01 t023-2-T THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING High Altitude Engineering Laboratory Departments of Aerospace Engineering Atmospheric and Oceanic Science Technical Report FEASIBILITY OF SATELLITE MEASUREMENT OF STRATOSPHERIC MINOR CONSTITUENTS BY'-SOLAR OCCULTATION... all.. raju S.d Drayt son F. 'I. Bartman W. R.< Kuhn R.'' Tall amraju ORA Project 011023 under contract with: NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION GRANT NG-10-72 administered through OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1973

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ABSTRACT The determination of stratospheric concentration of minor constituents by satellite solar occultation is examined. The method is shown feasible for ozone up to 50 km, water vapor up to 50 km, nitrous oxide up to 30 km, methane up to 50 km and carbon monoxide up to 20 km. Transmittance calculations for these and other gasses are presented for optimal spectral regions. Calculations of extinction by aerosols in the lower stratosphere show a dominant effect in the window regions near lOim. Several inversion techniques are developed and examples of profiles retrieved by different methods are compared. Computer programs are described to calculate the transmittances by the use of a band model and by the line-by-line integration technique. iii 111

TABLE OF CONTENTS Page ABSTRACT iii 1. INTRODUCTION REFERENCES 3 2. STRATOSPHERIC ABSORPTION BY MOLECULAR CONSTITUENTS. 4 2.1 Introduction 4 2.2 Band Model 4 2. 3 Ozone 7 2.4 Water Vapor 29 2.5 Methane 35 2. 6 Carbon Monoxide 38 2. 7 Nitrous Oxide N20 40 REFERENCES 43 3. EXTINCTION OF INFRARED SOLAR RADIATION BY AEROSOLS ON A TANGENT PATH THROUGH THE STRATOSPHERE. 44 3.1 Introduction 44 3.2 The Aerosol Model 44 3. 3 Optical Characteristics of Aqueous H2S04 48 3.4 Infrared TransmLssivities Along Tangent Paths 57 3. 5 The Extinction of Infrared Solar Radiation 57 3. 6 Additional Calculations and Improvement of the Model 62 REFERENCES 64 v

TABLE OF CONTENTS (co(ncltlded) Page 4. INVERSION PROCEDURES 65 4. 1 Introduction 65 4.2 Geometry and Technique 67 4. 3 Methods of Inversion 68 4.4 Results and Discussions 74 REFERENCES 86 5. DISCUSSION AND CALCULATIONS 89 APPENDIX 92 vi

Chapter 1. Introduction The purpose of the work performed under this grant is to examine the feasibility of determining stratospheric concentrations of minor constituents and/or pollutants from satellite measurements of infrared absorption during solar occultation. Before such a task can be undertaken it is important to have some estimates of typical distributions of the constituents either from actual measurements or from theoretical models involving chemistry, photochemistry, diffusion etc. It is also necessary to have the capability of calculating the infrared spectral absorption along a tangent path through the atmosphere once the distribution of the atmospheric constituents has been specified, i. e., we require to know the spectral line parameters for each constituent absorbing in a given spectral region. Our earlier report (Drayson, et al., 1972) contained a survey of the stratospheric distribution and the spectral properties of the minor constituents. For several molecules the optical masses along a tangent path were calculated for several tangent altitudes and for some molecules the absorptions along the tangent paths were roughly estimated from available laboratory data or approximate calculations. The report also considered stratospheric aerosols whose distributions are of considerable interest and which also may interfer with the absorption by molecular constituents. In this report we describe the progress made in the continuation of the investigation. In Chapter 2 we examine more closely the absorption by stratospheric molecules with the main emphasis on the more abundant of the minor constituents, for two reasons: i) The absorption by the more abundant minor constituents 1

is large even averaged over spectral intervals several wavenumbers wide, so that a comparitively simple instrument of medium spectral resolution could be employed for satellite measurement. ii) The spectral absorption properties of these molecules are known comparitively well. In particular several of them appear on the magnetic tape of line parameters compiled by AFCRC (McClatchey et al., 1973). Transmittance calculations have been made either by the use of a band model or by a line-by-line integration computer program (described in the Appendix). In general we have not attempted to determine the accuracy of the spectral line parameters, although an exception was made in the case of ozone. The extinction of solar radiation by stratospheric aerosols is examined in Chapter 3. The aerosols are important for two reasons: i) They may interfer with a sounding of a molecular constituent so that an accurate estimate of the extinction by the aerosol alone is required to determine the vertical profile of the molecular component. ii) The distribution of dust and aerosols in the stratosphere is known to be a factor that influences the climate at the surface and a continuous monitoring would aid our understanding of this problem as well as processes within the stratosphere itself. In Chapter 4 we consider the inversion problem, i. e. how to determine the vertical profile of the constituents from the occultation measurements. In the absense of noise the problem is compartively simple but in a realistic situation some smoothing constraints must be applied to prevent domination of the solution by the noise present. Several different techniques are examined and compared. 2

REFERENCES Drayson, S. R., F. L. Bartman, W. R. Kuhn and R. Tallanraja (1972), Satellite Measurement of Stlratospheric Pollutants and Minor Constituents by Solar Occultation: A Preliminary Report, University of Michigan Technical Report 011023-1-T. McClatchey, R. A., W. S. Benedict, S. A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. So Rothman and Jo S. Garing (1973), AFCRL Atmospheric Absorption Line Parameters Compilation, AFCRL Environmental Research Paper No. 434. 3

Chapter 2. Stratospheric Absorption by Molecular Constituents. 2. 1 Introduction The previous report dealt primarily with 1) the distributions of stratospheric minor constituents, 2) the locations of and available data on the infrared bands, and 3) a crude estimate of the maximum absorption one might expect from these various bands. These latter results were only very approximate, since they were obtained from absorption profiles given in McCaa and Shaw (1967) for the ozone bands, and in Burch et al. (1962) for other molecules, and the mass paths and pressures which were used generally did not correspond to stratospheric conditions. In addition, the measurements were all made at room temperature, and the absorption, especially in the wings of the bands, may depend strongly on temperature. The emphasis of the present investigation has been to construct transmission curves vs. height for stratospheric tangent paths corresponding to extreme or typical distributions of the minor constituents. These transmissions correspond to small wavenumber intervals (generally 1cm ) and were chosen to represent as nearly as possible the maximum absorption in each of the bands. A band model or direct line by line integration has been used to represent the absorption and the validity of the model and/or the band parameters have been compared with experimental results. Finally, the feasibility of using the various bands for tangentpath remote soundings is discussed. 2. 2 Band Model The band model is essentially a random model, i. e., the spectral lines are assumed to be randomly distributed. In addition, 4

the actual number of lines is used which is a feature of the quasi-random model developed by Wyatt et al. (1962). The lines are grouped into intensity intervals, and all lines in the intervals are given a strength equal to the average of the strengths of lines within that interval. The magnitude of the intensity interval is variable. For those spectral regions for which the lines have very similar intensities, the variation in intensity is chosen small, while for the intervals whose line intensities vary over many orders of magnitude, the lines are grouped into intensity decades. The transmission T. is given as, 1 T T 7 th Ti.f e p (-KPP^))4} (2. 1) where ALi is the width of the spectral interval over which the transmission is determined; R and nK are the mean line strength and number of lines in the intensity interval, P. is the profile function, and u is the mass path of the absorbing gas. The profile function which must be used for stratospheric tangent path studies is the Voigt profile, since both collision and Doppler broadening contribute to the line profile in this height range. This profile is given by, P= r 4 ) 2(2. 2) where a=o(,l/oc, with s<t and ot, being the collision and Doppler half-widths respectively. The collision half width is given for standard 5

temperature and pressure (t, P ) and is found for any other temperature O 0 and pressure by the relation ( L = <o( P It i/ P -F. The Doppler half width is calculated from 0( = (2ktln 2 / mc2) 1/2, where k is Boltzmann's constant, m is the molecular mass, and c the speed of light. w is actually line center but we assume a value equal to the center of the spectral interval; i. e., all lines in the interval are given the same Doppler half-width. This is also true for the collision halfwidth. The parameter f(x) = Ai4lnT 2- x/2CD, where x is essentially a measure of distance from line center given by x = 2 (o - o ) / A i If equations (2. 1) and (2. 2) are combined, the transmission for a single line (represented by the quantity in brackets in (2. 1), is, Tk = fexp[ — O p ( ))]dx (2.3) 0 Unfortunately equation (2. 3) has no analytical solution, and must be evaluated numerically. Accordingly, a three dimensional table was prepared for which the parameters and their ranges are, -6 ( log ( Sku /l w) 6 -7 4 log a)4 3 2 4 log(AD /c4J4 6 The first two are evaluated at intervals of 0. 1, while the interval for the latter is 1. The actual transmission for a given set of parameters is found by bilinear interpolation on the table. In order to reduce the computing time, yet maintain the necessary accuracy for this feasibility study, we have excluded the contribution to the absorption from spectral lines outside the interval of interest; this is equivalent to requiring 6

that the total line absorptivity fall within the specified spectral interval. Thus log A l/Ci was taken as 6 and the spectral interval adjusted accordingly. Uncertainties in the calculated transmission functions can occur from inaccurate line parameters from which the model is constructed, or from the actual model itself, i. e., the lines may not be randomly distributed. One would not expect the latter to be a major problem for the non-linear molecules since their vibration-rotation spectra are quite complex. As an example, Fig. 2. 1 compares a band model calculation with an "exact line by line" calculation averaged over one-tenth wavenumber intervals. While there is some "clustering" of the lines (at, e. g. 1124. 3 cm ) the band model represents an average transmission quite well. The averaging interval for the band model is also not critical as the average of the 0. 5 cm intervals gives nearly the same transmission as the average of the 1 cm- intervals. The line parameters used for both the band model and line by line integration are from McClatchey et al. (1973). A magnetic tape containing these data was graciously provided. This tape containing the most recently published tabulations provides line strengths, positions half widths, and ground state energies for selected bands of ozone, water vapor) carbon monoxide and dioxide, nitrous oxide, methane and oxygen. Approximately 110, 000 lines are included for wavelengths longer than 1 micrometer. 2. 3 Ozone Ozone bands and the corresponding spectral intervals chosen for the stratospheric tangent path calculations are given in Table 2. 1. 7

100 4 -C z 0 U) (L' n2 () CO 90L 70 - — r- EXACT 60 - n --- 0.5 cm' 1 E BAND MODEL --- I.0 cm' -1 PRESSURE = 2.0 mb MASS PATH= 1.3 atm cm 50 I I I I 1123 1124 1125 1126 WAVENUMBER (cm- ) Fig. 2. 1 Comparison of exact calculation and band model.

These intervals were chosen so that they give maximum or near maximum absorption for each band. McCaa and Shaw's (1967) absorption profiles were used to locate approximately these I'et ions,:1 the firll identification utilized the line positions and strengths as given by McClatchey et. al. (1973). It should be noted that the present analysis does not include three combination bands (1728, 1792 and 3181 cm ) discussed in the previous report. Their line parameters are not given by McClatchey et al., and we are not aware of any line listings for these bands. The 3181 cm ( 2 4J + V3 ) band is not important for this study since our earlier work indicated that even with the maximum realistic ozone distribution, maximum absorption within the band as estimated from the McCaa and Shaw data is only 9% at 24 km. However, the 1728 cm1 (<V +V3 ) band gives a comparable absorption but with the minimum ozone amount. The 1792 cm (S1 +T, ) band is intermediate to these two. Fortunately, it will be shown that the bands given in Table 2. 1 provide the necessary absorption throughout the stratosphere, and it is not necessary to consider these additional bands. Band |......3 |. +3. +1' 3.... 3 1 2 3 Wavenumber (cm 1) 1124.5 717 1025 2131 2795 30, Table 2.1 Centers of the 3 cm spectral intervals of tangent path calculations ozone bands selected for the stratospheric The model atmosphere from which the tangent path absorptivities were determined is fully explained in the previous report. The USSA (1962) was assumed and ozone distributions corresponding to the largest and smallest amounts one might expect to observe were constructed, These extreme ozone mass paths for these heights as well 9

as the tangent path pressures and temperatures are given in Table 2. 2. The latter were evaluated from a Curtis Godson approximation, i. e., they were weighted by the ozone mass path. Tangent Height jPressure Temperature Mass Path (km) (mb) (OK) (atm - cm) 12 88. 5- 110. 218 45. - 7. 1 20 37.9 - 37.7 220 24. 7- 5 6 30 8. 3 - 8.4 232 6. 8 - 1. 2 40 2.0 - 1.6 255 1. 6 - 0. 28 50.56 -.56 266 0. 43 - 0. 074 _ Table 2. 2 Curtis-Godson pressures and temperatures for selected tangent heights for extreme ozone amounts. The temperatures are the same for both extreme ozone mass paths. 1123 - 1126 cm1 ( 1 band) The 1123 - 1126 cm1 spectral interval contains 63 lines of the V1 band in the McClatchey et al. data. Line strengths range over four orders of magnitude at room temperature. A collision half width of 0. 110 cm is given. A comparision between the band model and experimental profiles of McCaa and Shaw (1967) is shown in Table 2. 3. Note that the transmissions, when averaged over a 3 cm interval which is Pressure (mb) 20 20 67 400 Mass Path (atm-cm) 1.3 2. 9 4. 2 12. 7 1123 - 1124.92.87.75.38 1124 - 1125.85.78. 53.23 1125 - 1126.90.86.76.31 Average.89 (.88).84(.83).68(.70).31(.29) Experimental.92.88 72. 3 4 (McCaa and Shaw) ____ __ Table 2. 3 A comparison of theoretical (band model) and experimental (McCaa and Shaw, 1967) transmissivities for the 1123 -1126 cm-l spectral interval. The transmissions shown. in parentheses refer to an averaging interval of 0. 5 cm rather than 1 cmr 10

E 1124-1125 T 302-1 1123-11241125-1126 I ~20 113-1123-1124 1125-1126 1124-1125 10 I I I I I I.. 0 0.2 0.4 0.6 0.8 1.0 TRANSMISSION Fig. 2. 2 Stratospheric tangent path transmissivities for the spectral region 1123-1126 cm- for extreme ozone mass paths.

close to the spectral resolution of McCaa and Shaw (1967), agree to within four percent of the experimental results. This accuracy is certainly adequate to estimate the feasibility of using this band for remote soundings. The importance of knowing the spectral resolution for these rather small intervals when comparing with experimental data is important, e.g. the absorption is larger in the 1124-1125 cm 1 interval than the adjacent intervals by as much as 19% in the one case. The band model calculation does not depend strongly on the width of the spectral subintervals over which the individual transmissions are calculated. For example, if a 0. 5 cm rather than a 1 cm interval is used, the average transmission is within 0. 02 of the 1 cm subinterval (shown in parentheses in Table 2. 3). The stratospheric tangent path transmissivities for the three 1 cm spectral intervals for extreme ozone amounts (Table 2. 1) are shown in Fig. 2. 2. If the spectral resolution is 3 cm or less, and if enough energy is available, then this V1 band in the spectral region near 1124. 5 cm should be adequate for tangent path remote soundings, at least to the midstratosphere. Of course, a larger spectral resolution would increase the transmission and lower the effective height. 715.5 - 718. 5 cm 1 ( - V band) 2 One would generally not consider the V2 band for remote sensing of ozone because of its strong overlap with the V2 band of carbon dioxide, centered about 667 cm. The integrated band strength is only about 18 atmcCaa and Shaw 1967) while the 2 band is only about 18 atm cm (McCaa and Shaw, 1967) while the V band 12 12

-1 -2 of carbon dioxide is about 214 atm cm (Drayson, 1973). Nevertheless, a comparison of the band model transmissions with the spectra of McCaa and Shaw (1967) was made for completeness. The spectral region of maximum absorption was taken as 715. 5 to 718. 5 cm. That this band is highly structured can be observed from both the experimental profiles of McCaa and Shaw and the data of McClatchey et al. In Table 2. 4 is given the number of lines for each 1 cm- interval as well as the mean line strengths for each intensity subinterval; these distributions were used in the band model calculations. 715. 5 - 716. 5 cm-1 716. 5 - 717. 5 cm-1 717. 5 - 718. 5cm.0327 (5).0254 (12).0163 (7) 0089 (9). 0085 (14).0035 (6).0020 (1). 0018 (4). 0010 (1) -1 -2 Table 2.4 Mean line strengths (atm -cm ) and numbers of lines (parentheses) used in the band model study for the 715. 5 to 718.5 cm1 spectral region of the -2 band of ozone. A comparison of the band model transmissivities with the McCaa and Shaw absorption profiles is given in Table 2. 5. The obvious clustering of the lines is apparent from the rather large variation in transmissions among the three spectral intervals. The average of these results gives a transmission about ten percent smaller than the experimental spectra indicate. However, because of the clustering, a small discrepancy in the location of band center or a spectral interval somewhat larger than 3 cm would increase the band model transmissions. We have not analyzed further this difference because, as explained previously, one would not choose this band for remote soundings. 13

Pressure (mb) 20 66.6 66. 6 533 533 Mass Path (atm-cm). 84 4. 20 2. 0 9. 4 3. 0 715. 5 - 716. 5cm.832.599.709. 164.508 716.5 - 717. 5cm-1.708.379.525.039.294 717.5 - 718. 5cm-1.896.703.771.329.661 Average.81.56.67.18.49 Experimental. 90. 67. 77. 27.59 (McCaa and Shaw) Table 2. 5 Comparison of band model transmissions with experimental spectra of McCaa and Shaw for the 715..5 to 718. 5 cm spectral interval of the V z ozone band. Tangent path transmissivities for only V band ozone lines 2 in the interval of 715. 5 to 718. 5 cm are shown in Fig. 2. 3. If one were to include the very strong absorption by carbon dioxide, the transmission values would be much reduced, and it is likely that only in the upper atmosphere, if at all, would the transmission be large enough for remote sounding studies. 1023.5 - 1026. 5 cm-1 ( j band) More studies have been devoted to the.3 and VA bands than any of the other ozone vibration bands. The spectra are complex and analysis is difficult because of the interaction of these bands. The line parameters of McClatchey et al. are the latest which appear in the open literature. These parameters are primarily from the work of Clough and Kneizys (1965, 1966) although some hot bands and isotopic bands have been added. Aida (1973) has also calculated line parameters but unfortunately the ground state energies are not available so that the line strengths cannot be calculated for arbitrary temperatures. The total number of lines which appears in his tabulation is about 10, 000. 14

-- Maximum ozone mass path Minimum ozone mass path U 30 1 716.5.5 717t / E 0" I 0715.51-716.5 -71.5 717.5-718.5 15^.5- 716.5-717.5 20 10 LI I I715. -716.5 0 0.2 0.4 0.6 0.8 1.0 TRANSMISSION Fig. 2. 3 Stratospheric tangent path transmissivities for the'2 band ozone lines in the spectral region of 715. 5 - 718. 5 cm-1.

