, University of Denver "Sunseeker" 1964 [128] Molecular sieve Steinberg and Rohrbough, General Mills, Inc. 1961 [129] 168

... — ----- WILLOW RUN LABORATORIESinstruments, principal investigators, and organizations making these nonsystematic stratospheric ascents yielding discrete moisture data at given altitudes. Two sets of snore or less systematic probes of stratospheric humidity conditions have been made. The British Meteorological Research Flights (MRF) used manually operated frostpoint hygrometers; maximum altitudes reached by the aircraft were about 13 km. Most ascents were over southern England [130], but some were as far away as northern Norway [131] and Kenya [132]. Nearly 400 ascents over southern England, well distributed throughout the year, reached or exceeded 9 km. The Japanese Meteorological Agency (JMA) utilized balloon-borne automatic irostpoint hygrometers at Sappora, Tateno, Hachejoshima, and Kagoshima [133). On several occasions two ascents were made in one day; on some days soundings were made at two or more stations simultaneously. About 100 ascents reached 9 km.; 2 ascents reached 31 km. Presented in figure 53 are the profiles determined by the authors cited in table 31. Also shown in this figure are the average profiles given by the aforementioned groups who have taken systematic measurements. The Barrett mean is a nonrigorous average of the original Barrett data derived at The University of Michigan in order to present these data in comparison with the other profiles. Figure 54 shows the original Barrett data. The other mean profiles (i.e., the JMA mean and the NRL mean) were taken from published works. The profiles in figure 53 show a decrease in mixing ratio with altitude to a minimum value several kilometers above the tropopause, the exact altitude differing from researcher to researcher. Above this point the mixing ratio increases to at least 32 km, the highest altitude to which analysis could be extended. Above 32 km most researchers merely use a constant mixing ratio, a imethod which yields adequate results. The points of greatest disagreement are: (1) the r':itude at which tht minimum occurs and (2) the magnitude of the mixing ratio at this minir ".... Researchers have reported altitudes -3 -2 of 12 to 18 km for this knee in the curve, with values of between 2 x 103 g/kg and 2 x 1i02 g/kg for the mixing ratio of the minimum point. As yet no noncontroversial theory has been advanced to explain the physical mechanism involved, so that no definitive statement can be given to indicate which values to use. Let us simply say that the most recent data [128] indicate -3 the knee at about 16.5 km with a mixing ratio of 2 x 10 g/kg. Since the mixing-ratio data are necessary prerequisites for the calculation of the amount of H20 in a particular atmospheric paih, many authors have found it advantageous to obtain simple functional approximations to a mixing-ratio profile. Figure 55 presents the approximation obtained by Zachor [27]. It can be seen that Zachor, for each profile under consideration, 169

WILLOW RUN LABORATORIES- i 30_ lBarklay * 28 * 26 _ Ha / 1t 26 / P' hKMolecular Steve 22' - i 20 // — oT Murcray (1964) [128] _18 - ~ mso Va~_h Murcray (1960) 127] X * f I I *i~,,,,.,,,l. MRF Mean [135] \BRL Mean [135] tr: ^. Xl^^ Barklay 14 Baray..um. JMA Meai [135] S12j _,~~. *~ ** * Barrett Mean |-. *~ % u,,_-::-:RI _ NRL or CBA Mean [125, 10 l 126] 8 6 - 2 0.001 5 0.01 5 0.1 5 1.0 5 10?!AIXNG RATIO (g/kg) FIGURE 53. WATER VAPOR MIXING-RATIO PROFILES 170

WILLOW RUN LABORATORIES 28- --- 26 --- nter-Da — Winter Data.. Summer Data 24 _.. _ _ 22 -. - -' 20 -- -_ -- 18 --- ^ - - -- -- -- 16 -,, 0.001 0.01 0.05 0.1 0.5 1.0 5.0 10 MIXING RATIO (g/kg) FIGURE 54. SUMMER AND WINTER MIXING RATIOS OF ~12 BARRETT [121] fO~~~~~~~~~~10.^. _.....171 2 0.001. 0.01 0.05 0.1 0.5 1,0 5.0 10 MNIDING RATIO (gkg) FIGURE 54. SUMMER AND WINTER MM. NG RATIOS OF BARRETT [ 121] 171

. —-..- WILLOW RUN LABORATORIES 30__fr- p 28 - - }-.... - I26 __ * __ 24 22 __ = —J -I July 1949 a Summer Data 20 7. -; - * —"s = 7 January 1950_ % - -; a/ \' L' Winter Data 16 E' 000 0.010.05 0.1'0'- 1.0-5.0 10 2g. _MIINGJRAT —-O ( -— / ISd-~ Approximation to i _.-__ _ ^.^' Winter Data { i...., Approximation to 8.. - _3Summer Data 4 -- -+ 0.001 0.01 0.05 0.1 0.5 1.0 5.0 10 MIXING RATIO (g/kg) FIGURE 55. ZACHOR'S SUMMER AND WINTER MIXINGRATIO APPROXIMATIONS [27] 172

.-... WILLOW RUN LASORATOR ESused two straight lines to approximate the data of Barrett. Since the data are on a log scale the work became merely a fitting of exponential curves to Bz: rett's results. Oppel [134], with his IRMA (Infrared Model Atmosl, erps) computer subroutine; has provided other convenient models for the H20 distribution. He derived tropical dry. Arctic dry, and temperate vet approximate H20 profiles. The basic data Oppel approxima.?ci vere Gutnick's work for the mid-latitude profile [135] and Mastenbrook's [125] stuldy for the tropical profile. As there have been no Arctic stratospt' -ric measurements, the Arctic model wis contrived by specifying the relative humidity at 60% in the tropospi;ere and utihzg a contant mixing ratio of approximately 0.0055 g/kg above the tropopause. Most researchers aplre that thrse are good assumptions. The profiles attained in this fashion are presented in figure b3. It can be seen that Oppel's work cannot be applied to a system where a high degree of accuracy is needed, as his IRMA profiles do rnt nearly conform to measured data from the altitude of minimum mixing ratio upward. However, because of their simplicity and because they yield somewhat reasonable results, they are very useful in giving thumbnail approximations that may be needed in less critical system applications. Lindquist [136] at The University of Michigan has also generated an approximate profile, based on the 2.7-t spectroscopic measurements made by CARDE (Canadian Armament Rezearch Defense Establishment). Lindquist, using wavelength intervals where there is only H20 absorption, calculated an approximate H20 profile from the measur.d transmission values. His results are shown in figure 57. Since the CARDE data vere obtained completely over Florida, this profile is characteristic of that particular geographical location. As with Oppel's work, Lindquist predicts a constant mixing ratio at high altitudes which is not in agreerlent with most researchers. Gutnick, selecting what he considered to be the most reliable data, has attempted to define an average profile under mid-latitude conditions (fig. 58). Gutnick strove to make his average profile (1) representative of all months or seasons, (2) representative of the climatic types comprising the selected area,- (3) based on an unbiased sample, i.e., the selected sample must represent average conditions and not some unusual year or years, (4) physically cr,nsi.-tent with known information, e.g., at any altitude,,he mean dewpoint cannot be warmer than the ~mean temperature. For study of the t-ropopapse, measurements made in Fuknaka (by JMA) and in Washington, D. C. (by CBA), were used in the averaging. In the stratosphere the following data were used for the averaging at 9-, 11-, 14-, 17-, 20-, 26-, 29-, and 31-km levels. 173

....... WILLOW RUN LABORATORIES 30 28 26 24 ~22'" ~-Temperate Wet 20 18.. 16' -- 2&~~~FILES 14 41 174 12 Tropical Dry' Arctic Dry o.0oo1 0.005 0.o0 o.os o. 0.5 1.0 5.0 10 MXING RATIO (g/kg) FIGURE 56. IRMIA APPROXIMATE WATER VAPOR PROFILES [134j 174

WILLOW RUN LABORATORIES.. --. 30 28 26 24 22 20 I ia E 16 14 < 12 10 4 - 2 \ 0 o3 2 10 102 10 t 1 10 100 WATER VAPOR MIXING RATIO (g/kg) FIGURE 57. AVERAGE WATER VAPOR PROFILE OVER FLORIDA, Current beat estimate, obtained from CARDE solar spectra [136]. 175

: —-- WILLOW RUN LABORATORIES - 0.010 0.100 1.00 30 mb.. E / ts BRLm-, 60 0 ):, I / i'/ i / M -'80 iSt. Q ir _t 5 i / 7 / /, to a - * ymwlxt~ 90 C g 100 mb 12 t- a, -i 7- / —-~ —------ 200 3 10f mb --'''J. Z7en,,30C MUNG~' "" ATIs~ RL ~6 *'' \ D/ /| o l -4D ree7 v00; 2 O, ——'.. —- - J --- --- ---—, -"'" -" —s nn.......oo[,:, - -.1 *..A. 0.010 0.100 1.00 MIXING RATIO (g/kg) FIGURE 50. GUTNICK'S AVERAGE WATER VAPOR PROFILE S / JSW -176

- WILLOW RUN LABORATORIES (1) Three NRL profiles of 8 February 1960, 8 April 1960, and 27 June 1960 were selected [126]. All were taken at the NRL Chesapeake Bay Annex (CBA) some 40 miles southeast of Washington, D. C. (2) Two Ballistic Research Laboratory (BRL) ascents, on 3 Aprii1960 at Aberdeen, Maryland (unpublished), and 29'April 1960 at A. Monmouth, N. J. [137], were selected by Brown and Pybus [124] from their extensive series of nonsystematic moisture ascents, from the Arctic to the Antarctic. (3) Two University of Denver ascents were chosen: an infrared spectrometer flight on 18 April 1960 and a hygrometric ascent on 1 March 1961, both at Holloman Air Force Base, Alamogordo, N. M. [12T]. (4) The UKAEA nitrogen-cooled vapor trap was flown over southern England seven times in the springor summer of 1958, 1959, and 1960 [123]. Samples were taken at elevations ranging from 24.4 to 30.2 km (mean height 27.5 km). (5) The General Mills molecular sieve ascent of 15 March 1961 was used, that in which two units were flown at San Angelo, Tex. (6) JMA data which were carefully screened and therefore v Wry reliable. (') The large number of MRF ascents. (The high scientific caliber of the personnel taking the observations and the carefully tested instrumentation made their inclusion a virtual necessity.) Ali stratospheric moisture ascents used in the averaging are presented in figure 58. The final averagecurve was plotted giving equal weight to each of the mixingSralio values. Note that all the data lie within an order of magnitude of the mean curve. The mixing-ratio profile shows an almost logarithmic decrease with height from the surface to about 7 km, then a very steep moisture gradient to 9 km. From 9 km the mixing ratio decreases less rapidly to 14 kn, then is almost constant to 17 km, reaching its minimum value in this layer.,From 17 to 31 km the mixing ratio increases logarithmically with height. Figure 59 shows a surprising comparison between Gutnick's average curve and Zachor's approximation to Barrett's summer profile. In order to obtain a feeli-, for the deviation of atmospheric transmission with the change of H20 profiles, a compari made between the transmission spectra from 1 to 10 p for two paths using two extreme,..es. The profiles were picked by encumpala.,i g ai kLw pLt Iils in figure 58. The results'are given in figure 60; the two profiles are designated as HI and LO. 17;

WILLOW RUN LABORATORIES 30 28 - Gutnick Profile 26 - 246F_ // Zachor Summer Profile 22- 20 16 _ E 14 12 10 8'.S 6 4J106 104 10 10' 10 WATER VAPOR MIXING RATIO (g/kg) FIGURE 59. COMPARISON OF GUTNICK'S AVERAGE PROFILE TO ZACHOR'S APPROXIMATION OF BARRETT'S DATA

WILLOW RUN LABORATORIES - 30 28 26 CS 12 O.o00 0.03 0.1 o.s 1 5 lo MIXING RATIO (gpkg) 12 1 FIGU0E 60. MIXING-RATO PROFILES BUNDING TE AT SO FI E 58 4 20.001 0.01 0.1 0.5 1 5 10 M;IXIG RATIO (g/kg) FIGURE 60. MIXING-RATIO PROFILES BOUNDING THE DATA SHOWN IN; FIGURE 58

.-' — WILLOW RUN LABORATORIES The paths chosen for consideration were (1) a vertical path straight through the atmosphere (0 to 100 km), and (2) a horizontal pat. at 15.5 km whose range extends approximately 19 km. The second path was chosen to show the maximum deviation of transmission, as it lies at the altitude-of approximately the largest spread between profiles HI and LO. Altshuler's method of computing the transmission was used and therefore it was required to calculate the value of W* for only two profiles for each path. The results of the investigation p..ve' ^ he rather enlightening. Fc: the iong vertical path W* was calculated to be 1.392 preit:iltt.ole cmr for HI and 1.387 p-ectpitable cm for LO. With this very small deviation in the W* values the e.pect-d.'asult oi small deviation in the transmission was obtained. In fact, the m dxilr'.,i ded lation at any point was between 1% and 2% transmission. When the horizontal path was rcmputed, values of W* were found to be 6.928 x 10' preciptirib!e cm Ifr HT rad 9.325 x 105 precipitable cm for LO. The transmission was calculated anJ ptt'!,. f r earl;,t the two profiles, with the comparison obtained shown'n figure 61. As can Wt -e;: i,, "fo *fi t4u,,he maximum dlfference between the curves is about 18%. This large difference, he ev-r, only.- niests itse;' in the center of the very strtng H20 absorbing bands at 2.i ( and 6.3,1. Ot.crwis,, the two.twrves seem to compare very well.1' can be inferre& by these comoirisons that for nearly vertical paths of any range, the.rarnsmi,'ion is not crii'cllv depe-ndent on the HO0 profile selected. However, when long slant patl'.3 t!arge zeni:h anglee or nearly horizontal paths are under consideration, more care must he taken in the selection of a profile; especially, if data are required near the strongly absorbing 2.7- and 6.3-pi bands. SUMMARY Since the H20 disributton has a significant temporal and spatial variation, choosing a published IH,0 profile to use for a particular application is a nonexact process. For highaltitude paths, above 20 k i, Gutnick's profile or Zachor's summer profile probably will yield the best results. At above 20 km their profiles are very representative of the decrease in concentration of water vapor with increasing altitude. For altitudes of between 10 and 20 km, near the;,cnd in the profile, the most recent measurements [128, 138j indicate the use of a profile whose minimum concentration is approximately 2 x 10' -/kg. Murcray's 1964 profile and the MRF data then would be the best to use. As for low-alt'ude paths, below 10 km, almost any profile can be used, yielding as good results as can be expected. The moisture profiles presented herein undoubtedly will yield better data as the state of the art advances. This is especially desirable as there is still a question as to the exactness 180

WILLOW RUN LABORATORIES of the stratospheric data. However, despite the limitations, it appears that the present knowledge of the H20 distribution will be sufficient until the expected revisions appear. 680 280 - 3S I 1-4 1 - 1 - 4 I 20 60 - - 40 - 20., WAVELENGTH..) FIGURE 61. COMPARISON OF SPECTRA FOR HORIZONTAL eATjH 19 KM LONG AT 15.5-KM ALTITUDE. Each spectrum was computed for the respective profile shomn in figure 60.

-- -: WILLOW RUN LABORATORIES _... SUMMARY OF OPEN-AIR FIELD MEASUREMENTS OF ABSORPTION SPECTRA As was mentioned in section 3.14, perhaps the best approach to evaluating the quality of methods for computing atmospheric absorption based on band models is a direct comparison of computed atmospheric attenuation spectra with spectra obtained from open-air measurements. In an attempt to facilitate such comparisons in this report, a bibliography was assembled which represents a reasonably complete listing of the measurements that have been made which were reduced to absolute values of absorption versus wavelength.'able 31 presents a summery of the measurements included in this bibliography. Such pertinent information as path description, spectral region considered, resolution, and instrumentation is included in order to give the researcher a basis for selecting open-air data to be used for comparison with spectra computed by any of the methods presented in section 3.

. —-----—. WILLOW RUN LABORATORIES _ O 1ESI! 8; D S v Ii,, J, v l. ".f pk,, tJ a.-..... & it co. 2 S ft^^g., o -'. "-.'.; *.1 o,,... a. a - a o o'iS -o) 1W 0'" I= o - I C;| es "c; s 0? & o..,!ii.,c..s o e g,~ iO ^ E car. -I,~ z I* 1E LU L l W"' Wa * ell o8 n 3 E.-* V) Ci).ft~ C o~ -; c: o A E4 83

-— ~- -~~ WILt IVW K U N L A WJ W A I IK I Cl J.: *:' a^ a *8f?8^ D S ~ PP~. S i j t i E C vg o Nb. * j 0.9 o. CA Co eA,v'3~. r.,9 iW b V V g 0' Br,'. i-.'".., * ID 14' 1P4ur

6::: 6: SUMMARY OF LABORATORY MEASUREMENIS OF HOMOGENEOUS-PATH SPEC.T'A As was menttoned in section 2.3, deriving a nethod for the computation of atmotphieric absorption spectra entails emIp;icatly fitting aband-modelfunction to a collection of homogeneous absorption spectra. Table 32 presents the results of researchers who have published the main body of competent laboratory measureme.,ts of absorption spectra, including the information necessary to facilitate choosing the best data for a given application. The purpose of the table is twofold. First, it is intcnded for tnose who wish to perform their own empiricp.l fits of existing band-model functions to laboratory data, or to empirically fit newly derived transmissivity functions. Second, It is intended for thore who wish to see whether or not a particular method based on a band model is consistent with the slant paths under consideration (i.e., to ensure that the slant-path Pbsorher concentrations are tounded by the laboratory data). In choosing a particular set of dta one is referred to the suggestions in section 2.3, with emphasis on one particular point. To obtain best results, the ranges of absorber concentration and equivalent pressure specified for the laboratory data bhcuid encompass the intended range of W arhd P fo; which the researcher intends to calculate atmospheric transmission. It,-an be observed fhcm table 32 that only three authors, Howard. Burch, arnd Waih.aw, present data of sufficient range of W and P that they can be used in A fitting procedure. In fact, fora great many?pplications, even their data are ot questionable utility. For ir.nstance, a slant path at a zenith angle of 60~ extending through the entire atmosphere coatai.s appr.xinately 30 pr cm H20 reduced to STP. Li the ta'..l, however, it is see, that no researcher has carried out measuiemer-ts with samples of H 0 in excesa of about 4 pr cm. Therefore, to predict absorption for such a path by -..:nd-.nodel method based on these laboratory data would requi extrapolaitor of the'data, which'p generPl yieidj poor.asults. It would be desirable to',-'. further laboratory measurements, particularly those measurements tontaining a large anmornt of abso"ber, for a broad ran.e of equh ilent bro:dening pressures. -lowever, because the p_'esent state:Nf! e art is inadequate in this area, t te researcher must ccr.tent himself with the available data until further roeasuremenis are performed. Tsaere is one final recornmendaticn fo: those wno,ish to use the data c'*.d 4n this section for performing empirica' fits. - Itho',gh the refer.nced works present c,:,ves that can be used directly,'. is suggested:..t an attempt It mad,!'o s cure better representazions of the dati by directl, contacting the people invoi'ed in t.king the 1atl.orat ry mxasuremtnts. 1S5.

