T HE UN IV ER SIT Y OF MI CHI GAN COLLEGE OF ENGINEERING Department of Meteorology and Oceanography Technical Report No. 2 A STUDY OF THE INFORMATION CONTENT OF UMKEHR OBSERVATIONS Carlton L. Mateer E. So Epstein Project Director ORA Project 04682 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. G19131 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April 1964

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1964.

ACKNOWLEDGMENTS The author wishes to express his gratitude to all who assisted him during the course of this studyo He is particularly appreciative of the advice and assistance rendered by Professors Edward So Epstein and Aksel CO Wiin-Nielsen, Co-Chairmen of his Doctoral Committee. The author also wishes to thank Professors Eo Wendell Hewson and Donald Ao Jones for serving as members of the committee and for the help they have giveno An expression of deepest gratitude is due Dro Hans Uo DUtsch, National Center for Atmospheric Research, for serving as a member of the committee, for his guidance and encouragement and his interest in the work, and for his generous provision of tables, data, and other information connected with his method of evaluating Umkehr observationso The tabular and graphical material related to Dr. DUtschVs method are reproduced here with his kind permission. The author also wishes to express his appreciation for many helpful discussions with, and the constant interest and encouragement of his associates, particularly Messrso Allan Ho Murphy, Chien-Hsiung Yang, and Charles Youngo Mro So Roland Drayson pointed out the approximate equivalence of the evaluation method developed by the author and that proposed by Twomeyo Dro Paul Ro Julian suggested the interpretation of the generating function curves in terms of the resolving power of the Umkehr observations O Acknowledgment is due to Professor Alan Wo Brewer, University of Toronto, and Mr. Wayne So Hering, Geophysics Research Directorate, for provision of unpublished ozone sonde data, and to the Meteorological Service of Canada and Mro Walter Do Komkyr, United States Weather Bureau, for provision of Umkehr datao While engaged in this work, the author has been supported by the Meteorological Service of Canada, and the National Science Foundation~ The support of the National Center for Atmospheric Research, where the author was a summer visitor in 1963, is also appreciatedo The author also wishes to acknowledge the support of The University of Michigan through the use of the IBM 7090 at the Computing Center, Professor R.C.F. Bartels, DirectorO ii

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES x ABSTRACT xiii 1. INTRODUCTION 1 1.1 A Brief Survey of the History of Ozone-Meteorological Research 1 1.2 A Description and Qualitative Explanation of the Umkehr Effect 4 1.3 The Problem and the Approach 8 2. REVIEW OF UMKEHR EVALUATION TECHNIQUES 11 2.1 Preliminary Remarks 11 2.2 Method A 14 2.3 Method B 19 2.4 Method of DUtsch 22 2.5 Sources of Error 25 2.5.1 Multiple scattering and reflected light 25 2.5.2 Empirical cloud-corrections and large particle scattering 31 3. THE CHARACTERISTIC PATTERNS OF UMKEHR CURVES 35 3.1 Preliminary Remarks 35 3.2 The Statistical Procedure 37 3.3 The Application of the Statistical Procedure 40 3.4 Physical Explanation 57 4. DISCUSSION OF THE LINEARIZED EVALUATION METHOD 68 4.1 Preliminary Remarks 68 4.2 The "Complete" Solution of the Linear Equations 70 4.3 The Problem of Information Versus Noise 76 4.3.1 The use of the characteristic patterns of the Umkehr curve 76 4.3.2 Stepwise solutions using the eigenvectors of A*A 78 iii

TABLE OF CONTENTS (Concluded) Page 4.4 Objective Methods of Smoothing the Solution 82 4.4.1 Truncation of the eigenvector expansion 82 4.4.2 Twomey's method 83 4.5 Solution Contributions Associated with the Characteristic Patterns of Umkehr Curves 89 5. FURTHER REMARKS ON EVALUATION METHODS AND PRESENTATION OF RESULTS 92 5.1 The Scaling Problem 92 5.2 The Effect of Adding Random Noise to the Observations 109 5.3 -The Need for More;than One Standard Distribution 109 5.3.1 Convergence of the iterative procedure using the second derivatives 109 5.3.2 Average solutions 111 5.4 The Need for Second Derivative Corrections 115 5.5 Comparison with DUtsch's Solutions 118 5.6 Another Orthogonal Vector Expansion for Solutions 121 5.7 Vertical Distributions Using Other Wavelength Pairs 129 6. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK 140 APPENDIX A. TERMINOLOGY AND UNITS USED IN OZONE-METEOROLOGICAL RESEARCH 143 B. FURTHER DETAILS OF DUTSCH'S EVALUATION METHOD 148 C. THE COMPUTATION OF THE GENERATING FUNCTION CURVES FOR PRIMARY SCATTERING 160 D. THE PROCEDURE USED WITH OZONE SONDE DATA 163 E. TABULATIONS OF INDIVIDUAL SOLUTIONS 168 BIBLIOGRAPHY 193 iv

LIST OF TABLES Table Page 1. Higher Order Scattering Corrections to be Subtracted From Observed N-Values at Various Zenith Angles 27 2. The First Eight Eigenvalues of the Correlation Matrix of the Points on the Umkehr Curve, Including (in parentheses) Correlation Between the Eigenvector Coefficient and Total Ozone 43 3. The First Eight Eigenvalues for the Correlation Matrix of the Points on the Umkehr Curve When the Instrument Constant is Eliminated and Total Ozone is Used 46 4. The First Eight Eigenvalues for the Covariance Matrix of the Points on the Umkehr Curve When the Instrument Constant is Eliminated and Total Ozone is Used 50 5. Characteristic Patterns for the Covariance Matrix of the Points on the Umkehr Curve When the Instrument Constant is Eliminated and Total Ozone is Used for the North American C Wavelength Data Sample 54 6. Frequency Distributions of Truncation Levels for Expansions of Umkehr Curves in Terms of Their Characteristic Patterns 56 7. Eigenvalues and Vectors for one Configuration of DUtsch's First Derivative Matrix for his Standard Distribution I 74 8. Averages of Smoothed and Error Dot Products and Fractional Variance Explained by Each and by the Combined Dot Products 77 9. Frequency Distribution of the Number of Eigenvectors Used in the Stepwise Solution Procedure and Average Fractional Umkehr Curve Variance Explained by Each Vector 79 10. Stepwise Solutions for March 21, 1962, Showing Individual Solution Contributions by Each Eigenvector 81

LIST OF TABLES (Continued) Table Page 11. Stepwise Solutions for March 21, 1962, Showing the Individual Solution Contributions by Each Eigenvector Using Twomey's Method With 7 = 0.5 87 12. Average Solution for Arosa Data Sample by Twomey's Method With Column Scaling of 10, WQ = 0.1, and y = 0.5 94 13. Average Solution for Arosa Data Sample by Twomey's Method With Column Scaling Equivalent to Standard Distribution Layer Partial Pressures, WQ = 0.ol (1.0) and y = 0.5 96 14. Solution Statistics for Arosa Data Sample for Solutions by Truncated Eigenvector Expansion Method With Column and Equation Scaling Vectors Determined From Derivative Matrix 97 15. Solution Statistics for Arosa Data Sample for Solutions by TEVE and Twomey Methods With Column Scaling Vector Determined From Derivative Matrix, WS2 = 0.1, and y = 0.1 99 16. Averages and Standard Deviations of Layer-Mean Partial Pressures for Balloon Soundings and Umkehr Data 101 17. Column Weighting Vectors Used With Derivative Matrices in Solutions 102 18. Statistics for Arosa Data Sample Solutions Carried out by TEVE and Twomey Methods Using Column Weighting Vector CI, W2 = 0.1, and SI 103 19. Statistics for Arosa Data Sample Solutions to Illustrate the Effect of Variations in the Weight on the Ozone Conservation Equation, with 7 = 0.5 104 20. Statistics for Arosa Data Sample Solutions Using CII and CIII with Twomey's Method and Using CI With the "Combined" TEVE-Twomey Method 105 21. Statistics for Arosa Data Sample Solutions With Rev spect to Standard Distribution II when Total Ozone is Less than 300 m atm-cm9With W2 = 0.5 107

LIST OF TABLES (Continued) Table Page 22. Statistics for Arosa Data Sample Solutions With Respect to Standard Distribution III When Total Ozone Exceeds 375 m atm-cm 108 23. Statistics for Arosa Data Sample Solutions When Random Noise Has Been Added to the Umkehr Curves 110 24. Frequency Distributions of the Number of Iterations Required for Convergence When the Second Order Partial Derivatives are Used 112 25. Average Solutions for Arosa Data Sample With Second Derivative Corrections Included when Total Ozone is Less Than 300 m atm-cm 114 26. Average Solutions for Arosa Data Sample With Second Derivative Corrections Included When Total Ozone Exceeds 375 m atm-cm 116 27. Average Solutions for Arosa Data Sample to Illustrate Differences Between Linear (L) and Nonlinear (NL) Solutions 117 28. Individual Solutions Chosen to Illustrate the Differences Between Linear (L) and Nonlinear (NL) Solutions 118 29. Solution Statistics for Arosa Data Sample to Compare the DUtsch, TEVE, and Twomey Methods of Solution With Second Derivative Corrections Applied 120 30. Correction Factors for AD Total Ozone Measurements Used in Umkehr Evaluations 131 31. First Four Eigenvalues of A*A With Mean and RMS Residuals for the Various Wavelength Combinations 138 B-1. Standard Vertical Distributions of Ozone Used by DUtsch 152 B-2. Umkehr Curve Points for the Various Standard Distributions and Wavelength Pairs, With Secondary Scattering Effects Included 153 vii

LIST OF TABLES (Continued) Table Page B-3. First Order Partial Derivatives for Standard Distribution I, A Wavelength Pair, With Secondary Scattering Effects Included 154 B-4. First Order Partial Derivatives for Standard Distribution I, C Wavelength Pair, With Secondary Scattering Effects Included 155 B-5. First Order Partial Derivatives for Standard Distribution I, D Wavelength Pair, With Secondary Scattering Effects Included 156 B-6. First Order Partial Derivatives for Standard Distribution II, C Wavelength Pair, With Secondary Scattering Effects Included 157 B-7. First Order Partial Derivatives for Standard Distribution III, C Wavelength Pair, With Secondary Scattering Effects Included 158 B-8. The Overlapping-Layer Zenith-Angle System Used by DUtsch to Obtain Smooth Solutions 159 D-1. Statistical Parameters Used in Estimating Layer-Mean Partial Pressures From Total Ozone 166 E-1. Individual Solutions for Arosa Data Sample, by Twomey's Method, With Respect to SI, and With Second Derivative Corrections Applied 169 E-2. Individual Solutions for 42 Low-Ozone Arosa Umkehrs, by Twomey's Method, With Respect to SII9 and With Second Derivative Corrections Applied 173 E-3. Individual Solutions for 29 High-Ozone Arosa Umkehrs, by Twomey's Method, With Respect to SIII, and With Second Derivative Corrections Applied 175 E-4. Individual Solutions for Arosa Data Sample, by TEVE Method, With Respect to SI, and With. Second Derivative Corrections Applied 177 viii

LIST OF TABLES (Concluded) Table Page E-5. Individual Solutions for 42 Low-Ozone Arosa Umkehrs, by TEVE Method, With Respect to SII, and With Second Derivative Corrections Applied 181 E-6. Individual Solutions for 29 High-Ozone Arosa Umkehrs, by TEVE Method, With Respect to SIII, and With Second Derivative Corrections Applied 183 E-7. Individual Solutions for 93 Arosa Umkehrs by DUtsch's Techni que 185 E-8. Individual Solutions for 98 North American Umkehrs on C Wavelengths, by Twomey's Method, With Respect to SI, and With Second Derivative Corrections Applied 189 ix

LIST OF FIGURES Figure Page 1. Umkehr curve for observations at Edmonton on the afternoon of May 22, 1961. 5 2. Average vertical distribution of ozone from ozone sonde and Umkehr data (Table 16). 7 3. Schematic diagram showing the path of the direct solar beam to the scattering point and that of the scattered beam from there to the instrument on the ground. 12 4. Schematic diagram showing the atmospheric layer division used in Method A. (after Walton). 15 5. Curves of t(Qk,Q,x1,x2) = N(@k,Q,x1,x2) for O1 = 800 and ~2 = 86.50~ on xl, x2 diagram. 18 6. Corrections (to be subtracted from observed N-values) for ground reflection in the case of 80% albedo. (After Dave and Furukawa.) 31 7. Cloud corrections (N-units to be subtracted from observed values) as a function of luxmeter ratio and solar zenith angle. 32 8. Sample Umkher curve for A.rosa, March 30, 1962, showing observed values and corrected values. Total ozone is 394 m atm-cm. 33 9. "Standard" Umkehr curves for A rosa and North America for the C wavelength pair. 45 10. First Characteristic Pattern of the correlation matrix, with the instrument constant eliminated. 48 11. Second Characteristic Pattern of the correlation matrix, with the instrument constant eliminated. 49 12. Third Characteristic Pattern of the correlation matrix, with the instrument constant eliminated. 49

LIST OF FIGURES (Continued) Figure Page 13. First Characteristic Pattern of the covariance matrix, with the instrument constant eliminated. Compare with Fig. 10 but note difference in scale. 51 14. Second Characteristic Pattern of the covariance matrix, with the instrument constant eliminated. 52 15. Third Characteristic Pattern of the covariance matrix, with the instrument constant eliminated. 53 16. Scatter diagram of first pattern vector coefficient plotted against total ozone deviation from mean for C wavelength pair, North America sample. 58 17. Scatter diagram of first pattern vector coefficient plotted against total ozone deviation from mean for C wavelength pair, A rosa sample. 59 18. Vertical distribution of ozone used in the computation of source function X(Gz). 61 19. Source functions X(Q,z) plotted against height for various zenith angles for the A wavelengths. 62 20. Source function X(O,z) plotted against height for various zenith angles for the C wavelengths. 63 21. Source functions X(O,z) plotted against height for various zenith angles for the D wavelengths. 64 22. Relative intensities and intensity ratios for wavelengths A, C and D plotted on a logarithmic scale against solar zenith angle. 65 23. Solution contribution in the various layers for each of the first four Characteristic Patterns. 91 24. Illustrating the smooth curve obtained from the original block distribution. 124 xi