Although much effort has been devoted to an analysis of the V1 and.3 bands, discrepancies still exist among the calculated line parameters. The half width as given by McClatchey et al. is 0. 11 cm (from Lichtenstein et. al., 1971), while Aida calculated a half width for each line adjusted to a mean line width of 0. 078 cm (from Walshaw, 1955). Also the band centers for the overtone band 2V3 -V and the 3 3 V3 band of the isotope 10 6010 are different for the two calculations. For the overtone band McClatchey et al. give a band center of 1027. 096 cm, while Aida (1973)gives 1012. 1 cm. Similarly McClatchey et al. -1 give 1028. 096 for the center of the isotopic band while Aida gives 1024. 1 cm -1 -2 Although the intensities of these bands are small (4. 26 atm cm2 -f1 -2 for the overtone and 1. 54 atm cm for the isotope), they can strongly influence the absorption in some parts of the spectrum. For example, -1 in a 1 cm interval, centered about 970 cm, the total line strength -3 -1 -2 as given by McClatchey et al. is about 13 x 10 atm cm, while Aida's -3 -1 -2 results give a strength of 40 x 10 atm cm. Much of this discrepancy can be resolved if the 2V3 - v3 band is given the same center in both calculations. For example if the band center in the McClatchey et al. data is shifted down 14.9 cm to agree with that of Aida, then the total -3 -2 -1 line strength from the McClatchey et al. data is about 41 x 10 cm atm, agreeing well with that of Aida. A comparison of transmissions with experimental data of McCaa and Shaw indicates better agreement when the 2 V3 -V band is centered at 1012. 1 cm; for example at 20 mb and a mass path of 4. 62 atm-cm, the transmission in the interval 969. 5 to 970. 5 is 0. 94 for the unshifted data, while a shift of the 213 -13 band 3 3 gives a transmission of 0. 83. The latter agrees much better with the value of 0. 80 as given by McCaa and Shaw (1967). 16

-1 Band model transmissions for the 1023. 5 to 1.026. 5 cm spectral interval are compared with experimental data from MVLcCaa and Shaw in Table 2. 6. The difference approaches ten percetlt in some cases. For this model the half width was chosen as 0. 08 cm and the distribution of mean line strengths (averaged over an intensity interval of 0. 4) is shown in Table 2. 7. Pressure (mb) 20 20 20 66. 6 66. 6 Mass Path (atm-cm) 12.6 2.6.45.41.87 1023.5 - 1024. 5.012. 228. 652.493.344 1024.5 - 1025.5.006.177.603.415.277 1025.5-1026.5.013.243.670.515.296 Average.010.216.641.474.305 Experimental 02 19.55.48 22 (McCaa and Shaw)... Table 2. 6 Comparison of band model and experimental transmissivities in the 1023. 5 to 1026. 5 cm'l intervals for the Y3 ozone band. 1023.5 - 1024. 5 1024.5 - 1025.5 1025. 5 - 1026. 5 ~ 739 (6). 797 (7). 783 (5). 192 (3). 229 (4). 260 (3).0090 (16).00986 (15).0129 (14).00088 (111).00101 (121).0011 (110).000045 (5).000055 (9).000049 (24) - ~ ~ ~~~ ~ '- '....!...... Table 2. 7 Distribution of mean line strengths and numbers of lines (in parentheses) as used in the band model calculation for the 1023. 5-1026. 5 cm-1 spectral region of the Y;3 ozone band. If the 2V- -Vband is shifted to 1012. 1 cm the transmis3 3 sions and mean line strengths are as given in Tables 2. 8 and 2. 9 respectively. While the agreement is somewhat better, it is not definitive. The weak overtone lines change only slightly the overall strength in this spectral region (Compare T.ables 2. 7 and 2. 9). 17

Pressure (mb) 20 20 20 66. 6 66. 6 Mass Path (atm-cm) 12. 6 2. 6.45.41.87 1023.5 - 1024. 5 cm-1.012.221.629.478.266 1024. 5 - 1025. 5 cm. 006. 166.578. 399. 206 -1 1025.5 - 1026. 5 cm.013.234.648.502.301 Average.010.207.618.459.257 Table 2. 8 Band model transmissions for the 1023. 5 to 1026. 5 cm1 spectral interval in the 3 ozone band. Center of the 2 3 - 3 band has been shifted to 1012.1 cm 1023. 5 - 1024. 5 cm1 1024. 5 - 1025. 5 cm-1 105. 5 - 1026. 5 cm-1 739 (6).797 (7).783 (5) 192 (3).229 (4).260 (3).0107 (21).0105 (23).0131 (19).000803 (96).000937 (99).000999 (100).000034 (11).000047 (14).000044 (25) -2 -1 Table 2. 9 Distributions of mean line strengths (cm atm ) and numbers of lines (in parentheses) as used in the band model calculations given in Table 2. 8 These calculations presented in Tables 2. 6 and 2. 8 were made with a half width of 0. 08 cm which corresponds to the mean half width given by Aida (1973). If the half width is 0. 11, the value given by McClatchey et al., then the transmissions are as shown in Table 2. 10. The distribution of line strengths corresponds to Table 2. 9. It is i-ot obvious which half width is more appropriate. For the mass paths of 2. 6, 0. 45, and 0. 87 atm-cm, the half width of 0.11 cm~1 gives better agreement while for the other mass paths, the best agreement is foulnd for a width of 0. 08 cm. Actually the only significant difference between the theoretical and experimental transmissivities is for the mass path of 0.41 atm-cm, where our band model value is about 0. 07 smaller than the value from McCaa and Shaw. 18

Pressure (mb) 20 20 20 66. 6 66. 6 Mass Path (atm-cm) 12.6 2.6.45.41.87 1023. 5 - 1024. 5cm-1. 007. 199.600. 429. 235 1024.5 - 1025. 5cm-1.003.14.525.35.168 1025.5 0 1026. 5cm1. 007 209.601.454.258 Average.005.182.575.411.220 Table 2. 10. Band model transmissions for the 1123. 5 to 1126. 5 cm1 interval in the Y3 ozone band corresponding to line strength parameters as given in Table 2. 9 with half width of 0. 11 cm A calculation was also made for the V3 isotope 160160180 3 -1 -1 shifted down 4 cm, to 1024. 1 cm, in agreement with Aida. The transmissions in the 1023. 5 to 1024. 5 cm interval changed by less than one percent. Stratospheric tangent path transmissivities for two 1 cm spectral intervals, at 1024. 5 - 1025. 5 cm1 corresponding to a region -1 of maximum absorption, and 969. 5 -970. 5 cm, located in the wing, are shown in Fig. 2. 4. The region of maximum absorption in the V3 band can be used for remote sounding from the midstratosphere at least to the stratopause. However in the low stratosphere, the transmission is small, and a spectral region in the wing of the band could be used. It appears that a wavenumber somewhat greater than 970 cmshould be appropriate. The agreement in transmission between experimental and our band model results is better for the V3 and 1 bands than for 3 1 the others investigated; one would expect this to be the case since more work, both theoretical and experimental, has been done on these bands. 19

E - I I o 0 0.2 0.4 0.6 0.8 TRANSMISSION 1.0 Fig. 2. 4 Stratospheric tangent path transmissions for the spectral regions 1.024. 5-1025. 5 cm 1 and 969. 5 - 970. 5 cm-1.

Fortunately the V band appears to be the most desirable for remote 3 sensing work. The band is strong enough so that one could choose a number of spectral intervals within the band so that soundings could be made throughout the stratosphere. There are a number of disadvantages to using this band which should be mentioned. The line density is so large that direct line by line calculations are probably not feasible, and some empirical or band model approach might have to be employed. Secondly, there are still discrepancies among the theoretical line parameter tabulations which need to be resolved, e. g., the half widths and centers of isotope and overtone bands. The weaker lines may not be important near band center, but in the wings these lines become extremely important because of their strong temperature dependence. To illustrate, we show in Table 2. 11, the transmission in two spectral intervals, the one at 1025 cm being a region of the band corresponding to maximum absorption and containing many strong lines, -1 and the 970 cm region on the wing of the band. For example, at 300K the strongest lines in the 1025 cm region are about one-hundred times stronger than in the 970 cm region and at 200 K this difference increases to about one-thousand. The strongest lines are less temperature dependent and this is clearly shown in Table 2. 11 where the transmission for -1 the 1025 cm region varies by less than one percent, while the variation for the wing region is about forty-eight percent. Thus the line parameters must be accurately known because the stratospheric temperature range is similar to that given in Table 2. 11. It is unfortunate that there are no low temperature ozone measurements against which we can compare our theory. 21

An additional disadvantage in using this spectral region is that the aerosol absorption especially in the low stratosphere is quite large. This problem is discussed in Chapter 3. Temperature 300 275 250 225 200 969.5 - 970.5.466.605.637.876.952 1024.5 - 1025.5.147.145.147.152.153 Table 2.11. Effect of temperature on transmission in the spectral intervals of 969. 5-970. 5 (20mb and 26 atm. cm) and 1024. 5-1025.5 cm-1 (20mb and 2. 6 atm. cm) 2129. 5 - 2132.5 cm1 ( + ) 1 3.. A comparison of the band model calculations for the + f3 1 3 band with the absorption profiles of McCaa and Shaw is shown in Table 2. 12. The band model transmissivities represent averages over 1 cm spectral intervals, centered about 2130, 2131 and 2132 cm. Note that in all but the last case, the model transmissivities are smaller than the experimental results, in one case by as much as twenty-eight percent. Mass Path (atm. cm) 13. 8 4. 5 0. 87 4. 8 0. 23 Pressure (mb) 20 20 20 66. 6 66. 6 2129.5 - 2130.5 cm-1.185.303.564.180.862 2130.5 - 2131.5 cm1. 172. 243.593. 214.873 2131.5 - 2132.5 cm-1. 251. 279.778.269.938 Average. 20(.23). 27(.29).64(.64).22(.25).89(89) Experimental. 33.55.82 30.85 (McCaa & Shaw) Table 2.12. Comparison of band model transmissivities with the experimental study of McCaa and Shaw (1967) for the spectral interval 2129. 5 to 2132. 5 cm-1 for the 'Y1 + 3 ozone band. The transmissions in parentheses refer to a single spectral interval extending from 2129. 5 cm-1 to 2132. 5 cm. 22

This discrepancy is probably not due to experimental resolution since the transmissions of all three spectral intervals are less than the experimental values. The discrepal(y is also not duie to l difference in band center, sinc e the.i1 ne parametc rs w(cr' c(,)hoS('r so 1as to give minimum transmission in the band and the experimental results do not yield such small values at any wavenumber within the band. In addition, the experimental results include the effect of the CO fundamental at 2143 cm; McCaa and Shaw state that the absorptance of this band was never greater then twenty percent. This effect, although small, would make the experimental ozone transmissivities even higher, giving a larger disagreement with the band model. The band model also does not appear responsible for the discrepancy. Two possible sources of error were the averaging interval and the assumed intensity subintervals. The former was checked by applying the band model to a single interval extending from 2129. 5 to 2132. 5 cm, and the transmissions are shown in parentheses in Table 2. 12; the agreement is within three percent. Also rather than grouping the lines into intensity decades, which yielded two intensity subintervals, they were grouped for intervals of 0. 2, 0. 4, 0. 6 and 0. 8, as well. The various intensity intervals are shown in Table 2. 13 for the 2129. 5 to 2130. 5 cm interval, and the effect on the transmission is shown in Table 2. 14. The uncertainty in the transmission is less than five percent. Similar results were found for the other two spectral intervals. For an intensity interval of 0.4, for example, and if S is the strongest line in the spectral interval, then the intensity intervals would be Sm to 0.4 Sm, 0. 4 S to (0.4) Sm, (0. 4)2Sm to (0. 4)3Sm, etc. 23

0.1 0. 2 '0. 4 0. 6 0. 8.002 (29).026 (24).030 (17).035 (10).044 (3).0028 (6).0045 (11).010 (14).020 (13).030 (10).0022 (4).0055 (10).018 (9) 0019 (2) 008 (6) l ___________ l____________ l__ l.003 (7).,. W.,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Table 2.1 3. Distributions of line intensities for various line intervals for the 35 spectral lines in the 2129. 5 spectral interval of the 1 + V3 ozone band. strength to 2130.5cm Mass Path (atm cm) 13.8 4.5 0.87 4.8 0.23 Pressure (mb) 20 20 20 66.6 66. 6 0.1.185.303.564.18.862 0.2.185.321.604.161.868 0.4.191.325.581.207.856 0.6.200.327.558.171.866 0.8.197.310.570.204.869 Table 2. 14. Band model transmissivities for the 2129. 5 to 2130. 5cm1 ozone band distributions of line intensities as given in Table 2. 13. Compare with Table 2. 12. There is still much uncertainty in the line parameters for this band and we suspect this is the reason for the discrepancy between the band model and experimental results. McClatchey et al. (1973) state -1 that the line positions up to J=20 and K =4 are accurate to 0. 3cm with awith "the error in line position significantly greater for higher quantum numbers. " Although the total band intensity as used by McClatchey et al. agrees with the experimentally determined value of 32 atm cm2 of McCaa and Shaw, nevertheless, an incorrect distribution of intensities of these lines could cause the lower transmission. In any case, it is o obvious that our present knowledge of the '] + Y3 band is not adequate for application to remote sensing. 24

Stratospheric tangent path transmissivities for the spectral interval 2129. 5 to 2132. 5 cm1 are presented in Fig. 2. 5. If the line parameters could be improved, then this band could be used for remote sensing at least in the lower stratosphere. 3044. 5 - 047. 5 cm1 (3 3044.5 - 3047. 5 cm (3 Vf3) A similar comparison of the 3044. 5 - 3047. 5 cm spectral interval with the McCaa and Shaw results is shown in Table 2. 15. Pressure (mb) 20 66 400 400 400 Masspath (atm- cm) 14. 6 1. 4 19. 1 6. 1 14. 3 3044.5 - 3045 5 58 92 24 67 33 3045.5 - 3047.5 51 90 22 64 32 3046.5 - 3047.5 53 90 16 62 28 Average.54.90.21.64.31 Experimental 76 93 49 76 56 (McCaa and Shaw) Table 2.1. Comparison of band model transmissivities with experimental results of McCaa and Shaw (1967) for the 3044. 5 to 3047. 5 cm-1 spectral interval for the 3-3 ozone band. As in the priorcase the band model transmissivities are much smaller and the discrepancy is probably also due to the uncertainties in the line parameters. Few details on the calculation of the line parameters for this band are given, but McClatchey et al. do state that the line positions -1 are accurate only to ~5 cm, which is certainly not adequate for application to remote sensing. Stratospheric tangent path transmissivities for the region of maximum absorption in this band is shown in Fig. 2. 6. If one takes into account that the computed transmissivities are probably smaller than is 25

E - (D I 2130.5-2131.5 I. * I I I. i.2 0.4 0.6 C TRANSMISSION Fig. 2. 5 Stratospheric tangent path transmissions for the spectral interval 3044. 5 - 3047. 5 cm-1.