\ ------— WILLOW RUN LABORATORIES —---- I~ ~~~ ~~ ~ L.2,-.,.,,. _~ i RAV 3 Bi rr ad0 ^tif^?^??'?!????? ^8!?? r 3; ~~.^.i~li~l^~lli~ l~iWi; -a "a ^ -a;S i0;;K-i., 1 iia ~ ~,7A!.&I 74i.A <' ^3 S'K " 6 4?,? - -? -" ~ S -~ - -;8 4rE 44w ~ a r~~~~~~~~~~~~~' &t -4 0 Zn.4 to, c ofl-lQ t f.~44 - 4 * 44 4 ^ -.^-. #4 H ^ ^ w w ^ ^ wt ^ r # 444 40 t a - 44 4 * P. Ay< uC.~* r 4 L~ O P o * I 0 ^ ^1. - *,..4 - -. 5 - - ~- et^ ^'^ I ^ ^ o- a 44^^^ 44 | u. # 54 4n * "'4 4-l^ 4'" 4 ~<: * *"* 4 ^ -*t ~ w * ~ <0.4 -4 *4 -. 4. - M.4 -i - ~ * ^ * < S a o i3C ~ 42: 44 -$pe~v ~ii 118 4:...~ 486g 96 6ag;, a pO ~ -~~'S4- I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x.4. a [^) &.4 "44'' 11f~~~~~~~~~~~~~~~~~~~~~. 4-' ~ ~ ~ ~ ~ ~ ~ ~~-'"4 44 ~' *4 4- 4) c, rli: tr *1 *S C ".4 -. 0 4 L. 44., -* #4...4.444 #4-44444-4-4', -* 5CiL4 44 3 P, - 4 - * < - - ^.4 -o -*,4) tOs o r'-4 4.,* 0 *. 0 4, -wz - #444 4.. **~~ ~ rasS i g~. jj g g S ~ ~ I^ 4- * 8 6' * ~^.<~ ^ (^ ~~.- *&* * * ~,< <*: r. ~ V' *8 8i 8 ~ f tl r gL~ F* S S 88 S g + i* CC^ S B^8. J g~~~~~~~~~~~~~~~~3 *r ri CP.~ ^n 43Y7 sM*4 F * < * 5 r ^.. g ^^ i~~~~~~~~~~~~~~~~~~~l 21 s? *' -^^ 1 * 3 ^ * * ~ * l ^* ^ *8 C* ** N P ~. 88 2 wY w * t ^~~~~~~~~~~~~~~~~4.3 ~*' -^I'J 3 40? -' <t 0 ~ ~~~~ ~ ~ ~ ~~~~~~~ ** co;;4s* ~t <>, ~> *( t(.,'I *O.9 *r cr s y e ~!86 ~ ~ ~ uq#w~

. —- -- WILLOW RUN LABORATORIES7 COMPARISON OF ATMOSP4E11C MOICULAR ABSORPTION SiMCT In this section, molecular absorption spectra computed by each of the tr;s..o. presented in ection 3, except that o( Elder and Strong, are compared with la'itspafh f eld merstrements' abaorption and with absorption spectra calculated by the rigorous n cth-,d. A14orptlon slpctra were also cor.pued by each of the bahi-model metihe s for five atmospheric pati;s common l.o many tntared alriations; these data wre plotted or a,:ommon graph. The comlparso. preserted are indicative e. bLh the degree of "spread" tn the data ard the accuracy h which "computed spectra" canr approxtnmat the true absorrtiot spectra for.n atmospherlc path. 7.1. COMPARISON OF COMPUTED SPECTtRA WITH' SF:'tA MEASUtRED BY TAYLOR AND YATES Figures 62 aj.d 63 are comparisons o- measured astI c. un.icd sptctra Lor a l000-ft, t)ris*ral, stea* tve path. The measured data were reitc.* it- ylor and Yates f42J ustng i Midel 83 Perkin-ltter norochromator with a ilsttum fluo.'t t, prtsm for the region from 1.0 to S.5 p and a Modei 12-C spectrometer wuth a x'tsum chlride prism for the region frotn 5 to i5 p. The average resolutton of the measured data, ^/,S Is at. 300. For this path or.'y CO. aftd HtO conrribuit signdtcartly, absorXton. Nitrous oxide abstorpton is observable as approxitmltely 4.5 i but ti only a r.;or cortributor. The concentration of HO wa, cetermtred bv'iavlor and Yates from average values of tempnrature. pressure, anrAd ntAtive humtdity measured along the path. The atmospheric centimeters of CO. and N,, 0 -e calculattFj assuming mixing ratios of 320 and 0.28 pprr. respectively.'the effective broadening pretsure for all gases was taken to be'46C mnm fg. Using the pith parameetrs nwx[ed on figures 62 a.x 63, -e calculated the absorption spectra by each of the rmetha;s of.ecttors 3- the results are supesrposed on the Taylor and Yates sectra. Nitrous oxide aoorption was computte only by the methods of Gppel, Green and Gritgs, ai Altshuler since the other researchers did not develop transmisstvtty functions!or thta gas. A cursory examinafton of ftgures 62 and C3 shows that the comparison is better In some spectral regions. than in others, but in general the spread tn the data is rather large. The best general -^,reement of the various spectra is displayed In the spectral region from 2.5 to 3.0 i and the greatest divergence is oted from 4.5 to 5.0 p. *The term "computed spectra" refers to spectra computed by the methods of section 3 and is not to be confused with the rigorous calculation. 187

* I * -,- a:>1 L ith sloI -::y, l*~A? ~. r', iF! 1 0 6 ) X$; n X c.' *. t L I; $'. i 20!2-1I{ O5a cmi',.:i~ N"0. 760!! / 1 = 1,8 t^ t r'A ~~- t'. > 80 \ I f 40. t' ^ */^^^ ^ \ *f;~~~~~~~~~~~~~~~~4 SZ.~ i I I <: 0.057 prcrr:-.O Pe r 60 mml 1 g t" 20 ) 0 0085 atm cm N.o() F 760 nizm Hg, r 8)ill~ f ~ ~~ T.~~~ S'ull, Wyatt, & Plass CO2Oly 1 Tay lor & C02-,.4 61) - Alt:huicler 4- yate. C ~',tr..,.. 5.0!) pi?~~ 20, I 20 WAVEL:NGTU ({i';E62COMAION OF M-EA:URED SPECTRA OFTYOM YATES WITH COMMiD'&'.AF,, A, SiULFR; ELSASSER.; ST(JLL. WYAT",T, AND PLASS; AND CARPEN'TER SHULER; EILIA~~ER; STUPL. UilYAT"JT, ASD PLA~~~; A!;a CARPENTE*

eo~ - V /\ f-/^t El 0~~ 10 52 o Gr re e'n I' ri4 ~Fi U U i m-.~~~~~~~~~~~~~0v~ ~ ~ ~~ ~~~~~. Z, G r |i t g0 $ ^1^^11)1^ IT^ lotl atm cmtCO.> -P TOm V: / ^P^ ( < ((? 1 ^ r 0'57 ~~pr c i 1120 P;0i m?- 0,* ( ItV g ^ M[" }^ [ ~~0.0085 at m car NNOP 70rmH; f\ ^ trdodj| ^ *t; W' tli'tv.' - 3i~ 10 20 7010 I 20 25 WAVELEANGTH (4/ F1GURE 63. COMPARISON OF MEASURED SPECT4A OF TAYLOR AYD YATES W TH COMPUTED SPECThA OF BRADF; GEEN AND IGS, OPPEL AND ZACHO AZs;,i,.f. I~~~~~~~~~~~~d t rbE f 2.~~~~~~~~~~~~~~~~~~~~~~~~J~ 5.0 10 20* f,,b; r WAVELENGTH ~ ~ ~ co IG.~RE63 C.MARSO, F EAURD IPTTIA F AY IAAk:. Y TE. ~Ti OMUTD PE GREEN AND GRIGGS, OPPEL; AND ZACHO

----... WILLOW RUN LABORATORIES —- - - Unfortunately, r., single menod yields results that compare tli tthe nimrasred data fto the entire spectral tnterval displayed; hcwever, each method compares well fr certain Intervals. For example, the results of Oppel are in extremely close agreement with the data of Taylor al.d Yats from 2..S to the center of the 4.3-p band but are ir. relatively poor agreement from 4.4 A to 5.0O. Oppel's data agree reasonalty well for each of thl three shortwavelength H20 bands but indicate HZ0 absorption centered at 2.S 4a that is nt present in the measurcrl data or olhir computational methods. Stull et al. show H20 absorption at slightly shorter wavelengths (apprx;tmately 2.1;) tht is stronger and has a different spectral eharacter. S(uP'f results for this rtgior also arc not in agreemert with other sources of data. Th4 reasons for these didferences are not known it the present time. After careful examination of the figures, similar general observations ca' be made c;ncernitr any oete ro the tmany methods. A critical analysis of the reasons for a particularlv good agreemerr or a oor agreement for a given method is difficilt to perfot: behcause of the many variables involved in its development. However, a general disagreement be'*ee:, merasured and omputed spectra can in home instances be attributed to the:aboratory data used at the development of the computational method selected. In certain cases the disagreement can be attributed specifically to he laboratory data. For example, for the three H b0 barxs certerred at 1.14, 1.38, ani; 1.83 i, respectively, there is noted a spectral hift between the ntmesured data and the spectra computed by the methsds of Altshuler, Zachor, and Green and Griggs.'The spectra of these researchers are consistent with the labortcry data u!.ed to spectiy the spectral absorption coefflc tents- each used the data ot Howard et al- therefore, lihe dit.agreemeMn is attributed to differences in two sets of measured data, that oLtained by Howard and that oblained by'aylor and Yates. Most researchers are tn gzneral agreement that the spectra of Taylor and Yates are spectrally correct for all three H20 bands, &ad Cppel hus shifted his results an appropriate amount to achieve a better spectral agreement for these bands. The c.mparison presented in figures 62 and 6, tndlcates that through a proper choice of computational nethods for each of the respective absorption bands a single spectrum could be corputed which Is in very'good agreem!?nt with the measured data of Taylor and Yates. The spectra previously discussed demonstrate, to a limited exten, the type of comparison obtainable fo,' each of the absorption bands througnout that portion of the intrared spectrum of interest to the infrared researcher. (The only band that was not represented Is the 9.6-p 05 band stice 03 does not exist in sufficient quantity at sea level to cause an observable amount of absorption for a 1000-ft path.) t would ldteed be enlightening if extenttve comparisons 190

-- WILLOW RUN LABORATORIES could tI- made between measured and computed spectra for the entire spectral Intervl for a vrriety rtI abhorber concentrations ard equtvalnt presslres. Unfortunately, measured data of such exient do not exist to facilitate such comparisons. The experimentalists have confined their extensive field-measurement programs almost exclusively to the measuremnnt of spectra for only the central portlon of the 2.7-t and 4.3., bands. Therefore, comparison between computed.an measured spectra of reasonable ext.*rit:old be made only for these spectral retglos. Future measurement prograz;. will hopefu'ly te concerneR with the other bands and the window regiortr 7.2 COMPAr'ISON OF COMPUTED SPECTRA WIrH THE RIG)tROUS CALCULATION FOR THE 2.';. p H2 BAND T, empha.' the appearance of H 0 comruteu spectr~ for the 2.7." band under conditions of high rhtr.b- ton, comparisons of spectra calc ilated by the rigorous aethod were made with compted spectra tor two homogeneous pAths which containLd H20 as the only absorbte. The spectrl region for which the comparisons were made exten-.s frrm 2. t p to approximately 2.86,. Gates et a!. [7 calculated the H20 at sorptton spectra at infinite resolution by summing the contribttion to absorption of approximately 4500 lines. These spectra were then degraded to a lower resolution ty scannlng the sF.l a with a tr:anrm. lar slit functto: which is given oy Sf() 3 a *, r V where a t 1.0 cm' ts o.ie-hal the spectral sllt width. Spertra were computEii by each of the band-*modCe of section 3.nethoSd and superpSsed on the Gates daia. The result 3 ire r.resented in figures 64 through 75. The aisorbr concentration used t'n calculati;g th,.Ffe\:t- preseited ti figures 64-59 is approximately ^.,at amount of i!,O which would exist in a sea-level path I m long with the relative humidity equal to 5C. F'or the same atmospheric codittions, the equivalnt path length for the spectra presented in figures 70-75 wotld be 10 m long. To further aid the comparison of the two sources of data, tne Gates spectra were smoothed to a resotluton approxinutely equal to the resolution of each respective computed curve. AMter a cursory examitation of the data presened in these figures it becomes obvious that th spectra computed by the band-model methods are of very tow resolutcon when one notes the extremely fine structure inherent In H20 molecular abrorpttoo. The resolution of the bandmolel spectra ranges from approximately 40 cm 1 for the methods of Altshuler, Howard et at., Green aid Griggs, and Lndquist, to approximately 10 cm for Oppet's method. Theoretically, the resolution of bald-model methods for. 20 is limited to a spectral interval which contains a sufficient number of lines to cause the infitnte product of th3 statistical model to converge 191

- —. —-. WILLOW RUN LABORATORIES — 0.001 pr cm H20 P * 760 mm Hg Slit Width 2.0 cnm'1 00s r w- i i WAV1EUMBER cm' 1 ) FIGURE 64. COMPAR3SON OF CALCULATFD (RIGOROUS} SPECTRA OF GATES WITHi COMPUTED SPECTRA OF ALTSHULER. Resolution of smoothed Gates curie is 40 cmI. 192 K 20 - r,'. PUTED SPECTRA OF ALTSHULER- Resolution of smoothed Gates cure is 40 cm-1 192

WILLOW RUN LABORATORIES ---- 0.001 pr cm H2' P u 760 mm Hg S1. Width a 2.0 cm' 1910 ~ *- - I- --.'' i I[ o! f o10.| Williams - 0 110,Howard, 0 S L I I, i.', s T 1 1,,i 4000 3600 390 0? 300 3750 WAVENltMBSER t to ) FIGU10E 65. COMPARISON OF CALCULATED (RIGOROUS) SPECTR OF GA'TS WlTi.. Gates cueI is 40 cm'i. itZ 70 i.:tIl3 I ~-,,; 1'i II 20K ii 3750 3700 3650 3600 3855 3500 WAVlUMBhMB (cm m1j FIGURE 65. COMPARISON OF CALCULATED (RIGOROUS) SPECTRA OF GATES WITH COMPL IED SPECTRA OF HOWARD, BURCri, AND WILLIAMS Resolution of smoothed Gates curve is 40 cm-1. 193

-WILLOW RUN LABORATORIES 0.C01 pr cm H20 P 760 mm Hg Slit Width - 2.0 cm'1,- il -/;ii i!!: Ad j _ f_ Eli "" L i 1 I'; j 4O 3 3800 3&800 37S t o. o...|iji^^^ t 1:1o^ii t jI t 4 40-I -' i ait 30 - j.I I I i 20- ||. * I - iS to0-^- j -T 3750 10 35350 3300 3S60 30X) WAVENUMBETR' erm' l FIGURE 66. COMPARISON OF CALCULATED (RIGOROUS) SPECTRA OF GATE?. WITH COMPUTED SPECTRA OF LtNDQUIST. Resolutlon of srioot'hed Gates curve's 40 era 1. 194 1;i 94i i:

-_ —-. — WILLOW RUN LAaORATO RES 0.001;,r cm H 0 p 760 M.mr Hg SlU Width 27.0 cm&' 0so - r - r n 1b. _ L t _ t _._11 - I::'', m' i: / 1 fI: 0 t G r t " 4000 3~5 0 00 35 3a,.0 3" l( WAV. NUMBER (cm' O, —. - *5 103, l?-, -'10FI E 67. C'PAr OF CTRA OF ATS WH C'*^r"'l, t'.,. I, ~ 37- 35, -.,, ~!"t-' -':50'1; 3,-'5 a l I'" WAV NJMBER I um 1l) FIGURE 67. COMPARIqON OF CALCULATED (RIGOROUS) SPFCTRA OF GATES WITH COMPUTED SIPECTRA OF GREEN AND GRIGGS. Resolution cf smoothed Gates i 40 cm". 195

WILLOW RUN iABORATORIES -. - 0.001 pr cm I 0 P - 760 mm Hg Slit Width a 2.0 cmr1'i._ I t, I,,T T,. T, I i t I 004 i ii l i ii''i l iAl! i d'Y 40( 39O 3800 a70 3750 WAVENUMBER! em FIGURE 68. COMPARISON OF CALCULATED (RIGOROUS) SPFCT.A OF GATES WITH COMPUTED SPECTRA OF STULL, WYAiTTl,.ID PLASS. Resolution of smoothed Gates curves is 20 cm 1. 196'196

- W'LLOW RUN LABORATORIES0.001 pr cm H 0 P 760 mm Hg Slit Width - 2.0 cm'1 l- -i!i i i iso _150 HBe -yA1 i A,100I.,,. -i WAVENUMBR i At l * 1 -r-.,'. 40-~1 3i0h-'' oL.- 1 ^ i -'.3P^7~ 50 3 5 370000 FIGUIR' 69. COMPAPISON O' CALCULAT (RIGOROUS) SPECTRA OF GATES WITH COMPUTED SPECTRA OF OPPEL. Resolution of smoothed Gates curve!, 10 crm 1. 1c7

_.... WILLOW RUN LABORATORIES - —.. -- 0.01 pr cm Ha0 P 760 mm Hg Sllt Width 2.0 cm1 _ w s4fo 3* ss Althuler3j iAR WAVEJMBE?i l t0 -.. -.a 0 li,. 0'i:; Ii i!'! /1 1,,! t. ~ u' ~ ~,,:''t t - J v I'iIL. -- I I I._' _iL. ^ i _ 3750 370 of0 3^X 33& WAVENUMBE R i t FIGUPE 70. COMPARISON OF CALCULATED (RIGOROUS) SPECTRA OF GATES WITH COMPUTED SPECTRA OF ALTSHULER. Resolution of smoothed Gates curve ts 40 cm-1, 198

- -— WILLOW RUN LABORATORIES 0.01 pr cm H0 P " 760 mm Hg Slit WLdth * 2.0 cm', ^'^ i \ Hw "1ard, fatrIHy Burch, |Sj 0.G-a ——'-s0t I a t I i| i, F; J'; i1:'... t" i ri:t Ii " l! j'i j i'*:1: i' ii'~/:: i t II: 4000 3 t60 34i0 3dCO 3MO 3;10 3 WAVMENUMB I cm t FIGURE 71. C:OMPARtON' OF CALCULATED (RRO(;ROUS) SPE:CTRA OF GATES WiTH WAV~.NUMBER t era' l I FIURE 71. COMPARISON OF CALCULATED (RIGOROUS) SPECTRA OF GATES W[TH COMPUTED SPECTRa OF HOWARD, BURCH, AND WILLIAMS. Resoluttio of the mnoothed Gates cirve is O0 rm-l. 199 194,

WILLOW RUN LABORATOR.S —- S 0.01 pr cm H 0 P = 760 mm Hg Slit Width " 2.0 cm1 too 2i-2. I - -7 - i- -, l I. —' — TI-.... 90 1.induquist Z 70 20 4000 390 3X)0 3860 30 3760 W, VENUMBER (cm 1 t,00 -- - r7-r -T-I 7 —i —r T - I i z 1 j I ^., c i;~~~~~~'. I Ii' j; io _1 s0L. ii 3750 3700 35 300 3650 3500 WAVE NUMBER (o om } FIGURE 72. COMPARISON OF CALCULATED (RIGOROUS) SPECTRA OF GATES WITH COMPUTED SPECTRA OF LINQUIST. Resolution of the smoothed Gates curve Is 40 cm-i. 200 FIUE 2 CMAMNOFCIULTD RGOOS)SETR F AESWTHCM PUE SETR F IDUIT Vsluinofte mote Gtscuv i 0 m, 2W -j IA t1i llk A Qnii i