LIST OF FIGURES (Concluded) Figure Page 25. Average solutions for 42 low-ozone cases at A.rosa using the Characteristic Pattern method, with TEVE solutions for comparison. 125 26. Average solutions for 29 high-ozone cases at Arosa using the Characteristic Pattern method, with TEVE solutions for comparison. 127 27. Average solutions for 100 A.rosa Umkehr curves using the Characteristic Pattern method, with TEVE solutions for comparison. 128 28. Average solutions for 98 North American Umkehr curves for the individual wavelength pairs. 13 2 29. Average solutions for 98 North American Umkehr curves for the double wavelength pairs. 133 30. Average solutions for 98 North American Umkehr curves for combined wavelength pairs and double pairs. 134 31. Average solutions for 98 North American Umkehr curves for the C wavelength pair by different methods. 135 xii

ABSTRACT The purpose of this study is to determine precisely how much information about the vertical distribution of ozone is contained in Umkehr observations, with particular reference to the predictability of the so-called secondary ozone maximum in the lower stratosphere. Since vertical ozone distributions are used in atmospheric circulation studies, it is particularly important to know the limitations of distributions determined from Umkehr observations. First, the observations are examined to determine how many linearly independent pieces of information may be derived from an Umkehr curve. Empirical orthogonal functions are used in this analysis and the characteristic patterns of the Umkehr curve, and their relative importance, are determined. When the noise of the Umkehr curve has been filtered out by expanding the curves in terms of their characteristic patterns, there are at most four linearly independent pieces of information contained in each curve. Moreover, little or no additional information is obtained when the observations are taken on more than a single wavelength pair. A simple physical explanation is given for these results. Second, the linearized evaluation method of DUtsch is examined by eigenvalue analysis to determine the number of pieces of information that may be determined from the system. The result of the purely statistical analysis is confirmed, viz., that there are at most four pieces of information about the vertical distribution of ozone to be determined. In addition, even when we solve only for four pieces of information, the result depends on the standard distribution from which the solution is computed and on the scaling of the system of linear equations used. By expanding the solution in terms of the eigenvectors of the system, it is determined that the wild oscillations in the complete solution of the system are introduced by those eigenvectors representing linear combinations of the unknowns about which the observations contain no information. By removing from the system these linear combinations whose amplitudes are determined by the noise of the Umkehr curve, a smooth, physically realistic solution is obtained. The equivalence of this solution method to one recently advanced by Twomey is demonstrated and the results of solutions by both methods are presented and compared with solutions obtained by Dfutsch's method. Finally, it is concluded that there is no possibility of determining from Umkehr observations whether or not there exists a distinct secondary maximum in the lower stratosphere. Moreover, the vertical distributions of ozone obtained from Umkehr observations are to be compared with each other only when determined by the same objective technique. Vertical distributions obtained xiii

by subjective methods depend on the opinion of the evaluator about what the vertical distribution "should" look like and on chance decisions he may make in his evaluation. It is clear that great care must be taken when making inferences about atmospheric motions from vertical distributions obtained from Umkehr observations. xiv

lo INTRODUCTION 1.1 A BRIEF SURVEY OF THE HISTORY OF OZONE-METEOROLOGICAL RESEARCH Ever since the first systematic measurements, by Dobson and his collaborators (1926, 1927, 1929, 1930), of the total amount of ozone in a vertical column of the atmosphere, meteorologists have been fascinated by the strong relationship between ozone amount and day-to-day weather variationso Equally interesting are the pronounced seasonal and latitudinal variations in ozone amount with a spring maximum and autumn minimum at all latitudes~ In all seasons, the ozone amount increases from the equator toward the poleso The above discoveries were followed immediately by a series of papers by Chapman (1930), Mecke (1931), Wulf (1932, 1934), and Wulf and Deming (1936a, 1936b), in which the photochemical theory of ozone formation in the upper atmosphere was advanced, and the vertical distribution of ozone calculated on the basis of an equilibrium between the reactions producing and destroying ozone under the influence of solar ultraviolet radiationo The photochemical equilibrium theory suggested a vertical distribution in qualitative agreement with that deduced from the observations of Eo Regener and Vo HO Regener (1934), who employed a small ultraviolet quartz spectrograph sent aloft on a balloon to measure the vertical distribution of ozone up to 32 km. At the same time, Go3tz, Meetham, and Dobson (1934) were able to infer the main features of the vertical dis

tribution from measurements of the Gotz (1931) Umkehr or inversion effect. These studies showed a maximum ozone density between 20 and 30 km, the latter study indicating a rapid decrease with height above 30 km, with most of the ozone in the atmosphere being below 30 km. In 1937, Wulf and Deming calculated an approximate "rate of maintenance" of the photochemical equilibrium. They found that the rate increased with height and suggested that ozone below 30 km was "protected" from the influence of the photochemical reactions and that air motions would be important in determining the vertical distribution in these layers. Taking advantage of later information available, Schroer (although not published until 1949, his work was completed in 1944), Dutsch (1946), and Craig (1948), independently, considered the effects of variation of solar elevation (and hence latitude) on the photochemical distribution, as well as the reaction times. They found a summer solstice maximum and a winter solstice minimum of total ozone, with a latitudinal gradient such that ozone increased from the poles to the equator. These predictions are in distinct contradiction of the observations. However,.the studies also showed that the time required to approach equilibrium decreased very rapidly as height increased, being of the order of a year or more at 20 km, of days at 30 km, and hours at 40 km (Craig's results)o Thus ozone is neither created nor destroyed at an appreciable rate below about 25 to 30 km. These results led to the conclusion that ozone mixing ratio was a conservative property of the atmosphere below these levels and, hence, that the explanation for the observed seasonal and latitudinal

distribution of ozone must be sought in the general circulation of the atmosphere. Conversely, any acceptable theory of the general circulation of the atmosphere must also be consistent with the observed distribution of ozoneo Thus ozone plays a unique role in general circulation research for not only does it influence, through its radiative properties, the distribution of heat sources and sinks in the stratosphere and mesosphere, but it is also important as a tracer in studies of atmospheric motions. The period following the second world war has seen many advances in ozone-meteorological research. The global network of stations measuring total atmospheric ozone has greatly increased. The balloon observation technique of Eo. and V. H. Regener was continued (Paetzold, 1954). The vertical distribution of ozone has been measured from rocket-borne equipment (Johnson et al., 1952). Instruments for use with balloons have been developed by Kulcke and Paetzold (1957), Vassy (1958), Brewer and Milford (1960), and Regener (1960). With regard to indirect methods, the Umkehr technique has been improved by Dutsch (1957, 1959a, 1959b) and infrared techniques have been developed by Epstein, Osterberg and Adel (1956), Goody and Roach (1956), and Vigroux (1959)o Other methods proposed and used include lunar eclipse measurements (Paetzold, 1952), twilight balloon or satellite photometry (Pittock, 1961, 1963; and Venkateswaran, Moore and Kreuger, 1961), and measurements from a satellite (Singer and Wentworth, 1957, Twomey, 1961, and Rawcliffe et al., 1963). During this period, further impetuws was given to ozone -meteorological research by the discovery, reported by Teweles and Finger (1958), of the large ins

creases in ozone amount which were associated with the explosive stratospheric warmings (Scherhag, 1952). Most of these developments, and many others, have been recorded in greater detail in the reviews of Craig (1950), Gotz (1951) and Taba (1961). 1.2 A DESCRIPTION AND QUALITATIVE EXPLANATION OF THE UMKEHR EFFECT The Umkehr effect is observed when measurements are made with an 1 ultraviolet spectrophotometer, of the ratio of the zenith sky light intensities of two wavelengths in the solar ultraviolet when the sun is near the horizon. The shorter of the two wavelengths (intensity I) is strongly absorbed by ozone, the other (intensity I') is weakly absorbed. If the value of log I/I' is plotted against the sun's zenith angle, it is observed that this log-intensity ratio decreases as the zenith angle increases until a minimum is reached for a zenith angle of about 85~ (when the wavelengths are 3114 and 3324 A). As the zenith angle increases further, the log-intensity ratio increases again. This effect, first noticed by G6tz (1931), is illustrated in Fig. 1, where the quantity (-100.0 log I/I' + constant, called the N-value) is plotted against the fourth power of the zenith angle. It is customary to use a high power of the zenith angle as abscissa so that the values obtained when the sun is close to the horizon are spread out on the graph. 1A brief description of the instrument, the terms, and the units used in ozone-meteorological research, insofar as used in this report, is given in Appendix A.

ZENITH ANGLE (DEGREES) 60 65 70 74 77 80 83 85 86.5 89 90 150 88 140 130 120 110 C X (3114/3324) 100 90 80 70 60 50 1 2 3 4 5 6 7x107 (ZENITH ANGLE)4 Fig. 1. Umkehr curve for observations at Edmonton on the afternoon of May 22, 1961.

The reason for the occurrence of this effect and for its sensitivity to the gross features of the vertical distribution of ozone were explained by G~otz, Meetham, and Dobson (1934) as follows. We first note the main features of the vertical distribution. of ozone as shown in Fig. 2, with a maximum of ozone partial pressure just above 20 km and lower values above and below this level. Considering light which is scattered only once in the atmosphere, the light received by the instrument is contributed by light scattered downwards from all levels in the atmosphere. The amount of light contributed by scattering at any particular level depends on (a) the number of air molecules available at that level to scatter the light, and (b) the absorption by ozone and the scattering by air molecules both before and after the scatteringo For any given zenith angle of the sun, the effect of (a) is to decrease the contribution as height increases, while the effect of (b) is to increase the contribution, since more and more of the longer slant path of the direct ray before the scattering event is replaced by the shorter vertical path after the scattering evento It turns out (see Figs. 19-21) that, for a given zenith angle, the light contributing to the intensity comes from a fairly well-defined layer of the atmosphere and that it is possible to consider an effective scattering height~ This effective scattering height depends on the ozone absorption coefficient and on the solar zenith angle, increasing, in fact, with each of theseo Hence, the effective scattering height will always be higher for the short wavelength which is more strongly absorbed. Thus, as the sun approaches the horn

45 2 AZ 5~~~~~~~~3 10 30 P R P I \\ \ I ~ I Y\ I 1 I E 20 s - 025 $ H U E 50 I E I II -20 G (mb) H 100 i15 (kin) km) 200. 10 500 5 1000 0 50 100 150 200 250 300 PARTIAL PRESSURE OF OZONE (pFmb) Fig. 2. Av~erage vertical, distribution of ozone from ozone sonde and Umk~ehr data (Table i6).

zon, the two intensities decrease, but I more rapidly than I', so that I/It decreases. However, when the effective scattering height for the short wavelength is above the ozone maximum, I decreases more slowly than I', because the ozone absorption occurs mostly in the shorter vertical path after the scattering event, and the ratio I/I' increases until the effective scattering height for I' is also above the ozone maximum. Then the ratio I/I' again decreases. For all pairs of wavelengths used with the ozone spectrophotometer, this second reversal occurs when the sun is below the horizon. It is clear that the existence of the reversal or inversion in the Urmkehr curve implies the existence of a maximum of ozone concentration at some level in the atmosphere. Moreover, one would expect the position of the reversal to be related to the total amount of ozone and to the position of this concentration maximum in the atmosphere. Thus, it is reasonable to expect to be able to infer some information about the vertical distribution of ozone from measurements of the Urmkehr effect. 1.3 THE PROBLEM AND THE APPROACH In recent years, ozone workers have sought to extract more and more information about the vertical distribution of ozone from Umkehr observations. In particular, there has been considerable interest in *the possibility that the main features of the lower stratospheric structure, viz., the existence or nonexistence of the so-called secondary maximum, might be inferred from such observations. The aim of the present study

was to determine precisely how much information about the vertical dis-~ tribution of ozone could be obtained from the Umkehr observations, with particular reference to the lower stratospheric structure. As a byproduct of this study, an extension of the Dutsch evaluation technique has led to the development of another method for estimating the vertical distribution. This method, which was developed independently of a very similar one proposed by Twomey (1963), takes into account the actual information content of the observations and, in addition, permits the incorporation of known facts about the vertical distribution and its variability as mathematical constraints on the solution. Following a brief review of the literature pertaining to research on Umkehr evaluation techniques in the second chapter, the problem of determining the information content of the observations is approached from two quite separate points of view. First, in the third chapter, the problem of statistically deriving a linear transformation between the points on the Umkehr curve and the vertical distribution is considered by examining the curve to determine how many linearly independent pieces of information may be derived from ito In this process, the characteristic patterns of the Umkehr curve, and their relative importance, are determinedo Second, in the fourth chapter, the linear physical-mathematical transformation of DUtsch is examined by eigenvalue analysis to determine the number of pieces of information that may be deduced from the system~ From this analysis, another method of solving for the vertical distribution is developed and its approximate equivalence to the

10 method proposed by Twomey is indicated. Finally, in the fifth chapter, the imposition of constraints consistent with our independent knowledge of the vertical distribution is considered and results are presented showing the effects of these constraints on the derived vertical distribution. The results are compared with those obtained by Dutscho In addition, other possible solution methods are discussed and some results presented.