E F( 30 3045.5- 3046.5 20 3044.5- 3045.43046 3046.5 -3047.5 — - m 3044.5 - 3045.5 3046.5 -304753046.5- 3047.5 10 - I I 0 0.2 0.4 0.6 0.8 I.C TRANSMISSION Fig. 2. ( Stratospheric tangent path transmissions for the spectral interval 3044. 5-3047. 5 cm-. )

E H- 34 I: Lii I IOL 0 0.2 0.4 0.6 0.8 1.0 TRANSMISSION Fig. 2. 7 Stratospheric tangent path transmissions for spectral interval 2793. 5 -2796. 5 cm1.

the actual case (see Table 2. 15), then only in the low stratosphere could this band be useful for remote sensing of ozone. 2793. 5 - 2796. 5 cm1 ( + 2 + 3 ) Band model transmissivities for this combination band are also much smaller than the experimental values, and are compared in Table 2. 16. The line parameters were calculated by McClatchey et al. from a rigid rotor analysis and include lines for J less than 25 and K a less than 6. The discrepancy may again be due to the distribution of intensities among the lines. Pressure (mb) 20 66. 6 66. 6 400 400 Mass Path (atm. cm) 14. 6 1. 2 4. 5 14.3 19.1 2793.5 - 2794.5 cm1. 82 1.00.95.83.79 2794.5 - 2795. 5 cm1. 73 1. 00.92.74.68 2795.5 - 2796.5 cm-1.70 1. 00.91.71.65 Average.74 1.0.93.76.71 Experimental ( aExperimetal) 90.99.97 88.83 (McCaa & Shaw)' Table 2. 16. Comparison of band model and experimental transmissivities for the 2793. 5 to 2796. 5 cm-1 spectral interval in the I1 + Z'2 + '3 ozone band. Stratospheric tangent path transmissivities are given in Fig. 2. 7. As indicated in Table 2. 16, these transmissions are probably too small; thus even in the region of maximum absorption of this band, the absorption would be small, making remote soundings difficult. 2. 4 Water Vapor Calculations similar to those for ozone have been carried out for regions of maximum or near maximum absorption in three water vapor bands. The spectral regions are given in Table 2. 17. 29

Table 2. 17. Spectral intervals of selected water vapor bands used in stratospheric tangent path transmission calculations. -1 -l The spectral intervals of 3743-3753 cm and 5335 - 5355 cm1 contain lines of numerous bands and a few of these are listed above. According to McClatchey, et al., the line positions are generally accurate to 0. 01 cm. Half widths for the individual lines are given, although for this analysis, we assumed a mean half width of 0. 08 cm. The spectral interval of 1660-1675 cm contains 59 lines and was subdivided into three 5 cm intervals. The spectral region around 3700 cm is highly structured and the interval was divided into two 5 cm subintervals; the total number of lines is 82. The 5335 to 5355 cm interval was divided into two 10 cm - subintervals containing 110 lines. These spectral intervals were chosen so as to be compatible with the experimental resolution of Burch et al. (1962), whose data were used for comparison with the band model calculations. The model atmosphere used for deducing the tangent path transmissivities is given in Table 2. 18.The extreme mass paths were taken from the previous report. Tangent Height Pressure Temperature Mass Path (km) (mb) OK (atm cm) 12 134- 113 217 - 218 72.0- 33. 1 20 38. 2 - 31.8 220 - 221 40. 9 - 9. 45 30 8.31 - 8.5 232 - 231 14. 5 - 2. 01 40 2. 0 - 2.06 255 2.85 -.46 50.56 -. 575 266 - 267.64 -.12 Table 2. 18. Curtis-Godson pressures and temperatures for selected tangent heights for extreme water vapor mass paths. 30

1660 - 1675 cm1 ( ) 2 A comparison of the band model transmissions with the experimental data of Burch et al. (1962) for the 1660 - 1675 cm1 spectral interval is shown in Table 2. 19. The agreement is excellent with the average of the band model results agreeing to within three percent of the experimental values. The absorption profiles are complex as can be seen from Table 2. 19; in the one case the absorption varies by twenty-six percent in adjacent spectral intervals. Thus the spectral interval must be carefully chosen if this band is to be used for remote sensing. Mass Path (atm. cm) 2. 12 21. 9 44. 7 43. 8 95. 8 Pressure (mb) 18. 7 140 50.6 140 1070 1660 - 1665.944.591.623.445.000 1665 - 1670.926.444.456. 275.000 1670 - 1675.946.665 o698.536.007 Experimental 94.57. 62.42. 0 (Burch et al. ) Table 2. 19. A comparison of band model transmissivities with data of Burch et al. (1962) for the water vapor spectral interval of 1660 to 1675 cm-1 ( ) The stratospheric tangent path transmissions for two spectral regions in the V, band are given in Fig. 2. 8. The 1538 - 1544 cm region is representative of maximum absorption in the band. The transmission is near one-hundred percent in the upper stratosphere and remote soundings are questionable. A smaller spectral interval would be desirable if the available energy is adequate. Numerous other spectral regions in this band would be adequate for remote soundings at least up to the midstratosphere, e. g., the 1660 - 1675 region, which is also shown in Fig. 2. 8. 31

H 30 '30 1538-1544 LI 20 1660-1665 Goo ~16651670X~~'"~1670-1675 1665-1670 10 0 0.2 0.4 0.6 0.8 1.0 TRANSMISSION Fig. 2. 8 Stratospheric tangent path transmissivities for water vapor for the spectral intervals of 1660 -1675 cm-1 and 1538 -1544 cm-1.

3743 - 3753cm1 spectral region As mentioned previously, this spectral region contains lines of many bands, and the absorption spectra are extremely complex (see, e. g, Burch et al., 1962). Also Burch et al. give the resolution of their absorption profiles as approximately 10 cm; thus there is probably an uncertainty of at least a few wavenumbers. Both these factors make comparison of the band model transmissions with their data rather difficult. However, we have attempted a comparison which is shown in Table 2. 20. The band model transmissions are significantly Pressure (mb) 1150 1043 151. 92. 36. 7 Mass Path (atm-cm) 135 4. 42 146. 4. 42 4. 42 -1 3743 - 3748cm 0. 09 0.51.64 3748 - 3753cm-1 0.10 0.51.60 Experimental 0 26.03 57.78 (Burch et al. ) Table 2.20 Comparison of band model and experimental transmissitivities for water vapor lines in the 3743 to 3753 cm-l spectral interval. lower but this was anticipated since the spectral interval was chosen to maximize the absorption. The spectral interval could be increased -1 about 2 cm without significantly increasing the absorption, and this increase would be well within the resolution range. Thus the comparison neither justifies nor negates the band model and/or line parameters. Tangent path transmissivities are shown in Fig. 2. 9. If the contribution to the transmission from other molecular species is not large, then this band could be used for remote sensing of water vapor, at least in the lower stratosphere. 33

50 I 30 3798-3753 20 3743-3748 10 I I I I 1I 1. I 0 0.2 0.4 0.6 0.8 1.0 TRANSMISSION Fig. 2. 9 Stratospheric tangent path transmissivities for the 3743-3753 cm-1 spectral region.

-1 5335 - 5355 cm spectral region Results of the analysis for this spectral region are quite similar to those for the 3743 - 3753 cm region. The band model transmissions are smaller than Burch et al's results, but again, the model was chosen to give maximum absorption, which appears as a narrow peak in the Burch et al. absorption profile. Also the resolution is given as "approximately 20 cm, and if it is only slightly larger, the transmission will decrease -significantly. The tangent path transmissivities are given in Fig. 2. 10, and indicate that this band is not suitable for remote sensing of water vapor. Only in the low stratosphere is there significant absorption and in this height range -1 the 1600 cm spectral region could be used which contains fewer lines from other molecular species. 2. 5 Methane Although an extensive analysis on the feasibility of detecting methane by the solar occultation technique has not as yet been completed, our preliminary study seems promising. The strongest bands are the V (3019 cm ) and (1306 cm 1) fundamentals with -1 -2 strengths of 370 and 204 atm cm respectively. A difficulty is that the line parameters need improvement, especially in extending the calculations to higher rotational quantium numbers and improving the intensities. Thus a detailed comparison with experimental transmissivities has not been made. The stratospheric tangent path transmission for a single spectral interval is given in Fig. 2. 11 which represents a region of maximum absorption in the fV band. This band model calculation 4 35

30 -LIJ 20 5335-5340" 5335-5340 - 5340-5345 5340-5335 10 I I I I I I I 0 0.2 0.4 0.6 0.8 1.0 TRANSMISSION Fig. 2. 10 Stratospheric tangent path transmissivities for water vapor in the 5335-5345 cm-1 region.

- 30 -I I 2,,,, — 1303-1304 1304-1305 1305-1307 10 0 0.2 0.4 0.6 0.8 I. TRANSMISSION Fig. 2. 11 Stratospheric tangent path transmissivities for the 1304-1307 cm- region of methane. 0

indicates that remote sensing may be feasible throughout much of the stratosphere. One must keep in mind, however, that the tabulation of McClatchey et al. excludes numerous weak lines whicih may signifi.canltly alter the atmospheric absorption. None of the stronger bands of methane is in a spectral region free from interference by other molecules. Nitrous oxide absorbs in the v4 band and ozone overlaps the v3 band so that these gasses would have to be sounded simultaneously. The best region would appear to be in the v4 Q-branch between 1304 and 1307 cm - (see Fig. 2. 11). The width of this channel is critical and there is overlap with N20. At 1350 cm the absorption by CH4 is less by about a factor of two, but the problem of overlap is less severe. Alternatively the -1 Q-branch of the v3 band (3016 - 3019 cm ) can be used while the region near 2960 cm1 has less absorption but also less interference by ozone. 2. 6 Carbon Monoxide Assuming various constituent profiles, calculations of atmospheric transmittances have been made. Table 2. 21 shows the effect of slight changes in concentration of CO on the atmospheric transmittance. The calculations shown are for 2 model CO profiles, at a resolution of 5 cm 1at the spectral region 2173 cm near the peak of the R branch of the fundamental 4. 6jam band. The second model constituent profile is 10% higher than the profile in the first model. The line parameters used in the calculations are from Kunde (1968). Column 4 in Table 2. 21 shows the transmission with the increased CO concentration. At low altitudes (where the spectral line. Ir 38

is essentially Lorentz broadened) the absorption has increased by about 5% and this is consistent with the result obtained using a strong line approximation, in which the absorption is proportional to the square root of the optical mass. At higher altitudes the increase in absorption is less than 5% because Doppler broadening becomes important, and as the line centers are completely absorbed, there is lesser wing contribution. This result can be expected to hold for other molecules in spectral regions where the lines are non-overlapping along the absorption paths. The region near 2173 cm also contains absorption lines of carbon dioxide and nitrous oxide. No attempt has been made to calculate the absorption by these gasses but some interference can be expected because of the low stratospheric mixing ratios of CO. The 2-0 band of CO is much weaker and would be difficult to use. Hence the region near 2173 cm appears to be the best choice for a CO measurement. Tangent CO concentration Atmospheric Atmospheric Perc.Increase Height Model 1 Transmission Transmission of km Vol. Mix.Ratio ppm with Model 1 with Model 2 Absorption (1 0%inc.of CO) 70 1.0 60 0. 08 - - - 50 0.05.99825.99821 2. 29 40 0.02.99784.99779 2.31 30 0.02.99618.99604 3.66 20 0.02.98384.98305 4. 89 10 0.1.85910.85196 5.07 Table 2. 21 Atmospheric Absorption by CO 39

2. 7 Nitrous Oxide N20 The line parameter tape (McClatchey et. al. 1973) contains spectral line parameters for N20. The molecule is linear. The theory of its spectrum is well understood. In addition, many experimenters have investigated the absorption properties of the stronger absorption bands, so that the parameters on the tape should be fully adequate for the calculations described here. The strongest of all the N20 bands is the v3 centered near 2223. 8 cm, and is a factor of seven more intense than the next strongest band, the v1 centered at 1284. 9 cm. The maximum stratospheric absorption by the v band occurs at the peak of the P and R-branches is near 2210 and 2236 cm respectively. Calculations in these two spectral regions were made using the line-by-line program described in the Appendix. Carbon dioxide also absorbs in this region and similar calculations were made of the transmittance of this gas. Carbon monoxide lines of the 1-0 band also occur near 2210 cm but these are weak and were not considered here. Water vapor absorption is small under stratospheric conditions. Figure 2. 12 shows the transmittances for N2 0 and CO2 for 4 cm intervals centered at 2210. 5 and 2236. 5 cm. The N20 concentrations are from Table 4. 1. 3 of our earlier report (Drayson et. al., 1972) and the U. S. Standard Atmosphere (1962) has been used. The CO2 concentration is assumed constant at 320 ppm V. The N20 transmittances are almost identical for the two intervals but the CO2 transmittance is remarkably higher at 2210. 5 cm making this the clear choice of a spectral region to sound N20. It should be pointed out that there may be some uncertainty in the line 40

N20 MIXING RATIO (PPMV) 0.18 0 0.06 0.12 0.24 0.30 ^ 30 E F(9 I 'rl 0 0.2 0.4 0.6 0.8 1.0 TRANSMISSION Fig. 2. 12 Stratospheric Tangent path transmittance of N20 and CO2.

parameters of CO2 since hot bands dominate the absorption in these regions and the intensities of the bands have a higher degree of uncertainty than the fundamentals. At 2236. 5 cm- the carbon dioxide band with the most intense lines appears to have been included twice on the tape so the transmittance may in fact be higher than indicated. Another possible spectral region for sounding N20 is -1 near 1270 cm (R-branch of the v, band) which gives considerably less absorption and has some interference by CH4 absorption. The V2 band is much weaker than either the vl or v3 bands but has the ad-1 vantage of closely spaced lines in the Q-branch near 590 cm. However absorption by CO2 is again strong and a rough estimate showed the CO2 absorption to be at least as large as the N20 throughout the stratosphere. The peak of the P-branch of the v3 band of N20 near 2210 cm appears to be the best spectral region for sounding N20. The instrument resolution and exact wavenumber are not critical factors and the interference by other molecules is minimal. 42

REFERENCES Aida, Masaru, 1973: The spectral absorptivity by the 9. 6 micron ozone band. Paper presented at the Radiation Commission in Sendai, Japan, Data tape containing Aida's band parameters was provided by Dr. James Russell, Langley Research Center. Burch, D. E., D. A. Gryvnak, E. B. Singleton, W. L. France and D. Williams, 1962: Infrared absorption by carbon dioxide, water vapro and minor atmospheric constituents. AFCRL62-698 Air Force Cambridge Res. Labs. Clough, S. A., and F. X. Kneizys, 1965: Ozone absorption in the 9.0 micron region. AFCRL-65-862 or Physical Sciences Research Paper no. 170, 79pp. Clough, S.A. and F.X. Kneizys, 1966: Coriolis interaction in the 1- and 5 fundamentals of ozone. J. Chem. Phys. 44, 18 55 -1861. Drayson, S. R., F. L. Bsrtman, W. R. Kuhn and R. Tallamraju, 1972: Satellite mieasurements of stratospheric pollutants and minor constituents by solar occultation: A preliminary report. High Altitude Eng. Lab. Tech. Report No. 011023-1-T. University of Michigan. 143 pp. Drayson, S. R., 1973: A listing of wavenumbers and intensities of carbon dioxide absorption lines between 12 and 20 pm. Technical Report 036350-4-T, University of Michigan. Kunde, V. G., 1967: Tables of Theoretical Line Positions and Intensities for the V = 1 V = 2 and V = 3 VibrationRotation Bands of C12016 and C13016. NASA TMX-63183. McCaa, D. J. and J. H. Shaw, 1967: The Infrared Absorption Bands of Ozone. AFCRL-67-0237, 93pp. Walshaw, C. D., 1955: Line widths in the 9. 6 P band of ozone. Proc. Phys. Soc. A, 68, 530-534. Walshaw, C. D. 1957: Integrated absorption by the 9. 6 - band of ozone. Quart. J. Roy. Meteor. Soc. 83, 315-321. Wyatt, P.J., V.R. Stull, and G. N. Plass, 1962: Quasi-random model of band absorption, J. Opt. Soc. Amer., 52, 1209-1217. 43

Chapter 3. Extinction of Infrared Solar Radiation by Aerosols on a Tangent Path Through the Stratosphere 3. 1 Introduction In order to determine the effect of aerosols on a beam of infrared radiation, it is necessary to know the complex index of refraction of the aerosol and its particle size distribution. Since the stratospheric aerosol probably contains several or all of the following constituents; HSO4, (NH4)2SO4, H202, HNO NOHSO and HNO H2S04 - H20 (Remsberg, 1971) a precise calculation would involve knowing the relative amounts, particle size distribution and complex index of refraction of each of these constituents. As an initial calculation, the infrared extinction due to two greatly simplified models of stratospheric aerosol, containing only aqueous solutions of sulfuric acid (Remsberg, 1973), was determined. 3. 2 The Aerosol Model The stratospheric aerosol model discussed by Remsberg (1973) was used in this calculation. It is a bi-model distribution described by the equations (see also fig. 3. 1): dN.= 10 0. 03 r < 0.05 um (3.1) d(log r) dN 10+3. 72 +4. 19 0. 1 - r 0. 3 mm (3. 2) d(log r) dN -100. 474 -3. 82 0. 3 ~ r 1 gm (3.3) d(log r) The particle number density in each size range, obtained by integrating the above equations, and the mass density in each size range, obtained from: 44

C-I cc E ^ol01 "{22 10-2 10 10 101 RADIUS, (MICROMETERS) Fig. 3. 1 Bi-model aerosol size distribution used for tangent path extinction calculations (after Remsberg, 1973). 45

r2 4 3 dN(r) dr M -- f r dN(r) dr (3. 4) r with the density of sulfuric acid, J= 1. 8 g/cm3, are: r,,um..03 -. 05.1 -.3.3 - 1 N, cm. ~ 2219.4 7.99 3.76 M, gr.cm. 1. 077 1012 0.414 10 12 2. 258 - 102 M, gr.cm. 1. 077.10 O. 414 10 2. 258 10.... s~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The model contains a fairly large number of Aitken nuclei, however not as many as indicated by DeBary and Rossler (1970). The mass density M is intermediate between values given by DeBary and Rossler and by Lazrus (1971). The model)thennay represent an average condition in the stratosphere. Larger numbers of particles would exist after volcanic explosions. The relative altitude distribution fk used for this calculation is shown in figure 3. 2 and is similar to that measured by Chagnon and Jung (1961). In the altitude range 17-22 km., the number of particles is the exact number(N ) given by the equations 3. 1 to 3. 3. At max all other altitudes the total number is scaled by the ratio given in figure 3. 1 (i. e. N = fk Nma) however the relative size distribution is not changed. Note that the aerosol density has been assumed to be constant in one kilometer layers. The length of a slant path through a 1 km spherical shell layer in the atmosphere is given by (neglecting refraction) (Drayson, et. al. 1972). AX 2 (y - R - ) (3.5) k 2 + 1 o k o 46

— 4 - 0 0 Lr\r~~~~~~~~r ~ O I I l I I l o -— ______ I __ICD L__ CD013 (W) '3anllnv Fig. 3. 2 Relative altitude distribution of aerosols, f, used for tangent path extinction calculations. 47

between the altitudes specified by radial distances Rk+l and Rk. The total amount of aerosol traversed along the tangent path is: D= Z fk A X (3.6) k where D has units equal to kmof aerosol of number density given by equations 3. 1 to 3. 3. That isjequivalent km. of aerosol at the concentration Nma. A curve of the values of D versus tangent altitude is given in figure 3.3. It can be noticed that the shape of D vs. tangent altitude is vaguely similar to the curve of f vs. altitude. 3. 3 Optical Characteristics of Aqueous H2 SO4 Values of the real and imaginary parts of the complex index of refraction of 75% and 90% aqueous solutions of H2SO4 (Remsberg, 1971) are given in table 3. 1 and figures 3. 4 and 3. 5 -1 for the wavenumber range 750 to 1570 cm Extinction coefficients for the aerosol of size distribution specified above, calculated from the relation: B X r' r 2 A dN(r) Bxt( f 7[ r A t(r,, dN(r) dr (3 7) Bext neX r dr rl A where Qext (r, X, n ) is the efficiency factor calculated from Mie theory of scattering, are given by Remsberg(1973) for 75% and 90% H2S04 for the wavenumber range 750 to 1150 cm1 (see figures 3. 6 and 3. 7). Approximate extinction coefficients for the range 1150 to 1570 cm were determined using relations given by Remsberg (1973), i.e.: 48