_. WILLOW RUN LABORATORIES - 0.01 pr cm H20 P - 760 ntr Hg Slit WIdth - 2.0 cm-1 0050 i t i j 0i U) Griggsp.i:,. 4000 395 38 3800 375 WAVENUMBER (cm' 1 100 -r —-- - -- - ~F 90-!: I i i (! - lA r S 80- i sr -,,, ___ j;4 iA ii c l [0j 30'i I' iI t'-l,::~ ~i,!, ~i]!;, I!P; iIn', 3750 3700 3650 3600 3550 3500 WAVkENUMBER (cm' L FIGURE 73. CI'MPARISON OF CALCULATED (RIGOROUS) SPECTRA OF GATES WITH COMPUTED SPECTRA OF GREEN AND GRIGGS. Rlsolution of the smoothed Gates curve is 40 cm-1. 201

, — -V- WILLOW RUN LABORATORIES -- 0.01 pr cm H 0 P - 760 mm Hg Slit Width " 2,0 cm'1 * 100 - I9o- l - tj - ~ 70 4 iN ip i A~& Ptlas tBEi i Icm 4000 3860 380 38SO.333O WAVENUMBER I cm' l) 202 WAVEN BER I cm Zc70 i 20 cm-1. 0 0i j 202

WILLOW RUN LABORATORIES 0.01 pr cm H,0 P - T60 mm Hg Slit Width - 2.0 cm-n U) Bj jt^it ~.oppe 1;i _ iN 71/1 ~ M'r'4000 395 a 395 0 3 50 30 3750 WAVEUMB cm 11 00 80 -'; I \;? i / " i ('' i t;' i. i 1~;!'' ~, |tXI - I. 14toi'! i 1 3750 37 0( 390 330 3 0 3555 0 5 WAVENUMBRh I cm' I FIGURE 75. COMPARISON OF CALCULATED (RIGOROUtS) SPECTRA OF GATES WITH COMPUTED SPECTRA OF OPPEL. Resolution of smoothed Gates curve is 10 cm- 1. 203 0I 37 f 3700 3 j 0'",a- d t'.,ii 1AV NtB''1'

.-WIL. WILLOW RUN LABORATORIES -- - to an exponential (see sec. 2.2.4). For n = 50, where n is the number of lines in a given spectral interval, the exponential approximates the infinite product to an error: 0.01. For the 2.7-4 H20 band a spectral interval of approximately 5 cm' would be required to include 50 lines, so the maximum resolution obtainable for this spectral region has nearly been achieved by Oppel. In general, it is not practical to attempt to achieve the maximum resolution obtainable from the statistical model since the computational labor required'in performing the empirical fit becomes excessive and the laboratory data must have a nearly precise spectral calibration. It is suggested that if extremelyhigh resolution spectra are required, the e-'ct calculation should ta. used rather than a band-model method. Note in t,',figures that, for each curve, as the spectral interval of comparison becomes larger the comparison between the average values of transmissions improves. To demonstrate the compariorn for the entire spectral region presented, each curve was integrated from 4000 to 3500 cm and the valuen of r and fAdv were deternm!ed. The results are presented In tables 33 and 34. Note that the maximum difference between the Gates data and computed data is less than 18r. One factor that contributes largely to this difference is that the computed spectra are cos.sistently higher than the Gates data for'he spectral region from 4000 to approximately 3800 cm. This difference results from the laboratory data used to determine the absorption coefficients used in the band-model methods. That is, the computed spectra are consistent with the laboratory data for this spectral region. Hence, the differences between the Gates data and the computed spectra result from a difference between thr laboratory data and the Gates spectra, and is therefore not characteristic of the band model. This difference suggests that even though spectra otL;ined by the rigorous method are of very high resolution and supposedly highly accurate, such results should not be accepted as correct until they are supported by experimental data. Unfortunately, an experimental measuremen. for this amount of H 0 is not available, so it is not possible to determine the corrections of any single spectrum. Howe\?r, the spread in the data is sufficiently small to indicate that the probable error between any of the given values and the true values would be tolerable for most broadband systems applications. 7.3. COMPARISON OF COMPUTED SPEC'TRA WITH SLANT-PATH FIELD MEASUREMENTS FOR THE 2.7-p BAND The Canadian Armament iesearc;. and Defense Establishment (CARDE) [1401 recorded many infrared solar spectra for the 2.7-1i H20 and CO2 bands at many different elevation angles 204

WILLOW RUN LABORA1 ORI ES TABLE 33. VALUES OF AVERAGE TRANSMISSION AN TOTAL-BAND ADSORPTION FOR THE SPECTRA PRESENTED IN FIGURES 64-69. Integrations were taken from 2.5 to 2.8571 {. Author 7 yAdr Altshuler 0.760 119.8 Howard, Burch, and 0.826 87.1 Williams Lindquist 0.796 102.1 Green and Griggs 0.702 149.2 Stull, Wyatt, and Plass 0.80. 98.4 Oppel 0.831 84.5 Average 0.786 106.9 Gates 0.765 117.5 TABLE 34. VALUES OF AVERAGE TRANSMISSION AND TOTAL-BAND ABSORPTION FOR THE SPECTRA PRESENTED IN FIGURES 70-75. Integrations were taken from 2.5 to 2.8571 u. Author 7 iA dv Altshuler 0.436 281.9 Howard, Burch, and 0.464 267,9 Williams Lindquist 0.431 284.7 Green and Griggs 0.378 310.8 Stull, Wyatt, and Plass 0.473 263.3 Oppel 0.415 292.6 Average 0.433 283.5 Gates 0.397 301.5 -2205

.-.. - WILLOW RUN LABORATORIES --- with electro-optical equipment installed In an RCAF C7-10O Mark 4 jet aircraft. Almost all measurements were made in the region of Cape Kennedy, Florida, at altitudes of 10.7, 12.2, and 13.7 km These data were reduced to absolute values of s"ant-path transmission from the aircraft to the top of the atmosphere. Three representative spectra were selected from the CARDE data and each is compared to spectra computed by the methods of section 3. Before the computations could be performed it was necessary to determine for each path the effective broadening pressure and absorber concentrations for the two absorbing gases, H2G and CO2. These slant-path parameters were calculated with the programn described in appendix I using the following input data: (1) Pressures and temperature profiles were taken from the t. S. Standard.Atmosphere 1i62 since the exact pressure and temperature distributions existing at the time ot the measurement are not known. (2) Carbon dioxide was assumed to be uniformly mixed at 320 ppm. Because of the uniformity in the CO2 measurements described in section 4.2, thi9 approximation is felt to be quite valid. (3) The amount of water vapor in each of the paths was taken fro r. the work.rf Lindquist [136]. He determined the amount of water vapor in each of the CARDE paths by comparing the CARDE spectia with infinitely resolved water vapor spectra in the 3847- to 3860-cm region (where no C02 absorption exists). The water vapor spectra were calculated by the rigorous method employing a direct slant-path pressure integration. For the rigorous calculation, Lingquist used the line positions and li.e-parameters of Gates et al. [7]. The slant-path parameters calculated by the computer program described in appendix l using the above meteorological data are presented on each of the figures in conjunction with the altitude, zenith angle, and resolution of the CARDE data. These slant-path parameters were used to compute the absorption spe tra by each of the methods discussed in section 3 which are applicable to the 2.7-a ba.,d, ar.d tne resuits of these computations are presented in figures 76 through 90. As for the Gates data, the CARDE data were smoothed to a resolution comparable to the resolution of the'computed spectra. In general., the comparison of computed and measured spectra is surprisingly close. The methods of Altshuler and Zachor yield spectra that display the greatest divergence from the measured data, and the methods of Stall et al., Green and Griggs, and Oprel show the best agreement. Oppel's Jre:tra for all three paths snow remarkable agreement considering the procedures involved. All authors, however, are in general disagreement with the CARDE data 206

..- WILLOW RUN LABORATORIES. —. l10 atm m 00C2 PCO 56.1 mm Hg O2 - Altitude: 13.7 km 4 Zeith: 68.70 ^ 4 [ —F Resoluttion: 0.84 cn- I 30 S -wt..,' d Resolution: 40 cm-1 20o 10 _, * 4COO _ 330 300 38,50 380W) 3750 WAVENUMBETR (cm 1 } 2.26 x 103 pr cm H^O P = 26.1 mm Hg H O *' }' bpo 80 i J —i 70Z a Aashuler I70.4I fCARD2 4 0"wvo- IIr^!t S 1ii,, o2 - I,\:. J? Ivl. $ 3r - i\ Ni *.y^ __I p^ J750 3700 3850 3(300 3B-(. 3300 WYAVENUMB3E aDr' *l FIGURE C6. COMPARISON OF CARDE SOLAR SPECTP.A WITH COMPUTED SPECTRA OF ALTSHULER 207

..WI.LLOW RUN LABORATORIES —-... 110 atm cm CO2 PC " 56.1 mm Hg t - A mm *_ 0 - Altitude: 13.7 kmra Zenith: 68,70: 40K — < Resolution: 0.84 cm-1 I 0 308 Smoothed Resolution: 40 cm" 20 - O. — 4000 4000 3.50 3900 3.,50'AO 3750 WAVENUMBER cm'!! 2.26 x 10'3 p'r cm 112 2.26,> 10o ptr cm It20 PQ 26.1 mm Hg 100 -' T H7;,.''' ~, kf'i' ~ i''j - bi 40 4 — 375 3700 530 50 WAVIMUMBEB (rm ^t FCIGURE 77. COMPARISON OF CARDf SOLAR SPECTRA WITH COMPUTED SPECTRA OF 2ACHOR

------ -WILLOW RUN LABORATORIES -- 110 atm cm C02 PC2 56.1 mm Hg Altitude: 13.7 km Z 4 enith: 68.70 jz - z Resolution: 0.84 cm-l a; 3 Smoothed Resolution: 40 cm'l 4000 3950 3O 1:K) 3O a7tS WAVENt)MBUIR B mT l) 2.26 x 10'3 pr cm H2O P1 = 26,1 m;i Hg V80HGreen ~< ^'.11,'_n ('1 _. __ _ gl out- 6 i d \ Grlgg8 if. 37 &0 37(0 3650 360(* 3S0 3300 WAVENUMBEFR, cm l, FIGURE 78. COMPARISON OF CARDE SOLAR SPECTRA WITH COMPUTED SPECTRA OF GREEN AND GR!GGS 209

.-... ---- WILLOW RUN LABORATORIES ---- - 110 atm cm CO2 PCo2 5.1 mm Hg 100 *. ~ e. I io' w 60 Altitude: 13.7 m k ~:! Zenith: 68.70 Resolution: O.e4 cm- - 30 Smoothed Resolution: 20 cnm-1;o f:-: - 40.0 3'eo5 3t>00 3850. 3800 3'75t0 WAVKNUM.BER' X 2.26> 10'3 pr cm l,0 p P 26,1 mm HM ~Z 7 8r ^}l9 | nCARDE 1.X. 70- ij tu.l, Wyatt,,1 W^AV"~UMB] ( cm' i } & Plass I E.40fFIGURE 79. COMPARISON OF CARDE SOLAR SPECTRA WITH COMPUTED SPECTRA OF 210 210

. —--—.. WILLOW RUN LABORATORIES 110 atm cm CO2 PCO 56. mm Hg C. C Zenith: 68.7~0 J -- Resolution: 0.84 cm'1 I I d 3C Smoothed Resolution: 20 cm' \ 10 _ O -' - - 1 - ~',, i,. _ ^ -L 1 i S i t,,;, i i 4000 ) 385 300 3760 WAVEMTnaBER ( cm' 1 2.26 * 10'3 pr em H.2 P. 26.1 mm Hg _ "2 10'I 374 0 3^0o 36.0 3 560 3307 WAVENflM { cm R1 FIGURE 80. COMPARISON OF CARDE SOLAR SPECTRA WITH COMPUTED SPECTRA OF OPPEL 211 i~~~~~~~~~~~0T

... WILLOW RUN LABORATORIES -- 66.5 atm cm Pco - 89. mm Hg Z 4o -- Altitude: 10.7 I km l-_ Smoothe Reso-utort: 40 cm' i —- 1 0 1 0 i, 4070.379)0'300 3a3 0o 3760 WAVENUMBE ( cm' 2l 6.1 2 10 pr cm H2O PH = 54.4 mm Hg 2ta UT7TT — T -T - T I - i I I I I I 7 70 3#35 3 ^ AlT< y~iAts hu ler k i ^ N. ijsjb^ " y=CARD 3760 3700 3650 36500 335 3600 WAVMIJMBE:R ( cmt'l FIGURE 81. COMPARISON OF CARDE SOLAR SPECTRA WITH COMPUTED SPECTRA OF ALTSHULER 212

WILLOW RUN LABORATORIES 66.5 atm cm CO2 PCO = 89.5 mm Ig *90 180 Z 70 Z Altitude: 10.7 km 30 - ZenitULh: 6~ 0' 20 Resolution: 0.76 cm'-1 i 2 Smoothed Resolution: 40 cm 4000 3950 3900 3850 3800 3750 WAVENUMBER I m *1 6.1 x 103 pr cm H20 PH = 54.4 mm Hg 310 370 [3650 3600 350 3 W AVE~~! am' ~ eo H. r'ViI1 " 90W^ ^i ftlnw'0 ^|| to (\<l6 tm CARDE FIGURE 82. COMPARLSON OF CARDE SOLAR SPECTRA WITH COMPUTED SPECTRA OF ZACHOR 213 3750 3700 34350 360 3550 3500 ul er, j f, i~~~~~~~~~~l 21

W8LLOW RUN LABORATORIES --- 66.5 atm cm CO2 PCO 5 89.5 mm Hg 80 0 40 Altitude: 10.7 km jl 30 - Zenith: 60 o _ Resolution: 0.76 cm1!' Smoothed Resolution: 40 cm~'1! ) ji 0i i I i t I 1 l 1 ti| } s ~ l't.! I I i t_ 4000 3960 390 38,50 3 3760 WAVE3JMBER I cm' 1) 6.1 10'3 pr cm H20 PHO 54.4 mm Hg sOO! r' n- f 1 -T j r _i -1 ~ r-VT —F-T-r - T -T -TTT 0, ~~zo~~~~ot~- - I~ ~ ~ 0",.^ iAR DE 3760 3700 3650 3000 3550 3600 WAVENUUBER (R m' 1 FIGUR-E 83. COMPARISON OF CARDE SOLAR SPECTRA WITH COMPUTED SPECTRA OF GREEN AND GRIGGS 214

----:..WILLOW RUN LABORATORIES - -..-...- 66.5 atm cm CO2 PC2 89.5 mm Hg rO 40 z wt Zenl~z fi~ i i i1t i 1 i Q 4 - Altitude: 10.7 km I I f 30 -- Zenith: 6~ 2- _ Resolution: 0.76 cm1 Q...~~~!...i, i.' L },!-^.l L,t i Smoothed Resolution: 20 cm " 4000 3050 3900 3& 50 3 780 0 WAVENUMBER (cm' I 6.1 103 pr cm H20 PH O h 54.4 mm Hg 2ioo'-rr-In-0-11 -- - 10 aC~~~~1. 3205LL 3700 365 3600 35,0 0 STULL, \WYATT, AND a P tASS 215 I so tjy( i; )I^ & PPlas81 215

-----..WILLOW RUN LABORATORIES —-. 66.5 atm cm CO2 PC 89.5 mm Hg lo00~C b 80' i f J~t 7' iR 601~~~~~~~~~~~~~~~~~~~~~~30 - Zenith: 6~! Io - Smoothed Resolution 10 cm4 4000 3950 3900 3850 380 3750 WAVENUMFBER cm' l 61 x 10'3 pr cm H20 P.O =54.4 mm Hg 100o - -T" — -— T — -' —— r1 00!'C ij; OPPEL

_WILLOW RUN LABORATORIES 350.6 itm cm CO2 84.3 mm Hg 2 CO2 100 0 ~0-I - o4000 3950 3900 3860 3800 3750 WAVENUMBER I cm'1) 3 70 1 3.32 X 10" pr cm K^O 2Q = 56.5 mn H OG -i-j-i —-r- I 1"m-T'-r-'-F-T-\-\ —r-T —r mr-w-ir 3 4000 39 300 38 383 37 - 2o~if Altshude:r L A.1.7k m WAVENUMiBER ( cm -l FIGURE 86. COMPARISON F ARDE SOLA SPECTA WIT COMPTED STRA F ALTSHIULER 217 100 T --- W~ 80 FIUR0. CMRS O F CARDE ALTSHULER 217

-- --- -,.WILLOW RUN LABORATORIES -- 350.6 atm cm C02 PO = 84.3 mm Hg 12 9 o i l, l..., J"I - ~ tI i, I''..... ~, ^ — J — 4A000 00 385k 3m 20 Resolution: 07 In — Smoothed Resolution 40 cm oo. 4000 3 70 3900 38560 3800 3?50 WAVENUMBER cm' 1 3.32 x 10prcmH20 PH - 56.5 mm Hg 218 0 H 3753 3700 3650 3000 355 3530:) WAVENUMLEL!'1) FIGURE 87. COMPARISONl OF CARDE SOLAR SFP'CGTRA WITH COMPUTED SPECTRA OF ZaCHOR 218

_ --.-,WILLOW RUN LABORATORIES. 350.6 atm cnm,2 PC = 84.3 mm Hg 700Fo — sooteed Resolution: 40'tll cm 1 w i t 9o 8 50 - * *''"**t~ isj I~j^<", ^n<^ ""'. 0. so 1 4000 s060 390C 3850 3900 3750 WAVtNLMtBE {( cm' 1 22 3.32 > 10U2 pa cm H20 PH> 56.5 mm Hg 375 3700 3650 i6T 3t0 35 GREE 40AN GRICS z TO 4- ii i i20' 219 3750 3700 3650 3600 3550" 3500 WAVENUhltBER {crm - FIGURE 88. COMPARISO1N OF CARDE SOLAR SrECTRA WITH COMPUTED SPECTRA OF GREEN AND GRIGCS 219

-WILLOW RUN LABORATORIES. 350.6 atm cm CO PO a 84.3 mm Hg =,o!'i. m j' T Altitude: 10o km | S f! _I.I.l,_ {J:,':s' 10.7km' 4000 39SO 3900 3&60 3$00 3750 WAVENUMBRE ( cm ml ) 3,32 x 10Q2 prcm H2O PH 56.5 mm Hg Z 70 CARDFl { Stull, Wyatt, & Plass: 3750 3700 34,0 3 5.50 3500 29i0 Re 0.76 m WAVNB.1) i FIGURE 89. COMPARISON OF CARDE SOLAR SPECTRA WITH COMPUTED SPFCTRP. OF ~ oSoo5R STULL, YATT, AND P20 A 220o t 3750 37W 3prc0 3tHX) 3560 36 00 SCtRLL, WYATT, AND P PLA "o.~X

- --.. WILLOW R4N LABORATORIES - - -. 350.6 atm cm CO2 PC O 84.3 mm Hg 2 10Altitud: 1 m' l-Resolution 0.76 cra" 1 I [I JLi I ^i KNI[ 10 Smoothed Resolution: 10 cm4''j V!'1l 1 I Ig 4ooo000 3 36 38C 7 WAV-NUMBE3t em 1 } 3.32 X 10 pr cm H20 PH20 56 5 mm Hg 90 ude:10.7 kmi 1 / leolut 0.76^ cm-Oppel t t10 -- = thed Resolutio- -- cm37S0 -^ 3700 3650 30 36 3 WAVENUMBIER em I FIGURE 90. COMPARISON OF CARDE SOLAh SPECTRA. WITH COMPUTED SPECTRA OF OPPEL 221 03 Cd~~~ 60t peti~