2. REVIEW OF UMKEHR EVALUATION TECHNIQUES 2.1 PRELIMINARY REMARKS Certain basic principles are common to all evaluation techniques and it will be convenient to discuss these first. The evaluation of the Umkehr effect involves the computation, by numerical quadrature, of the quantity log I'/I plus an unknown instrumental constant. This constant is usually eliminated by taking a further ratio. Thus, if Q is the solar zenith angle, we use (log I'/I)Q - (log I'/I)O, where Go is a zenith angle such that the quantity (log I'/I)o~ depends mostly on the total amount of ozone and very little on the vertical distribution. Referring to Fig. 3, if Io is the intensity in a narrow spectral region outside the earth's atmosphere, then the intensity, Is, at the scattering point will be, Is = Ioexp -z (r3+P)(sec ~)pdh (1) where depletion by ozone absorption and molecular scattering only are considered, and where a = ozone absorption coefficient (gm-1) r3 = ozone mixing ratio (mass of ozone/mass of air containing the ozone) at height h = Rayleigh scattering coefficient (gm'l) = angle of incidence of direct beam at height h, and p = air density at height h. 11

12 ZENITH SKY DIRECT SOLAR BEAM POINT EARTH'S SURFACE CENTER OF EARTH Fig. 3. Schematic diagram showing the path of the direct solar beam to the scattering point and that of the scattered beam from there to the instrument on the ground. The amount of energy scattered downwards. in the direction of the instrument from the air molecules in a layer of thickness dz is just dIs = Kp(l+cos20)Ispdz (2) where K is a constant. This energy will undergo further depletion in its vertical path from the scattering point to the instrument. The amount of this energy finally received at the instrument is Z dI = dIs exp -( (ar3+))pdh The total intensity received at the instrument, for light scattered down

13 wards at all heights in the atmosphere, is obtained by integrating (3) over the entire vertical column. Combining (1), (2), and (3), the total intensity I is then given by 00 ' Z 00 ' I = IoKP(l+cos2Q) exp - (ar3+P)pdh- z (ar3+P)(sec ~)pdh pdz A similar expression is valid for I', the intensity of the longer wavelength~ The evaluation of (4) involves a double quadrature, since p, r3, and ~ are all functions of height. First, we have to evaluate the exponent in the exponential term by a quadrature. We may compute a source function: Z' co x(z) = p(z)oexp ( ar3+)pdh - (r3+)(sec )pdh (5) and then perform a second quadrature to evaluate the integral of X(z) over all heights~ The above expression may be simplified slightly by noting that multiplying factors common to both I and I' will cancel out when the ratio is taken. Moreover, multiplying factors common to all zenith angles will cancel out when the instrumental constant is eliminatedo Thus we may omit from further consideration the quantity KP(l+cos 2)exp io (ar3+p)pdh The quantity remaining may be written as ( =p (r+)sec)-l]0 (6) Q(G) = exp 4-$.43fi)(e)]pdh} pdz (6)

14 and the source function of (5) redefined as 00 X(0,z) = p(z)exp i (ar3+)[(sec5)-1]pdh. (7) tz The quantity required for comparison with the instrumental observations is 100 slog -og 2.2 METHOD A The classical method A was developed by Gbtz, Meetham, and Dobson (1934) and yields an approximate picture of the vertical distribution of ozone. The method has also been used by T/nsberg and Langlo (1944)o More recently-, Walton (1957) compiled instructions for use during the International Geophysical Year (IGY). The method described here is that of Walton, which differs only slightly from the earlier methodso The atmosphere is divided up into five layers as shown in Fig. 4, which also indicates the symbol used for the amount of ozone in each layer. The symbol Q is used for the total amount of ozone and k is a constant derived from aircraft measurements, which have shown that ozone concentration in the troposphere is nearly proportional to the total amount (Kay et al., 1954)o The uppermost ozone-bearing layer, in which ozone decreases rapidly with height, is split up into three sublayers for improved accuracyo The ozone concentration is assumed to be uniform in each of the layers or sublayerso For all rays scattered downwards within each layer, a mean ozone absorption path through each layer is computed by taking

LAYER HEIGHT OZONE CONTENT 0 x =0 54 km 1.1 48 km.057 x 1 I 1.2 42km.204x; x 1.3 36 km.739 x 2 x2 2 24 km 3 x3=Q (1-k)-x -x2 12 km 4 x =kQ 0 km Fig. 4. Schematic diagram showing the atmospheric layer division used in Method A (after Walton).

16 the average of the geometric paths for rays separated by 1-km intervals in the vertical. Thus, if i refers to the layer in which the scattering event takes place, and j to a layer in which absorption occurs, we may define an ozone absorbing mass, ~i, as follows: i ~i = [(sec )-l]ijxj (8) j=O where the bar refers to an average for rays, through the jth layer, which are scattered downward in the ith layer. More generally, an ozone absorbing mass, 2, and a Rayleigh scattering mass, L, may be defined as follows: 00 2 = [(sec )-l]r3pdh Jz 00 (9) L = [(secO)-l]pdh Substituting these in (6), we get 00 Q(g) = e-U'-Lpdz 4 Zi-l =~ee e e pdz i=O or 4 Q() = X Aie -7 (lo) i=O where a is now in (m atm-cm)-l, because ~i is expressed in m atm-cmo Since pe-3L is always positive, the approximation of (10) is a legit

17 imate one, provided the correct value of ~i is usedo However, ~i as defined by (8) is not, in general, the correct value~ Indeed, since the averaging is carried out over rather deep layers of the atmosphere, one would expect a not inappreciable error to be introduced. It is customary to use tabulations of Bemporad's function, or Chapman's grazing incidence integral (Wilkes, 1954), to get Lo Finally, we may compute the quantity Q(@kOxl,xx2) = 100 log Q(k) - log Q(@0) (11) which we wish to have agree with the observed quantity N(@k,A,xl,x2) = 100 g {() -k (log I (12) Walton provides tables for calculating ~ for go = 600, G1 = 800, and @2 = 8605o0 In their original paper, G~otz et al., recommended the use of an additional angle as a check~ In practice, a series of values of xl is chosen, then calculations are performed to find, by successive approximations, the value of x2, such that -)(GkR,x,1,x2) N(Qk~J,xl,x2), k 1,2 The results are plotted in a graph as in Fig 5. The intersection of the two curves gives the desired solution~ The calculations are somewhat tedious and Walton (1959) has essentially pre-computed all possible solutions and plotted the results on a series of graphs so that the points required to plot the curves in

18.15 x2.10 X2.10 0 X v,(80) = N(80), (86.5) = N (86.5).05 0.05.10 xl Fig. 5. Curves of r(Ok,S,Xl,x2) = N(Qk,y,Xl,x2) for @1 = 800 and @2 = 86.5~0 on xl, x2 diagram. Fig. 5 may be read directly without computation. The resulting solution for the vertical distribution is usually plotted as a block diagram, and a smooth curve, which leaves the same amount of ozone in each layer, is drawn through the block distribution. Method A serves to give a rather crude picture of the vertical distribution. However, although the physical-mathematical model of the atmosphere is a little crude and the final solution depends on the layer division chosen, the method certainly has the advantage of pbjectivity in

19 that a "unique" solution is obtained. 2.3 METHOD B The classical method B was also developed by Gotz et al., in an attempt to use more information from the Umkehr curve to obtain greater detail in the vertical distributiono The method has also been used by Karandikar and Ramanathan (1949). Instructions for use during the IGY were prepared by Ramanathan and Dave (1957) and the summary given below follows these. The ozone-bearing atmosphere is divided into nine layers, each 6 km thick, with ozone density assumed constant in each layer. In order to calculate the light scattered downward into the instrument, the entire mass of air in each layer is assumed to be concentrated at a height of 2 km above the base of the layer. Thus only a single ray is traced for each scattering layer and, in the notation of this study, we have for the ozone absorbing mass 9 i = [(secl)-l]ijxj (15) j=i for light scattered downward from the ith layer. In this case, as opposed to method A, the layer number increases upwards. The quantity Li is determined from Wilkes tabulation and we get the equivalent of (10) to be 10 Q(G) = mieiLi (14) i=l

20 where mi is the mass of air in the ith layer, and the 10th layer embraces the entire atmosphere above 54 km. The final quantities to be compared are i(Gknxl,..x9) = 100Log f- logQ(@o) (15) and the observed quantity N(~@kX,l, x9) = 100 og k - og (16) Ramanathan and Dave present tables to assist in the computation of q for (G,...,07) = (600, 700, 750 800, 84 865, 880, 90~). An additional requirement is, of course, that the sum of the layer amounts equal the observed total amount of ozone. In the suggested method of solution, a trial distribution is assumed and the associated Umkehr curve is calculated and compared with the observed one. Then adjustments are made in the distribution until the calculated and observed curves agree within experimental error. To facilitate this adjustment process, Mateer (1960) calculated values of ~k = 7(@k,xls,...,x9s) and of the first order partial derivatives qk/ xi for three "standard" vertical distributions of ozone. (The subscript s refers to layer ozone amounts in the standard distribution of ozone.) The evaluation then consisted of sdlving, by hand relaxation, a set of eight linear equations in nine unknowns, viz.,

21 xi = S ~~~~~~i=l ( ~(17) i=l, 3xi i where sxi = xi-xis represents the deviation of the solution from the standard distribution. This method has also been used by Muramatsu (1961) who calculated similar tables for three additional standard distributions. Since the above system of equations is underdetermined, method B is a subjective one and "unique" solutions are not possible, regardless of whether the approximate linear method or the successive approximations method is usedo This nonuniqueness was clearly recognized by Gotz et al., in the original paper on this method. They noted, for example, "we may conclude that the shape of the Umkehr curve depends mainly on the value of" o. Moreover, they found that there was no advantage in using many points from the Umkehr curve because "more than six points would be much more interdependent." They worked mainly with mean Umkehr curves appropriate to a specific small range of values of total ozone, thereby minimizing random errors. Using fewer layers than Ramanathan~ and Dave, they essentially attempted to obtain five unknowns from seven equations by a "least square" solution method. For a single Umkehr curve, they note that "the probable errors of values of xi... are so large that the resulting ozone distribution is almost meaningless." These "words of warning" should be' kept in mind when reading the later chapters of this

22 report. 2.4 METHOD OF DUTSCH In a series of reports, DUtsch (1957, 1959a, 1959b, 1963) has introduced a measure of objectivity into Umkehr evaluations by method B. In fact, the method is completely objective once the basic solution system has been selected. DUtsch divided the atmosphere up into layers approximately 2.5 km thick, such that the pressure at the bottom of each layer was W2 times that at the top. He used three standard vertical distributions of ozone, with ozone in each layer up to 72 km. The ozone amounts in the upper layers were based on his photochemical calculations. With the assumption that ozone density was constant in each layer, and using Bemporad's function, his quadrature formula was I = IoKP(l +cos2) X P e L exp X5k+ ik-k (18) i=l k=i+l where aik is the relative slant path through the kth layer for the ray scattered downward in the ith layer, APi is the pressure difference between the top and bottom of the ith layer, and Po = 10130250 mb, the surface pressure in the standard atmosphere. In calculating values of Qk for comparison with the observations Nk, Diutsch does not eliminate the instrumental constant but estimates it empirically. Moreover, he does not use directly in the solution system the requirement that the vertical distribution be exactly equivalent to the

25 measured total amount of ozone. Instead, he uses the comparison between computed total ozone and observed total ozone as a check on the accuracy of his solutiono He defines x. f. = 1 Xis and xi-Xis Af = fi 1 X (19) 1i 1~ Xis Then, using Taylor's expansion, we have n an n 2 N k Af + AfAfj (20)k+ Nk - k + Xafi fi + fifj. ()fj i=l i=l j=l If we ignore the last term on the right-hand side of (20), the problem is reduced to one of finding the solution of a set of simultaneous linear algebraic equations. Assuming such a solution exists, using superscript m to denote the mth iteration, and defining, (o) Sk = 0 n n S(m) =f1 j Afm) sAf (21) i=l j=l then we have n 7 k (m) (m-1) L 3f Afi = Nk - k - k, k=l,. o,12 (22) i=l

24 as the basis for an iterative solution procedure. The iteration may be stopped whenever 12 (m) (m-l) Sk - s < C (23) k=l where c is some suitably small numbero Dutsch found that the tropospheric layers could not be subdivided because the derivatives were too similar for these layers. He found the same to be true for the layers above 2 mb. Hence, he combined the tropospheric derivatives into a set for a single layer and did the same thing for the Sayers above 2 mb. In these combined layers, he assumed that ozone always appeared in the same relative proportions as in the standard distribution. This left the unknown vertical distribution as a set of nine quantities to be determined. DUtsch used a total of 12 zenith angles (60, 65, 70, 74, 77, 80, 83,,k a2 krt 85, 86.5, 88, 89, 90) and computed k, k and, for these 12 zenith angles and for each of three standard distributions. In attempting to solve for the vertical distribution in the nine layers directly from a set of nine linear equations, he obtained physically unrealistic results, including negative ozone densities in certain layers, and solutions that exhibited large variations in ozone concentration from one layer to the nexto He attributed this difficulty to the inaccuracies of the measurement and to the linearity imposed on a nonlinear problems To get around these difficulties, he gradually evolved a system whereby the nine atmospheric layers were combined into sets and subsets of over

25 lapping layerso In each overlapping layer, he assumed.that the fractional change in ozone content was the same in each of the original layers comprising the larger overlapping layer. In each subset, he effectively had a set ofat.most five linear equations in five unknowns. The solutions for the subsets were averaged to get an average for each set and, finally, the set solutions were averaged to obtain the final solution for the linear system. The second order derivative corrections Sk) were then calculated, the right-hand side of (22) adjusted, and the linear solution repeated, the iteration being stopped when the condition of (23) was met. The entire solution procedure as now used requires approximately 105 seconds of computer time on the IBM 7090, most of the (m) time being taken up in the computation of the Sk o Further details of the DUtsch method are given in Appendix B. The procedure is clearly an objective one, once the basic systems of overlapping layers have been selected, and provides a smoothed picture of the vertical distribution. 2.5 SOURCES OF ERROR 205.1 Multiple Scattering and Reflected Light Quite apart from the fundamental mathematical difficulties in the solution of Eq. (22), which will be discussed later, there are a number of errors inherent in the physical-mathematical model used to compute the synthetic Umkehr curves. The major source of error, which has received much attention during the past decade, is the effect of multiple scattered

26 light which contributes to the intensity of the light entering the instrument. Following Ramanathan and Dave, if we let P, Pt be the intensities of primary scattered light and M, MI the intensities of multiple scattered light for.the short and long wavelengths of a pair, then we have that IfI P+MI PI P+ (24) I P+M P M P Hence, if M/P and M'/P' were known, the observed Umkehr curve could be "corrected" to a basis of primary scattering and the computation, using P, P', could proceed as previously~ Walton (1953) calculated the ratios S/P, S'/P', where S,ST are the intensities of secondary scattered light. He assumed a plane parallel atmosphere with the total atmospheric ozone content concentrated in a thin layer at the center of gravity of the ozone distributiono He found that the corrections varied with solar zenith angle but were nearly constant for zenith angles between 80 and 88~. The effect of the secondary scattering correction was to decrease the computed amount of ozone at higher levels and to increase it at lower levels. That is to say, the center of gravity was lowered by some 2-3 km. Walton's corrections are given in the first column of Table 1l Ramanathan, Moorthy, and Kulkarni (1952) used simultaneous Umkehr curves for two pairs of wavelengths (3112/5332 A) and (3075/3278 A), to