TABLE 3.1 Optical Constants (n = n - i k) of Aqueous H2SO4 75% H2SO4 90% H2SO4 FREQ k n FREQ k n FREQ k n 1571.0.14352 1.29568 1570.3.06519 1.32194 1033.4.37559 1.51173 1508.3.13326 1. 34542 1530.5.06004 1. 29689 1023. 7.35152 1. 47172 1433. 1.14802 1. 26802 1493. 6 05938 1. 25767 1014.4.36357 1. 40098 1383. 1.18988 1. 19644 1456. 8 07099 1. 18325 1007.7.39439 1.35505 1338.0.27709 1. 12007 1434.5.08932 1. 11536 1002.8.44602 1.30615 1297. 2.42056 1. 11362 1415.5.16894 1.01961 997. 2.53795 1.27432 1255. 1.52732 1. 13369 1408.6.22500.99501 992.4.63043 1. 27656 1231.7.62023 1.16010 1395.1.35646 1.01282 987.8.72579 1.31102 1209.7.69276 1.21735 1384.1.43227 1.06756 981.6.82897 1.39028 1184.8.74664 1.31945 1377.7.46363 1.10853 977. 1.90717 1.46549 1165. 2.79953 1.40849 1371.3.48821 1.15374 972.7.90743 1.56620 1150.2.79673 1.49183 1358.9.47339 1.24597 966.1.93171 1.69371 1131.1.74543 1.59253 1352.8.46908 1.27375 961.1.87506 1.79688 1114.0.68356 1.63027 1342.8.43757 1.30709 956.8.83078 1.85250 1104. 2.64292 1.62611 1333 0.38449 1.32172 949.9.72717 1.92317 1094.6.62505 1.59725 1321 6.35226 1.30987 944.5.61114 1.94337 1085.2.61812 1. 57582 1310.4.33114 1.29498 938.5.52953 1.90728 1079.1.63433 1.55688 1285.5.34575 1.25624 933.2.47916 1.86843 A4 1072. 2 66775 1. 56030 1266.9.38496 1.24231 929.4.44649 1. 83665 cD 1059.6.76254 1.60811 1247.5.42762 1.23318 932.7.43794 1.79859 1053.8.79410 1.63656 1224.5.49547 1.21762 918.7.47298 1.77194 1041.8.78138 1.76056 1214. 2. 54171 1.22111 914.4.50752 1.77024 1032.8.74167 1.82434 1202.8.60234 1.24946 911.4.53047 1.78253 1019.8.63816 1.88088 1195.8.63789 1.27660 904.8.55422 1.83406 1011.4.57044 1.88976 1190.3.67201 1.31293 901.9.54875 1.85984 1004.8.53086 1.87327 1180.8.69002 1.38008 899.0.52618 1.89470 988.9.45388 1.83372 1172.9.70626 1.43635 891.6.45885 1.93520 971.6.41524 1.77328 1163.9.68149 1.49979 887.7.41549 1.94322 951.0.38927 1.73457 1156.3.67557 1.54355 883.8.36516 1.94595 934.9.39814 1.67112 1148.9.63875 1.58053 880.0.33020 1.92861 927.2.42728 1.64740 1140.4.62270 1.60280 874.7.28120 1.90672 920.3.48341 1.62958 1136.8.60050 1.61745 865.3.23288 1.86844 907.6.57254 1.67483 1133.3.56369 1.63732 859.7.20708 1.84727 901.7.59737 1.73149 1120.6.49683 1.66145 853.8.20083 1.81158 896.0.58872 1. 78764 1111.6.45509 1.67053 843.6.18622 1.76891 887.5.55768 1.84973 1102.9.41969 1.66178 830.1.16132 1.73590 871.9.44108 1.86926 1094.3.38950 1.64046 819.1.16432 1.70130 856.7.32979 1.83481 1082.9.36166 1.61575 802.7.15708 1.66877 844.5.28252 1.80852 1072. 9 35821 1.57053 787. 7.16121 1.63697 840.3.31144 1.80826 1065. 1.39423 1.52436 770.0.15996 1.61359 828.3.26172 1.77577 1056.4.40302 1.51570 754.5.16197 1.59681 818. 2 25650 1 751f7 1048.9.42562 1.51649 747.3.17297 1.58389 804.4.23696 1. 72780 1040.6.40177 1. 52266 786.2.23446 1.69800 765.4.23902 1. 66782 748.2.25159 1.64731

25 [ 20 E uLi o 0 I10 -- 1 10 100 1000 D, TOTAL AEROSOL ALONG A TANGENT PATH, (km at Nmax) Fig. 3. 3 Total aerosol, D, traversed along a tangent path.

WAVELENGTH 7 (microns) 7.5 8 WAVELENGTH (microns) 6.5 8.5 9 10 11 12 13 14 2.0 2.0 1.8 r 1.6 S 1.4 -0 cc 1.2 ' 0.6 - ( 0.4 Fo 0.2 0.0 1600;1.8 1.6 - 1.4 o 1.2 e 1.0 80.8 0.6 C 0.4 0. 2 0.2 0.0 1500 1400 1300 FREQUENCY (cm-1) 1200 1100 1000 900 800 700 FREQUENCY (cm-1) Fig. 3.4 Optical constants for 75% H2SO4.

WAVELENGTH (microns) 7 7.5 WAVELENGTH (microns) 2.0.1.8 - 1. 6 C1.4 Ln21. 2 _- 1.2 L) _ 0.8 c.Z en <-) '0.6 a 0.4 ~0.2 9 10 11 12 13 14 2.0 -.1.8 1.6 1.4 V 1.2 L1.0, 0.8 0.6 E 0.4 o0.2 0.0 1600 1500 1400 i300 FREQUENCY (cm-1) 1200 0.0' L-L 1150 1100 1000 900 FREQUENCY (cml1) 800 700 Fig. 3. 5 Optical constants for 90% H2SO4.

WAVELENGTH (p m) 9 10 11 12 13 I E _. 1.5! 1.0 0.5 4-, x WC Fig. 3. 6 Spectral extinction coefficient for 75% H2S04 (Remsberg 1971). WAVELENGTH (u m) E 1. 5 A I0.5 -'0 Wt Ca 1100 1000 900 800 FREQUENCY (cm1 ) Fig. 3. 7 Spectral extinction coefficient for 90% H2S04 (Remsberg 1971). 53

Qext Qabs + Qsca abs s ca e- ^ abs (3. 8) since: Qa< Qab sca abs (3. 9) in the wavenumber range of interest. Also: Qabs 24nk ( 7r ) n2 -k2+2) +4n2k abs A k Be 48i7xtk ~~k-1 4r ^-2 2+ B... (n -k 2) +4n k extk - x (3. 10) _r3 dN(r) dr fr - dr (3. 11) then: Bext(X1) nk Fn 2+2)2 + 4n2k.............. ~ i (3. 12) B k2) i nk I2 2 2n 2 WI' ext(2) n2-k+2) + 4n2k2 Approximate extinction coefficients for 75% and 90% H2S04 were calculated from equation 3. 12, using Remsbergs Bext at 1150 cm1 as a reference value. The approximate extinction coefficients calculated in this fashion are shown in figures 3. 8 and 3. 9, Remsbergs values and the approximate values are compared in the wavelength range 1000-1150 cm. Note the excellent agreement in the range of overlap between the two sets of values. 54

2 2.0 1.8 X/ x X x X X Approximate Values I — E r:2 o 0 -x elm 1.6 1.4 1.2 1.0 0.8 xI e Remsberg - t en \x 0.6 \X,X _x_ x x 0.4 - 0.2 L 1000 1 100 1200 1300 1400 (WAVENUMBER, cm-1) Fig. 3. 8 Approximate extinction coefficients for 75% H2SO4. 1600 1500

II E m 1. o CI a1 0a 1300 (WAVENUMBER, cm'1) 1600 Fig. 3. 9 Approximate extinction 90% H2S04. coefficients for

3. 4 Infrared Transmissivities along Tangent Paths The infrared transmissivity of a tangent path through the aerosol layer was calculated by the equation: T(\) = exp / - ex () Z f A X / (3. 13) ext. k k for several paths tangent in the range of altitudes 16-26. 6 km. The results are shown in figures 3. 10 and 3. 11 for 75% and 90% H2SO4, respectively. 3. 5 The Extinction of Infrared Solar Radiation The extinction, due to the aerosol models described above, of solar infrared radiation along paths through the earth's atmosphere was calculated from the equations I (X) = T() B(X, Ts) (3. 14) where I(X) is the spectral intensity (spectral radiance) of the solar radiation after traversing a tangent path through the earth's atmosphere, T(X) is the transmissivity defined above and B(X, Ts) is the radiance of the photosphere of the sun, assumed to be a blackbody at temperature T = 5036 K (Saiedy and Goody, 1959). Figures 3. 12 and 3. 13 show s the results for tangent paths at altitudes of 16 to 26. 6 km for 75%0 H2S04 and 90% H2SO4. Discussion of Results: The results shown above indicate the significant difference between the absorption of 75% and 90% aqueous solutions of H2SO4, each with the same particle size distribution and with the same relative 57

1. 0. vI 0.9 0.8 0.7 26. 6 km - 24.4 \ ~~ --- —~~~~~~~~~~~~~~~~ —~~~~23.05 22. 05 20.08 19. 5 18. 0 16.0 0.6k >-!2 0.5 z -. Cn 00 0.4 - a3 - 0.2 0.1 n,-,. I 1 I I I I I I I 1 500 600 700 800 900 1000 1100 1200 1300 1400 (WAVENUMBER, cm1) Fig. 3. 10 Tangent path transmissivities for 75% H2S04. 1500 1600 1700

1. 0. 0. >uQ H CO c0 0. 0. 0. 0. 500 600 700 800 900 1000 1100 1200 1300 1400 ( WAVENUMBER, cm-1) Fig. 3. 11 Tangent path transmissivities for 90% H2SO4.

4. 4.( 3.1 3.( 3.1 9n nc E LO. U),' 3.2/ E 3.0 22. 05 2. 8 2.6 20. 8 2. 4 19. 5 2. 2 18. 0 2. 0 1. 6 750 800 850 900 950 1000 1050 1100 (WAVENUMBER, cm l) Fig. 3. 12 Extinction of solar infrared radiation, by aerosols, on tangent paths through the earth's atmosphere (75% H2SO4). 60

4. 3. 3. 3. 3.; 3.( Ef J^.-1~~~~~~~ /y \ ^-/ / ~~~~~~~~~~~19. 5 '. 8 8 / E. 18 0 E 2 66~(WAVENUMBER, cm2) 2. 4 2. 2 2. 0 1. 8 1. 6 1. 4 1. 2.50 800 850 900 950 1000 1050 1100 (WAVENUMBER, cm-1) Fig. 3. 13 Extinction of solar infrared radiation, by aerosols, on tangent paths through the earth's atmosphere (90% H2S04). 61

number distribution as a function of altitude. The differences in complex index of refraction are reflected in turn in extinction coefficients transmissivities for tangent paths through the atmosphere and in solar radiation intensities after traversing tangent paths. The 75% H2S04 aerosol shows strong absorption bands centered at 900 cm, 1060 cm and 1200 cm. There is strong absorption near these wavenumbers for 90% H2S04 as well (the bands are shifted slightly, they appear to be centered at 908 cm, 1050 cm and 1150 cm ). addition, 90% H o -1 01 In addition, 90% H2SO4 shows great absorption at 980 cm1 and 1375 cm0 Measurements of solar radiation in the window region of the spectrum through tangent paths in the atmosphere should provide an excellent measurement of the extinction of stratospheric aerosols, although the absorption at 1050 cm will be almost completely masked by absorption due to the v3 band of 03 at 1042 cm The absorption of either 75% H2SO4 or 90% H2SO4 at the 900 cm 1 wavenumber region should be clearly noticeable in window region measurements. The 980 cm and 1375 cm absorption may also be recognizable in tangent path spectra although the former may be interfered with by the 1042 cm 03, 961 cm" CO2 and the 884 cm HNO3 absorption regions, and the latter may conflict with 1594 H20, 1306 CH4, 1285 N20 and 1333 HNO3. 3. 6 Additional Calculations and Improvement of the Model Additional calculations should be rmade for 75% and 90% H2S04 with other aerosol size distributions. A mixture, in equal amounts, of 75% and 90% H2SO4 should also be considered. 62

The exact nature of this mixture is an interesting problem in itself. Would such a mixture contain 50% of each 75% and 9!0%o aqueous H2SO4 for each particle size? Or would the smaller sizes tend to be mostly or all 90% H2SO4, with larger sizes being 75% H2SO4? Calculations can be made for both possibilities, however the physical process describing the formation of the aerosol particles should shed some light on this question. The aerosol model should be improved by adding the effects of the other most likely constituents (NH4)2SO4, H202 HNO3, NOHSO4, and HNO3 - H2SO4 - H20. The altitude range of the model should be increased to include effects at 50 km, where a secondary aerosol layer may cause noticeable absorption on tangent paths through the atmosphere (Elliott, 1970). 63

REFERENCES Chagnon, Co W. and Co E. Junge, (1961), The Vertical I)istr ilbutior of Sub-Micron Particles in the Stratosphere, J. Mlcrteorol, 18, 746-752. de Bary, E. and F. Rossler, (1966), Size Disbributions of Atmospheric Aerosols Derived from Scattered Radiation Measurements Aloft, J. Geophys. Res., 71, 1011. Drayson, S. R., F. L. Bartman, W. R. Kuhn and R. Tallamraju, (1972), Satellite Measurements of Stratospheric Pollutants and Minor Constituents by Solar Occultation: A Preliminary Report, Univ. of Michigan Report No. 011023-1-T, Final Report on N. O. A. A. Grant NG-10-72, High Altitude Engineering Laboratory, Departments of Aerospace Engineering and Atmospheric and Oceanic Science. Elliott, D. D., (1970), Effect of a High Altitude (50 Km) Aerosol Layer on Topside Ozone Sounding, Space Research XI, Proc. of 13th Plenary Meeting of Cospar, Leningrad, 1970, Akademic-Verlag, Berlin 1971. Lazrus, A. L., B. Gandrud, and R. D. Cadle, (1971), Chemical Composition of Air Filtration Samples of the Stratospheric Sulfate Layer, J. Geophys. Res., 76, 8083-8088. Remsberg, Eo, (1971), Radiative Properties of Several Probable Constituents of Atmospheric Aerosols, Pho D. Thesis, University of Wisconsin, Madison. Remsberg, E., (1973), Stratospheric Aerosol Properties and Their Effects on Infrared Radiation, J. Geophys. Res., 78, No. 9, 1401-1408. Saiedy, F. and Ro M. Goody, (1959), The Solar Emission Intensity at 11 Microns, Monthly Notices, Roy. Astron. Soc., 119, 213. 64

Chapter 4. INVERSION PROCE( I )U[ES 4. 1 Introduction This chapter deals with the inversion procedures, by which the atmospheric concentration of the absorbing species can be obtained from the solar occultation measurements of the radiant intensity I. The problems of inversion for atmospheric temperature from radiance measurements have been studied by many authors. (Kaplan 1959, King 1964, Wark and Fleming 1966, Smith 1972, Chahine 1968, Gille 1968, Burn and Uplinger 1970, Rodgers 1970, Wark 1970, McKee and Cox 1973). Inference of water vapor and ozone from radiance measurements have been reported by Yamamoto and Tanaka (1966), Venkateswaran et al. (1961), Conrath (1969), House and Ohring (1969), Prabhakara et al. (1970), Smith (1970). For the solar occultation experiment the maximum intensity (Io ) is obtained when the measurement is made at maximum tangent heights and there is essentially no atmospheric absorption. The problem reduces to one of obtaining the concentration of species from the atmospheric transmittances (T = I/Io ), since emission from the atmosphere is very small and can be safely neglected in comparison to the total intensity. Rayleigh scattering at the spectral regions we are interested in is also small. Mie scattering by aerosols becomes important below tangent heights of approximately 30 km. and has to be included in the calculations. (See Chapter 3) We have seen from the previous chapters how the calculation of the atmospheric transmittance is very complicated. Not only is the distribution of absorbing constituents required for such a calculation but also the atmospheric pressure and temperature 65

distribution is required. Calculations of transmittances for the 15um band of CO2 have been published by Drayson (1966) and Kunde (1967). These calculations require a detailed knowledge of the line positions, strengths, widths of the spectral lines in the region studied and also involve a large amount of computer time for the calculations. Therefore in cases where no such detailed knowledge of the spectral band is available or when use of large amounts of computer times is prohibitive, simplified calculations of atmospheric transmittance using Band Models have been used. (Goody 1964). Because of this factor we have decided to try out the inversion procedures assuming that the transmittances can be calculated by the strong line approximation where the absorption is proportional to the square root of the optical mass. T= l-2. Js.ua-! (4.1) where S, a and 8 are the strength, half-width and spacing between lines and u the optical mass, is the sum over all layers along line of sight u = Z u (4.2) The half-width is proportional to pressure a a= jo P (4. 3) where p is the equivalent or mean pressure for the entire path p-E e U (4.4) and similarly T is the equivalent temperature t =t - u/ (4.5) 6 u 66