-- WILLOW RUN LABORATORIES ---- for the spectra region from 3900 to 3800 cm' The computed data are high on the average by approximately 10 percentage points in transmission. Oppel's spectra show better agreement for this spectral region than do any of the other data. Since general disagreement for broad spectral intervals is usually related to the laboratory data, indications are that the data of Burch et al. [1] are in closest agreement to the CARDE data. To further compare the computed and measured data, average transmission and integrated absorption values were determined for each curve. The results are presented in tables 35, 3C, and 37. The path containing the least amount of absorber represented in figures 76-80 shows the best comparison with the CARDE data. The values of Oppel and Stull et al. compare almost exactly. As the quantity of absorber increases, the comparison is slightly poorer, as noted for the paths shown in figures 81-s0. However, in all! cases, the difference is no greater than 8 percentage points and in general is much less. For most applications, such results would be more than adequate. 7.4. COMPARISON OF COMPUTED SPECTRA WITH SLANT-PATH FIELD MEASUREMENTS FOR THE 4.3-g BAND The next series of comparisons is indicative of the degree of accuracy with which the bandmodel methods can predict absorption for the 4.3-p CO2 absorption band for high-altitude slant paths. Kyle et al. [149J made a series of high-altitude, balloon-borne solar-spectra measurements for the 4.3-. CO2 band which were reduced to absolute values of transmission. Three spectra were selected from Kyle's work.;epresenting slant paths extendirg from 30, 25.6, and 15.2 km, respectively, to the limit of the atmosphere.- These three paths were selected to facilitate comparisons for levels of absorption that extend from that which is re!atively weak to that which is relatively strong. To compare these spectra with spectra calculated by the bWnd-niodel methods, it was first necessary to determine the eouivalent path parameters for each of the Kyle:aths. These parameters were computed with the computer program previously described, using the same input data that were used for the determination of the equivalent parameters for the CARDE paths. Water vapor data were not required for this computation since H20 does not absorb in the 4.3-p band. The altitude, ienith, CO2 absorber concentration, and equivalent pressure are noted on each figure. The first of these comparisons (fig. 91) demonstrates tht..rrm.rison between measured andi computed'pectra for an extremely high altitude slant path. It is noted that the results of 222

..... WWILLOW RUN LABORATORIES -- TABLE 35. VALUES OF AVERAGE TRANS- TABLE 36. VALUES OF AVERAGE TRANSMISSION AND TOTAL-BAND ABSORPTION MISSION AND TOTAL-BAND ABSORPTION FOR THE SPECTRA PRESENTED IN FOR THE SPECT-A' PRESENTED IN FIGURES 76-80. Integrations were FIGURES 81-85. Integrations were taken from 2.5 to 2.8571 p. taken from 2.5 to 2.8571 {l. Author r JAdv Author JfAdv Altshuler 0.683 158.6 Altshuler 0.626 18C.9 Zachor 0.654 172.9 Zachor 0.596 201.9 Green and Griggs 0.681 159.7 Green and Grlggs 0.606 - 197.1 Stull, Wyatt, and Plass 0.663 167.1 Stull, Wyatt, and Plasn 0.613 193.3 Oppel 0.669 165.3 Oppel 0.595 202.3 Average 0.671 164.7 Average 0.607 196.3 CARDE 0.668 166.0 CARDE 0.557 221.5 TABLE 37. VALUES OF AVERAGE TRANSMISSION A1D TOTAL-BAND ABSORPTION FOR THE SPECTRA PRESENTED IN FIGURES 86-90. Integrations were taken from 2.5 to 2.8571,. Author 7 JA dv Altshuler 0.415 292.5 Zachor 0.100 300.1 Green and Criggs 0.400 300.1 Stull, Wyatt, and Plass 0.414 293.1 O pel 0.392 303.1 Average 0.404 297.8 CARDE 0.368 315.8 223

S0o19 aNIV N33IHO QN~V H3'flHS&LIV:'E3dd0:H.N3gd8HV3:SSV'Id;GHOJ.VIQi.0 VUHi33dS a3LdfldO3 HMLA 3IAX: 3O VHLad33d HS HVIOS IO NOTHVYdi03'16 iSHflID (rt),lUOIq~A^'AdM or- o..a -- --- --— Ol —— X: —- > —---. -----, -----. -— I — 0{; QL 0 - - o/ z oz Os la^~ t l}ddo~J l a 06.S.HtU {',t= Od ZO 3; tu 0,'g olS:tnIluoz uI{j 0:apnlllv gJ QL ssP~~~~~~~~~~~~~~~~~ld( 09~~~~~~~~~~~?

------...WILLOW RUN LABORATORIES ------ Altehuler compare rather poorly with the Kyle data, possibly because the laboraitorr'*.. Howard et al. used by Altshuler to detex mile the absorption coefficient did -lot;.c';uO r 4 containing quantities of absorber as small as those contained in this partictJar palh, h.;,., the results of Altshuler are extrapolations of the laboratory data rather than,nte;'olrLoa. ^ condition which should be avoided. The remaining five methods all compare reasonably well, showing a maximum difference in the center of the band of approximately 20 percentage points in transmission. The spectra of Carpenter show the greatest amount of absorption and Bradford's spectra show the closest agreement. Each of the spectra was integrated to determine values of average transmission, and it is noted from the data presented in table 38 that all of the integrated values compare to the Kyle data within 7 percentage points, the methods of Bradford, Oppel, and Plass compare almost exactly. The next comparison (fig. 92) has an absorption level approximately 25% higher than that displayed in figure 91. Altshuler's spectra are high by an amount approximately equal to that shown in the previous comparison. The spectra of Bradford, Oppel, Plass, and Carpenter sh5ow a slightly better agreement in the central portion of the band, but their integrated values (note table 39) compare as those shown in table 38. The spectrum of Green and Griggs does not compare as well as that shown previously, the integrated value being less than the measured data by approximately 10t%, as compared to a difference of less than 4% for the spectrum shown in figure 91..Figure 93 emphasizes the comparison of the Kyle and computed spectra in the wings of the band, the absorber concentration being sufficient to cause total absorption in the center of the band. The spectrum of Carpenter is the only one which compares well for the short-wavelength edge of the Kyle spectrum. For the long-wavelength edge of the band, the spectrum of Piass compares best. The integrated values of absorption given in table 40 demonstrate, as for the two previous comparisons, that the methods of Bradford, Oppel, Plass, and Carpenter yield values of average transmission that compare extremely well- o the Kyle values, and the values computed by the method of Green and Griggs are consistently nigh. It is interesting to note that Altshuler's results were high for the two previous comparisons, b'tt are low by approximately 20% for the high-absorption path. In section 3 it was shown that Altshuler used a transmissivity function for the 4.3-0 C02 band-that was empirically derived and that Green and GrIggs employed the statistical model, whereas the methods of the other four authors were based on the Elsasser model. Since the line structure of C0,i-,sassurred to be more regular than random, the Elaasser model reportedly 225

-, —WILLOW RUN LABORATORIES -- TABLE 38. VALtPo OF AVERAGE TRANS- TABLE 39. VALUES OF AVERAGE THANSMISSION AND TOTAL-BAND ABSORPTION MISSION AND TOTAL-BAND ABSORPTION FOR THE SPECTRA PRESENTED IN FOR THE SPECTRA PRESENTED IN FIGURE 91. Integrations were taken FIGURE 92. Integrations were taken from 4.1 to 4.5 A. from 4.1 to 4.5 i. Author r fAdv Author 7 IAdv P sh:; r 0.856 31.3 Altshuler 0.698 65i.4 C. un uitd Grlggs 0.830 38.8 Green aud Grlggs 0.714 62.2 3radforl 0.780 48.4 Bradford 0.647 71.7 Op.*el 0.791 46.0 Oppel 0.646 77.9 Plass 0.796 44.9 Plass 0.631 81.2 Carpenter 0.748 55.4 Carpenter 0.619 83.8 Average 0.P00 43.8 Average 0.0i54 74.7 Murcray 0,793 45.6 Murcray 0.643 78.5 TABLE 40. VALUES OF AVERAGE TRANSMISSION AND TOTAL-BAND ABSORPTION FOR THE SPECTRA PRESENTED IN FIGURE 93. Integ;rations were taken from 4.1 to 4.5 /i, Author T.fAdl Altshuler 0.302 131.2 Green and Griggs 0.405 132.0 Bradford 0.360 140.9 Oppel 0.341 145.0 Plass 0.339 145.4 Carpenter 0.377 137.1 Average 0.354 140.2 Murcray 0.367 137.4 226

-.. WILLOW RUN LABRAORATORIES-. 0 | L..' ^ C S O o < ) Oo-o,o ^o ~ X' o 2 O~ cu, O 80 0 0 0 0 0 0 0 0 0 (%).~oI-'s-mSVH k E4 ~~~~~~~~~~~~~~~~~~~~~~~n Z~~~~~~~~~~r w-41. C? =) ~22

WILLOW RUN LABORATORIES - ^ ~~~~~~~~ --—.. -,.... X I C.,rsr' o u. o o Z... *$ ~ f" -i -. 5' $4 -tol _ x _ o X.. I I 0 r4 44 -' c^ y1 ^ M e,,,,~ 1 1 1 l I L c j - I ~> CV -(%) NOISSISNVHJX 228

— WILLOW RUN LABORATORIES — describes the line structure more accurately than does the statistical model. The spectra presented in figures 91, em, and 93 seem to support this statement. Hence, the divergence of the results of Green and Griggs and Atshuler from the Kyle data can possibly be attributed to the model selection. 7.5. COMPARISON OF COMPUTED SPECTRA WITH THE RIGOROUS CALCULATION FOR THE 15- A BAND The next absorption band that was selected for comparison was the 15-. CO2 band. Unfortunately, experimental field measurenents for atmospheric slant paths do not exist for this spectral region; therefore, comparison could be made only among the various calculated sectra. Drayson [5], uing the rigorous method, calculated the spectral transmission from 500 to 059 cm at infinite resolution, then averaged his results over various wavenumber intervals The spectra presented here were averaged over 5-cm intervals. His calculations were nmade for slant paths that extend from a point outside the earth's atmosphere down to 34 different pressure levels, ranging from 0.3 mb (0.0223 mm Hg) to 1013.25 mb (760 mm Hg), for six zenith angles, 0~, 15~, 30~, 450, 600, and 75~. Drayson integrated the exact transmisoivity function directly with respect to pressure —eliminating the need for the Curtis-Godson approximation-assuming a Lorentz pressure-broadened line shape for high pressures and a mixed Doppler:Lorentz line shape for pressures lower than 100 mb (7.51 mm Hg). The temperature and pressure profiles given in the U. S. Standard Atmosphere 1962 were assumed throughout, and the mixing ratio of CO2 was considered by Drayson to be constant at 314 ppm. Four paths were selected from Drayson's work and his results are compared with spectra computed by the methods of Altshuler, Elsasser, and Plass, which are presented in figures 94-97. The spectra of Altshuler and Elsasser were computed using the equivalent path parameters noted on each curve and the model atmosphere us.d by Drayson. The spectra of Plass were taken directly from his published tables [63]. which were based on the work of Stull et al. [62]. Plass used values of pressure and temperature taken from the ARDC 1959 model!ttnosphere and a C02 mixing ratio of 330 ppm compared to 314 ppm used by Drayson. However, the differences in the model atmospheres used by the two researchers has a small effect on the final transmission values. The first two of the four comparisons (figs. 94 and 95) are tor slant paths extending from 15 km to the liumit of the atmosphere at zenith angles of 750 and 00, respectively. The amounts of CO2 given on the figures,re the equivalent amounts of CO for sea-level paths (W*}, which explains the values of equivalent pressure. The high-resolution spectra indicate that there 229

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_... WILLOW RUN LABORATORIES.-; are four distinct regions of relatively high absorption. These are the main P branch and the three main Q branches. As the quantity of absorber and the effective broadenilu pressure changes, the contribution to absorption of the spectral lines whose centers lie within these main branches varies such that the relk.ive spectral character of the 15-M CO2 band changes marKedly. It is for this reason that the use of a band model for this band is unsatisfactory If highly accurate results are requfled. Thts is borne out particularly by figire 95. Plass's spectrum shows a maximum absorption between the main P and Q branches while the rigorous calculation shows a definite minimum. Also, it is noted that the spectra based on band models show a level of absorption that is consistently higher than the spectra calculated by Drayson. Elsasser gives an average band transmission that is approximately 18 p "entage points lower than the Drayson value. The values of.ltshulcr and Flass compare somewnat better. The last two comparisons (figs. 98 and 97) were made for slant paths that extend through the entire earth's atmosphere to demonstrate the comparison under conditions of high absorption. Plass did not include low-altitude paths in his tables, so his results could not be compared. For these paths the comparisons between computed spectra and the rigo, ous calculation are considerably better because the greatest contribution to absorption occurs at low altitudes where the pressures are relatively high. For this band, because of the nature of the spectral line absorption, the Eisasscr model and the Curtis-Godson equivalent pressure become better approximat;ons as the pressure and absorber concentration increase. It ti impossible to state conclusively that any one spectrum is more correct than some other spectrum; however, because of the apparent inadequacies in the barid-model approach. the spectra of Drayson are more probably correct. Indeed, a series - silAru -path field measurements ior this spectral region would be enlightening. 7,6. GENERAL COMPARISON OF COMPUTED SPECTRA All of the figures presented thus far present comparisons of computed spectra with either experimental field measurements or with spectra obtained using the rigorous calculation, The next series of curves include only spectra computed by the methods of section 3 for five slant paths common to a variety of infrared applications. For each path, the tnree parameters W*, W, and I' were ccnputed for each gas, using the model atmosphere described in sections 7.1 and 7.3 for temperatu:re, pressure, and CO,, and IO20 mixing ratios. The mixing ratio used for H O was the IRMA wet-temperate model [1341. The mixing ratio used for 03 was a shifted Elterman profile (see sec. 4.5) that yields 0.44.tm cm 03 for a vertical path through the entire atmosphere. Using these. xrameters, spectra were computed from 1 to 20 i by each 234

. —-— WILLOW RUN LABORATORIES author's method, except those of Elder and Strong, Bradfo-d, Carpenter, Llndquist, and Howard. et al. These were not considered because each method ti applicrble only to a single absorbing gas. The results of the spectral computations are presented in figures 98 through 102. The path parameters W*, W, and P for each gas are presented on each figure in conjunction with the geometry of the slant path. Thb spectra presented in these five figures were included for two reasons: (1) to demonstrate the general nature of absorption for each of the given slant-path geometries, and (2) to indicate the spread in the data for each of the H20 bands, the N20 band at 4.5., and the 0 band centered at approximately 9.6 p. The 2.7-, 4.3-, and 15-pi bands were previously analyzed, so they are not discussed in this sectioa;. Unfortunately, there is not an experimental spectrum available for any of the slant paths considered in these comparisons; therefore, the correctness of any given Sp:trum or collection of spectra can be neither substantiates nor refuted. It is felt, however, that the spread in the H20, N20,.-.d 03 absorption spectra and variation of this spread in the data as a function of absorber concentration are particularly enlightening. It is noted in figures`8 through 102 that three of the slant paths begin at the surface of the earth and the remaining two have a minimu n., <'titv-e of 10.7 km. The most contrasting results observable from these two distinct classes of slwat paths is that H20 shows a relatively small amount of absorption for all oards except the 6.3-f band, which remains almost opaque at the band center for the path of figure 102. Furthermore, the difference in H20 absorption between the paths of figures 98 and 99 is quite smali, indicating that H 20 does not absorb strongly for altitudes higher than approximately 10,000 ft except for the 3.3-pI band or for slant paths at large zenith angles. Spectra computed by each of the metnods presented in these figures for slant paths extending from the limit of the atmosphere down to some 50 different altitudes for a variety of zenith angles would be required to give a reasonably complete description of H 0 absorption. Since our objective was only to indicate the type of comparison obtainable, such extensive computations were conbidered unnecessary. The H20 sea-level equivalent concentrations for the paths presented in figures 98 through 102 are, respectively, 4.46, 3.09, 27.2, 4.82 x 103, and 4.77 x 102 pr cm. Comparing each of these values with the dait presented in table 32 shows that all but two of these concentrations are bounded by the laboratory data. Therefore, the H20 spectra presented in figures PS thrcvgh 100 represent extrapolations of the laboratory data. As was mentioned earlier, such cont;tiijns should be avoided to increase "he probability of reliable results. However, absorption spectra for these high H20 concentrations were included to erp;iztAe tite fa'4 -.ls i;c i,pi.h t a u L 235

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WILLOW RUN LABORATORIES data becomes large as the H 0 concentration increases beyond con;-,etrations that are bounded 2 by the laboratory data. In figures 98 through 102, it is noted that the largest spread in the data exists for the path containing 27.2 pr cm of H20. Particularly nrie the close comparison between the spectra of Zachor, Altshuler, and Green and Griggs and how these data differ from the results of Oppel and Stull et al. The close comparison among these researchers undoubtedly results from the fact that each used the laboratory data of Howard et al. to specify the spectral absorption coefficients. Also, each used the same form of the Goody model in the development of the transmissivity functions presented in section 3. Oppel used the same model as Zacsor, Altshuler, and Green and Griggs, but used a different collection of laboratory data. Stull et al., in contrast to the other authors, employed the quasi-random model, d used the data of loward et al. only to normalize the equivalent line intensities. The H20 spectra presented and the above discussion seem to indicate that the present state of the art of methods for calculating H20 spectral absorption are rather poor, particularly for high H20 concentrations. An enhancement of the present state of the art would require laboratory data at sufficiently high jncentrations to bound the H20 concentrations which exist for long low-altitude paths and atmos.,heric field measurements to provide a basis for comparison. Until the advent of such data, it is recommended that tliese methods be applied only to those paths which yield concentrations that are bounded by the laboratory data used to define the absnrption coefficient. Under these conditions the spectra presented indicate a spread in the data that is relatively small. A final point, mentioned in section 7.1, is the apparent spectral shift between the spectra computed by the methods of Zachor, Altshuler, and Green and Griggs and the spectra determined by the methods of Oppel and Stull et al. for the H20 bands centered at 1.14, 1.38, and 1.88 {J. Other experimental sources indicated that the data of Howard et al contained errors in spectral calibration, and based on this fact it is recommended that the absorption coefficients determined by Zachor, Altshuler, and Green and Griggs be shifted spectrally to agree with the spectra of Oppel, if their methods are to be employed. Note in figures 98 through 102 that there are significant differences in the spectra for the spectral regiop from 4.4 to 4.6 A. Absorption in this spectral region is caused by the wings of ith 4.3 CO2 band and the 6.3 H20 band, and the N 0 band which is centered at approximately 4.5 l with N20 giving the longest contribution to absorption. Only three researchers developed methods for computing N20 absorption: Oppel, Green and Griggs, and Altshuler. Therefore, the spectra of Stull et al., Zachor, and Elsasser do not include the effects of this gas. The spectra that include N90'absorption demonstrate a spread, in the data as large as 241

-. ----- WILLOW RUN LABORATORIES 20 percentage points In transmission at 4.5 i with no two spectra being in close agreement. The reasons for these differences are difficult to assess, and until field measurements of absorption which can be used ai a basis for comparison are made, the tools for evaluating the methods for predicting N20 absorption are not available. Figures 98-102 demonstrate that 03 is a significant absorber at 9.6 4, particularly if the slant path passes through that region of the atmosphere at which the 03 concentration is. maximum (see sec. 4.5). The four methods for computing 03 absorption yield results that are significantly different, the spread in the data becoming greater as the equivalent sea.-l.vel path concentration decreases. The spectra for the path containing the least amount of 0, is shown i,;igure 99. For this case, the difference between the method which yields the greatest amount of absorption (i.e., Green and Griggs) and the method which yields the least (i.e., Altshuler) is appro^.mlitely 30 percentage points in transmission. These differences are attributed to a variety of inputs: Zachor, Elsasser, and Altshuler used the Elsasser model; Green and Griggs used the statistical model. Zachor and Green and Griggs used Walshaw's data, Altshuler used three different sources of data, and Elsasser used Summerfield's data originally and later modified his resui;t based on an analysis of Walshaw's data. Elsasser and Green and Griggs used a linear pressure correction on half-width and Zachor and Altshuler used pressure correc'iciis that were highly nonlinear (see sec. 3). Such diversified approaches to the problem and the fact that experimental field measurements of 03 absorption are not available make it impossible to determine whether one set of results is more correct than another. Obviously, further research is needed for this spectral region. Considering the spectra presented in figures 98 to 1C2 in toto, the state of the art does not appear to be particularly good' especially in the wings of the bands, and hence in the window regions, for absorber concentration of sufficient magnitude to cause some absorption in the window regions (i;ote figs. 98 and,100). As the slant-path parameters W and P are dec eased to values that are bounded by the laboratory data (under these conditions attenuation in the window regions issmall), the spread in the computed data becomes small enough to ensure some measure of reliability in absorption values for the spectral region from 1.0 to approximately 4.4,; but there is still a significaan spread in the data for the spectral region from 4.4 g tnrough the 15-M CO2 band. These differences are in general caused by the fact that the,various authors used different collections of laboratory data and varied methods of fitting the band models to the laboratory data. It. appears, therefore, that in order to improve the methods for computing absorption for the wings of the H 20 bands, the N2 band, the 9.6-p 03 band, and the 15-j CO2 band more 242

-- ---— WIL LOV. RUN LABORATORIES - research is requtreu, particular)v the measurement of homogcneous absorption spectra to update and close the gap in the presently available spectra and open-air field measuzemrals of absorption %':ich can;e used as a basts for comparison Untt: the advent of such reeirch, the infrared researcher must be content with tl'r present state of the art, realizing that the greatest reliability is achieved for values c,' \V and P which are bounded by the laboratory data.