TABLE 1 HIGHER ORDER SCATTERING CORRECTIONS TO BE SUBTRACTED FROM OBSERVED NAVALUES AT VARIOUS ZENITH ANGLES Ramanathan Sekera Investigatoro Walton Dutsch Larsen & Dave & Dave Wavelengths~ C C A C D C C C Total Ozone400 --- 336 336 336 336 360 250 (m atm-cm) Zenith Angle (degree) 60o 0 0 3.6 0.5 -1.8 0,4 3 2 70 1o5 1 5.9 0.3 -3.0 0.3 4 4 75 3 2 10.7 1.5 -3.6 o.6 5 6 80 6 4.5 18,2 6.0 -3.2 2 o3 10 11 84 6 6 22. 2 13.6 11 5.0 14 86.5 6 6 21.7 16.2 439 6.2 10 13 88 6 20.1 15.6 6.9 6.2 -= 11 90 6 13.7 8.6 2.2 4.0

28 deduce an Umkehr curve for (3075/3112 A) by eliminating the effects of (3278/3323). They do not state how the effects of the latter pair were eliminated, but it appears likely that they used the "double" wavelength pair obtained by subtracting the N-values for the first pair from those for the second and assumed this to be equivalent to using (3075/3112 A). They reasoned that (3075/3112) are both strongly absorbed by ozone and, hence, that the effects of multiple scattering would roughly balance out. Similar considerations should apply when the weakly-absorbed wavelengths (3278/3323) are compared. They concluded that vertical distributions calculated from (3075/3112) should, therefore, eliminate much of the effect of multiple scattering. They show results for Umkehr curves on three days and confirm Walton's result, namely that the effect of correcting for the higher order scattering is to lower the center of gravity of the derived vertical distribution. The corrections suggested for secondary scattering by Ramanathan and Dave are listed in the second column of Table 1L DUtsch has also incorporated corrections for higher order scattering in his evaluation procedure, the effect being incorporated both in the standard Umkehr curves and in the first order partial derivatives. His quadrature formulation for computing the secondary scattering is given in Appendix B. If we let Ik represent the contribution to I due to kth order scattered light, then DUtsch assumed Ik/Ikl = C, a constant being determined as I2/I1e It follows then that

29 I = I1 + oo. = I1. (25) Multiple scattering corrections based on this assumption for Dutsch's first standard distribution, for the.A,-C, and D wavelength- pairs, are listed in columns 3 to 5 of Table 1. This procedure appears to give too large a correction, that is to say, I2/I1 > I3/I4 >... Ik/Ik-l. Duitsch's current procedure is to assume I = Il+I2-only. These corrections, for the — C wavelength pair, are listed in the sixth column of Table 1. Larsen (1959) has determined values of (l+M'/P')/(l+M/P) based on an empirical- analysis of skylight observations at different. wavelengths. His results necessarily incorporate the effects of all orders of scattering and are listed in the seventh column of Table 1. Sekera and Dave (1961) have computed the effects of multiple scattering for a plane parallel atmosphere using the C wavelength pairo They reason that the effective height of secondary scattered light (and also higher orders) is situated very near the ground and that the primary scattered radiation giving rise to the secondary scattering originates from a relatively narrow cone with its axis along the zenitho Consequently, they divide the atmosphere into two layers, vizo, an upper layer containing all the ozone in which only primary scattering is considered and a lower layer in which no ozone is present but all orders of scattering are consideredO Their results are presented in the last column of Table lo

In comparing columns 1, 4, 6, 7, and 8 of Table 1, we note that there is fairly good agreement between Walton's results and those of DUtsch, in which only secondary scattering is considered. There is also moderately good agreement between the last two columns in which all orders of scattering seem to be included. There appear, however, to be moderate differences between DUtsch's results and the other,s. for zenith angles between 700 and 800. However, there is good agreement that the effect of multiple scattering corrections on the derived vertical distribution is to decrease the ozone content at high levels (above about 20 kmin) and to increase the ozone content of the atmosphere below this level. Dave and Furukawa (1964) have examined the effect of ground albedo on Umkehr observations. They consider only primary scattering and calculate corrections applicable to the A and C wavelength pairs for a surface albedo of 80%, and total ozone amounts of 260 and 400 m atm-cm. Their corrections are shown in Fig. 6. They also show how the effect of low level clouds can be estimated from these results, when the clouds remain scattered to broken, so that zenith measurements can still be taken on blue sky. The corrections for ground reflection are somewhat smaller than those for higher order scattering and act in the opposite sense in that they are largest at 600 to 700, and smallest when the sun is near the horizon, whereas the higher order scattering corrections are largest when the sun is near the horizon.

7 6 6 A, 400 m atm-cm 5 3 ) C, 260 m atm-cm AC,260 m atm-cm Oj7 XC, 26 m atm-cm 60 70 80 90 SOLAR ZENITH ANGLE (Degrees) Fig. 6. Corrections (to be subtracted from observed N-values) for ground reflection in the case of 80% albedo. (After Dave and Furukawa.) 2.5.2 Empirical Cloud Corrections and Large Particle Scattering Diitsch has devised an empirical method for correcting Umkehr curves taken on cloudy zenith sky, based on a comparison of the intensities of visible to ultraviolet skylight. The corrections, which are always subtracted from the observed values, are shown in Fig. 7 as a function of

32 7 \16 \18 1 20 _\ \ \ \ \ 5 10 12 14 \ 607 \ \ < 8 c/4 LLU -3 4 \ 2 2 60 70 80 90 SOLAR ZENITH ANGLE (Degrees) Fig. 7. Cloud corrections (N-units to be subtracted from observed values) as a function of luxmeter ratio and solar zenith angle. luxmeter reading and solar zenith angle. The luxmeter readings used in the graph are deviations of the observed visible/ultraviolet light ratio from that obtaining under very clear sky conditions at the same zenith angle. The corrections are derived empirically. A sample Umkehr curve showing both the observed and corrected points is plotted in Fig. 8. DUtsch finds this correction procedure satisfactory provided the cloud interference (the magnitude and variability of the corrections) is not too great.

33 90 0o 100 -o Observed points Corrected points 110 I10 -120 0? 130 * 140 1500 0 160 170 - 60 70 80 90 Solar Zenith Angle (Degrees) Fig. 8. Sample Umkher curve for Arosa, March 30, 1962, showing observed values and corrected values. Total ozone is 394 m atm-cm.

The effects of large particle scattering on the Umkehr curve have not yet been considered in detail. By comparing Umkehr curves for Oxford and TromsO, Larsen (1959) found persistent differences of 7 and 8 N-units at sec 0 = 4 and 8, respectively. However, at least part of this difference is undoubtedly due to differences in the mean vertical distributions over these two stations. DUtsch's cloud corrections include some large particle scattering effect, since he occasionally should apply a cloud correction on apparently clear sky, according to his luxmeter readings.

35 THE CHARACTERISTIC PATTERNS OF-UMKEHR CURVES 3o1 PRELIMINARY REMARKS It is a natural property of a geophysical variable, which is measured sequentially in time at a network of points, that a high degree of spatial correlation will exist between the individual measurements over the network of pointso In addition, a high degree of serial correlation will exist in the individual series of measurements at each of the fixed pointso In the case of Umkehr observations, it is natural to inquire into the degree of independence of the individual measurements in a single series of measurements on a given half-day. For example, is the measurement at a solar elevation of 1~ independent of the observations at 0~ and 2~? As noted earlier, Gotz et alo, recognized a strong degree of interdependence between the points on the Umkehr curve and also remarked that the main features of its shape seemed strongly related to the total amount of ozone. There were, however, certain variations in the curve, total ozone remaining constant, that suggested a variability in the vertical distributiono The question of interdependence also arises in the formulation of a purely statistical technique for evaluating the Umkehr effecto Suppose we let Pljooo,p9 be the mean ozone partial pressures in nine layers of the atmosphere, such that these nine numbers specify the complete vertical distributiono Suppose further than we have measurements at 12 35

36 points on the Umkehr curve ul,o o.,ul2. If we have a number of sets of observations of both these quantities, we may attempt to derive a linear statistical transformation to predict the Pi from the Ujo Thus 12 Pi = aijj (26) j=1 where the Pi are the estimates of the mean ozone partial pressures in the nine atmospheric layers and the aij are the elements of the coefficient matrix of the transformation. In matrix notation P = AU (27) where the elements of P are pik representing the estimate for the ith layer and the kth observation and the elements of U are ujk representing the jth point on the kth Umkehr curveo There is no loss in generality in letting the Pi and uj be measured from their respective means, in which case the least squares solution for A is simply A = (PU*)(UU*)- (28) where U* is the transpose of U. The transformation matrix A exists if and only if the inverse of UU* exists, that is to say, UU* must be nonsingular. If there are strong linear interdependencies between the points on the Umkehr curve, the matrix UU* will be singular or very nearly so and the matrix A will, for all practical purposes, not exist.

37 3~2 THE STATISTICAL PROCEDURE To investigate the degree of independence of the observations, we shall use the empirical orthogonal functions introduced to meteorologists by Lorenz (1956)o According to Lawley and Maxwell (1963), the technique was put forward by Pearson in 1901, and later developed by Spearman in 1904 as Factor Analysis, which is much used by psychologists, and by Hotelling in 1933 as Principal Component Analysis~ There is evidence of some disagreement between statisticians (compare, for example, Kendall (1957) and Lawley and Maxwell) as to the precise differences between these analysis techniqueso These differences need not concern us here and, following meteorological practice, we shall use the terms "empirical orthogonal functions" (Lorenz) or "characteristic patterns" (Grimmer, 1963)e What we seek to do in this procedure is to effect a reduction in the number of variables required to describe the Umkehr curve so that the main features of the curve are retained and the random errors of measurement, the noise of the curve, are eliminated. This is essentially a filtering problem~ To achieve this, we seek linear transformations, to a new set of variables Yi, of the form 12 Yik = X bijujk (29) j = where, as before, k represents the kth set of observationso In matrix notation

Y BU (30) It follows immediately that, if uj = O, then also yi = 0. We shall now require that the new variables yi be uncorrelated. In matrix notation, this requirement may be written as YY* =A (31) where A is a completely diagonal matrix having nonzero elements on the diagonal only. Introducing (30) into (31), we have B(UU*)B* = A (32) This is the well-known problem of determining the eigenvalues and vectors (latent roots and vectors or characteristic roots and vectors) of the real symmetric matrix UU*, in which the elements are proportional to those in the covariance matrix of the points on the Umkehr curve. It can be shown that a solution exists in which the eigenvalues, the diagonal elements of A, are all real and nonnegative and the eigenvectors are stored in the rows of B. There is no loss in generality in assuming that the eigenvalues, ki, i = 1,...,12, are stored in the diagonal elements of A in order of decreasing magnitude, provided that the rows of B are arranged accordingly, nor in requiring that the eigenvectors, which are orthogonal, be also orthonormal. That is to say, BB* = B*B = I ()

39 where I is the identity matrix. These eigen, or characteristic, vectors are the spatial empirical orthogonal functions of Lorenz and have been dubbed "characteristic patterns" by Grimmer. We shall hereinafter refer to them as the characteristic patterns (C. Po's) of the Umkehr curve. The reason for this nomenclature becomes clear if we expand the vector of points from each Umkehr curve in terms of these characteristic patterns~ Thus, from (30) and (33), we have U = B*Y (34) It is useful now to introduce the concept of the "total variance" of the Umkehr curve as Vk = ujk (35) j-= It follows directly, from (33) and (34), that 12 k = Yik (36) i=l Since Yik is the coefficient of the ith pattern vector of B in the expan2 2 sion (34), we have the result that the ith pattern "explains" Yik/vk of the total variance of the Umkehr curve~ Moreover, since n Yik = (37) k=l and n n 12 n 12 V= Xv = X Ujk = XX j k (38) kal k=l jo= k=l i=l

40 we may say that the fraction of the total variance, V, of the Umkehr curves explained by the ith pattern is just ki/V. Hence the patterns occur in B in the order of their ability to explain total variance of Umkehr curves. It can be shown (see Lorenz, 1956 or 1959, or Kendall, 1957) that the representation is an optimum one in the sense that among all possible linear combinations of the points on the Umkehr curve, these patterns account, successively, for the largest possible proportions of the total original variance. 3.3 THE APPLICATION OF THE STATISTICAL PROCEDURE It was noted above that the solution procedure of DUtsch involves the selection of 12 points, U1l,-,u12, from each Umkehr curve. It was further noted that the more usual solution procedure of others would be, if u12 is the curve point corresponding to the greatest solar elevation, to use Ul-U12, U2-U12.,*e u11-u12, with the total amount of ozone as the last numbero Both of these procedures are used to represent the Umkehr curve in the present study. In addition, there are two possible choices for the matrix UU*, vizo, the covariance matrix or the correlation matrix of the points of the Umkehr curveo The covariance matrix is UU* with each element divided by n, the number of Umkehr curves usedo The correlation matrix is obtained from the covariance matrix by dividing the (i,j) element of the latter by the quantity tV _1/2 ikn /2