However it is easy to incorporate the detailed calculation of transmittances instead of the above strong line approximation at a latter stage. 4. 2 Geometry and Technique Consider one absorbing constituent in the spectral region of interest and the atmosphere, spherically stratified and symmetric is divided into n thin concentric spherical shells. Horizontal gradients in temperature and pressure have been considered in the calculations by Davis (1969). Measurements of radiant intensity are made at n tangent heights where the line of sight from satellite to the sun passes through an increasing amount of atmosphere and number of shells as occultation proceeds. The tangent height is defined as the minimum altitude of the line of sight from the earth's surface. The pressure temperature and mixing ratio of constituent is assumed constant within each shell. For all inversion methods an assumption of the absorber concentration above the top most shell is necessary for accurate inversion at the upper tangent heights. We have measured transmittances Tm, i=l, n (4. 6) 1 and we are to determine the mixing ratio in each of these shells. Ci i =, n (4. 7) One can formulate mathematically T = F. (C., j i) i = 1, n (4. 8) i 1 J where the function Fi depends on tangent height and hence the temperature and pressure variations in the atmosphere. There is not contribution from the atmosphere below the tangent ray and thus for the shell 67

at the top of the atmosphere Tm= F1(C) (4. 9) 1 1 Figure 4. 1 shows the geometry of the occultation experiment. 4. 3 Methods of Inversion "Onion Peeling" The onion peeling method developed by Russell (1970), McKee et al. (1969), Russell and Drayson (1972) starts the inversion at the topmost layer, and after the concentration in the first shell is obtained the procedure is carried out for the next lower shell and so on until the lowest shell. We assume a concentration C1 for the topmost shell and calculate T. If the assumed concentration is close to the C actual, that we can make a linear approximation between measured transmittance T and calculated transmittance T ml r a~clr cl c)T1 Tm = TI + -— |l AC1c (4.10) M. 11 a 1 The partial derivative is calculated and the perturbation parameter is determined from L r aTc 1 A~C1 = A~T1e J~ (4. 11) where AT = T T I mlc1 1 This value of AC1 is added to the original assumed concentration C1 and the process if repeated until a desired convergence criteria is met. The procedure is next carried out for shell 2. It is convenient to assume the initial guess for C2 equal to the above C1 obtained 68

after inversion. The process is repeated and continues downwards one shell at a time to the lowest shell. The measurements T inevitably contain errors, mi like radiant intensity bias errors, scale errors and random noise errors. Besides these there are errors in the determination of tangent heights, atmospheric pressures, temperatures and knowledge of absorption line parameters, etc. The effect of random noise errors can be reduced by either smoothing the input transmittances or smoothing the retrieved profile. Eigenvectors and Smoothing Matrix Methods The method of eigenvectors has been developed by Mateer (1964) and used by Russell and Drayson (1973). We can write n equations. n Tc ki 3Cck k=l or in matrix form AT= B- AC (4.13) The least squares solution is given by AC = (B'B) (B'. AT) (4. 14) where B' is transpose of matrix B. The above equation is unstable and leads to erroneous results when noise is present. The matrix B'. B is symmetric and has real non-negative eigenvalues The eigenvalues \9 X2... n have corresponding orthonormal eigenvectors ' v. vn. The column matrix B'. AT is now expressed as n B'.AT = bi. vi (4.15) i=l 69

where b. are constants for each eigenvector. The constants b. are 1 1 determined by solving the n above equations. We have AC = (B'. B) B'.T (4. 16) in n b. = (B'- B)-E ii = b 1 v V i=l i=l i Now the errors in measurements incorporated in T and the mi errors in calculations incorporated in T, the partial derivatives T Ci c. C, B, B' are all contained in the constants b.. Therefore 3Ck1 we can write n b.' n 1 i i= 1 where b' are values of constants with no error, and e. are the error 1 1 terms. If X. are ordered in decreasing magnitude, for larger i, X. may be very small, and error terms will then be large. Inclusion of eigen-vectors vi corresponding to the smaller X. gives the details of the solution which may or may not be due to noise, and cause instability or erroneous results when noise is present. We therefore truncate and use only a limited number of terms, so as to control the noise. The selection of where to truncate depends on a priori knowledge of the amount of noise present. To trucate too much would mean inferior results due to elimination of valid information. Another technique of smoothing (Wark and Fleming 1966) is to introduce a smoothing matrix H and a smoothing parameter Y The selection of appropriate values for H and Y has been discussed in papers by Phillips (1962) and Twomey (1963). The solution is given by AC = (B'.B + yH)-. B'1 AT (4. 18) 70

Increasing smoothing by increasing Y, makes the solution more dependent on matrix y H. If H is the identity matrix, the method is related to the eigenvector expansion since the eigenvectors of B'B + Y H are the same as those of B'B and the eigenvalues are X. + Y. The eigenvector expansion is not truncated but is replaced by the expression n bi A C = E -i ~. v. (4. 19) i=l Xi +Y However the eigenvalues never become smaller than y and the problem of amplification of the noise may be controlled by choosing a sufficiently large value of y If the matrix B'B is non-singular, solution with y = 0 corresponds exactly to the eigenvector method with all terms included. Kalman-Bucy Filter Gray et al. (1973) have discussed a technique of smoothing using the Kalman Bucy filter. This technique (as also the above two methods) is efficient when the initial guess of constituent density C. is close to the actual value and the linear relation between T i 1 mi and Tc is valid. The method assumes an initial guess of concentration C. and updates this state vector after each measurement by calculating the "filter gain" (Newell and Gray 1972) vector K. 1 K P.-'B' [B P.i -B'i + R (4. 20) ci i CTci where B'i is the transpose of vector Bi =, -,... 1 L 2 R is the noise covariance and P. 1 is the covariance matrix which is originally assumed and updated after each measurement by P. = Pi-i -K. B P (4.21) 1 i 1 i-1. 71

The state vector is updated C = C. +Ki A T (4. 22) The three matrix methods, i. e. the truncated eigenvector expansion, the smoothing matrix and the Kalman Bucy filter, have a common characteristic. The initial guess of the concentration profile is modified only if the measurements indicate a real deviation from the initial profile. Thus if the absorption is very small at the upper levels the measurements contain mostly noise and little information and the inversions show little modification of the initial guess at these levels. Similarly if the absorption is almost complete at the lower levels the measurements also contain little information on the concentrations and the initial guess again retained. Abel Equation The Abel integral equation OD X. 2f x(r) r- dr (4.23) l - 2/2 ri r - ri has been inverted (Roble and Hays 1972) and the inversion for constituent profiles has been tried. The values of Xi = (i. pi ) are determined from the measured transmittances T 2 Xi U.. Pi (1- Tm 8/2} / (S- ) (4. 24) The solution for x (r ) is given by 1 f dXZ!dri x (r)= - 1 / dr. dri (4.25) where x (r) is the product of C and p. 72

The values for X. are discrete, since measurements are made at 1 selected tangent heights only. Between individual data points, Xi is assumed to have an exponential variation given by X. (r) = A exp [- Bi (r-ri)] (4. 26) where the coefficients A. and B. are chosen to fit the data with desired 1 1 amount of smoothing incorporated. Thus for smoothing M/2 data points on either side of the i-th data point, A. and B. are calculated so as to fit the data from (i-M/2 ) to (i + M/2) in a least squares sense with a minimum variance. (i+M/2) 2 i = E Xk- Ai exp - Bi (rk -i (4. 27) k = (i-M/2) Now the derivative dX/dr near the i-th data point is given by dX = A.'B. exp - Bi (r-ri) (4. 28) dr 1 1 1 Substituting the above expression and replacing the integral by a finite sum, the solution for x (r ) can be written as 1 r r 1 r exp - B (r-r.) x (r9)- IZ Ai.- Bi ~. dr (4. 29) - ITi i 2.d ri - r r _here _ + where r=r r.r, =(r + r )/2 and r. (r r /2 Roble and Norton (1972) have discussed the evaluation of the integral in the above equation. From the values of x, the constituent profile C is obtained. It is important to note that the use of the Abel Integral equation as developed in this section is dependent on the strong line approximation contained in equation (4. 24). In the other inversion 73

methods the approximation was used as a convenience in testing the procedures to avoid lengthy transmittance calculations but is an essential part of the development of the Abel equation inversion. The prospects appear poor for modification to the more general situation where the strong line approximation is not valid. In order to do this we have to find an Xi which is a function of the measured transmittance and an x(r) which is a function of the concentration to use in equation (4. 23). However transmittance is expressed as an integral over wavenumber of the monochromatic transmittance, which is itself an exponential of integral along the line of sight. Robel and Hays (1972), working in the UV spectral region, were able to overcome this difficulty by assuming that the monochromatic transmittance was independent of wavelength over small wavelength regions and that other spectral parameters were independent of altitude. Both these approximations are invalid in the infrared region of the spectrum. The errors in Tmi will cause Xi and correspondingly the coefficients Ai and Bi to have included error terms. Roble and Hays (1972) have given expressions for the standard deviation of the retrieved constituent density errors due to statistical errors in the measurements. 4. 4 Results and Discussion Inversion for CO2 concentration in a 10 shell model atmosphere is tried as a first step. The atmosphere between 70 and 20 km. is divided into 10 shells each 5 km. thick. The atmospheric pressure and temperature at the mid-altitude of shell (from standard atmospheric tables) is used as the assumed constant pressure and 74

temperature in each shell. Assuming CO2 mixing ratio of 320 ppm., "measured" transmittances T are generated at spectral region -1..f -1 -1 -1 of 655 cm where S= 3. 2 cm (atm' cm) and a = 0. 07 atm cm 0 The Ti are used in the inversion procedures to obtain the CO2 mixing ratio in the atmosphere. The results are shown in Table 1. The onion peeling method requires an initial guess of mixing ratio at the topmost shell and is taken as 300 ppm. The other methods shown in Table 1 require an initial guess of the concentrations in all shells, and this guess is taken as 300 ppm. All techniques retrieve the 320 ppm. mixing ratio of CO2 above 30 km. In the Kalman-Bucy filter technique a noise R= 1. E-14 is used to obtain results shown in Table 1. In the eigenvector method, the last two eigenvalues are zero. As no noise in measurements (Ti ) is assumed, we use the first eight eigenvectors for the calculation. In the smoothing matrix Y = 160. is used to obtain solution shown in Table 4. 1. Further decrease in y will improve the solution for the top 3 levels. At tangent heights of 25 km. and below the atmosphere is opaque. (Tm=0) and retrievel of concentration of CO2 in the last two shells is not possible because of lack of information. Using the onion peeling technique, there is no retrievel for the bottom two shells because the partial derivative in the expression for C. is zero. In the matrix methods as discussed earlier, the initial guess of concentration in the last two shells is retained as the measurements contain no information on the concentration. To study the effect of random noise, the generated T are rounded off to the second decimal place, and the inversion procedures are rounded off to the second decimal place, and the inversion procedures 75

are tried using the same initial guesses of CO2 mixing ratio. The onion peeling method retrieves the concentration profile fairly accurately between altitudes 50 and 30 km. At the top of the atmosphere there is larger error because here even for comparatively large changes in mixing ratio of CO2, the transmittance does not change very significantly. No smoothing of retrieved profile is incorporated in the results shown for the onion peeling method, so that the noise dominates. The results from the eigenvectors method using 5 eigenvectors to calculate the solution are shown in Table 1. The 10 eigenvalues and eigenvectors of matrix B'B are shown in Table 4. 2. We see that eigenvalues X9 and X10 are 0, and X8 is approximately 104 times smaller than \1 and the inclusion of the smaller eigenvalues X8 and \7 in the calculation contributes little to actual solution but will greatly increase the error terms. Truncation after 5 eigenvectors gives a solution which appears the closest to the actual solution we desire of 320 ppm., the maximum deviations being + 19 and -20 ppm. The improvement and subsequent worsening of solution as the number of eigenvectors(and eigenvalues) used is increased is Shown in Table 3. The retrievel of CO2 profile is best between 50 and 30 km. Where the information content of data is maximum. An idea of the information content of the measurements can be obtained by studying the eigenvectors and locating the largest terms in each vector. Results from the smoothing matrix method, using the identity matrix for H and y = 6800. are shown in Table 1. Various other values for Y are tried. The value 6800. corresponds to the 5-th eigenvalue \5 of (B'B) and we see the solution is very close to that obtained from the eigenvector method truncated after 5 eigenvector terms. 76

In performing the eigenvector and smothing matrix methods we tacitly have assumed a linear approximation for the relationship between Tmi Tc and C. If our initial guess of constituent profile is not close to the actual, such an approximation is not valid. It is desirable to perform an itterative procedure similar to the one carried out in the onion-peeling method. For each new i.tteration, the initial guess of concentration is the one calculated in the previous itteration. The itterative procedure is continued until a desired convergence criteria is met. The solutions from the Kalman Bucy filter technique are consistent with results from the other methods (Table 1). A noise of R=1. E-12 is used in Table 1 for inversion using Tm with random error. The selection of the initial covariance matrix and the value of R for given data and the effect of these quantities on the inversion is being studied. A similar set of calculations for CO inversion in a 10 shell model atmosphere from 55 to 5 km. with 5 km. thick shells are shown in Table 4. Results of inversion for CO2 in a 22 shell atmosphere using the Abel equation technique are shown in Table 5. Results from Abel equation inversion for CO2 using a 10 shell atmosphere with 5 km. thick shells (not included) are poor because of the assumptions made in calculating Tmi By using the simple closed form expression for calculation of transmittances Tc, we were able to calculate the partial derivatives T Tci /ICK quite easily. When using the more detailed methods for calculating Tci, the partial derivatives have to be calculated bunhfid Tci lated by using the finite defference approximation CK. This 77

requires calculating the transmittance with a small perturbation of assumed concentration in the shells. This procedure is very time consuming when the exact expressions for transmittance are used. As seen above there are a variety of methods for inverting and smoothing of data. A particular technique may be ideal in some situations. In the inversion using the Abel equation no initial estimate of the constituent profile is necessary. This method although has some advantages is very difficult to apply in the case where we do not assume a simple closed form expression for transmittance. The technique using the Kalman Busy filter requires the additional calculations of the covariance matrices which is a drawback for quick calculations, although providing a more detailed description of the information content of the measurements. The smoothing matrix and eigenvector methods are very similar. The smoothing matrix method is the faster and more efficient technique while the eigenvector method gives more insight into the information content of the measurements. The onion reeling method is very simple and easy to apply. Incorporating smoothing of input transmittances or smoothing of retrieved constituent profiles, reduces the effect of random errors on the inversion. The effect of other errors like bias and scale errors etc. will be further studied. Selection of an initial estimate of the constituent profile, which should be close to actual constituent profile, required for some of the inversion techniques is sometimes difficult to make. We have assumed n measurements, and n tangent heights. It would be preferable to have more measurements, from which the 78

random error component can be minimized. The selection of a model atmosphere and especially the thickness of shells is being further studied. Thick shells give coarse results with all the atmospheric fine structure of constituent profile hidden. Dividing the atmosphere into very thin shells is preferred but in some of the methods this could cause instability. Depending on the constituent profile, a model atmosphere consisting of shells of suitably varying thickness can be constructed, giving the fine structure by having thin shells where required, and eliminating problems of instability by having thick shells where necessary. Newell and Gray (1972) using the KalmanBucy filter point that in some cases to prevent premature convergence, the tangent height data was sampled in coarse intervals repeatedly. The above mentioned techniques can easily be extended to retrieve two and more constituents simultaneously. Study of inversion of aerosols and selection of the optimum amount of smoothing for given data and separation of noise from measurements will be continued. 79

TABLE 4.1 Inversion for CO2 Mixing Ratio T Calculated at 655 cm 1 Using S = 3.2 cm 1 (atm. cm)1, m. a = 0. 07 and Mixing Ratio of CO2 of 320 ppm. Mixing Ratio ppm Mixing Ratio ppm T mi Eigen- Smoothing Kalman with Eigen- Smoothing Kalman Tangent T. Error Shell Ht Onion Vector Matrix Filter Onion Vector Matrix Filter i km Peeling (8 terms) Y=160 R=. E-14 Peeling (5 terms) Y=6800 R=l.E-12 1 65.99204 320 320 311 320.99 505 300 306 478 2 60.98389 320 319 316 322.98 492 301 314 488 3 55.96887 320 320 319 322.97 270 307 301 297 4 50.94144 320 320 320 323.94 339 339 321 340 5 45.89160 320 320 320 323.89 32.9 328 324 333 6 40.78849 320 320 320 322.79 313 313 313 317 7 35.56759 320 320 320 322.57 316 316 316 318 8 30.08438 320 320 320 322.08 324 323 323 325 9 25 0 - 300 300 300 0 - 300 300 300 10 20 0 - 300 300 300 0 - 300 300 300

TABLE 4.2 Eigen Values and Eigen Vectors of Matrix B'B CO2 Inversion i 1 2 3 4 5 6 7 8 9 10 Eigen values i 1.8 x 106 3.9 x 10+5 9.0 x 104 2.3 x 104 6.8 x 103 1.9 x 103 5.5x102 1.6 x 102 0 0 0.000 0.000 0.000 0.002 0.007 0. 030 0.155 0. 988 0.0 0.0 0.000 0.000 0.002 0.008 0.032 0. 152 0. 975 -0. 157 0.0 0.0 0.000 0. 001 0.007 0.035 0.160 0. 974 -0. 158 -0. 006 0.0 0.0 Eigen Vectors 0.001 0.006 0.030 0.167 0.971 -0.167 -0.007 -0.001 0.0 0.0 V.i 0. 004 0. 024 0.141 0.974 -0. 174 -0. 007 -0. 001 -0. 000 a. 0 0.0 0. 020 0. 125 0. 981 -0.146 -0.006 -0. 001 -0. 000 -0.000 0. 0. 0 0.117 0. 985 -0. 129 -0.006 -0. 001 -0. 000 -0. 000 -0. 000 0.0 0.0 0. 993 -0.118 -0.005 -0.001 -0. 000 -0.000 -0. 000 -0.000 0.0 0.0 0.0.0 -..0 0.0 -0.0 0.0 -0. 0 -.0 1.0. 0.0 0.0 -0.0 0.0 - 0. 0. -0. 0. - -. 0.0 1. 000