"a -. WILLOW RUN LABORATORIES - SUMMARY AND CONCLUSIONS Two general methods for computirg the absorption spectra for an atmospheric slant path have been discussed: the "rigorous" method and the method which is ba.ed on a model of the band structure. The rigorous method involves summing the contribution trc absorption of each spectral line and hence is a somewhat exact evaluation of the general transraissivity function for spectral absorption. This direct integration has been performed by Drayson [5] and by Gates et al. [7] for the 15-p CO2 band ard the 2.7-4 H20 band, respectively. The rigorous method is desirable since any resolution is attainable and for slant-path calculations of absorption the Curtis-Godson approximation is not required. The method has limited spectral application, however, since all of the band parameters for CO2 and;i2O are known accurately for only certain wavelength intervals. Also, the evaluation of the general function requires extensive computational lasor and is therefore considered impractical for many infrared applications. The second method, that which is based on assuming a model of.he band structure, has been used by many researchers, rXsulting in methcds for computing slant-path spectral absorption for every atmospheric absorbing gas for the wavelength region fro n 1.0 through tht long-wavelength rotational H20 band. The reason for using band models was to develop methods which involve a minimum amount of computation and yet yield good firs:-order approximations to the true spectra. The methods of twelve authors have been presented and discussed, and spectra computed by each author's method were compared with spectra determined from open-air field measurements, spectra calculated by the rigorous method, and spectra computed by other band-model methods. The most indicative of these comparisons is that of "computed" spectra with slant-path open-air field measurements. Unfortunately, such comparisons were limited to the 2.7- and 4.3-p bands, since these are the only spectral regions for which measurements were made of slant paths.under such conditions that the absorber concentrations could be determined with reasonable accuracy. After a careful examination of the comparative spectra presented in section 7 and the data summarized in tables 31 and 32, the following general observations were made concerning the methods based on band models. (I) The 1.14-, 1.38-, 1.88-, 3.2-, and 6.3-p H20 Bards All nmethods, except Oppel's, weire based on laboratory homogeneous spectra that were measured prior to 19ja and have a maximum H20 concentration of less than 4.0 pr cm. The 244

- WILL OW RUN LABORA TORIES d&aa of Howard et al. are apparently shifted to shorter wavelengths for the 1.14-, 1.38-, and 1.88-, H20 bands, which is inherent in the methods of Altshuler, Zachor, and Green and Griggs. The only existing comparative open-air field measurements are for homogeneous, paths, and therefore the W and P approximations could not be evaluated. The comparisons presented for each of the five spectral regions (note figs. 98 and 100) indicate large differences in absorption in the wings of the bands and the window regions for absorber concentrations that are not bounded by the laboratory data. For concentrations which are bounded by the laboratory data the comparisons are greatly improved. (2) 2.7- and 4.3-: Bands For each of these spectral regions extensive laboratory homogeneous-path spectra exist for broad ranges of W and P at relatively high resolution. Reliable slant-path open-air field measurements of absorption are available for the central portion of the 2.7-ji band and for the entire 4.3-p band. The comparisons (note figs. 76-93) indicate that band-model methods are capable of yielding good first-order approximations to measured data when the spectra are averaged over spectral intervals on the order of 0.04 A or larger. At significantly higher resolution, the band-model spectra in general display values which differ significantly from measured data. These comrarisons further indicate that the equivalent sea-level path and Curtis-Godson approximations can be used effectively. (3) 4.5-1 N20 Band N,0 absorptioni is the spectral region from 4.4- to 4.6-p is imtiners-d in the wings oi the two adjacent bands of CO2 and H 0, which makes it difficult to evaluate the state of the art of methods for predicting absorption due to this gas. iowever, the differences in the spectra noted in figures 98 through 102 indicate that the s*ate of the art is relatively poor. There exist many homogeneous-path spectra (note table 32) for this gas, but only three researchers have developed computational methods. Since the results of each if these three researchers are in disagreement, more computational methods and open-air measurements of absorption: spectra will be required before the state of the art can be improved, (4) 9.6-O 0 Band For this absorption band the available laboratory homogeneous-path data are extremely meager. The most recent a.id only relatively extensive measurements have been made by Walshaw [341; these were used by Zachor and Green and Griggs. The methods of Elsasser and Altshuler were based on data that were less extensive and much older. Open-air field measurements of 03 absorption for paths for which the 0O concentration is known do not exist. Therefore, comparisons between computed and measured spectra-could not be made.. The 245

- --... - WILLOW RUN LABORATORIES intercomparison of computed spectra for t3ts spectral region (note figs. t'-I1 0,) displays results which are significantly different; the greatest amount of divergencn is noied for those p'.hs which contain the least amount of 03. The uncertainties in 03 self-b-oads.t'.ing, which resulted in slightly different treatment by the four researchers, ir one factor which contributed largely to this difference. (5) 15-, CO2 Band The 15-p CO2 band has not been considered bY most researchers; consequently, only three methods for computing absorption spectra are availal;e. Two of these methods employ the Elsasser model and one the quasi-random model. Open-air slant-path field measurements do not exist for this spectral region, so comparisons were made between computed spectra and spectra calculated by the rigorous method. In general, band- model methods tend to overpredict absorption compared to that calculated by the rigorous method, with the spectra of Stull et al. which are based on the quast-random model, being in closest agreement.'These general observations conclusively indicate that the state of the art of methods for comtputing atmospheric absorption is far from saturated. Indications are that the concept of band models and such concepts as the "equivalent sea-level path" and the Curtis-Godson approximation are theoretically sound, and when coupled with reliable laboratory homogeneouspath spectra and sound empirical procedures, computational methods can be developed which yield results that are comparable to measured data. However, inconsistencies are observable when the band-model methods are used to compute absorption for absorber concentrations that are net bounded by the laboratory data. Therefore, to enhance the present state of the art of band-model methods, it is suggested that research should be continued with emphasis on the following four areas: (1) Obtain high-resolution laboratory data for all absorbers and all absorption bands for ranges of cor.c ntration and pressure sufficiently broad to bound the slant-path parameters encountered in practice. This would not be a comprehensive program, but one which would close the gaps in present laboratory data. Such data would thus provide a basis for developing a consistent set of band-model methods for computing absorption which would not require cxtrapolations. (2) Investigate the various methods used for empirically fitting band models to laboratory data and extend such methods, if necessary, to develop a standard method for use by all research ~.. (3) Determine the band parameters for all absorption bands for the major absorbing constituents and eventually for all absorbers. This would ao.'. for to.e Zcr- "'.I-cZ i..... the rigorous method to any band. 246

-- --. WILLOW RUN LABORATORiES (4) Obtain atmospheric slant-path, high-resolution spectra under such conditions that the quantity of each absorber in the path could be determined. Such data would provide a true test of the degree of correctness of both the exact method and the methods based on band models. The author feels that the results of the above program should be controlled by a single agency and utilized so that a general program for computing atmospheric transmission could be compiled and made available for general dissemination. A computer program is being written which utilizes what we consider to be the optLmum band-model method for each spectral?'%nd, selected from the twelve methods presented in this report. This program, coupled with the slant-path program described in appendix I, will then provide for comF.ating slant-path transmission spectra which are consistent with the present state of the art. The program is being written so that it can be easily modified as the state of the art advances. It will be published as an addencdum to this report.

.... — WILLOW RUN LA-BORATORiES -.... Appendix I COMPUTER PROGRAM FOR CALCULATING W, W", AND F The three parameters W*, W, and P are defined by the following relationships: w*= -j I.( P (x)x (100o X. W,p(x)dx (101) j P(x)p(x) dx P....... (102) P(x) d 0 wih.?e x = distance along path in meters P pressuz T temperature p =.rtial density of absorbing gas Exprising equations 100-102 in terms of pressure, temperature, and mixing ratio, we have - w = M(x) I( ) dx (103) W X M(X) jP( ))()dx (104).J.M(X )' ~ -0 ) dx * P: (M>>(IXMO) (A) (105) where M(x) is the mixing ratio of a given absorber expressed in moles of gas per mole of air. Equations 103-105 are then used directly to compute the values of the three parameters for all 249

-..-WILLOW RUN LABORATORIES gases for which the mixing ratio is given in parts per million (ppm) X 10'. CO2 0O3 N20, CO, and CH1 will be in these units. W* and W as determined by equations 103 and 104 will be in atmospheric centimeters. To compute the values of the three parameters for water vapor, theb mixing ratio in grams of H20 per kilogram of air is multiplied by the factor 1.225 x 104. Equations 103-105 will then yield the proper values, with W* and W expressed in precipitable centimeters. It is noted that the integral I(a, b) M(x) (p dx (106) is common to equations 103, 104, and 105. Therefore, each of the parameters may be determined by simply computing equation 106 for different values of a and b. Or, W*= 1(2, n + 1) W= (1, 1) p (2, 1) A computer program has been written in FORTRAN for The University of Micligan IBM 7C90 to evaluate I(a, b) for any slant path for which the mixing ratio, temperature, and pressure profiles are known as functions of altitude Z. In the following discussion it is assumed, therefore, that the parameter M(Z) is specified for each absorbing gas and P(Z) and T(Z) are given. The first step in evaluating I(a, b) is to determine the refracted path through the atmosphere. Consider a spherical earth surrounded by a spherical atmosphere. i: is assumed for this calculation that the atmosphere exists in spherical shells, each shell being homogeneous. Under this assumption the refractive index will be constant throughout a given shell, changing abruptly as one passes from one shell to the next. The slant path is defined by the lowest point in the path Z1, the highest point in the path Z2, and the angle subtended by Z1 and Z2 at the irth's center (see fig. 103), The first approxirnation to the true path is the straight line joining Z1 and Z2 (see fig. 104), The angle 4, is divided into N equal angular increments A01, AQ2..., a, where N is determined from the equation 250

WILLOW RUN LABORATORIES Z2' Zi,// /,' ( Earth FIGURE 103. PARA-METERS DEFINING GEOMETRY OF SLANT PATH

-— _... WILLOW RUN LABORATORIES..22 r 3~~~~~P/ \ M / rr 2 1~~~~~~~~~~~~~ r2 3 \ rl 82 v-X O....................... ~?..:.:.......:.'.:.."~.'.v.'E rt 2~.....:.......,:....;..:.....;.- E i:::.;:.::;:....\..... j..,........'.......... I~~~~~~~~:::..................... ~ ",'::"::':':"o~~ ~ ~ ~~ ~~~~...v::'.':'.':,". ~,',,:c,:...........":!:!::...............' FIGURE 104. SCHEMATIC DIAGr-AM OPF ATMOSPHERIC REFRACTED PATH 252~..~~~~~~~~,...eae-e#* — ** —. —— * ~.-. --.... rr~ r.-. —>- t. —$.~..-~~s-..w..* ~r @-..-~ — rt *.-~r~~~~~ ~ ~~~,-.rr~~ ~, —-.-a-.-t —.#4. —.-.vr ~~ ~ ~~ ~~ ~~~v r _ ^*.....* *. 4'. X..X' ]. ~- 4 *,^,rr~~.~~~ o,.* *-. ~ ~.. @e*. |.^, ^ ^~~ *@*.g.. -w.... -.w*&.\.\* *. e,,r t r. *':.'::.'::.*L':::.'::.-::lr~~.'J':-.-..:-.-; ~.:-;..-2Y..:-.:;.-.''.-.......,~~~~1~~1~~~ *~~~ ~1~~~ -.~~~~.~~:,:.:w.:::;'.~~~::.'::.^S'.'.';:.-.\'::: rr~ "-'-~~~~~ r~. —-'-:' - "':- *'.'~~rt~i~~ r~~~t rl:. l~~,::,:.:,.':..:~~rr~~ ~~r~ i~L~GURr~E:~~ 104.~r~i~~ SCHE:MATIC ~ O>GRAM ~~~~~~r OFAT ~PHl I EPACE P 2S2~~~~~~~~~~~~C~~~.rr~~~trr~,~lr~.rr~) ~(~~~~~

WILLOW RUN LABORATORIES N max{x 2"; 20oo 00 where Z1, Z2 and range are expressed in meters. This method of incrementing the path insured that a point wouid be calculated at least every 200 m altitude. a1 is defined as the angle between the line of sight and the radial line r1. Point b is the point where the line of sight intersects the second radial line r2. If a tangent line t1 is dramw through b which is normal to r2, it is obvious that the angle 02 will be given by (P. + Z2) 0 (107).2arc sin F j(e 2 )^'sin (107) when nl and n are the refractive indexes of the lower shell and upper shell, respectively, and R is the radius of the earth. To determine the refractive index of any given shell the results of Edlen [151] and Penndorf [152] are used. Edlen derived an empirical formula for the refractive index of air near the earth's surface: 294910 25540 s.i., I (ux.- ~'" 4i. 146 - - 41 -- 2 2 where n = refractive index of air at 28S~K and 760 mm Hg X = avelergth in microns The refract.ve index is also a function of temperature and pressure which is given by Penndorf's relationship / T /T s i P(Z) n(Z)= 1 + (n - ) T(Z)/P s) where T =?880K s' = 760 mm Hg ri ~ ice temperature P'Z) ambient pressure T(Z) = ambient temperature As was stated previously, the refractive index of a given ehell is aesumed to be constant throughout the shell, Also, for purposes of this calculation it is assumed that the value of ni for the ith shell is the value calculated for the lowest altitude of the shell, 253

-.-. — WILLOW RUN LABORATORIES Th, angle 82 in equation 107 defines a new line of sight which intersects the radial line rl at point C. A tangent line t2 is drawn normal to r3. Then the refracted angle 0P is give.. by Snell's law as before. This procedure is continued until the path intersects the rdiial line passing through Z2 at point d. The altitude difference between Z2 and d is used to increase the angle a. The entire procedure is then repeated until the point d converges to Z2 within less than!0 m. When the final refracted path is calculated, a range-altitude table is compiled where the rapge is measured from the lowest point in the path. This table, in conjunction with M(Z), P(Z), and T(Z), therefore specifies M(x), P(x), and T(x). I(a, b) is then evaluated using Simpson's rule to perform the integration over x. By simply changing the values of a and b, each of the path parameters W, W*, and P can be calculated. The program has the flexibility of evaluating I(a, b) simultaneously for as many as four absorbing gases. A listing of the complete program is presented below. The main prog'arn, all subroutines. definition of symbols, and a step-by-step procedure for inputing the data are included, The main program uses four subroutines: (1) ATMO. This subroutine calculates the refracted path through the atmosphez e between the source and the receiver. The method used is that which was described previously. The input data required are pressure, temperature, and wavelength. To be completely rigorous a new path should be calculated for each wavelength for which transmission values are desired. However, the refractive index of the atmosphere is a slow-varying function of wavelength, so only one path is.calculated. The wavelength used is the mean value of the wavelength being considered for a transmission calculation. (2) UGRAND. The subroutine UGRAND is a function which calculates the integrand of I(a, b) wherei(a-,b) M= M(x)j ( + dx. The input to the function is the value of x at which a value of I(a b is desied. The function uses The University of Michigan subroutine T to erpolte for a value of (altitu) to correspond to the desired. range)u The ssUniversity of Michigan subroutine TAB to interpol1ae for a value of Z (altitude) to correspond to the desired x (range). The samre routine is called upon for values to be selected from T (temperature) table, P (pressure) taLle, and M (mixing ratio) table. The integrand is evaluated and returned to the integration routine (SIMPSN). -. (3) SIMPSN. This routine integrates a single function and the absolute value of the function by Simpson's rule, the maximum number ox intervals being set by the user. The integration process is repeated until eithei the values of both integrals converge to the true values, the 254

- X... v-WILLOW RUN LABORATORIES -- tolerance being set by the user, or until the integration has been performed using the maximum number of intervals. The calling sequences for SIMPSN are: CALL SETSMP (AL WERR, XMAX, DEBUG) and CALL SIMPSN (FUNCIN, XCOWLM, UPPLIM, XINTGL, AINTGL) The first CALL of SIMPSN should be preceded by a call of SETSMP. ALWERR is the tolerance and is defined as the ratio of the computed error to the value of the integral. XMAX is the maximum number (floated) of equally spaced roints which the routine will use in attempting to satisfy the tolerance. This number will be used by the routine as set by the user if the number is even and in the range 6 to 32766. If XMAX is odd and in thet range, it will be treated as if it had been XMAX + 1. If XMAX is greater than 32766, it will be treated as if it had been 32766, and if it is less than 6 it will be treated as if it had been 600. DEBUG is a switch which is set to a nonzero value if the user desires to have the Simpson routine printout interrrediate information. i UNCTN is the address of the function which is to compute values of the integrand. XLOWvLM is the floated value of the lower limit of integration. tUPPLIM is the floated value of the upper limit of integration. XINTGL is the value of the integral obtained. AINTGL is the value of the integral of the absolute value obtained. ERROR is the estimated value of the error obtained. AEPROR is the estimated value of tie error of the integral of the absolute value obtained. (4) SCAN. The subroutine SCAN is used tn find the location of somc number X in a linearly ordered table. The arguments are: UN the value whose location is being sought J if J = 0 the table is considered one dimensional if J = 1 the table is dimensioned (4, 1) XLOW the table entry which is closest to, but less than, UN HIGH the table entry which is closest to, but greater than, UN KN the index location of HIGH in TAD LBHI the number of values in TAB 255

--- WILLOW RUN LABORATORIES ---- O10 if NO - 0, normal exit if NO = 1, the UN wab not found to be within the limits of TAB (1) - T.B (LBH) iU UN is equal to a value in the table, XLOW equals that value, HIGH is set to zero and KN to the index of XLO)W. A detailed procehd-re for inputing the data to tha path program is given bslow which is preceded by a list of input data symbols: TMN =Name of temperature table T Temperature (OK) PRS = Name of pressure table P = Pressure (mm Hg) NG = Number of gases XM = Name of gas AN = a coefficient of!(a, b) AM = b coefficient of I(a, b) XMR = Mixing ratio VE - Mean latitude of path (degrees) C = Highest altitude of path (meters) D = Lowest aititude of path (meters) -HI - Angle subtended by C and D at earth's center (radians) IDB = Debugging switch Input Data Format Temperature Table Card 1 Column 1-3 The rnante o1 the table All remaining cards 4-63 Contain 5-12 colu:mn fielde in floati.g poir.. The fiwst represents the temperature at tl- earth's;urface, with a total of 261 values.* Pressure Table Same as temperature Mixing-Ratio Data Card 1 Column 1 The number of gases being used - 4 Flxed point. The following cards through and including mixing rati' are to be reocated for each gas in order of assigned number (i-4! ~The following tables must have 261 vdaues: temperature, pressure, and mixing ratio. Entries 1 through 200 are ever' 290 r.; 201 through 261 are _v'.ry 1000 m. 256