41 When the covariance matrix is used and total ozone is one of the "points" of the Umkehr curve, some choice must be made with respect to the weight with which total ozone enterse In the studies reported here, total ozone was entered in m atm-cm and the Umkehr curve points in 1000 logunits (as opposed to the customary 100 log-units)o Two basic data samples have been used. The first sample is from observations taken at North American stations, vizo, Edmonton, Moosonee and Toronto, Canada, and Sterling, Virginia, in the United States' The sample comprises data for 98 Umkehr curves on wavelength pairs A, C, and Do. The second sample is from Arosa, Switzerland, and comprises 100 Umkehr curves for wavelength pair C. Both data sets cover a wide range of total ozone amountso The North American total ozone data are based on AD double-pair direct-sun measurements, while those from Arosa are based on direct-sun measurements with the C wavelength pairo To make these data comparable,:the Arosa measurements have been increased by 6% (a value suggested by DUtsch as appropriate for Arosa measurements), where it was necessary to do soo Both sets of data have been carefully scrutinized (the former by the writer, the latter by DUtsch) to eliminate the spurious clerical errors which often arise in processing the instrument readings through to the final data forms. In the case of the North American curves, the sample represents virtually all of the available data for three wavelength pairs when measurements were possible on all 12 zenith angleso In the case of the Arosa sample, curves were selected from

a large sample of over 500 taken during the period March, 1961, through July, 1962, inclusive. The curves were selected to include most of the low and high ozone cases from the larger sample. Finally, cards having odd numbers in the last position of two of the measurements were selected to represent intermediate ozone values. The determination of the eigenvalues and vectors of the correlation and covariance matrices were carried out by the Jacobi method, more specifically using subroutine EIGN, which is directly available as a system subroutine on the IBM 7090, Computing Center, The University of Michigan. The first eight eigenvalues, for the correlation matrices obtained when the entire Umkehr curve is used (i.e., without total ozone), are listed in Table 2. The computations have been carried out for the Arosa sample, for the North American sample on each wavelength pair separately, and for the latter sample with the wavelength pairs combined into one vector of 36 elements. The result is quite clear: some 95% or more of the normalized variance of the Umkehr curves is explained by a single characteristic patterno Moreover, this pattern is in all cases nothing more than a simple shift of the entire Umkehr curve. The correlation between total ozone and the coefficients of the characteristic patterns (when the Umkehr curve is expanded with respect to these patterns) is also given in Table 2 in brackets. We note that the coefficient of the first C. P. is highly correlated with total ozone and, hence, the amount that the Umkehr curve is shifted depends strongly on total ozoneo Thus the statement of Gdtz et al., that the shape of the curve depends mostly on the to

45 TABLE 2 THE FIRST EIGHT EIGENVALUES OF THE CORRELATION MATRIX OF THE POINTS ON THE UMKEHR CURVE, INCLUDING (IN PARENTHESES) CORRELATIONS BETWEEN THE EIGENVECTOR COEFFICIENT AND TOTAL OZONE Eigen- North America Arosa value wNumb er A C D A-C-D C.96716.96922.96518.94844.97767 (.916) (.887) (.799) (.876) (.988).02131.01870.02225.02088.01505 (-.279) (-.317) (-.030) (-.430) (-.056).00oo858.00814.01007.01258.00529 (.098) (-.lo6) (.485) (,.012) (-.074).00144.00264.00110.01002.00114 (.058) (.191) (.016) (-.105) (-.032).00063.00050.00055.00264.00034 5 ~( -.041) (.013) (-.022) (.o002) (. oo8) 6.00029.00023.00023.00234.00020 (.092) (-.014) (.021) (-.077) (.014).00016.00018.00018.00076.00009 (-.023) (-.044) (.ooo) (.039) (-.026) 8.ooo00016.00011.00017.ooo45.00007 (.003) (-.021) (-.025) (.005 (-.015)

44 tal amount of ozone, is quantitatively confirmed. The essential result of this analysis is shown graphically in Fig. 9 as "standard" Umkehr curves for the C wavelength pair for total ozone amounts of 300, 350, and 400 m atm-cm. The correction of 6% has been applied to obtain the Arosa curves, and the curves for 350 m atm-cm have been made to coincide at a zenith angle of 600o In examining Fig. 9, we note in particular the difference in the position of the reversal which occurs, at Arosa, when the sun is closer to the horizon. This suggests that there is a difference in the mean vertical distribution of ozone over Arosa compared to North American. However, neither of the two samples is completely representative. In particular, we note from Figs. 16 and 17 that although the two samples have about the same mean total ozone (near 350 m atm-cm), the standard deviation is 63 m atm-cm for the Arosa sample compared to only 42 m atm-cm for the North American one. This undoubtedly plays some role in determining the difference in the mean curves and, as noted earlier, the curves of Figo 9 are relatively simple shifts of the mean curveso Results for the Umkehr curves with the instrument constant removed (i.e., including total ozone) and using correlation matrices are given in Table 3o In this case the correlations between total ozone and the pattern vector coefficients have not been computedo Results are included for the "double" pairs, AD, AC, and CD. A somewhat different picture emerges now because the variability of total ozone has been added to that of the Umkehr curve, but shifts of the entire curve have been eliminatedo

ZENITH ANGLE (Degrees) 88 60 65 70 74 77 80 83 85 86.5 89 90 150 /140 l l I, /, 100 ~~ 2345 /, I ZE I A _/ I' // - 90 1 / I/ Fig. 9 "Sadad North America g 90i.,',Arosa 70 60 50 wa/elength pair. 2 671,~~(ZNIH North Amrc

TABLE 3 THE FIRST EIGHT EIGENVALUES FOR THE CORRELATION MATRIX OF THE POINTS ON THE UMKEHR CURVE WHEN THE INSTRUMENT CONSTANT IS ELIMINATED AND TOTAL OZONE IS USED Eigen- North America Arosa value Number A AC C CD D AD A-C-D C 1.73729.76477.62100.68504.67945.75385.59495.71067 2.20098.12204.31185.15550.25814.15581.29177.21543 3.03175.06311.03093.08378.02729.05029.02876.05687 4.01453.03098.02173.04386.01553.02288.02563.00530 5.00599.00768.00555.01148.00789.00796.01295.00441 6.00482.00539.00289.00660.00503.00442.01157.00286 7.00202.00257.00210.00492.00284.00219.00909.00177 8.00149.00162.00132.00312.00135.00140.00602.00093 3.001.00135.

We find that three C Po I s are required to explain 97% of the normalized variance of the single pair curves. The first three Co Po's for each data batch are shown in Figs~ 10-12, inclusive (excluding the cases where the A, C, and D wavelength pairs are combined and the double pairs)o In order to have something interpretable in terms of N-values, the pattern vector elements have been multiplied by the standard deviation of the appropriate curve point and then the entire vector has been multiplied by the root mean square coefficient the Co P. would have if the Umkehr curves were expanded in terms of these patterns. The "point" corresponding to total ozone is not plotted on these diagrams. The dot product of the original orthonormal pattern vectors for the two C wavelength pair samples is shown on each of the figures. The high values of this product indicate that the patterns are essentially the same and strongly suggest that these patterns are fundamental properties of the Umkehr curveo Finally, results for the Umkehr curves with the instrument constant removed (total ozone included) and using the covariance matrices are given in Table 4a- In this case, the Arosa total ozone values were increased by 6% so that the results would be more nearly comparable. The mean Umkehr curve variance and the root mean square curve-point deviation from the mean are also listed in Table 4o It should be remembered that these deviations have to do with the shape of the curve since the shift of the entire curve with total ozone has been eliminated except insofar as total ozone itself is concernedo

ZENITH ANGLE (Degrees) 60 65 70 74 77 80 83 85 88 90 86.5 89 _4 Dot \ Arosa C 4 C / Dot Product 3 2 D -3 -4 -5 -6 -7 1 2 3 4 5 6 7x10 (ZENITH ANGLE)4 Fig. 10. First Characteristic Pattern of the correlation matrix, with the instrumennt constant eliminated.

ZENITH ANGLE (Degrees) 60 65 70 74 77 80 83 85 '88 90 3 I { II1I I I I I I I I I 1 2 3 4 5 6 7x107 (ZENITH ANGsLE)4 Fig. 11. Second characteristic pattern of the correlation matrix, with the instrument constant eliminated. ZENITH ANGLE (Degrees) 60 65 70 74 77 80 83 85 88 90 2 86.5 89. 1 - Dot Product.973 A -1 I I I I 1 2 3 4 5 6 7x107 (ZENITH ANGLE)4 Fig. 12. Third characteristic pattern of the correlation matrix, with the instrument constant eliminated. 2~~~~ 37 'I 5 7l

TABLE 4 THE FIRST EIGHT EIGENVALUES FOR THE COVARIANCE MATRIX OF THE POINTS ON THE UMKEHR CURVE WHEN THE INSTRUMENT CONSTANT IS ELIMINATED AND TOTAL OZONE IS USED Eigen- North America Arosa value Number A C D A-C-D C 1.90699.65569.70521.77319 o80605 2 o.06070.29271.25587.15858.13376 3.02022.02975 o 01858.02765.04425 4 o00449 oo00890.00903.01633 00oo849 5 o.00239.00570 o00446.00715.00289 6 o00202.00192.00190.00330 000143 7 oo00114.00oo180.00152.00246 o00100 8 o00082,OOi18.00127.00214 oo00067 Mean Curve Variance (N-units)2 408, 8 124o2 89.5 589.5 185o 9 RMS Point Deviation (N-units) 5.8 3,2 2.7 4.2 309

As before, we find that most of the curve variance is explained by the first three pattern vectors. These patterns, suitably scaled, are shown in Figs. 13, 14, and 15, respectively. The dot product of the ZEN ITH ANG LE ( Degrees) 60 65 70 74 77 80 83 85 88 90 6 I I Arosa C 86.5 89 4 2 -4 -6 Dot Product.953 -8 I I I I I 1 2 3 4 5 6 7x107 (ZENITH ANGLE) 4 Fig. 13. First characteristic pattern of the covariance matrix, with the instrument constant eliminated. Compare with Fig. 10 but note difference in scale.

52 ZENITH ANGLE (Degrees) 60 65 70 74 77 80 83 85 88 90 3 I I I I I I I I l C 86.5 89 Dorosa C LU._J >0 -1 Dot Product.950 -2 I I, 1 2 3 4 5 6 7x10 (ZENITH ANGLE) 4 Fig. 14. Second characteristic pattern of the covariance matrix, with the instrument constant eliminated. original orthonormal pattern vectors for the two C wavelength pair samples is shown on each of the figures. The high values of the product again attest to the similarity of the respective patterns. We note further that the patterns are much the same, regardless of whether the correlation or covariance matrices are used. As a matter of interest, the complete set of 12 characteristic patterns for the covariance matrix of the North American data sample C wavelengths are given in Table 5.

53 ZENITH ANGLE (Degrees) 60 65 70 74 77 80 83 85 88 90 2 1 '8d58 I9 - A <0 ~z ~~D -1-~Arosa C Dot Product.965 -2 X l 1 2 3 4 5 6 7x 10 (ZENITH ANGLE) 4 Fig. 15. Third characteristic pattern of the covariance matrix, with the instrument constant eliminated. In the 12 columns are listed the vector points corresponding to total ozone, and @ = 90, 89,..., 650, respectively. The root mean square coefficient for each vector is obtained by multiplying the mean curve variance (second last item of third column, Table 4), by the appropriate fractional eigenvalue, then taking the square root. For example, the rms coefficient for the first pattern vector is [(.65569)(124.2)]1/2 = 9.02. If we multiply the element of the first C. P. corresponding to 770, by this number we get (9.02)(.36049) = 3.25, which is the value plotted in Fig. 13.

TABLE 5 CHARACTERISTIC PATTERNS FOR THE COVARIANCE MATRIX OF THE POINTS ON TBE UMKEHR CURVE WREN THE INSTRUMENT CONSTANT IS ELIMINATED AND TOTAL OZONE IS USED FOR THE NORTH AMERICAN C WAVELENGTH DATA SAMPLE Pattern Characteristic Pattern Points Number 1 2 3 4 5 6 7 8 9 10 11 12 1.44756 -.32217 -. 31989 -.31970 -.27332 -.16005.04732.28913.36049.33195.236(38. 09660 2.11456.22497.25767.29777.34968.40192.41605.38990.31687.22026.14114.05846 3.19679.62426.37491.09812 -.24682 -.42012 -.32447 -.01912.14929.16852.16520.01834 4.84447.08037.04244 -.01011.08099.17243.04745 -.12274 -.31746 -.24298 -.21075 -.14372 5.13689 -.27764.10991.25992.19351.05772 -.16470 -.41511 -.17384.16730.47192.55082 6 -.01676 -.15652.23137.14620 -.23669 -.41933.47743.35080 -.35800 -.34570.21730.15018 7.07968 -.47737.20370.43829.24123 -.33606 -.19566.05730.08888.18600 -.09816 -.51707 8 -.02278 -.00295.16477 -.08376 -.20654 -.10094.59317 -.53735 -.02567.46931 -.19050 -.12107 L 9.06549 -.21388.20688.26346 -.35352.09671 -.00304 -.16955.56319 -.38551 -.35171.29185 10 -.03752 -.07109.16233 -.05963 -.31720.35684.00487 -.19240.10720 -.23969.60575 -.51260 11.05274.16239 -.29131 -.06523.47262 -.40744.25722 -.30743.37848 -.37397.20989 -.06726 12 -.01849 -.20686.63297 -.65888.30375 -.04018 -.07569.03552.08780 -.07478 -.04586.07628

55 The Umkehr curves have been expanded in terms of the Co P, s of the covariance matrix in an attempt to find out how many patterns are required to "explain" everything but the experimental error which might be expected in the data. In the present case, it was assumed that, when the total residual curve variance was less than 6.0 (N-units)2, we were down to the level of experimental error~ (The total ozone residuals were in units of 10 m atm-cm.) In each expansion, the coefficients of all patterns and the variance explained by each were computed and the total variance explained was summed. The sum was tested after the addition of the variance explained by each C. PO and the series expansion was truncated when the residual variance was less than 6000 In the case of the A wavelength pairs, a value of 12.0 (N-units) was used and, for the combined pairs (A-C-D), a value of 34.0 (N-units)2 was used for testing purposeso These higher testing values were used in the latter cases because of an apparently higher noise level. In addition, for the C wavelength pairs, the Arosa curves were expanded in terms of the North American patterns, and vice versa. The results are listed in Table 6, which gives the frequency distributions of truncation levels for the various expansionso The bracketed numbers in the C columns are the truncation levels when the Arosa curves are expanded in terms of the North American patterns, and vice versao In general, most of the curves require only three characteristic patterns to explain all the variance except that attributable to experimental error. Most of the remaining curves require only one additional characteristic pattern.