TABLE 4.3 Inversion of CO2 Using Method of Eigen Vectors T. With Random Error mi Mixing Ratio ppm. No. Shell Of Eigen 1 2 3 4 5 6 7 8 9 10 vectors included co0 1 300 300 300 300 300 301 303 325 300 300 2 300 300 300 300 300 302 318 323 300 300 3 300 300 300 301 303 318 316 323 300 300 4 300 300 301 306 334 314 316 323 300 300 5 300 301 307 339 328 313 316 323 300 300 6 1 300 301 305 340 328 313 316 323 300 300 7 330 492 274 338 328 313 316 323 300 300 8 479 468 273 338 328 313 316 323 300 300 10

TABLE 4.4 Inversion for CO Mixing Ratio T -1 Calculated at 213 using cm (atm cm ) and.065 Tmi Calculated at 2173 cm using S = 9.0 cm (atm cm ) and a =.065 Shell Tangent Mixing ratio ppm ht assumed for calculating T. mi 00 CA 1 2 3 4 5 6 7 8 9 10 50 45 40 35 30 25 20 15 10 5.12.10.08.05.02.03.04. 05.06. 08 T. mi.99799.99611.99310. 98905.98507. 96582.91533.78508. 52498 0 T. mi with error used in inversion methods. 998. 996. 993. 989. 985.97. 92.79.52 0 Initial guess of mixing ratio.13.11.081.051. 021.03.041.051.06.081 Onion Peeling.119.107. 0821.0502.0202.0223.0360. 048.0617 Eigen Vectors Method (9 terms).129.105.082.0502.0201.0218.0359.0479.0617.081 Kalman Bucy Filter R = 1.E - 17.118.105.0815.0502.0201.0217.0351.0474.0613.081

TABLE 4.5 Inversion Using the Abel Equation 22 Shell Model Atmosphere Smoothing Parameter M = 2 Shell Tangent Tmi CO2 mixing ratio ht km ppm 1 69.997 2 68.996 3 67.995 4 66.994 74 5 65.993 165 6 64.992 232 *^b~p~~~ ~7 63.991 183 8 62.989 158 9 61.987 294 10 60.986 350 11 59.984 273 12 58.981 304 13 57.979 332 14 56.976 331 15 55.973 339 16 54.969 340 17 53.965 336 18 52.960 339 19 51.955 338 20 50.949 21 49.942 22 48.934

Shell 1 SUN / H _= I --— ^ ^ ^ - -- - ~, ^,",S A T E L L I T 'E r. 1 Shell i n Figure 4. 1 Geometry of Solar Occultation Experiment

REFERENCES Burn, J. W. and W. G. Uplinger (1970), The Determination of Atmospheric Temperature Profiles From Planetary Limb Radiance Profile, NASA CR-1513. Chahine, M. T. (1968), Determination of the Temperature Profile in an Atmosphere From its Outgoing Radiance, J. Opt. Soc. of AM. 58, 1634. Conrath, B. J. (1969), On the Estimation of Relative Humidity Profiles from Medium Resolution Spectra Obtained form a Satellite. J. G. R. 74, 3347. Davis, R. E. (1969), A Limb Radiance Calculation Approach for Model Atmospheres Containing Horizontal Gradients of Temperature and Pressure, NASA TN-D 5495. Drayson, S. R. (1966), Atmospheric Transmission in the CO2 Bands Between 12 and 18 gm., App. Optics. 5, 385. Gille, J. C. (1968), On the Possibility of Estimating Diurnal Temperature Variation at the Stratopause from Horizon Radiance Profiles, J. G. R. 73, 1863. Goody, R. M. (1964), Atmospheric Radiation, I-Theoretical Basis., Oxford at the Clarendon Press. Gray, C. R., H. L. Malchow, D. C. Merritt, R. E. Var and C. K. Whitney (1973), Aerosol Physical Properties from Satellite Horizon Inversion, NASA CR-112311. House, F. B. and G. Ohring (1969), Inference of Stratospheric Temperature and Moisture Profiles from Observations of the Infrared Horizon NASA CR-1419. Kaplan, L. D. (1959), Inference of Atmospheric Structure from Remote Radiation Measurements, J. Opt. Soc. of Am. 49, 1004. King, J. I. F. (1964), Inversion by Slabs of Varying Thickness, J. Atm. Sc. 21, 324. Kunde, V. G. ('1967), Theoretical Computations of the Outgoing Infrared Radiance from a Planetary Atmosphere. NASA TN-D-4045. Mateer, C. L. (1964), A Study of the Information Content of Umkehr Observations, Tech. Rep. No. 2., NSF Grant No. G-19131., Coll. of Eng., University of Michigan McKee, T. B. and S. K. Cox (1973), Stratospheric Temperature Profiles from Limb Radiance Measurements., J. App. Meteor. 12, 867. 86

REFERENCES (continued) McKee, T. B., R.I. Whitman and J. J. Lambiotte (1969), A Technique to Infer Atmospheric Water Vapor Mixing Ratio from Measured Horizon Radiance Profiles, NASA TN-D-5252. Newell, R. E. and C. R. Gray (1972), Meteorological and Ecological Monitoring of the Stratosphere and Mesosphere, NASA CR-2094. Phillips, D. L. (1962), A Technique for the Numerical Solution of Certain Integral Equations of the First Kind., J. Assoc. Comp. Mach 9, 84. Prabhakara, C., B. J. Conrath, R. A. Hanel and E. J. Williamson (1970), Remote Sensing of Atmospheric Ozone Using the 9. 6 tm Band. Roble, R. G. and P. B. Hays (1972), A Technique for Recovering the Vertical Number Density Profile of Atmospheric Gases from Planetary Occultation Data., Planet. Sp. Sci., 20, 1727. Roble, R. G. and R. B. Norton (1972), Thermospheric Molecular Oxygen From Solar u-v. Occultation Data., J. G. R. 77, 3524. Rodgers, C. R. (1970), Remote Soundings of the Atmospheric Temperature Profile in the Presence of Could., Quat. J. Roy. Met. Soc. 96, 102. Russell, J. M. (1970), The Measurement of Atmospheric Ozone Using Satellite Infrared Observations in the 9. 6 /m Band, Rep. No. 036350-1-T, High Alt. Eng. Lab., Univ. of Mich. Russell, J. M. and S. R. Drayson (1972), The Inference of Atmospheric Ozone Using Satellite Horizon Measurements in the 1042 cm-1Band, J.Atm. Sci. 29, 376. Russell, J. M. and S. R. Drayson (1973), The Inference of Atmospheric Ozone Using Satellite Nadir Measurements in the 1042 cm- Band, NASA TR-R-399. Smith, W. L. (1970), Iterative Solution of the Radiative Transfer Equation for the Temperature and Absorbing Gas Profile of an Atmosphere, App. Optics. 9, 1993. Smith, W. L. (1972), Satellite Techniques for Observing the Temperature Structure of the Atmosphere, Bull. A. M. S. 53, 1074. Twomey, S. (1963), On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature, J. Ass. Comp. Mach. 10, 97. Venkateswaran, S.V., J. G. Moore and A. J. Krueger (1961), Determination of the Vertical Distribution of Ozone by Satellite Photometry, J.G. R. 66, 1751. 87

REFERENCES (continued) Wark. D. Q. (1970), SIRS: An Experiment to Measure the Free Air Temperature From a Satellite, App. Optics. 9, 1761. Wark, D. Q. and H. E. Fleming (1966), Indirect Measurements of Atmospheric Temperature Profiles from Satellite: I Introduction Mon. Wea. Rev. 94, 351. Yamamoto, G. and M. Tanaka (1967), Estimation of Water Vapor Distribution in the Atmosphere from Satellite Measurements, Application Satellites Edited by M. Lunc., Gordon and Breach Inc., New York. 88

Chapter 5. Discussion and Calculations. Although this study is not yet complete it is already evident that useful measurements of stratospheric distribution of some minor constituents can be made from a satellite using the solar occultation technique in the infrared spectral region. Furthermore comparatively simple instrumentation of medium spectral resolution (a few wavenumbers) may be employed for the more abundant of the minor molecular constituents such as water vapor, carbon dioxide, methane, nitrous oxide, ozone and perhaps carbon monoxide and nitric acid. Details of the absorption in different spectral regions have been given in the second chapter and recommendations on the spectral intervals to make measurements have been made for most of the molecules. Carbon dioxide has not been included in the chapter as it would probably be difficult to improve on our present knowledge of its stratospheric concentration. It is possible, however, that measurements in the Q-branch near 668 cm or in the 4. 3 gm band could yield information on the lower mesospheric distribution. Similarly nitric acid vapor has not been included, in this case because of the difficulty in determining its stratospheric absorption. The most promising spectral regions appear to be the Q-branches near 879 and 897 cm, although special care is needed to distinguish between nitric acid absorption and extinction by aerosols. In most of the stratospheric absorption calculations a band model was employed. In the majority of our future calculations we expect to use the line-by-line integration method for greater accuracy, although errors introduced by the band model do not affect the feasibility aspect of the study. For some of the molecules the more sophisticated technique may not be justified at the present time because of the inadequacy 89

of the spectral line parameters needed as input to the computer programs. This is certainly true for most of the ozone bands and probably for methane also. It is clear that a careful and comprehensive comparison between theoretical and laboratory data is required, not only for room temperature measurements but also for measurements taken at stratospheric temperatures. 220 K is a representative temperature in the lower stratosphere and few absorption measurements have been taken under these conditions. We recommend that laboratory absorption measurements of this nature be undertaken. Sulphuric acid aerosols in the lower stratosphere have been shown to give large values of extinction between about 800 and 1600 cm the exact characteristics depending on their concentration, size distribution, etc. This is encouraging for those who would like to study the aerosols, but adds another uncertainty to the determination of the concentrations of the molecular constituents. We need to be able to predict the extinction by the aerosols, possible by measureing the extinction in a spectral region close to that chosen for the molecule. The extinction calculations described are for realistic values of parameters but need to be extended to different ranges of size, concentration, composition, height distributions and spectral regions. Several different inversion techniques have been developed and tested but it is not yet apparent how best to incorporate the smoothing which is necessary to prevent the domination of noise. We expect to use the inversion programs to study the effect of many sources of error and will eventually be able to predict uncertainties in the concentrations of the retrieved profiles. A major problem will be the computation of the transmittances in the inversion program, made difficult by the large amount of computer time needed to calculate them accurately. 90

For the reasons stated in the first chapter we have largely confined our discussions to the more abundant of the minor constituents. Other molecules such as NO and N02 are of great interest but their absorption is smaller and would require more sophisticated instrumentation. We plan to examine some of these but may be limited by the absence of adequate spectral data, both theoretical and experimental. 91

APPENDIX Computer Programs for Line-by-Line Transmittance Calculations The computer programs written for the line-by-line calculation of atmospheric transmittances for the occultation geometry have been adapted from the already existing programs for slant path transmittance which assumed a plane parallel atmosphere. The atmosphere is divided into concentric shells and the regions between shells are assumed to be homogeneous. The program calculates the optical masses for each tangent height path in each region and uses them to compute the monochromatic transmittance along the tangent path. Refractive effects have not been incorporated but this would be easy to do since they would affect slightly the optical masses. Integration over wavenumber intervals of 0. 1 cm1 is accomplished by numerical quadrature. The following is a brief description of the input/output of the programs which are written in Fortran IV. Two programs are used: 1. SETUPV This program processes the line parameter data and outputs quadrature intervals, wing contributions, etc. 2. HSLANV computes the transmittances using the data out putted by SETUPV. Input /Output of SETUPV Line 23 Input of line parameters from binary file or tape. BNUS (I) Line wavenumber (cm ), rounded to two decimal places TS (J, I) Line intensities at 6 temperatures 300, 275, 250, 225, 200, 175 K corresponding to J = 1, 6. AR (I) Line halfwidth (cm ) at 1 atm. and 300K. 92

Line 34. ANUZ The starting wavenumber (cm ) NUMBER The number of 1 cm intervals for which transmittance calculations are to be made. All output statements write data to be used by HISLANV Input/Output of HSLANV Line 29. Namelist NAM1. PCRIT Pressure (mb) below which Voigt profile is used WTM Molecular weight of the molecule NQI, NQC Gaussian quadrature parameters, have value of 2 or 4 giving maximum efficiency and lowest accuracy or less efficiency and greater accuracy respectively. KMAX The number of tangent heights calculated. Altitude Z (km), pressure P(mb), temperature T(K) and concentration CONC must be specified at the (K + 1) bounding shells, starting with the highest altitude. CONC is mixing ratio by volume for IS = 1, or number density mol cm3 for all other values of IS. Line 49. 5. Prints the parameters inputted in line 29. Line 82. Namelist NAM2. ANUZ and NUMBA correspond to ANUZ and NUMBER in program SETUPV. All other input statements use the output from SETUPV. Line 397. ANUZ center of 1 cm interval for which calculation is made. TRAN (J, K) transmittance for tangent altitude Z(K + 1) and pressure P (K + 1) for 10 intervals averaged over 0. 1 cm. J = 1 corresponds to interval (ANUZ - 0. 5, ANUZ - 0. 4), J = 2 to the next highest etc. 93

1 C THIS PROGRAM I S CALLED SETUPV FCRTRAN IV 2 COMMCN NCSTRG,L,N NNCINT, NCN, NEND, IWARN 3 EECUIVALENCE (ST (1 ),TS( 1 ) ) 4 1)IMENSION NOSTRG{ 1C),L( 100),NOINT(72),N N0(144),IWARN(10) S DIMENS IN ISTRGL( 100 ), [NUS( '200), INUS(2200),I ABOVE (100), 6 1IBELOW( CO),D(2200),SM(18),ISS(2200) ST(13200),AR(2200),TS(6,2200) 7 1,IVST(2200) 8 DATA I 1 I2 IONE,13,KID, ISTC, IC,JJ/8*1/ 9 DATA JJJ, ITWO/4,0/ 10 200 FORMAT (F6.2,6E10.*4 11 201 FORMAT(F7.2,I5) 12 202 FORMAT (F7.2,6E10.4,F6.4, I9) 13 281 FORMAT (215,F6.1) 14 250 FORMAT(F7.2,I2,214,1112, 10I1/(6E104)) 15 252 FCPMAT (9(212,14)) 16 253 FORMAT (2413) 17 203 FORMAT (I2) 18 DIST=3.5 19 MS=2200 20 IDIST=D I ST*100. 22 00 301 I=1,MS 23 READ(7,END=1301) BNUSI) (TS(J,I ),J=1,6),AR I) 24 301 ISS(I)=(I-1)*6 25 CALL ERPOR 25.5 1301 MS=I-1 25.6 FSP=MS+1 26 INUS(MSP)=10COOOOOO 27 DO 303 I=1,MS 28 IF (ST(JJJ)-.1) 304,304,305 29 305 IVST(JJ) = I 30 JJ = JJ+1 31 304 JJ = JJJ+6 32 303 INUS(I )=(BNUS(I)+.001)*1CC. 33 JJ = JJ-1 34 802 READ (5,201,END=1 COC) ANUZ.NUMBER 35 1802 CONTINUE 36 NU2=ANUZ+.001 37 NUZZ=(NUZ/10)*10 38 AVNU=NUZZ 39 AVKU=4.5+AVNU 40 NLZY=NUZZ*100 41 NUZX=NUZY-I DIST 42 I S=IONE 43 DO 600 I=IS,MS 44 IF (NUZX-INUS (I)) 601,601t600 45 601 ICNE=I 46 GO TO 602 47 600 CCNTINUE 48 IONE=MSP 49 ITiO1=MS 50 GO TO 607 51 602 NUZV=NUZY+900+IDI ST 52 I'S=MAXO (ITWO 1 ) 53 DO 603 I=IS,MS 54 IF (NUZV-INUS(I)) 604,6C3,6C3 55 604 ITI01=I-1 56 GO TO 607 57 603 CONTINUE 58 I T 01=MS 59 607 CONTINUE 94

60 IOUT=MAXO( ITWO+,IONE) 61 I TO= I TWO 1 62 NUZY=NUZY-50 63 WRITE (6,281) ICNE,ITWO,AVNU 64 -IF (IOUT-ITWO) 611,611,610 65 611 WRITE (6,202) (8NLS(I), (TS(J,I),J=1,6),AR( I),II=IOUT,ITWO) 66 610 CONTINUE 67 801 NUZ= ( NUZ-.499 )*100. 68 DO 311 K=ISTO,MS 69 IF (INUS(K)-NUZ) 311,313,313 70 313 IST = K 71 GO TO 312 72 311 CCNTINUE 73 IST=MSP 74 312 NUZ = ANUZ+.001 75 NUZM=NUZ*100-60 76 IF (KID.GT.JJ) GO TO 762 77 DO 320 K = KID,JJ 78 KK = IVST(K) 79 KD = K 80 IF (INUS(KK)-NUZM) 32C,32C,322 81 320 CC:TINUE 82 KD=JJ+1 83 322 KID = KD-1 84 IF (KID.EQ. 0) KID=1 85 762 II=1 86 ITOTAL = 0 87 IIS = 1 88 JTOTAL = 0 89 DO 351 J=l,10 90 INUZM = NUZM + 10*J 91 MICNU = INUZM+5 92 331 IF (JJ-KID) 445,446,446 93 445 IWARN(J)=l 94 GO TO 336 95 446 KAD=IVST(KID) 96 IF (MIDNU-INUS(KAD)-10) 332,332,333 97 333 KID = KID+1 98 GO TO 331 99 332 IWARN(J) = 1 100 IF (IABS(MIDNU-INUS(KAD))-10) 335,336,336 101 335 IWARN(J) = 0 102 336 CCKTINUE 103 NOSTRG(J) = 0 104 I0 = I 105 400 IF (IST-MSP) 410,351,351 106 410 JALFA=INlS(IST)-INUZM 107 IF (JALFA-10) 350,350,351 108 350 L(IIS) = JALFA 109 NOSTRG(J) = NOS'TRG(J)+1 110 JTCTAL = JTOTAL+1 111 ISTRGL(I S)=IST 112 IF (JALFA-1O) 352,353,352 113 353 IABOVE(IIS)=-1 114 IBELOW( I IS) = 1 115 IF (INUS(IST)-INUS(IST-1)-1l 354,354,355 116 354 IBELOW(IIS) = 0 117 355 IIS = IIS+1 118 GC TO 351 119 352 IF (JALFA) 356,356,357 95