.- WILLOW RUN LABORATORIES Card 2 Column 1-3 Name of gas 1 Right adjusted, 4-8 a coefficient corresponding to gas 1 in I (a, b). Floating point. 9-13 b coefficient corresponding to gas 1 in I (a, b). Floating point. All remaining cards 4-63 Mixing ratio table for gas 1. 12 Column fields, floating point, 261 vidues as for T and P tables. For CO2, 0 N 0, -4 CO, and CH4 values should be ppm x 10. For H20 values should be 1.225 x 10 x g/kg. The above, starting at card 2, is repeated for each gas used. Wavelength Data Card 1 Column 1-4 The value of wavelength for which these calculations are to be made. Floating point. Path Data (to be repeated for each path) Card 1 Column. 1-4 "Data" indicating following data as path information. Card 2 1-12 Latitude (in degrees) 13-24 Highest path altitude (in meters) measured from earth's surface. Data in columns 1-48 are 25-36 Lowest altitude (in meters) mea -ured from earth's surface. in floating point. Data switch in fixed point. 37-48 Angle subtended at earth's center (in radians). 50 Data switch right adjusted. If zero, data consist of reduced optical path for each gas. If 1, it will include the maximum value of the range. A valve of 2 will include information from the path calculation. A complete listing of the main program and all subroutines is presented below. Except for minor modifications for different executive systems, the program can be used directly on any computer capable of handling the FORTRAN language. 257

-..- WILLOW RUN LABORt TORIES --- MAIN PROGRAM COMMON ZO,ZR,ZS,PHI,NG,XLAM,NF, JG, RS,AN,AM,Z,ZZ,ZZZ,TXMR,P,R,IDB, 1C,D DIMENSION R(5.,),ZZ(501),Z(261),T(261),P(261),XMR(4,261),AN(6),AM(16), ZZZ(501),U(4),XM(4) 1000 FORMAT (4E12.7,12) 1001 FORMAT (E20.8) 1002 FORMAT (A3,5E12.7/(3X,5E12.7)) 1004 FORMAT (I1) 1005 FORMAT (A3,2E5.3/(3X,5E12.7)) 1006 FORMAT (H1,40X,34HSPECTRAL RADIANCE AND TRANSMISSION) 1008 FORMAT (1H0,50X,17HTADLES BEING USED//51X,8HPRESSURE,5X,A~/51X, 1H 1'TEMPERATURE,2X,A3/(5iX,3HGAS1X,A3)) 1013 FORMAT (A6) 1015 FORMAT (lH-57X4HPATH/ /42x8iLATITUDET75,E1 5.6/42x17HREC EIVER ALTIT J 1DET75tE1 5.6/42x 15HSOURCE ALTITUDET75,E15.6/42x 15HANGLE SUBTENDEDT7 25,E15.6/42x24HEMISSIVITY AT THE SOURCET75,E15.6) F UGRAND 4 READ 1002, TNM.(T(K), K=1,261) READ 1002, (PRS, (P(K),K-1,261)) READ INPUT TAPE 7, 1004, NG DC I J- 1,NG 1 READ INPUT TAPE 7, 1005, (XM(J),AN(J),AM(J),(XMR(J,K),K=1,261)) READ INPUT TAPE 7, 1001, XLAM C'DATA' IN COLUMNS 1-4 IMPLIES A NEW PATH C IS TO BE CALCULATED WITH GEOMETRY AND WAVE C LENGTHS FOLLOWING WRITE OUTPUT TAPE 6,1e,06 WRITE OUTPUT TAPE 6,1008, PRS,TNM,(XM(J),= 1,NG) 100 READ INPUT TAPE 7, 1013,'DATA IF(DATA - 6HDATA) 4,2,4 2 READ INPUT TAPE 7,1000, FE,C,D,PHI,IDB 23 ZO 19.61330/(3'085462E-6 + 2.27E-7*COS(FE) - 2.E-12*COS(4.*FE)) ZR - C+ZO ZS D= +Z WRITE OUTPUT TAPE 6, 1015,FE,C,D,PHI EXT= 0.0 Z(1)=EXT DO 5 3=2,201,1 EXT=-EXT+200.0 5 Z(J) = EXT DO 6 J202,261,1 EXT= EXT+1000. 6 Z(>J) EXT 258

-- ----- WILLOW RUN LABORATORIES- - C C CALCULATE PATH 7 CALL ATMO NF= NF RS=R(NF) DO 9 J =1,NF,1 9. Z(J) =ZZZ(J)-ZO N1 = RS/200, "=- NP IF (500. - XP) 10,11,11 11 IF (XP - 10.) 13,12,1. 13 XP= 10. GO TO 12 10 XP = 500. 12 XMAX = XP*6. C R N 4 C EVALUATE U (R) INTG(M(NG)*.P/DO) (TO/T) DR) C NG O C AT'NP' POINTS FOR EACH GAS C 203 CALL SETSMP(1.E-15,XMAX,0.) DO 204 II1,NG,1 JG = I CALL SIMPSN (UCRAND,R(1),R(NF).,UIT,ABS) 204 U(I) = UNIT IF(IDB) 205,206,205 205 WRITE OUTPUT TAPE 6, 3002, R(NF),JG,(I,U(I), I=1,N3) 3002 FORMAT (81iORANGE E14.7,14H INTEGRATION I3,6H GAS I1,8H VALUE E 114.7,(/4 5x 1,8x E l4.7)) 206 WRITE OUTPUT TAPE 5, 2000, NG,(XM(K),U(K),K=l,NG) 2000 FORMAT (I1,4(A3,E16.8)) WRITE OUTPUT TAPE 6, 2001, NG,(XM(K),U(K),K=1,NG) 2001 FORMAT (1H0,1,15X,4(A3,E116,8,6X)) GO TO 100 END

.-..... WILLOW RUN LABORATORIES -- SUBROUTINE UCRAND FUNCTION UGRAND(X) COMMON ZO,ZR,ZS,PI.EPS,NG,XLAM,NF,JG,RS,OIUT,NMNN,IIBEX,FAC,DELR, 1 BC,D,E,AN,AM,Z,ZZ.ZZ Z,T,XMR,P,R AMD,VK,UK,TAU.FTAU,TA'tR,RIN,IDB DIMENSION AN(6),AMh(6),Z(261),ZZ(501),ZZ Z(501),T(261),XMR(4,261), 1 P(261),P(501),AMD(4,200),VK(4,200),UK(4,150),TAU(4,6,150), 2 FTAU(4,505),TAVUR(505).RIN(505) DIMENSION SW(4) DZ = TAF'OX,R(1).ZZ(I),1,1,,NF,SW(1)) IF(DZ-1.ES) 4,6,6 4 IF (DZ) 20.2,2 2 DT = TAB(DZ,Z(1),T(1),1,1,l,261,SW(2)) DP = TAB(DZ,Z(J),P(1),1.,1,261,SW(3)) DM = TAB(DZ,Z(1),XMR(JG,.),1, 4,1,261,SW(4)) DO 3 J=1,4.1 J J IF(SW(J)- 1.0)3,3,7 3 CONTINUE 5 UGRAND= DM*((DP/P(1l)),AN(JG)*(T()/DT)* *AM(JG)) RETURN 6 = 261 GO TO 8 7 CALL SCAN (DZ,Z,O,X,Y,1,261,XS) GO TO (8,9.10,11), J 8 DZ = Z(I) S DT = T(l) 10 DP = P() 11 DM = XMR(JG,I) 13 WRITE OUTPUT TAPE 6. 1000, J 1000 FORMAT (iHO,12HCHECK SWITCH,12) 12 GO TO 5 20 UGRAND = 0. RETURN END WRITE OUTPUT TAPE 6, 1001, X,DZ,DP,DM,UGRAND 1001 FORMAT (5E20.A) 260

-.. WILLOW RUN LABORATORIES --—. SUBROUTLNE ATMO COMMON R,ZH,ZL,HIT,EPS,NG,XLAM,NN,JG,RS,OUT,NMN,n,BEX,FAC,A,B, 1C,D,E.AN,AM,ZZ,Q,Z,T,XMR,P,RH,AMD,VK,UK,T FTAU,FTAUTR,RIN,UB DIMENSION AN(6),AM(6),ZZ(261),Q(501), Z(501),T(231),XMR(4,261), 1 P(261),RH(501),AMD(4 200),VK(4,200),UK(4,1 50),TAU(4,6,150), 2 FTAUi4,505),TAUR(505),RIN(505),U(4,505) DIMFNSION THT(2), FM(2) 52 FORMAT(10X,9HPRESSURE=,E12.7,22HERROR IN INTEWPOLATION) 53 FORMAT(10X,12HTEMPERATURE=,E12.7,22HERROR IN INTERPOLATION) CALL FTRAP TEST = 0. IF (ZH-ZL) 142,143,143 142 ZX = ZL ZL = ZH ZH- ZX 142 IF (DB - 1) 26,26,25 25 TEST= 1. 26 CD = ABSF(C-D) N CD/200. RNG SQRT(ZL**2+Z Ft*2- 2.*ZLZ H COS(PHIT)) M=RNG/2000. IF(M-N)2,3,3 2 FN=N GO TO 4 3 FN=M 4 TIITH=ARCSIN(ZL*SIN(PHIT)/RNG) IF (THTH) 216,205,216 2)5 RNG - CD 216 I1 (500.-FN) 215,206,206 215 FN = 500. GO TO 201 206 IF (FN-10.) 200,201,201 200 FN - 10. 201 IF(TEST) 41,41,100 100 n WITE OUTPUT TAPE 6,101,RNG,FN,THTH 161 FORMAT(1 H1,3(F.14.6,2X)) 41 DO 15 KK=1,10 W-RNG/FN PH1S=0 RH(1)=0 Z(1)=ZH THT(1)=THTH K=2.* FN NN=1 Z(1)=Z(~)-R EXT-0.0 DO 1'0 J=2,201,1 EXT=EXT+2C0.0 261

...WILLOW RUN LABORATORIES — 150 ZZ(J)-EXT DO 160 J=202,261,1 EXT=EXT+1000G. 160 ZZ(J)=EXT LQ=(Z(1*261,.)/.1E6 54 PP-TAB(Z(1),ZZ(1),P(1),l,,1,261,SW) IF(W- 1.0)56,56,55 55 PP-P(LQ) WRITE OUTPUT TAPE 6,52.P? 56 fT=TAB(Z(1),Z(1),T(1),1,1,1,261,SW) IF(SW- 1.0)58,i8,57 57 TT=T(LQ) WRITE OUTPUT TAPE 6,53,TT 58 FMS=1.+(6432.8+294910./(146.-(1./XLAM* * 2))+25540./(41.-(1./XLAM* *2 1)))*.1E-7 FM(i)=I.+(FMS- 1.) (1.+(T(1)/273.15))* PP/((1.+(TT/273.15))*P(1)) Z(1)-ZX(1+R CO 11 I=',K t4=NN- 1 RH(I+I) —RH(I)+W Z(I+1)=SQRT(W**2+Z(I)**2-2.*W*Z(I)*COS(THT(1))) 5 PHI=ARCSIN(SIN(THI (1))*W,'Z(I+ 1)) 99 IF(TEST)69,69,110 L10 WRITE OUTPUT TAPE 6,111,W,PHI,FM(l),RH(A),Z(I) 111 FORMAT(5(E14.6,1X)) 69 tF(PHI)7l,71,40 71 ZDIWF=Z(I+1)-ZL IF(TEST)72,72,140 L40 WRITE OUTPUT TAPE 6.141,W.PHt,FM(1),THT(1),PH(I),Z(I) L41 FORMArT t6(E14.6,1X)) 72 IF(ZDIFF-10.)20,20,73 73 IF(W-ZDIFF)11,11.74 74 W=ZDIIFF GO TO 11 40 Z(I+1-=Z(1+1)-R LQ-(Z(I+1)* 261.)/.1E6 PP=TAB(Z(1i+1),ZZ() ),P(1I,,91,1,261,SW) IF(SW-1.0)62,62,61 61 PP=P(LQ) WRITE OUTPUT TAPE 6,52,PP 62 TT=TAB(Z(I+1),ZZ(1),T(1),1,1,261,SW) IF(SW- 1.0)64,64,63 63 TT=T(LQ) WRITE OUTPUT TAPE 6,53,TT 64 FMS=1,+(6432.8+294910./(146.-(I./XLAM* *2))+25540./(41.-'./XLAM* * 12)))*.1E-7 FM(2)=1.+(FMS- 1.)* (1.+(T(1)/273.15))*PP/((1.+(TT/273.15))*P(1)) Z(I+1)=Z(I+1)+R THT(2)=THT(1)+PHI IF(1.570796-THT(2))85,81,81 81 GMA2 =ARCSIN(FM(l)*SIN(TiHT(2))/FM(2)) GC TO 80 85 THT(2)-3.1415926-THT(2) IF( (I)- Z(I+ 1))86,88,87 262.

.WILLOW RUN LABORATORIES 86 GMA2 =ARCSIN(FM(1)*iN(TrHT(2))/FM(2)) GMA2 =3.141592-GMA2 GO TO 89 87 GMA2 =ARCS1N(FM(2)*SIN(THT(2))/FM(1)) GMA2 =3.1415V2-GMA2 TO TO 80 88 GMA2=3.141592s-THT(1) 80 PHIS=PHIS+PHI DELPHI=PHIT- P;iS Y=(PHI*10.)/W!F(DELPHI-Y)9,9,4 7 IF(DELPHI- HI)8,8,10 8 W=DELPHI*W/PHI GO TO 10 9 ZDIFF-ZL-Z(I+1) IF(ZDIFF)79,12,12 10 FM(1)= FM(2) THT(1, GMA2 IF(TEST)11, 1,120 120 WRITE OUTPUT TAPE 6,121,PHLS,DELPHI,W,FM(1),THT(1) 121 FORMAT(5(El4.G,2X)) 11 CONTINUE 12 IF(ZDIFF-10.)20,20,i3 13 THTL=3.1415926-(THTH+PHIT) X=SQRT(ZDIFF**2+RNG** 2-2.*RNGJ*ZDIFF*COS(THTL)) ALPH=ARCSIN(7')IFF*SIN(TTL)/X) THTH=THTH+ ALPH GO TO 89 79 ZDI FF=-ZDIFF I(ZDIFF- 10.)2C,2T,70 70 THTH-THTH-.1*ALPH 89 IF(TEST)15, 5,15,S0 130 WRITE OUTPUT'APE 6,131,TH'TL ALPH,THTH,THT(1),FM(),W,ZDIFF 131 FORMAT(7(I1f.4,1X)) 15 CONTINUE 20 WRITE OUTPUT TAPE 6,30,THTH,THT(2) 30 FORMAT(1H-,37X,9HTHETA(1)=,E12.6.12H THETA(NN)=,E12.6/8H INDEX, 12 5X,8HALTITUDE,18X,24HRANGE FROM HIGHEST POINT,//) IF(D1B) 34,38,34 38 I 1 WRITE OUTPUT TAPE 6,37. 1,Z(1)RH(1),NN,Z(NN),RH(NN) 37 FORMAT(3XI4,24XE16.8,23XE16.8/12H.,./. 3XI4,24XE14.8,23XE16 1.8) RETURN 34 DO 35 1=1,NN K=NN+i I 35 WRITE OUTPUT TAPE 6,36,I,Z(I),H(I) 36 FORMAT(3X,I4,24X,E16.8,23X,E16.8) RETURN END 263

WILLOW RUN LAB RRATORIES SUBROUTINE SCAN SUBROUTINE SCAN (UN,TAB,J,XLOW,HIGH,KN,LBH, NO) NO= 0 IF (J) 100,100,200 DIMENSION TAB (4,1) 100 IF (TAB(1)-UN) 1,28,50 1 IF kTAB(LBH)-UN) 50,29,2 2 LHP = LBH LLP = 1 3 LTEST - (LHP-LLP)/2 + LLP IF (TAB(LTEST) - UN) 5,30,10 5 LLP = LTEST GO TO 15 10 LHP = LTEST 15 IF (LHP-LLP-1)50,40,3 28 LTEST = 1 GO TO 30 29 LTEST = L3H 30 KN = LTEST XLOW = TAB(LTEST) HIGH = 0. RETURN 40 KN = LHP XLOW = TAB(LLP) HIGH = TAD(LHP) RETURN 50 NO= 1 RETURN 200 IF (TAB(J,1)-UN) 101,128.50 101 IF (TAB(J,LBH) - UN) 50,129,) 02 102 LHP = LBH LLP = 1 103 LTEST = (LHP-LLP),'2 + LLP IF (TAB(J,LTEST)-UN) 105,1 30,110 105 LLP = LTEST GO TO 115 110 LHP - LTEST 115 IF (LHP-LLP-1) 50,140,103 128 LTEST =1 GO TO 130 129 LTEST a LBH 130 KN = L'FST XLOW =.TAB(J,LTEST) HIGH = 0. RETURN 140 IM = LHP XLOW -A TA(J,LLP) HIGH = TAB(J,LHP) RETURN 264

WILLOW RUN LAB RA O)RIES - Appendix UI COMPUTER PROGRAM FOR CALCULATING ELSASSER BAND.MODEL TRANSMISSIVITY FUNCTION * Absorption by aa Elsasser model h-s been given as: A sinh; exp (-y cosh 3)l(y)dy (0s) where y s is a function of tt.e properties of an individual line. / simpie algorithm, consinh 1 verging for all values of X and,3, can be del.ved as follows. Let y J exp (-t cosh 3)IU(t)dt Then by expanding the Bessel function into its power series, we obtain y co J = exp (-t cosh n) -b tndt n —0 or J - l'hm tM mexp (at) dt In= where a cosh 2. Let a = f tm exp (-at dt Then 00 C * n nCO *This expansion was derived by D. Lundholm of Lockheed Missiles tad Space Company. 265

-... - - WILLOW R UN LA5ORATORIES Integrating by parts gives the recurs.on formula for a as n y- y a n-If. a a -:j;R ] - Y exp -ay) and'n(n- 1)," ny2 n-2 + a \ a because b = —. - forn even (110) n [(m^): * f(m/2)!j 2' and b - 3 for n dd Then o a2; J ) —~ 2,_ Li (P.!)222 n;=0. where V 1 a0 =,J exp (-at) C= _ 1 - exp,-ayi) and from the recursion formulas, a2 = " - x aP (-ay + + 2L 4a4 =- e ( a2 + a a a This suggests the following formula fo, an: n an l a - exp — ay-(' 266

......-WILLOW RUN LABORATORIES - Thus the sum approaches exp (ay) as n approaches infinity and the term in brackets approac hes zero. Equations 109-111 then'furnish an algorithm solution to equation 108; this algorithm ia in convenient form for modern digital computers. George Oppel wrote a subroutine, ELSR, in Fortran IV for cither the IBM 7094 or Sperry R'.id 1108. The program uses the above theory and is used by the following sequence of FORTRAN statements to clculate transmission. B2X — * W (112) d BX = S/d * W (113) T = 1. - ELSR(B2X, BX) (114) Equations 112-114 can be used to compute transmission for any gas for which the Elsasser model is applicable.and the absorption coefficients are in the form noted in equation 112 and 113. Unfortunately, only the methods of Oppel and Bradford are in the required form. Hence, these equations cannot be applied to other methods without some modification of the ELSR program.