TABLE 6 FREQUENCY DISTRIBUTIONS OF TRUNCATION LEVELS FOR EXPANSIONS OF UMKEHR CURVES IN TERMS OF THEIR CHARACTERISTIC PATTERNS Characteristic North America Arosa Pattern Number A C D A-C-D C 1 25 14 (0) 21 10 9 (7) 2 40 46 (8) 61 43 29 (12) 3 25 29 (73) 13 26 50 (59) 4 6 6 (15) 3 13 11 (19) 5 2 3 (1) 0 4 1 (3) 6 o o (o) o 1 0 (0) 7 0 0 (1) 0 1 0 (0) 8 o 0 (o) 0 0 0 (0) We may conclude from the above results that at most four characteristic patterns are required to explain variations in the shape of the Umkehr curve when the instrument constant is eliminated and total ozone is included. The first three patterns at least are fundamental properties of the Umkehr curve and there is some possibility that the fourth pattern is also important. The remaining patterns are mostly noise, particularly the higher order vectors, which exhibit sign changes from one angle to the next and explain virtually no variance. If we assume that the main information content of the Umkehr curve has to do with variations in the vertical distribution of ozone, then we may further conclude that there are, at most, four pieces of information about these variations that may be inferred from Umkehr observations. Fron the evidence presented here, it appears that little or no additional information is to be obtained from observations on more than a single wavelength pair,

57 at least when observations are available for all 12 zenith angles. Thus the existence of strong interdependence between the points on the Umkehr curve, first recognized by G-tz et al., has been quantitatively confirmed. Pictorial evidence is presented in Figs. 16 and 17 for the strong control of total ozone in determining the shape of the Umkehr curveo These figures are scatter diagrams in which the coefficients of the first CO PO (of the covariance matrix with total ozone included) are plotted against total ozone for the C wavelengths and for the North America and Arosa data samples, respectivelyo There is some redundancy here in that total ozone is also included in the first "point" of the pattern vector However, total ozone does not dominate this pattern vector, and the high degree of correlation between the coefficient and total ozone could not exist unless total ozone also played a dominant role in determining the shape of the Umkehr curve~ 354 PHYSICAL EXPLANATION A simplified explanation of the above results, vizo, the strong colinearities existing between the points on the Umkehr curve, may be found by referring to the development of Section 201, which considers only primary scattering. We may perform the numerical integration indicated by Eq. (7) and plot the "source function" X(Q,z) as a function of height to see how broad a layer of the atmosphere actually contributes to the primary scattered intensity at the ground for each wavelengtho This has been done for each of the 12 zenith angles used by Duitsch, for each of the wave

25 20- o_0 20 0 0 Oo~~~~ 0 o ~0 ~~~~~o 0 0 L 100 -._ 00 0~~~ ___~~~~~~~~~~d o 5' - 0 I,, o 0 0 O~ Oi co f~O~rlto 09 r~~~~~~~~- o~0 00 o ()~~ 0 Co 0 LL_ ~ ~ 0 Qo 0 t= 0 0~~~~~~~~~~~~~- > ~ ~ SADR EITON DOmat - 0 0 correlation:= 0.96 0.. 0~~~~~~ 5 — 10 -F- 20 -100 -5 C0 2.05 A1 15 i — ~~~~~~0 0 - 15 - 0~ AVERAGE TOTAL OZONE 354 m atm -cm o -i 0 ~~~~~~~~~STANDARD DEVIATION 42 m atm -cm -20 I I - 100 - 50 0 50 100 150 TOTAL OZONE DEVIATION FROM MEAN An (m atm-cm) Fig. 16. Scatter diagram of first pattern vector coefficient plotted against total ozone deviation from mean for C wavelength pair, North America sample.

0 00 'o 20 0 0 0 0 0 0 0 1000 00 Lo y C~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-)' 0 5 0 ~~~~~~~~~~~0 LU ~~~~~~~~~~0 0 0~~~~ 0 5 0 00C o 0 0 U_~~~~~~~~~c ~~ ~ooo ~~~~~~0 c~ -5- V -10 o C. = 1.90 Af - 0~~~~~~~~~ P / 5: %15 o~b oo AVERAGE TOTAL OZONE 352m atm-cm STANDARD DEVIATION 63 m atm-cm 0~~~~~ 20 F_~~ -100 - 50 0 50 100 150 TOTAL OZONE DEVIATION FROM MEAN A-c (m atm-cm) Fig. 17. Scatter diagram of first pattern vector coefficient plotted against total ozone deviation from mean for C wavelength pair, Arosa sample.

lengths of the three pairs A, C, and D, for the vertical distribution of ozone shown in Figo 18, which corresponds to about 360 m atm-cm of ozone. The quadrature formulation used is described in Appendix Co It will suffice here to note that both refraction and the sphericity of the atmosphere have been taken into account in the calculationso The results are plotted in Figs. 19-21, inclusive, wherein the source functions have in each case been normalized so that the greatest computed value is unity~ Since the source function curves overlap so strongly, only seven of each have been drawn. The resulting relative Umkehr curves and the relative intensities of the individual wavelengths are plotted in Figo 220 Referring to the source function curves we note that, as stated earlier, the "return" does indeed come from a definite layer of the atmosphere. However, the layer is an extremely broad one, the half-height (abscissae =0~5) points on each curve being separated by at least 10 km and as much as 30 kmo On this basis, we could say that at most three zenith angles provide return from the entire layer of atmosphere that is "sensed" in the zenith angle range 600 < g < 90Q0 If we are more liberal and take the 3/4 height points, we might say that at most five zenith angles are required. Since the long and short wavelength curves for 600 very nearly coincide, we may conclude that variations in the vertical distribution have little effect on the intensity ratio at 600~ In view of the classical explanation for the Umkehr effect of G3tz et alo, in Section 1o2, it is interesting to note the essentially double peaked nature of the source function curves for 70~ and 74~ for

45 /5 - 35 10 R E 20 S U 50 E 20 (mb) HI O O N j.mb. 100 T 15 (kin) 200 500 5 1000 N_ 0 50 100 150 200 250 300 PARTIAL PRESSURE OF OZONE (,u mb) Fig. 18. Vertical distribution of ozone used in the computation of source function X(Q,z).

62 80 80 SHORT A WAVELENGTH LONG A WAVELENGTH 3055 A 3254 A 70 B 70 60 00 60 88 50 85 0 50 900 40 40 -X,) ( X 880 30 30 - 85o 20- 20 700 10 10 0 0 0.2.4.6.8 1.0 0.2.4.6.8 1.0 X (z) X (z) Fig. 19. Source functions X(;,z) plotted against height for various zenith angles for the A wavelengths.

63 80 80 SHORT C WAVELENGTH LONG C WAVELENCTH 3114 A 3324 A 70 70 60 60 90 -50 50 880 E 40 40 10~~~~~~~~~~40 600 30 770 30 ~~~~~~~~~~~~~88* 20 111 20 C ~ ~ 8 70 0 lo 10 600 () ~~ / 0.2.4.6.8 1.0 0.2.4.6.8 1.0 X(z) X(z) Fig. 20. Source function X(Q,z) plotted against height for various zenith angles for the C wavelengths.

64 80 80 SHORT D WAVELENGTH LONG D WAVELENGTH 3176 A 3398 A 70 70 60 60 50 900 50 'E ~~ ~880 40 40 85 0 20 7 20 l 0 0 0.2.4.6.8 1.0 0.2.4.6.8 1.0 X (z) X(z) Fig. 21. Source functions X(Qz) plotted against height for various zenith angles for the D wavelengths.

ZENITH ANGLE (Degrees) 60 65 70 74 77 80 83 85 86.5 89 90 100 I I 88 50 20 10 5.02 1A. 10 * 05 ~02.01 Ll I 4 I I i 1 2 3 4 5 6 7x10? (ZENITH ANGLE)4 Fig. 22. Relative intensities and intensity ratios for wavelengths A, C and D plotted on a logarithmic scale against solar zenith angle.

the short A wavelength. The upper peak of these undoubtedly represents the predominating effect of decreased absorption path, whereas the lower one represents the predominating effect of greater atmospheric density. This effect is also to some extent observed on the 770 and 800 curves for the short C wavelength, but is heavily smoothed out on the short D wavelength curveo We may note further that the Umkehr effect does not occur until the scattering layer for the short wavelength is almost completely above the ozone maximumo Because of the great similarity in the shape of the source function curves and the strong overlapping, it is not at all surprising that strong colinearities exist between the points on the Umkehr curve. Looking at it in another way, we may say that the Umkehr effect is an integrating effect and, since information is always lost when we integrate, the Umkehr effect can inevitably provide us with no more than a smoothed vertical distribution of ozone. A measure of the vertical resolution attainable in these solutions is obtained from the half-width of the source function curves o We may note that measurements on the A wavelength pair are probably equivalent to observing on the C pair when the sun is about 2~ below the horizon and that measurements on the D pair are equivalent to stopping C observations when the sun is still 20 above the horizon. Since the "return'" for the C pair at 90~ is already giving information about ozone content in an atmospheric layer which is very nearly in photochemical equilibrium, we may expect that the ozone layer sensed by the A pair at

67 90 will have an ozone content considerably dependent on the ozone content of the lower layer sensed by the C pair at 900~ Hence, in effect, there is likely very little additional information to be gained by A pair measurements~ The possibility of obtaining additional information by taking observations on more than one wavelength pair is discussed in a later sectiono Since multiple scattered radiation originates mostly in the troposphere from primary scattered radiation arriving from the ozone layers above (Sekera and Dave, 1961), we may conclude that the incorporation of multiple scattering in the source function curves will certainly not improve the independence of the points on the Umkehr curve~ This is confirmed by the statistical analysis of the previous sectiono

4. DISCUSSION OF THE LINEARIZED EVALUATION METHOD 4ol PRELIMINARY REMARKS In the preceding chapter, it was shown that a linear statistical transformation between points on the Umkehr curve and the ozone content in a number of atmospheric layers can yield information on not more than four layers which should encompass the entire atmosphere up to about 50 km. We saw also that, at least qualitatively, such a result is to be expected from consideration of a simple physical -mathematical model in which only primary scattering is consideredo Moreover, we considered that the inclusion of multiple scattering was not likely to improve the resolving power, even though an increase in absolute accuracy might be expected~ We may, therefore, anticipate some difficulty with the linearized evaluation equations of Dutsch, which may be written X dijpj = ui, i=l....12 (39) j=1 where pj is now the layer-mean ozone partial pressure deviation from the standard distribution value in layer j, ui is the difference, N(@i)-1(Oi), between the observed N-value and the standard distribution rN-value at zenith angle Gi, and dij is aqi/6pj In practice, the ui are not known exactly because of instrumental and "modeling" errors. The latter will include inadequate allowances for multiple scattering, no allowance for aerosol effects, errors due to

truncation of the Taylor expansion leading to Eqo (39):(these may be considered' as errors due 'to the linearization of the basically nonlinear problem), errors in the quadrature formulation, and changes in the vertical distribution over the period of observationo If we represent the instrumental errors by ci and the modeling errors by 3i then we have j dijpj = -i - ii (40) j=l On the surface, it would appear reasonable to assume that the Ei are random variables with zero mean and are uncorrelatedo We should note, however, that Ei is not the instrumental error for a single observation. Each Umkehr curve consists of from 50 to 150 observations, and considerable smoothing of random errors is effected in the process of selecting the N-values for the 12 zenith angles to represent the curve. In addition, some of the random effects of aerosol scattering will be smoothed out in this processo If we lump together these residual errors into one error vector, eik = Cik+3ik, where k, as previously, represents the kth of n Umkehr curves, then it does not necessarily follow that n X eik = 0 k=l or that n X (eik-ei)(ek-) =, ji (41) We may, of course, consider that some of the nonlinear effects are in

70 eluded in our system, at least in the final iterative solution of the equivalent of Eq. (22), where an allowance is made for the second order derivative terms in the Taylor expansion. 4.2 THE "COMPLETE" SOLUTION OF THE LINEAR EQUATIONS Let us now write Eq. (39) in matrix notation as Dp = u (42) In the DUtsch system, this is a set of 12 equations in nine unknowns, and the least squares solution may be written directly as p (D*D)'-D* D (45) A unique solution exists if and only if the matrix D*D is nonsingular. In practice, however, difficulties arise when the matrix is nearly singular. In a recent paper of fundamental importance to the present discussion, Twomey and Howell (1963) have discussed the stability of solutions of equations of this type, which arise in the attempt to evaluate indirect soundings of the atmosphere. They show, as found by DIitsch (1957), that the complete solution of the system leads to wildly oscillating solutions, including the physically unacceptable result of negative ozone concentrations in some layers of the atmosphere. In arriving at the result of Twomey and Howell, we shall follow a somewhat different procedure. First, we shall "normalize" or scale the solution vector by setting

71 /p He (44) where Z may be considered as a completely diagonal matrix, having as its diagonal elements the standard deviations of the ozone partial pressures in the appropriate layerso Then we have that DZa = = u (45) where A = D.o The need for such a normalization or scaling is discussed later. Next, we shall expand the normalized solution vector in terms of a set of orthonormal vectors as follows: = * (46) where the orthonormal vectors are in the rows of the matrix W and b is the vector of coefficients of the vectors in the expansiono This is perfectly general in the sense that any arbitrary vertical distribution of ozone (expressed in terms of the mean partial pressures in nine layers) can be expanded exactly in terms of such a set of vectors. If we now substitute (46) into (45), we have that AW*b = u (47) which has the least squares solution for the vector coefficients b (WAAW L*WhA* t (48) Let us next require that the contribution to the solution by each vector

72 shall also "explain" an independent portion of the total variance of the Umkehr curveo The residual variance of the Umkehr curve is just (u-AW*b) *(u-AW*b) = u*u b*(WA*AW*)b (49) when we make the substitution of (47) for u and note that each term in the expansion is a scalar, and that the transpose of a scalar is just the scalar itself. Thus, the solution contribution of each vector will explain an independent portion of the total variance of the Umkehr curve provided that W(A*A)W* = A (50) where A is a completely diagonal matrixo As in the previous chapter, this is just the problem of determination of the eigenvalues and vectors of the real symmetric matrix A*A, As before, we shall specify that the eigenvalues, Xi, i = 1,oo..,9, occur on the diagonal of A in order of decreasing magnitude and that the rows of W are numbered accordingly. Equation (48) now becomes b A -w(A*) (51) We note further that the coefficients bj are determined independently of each other, being just the product of the reciprocal of the corresponding eigenvalue, kj, and the dot product of the vectors wj and (A*u), where ij

is the jth row of Wo That is to say, b. - X. w(A*u) (52) In addition, it follows directly from (49) and (50) that the variance explained by the jth vector is \j.b2 aJ The eigenvalues and vectors for A*A for one configuration of DUtsch's first derivative matrix, in which the instrument constant is eliminated, total ozone is used, and the scaling vector CI shown in the first column of Table 17 is used in the diagonal of Z, are shown in Table 7. We note the following: (i) There is a tremendous range in the eigenvalues of the matrix, from 46~8 to about 10, the ratio being 4-5 x 106 (ii) The low order eigenvectors, corresponding to the larger eigenvalues. exhibit very few changes in algebraic sign from one element to the nexto Thus their contributions to the solution vector A will be "smootho" (iii) The high order eigenvectors, corresponding to the very small eigenvalues, exhibit frequent changes in sign from one element to the nexto Their solution contributions will not be "tsmooth " It is pertinent to consider what might be termed the "instability ratio" (Rj) for the normalized solution contribution by the jth vector~ We may define this as the sum of the squares of the normalized solution contributions divided by the amount of the variance of the Umkehr curve