120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 356 IABOVE(IIS)=1 IBELOW(I I S) =-1 IF (INUS(IST+1)-INUS( IST)-) 358 IABOVE(IIS)=0 359 GO TO 360 357 IABOVE(IIS)=1 IBELOW(I IS)=1 IF (INUS(IST+1)-INUS(IST)-1) 361 IABOVE(IIS)=0 362 IF (INUS(IST)-INUS(IST-1)-l) 363 IBELOW IIS)=O 364 CONTINUE 360 IIS = II S+1 IST = IST+1 GO TO 400 351 CONTINUE IS TO=IST BNU=ANUZ-DI ST IS=MAXO( 1, I1) DO 460 I=IS,MS IF (BNU-BNUS( )) 461,460,46C 461 I1 = 1-1 GO TO 462 460 CONTINUE I1=MS 462 BNU=ANUZ+DIST IS=I2 DO 463 I=ISMS IF (BNU-BNUS(I)) 464,464,463 464 12 = I GO TO 465 463 CONTINUE I2=MSP 465 IFIRST = I1+1 ILAST = 12-1 KK = 0 DO 370 M1=1,3 Y=M1-2 BNL = ANUZ+Y/2. DATA DXX/50./ IS IS=0 IF ('1)401,401,402 402 DO 371 I=1, I1 DXY=BNUS (I)-BNU D(I )=DXY*DXY IF(C(I).GT.DXX) ISIS=I 371 CONTINUE 401 ISIS=ISIS+1 IF (MS-I2) 403,1401,1401 403 ISIT=MS GO TO 1403 1401 DO 372 I=I2,MS DXY=BNUS( I )-BNU D( I)=DXY*DXY ISIT=I IF (D(I).GT.DXX) GO TO 1404 372 CCNTINUE GO TO 1403 1404 ISIT=ISIT-1 1403 DO 370 K=1,6 358,358,359 361,361,362 363 t363,364 96

180 S=0. 181 KK=KK+1 182 IF (I 1-I SIS) 1405,4C6,4C6 183 406 DO 373 I=ISIS, I1 184 ISUB=ISS(I)+K 185 373 S=S+ST( I SUB ) AR (I)/D( I) 186 1405 IF (IS IT-12)370,4C5,405 187 405 DO 374 I=12,ISIT 188 ISUB = ISS([)+K 189 374 S=+ST(I SUB)*AR( I)/( I ) 190 370 SM(KK) = S 191 CALL GRGNK 192 WRITE (6,250) ANUZJTOTAL,IFIRST, ILAST,(NOSTRG(I),I=1, 10 193 1 ),NOIN (NOINT( I ),I=,10)O,(SM(I),1=1,18) 194 380 IF (JTOTAL) 383,383,382 195 382 WRITE (6,252) (IBELOW(I),I ABOVE(I),ISTRGL(I, 1,JTOTAL) 196 383 IF (NOIN) 385,385,384 197 384 NOb = NOIN*2 198 WRITE (6,253) (NEND(I),I=1,NCN) 199 385 CCNTINUE 200 NIMBER=NUMBER-1 201 IF (NUMBER) 802,802,306 202 306 ANtUZ=ANUZ+1.0 203 NUZ=ANUZ+O. 1 204 IF (NUZ-(NUZ/10)*10) 1802,1602,801 205 1000 STOP 206 END 207 SUPROUTINE GRONK GRONK 01 208 CCMMON NCSTRGL,NOINT,NOII,NEND,IWARN 209 DIMENSION NOSTRG(10),L(100),NOINT(72),NEND(144),IWARN(10) 210 KK = 1 211 M1 = 1 212 DO 300 I = 1,10 213 MINIT = M1 214 M = 0 215 J = NOSTRG(I) 216 IF (J) 301,301,302 217 301 NENDIM1) = 0 218 NENC(MI+1) = 10 219 Ml = M1+2 220 M = 1 221 GO TO 320 222 302 K = 1 223 IF (L(KK)-2) 330,303,303 224 303 NEND(Ml) = 0 225 NEND(M1+1) = L(KK)-1 226 M = M1+2 227 = M+l 228 330 IF(K-J) 304,34C,304 229 304 IF (L(KK+1)-L(KK)-2) 305,305,306 230 306 NEND(MI) = L(KK)+1 231 NEND(M1+1) = L(KK+1)-1 232 Ml = M1+2 233: = M+1 234 305 K = K+1 235 KK = KK+1 236 IF(K-J) 330,340,330 237 340 IF (L(KK)-8) 307,307,350 238 307 NEND(M1) = L(KK)+1 239 NENC(Ml+l) = 10 97

240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 END OF FILE I = M+1 M1 = M1+2 350 KK = KK+1 320 CONTINUE IF (IWARN(I)) 311,311,312 311 MZERO = M JJ=1 201 IF {JJ-MZERO) 321,321,380 321 IF (NEND(MINIT+1}-NEND(MINIT)-3) 360,360,323 323 M=M+2 IX=NEND( MINIT+1 )-NENC(M INIT ) I 1=( IX+ 1)/3 12=(2*IX+1)/3 MALL = M1 - MINIT -1 DO 37C I =1,MALL MSUB = Ml-II 370 NENO(MSUB+4)=NEND(MSUB) NEND(M IN IT+1 )=NENO(M IN IT)+I 1 NEND (MINIT+2)=NEND(M IN IT+1 ) NEND(MINIT+3) =NEND(MINIT)+I2 NENC(M IN IT+4)=NEND( MINIT+3) 1P=M1+4 MINIT=M INIT+4 360 MINIT = MINIT+2 JJ=JJ+ 1 GO TO 201 380 CChTINUE 312 NOINT(I) = M 300 CONTINUE Ml = MI-1 NOIN = M1/2 RETURN END 98

1 C PRCGRAM PSLANV FORTRAN IV 2 C MOCIFIED FOR 2 OR 4 POINT CUACRATURE 3 CC MON ANU, ANIUZ,SECSN,TRAN,WWA, I I IST,JMAX.KADDKSLAKMAX,M, 4 1 KMESS,KSTOP,IE,AVNUKCFIT,K,P,ZENtAL2,ALALP,X,Y,ZERO,LSANZ, 5 2 ANY, CJUMPAR,GNU,ARR,GNUU,ST 5.5 CCPMCN /ADD/ CC(630) 6 DIMENSION ANU(250),IST(250),ANZ(250),ANY( 250 ) C(35 ) SEC (10), 7 1 SN(150),ST(8750),TRAN(2800),IE (8),P(36),AL2(35),AL(35), 8 2 ALP(35),X(35),Y(35) ZEFO (35) JUMP(35) 9 DIMENSION BNU(250),IBELOW(2CC),ENDPT(200),ENGTH(36),SM(18),T(36)t 10 1 TM(35),WAB(4), ISTRGL(100), IA(250),ITN(35),NO[NT(10}). 11 2 NOSTRG( 10), IABOVE (100),X(35) Yl(35),ZEROI(35),B(6 )WA(4) 12 DIMENSION AR{250),GNUU(20),ARR(250),GNU(20) 12.5 DI MENSIC\ CONC(36),Z(36) XX(35) 13 NAMELIST /NAM1/ PCRIT,WTMKMAXINQI,NQCCONC,ZlIS,PT 14 NAMELIST /NAM2/ NUMBAANUZ 15 902 FORMAT (2I5,F6.1) 16 904 FOPMAT(F7.2, 6E10.4,F6.4) 17 905 FORMAT(F7.2, 12,2 14,i11 2,10 1/ (6E10.4)) 18 906 FORMAT (S(212,I4)) 19 907 FORMAT ( 24F3.2) 20 800 FORMAT(F 10. 1,F 0.2,4 16/(F8.1, F1O.2,2E14.4)) 21 801 FORMAT (216/(10F8.3)) 21.4 DATA NQI/2/,NQC/2/,WTM/44./ 21.5 DATA ENDPT/. C. 001,.002,.003,.005,.01/ 22 DATA A CIST/.8/,ANDIST/.099/ PCRIT/100./,PPP/1013.25/,PC02/.032/ 23 KCRIT=O 24 TTT=298.0 25 AAA=l.O 26 I SCH=1 27 I1=0 28 PI E=3.14 15927 29 READ (5,NAM1) 29.5 KSLA=1 30.4 NCC=NOC*5 30.5 NCB=NQC*4 31 II=1 32 DO 100 I=1,5 33 CALL GAUSSN (ENDPT I ),ENDPT( I+1),NU(II),ENGTH (I ) NQC) 34 100 II= I+NQC 35 DO C11 I=1,NCC 36 GNUU(I)=CNU( )*GNU(I) 37 101 ENGTH(I)=ENGTH( I)*10.0 38 DOP= 3.581136E-7/SQR T( WTM) 39 ALCG2=ALCG(2.) 40 ALOG2=SORT(ALOG2) 41 RCOTPI=SQRT(PIE) 45 KME SS=1 0* KM AX-1 ) 46 KO=KMAX*KSLA 47 KLCT=KO*10 48 KADD=KMAX*10 49 KPLUS=KMAX+1 49.5 WRITE(6,800) PCRIT,WTM,NQI,NCCKMAX, IS, (Z(K),T(K),P(K), 49.6 lCONC(K ),<=1,KPLUS) 50 AAA=AAA/PPP 51 ALPHA=AAA*AAA*TTT 52 DO 102 1=1,225 53 JA=I-1 54 I ST ( I ) =K AX* JA 55 102 I (I )=6*JA 99

55.1 56 57 58 59 60 61 62 62. 1 62.2 62.3 62.4 62.5 62.6 62.7 62.8 62.9 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 KZ=1 DO 103 K=1,KMAX TA=(T (K) +T (K+l))/2. PA=(P(K)+P( K+1))/2. AL 2 ( K)= ALPHA/TA*PA*PA ALK) =SQRT(AL2 (K)) ALP(K)=AL(K)/PI E C (K)=(CCIC(K)+CCNC(K+1) )*.5 IF (I S.EC. 1) C(K)=C(K)*PA/(TA*.38054E-19) ZZ=(6378.+Z(K+1) )*2. DO 1033 I=1,K hH=Z( I)-Z(K+1) 1033 XX(K)=SQRT(HH* (ZZ+P1H)) XX K+1)=0. CO 1034 I=1,K CC (KZ) =(XX(I )-XX( I+1 ) )*C( I *8.176E-15 1034 KZ=KZ+1 IF (PA-PCRIT) 104,105,105 104 JUYP(K)=1 KCRIT=K AD=DOP*SCRT (TA) X1(K)=ALCG2/AD Y1(K)=AL(K)*X1(K) ZER01 (K) =X1 (K)/ROOTPI GO TO 106 105 JUM1P(K)=0 106 TAA=TA-275. DO 107 N=2,5 IF (TAA) 107,107,108 108 ITN(K)=N GO TO 109 107 TAA=TAA+25.0 ITN(K)=5 109 TN=ITN(K) TP=325.-25.*TN 103 TM(K)=(TA-TP )/25. 190 READ (5,NAM2) WRITE (6,NAM2) IS5=4 110 READ (5,902) IFIRST,ILASTAVNU DO 111 K=1,KCRIT X(K)=X1 (K)/AVNU Y K) =Y1 (K) /AVNU 111 ZERO (K)=ZERO1(K )/AVNU IJ=IFIRST-1 IMAX=ILAST-I J GO TO (401,402),ISWCH 401 IJKL=1 GO TO 403 402 IJK=ILASTA-IFIRST+1 IF (I JK) 401,401,404 404 ISHIFT= IMAXA-IJK IF (ISHIFT) 410,410,411 411 DO 405 I=1,IJK J=I+ISHIFT AR(I )=AR(J) ARR(I )=ARR(J) 405 BNU(I)=BNU(J) [ SHIFT=I SHIFT*KMAX IJKJ=IJK*KMAX 100

106 DO 406 I=1,IJKJ 107 J=I+ISHIFT 108 406 ST(I)=ST(J) 109 410 IJKL=IJK+1 110 403 IF (IMAX-IJKL) 412,413,413 111 413 ISUB=IST(IJKL) 112 DO 450 I=IJKL,IMAX 113 READ (5,904) BNU(I), (( J),J= 1,6), AR(I) 114 ARR(I)=AR(I)*AR(I) 115 DO 451 K=l,6 116 B(K)=B(K)*AR(I) 117 451 B(K)=ALOG(B(K)) 118 DO 450 K=t1KMAX 119 ISUB=ISUE+1 120 JA=ITN(K) 121 S1=B(JA-1) 122 S2=B(JA) 123 S3=B(JA+1) 124 450 ST (I SUB) = EXP (S2+ ( (S1 S3-S2-S2 )*TM (K )+S1-S3)*TM( K)/2 ) )*ALP(K) 125 412 ILASTA=ILAST 126 IMAXA=IMAX 127 IShCH=2 128 120 READ (5,905) ANUZ,JTOTAL,I 1,12,(NOSTRG(I),1=1,10),NOIN,(NOINT (I), 129 1 I=1,10),(SM( I ),I=1, 18) 130 IF (JTOTAL) 600,600,601 131 601 READ (5,906) (IBELOW(I),IABCVE(I),ISTRGL(I),1=1,JTOTAL) 132 600 hCN=NOIN*2 133 IF (NOIN) 602,602,603 134 603 READ (5,907) (ENDPT(I),I=1,NON) 135 602 CONTINUE 136 I 1=I1- I J 137 I2=I2-IJ 138 IE(1)=I1 139 IE(6)=I2 140 DO 130 K=1,KMAX 141 DO 130 N=1,3 142 JA=IA(N ) + ITN(K) 143 SU Fl=SM(JA-1 ) 144 SUM2=SM(JA) 145 SUP3=SM(JA+1) 146 ISLB=IST(N)+K 147 130 SN( ISUB)=(SUM2+ ( SUM3+SUM1-2.*SUM2 )*TM(K)+SUMl-SUM3)*TM(K)/2.0) 148 1 *ALP(K) 149 MI=1 150 M2=1 151 IF (I1-I2) 170,170,171 152 170 CONTINUE 153 CO 121 I=11,12 154 A=NU( I)-ANUZ 155 IF (A) 122,122,123 156 122 N=(A-.005)*100. 157 GO TO 125 158 123 N=(A+.005)*100. 159 125 A=N 160 121 ANL(I)=A/100. 161 171 CONTINUE 162 00 124 N=1,KLOT 163 124 TRAN(N)=C.O 164 DO 303 M=l,10 165 ANL O=M-6 101

166 ANUO=ANUO/1C. 167 ANUA=ANUC+.05 168 IEA=LLOP( I1,12,ANUA-ADIST,ANU) 169 IE(1)=IEA-1 170 IE(6)=LOP(I EAI2,ANUA+ACIST,ANU) 171 CALL LUOKBY(ANUA, I 1, 2,3,ANUA) 172 IE(1)=IE(1)+1 173 IE(6)=IE(6)-1 174 NOIE=NOINT(M) 175 IF (NCIE) 180,180,181 176 181 CONTINUE 177 00 306 MM=1,NOIE 178 KS TOP=KMAX+1 179 CALL GAUSSN(ENDPT(Ml),ENDPT(M1+ ),WAB, hA,NQI) 180 DO 200 III=1,NCI 181 I=I I I 182 BA =WAAB( II)+ANUO 183 hW'A=WA (II)*10. 184 IEA=LOOP(IE(1),IF(6),BAD-.2,ANU) 185 IE(2)=IEA-1 186 IE (5)=LOCP( IFA,IE(6),BAD+.2,ANU) 187 200 CALL LOOKBY(BAD, IE(1),IE(6),1,lO.*WAB(II )-.5) 18b 306 MI=M1+2 189 180 CONTINUE 190 NOIE=NOSTRG(M) 191 IF(NCIE)182,182,183 192 183 CCNTINUE 193 00 309 MM=1,NOIE 194 LS=I STRGL(M2)-IJ 195 FREQ=ANU(LS) 196 IE(3)=LOCP( IE (1), IE(6),FREQ-.199,ANU) 197 IE(2)=IE(3)-1 198 IE( 7)=LOOP( IE (3 ), IE( 6 ),FREQ-ANDIST,ANU)-1 199 IE(8)=LOCP(LS+1 IE(6),FPEC+ANDIST,ANU 200 IE(5)=LOCP(IE(8),IE(6),FRE+. 199,ANU) 201 IE(4)=IE(5)-1 202 I X I= (FREC-ANUA)* 1CC.1 203 BAD=IXI 204 CALL LCCKEY (FREQ, IE(1), 1E(6),0,BAD/10.) 205 IE(7)=IE(7)+1 206 IE(8)=IE(8)-1 207 IE(3)=LS-1 2C8 IE(4 =LS+1 209 IF ( IRELCW(M2)) 141, 142, 143 21C 142 I SLBB=NCB 211 GO TO 144 212 143 ISLBB=NCC 213 144 lO 145 I I =1,ISUBB 214 11=111 215 BAC=FPEO-GNU(II) 216 WWA=ENGTH( I I ) 217 145 CALL LCOKBY(f.AD, IE(7),IE(8),2,-GNU(II)*50.) 218 141 IF (I AlA0\E(M2)) 146,147,148 219 147 ISUBB=NCB 220 GC TO 149 221 143 ISLBB.=NCC 222 149 DCO 150 1 I I=1, IStBB 223 11=111 224 BAC=FREQ+GNU( II ) 22 5 IW A= ENGT H( I ) 102