_-___- WILL CW RUN LABORATORIES —-- SUBROUTINE ELSR ABSORPTION BY UNIFORMLY SPACED LINES- ELSASSER MODEL DIRECT INQUIRIES TO CEO OPPEL, LOCKHEED MISSILES AND SPACE COMPANY; SUNNYVALE, CALIF. FUNCTION ELSR(Z,BX) IF(Z) 3,3,4 3 ELSR0O. RETURN 4 IF(BX)14,14,2 2 BETAZ/BX X=BX/BETA Pl34.1415926 IF(X-.01 )5,5,6 5 IF(BETA-1.)7,20,20 6 IF(BETA**2/X- 100.)22,22,20 22 IF(X-20.)9,11,11 9 IF(Z-.01)10,13,13 11 IF(BX-.9)10,14,14 20 ELSRlI.-EXP(-BX) RETURN 21 ELSR SQRTI(2.* Z/PI) RETURN 14 U=SQRT(.5*Z) ELSR=I.-(1./(1.+U*(. 14112821+U*(.08864027+U*(.02743349+U*(-. 000394) 146+U*.00328975)))))**8) RETURN 10 IF(X-150.)7,21,21 7 FACT-1. BES=1. BEA=2. A=l. B=I. 18 DO 7 I —1,10 FACT-FACT* FLOAT(I) J=2*I- 1 IF(X-5.)23,23,12 23 BES=BES+(X/2 )**J*(FLOAT()+X/2. )/FACT/FACT IF(ABS(I.-BEA/BES)-1.E-3)15,16,16 12 A=A*FLOAT(J)**2 B=B*(4.- FLOAT(J)**2) BES=BEA+(A+(- 1. )** I B)/FACT/(8. X)*.I IF(ABSl.- BEA/BES)- I.E-3)27,16,16 16 BEA=BES 17 CONTINUE 27 ELSR=BES* BETA*SQRT(X/2./PI) RETURN 15 ELSR=BETA*X*EXP(-X)*BES RETURN 13 BA=.5*(1./EXP(-BETA)-EXP(-BETA)) AB=.5*(: /EXP(-BETA)+EXP(-BETA)) FACT-l Y=BX/BA R=EXP(-Ab*Y) 268

-.. WILLOW RUN LABORATORIES; -- - - A2M(<1.-R)/AB A=A2M AM,2. DO 24 IM,24 SFLOAT(1) T=2.*S FACTZFACT*S XYZZt120.*ALOC(2.) IF(S*ALOG(Y)-XYZZ/2,)103,103,102 102 XYZZ,120./T Yr2.**XYZZZ 103 CONTINUE A2M-T*(T- 1.)*A2M/AB**2.R*(Y**T/AB+T*Y**(T- 1. )/AB**2) AtA+A2 M/((FAC'T)**2 2. **.T) IF(.002-ABS(A-AM))24,24,25 24 AM=A 25 ELSR=A*BA RETURN J~~~E~~~~~~ND~~269 269

WILLOW RUN LABORATORIESREFERENCES 1. D. E. Burch et al., Infrared Absorption by Carbon Dioxide, Water Vapor, and Minor Atmospheric Constituents, AFCRL Report No. 6-698, The"hio State University, Columbus, 1962. 2. W. S. Benedict and L. D. Kaplan, "Calculations of Line Widths in H20-N2 Collisions," J. Chem._Phs., Vol. 30, 1959, pp. 388-399. 3. B. H. Winters, S. Silverman, and W. S. Benedict, "Line Shape in the Wing Beyond the Band Head of the 4.3 g1 CO2 Band," ant. Spectry. Radiative Transfer, Vol. 4, 1964, p. 527. 4. G. Plass, "STctral Band Absorptance for Atmospheric Slant Paths," Apl. Ojt., Vol. 2, May 1963. 5. S. R. Drayson, Atmospheric Slant Path Transmission in the 15-/I CO2 Band, Report No. 5683-6-T, High Altitude Engineering Laboratory, The University of Michigan, Ann Arbor, November 1964. 6. C. Young, "Calculation of the Absorption Coefficient for Lines with Combined Doppler and Lorentz Broadening," J. Quant. Spectry. Radiative Transfer, Vol. 5, 1965, pp. 549-552. 7. D. M. Gates et al., "Line Parameters and Co:nputed Spectra for Water Vapor Bands at 2.7-p," Nat. Bur. Std. (U. S.). Monograph No. 71, August 1964. 8. W. M. Elsasser, "Mean Absorption and Eqaivalent Absorption Coefficient of a Band Spectrum," Phys. Rev., Vol. 34, 1938, p. 126. 9. W. M. Elsasser and M. F. Culbertson, "Atmospheric Radiation Tables," Meteorol. Monographs, Vol. 4, No. 23, 1960. 10. G. N. Plass, "Models for Spectral Band Absorption," J. Opt. Soc. ATOI., Vol. 48,' 1958, p. 690. 1!. H. Mayer, Methods of Opacity Calculations, unpublished, 1947 12. R M. Goody, "A Statistical Model for H20 Absorption," Quart. J. Roy. Meteorol. Sec., Vol. 78, 1952, p. 165. 13. P. J. WVyatt, V. 1. Stull, and G. N. Plass, "Quasi-Random Model of Band Absorption," J. Opt. Soc. Am., Vol. 52, 1962, p. 1209. 14. J. King, "Modulated Band-Absorption Model," J. Opt. Soc. Am., Vol. 55, 1965, p. 1498. 15. R. ladenburg and F. Relche, "Ueber Selektive Absorption," Ann. Physik, Vol. 42, 1911, pp. 181-203. 16. W. M. Elsasser, Heat Transfer by Infrared Radiation in the Atmosphere. Harvard Meteorological Studies No. 6, Ha.rvw.rd University Press, Cambridge, Mass., 1942. 17. L. D. Kaplan, "Regions of Validity of Various Absorption-Coefficient Approximations," J. Meteorol., Vol. 1t~, 1953. 270'

..WILLO"'VO RUN LABORATORIES — 18. G. E. Oppel, (Classified Title), Technical Report No. LMSC-A325516, Lockheed Missiles and Space Co., Sunnyvale, Calif., June 1963 (CONFIDENTIAL). 19. P. J. Wyatt, V. R. Stull, and G. N. Plass, The Infrared Absorption of Water Vapor (Final Report), Report No. SSD-TDR-62-127, Vol. II, Aeronutronic Division of Ford Motor Co., Newport Beach, Calif., 1962, AD 297 458. 20. G. N. Plass and D. I. Fivel, "Influence of Doppler Effect and Damping Jn LineAbsorption Coefficient and Atmospheric Radiation Transfer," Astrophys. J., Vol. 117, 1953. 21. G. E. Oppel, Lockheed Missile and Space Co., Sunnyvale, Calif., private communications, 1966. 22. G. N. Plass, "Useful Representations for Measurements of Spectral Band Absorption," J. Opt. Soc. Am., Vol. 50, 1960, pp. 868-875. 23. A FORTRAN Program, "Least-Squares Estimation of Nonlinear Parameters," available as IBM Share Program No. 1428. 24. A. R. Curtis, "A Statistical Model for Water Vapor Absorption (')iscussion of Paper by Goody)," Quart. J. Roy. Meteorol. Soc., Vol. o7, 152. 25. W. L. Godson, "The Computation of Infrared Transmission by Atmospheric Water Vapor," J. Meteorol., Vol. 12, 1955. 26. T. L. Altshuier, Infrared Transmission.nd Background Radiation by Clear Atmospheres, Report No. 615r199, General Electric Co., Philadelphia, December 1961. 27. A. Zachor, Near Infrared Transmission over Atmospheric Slant Paths, Report No. R-328, Instrument Laboratory, Massachusetts Institute of Technology, Cambridge, July 1961. 28. R. O. B. Carpenter, Predicting Infrared Molecular Attenuation for Long Slant Paths in the Upper Atmosphere, Scientific Report No.?, Geophysical Research Directorate, Air Force Cambridge Research Center, Bedford, Mass., November 1957. 29. W. H. Cloud, The 15 Micron Band of C02 Broadened by Nitrogen and Helium (Progress Reporton Contract NOnr 248-01), Johns Hopkins University, Baltimore, 1952. 30. J. N. Howard, D. E. Burch, and D. Williams, Near Infrared Transmission through Synthetic Atmospheres, Geophysics Research Paper No. 40, Report No. AFCRC-TR-55-213, Ohio State University, Columbus, November 1955. 31. L. D. Kaplan and D. F. Eggers, "Intensity and Line Width of the i5 Micron CO2 Band Determined by a Curve-of-Growth Method," J. Chem. Phys., Vol. 25, 1956, pp. 876-883. 32. G. Yamam6to and T. Sasamort, "Calculation of the kbsorption of the 15 Micrnn Carbon Dioxide Band," Sci. Rept. Tohoku Univ., Fifth Ser., Vol. 10, 1959, pp. 37-45. 33. M. Summerfield, "Pressure Dependence of the Absorption in the 9.6 Micron Band of Ozone," thesis, California Institute of Technology, Pasadena,. 1941. 34.. C.D. Walshaw, "Integrated Absorption by the 9.6 Mi!Crn' n: d ~,..,'" A - "~.. J. Roy. Meteorol. Soc., Vol. 93, 1957, pp. 315-321. 271

- -- WILLOW RUN LABORATORIES.... 35. r,. Yamamoto, "On a Radiation Chart;" St. Rept. Tohoku Univ., Fifth Ser., Vol. 4, 1952, pp. 9-23. 36. H. A. Daw, Transmission of Radiation through Water Vapor Subject to Pressure Broadening in the Region 4.2 Mic,ns to 23 Microns, Technical Report No. 10, University of Utah, Salt Lake City, 1956. 37. C. H. Palrer, private communication to W. M. Elsasser based on data obtained at the Laboratory of Astrophysir at Johns Hopkins University, 1958. 38. E. E. Bell, Infrared Techniques and Measurements (Interim Engineering Report for Period July-September 1956 on Contract AF 33(616)-3312), Ohio State University, Columiuis, 1956. 39. W. T. Roach and R. M. Goody, "Absorption and Emission in the Atmospheric Window from 770 to 1250 cm-l," Quart. J. Roy. Meteorol. Soc., Vol. 84, 1958, pp. 319-333. 40. E. F. Barker and A. Adel, "Resolution of Two Difference Bands of CO2 Near 10 A," Phys. Rev., Vol. 44, 1933, p. 185. 41. C. Schaefer and B. Phillips, "The Absorption Spectrum of Carbon Dioxide and tne Shape of the C02 Molecule," Z. Physik, Vol. 36, 1926, p. 641. 42. H. W. Yates and J. H. Taylor, Infrared Transmission of the Atmosphere, NRL Report No. 5453, Naval Research Laboratory, Washington, 8 June 1960. 43. C. H. Palmer, "Water Vapor Spectra with Pressure Broadening," J. Opt. Soc. Am., Vol. 47, 1957, p. 1024. 44. N. C. Yaroslavsky and A. E. Stanevich, "The Long Wavelength Infrared Spectrum of H20 Vapor and ihe Absorption Spectrum of Atmospheric Air in the Region 20-2500 p (500 - 4 cm-)," Opt. pectry., Vol. 7, 1959, p. 380. i5. G. Hettner, R. Pohlman, and H. J. SchNmacher, "The Structure of the Ozone Molecule and Its Spectrum in the Infrared," Z. Physik, Vol. 91, 1934, p. 372. 46. H. S. Gutowsky and E. M. Petersen, "The Infrared Spectrum and Structure of Ozone," J. Chem. Phys., Vol. 18, 195:, p. 564. 47. J. Strong, "On a New Method jo Measuring the Mean Height of the Ozone in the Atmosphere," J. Franlin Inst., Vol. 231, 1941, p. 121. 48. C. W. Allen, Astrophysical Quantities, University of London Athlone Press, 1955, pp. 117-119. 49. J. H. Shaw, G. B. B. M. Sutherland, and T. W. Wormell, "Nitrous Oxide in the Earth's Atmosphere," Phys. Rev., Vol. 74, 1948, p. 978. 50. S. A. Clo:;gh, D. E. McCarthy, and J. N. Howard, "4v2 Band of Nitrous Oxide," J. Crem. Phys., Vol. 30, 1959, p. 1359. 51. E. K. Plyer and E. F. Barker, "Infrared Spectrum and Molecular Configuration of N20," Phys. Rev., Vol. 38, 1931, p. 1827. 52. D. E. Burch, Ohio State University, Columbus. private communication, 1960. 53. G. Lindquibs, Willow Run Laboratories of the Institute of Science and Technology, The University of Michigan, Ann Arbor, priva* communication, 1966. 272

--- WILLOW RUN LABORATORIES - 54. G. E. Oppel and,L. L. List, Atmospheric Transmission from 4 to 5 Microns. Report No. LMSC-A667599, Lockheed Missiles and Space Co., Sunnyvaie, Calif., August 1964. 55. W. R. Bradford, Predicting the Molecular Absorption of Infrared Radiation over Atmospheric Paths, Report No. DMP 1422, EMI Eltetronics, Hayes, Middlesex, England, 1963. 56. W. R. Bradford, T. M. McCormick, and J. A. Selby, Laboratory Representation' of Atmospheric Paths for Infrared Absorption, Report No. DMP 1431, EMI1 Electronics, Hayes, Middlesex, England, 1063. 57. A. S. Green, C. S. Lindenmeyer, and M. Griggs, Molecular Absorption in Plan. etary Atmospheres, Report No. GOA63-0204, Space Science Laboratory, General Dynamics Corp., San Diego, May 1963. 58. A. S. Green, A Semi-Empirical Formula for Infrared Transmission through the Atmosphere, Report No. ZPH-115, Physics Section, Convair Division, General Dynamics Corp., San Diego, 1961. 59. A. S. Green, Atmospheric Attenuation over Finite Paths, Report No. TDR63-174, Aerospace Corp., San Bernardino, Calif., August 1963. - 60. J. A. Rowe, Aerospace Corp., private communication, 1966. 61. R. 0. Carpenter, "The Transmisailon of Hot CO2 Radiation through Cold CO~ Gas" (1J), Proc. IRIS, Vol. 4, No. 2, May 1959 (CONFIDENTIAL). 62. V. R. Stull, P. J. Wyatt, and G. N. Plass, The Infrared Absorption of Carbon Dioxide (Final Report), Report No. SSD-TDR-62-127, Vol. III, Aeronutronic Division of Ford Motor Co., Newport Beach, Calif., 1963, AD 400 959. 63. G. N. Plass, Transmittance Tables for Slant Paths in the Stratosphere (Final Report), Report No. SSD-TDR-62-127, Vol. V, Aeronutronic Division of Ford Motor Co., Newport Beach, Calif., May 1963, AD 415 207. 64. M. Gutnick, "How Dry Is the Sky?" J. GeGpjhys. Res., Vol. 63, 1961, pp. 286-287. 65. M. Gutnick, "Mean Atmospheric Moisture Profiles to 31 km for Middle Latitudes," Appl. Opt., Vol. 1, 1962, p. 670. 66. J. A. Thomson, "Emissivities and Absorptivities of Gases," Ph.D. dissertation, California Institute of Technology, Pasadena, 1958. 67. A. Thomson and M. Downing, An Investigation of Two Factors in the Infrared Environment of an Artificial Satellite: Earth Backgr.ound and Atmospheric Transmission in the 2 to 5 Micron Region, Report iNo. Ph-069-M, Physics Section, Convair Division, General Dynamics Corp., San Diego, 10 April 1960. 68. G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 2: Infi ared and Raman Spectra of Pol'atomic Molecules, Van Nostrand, 1954, p. 21. 69. T. Elder and J. Strong, "The Infrared Transmission of Atmospheric Windows," J. Franklin Inst., Vol. 255, 1953. 70. P. W. Fowle, "Water Vapor Transparency to Low Temperature Radiation," Smithsonian Inst. Misc. Collections, Vol. 68, No. 8, 1917. 71. F. E. Fowle, "The Spectroscopic Determination of Aqueous Vapors," Astrophys. J., Vol. 35, 1912, p. 149. 273

- - WILLOW RUN LABORATORIES 79.. F. E. Fowle, "Redetermination of Aqueous Vapor above Mount Wilson," Astrophys. J., Vol. 37, 1913, p. 359.'3. F. E. Fowle, "The Nonselec;ive Transmissibility of Radiation through Dry and Moist Air," Astrophys. J., Vol. 38, 1S3, p. 392. 74. F. E. Fowie, "Avogadro's Constant and Atmospheric Transparency," Astrophys. J., Vol. 40, 1914, p. 435. 75. F. E. Fowle, "The Transparency of Aqueous Vapor," Astrop.ys. J., Vol. 42, 1915, p. 394. 76. J. N. Howard, The Absorption of Near Infrared Blackbody Rad ation by Atmospheric Carbcn Dioxide and Water Vaor (Report No. 1 ci Contr ct DA-44-009eng-12), Ohio State University Research Fcundation, Colt.mbus, 51 March 1950. 77. G. Hettner, "Uber das Ultrarc't Absorp.ionspecktrum de Wasserdampfes," Ann. Physik, Vol. 55, 1918, p. 476. 78. A. Elliott, G. G. MacNeice, and E. A. Jones, Trar.:missicn o~ the Atmosphere in the Near Infrared, Report No. A.R.L./R.2/EL50, Admiralty Research Latoratory, Teddingten, Middlesex, England, I February 194G. 79: A. Elliott and G. G. MacNeice, "The Transmission of t:e Atmosphere in the 1-5 Micron Region," Nature, Vol. 161, 158, p. 516. 80. D. F. Fischer and R. Heinz, The Influence of the Spectral'transmission of Optics and Atmosphere on the Sensitivity of Infrared Detectirs, USAF Technical Report No. F-TR-2104-ND, Wright Field, October 1946. 81. J. D. Strong, Atmospheric Attenuation of Infrared Ra.liations (Final Report on Contract OEMNsr-t)-, OSP Report No. 59d6, Cruft Laborator-y, Harvard University, Cambridge, Mass., 30 November 1945. 82. H.A. Gebbie et al.,'Atmospheric Transmission in th" I to 14 Ai Region," Proc. Roy. Soc. (London), Ser. A, Vol. 20, 1951, pp.:. 107. 83. H. A. Gebbie et al., Atmospheric Transmission in the 1 -4 Micron Region, Report No. A.R.L./R.4O/E60, T.R.E./T.2129, Admiralty iesearci Labratory, Teddington, Middlesex, England, ATI 36 135, Decemi)er 1949. 84. R. M. Chaurnrma, J. N. Howard, and V. A. Miller, Atmospheric Transmission of Infrarel (Report No. 18 on Contract W-44-099-eng-4J00, Ohio State University Research Foundation, Columbus, 30 June 1949. 85. R. M. Chapman, J. N. Howard, and V. A. Miller, Atmospheric Transmission of Infrared (Report No. 20 on Contract W-44-099-er.g-40T,0Ohio State University Research Foundation, Columbus, 15 October 1949. 86. R. M. Chapman, J. N. Howard, and V. A. Miller, Atmospheric Transmission of Infrared (Report No. 21 on Contract W-44-099-eng-400), Ohio State University Research Foundation, Columbus, 15 December 1949. 87. R. M. Chapman, J. N. Howard,: —d V- A. Miller, "The Pressure Dependence of the Absorption by Entire Bands of Water Vapor in the Near Infrared," J. OpC. Soc. Am., Vol. 42, 1952, p. 423. 88. H. Yates, Total Transmission of the Atmosphere in the Njear-Infrared, NRL Report No. 3858, Naval Research tLboratoiy,;'c,>'iii.t;;0, i, 1 i moer 151. 274