TABLE 7 EIGENVALUES AND VECTORS FOR ONE CONFIGURATION OF DUTSCH'S FIRST DERIVATIVE MATRIX FOR HIS STANDARD DISTRIBUTION I Vector Fractional Eigenvector Points Number Eigenvalue Eigenvalues 1 2 3 4 5 6 7 8 9 1.468C5E 02.69347 -.17969 -.47106 -.4618C -.40548 -.32145 -.12481.13566.30839.36841 2.15492E 02.22953 -.C8719 -.02204.16043.262C5.37448.44135.5i507.39917.371~7 3.41522E 01.06152.33405.47201.2481 7 - -.23142 -.35635 -.16788.33450.53235 4.917C6E 00.01359.68202.18706 -.21922 -.29723 -.16631.22658.46846 -.02005 -.25415.11645E 00.00CC173 -.49336.28229.26323 -.04507 -.39135 -.26618.52828.09560 -.30554 6.94714E-02.CCC014.26262 -.29892 -.08167.28847.34656 -.65079.11957.30991 -.28262 7.158C4E-02.OCCC2 -.23646.45609 -.27251 -.48948.52067 -.03103 -.15579.31067 -.17138 8.34137E-03.OCCC1 -.12243.37781 -.6985C.50999 -.06399 -.08045.11768 -.21095.15926 9.IC342E-C4.OCCCC.C0895.00577 -.11505.30598 -.35605.33088 -.34529.62215 -.38791

75 that is "explained" by this solution contribution. It follows that 9 X (bjwji) i-=l 1 Ji Thus the vectors corresponding to the large eigenvalues have a relatively small instability ratio, providing a large reduction in Umkehr curve variance for relatively small contributions to the solution~ Moreover, as noted above, these solution contributions are "smooth." On the other hand, the vectors corresponding to the very small eigenvalues have a very large instability ratio, providing always a relatively small or negligible reduction in curve variance while introducing very large nonsmooth or oscillatory contributions to the solution. We may conclude that it is the solution contributions of these vectors which introduce the large oscillations into the "complete"' solution of Eqo (45)~ Looking at the problem in another way, we might consider the "predictability," P., of the coefficient of the jth vector as the inverse of the instability ratio~ Thus, those vectors which explain little or no curve variance. but contribute much to the solution, are not really predictable from Umkehr observationso That is to say, the eigenvectors associated with the very small eigenvalues represent linear combinations of the unknown variables about which Umkehr observations contain no information (Lanczos, 1956)o The problem of deciding where the information ends and the noise begins must be decided by numerical experiment coupled with information about the probable experimental errorso

76 403 THE PROBLEM OF INFORMATION VERSUS NOISE 4o3.1 The Use of the Characteristic Patterns of the Umkehr Curve In attempting to separate the noise of the Umkehr curve from the basic information, we may use the filtering device of the Characteristic Patterns introduced in Chapter 3. Let us expand the Umkehr curve points ui in terms of these Co P.o s, truncating the expansion by using only the first four patterns. Thus A u B*y + u (54) where now B is a matrix with only four rows each comprising 12 elements, u are the points of the smoothed Umkehr curve, u is the average Umkehr curve for the sample considered and y B(u-U) (55) The noise of the Umkehr curve may be defined as u& where A u = u+u' (56) Substituting in (51) we get b + b = A WA*(u+u') (57) where -= AJ w(A*) (58)

77 and = A'lW(A*u') (59) The above quantities have been calculated for the Arosa data sample and are given in Table 8 as averages of wiA*u (mean smoothed dot product), wiA*uJ (mean error dot product), Xibi/v2 (mean fractional variance exi2 2 plained by smoothed curve coefficient), and Xibi/v (mean fractional variance explained by error curve coefficient), and ki(bi+bi) /v2 (mean fractional variance explained by combined curve coefficient) for each of the vectors wi, i - 1,...,9. We note- that there is certainly no question about the information content of the smooth Umkehr curve as provided by the-first three eigenvectors of the matrix A*A. Moreover, the error TABLE 8 AVERAGES OF SMOOTHED AND ER rOR DOT PRODUCTS AND FRACTIONAL VARIANCE EXPLAINED BY EACH AND BY THE COMBINED DOT PRODUCTS Vector.,2 Vector _ Xi _i__i _i(ti+bi)2 Number Wi ' w. A* i A* - --....... v2 v2 1 6.758 (1) -8.68 (-4).51553.00000.51552 2 -1.755 (1) -2.74 (-3).25212.00000.25214 3 1.718 (1) -3.53 (-3).21634.00001.21627 4 -1.379 (0) 2.68 ( - 3).01023.oooo8.01029 5 1.655 (-2) 1.40 (-2).000o 4.00124.00179 6 -4.016 (-2) 2.84 (-4).000o 8.00072.00129 7 -1.059 (-2) 4.27 (-4).00037.00037.00076 8 4.450 (-3)) 2.77 (-4).00017.00023.00038 9 -6.191 (-4) 3.84 (-5).00012.00015.00023 Number in parentheses is power of 10 by which preceding number is to be multiplied.

78 curve contribution is much smaller than that of the smoothed curve for these eigenvectors. The smoothed curve also provides the bulk of the contribution to the fourth coefficiento However, on the average, the fourth eigenvector explains a rather small fraction of the total curve variance. For the fifth and higher order eigenvectors, the variance explained is negligibly small and more is explained by the error curve than by the smoothed curveo The mean total curve variance is 372.7 (N-units)2; after the first three eigenvectors have been used, the mean residual variance is 6.0, after the first four, it is 1.2. In view of the uncertainties involved, we can certainly expect to obtain no additional real information by explaining this residual variance. We conclude that most of the information content of the Umkehr curve about the vertical distribution of ozone is obtained when we solve for the coefficients of the first three eigenvectorso Perhaps there is some further information to be obtained by solving for the coefficient of the fourth eigenvector; at least we are sure that the fourth vector coefficient is not much contaminated by noiseo However, there is no basis for inclusion of the fifth and higher order eigenvectors in the solution system. Thus there are, at most, four pieces of information about the vertical distribution of ozone in the Umkehr observations. 4-3.2 Stepwise Solutions Using the Eigenvectors of A*A We may, of course, proceed directly from Eqo (52), determining the coefficients bj and the variance explained \jbj for each vector in turn.

79 We test the residual variance after each.calculation and, when it drops below some specified limit, truncate the solution procedure, considering that we have extracted all the available information about the vertical distribution. This has been done for the Arosa data sample, using a residual variance testing limit of 6.0 (N-units)2, although the solution contributions and explained variance were computed 7for all vectors. The frequency distribution of the number of vectors used in the stepwise solution.procedure, plus the average fractional curve variance explained by each eigenvector, are given in Table 9. The numbers in the third column of Table 9.represent the same thing as the numbers in the last column of Table 8. In the present case, however, the fraction is taken with each curve and the fractions are averaged. In the previous case, the exTABLE 9 FREQUENCY DISTRIBUTION OF THE NUMBER OF EIGENVECTORS USED IN THE STEPWISE SOLUTION PROCEDURE AND AVERAGE FRACTIONAL UMKEHR CURVE VARIANCE EXPLAINED BY EACH VECTOR Eigen- Frequency Distribution Average Fractional vector of Solution Truncation Variance Explained Number Levels by Each Vector 1 0.4156 2 1.2241 3 61.3313 4 33.0166 5 4.0035 6 0.0030 7 1.0016 8 0.0013 9 0.0007

80 plained variance was- summed directly and the fraction taken at the end. To achieve the selected level for explanation of curve variance, we find that only five cases require more than four vectors, four of these requiring five vectors and the remaining one requiring seven. In the last case, the solution is quite ridiculous, the difficulty apparently being that there is a large residual for G = 650, which is not satisfactorily explained by any number of vectors, but the residual variance is brought down just below the limit when the seventh vector is included. In the four cases requiring the fifth vector, there are a few moderately large residuals after four vectors have been used, such that the fifth vector was required to bring the overall residual variance down below the selected limit. As might be expected, all these solutions look a little queer, three of the four having negative tropospheric ozone. The solution for March 21, 1962, is shown in Table 10. In this particular case, the inclusion of the fifth and sixth vector contributions would still leave a realistic looking solution. However, since very little variance is explained by these contributions, there is no reason for their inclusion. As indicated earlier, the seventh, eighth, and ninth vectors introduce the wild oscillations which must be excluded from the solution system. The nonuniqueness of the "complete" solution arises from these high order vectors. They represent linear combinations of the unknowns which may be added to the solutions, without affecting, to any appreciable degree, the residuals. Thus we have a triple infinity of solutions that

TABLE 10 STEPWISE SOLUTIONS FOR MARCH 21, 1962, SHOWING INDIVIDUAL SOLUTION CONTRIBUTIONS BY EACH EIGENVECTOR (Total ozone is 403 m atm-cm) Eigen- Vector Explained Solution Contributions and Summed Solutions vector Coefficient Variance Residual for the Various Vectors Number bi Variance Layer-Mean Partial Pressures (4mb) b %i b2 1i 1 2 3 4 5s: 6 7 8 9 204.4 23o5 42o1 84.3 132.6 13359 95.2 53.4 20.1 7o 0 1 0.225 2S4 202.0 0.a5- 3.2 25 - 1.6 - 1O -,3 0.2 0.2 0ol 2350 38.9 81.8 131.0 132.9 94o9 53o6 20.3 7.1 - 2.2 1.4 83 101ol 11.2 8.5 6.6 2.6 1.2 20,8 37.5 90.1 141.1 144.1 103.4 6002 22.9 8,3 5 5.433 122.5 8.2 21,8 76,9 32~4 - 0Q4 - 17.6 - 17.4 -5.5 55 4.3 42,6 114'4 122.5 140.7 1265 86.o0 54.7 28,4 12,6 21.2 14.5 13.6 13.9 6.o0 5.3 -7.3 0.2 1.0 4 - 2.590 6ol 2ol 21.4 99o9 136.1 154.6 132o5 80,7 47.4 28,6 1356 123 02 72 103 77 - 10 6.6 - 2,9 3.8 053 0 o.6 14.2 110.2 143.8 153.6 125o9 77.8 5102 28.9 1300 6 0X 9817 OoO09 1.9 35.1 - 8.9 - 1.9 5.1 4e8 - 5.8 1.1 0.9 -0.4 17 3 101.3 141.9 158.7 130.7 72.0 52,3 29.8 12 6 7 =15. 49 o.4 1.5 44 -212 101 136 -113 4 14 =14 4 8 55.3387 0o4 1 ol 50 384 -568 311 30 -24 24 =21 8 9 -43579 0'.05 10 - 5 - 8 121 -241 1218 130 91 -82 25

82 are "plausible" from the point of view of explained curve variance by choosing 0 > b7 > - 15o49, 0 < b8 < 33087, and also 0 > b9 > - 43~79, in the particular case of Table 10 In general, similar remarks apply to the choice of b5 and b60 Since there are at least an infinity of solutions within the above framework that are physically acceptable in that they "look reasonable," we can see clearly why solutions carried out subjectively (by, for example, solving (42) by hand relaxation) are not comparable with each other. Such solutions do not, in fact, obtain from the Umkehr curve the information that is really there, but have "noise" introduced, to an unknown degree, by the personal ideas of the evaluator about what the vertical distribution should look like. 4.4 OBJECTIVE METHODS OF SMOOTHING THE SOLUTION 4.41 Truncation of the Eigenvector Expansion By now it should be clear that the Umkehr observations contain no information about variations in ozone content from one layer to the next. This information must be given by the higher order eigenvectors whose coefficients are not predictable from the observations~ As noted in the last section, there is a multiple infinity of solutions which will satisfy Eqo (42). Most of these solutions are physically unrealistic, but there remain at least an infinity of solutions which are physically plausibleo It is necessary, therefore, to devise some objective method for selecting a "best" solution, which is consistent with the information that we know we may infer from the Umkehr observations~

One way of obtaining such a solution system is to expand the solution in terms of the eigenvectors of the matrix A*A, truncating the solution with either three or four vectorso It is evident from the preceding discussion that such a procedure will lead to the explanation of a satisfactory amount of curve variance. Moreover, the solutions so obtained are physically plausible when a suitable normalizing procedure has been chosen. A discussion of the scaling problem (i.e., choosing a suitable normalizing procedure) and a more complete discussion of solutions actually obtained by this method are given in the next chapter. 4.4.2 Twomey's Method In extending a paper by Phillips (1962) which dealt with a similar system of equations, Twomey (1963) introduced a method of objective smoothing which is particularly appropriate to the present case. Twomey and Howell (1963) have also discussed the application of this method in the evaluation of indirect soundings of the atmosphere. Twomey starts with Eq. (45) with an error vector, A, added as follows "A = u + e ' (60) He imposes the condition that 12 7 ei = constant (61) i=l and applies the constraint that the sum of the squares of the solution

84 deviations from a trial solution shall be a minimum. In our particular problem, an obvious choice for a trial solution is just the standard distribution for which the matrix D of partial derivatives has been calculated. This is equivalent to requiring minimization of the following quantity. 9 12 i + Y1 ei (62) j=1 i=l where y is an undetermined Lagrange multiplier. To find the minimum, Twomey differentiates (62) with respect to the (jo Noting from (60) that dei = Aij (63) we get 12 yj + X eibij = (64) i=l or A*e = -t (65) Premultiplying both sides of (60) by A* and introducing (65), we have (A*A+yI)s= A*u (66) which has the solution = (A*A+YI) a*, (67)