226 150 CALL LOOKBY (BAD,IE(7),IE(8),2,GNU(II)*50.) 227 146 CONTINUE 228 309 F2=M2+1 229 182 CONTINUE 230 303 CONTINUE 231 CALL PPIPUNCISS) 232 NUMBA=NUMBA-1 233 IF (NUMBA) 151,151,152 234 151 GO TO 190 235 152 NUZ=ANUZ+1.1 236 IF (NUZ-(NUZ/*10)10)110,110,120 237 END 238 SUBROUTINE GAUSS(AA,BB,C,D) 239 DIMENSION C(4),D(4) 240 8=BB 241 A=AA 242 X1=.8611363116 243 X2=.3399810436 244 1l=.3478548451 245 Y2=.6521 451549 246 X=(1.-Xl)*.5 247 Y=(l.+X1)*.5 248 C (1)=B*X+A*Y 249 C(4) =B*Y+A*X 250 X=(1.-X2)*. 5 251 Y= 1.+X2)*.5 252 C(2)=B*X+A*Y 253 C(3)=B*Y+A*X 254 = (B-A)*.5 255 D.1)=Y1*A 256 0(4)=D(1) 257 D(2)=Y2*A 258 D(3)=D(2) 259 RETURN 260 END 261 FUNCTION LOOP( IA,IR,ZA,Z8) 262 DIMENSION ZB(250) 263 IF (IA-IB) 100,100,101 264 101 LOOP=IA 265 RETURN 266 100 CONTINUE 267 CO 202 I=IA,IB 268 IF (Ze(I)-ZA) 202,202,103 269 103 LOOP=I 270 RETURN 271 202 CONTINUE 272 LOOP=IB+1 273 FETURN 274 END 275 FUNCTION VOIGT (XIN,YIN) 276 REAL*4 HH(2)/.8049141,.8121283E-01/,XX(2)/.5246476, 1.650680/,A(42) 277 1 /0.0,. 2,0. -.184,0 0,. 15584,0.0,-. 121664O0.0,.8770816E-1,0.0,-.5 278 2 851412E-1,0.0,.3621573E-1,0.0,-.2084976E-1,0.0,.1119601E-1,0O.,279 3.5623190E-2,0.0,.264E763E-2,0.0,-.1173267E-2,0.0.4899520E-3,0.0 280 4,-.1933631E-3,0.C,.72287 5E-4,0. O0,-.2565551E-4,0.0,8662074E-5,0. 281 5 0,-.2787638E-5,0.0.8566874E-6,0.0,-.2518434E-6,0. O,.7093602E-7/ 282 DIMENSION RA(32),CA(32) RB(32),C8(32),B(44),AK(5 ),AM(5),DY(4) VOIGTO03 283 X=XIN 284 Y = YIN VOIGTO10 285 X2 = X*X VOIGTO11 103

286 Y2 = Y*Y VOIGTO12 287 IF (X-7.0) 200,201,201 VOIGTO13 288 200 IF (Y-1.) 202,202,203 VOIGTO14 289 203 RA(1) C. VO GT015 290 C(1) = O. VOIGT016 291 Re () = 1. VOIGT017 292 CB(1) = O. VOIGT018 293 RA(2) = X VOIGTO19 294 CA2) = Y VOIGT020 295 RB(2) =.5-X2+Y2 VOIGT021 296 CB(2) = -2.*X*Y VOIGT022 297 CB1 = CB(2) VOIGT023 298 UV1=0. VOIGT025 299 DO 250 J=2,31 VOIGT026 300 JMINUS = J-1 VOIGT027 301 JPLUS = J+1 VOIGT028 302 FLOATJ = JMINUS VOIGT029 303 RB1 = 2.*FLOATJ+RB(2) VOIGT030 304 RA = -FLOATJ*(2.*FLCATJ-1.)/2. VOIGTO31 305 RA(J+1) =RB1*RA(J)-CB1*CA J)+RAI*RA RJ-1 ) 306 C (J+1)=RB1*CA(J)+CB 1*RA( J)+RA1*CA(JMINUS) 307 RB (J+1) =RBI*RB( J)-CB1*CB J) RA1*RB(J-1) 308 CB (J +1 )=RB1*CB( J) +CB1*RB (J) +RA1*CB { J-1) 309 UV= CA ( J PLUS )*RB(JPLUS)-RA(JPLUS)*CB(JPLUS))/(R8(JPLUS)*RB(JPLUS)+VOIGT036 310 1CB(JPLUS)*C8(JPLUS)) VOIGT037 311 IF (ABS(UV-UV1)-1.E-6) 251,25C,250 312 250 UV1=UV VOIGT039 313 251 VOIGT=UV/1.772454 VOI GTO40 314 RETURN VOIGT041 315 202 IF (X-2.) 301 301,302 VOIGT042 316 301 AINT = 1. VOIGT043 317 MAX = 12.+5.*X2 VOIGTO44 318 KMAX = MAX-1 VO IGT045 319 KO=O 320 DO 303 K=KO,KMAX 321 AJ = MAX-K VOIGT047 322 303 AINT = AINT*(-2.*X2)/(2.*AJ+1.)+1. VOIGT048 323 U = -2.*X*AINT VOIGT049 324 GO TO 304 VOIGT050 325 302 IF (X-4.5) 305,306,306 VOIGT051 326 305 B(43}=0. VOIGT052 327 e(44) = 0. VOIGT053 328 J = 42 VOIGT054 329 00 307 K = 1,42 VOIGT055 330 B(J) =.4*X*B(J+1)-B(J+2)+A(J) VOIGT056 331 307 J = J-1 VOIGT057 332 U = B(3)-B(1) VOIGT058 333 GO TO 304 VOIGT059 334 306 AINT = 1.0 VOIGTO60 335 MAX = 2.+40./X VOIGT061 336 AMAX = MAX VOIGT062 337 00 308 K=1,MAX VO GT063 338 AINT AINT*(2.*AMAX-1.)/(2.*X2)+1. VOIGT064 339 308 AMAX = AMAX -1. VOIGT065 340 U = -AINT/X VOIGT066 341 304 V = 1.772454*EXP(-X2) VOIGT067 342 H =.02 VOIGT068 343 JM = Y/H VOIGT069 344 IF (JM) 310,311,310 VOIGT070 345 311 H=Y VOI GT071 104

346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 310 Z = 0. VOIGT072 L = 0 VOIGT073 CY(1) = 0. VOIGT074 312 DY(2) = H/2. VOIGT075 DY(3) = DY(21 VOIGT076 DY(4) = H VOIGT077 318 AK(1) = C. VOIGT078 AM(1) = 0. VOIGT079 DC 313 J=1,4 VOIGT080 YY = Z+DY(J) VOIGTO81 UU = U+.5*AK(J) VOIGTO82 VV = V+.5*AM(J) VOIGT083 AK(J+1) = 2.*(YY*UU+X*VV)*H VOIGT084 AM(J+1) = -2.*(1.+X*UU-YY*VV)*H VOIGT085 IF (J-3) 213,314,313 VOIGT086 314 AK(4)=2. AK(4) VOIGT087 AM(4) = AM(4)+AM(4) VOIGT088 313 CONTINUE VOIGT089 Z=Z+H VOIGT090 L = L+1 VOIGTO91 U = U+.1666667*(AK(2)+2.*AK(AK(3)+AK( AK(5)) VOIGT092 V = V+.1666667*(AM(2)+AM(3)+AM(+AMAM(4)+AM(5)) VOIGT093 IF(JM) 315,320,315 VOIGT094 315 IF (L-JM) 318,317,320 VOIGTO95 317 AJM = JM VOIGT096 H=Y-AJM*H VO IGT097 GC TO 312 VOIGT098 320 VOIGT= V/1.772454 VOIGT099 RETURN VOIGT100 201 Fl = 0. VOIGT101 DO 330 J=1,2 VOIGT102 330 F=Fl+HH (J)/(Y2+(X-XX(J))*(X-XX(J)))+HH(J)/(Y2+(X+XXJ) )*(X+XX(J)) VCIGT103 1) VOIGT104 VOIGT=Y*F1/3.1415927 VOIGT105 RETURN VOIGT106 END VOIGT 107 C PRIPLN HAS BEEN MODIFIED FOR USE WITH SLANTV SIBROUTINE PRIPUN(IS5) COMMON ANU,ANUZSEC,SN,TRAN,WWA,II,IST,JMAXKADD,KSLAKMAXM, 1 KMESS, KST OP, I E, AVNU, KCR IT,K,P, ZEN, AL2,AL,ALP, X, Y,ZEROL S,ANZ 2 ANYC,JUMP AR,GNU,ARR,GhUU, ST DIMENSION ANU(250),IST(250),ANZ(250),ANY(250),C(35) SEC(1O), 1 SN(150),ST(8750),TRAN(10,280),IE(8),P(36),AL2(35),AL(35), 2 ALP(35),X(35),Y(35),ZERO(35),JUMP(35) DIMENSION BNU(250),IBELOW(200),ENDPT(200),ENGTH(36),SMn18),T(36), 1 TM(35),WAB(4),ISTRGL(10O),IA(250),ITN(35),N[NT (10), 2 NOSTRG(10),IABOVE(1CC),X1(35),Yl(35),ZEROI 35),B(6) WA(4) DIMENSION AR(250),GNUU(20),ARR(250),GNU(20) 900 FORMAT (16H1 INTERVAL IS,F6.1//(1H,10F9.6,Fl0.2,15)) IS5=1 KMID=KMAX*KSLA WRITE (6,900) ANUZ,((TRAN(JK),J=,10),P(K+1 ),K,K=1,KMID) WRITE(8) ANUZ,((TRANIJ,K),K=1,KMID),J=1, 10) RETURN ENC C LOCKBY HAS BEEN MODIFIED FOR USE WITH SLANTV SUBROUTINE LOOKBY(FRECUE t II I1,112,II14 Z) COMMON ANU, ANUZ,SEC,SN,TRAN,WWA,II I STJMAXKADDKSLA,KMAXM, 1 KME SSKSTOPIE,AVNU, KCRIT,K,P, ZEN, AL2,AL,ALP.X,YZEROLS,ANZ, 2 ANY,C,JUMP,AR GNU,ARR,GNUUST 105

405.5 COMMON /ACO/CC(630) 406 DIMENSION ANUI 250),IST(25C),ANZ(250),ANY(250),C(35),SEC(10), 407 1 SN(150),ST(8750),TRAN(2800), IE(8),P(36tAL2(35),AL(35), 4C8 2 ALP(35),X(35),Y(35),ZERO(35)J35JUMP35) 409 DIMENSION AR(250),GNUU(2C) ARR(250) GNU(20) 409.5 DIMENSION COEF(35) 410 DIMENSION SZ(105),SA(105) 411 FREQ=FREQUE 412 111=1111 413 112=1112 414 14=114+1 415 ZZ=Z 416 ZZZ=ZZ*2. 417 KP=2 418 GO TO (1C0,101,102,.103),I4 419 100 CELTA=.O1 420 GO TO 104 421 103 DELTA=.05 '422 104 IC=-1 423 GO TO 403 424 101 IY=IE(2)+1 425 IZ=IE(5)-1 426 102 JSLANT=M+KMESS 427 F=0. 428 403 IF (II1-I12) 180,180,181 429 180 00 200 I=Il,II2 430 ANZ(I)=AS (ANU(I)-FREQ) 431 200 ANY(I)=ANZ(I)*ANZ(I) 432 181 CONTINUE 433 KPP=2 434 D00 202 KK=1,KMAX 435 K=KK 436 GO TO (110,111,111,113),14 437 111 CONTINUE 438 501 GO TO (121,121,122), 4 439 113 SN1=SN(K) 440 JA=K+KMAX 441 Sh2=SN(JA) 442 JA=JA+KMAX 443 SN3=SN(JA) 444 GO TO 105 445 122 SN1=SZ(KP-1) 446 SN2=SZ(KP) 447 SN3=SZ(KP+1) 448 GO TO 105 449 121 CCNTINUE 450 110 SN1=SA(KPP-1) 451 SN2=SA(KPP) 452 SN3=SA(KPP+1) 453 KPP=KPP+3 454 105 S RE=( (SN3+SN1-S N2-SN2)*ZZZ+SN3-SN1 ) *ZZ+SN2 455 GO TO (130,131,132,133),I4 456 131 IF (JUMP(K))134,134,135 457 134 SRE=SRE+PNTZ(II 1 12) 458 GO TO 136 459 135 SRE=SRE+RNTZ( II1,IE(2))+RNTZ(IE(5), I112)+XED(IY,IZ) 460 GO TO 136 461 132 IF (JUMP K)) 137,137,138 462 137 SRE=SRE+PNTZ(II1, IE(3)) RNTZ( IE4), I12) 463 AKNU=AR( LS)*AL( K) 106

464 I =IST(LS)+K 465 SRE=SREST( 18)/(GNLU(II)+AL2(K)*ARR(LS)) 466 GO TO 136 467 138 SRE=SRE+XED IIl,IE(3) )+XEC( IE(4), 112) 468 IB=IST(LS)+K 469 SRE=SRE+ST( IBA)/ALP(K)/AR(LS)*VOIGT(GNU( I *X(K),Y(K)*AR(LS))* 4701 ZERO(K) 471 136 COEF(K)=SRE 472 473 474 475 476 477 GO TO 202 478 130 IF (JUMP(K)) 150,150,151 479 150 SZZ=RNTZ ( II1, IE(7 ) )+RNTZ(IE(8), I 12) 480 GO TO 152 481 151 SZZ=RNTZ (I II, IE ( 2 ) +RNTZ( IE( 5) I I 2)+XED( IE (3) tIE(7) +XED( IE(8),IE( 482 14)) 483 152 SZ(KP)=SZZ+SRE 484 GO TO 202 485 133 SA(KP)=SRE+RNTZ(I II1, IE1 ) +RNTZ(IE(6),1 I12) 486 202 KP=KP+3 487 GC TO (160,1661,1661,160),14 487.1 1661 JSLANT=M 487.2 KX=1 487.3 00 503 J=1,KMAX 487.4 F=O. 487.5 DO 502 K=1,J 487.6 F=F-CC ( KX)*COEF (K) 487.7 502 KX=KX+1 48'7.8 TRAN (JSLANT) =T RAN (JSLANT) +WWA*EXP (F) 487.9 503 JSLANT=JSLANT+10 487.95 GO TO 161 488 160 IC=IC+1 489 ID=IC+1 490 GO TO (170,171,172), IO 491 170 KP=1 492 FREQ=FREQ-DEL TA 493 IF (14-1) 173,174,173 494 174 ZZ=ZZ-.1 495 GO TO 175 496 173 ZZ=FREQ 497 175 ZZZ=ZZ*2. 498 GO TO 403 499 171 KP=3 500 FREQ=FREQUE+DELTA 501 IF (14-1) 176,177,176 502 177 ZZ=Z+. 1 503 GO TO 178 504 176 ZZ=FREQ 505 178 ZZ=ZZ*2. 506 GO TO 403 507 172 CONTINUE 508 161 RETURN 509 ENO 510 C XED HAS BEEN MODIFIED FOR USE WITH SLANTV 511 FUNCTION XED(IAIB) 512 COMMON AKLANUZ,SECSNTRANWWAI I ISTJMAXKADDKSLAKMAXHM 513 1 KMESSKSTOP,IE,AVhU, KCRI T,K,P, ZEN AL2,AL,ALP, X, YZEROtLSANZ, 107

514 2 ANY, CJUMPARGNUARR, GNUUST 515 DIMENSION ANU(250),IST(250),ANZ(250),ANY(250),C(35),SEC(10), 516 1 SN(150 ),ST(8750),TRAN(2800),IE(8),P( 6),AL2(35),AL(35), 517 2 ALP(35),X(35),Y(35), ZERO(35),tJUMP 35) 518 DIMENSION AR(250),GNUU(20),ARR(250),GNU(20) 519 IF (IA-I ) 100,1o00,101 520 100 KUP=IST(IA)+K 521 SUM=0.0 522 XA=X(K) 523 YA=Y(K) 524 00 102 I=IA,IB 525 SLM=SUM+ ST(KUP) *VOIGT (ANZ (I )*XA,Y A*AR ( ) )/AR (I )/ALP(K) 526 102 KUP=KUP+KMAX 527 XED=SUM*ZERG(K) 528 RETURN 529 101 XEC=O.O 530 PE TURN 531 END 532 C RNTZ HAS BEEN MODIFIED FOR USE WITH SLANTV 533 FUNCTION RNTZ(IA,IB) 534 COMMON ANU,ANUZ,SEC,SN,TRAN,WWA1 I, I ST,JMAX,KADD,KSLA,KPAX M, 535 1 KMESS,KSTOP, IEAVNUKCRIT,K,P,ZENAL2,AL,ALP,X,Y,ZEROLS,ANZ, 536 2 ANY,C,JUMP AR GNU,ARR,GNUU ST 537 DIMENSION ANU(250),IST(250),ANZ(250),ANY(250),C(35),SEC(10), 538 1 SN(150),ST(8750),TRAN(2800),IE(8),P(36),AL2(35),AL(35), 539 2 ALP(35) X(35),Y(35),ZEPO(35) JUMP(35) 540 DIMENSION AR(250),GNUU(20),ARR(250),GNU(20) 541 IF (IA-IB) 100,100,101 542 100 KUP=IST( IA)+K 543 SUM=0.0 544 A=AL2(K) 545 00D 102 I=IA,IB 546 SLt=SUM+ST(KUP)/(ANY(I)+A*APR(I) 547 102 KUP=KUP+KMAX 548 RNTZ=SUM 549 RETURN 550 101 RNTZ=0.0 551 RETURN 552 END 553 SUBROUTINE GAUSSN(AA,BB,C,D,N) 554 DI ENSIC: C(4),D(4) 555 DATA A/.2886751/ 556 IF (N.EQ.4) GO TO 100 557 IF (N.EQ.2) GO TO 200 558 WRITE(6,950) N 559 STCP 560 S50 FORMAT'O WRONG QUADRATURE: K=',14) 561 200 CC=(AA+BB)*.5 562 OC=Be-AA 563 DD=DC*A 564 C(1)=CC-DD 565 C(2)=CC+DD 566 1)()=DC*.5 567 C(2)=D(1) 568 RETURN 569 100 CALL GAUSS(AA,BB,C,D) 570 RETURN 571 END END OF FILE 1 NIVERSIY OF MICHIGAN 3 9015 02844 0488 108