-- -- WILLOW RUN LABORATORIES 89. E, 0. Hulbert, Atmospheric TrPnsmission of Infrared Radlarlon, NRL Report No. 234b, Naval Research Laboratory, Washington, 22 July l544. 90. Handbook of Geophysics and Space Environments, Air Force Cambridge Research Laboratories, Bedford, Mass., 1965. 91. C. D. Keeling, "The Concentration and Isotopic Abundance of Ca.rbon Dic 4c in the Atmosphere," Tellus, Vol. 12, 1960. 92. G. S. Callendar, "On the Amount of Carbon Dioxide in tb.,^mosphere," Tellus, Vol. 10, 1958, pp. 243-248. 93. E. G. Glueckauff, "CO2 Con'ent of the Atmosphere," Nature, Vol. 153, 1944, pp. 620-621. 94. S. Fonsellous, F. Koroleff, and K. Burch, "Microdeterinination of CO2 in the Air with Current Data for Scandinavia," Tellus, Vol. 7, 1955, pp. 258-265. 95. J. R. Bray, "Ai Analysis of the Possible Recent Change in Atmospheric Carbon Dioxide," Tellus, Vol. 11, 1959, pp. 2.20-230. 96. G. S. Callendar, "The Effect of Fuel Combustion on the Amount of Carbon Dioxide in the Atmosphere," Tellus, Vol. 9, 1957, pp. 421-422. 97. A. Adel, "Equivalent Thickness of the Atmospheric Nitrous Oxide Layer," PYys. Rev., Vol. 59, 1i41, p. 944. 98. L. Goldberg and I'. A-. Mller, "The Vertical Distribution of Nitrous Oxide and Methane," J. Opt. Soc. Am., Vol. 43, 1953, p. 1033. 99. R. M. Goody and C. D. Walshaw, "The Origin of Atmospheric Nitrous Oxide," Quart. J. Roy. Meteorol. Soc., Vol. 79, 19-'6, p. 496. 100. R. McMath and L. Goldberg, "The Abundance and Tenperature of viethane in the Earth's Atmosphere," Proc. Am. Pny~. Soc., VilA, 4, 1948, p. 623. 101. R. L. Slobod and M. E. Krogh, "Nitrous Oxia? as a Constituent of the Atmosphere," J. Am. Chem. Soc., Vol. 72, 1950, p.). 1175-1177. 102. J. W. Birkelandd, D. E Burch, and J. Ii. Shaw, "Some Comments of Two Articles by Taylor and Yates," J. Opt. Soc. Am., Vol. 4', 1957, p. 441. 103. J. W. Birkeland,'Determination of Ground LevtIl N'0," M.S. thesis, Ohio State University, Columbus,'1957. 104. A. L. Bowmar, A Determination of the Abundance of Nitrous Oxide, Carbon Monoxide and Methane in Gr )u'.- Level Air at Several Locations near Columbos, Ohio (Scientific Report No. 1.tract A} 19(604)-2259), Ohio State University, olumbus, 1959. 105. W. E. Groth and H. Schlerholz, "Photochemical Formation of Nitrous Oxide," J. Chem. Phys., Vol. 27, 1957, p. 973. 106.. D. R. Bates and A. E. Witherspoon, "The Photochemistry of Some Minor. Constituents of the Earth's Atmosphere (CO2, CO, CH4, N20)," Monthly.Notices Roy. Astron. Soc., Vol. 112, Nc. 1, pp. 101-124. 107. M. V. Migeotte,'The Fundamental Band of Carbon Monoxide at 4.7;A in the Solar Spectrum," Phys. Rev., Vol. 75, 1940, p. 1108. 275

--— WILLOW RUN LABORATORIES —--------- 108. J. H. Shaw and J. N. Howard, "A Quantitative Determination of the Abundance of Telluric CO Above bolumbus, Ohio," Phys. Rev., Vol. 87, 195., p. 380t. 109. J. H. Shaw and N. Howard, "Absorption of Telluric CO in the 23 pL Region," Phys Rev., Vol. 87, 1952, p. 679. 11(. J. H. Shaw and H. F. Nielson, Infrared udies of the Atmosphere (Final Report on Contract AF 19(122)-65), Ohio State University Research Foundation, Columbus, 1954. 111. J. H. Shaw, The Abundance of Atmospheric CO above Columbus, Ohio (Contract AF 19(c04)-1003). Report No. AFCRC TN 57-212, Ohio State University Researcn Foundation, Columbus, 1957. 112. J. L. Locke and L. Herzberg, "The Absorption due to Carbon Moncxide in the Infrared Solar Spectrum," Can. J. Phys, Vol. 31, 1953, p. 504. 113. W. Benesch, M. V. Migeotte, and L. Neven, "Investigations of Atmospheric CO at the Jungfraujoch," J. Opt. Soc. Am.. Vol. 43, 1953, p. 1119. 114. A. Adel, "Identification of Carbon Monoxide in the Atmosphere above Flagstaff, Arizona," Atrophys. J, Vol. 116, 1952, p. 442. 115. L. Goldberg, "The Abundance and Vertical Distribution of Methane in the -- Earth's Atmosphere," Astrophys.J., Vol. 113, 1951, p. 567. 116. L. Elterman, "Comparison of 03 Concentrations," Appl. Opt., Vol. 3, 1964, p. 641. 117. I. London, K. Ooyama, and C. Prabhakara, Mesosphere Dynamics, Geophysics Research Directorate, Air Force Cambridge Research Laboratories, Bedford, Mass., May 1962. 118. Ozone Observations over North America, ed. by W. S. Hering and T. R. Borden, Jr., Environmental RLsearch Paper No. 38, Report No. AFCRL-64-30(II), Air Force Cambridge Research Laboratories, Bedford, Mass., July 1964. 119. T. J. Kowall, Atmospheric Infrared Attenuation Coefficient (AIRAC) Studies (Final Report). Report No. LMSC-805146, R/C N1-175 (65-1), Lockheed Missiles and Space Co., Sunnyvale, Calif., 1 June 1965, 120. B. A. Bannon and L. P. Steele, Average Water-Vapor Content of the Air, Geophysical Memoirs No. 102, Meteorological Office, Air Ministry, England, 1960. 121. E. W. Barrett, E. A. Herndon, and H. J. Carter, "Some Measurements of the Distribution of Water Vapor in the Stratosphere," Tellus, Vol. 2, No. 4, 1950, pp. 302-311..122. F. R. Barclay et al., "A Direct Measurement of the Humidity in the Stratosphere Using a Cooled Vapor fTrap," Qu.rt. J. Roy. Meteorol. Soc, Vol. 86, No. 368, 1960, pp. 259-264. 123. F. Brown et al., "Measurenments of Water Vapor, Tritium, and Carbon-14 Content of the Mhldle Stratocphere over Southern England," Tellus, Vol. 13, No. 3, 1961, pp. 407-416. 124. J. A. Brown and E. G. Py~,us, Stratospheric Water Vapor Measurements by Means of a Dew Point Hygrometer, Baliistic Research Laboratories, Aberdeen Proving Ground, Md., 1960. 276

- WILLOW RUN LABORATORIES125. H. J. Mastenbrook and J. E. Dinger, Measurement of Water Vapor Dtstribution in the tratushere, Report No. 5551, Naval Research Laboratory,,ashington, 1960. 126. H. J. Mastenbrook and J. E. Dinger,'Distribution of Water Vapor in the Stratosphere," J. Geophys. Res., Vol. 66, No. 5, 1961, pp. 1437-1444. 127. D. C. Murcray, F. H. Mtrcray, and'V. J. Williams, Distribution of Water ap in the Stratosphere As Determined from Infrared Absorption Measurements (Scientific Rer.rtt No. I on Contract AF 19(604)-7429, University of Denver, Denver, 1961. 128. D. G. Murcray, F. H. Murcray, and W. J. Williams, "Further Data Concerning the Distribution of Water Vapor in the Stratosphere," Quart. J. Roy. Meteorol. Soc., Vol. 92, 1966, p. 391. 129. S. Steinberg and S. F. Rohrbough, The Collection and Measurement of Carbon Dioxide and Water Vapor in the Ipper Atmosphere, Research Department, Electronics Group, General Mills, Inc., Minneapolis, 1961..30. G. B. Tucker, An Analysis of Humidity Measurements in the Upper Troposphere.nd Lower Stratosphere over Southern Englard, Report No. M.R.P. 1052, Meteorological Office, Air Ministry, England, 1957. 131. E. W. Brewer, Ozone Concentration Measurements from an Aircraft in N. Norway, Report Nu. M.R.P. 946, Meteorological Office, Air Ministry, Engiand, 1955. 132. N. J. Kerley, "High-Altitude Observations between the United Kingdom and Nairobi," Meteorol. Mag., Vol. 90, No. 1062, 1961, pp. 3-17. 133. Japanese Meteorological Agency, IGY Data on Upper Air (Radiosonde) Observations during World Meteorologic?.1 Inter,'als, March 1960. 134. G. E. Oppel and I. A. Pearson, Infrared M3del Atmospheres, Lockheed Missile and Space Co., Sunnyvale, Calif., 1963. 135. M. Gutnick, Mean Moisture Profiles to 31 m for Middle Latitudes, Interim Notes on Atriospheric Prop rties No. 22, Geophysical Research Directorate, Air Force Cambridge Hesearch Laboratories, Waltham, Mass., 1962. 136. G. Lindquist, "A Water Vapor Profile from the CARDI Solar Spectra" (UNCLASSIFIED), in Semia nual Report of the Ballistic Missile Radiation Analysis Center (U) (1 July tnrough 31 December f964y Volume I1 General Reiew, Report No7. 4613-83-P(I), Institute of Science and Technology, The Univcrsity of Michigan, Ann Arbor, March 1965, pp. 23-47 (SECRET). 137. S. T. Marks, Summary Report on BR.-IGY Activities, BRL Report No. 11C4, Ballistic Research Laboratories, Aberdeen Proving Ground, Md., 1960. 138. D. G. Murcray, University of Denver, private correspondence, September 1966. 139. J. H. Tayler and H. W. Yates, J. Oot. Soc. Am., 1957, Vol. 47, No. 3. 140. C. Cumming et al., Quantitative Atlas of Infrared Stratospheric Transmission in the 2.7 I Region, Report No.'IR 546/65, Canadian Armament Research and Development Establishment, Valcarticr, Quebec, Canaaa. 277

-I- -'L t LOW R UN LABORATORIES ------- 141. C. P. Farmer, P. J. Perry,.nd D. B. Lloyd, Atmospherin lra^,,:inisson Measurements in the 3.5 to 5.5 j Band at 5200 Meters AltitcJe, Ri;ort No. DMli-a78, EMI Electronics, Ltd. London, England, AD 420 215. 1963. 142. G. E. Berlinquetite,.nd P. A. Tate, Some Short Range Narro* Beam Atmospheric Transmission Measurements in the Near Infrared, Report No. 420, Defense Research Board, Valcartier, Quebec, Canada, December 1963. 143. D. Markle, Project Lookout II Stratosphlric Infrared Transm'ssion from Airtorne Solar pectra, Part I, Re{ort No. CARDE T.M. 708/62, Defense Research Board, Valcartier, Quebec, Canada, August 1962. 144. D. Murcray, F. M'r.cray, and W. Williams, Variation of the Infrared Solar' Spectra between 2800 cmr1 with Altitude, Scientific Report No. 2, University of Denver, Denver, AD 414 638, July 1963. 145. D. G. Murcr.;y, Infrared Atnospheric Transmittance and Flux Measurements (Six-Month Technical Report No. 6), University of Denver, Denver, AD 416 813, July 1963. 146. D. G. Murcray, Infrared Atmospheric Transmittance and Flux Measurements, University cf Denver, Denver, AFCPL, January 1964. 147. D. G. Murcray, F. H. M',rcray, and W. Williams, "Comparison of.xperimental and Theoretical Slant Path Absorptions in'he Region from 1400-2590 cm-l," J. Opt. Soc. Am., Vol. 55, N-.; 10, October 1965. 148. D. G. Murcray, F. H. Murcray, and W. J. Williams, Va;.iation with Alti(ude of the Transmittance of the Earth's Atmosphere with Grating Resolution, Report No. AFCRL-65-854, University of Denver, Denver, November l965. 149. T. G. Kle et al., Absdrntion of Solar Radiation by Atmospheric CO2, Report No. AFCRL-65-290, University of Oenver, Denver, April 1965. 150. L L. Abes, A Study o the Total Absorption near 4.5 by Two Samples of N20, as Their Total Pressure and N20 Concentrations were Independently Varied, Scientific Report No. 3, AFCRL-62-236, Ohio State University, Columnbus, January 1962. 151. B. Edien, "The Dispersion of Standard Air," J. Opt. Soc. Am., Vol. 43, 1953, pp. 339-344. 152. R. Penndorf, "Tables of the Refractive Index fur Standard Air and Rayieigh Scattering Coefficients for the Spectral Region Between 0.2 and 20 Microns and Their Application to Atmospheric Optics," J. Opt. Soc. Am., Vol. 47, 1957, pp. 176-185. 278

Security Classification, DOCUMENT CONTROL DATA R&D ($e4Uft? c>Ieeilfction ot tilft. body -of abbertct end tnrde*rmgl *mnrt*on mul 4be ntered wAen the oewvrf report to cteie'fIed) - OItCNINAtING ACTITTY (Corporete eutho)l 2-.'CPORT IKCUtV T C L.ArSIVICATION Willow Run Laboratories, Institute of Science and Technology, The Unclastfied University of Michigan, Ann Arbor, Michigan t b G.ou 3 t -PORT TITLC "'. BAND-MODEL METHODS FOR COMPUTING ATMOSPHERIC SLANT-PATH MOLECULAR ABSORPTION 4 DESCRIPTIVE NOtES (Type of report end Inclusive deri)c IRA State-of-the-Art Report S AUTHNGO(S (Leel nrme. Irast nom". inti^el)' Anding, David * REPOPT OATC?a?O %L NO op PAce~ bNO r 0' bnErv _ 1967._ — ___ -— __ _ XtV+ 278 152 t*.COENTRACI OR1 rt AN tNO. 9 ORIGINAlTOR'S 0tt RORT NUIIEt (S) NOnr 1224(52) 7142-21-T b PROJC T NO:. C Sb OTit Pt P^ONT NO(S) (Any oather numbto thae may be 6s~iend this report~ Jr.. -. *: 0., 1. AVA IL AB Il /MtT AtON NOTriCs -& - This document is subject to speclal export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of the office cf Naval Research (Code 421), Washington, D, C. 20360. I1 SU PLtEMNTARY NOTES t SPONSOINWG MILITARY ACTIVITY Office of Naval Research, Physics Branch Washington, D. C. 13 ABSTRACT The general transmissivity equation for computing slant-path molecular absorption spectra is developed and two methods for evaluating this equation, the direct integration and that which assumes a model of the band structure, are discussed. Five band models are discussed and twelve methods for computing molecular absorption based on these band models are presented. Spectra computed by bandmodel methods are compare' with spectra calculated by direct integration of the general transmiss:lvity equation and with open-air field measurements of absorption spectra. Conclusions concerning the capability of band-model methods for predicting slant-path absorption spect-* are stated and recommendations for future research are outlined. A summary of open-air field nm' jrements; of absorption spectra and laboratory measurements of absorption spectra for homogeneous paths is prsented and a computer program for computing the equivalent sea-level path, the Curtis^Godson equivalent pressure, and the absorber concentration for atmospheric slant paths for any model atmosphere is given in appendix I. DSty JAN 414 7 Stuy Classification C —cu, ty ClasAsification

,ecurity Classificatimt. __ cLINK A LINK 6 LINK C K-Y WOtOIS - l. A oL: WvT OL. Wv ~s. a. Atmospheric transmission Molecular Absorption Radiation Attenuation Spectra -.':' -. INSTRUCTIONS l1 ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements of the contractor, subcontractor, grantee, Department of De- such as: f(nset activity or other organizatiotn t:crportte author) issuing ( liiedreqesters may obtain copies of his t~e report. t^r tepQort. -. report Irom DDC."'2a. REPORT SECUIZ'Y CLASSlFICATION: Enter the oversll security classtifiction of the report. tIudicate whethert u r re xprt by DDC is not authtorized.* "Restricted Data" is included. Marking is to be in accord. since with appropriate security regulations. (3) "*U. S. Government agencies may obtain copies of this report directly from [DC. Other qualified DL C 2b. GROUP; Automatic downcading is specified in DoD u s hall request through rective 5200. 10 and Armed Forces Industrial Manual.'Enter -u the group number. Also. when applicable, show that optional' markings bave been used for Group 3 and Group 4 at author- (4) U. S military agencies.my obtain copies v this ited.!~i~~~ z**t~~~ed. -Jc~ nreport directly from DDC Other qualified users 3. REPORT TITLE: Enter the complete report title in all shall request through capital lettetr Titles in all cases should be unclassiflrd. If a meaningful title cannot be selected without classafica- tion, show title classification in all capitals in parenthesis (5) "All distribution of thi. report is controlled. Qualimmediately foltowing the title. ified DDC users shall request through 4. DESCRIPTIVE NOTES: If appropriate, enter the type of..... report- e.g., interim, ptroireass. summay, annual, or final. If the repor has been frnished to the Office of Technical Give the inclusive dates when a specific reporting period is Services. Departnent uf Commerce, for sale to the pubic, idicovered. cate this fact and enter the price, if known, 5. AUTHORIS): Enter the namee(s) of aae'cor(s) as shonn on t1. SUPPLEMENTARY NOTFS: lUse for additonal exptlanor in the report. Entet last name. first nratle, nridile initial, to note. If -r.iitary, show rank and. branch of service. The riame of the principal.,thor is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory rponsoring (par 6. REPORT DATI. Enter the date of the report as day, fo the'eserc) and developrent. Inctude address month. year, or month, year. If more )han cne date appears on the report, use date of publication. 13 ABSTRACT: Elntr an abstract givin, a brief and factual 7^. TOTAL NtUMBER OF PAGF5: The tot sumimary of the document Indicatite of the report, ever. though Ur it may also appear elsewhere in the body of the technical reshould follow normal paginatiup procedures, i.e. enter the port. If additional space is required, a contin.uattiu- sheet shall number of pages containing itnortation. - attach o 7b. NUMBER OF REFERENCES; Enter the total number of It is hghly desirable that the abitract of classifed reports references cited in the report. be unclassified. Each paragraph of:he abstract shali nd with 8. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the ltitary security classsificaton.cf the inthe applicable number f the contract orsgrint under which formation in the paragraph, represented as frS). (Si. rc). oa (U) tht repon was written.. There is no lhmistatin on th.e lengh of the abst.,ct. How. 86, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from ISO to 225 woeds. military depwrtment identification, suCh as project number,: Ky o a bpeet number. ytem number~ task nnber, etrc. 14. KEY WORDtS: K.ey words re technically mentina. te rrn si. tnumbersytem task b er etc or short phrases that characterize a report and may b.sed as 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offt* inde- entriest for rataloging the report. Key words must be cial rteport number by which the document will be identified selected so that no security classification is required. Identi. and controlled by the originating activity.'This number must tiers, such as equipment model designation, trade name, nilitary be u:niuue to this report. project code name, geographic location. may be used as key 9b. OTllEni REPORT1 NUMBER(CS): If the report has been words hut will be followed by an indication of technical cor.arsaigned any o.;er repcrt numbers (either by the orimnator text. The assignmnent of links, rules, and weights is optional. or by the sponsor,, also enter this number(s). t. AVAI AP?!I IT Vrt/!TATIOO HTt E-,tr any In — Ittions on further dissemination of the report, other than those *.S. covs N.R:NTr PRINTliN orc- tcr. o. -4.t08 Security Classific.adion

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