To determine the precise meaning of this equation in terms of our previous discussion, it is instructive to expand the solution in terms of the eigenvectors of the matrix (A*A+7I). Let V(A*A+yI)V* = A' (68) ButV(yI)V* = yI whence-. V(A*A)V* = ' - yI However, WA*AW* = A and therefore A' = A+ I and W = V provided we require that the eigenvectors- in the rows of V be orthonormal~ Thus we are expanding our solution in terms of the same eigenvectors as before, but now each of the eigenvalues has been increased by 7. The solution for the eigenvector coefficients may now be written

as b (. +)') -Ilj (A*u) (69) Thus, in the determination of bj, we now multiply by (kj+7)-1 instead of o kjm 0The quantity y is, therefore, a smoothing factor which must be chosen sufficiently large that the high order eigenvector coefficients are effectively reduced to zero, but the low order vector coefficients remain essentially unchanged. The stepwise solution procedure discussed in Section 4-3.2 and illustrated in Table 10 may now be repeated for Twomey's method. The new vector coefficients are obtained from those in Table 10 upon multiplication by %j/(7+kj). The same multiplying factor is required for the variance explained by each vector. In Table 11, the solution of Table 10 is repeated using y = 005o This value of y is sufficiently large to eliminate effectively the solution contributions of eigenvectors six through nine, inclusive. The contribution of the fifth vector is decreased to 20% of its original value, that of the fourth to 65%, while the contributions of the first three vectors are relatively unchanged. We may note that the smoothing accomplished by Twomey's method is achieved at the expense of some loss in explained variance. For example, comparing the method of Table 10 with Twomeyts, we find the re2 sidual curve variance here is 19.6 (N-units), whereas three vectors left 8.2 and four vectors 2.1 in the previous method. The variance explained can, of course, be increased by decreasing y, at some loss in

TABLE 11 STEPWISE SOLUTIONS FOR MARCH 21, 1962, SHOWING THE INDIVIDUAL SOLUTION CONTRIBUTIONS BY EACH EIGErVECTOR USING TWOBMEYS METHOD WITH Y = 0.5 Eigen.- Vector Solution Contributions and Summed Solutions vector Coefficient Explained Residual for the Various Vectors Number Variance Variance Layer-Mean Partial Pressures (imb) bi. i 1 2 3 4 5 6 7 8 9 204.4 23.5 42.1 84 3 132 6 13359 95 2 53.4 20. 1 7 o: 1 0.224 2,* 4 ~202- 0.,05 - 3.2 - 25 1.6 1- 0 1 0.2 0.2 0.2.1 23~0 38 9 81.8 131.0 132.9 94.9 53 o 6 20.5 3 7 1 9 2.1 1,4 8.0 9 8 10.9 8.2 6.4 2.5 1,2.2 2,o078 69~1 132.9 2 6 20,9 40,3 89.8 140 8 143.8 103.1 60o.0 22,8 8~ 3 3 4,o849 109. 35 253.6 19o5 68.6 28 9 - 0,4 -15.7 -15.5 -4 9 4.9 3 8 40o4 108.9 118.7 140.4 128.1 87.6 55.1 27.7 12.1 4 -1.o676 4.0e 196 -13.7 - 9.4 8.8 9,0 3.9 - 3.4 -4.7 0.1 0.6 26.7 99o5 127.5 149.4 132.0 84.2 50.4 27.8 12.7 0.229 0~03 502319 14 1.9 1o5 - 0~2 - 1.2 - 0.5 0.7 0.1 -0.1 25.3 101 4 129.0 149.2 130.8' 83.7 51.1 27.9 12.6 6 0o018 0o001 - 0.1 0 2 0.1 0.0 - 0.1 - 0o1 0.1 0.0 0.0 25.2 101.6 129,1 149.2 130.7 83.6 51.2 27o9 12.6 7 -0.049 - 196 0 o 1 0.7 0o3 0o4 - 0.3 o0 0o0 0.0.0 o 8 0.023 19.6 0.0 0.1 - 0.2 0.1 0.0 0.0 0.0 0.0 0O0 0.0 0.0 0.0 0 00 0.0 00 0.0 0.0 00 9 o~oo8 e 19,6 25o3 101.0 129.o2 149.o7 130.o4 83o6 51o2 27.9 12.6

88 the smoothness of the solution. The example presented here is an extreme one insofar as explained variance is concerned, since about 60% of the curve variance is explained by the third eigenvector. Normally, only about one third of the curve variance is explained by this vector so that the loss is somewhat less. As will be demonstrated later, Twomey's method is an extremely good one, providing good smoothing of the solution without a too large loss in explained variance. It has the distinct advantage over the method proposed in the previous section that the mathematical constraints applied to the solution are more clearly understoodo In particular, the solution contributions by the various vectors are diminished in accordance with the predictability of the vector coefficients. In the method proposed in the last section, we would like to include the fourth vector in the solution, yet the solution contributions seem rather large compared to the amount of variance explainedo We should probably eliminate completely the contributions of the fifth and higher order vectors. It also appears reasonable to retain the first three vector contributions without reductiono Thus a combination of the two methods, wherein the first three vector contributions are retained with full weight, the fourth vector contribution is retained at some reduced weight, and the remaining vectors eliminated completely, might prove superior to either of the above methods. Our main concern here is to determine what information is contained in the Umkehr observations and how this information can best be inter

89 preted in terms of the vertical distribution of ozoneo It may, therefore, be unwise to weight the eigenvector solution contributions according to the size of the eigenvalue as in Twomey's method, since the linear combinations represented by the eigenvectors almost certainly do not occur in the atmosphere. with the same proportionate strength as indicated by the respective eigenvalueso Indeed, there is good numerical evidence, comparing the last columns of Tables 8 and 9 with the eigenvalues of Table 7, that they do not..Solutions computed according to the combined method are presented in the next chapter. 4.5 SOLUTION CONTRIBUTIONS ASSOCIATED WITH THE CHARACTERISTIC PATTERNS OF UMKEHR CURVES It is of interest to consider the solution contribution associated with each of the Characteristic Patternso Since these patterns represent deviations from the mean Umkehr curve, we may consider the associated solution contributions as deviations from the mean solution for the sampleo If we combine Eqo (34), (44), (46), and (51), the solution contribution vector Api associated with the ith C. P. is given by where &*i is the Co Po and yi is the coefficient of this C. P. when the Umkehr curve is expanded in terms of the C. P os. The APi have been computed using the Co P.o s and the derivative matrix configuration used in Section 403.2. The truncated eigenvector expansion method of solution has been assumed with four eigenvectors usedo Root-mean-square values

90 were used for the C. P. coefficients, yio Results are plotted in Fig. 23 for i = 1l2,3,4. Very similar solution contributions are obtained when TwomeyVs method is used. It has to be remembered that the contributions represent deviations from the mean solution. These solution contribution vectors appear with both positive and negative signs and, on the average for the entire sample, each contribution vector has zero mean. Since the coefficient of the first C. P. is strongly correlated with total ozone, we see that above-average ozone results in addition of ozone in the lower stratosphere (layers 2 and 3) and below-average ozone results in subtraction of ozone from the lower stratosphere.

9 8 I 1/~~~~~~~ 5 \ I:U I I~~~~~~~~~~~~~~~~~~~~~~~ 4 3, 1 goo 4~~~ -10 0 10 20 30 Solution Contribution Ap (,umb) Fig. 23. Solution contribution in the various layers for each of the first four Charactertic Patterns.

50 FURTHER REMARKS ON EVALUATION METHODS AND PRESENTATION OF RESULTS 5do THE SCALING PROBLEM There are two scaling problems to be considered, Viz., the weighting vector for the equations (the rows of the matrix D and the corresponding elements of u) and the weighting vector for the variables (the columns of D)o In the customary configuration of the matrix D as used in this report, the first equation (row) represents the conservation of ozone, i.e oe the requirement that the vertical distribution should be nearly equivalent to the measured total amounto The remaining rows of the matrix D are those of Dfitsch's matrix (Tables B-3 to B-7), but with the first row subtracted from each of the otherso This is equivalent' to elimination of the instrument constant. In addition, since he solves for the fractional ozone change in each layer and we solve for the mean partial pressure change from the standard distribution, the elements of each column have to be divided by the mean ozone partial pressure in the various layers in the standard distributiono The problem of scaling the ozone conservation equation deserves some discussiono If the equation is used in such a way that the observed residual entered in the vector u is in m atm-cm, then ozone conservation dominates the solution procedure in such a way that solution total ozone is exactly (to the nearest m atm-cm) the same as the observed total amount o This is equivalent to claiming a priori knowledge that the total 92

93 amount of ozone is known without error~ However, actual a priori knowledge is that measured total ozone has a standard error of estimate of about 2 m atm-cmo It has been found by numerical experiment that, in the case of the C wavelength pair, agreement within a standard deviation of about 2 m atm-cm is achieved by using a total ozone weighting factor Wg = 0olo In all cases, the weighting factor used will be stated. Apart from the ozone conservation equation, do we have a need for scaling the problem? There are two simple direct approaches~ First, since we are solving for mean partial pressures in each layer, w4y not solve directly for these without scaling? This is equivalent to choosing column and equation scaling vectors with all elements equal~ The average solution and associated statistical data, for the Arosa data sample are given in Table 120 The solutions have been carried out using Twomey's method with a column scaling vector of 10 units, W Q = 0-o.1 a scaling factor of unity for the remaining equations, and y = 0.50 The solution residuals are moderately large; moreover the ozone mixing ratios appropriate to layers 8 and 9 indicate a rather large increase of mixing ratio with height, which is not in conformity with photochemical theory~ A second direct approach is to solve, as DU-tsch does, for the fractional change in the ozone amount in each layer~ This is equivalent to using, in the present context, a column weighting vector having elements proportional to the ozone partial pressures of the standard distribution in the respective layerso The average solution, with associated statis

94 TABLE 12 AVERAGE SOLUTION FOR AROSA DATA SAMPLE BY TWOMEY'S METHOD WITH COLUMN SCALING OF 10, WQ = 0.1, AND y = 0.5 Layer-Mean Standard Deviations Partial of Layer-Mean Pressures Partial Pressures (pmb) (pmb) 1 34 1307 2 48 14 9 3 86 18o 4 4 129 19o 5 5 12 3 17 8 6 77 11.8 7 40 6.4 8 18 3~0 9 14 2~6 Mean Residual Variance (N-units)2 4.8 RMS Residual (N-units) o 063 Mean Total Ozone Residual (m atm-cm) 3.0 RMS Total Ozone Residual (m atm-cm) 6~3

tical data for solutions by Twomey's method, with y = 0.5, Wo = 0.2 (0.1 was too low), and other equations scaled with unity, are given in Table 135. The bracketed numbers in the table are for the case-where W= 1.0. We note the improvement in the total ozone "fit" when the conservation equation dominates the solution with no real change in the mean solution. A positive total ozone residual means that the observed value exceeds the solution value. Thus the difference between the two solutions, in the mean, is that the ozone solution deficit of 2.2 m atm-cm when Wo = 0.2 has been put into the lower atmosphere when WQ = 1.0. Another obvious way of scaling the problem is to divide the elements of the ith row of D (and the ith residual) by the quantity (. )/ to get matrix D'. Next, we may scale the jth column-of D' by dividing by the quantity (. d,2)1/2 to get the matrix A. Then the matrix (A*A) is i ij like a correlation matrix in the sense that the diagonal elements are all unity and the off-diagonal elements are less than unity in absolute value. The avera eas obtained in this manner are given in Table 14. The solutions are unrealistic in that tropospheric ozone concentrations are frequently negative and the concentration in layer 7 is sometimes less than that in layer 8, something not anticipated from photochemical theory. In addition, the residuals are somewhat larger than we would care to accept. Otherwise the solutions seem not unreasonable. The individual solutions show a strong increase in lower stratospheric ozone when the total amount is large. These solutions have been carried out with the truncated eigenvector expansion (hereinafter TEVE) method using

TABLE 13 AVERAGE SOLUTION FOR AROSA DATA SAMPLE BY TWOMEY'S METHOD WITH COLUMN SCALING EQUIVALENT TO STANDARD DISTRIBUTION LAYER PARTIAL PRESSURES, WQ = 0.2 (1.0), AND y = 0.5 Layer-Mean Standard Deviations Partial of Layer-Mean Pressures Partial Pressures (tmb).(!mb) 1 29 (30) 9.1 (10.9) 2 48 (49) 8.6 (9.7) 3 100 (100) 25.2 (26.8) 4 144 (144) 39.2 (39.5) 5 108 (107) 19,1 (19o0) 6 63 (64) 11.9 (11o9) 7 47 (47) 7.1 (704) 8 26 (27) 3.7 (3,7) 9 9 (10) 1.! (1.2) Mean Residual Variance (N-units)2 3.8 (3.4) RMS Residual (N-units) o.56 (0o53) Mean Total Ozone Residual (m atm-cm) 2,2 (0.1) RMS Total Ozone Residual (m atm-cm) 4.1 (0,2)

97 TABLE 14 SOLUTION STATISTICS FOR AROSA DATA SAMPLE FOR SOLUTIONS BY TRUNCATED EIGENVECTOR EXPANSION METHOD WITH COLUMN AND EQUATION SCALING VECTORS DETERMINED FROM DERIVATIVE MATRIX Equation Column Layer-Mean Standard Deviations Weights Weights Layer or W Partial of Layer-Mean Equation (Z i (L / Pressures Partial Pressures iij i (ptmb) (tmb) 1. 0.52 (=wn,) 1.91 30 21 1 2 o069 2,39 63 40oo 3 o0,88 1.81 93 3359 4 1.09 1.52 130 24.6 5 1.41 1.532 120 15. 3 6 1,78 1o15 73 10.1ol 7 2,29 0.95 37 8,5 8 3.08 0.72 23 300 9 4.14 0.39 13 3,4 l0 5.73 1.1 9.38 12 22,44 Mean Residual Variance (N-units)2 13o.8 RMS Residual (N-units) 1o08 Mean Total Ozone Residual (m atm-cm) -0.8 RMS Total Ozone Residual (m atmr-cm) 3-,8

98 the first four eigenvectors. Still another method of scaling would be to introduce an arbitrary value for WQ, but otherwise to scale only the columns of D by the method of the previous paragraph. Solution statistics for this procedure are given in Table 15. Both TEVE and Twomey methods have been used with W = 0G1, and y = 0.1 for Twomey's method. The solutions by the TEVE method are fairly reasonable, there being a few small negative ozone values in the troposphere and occasional low values in layer 6. The solution residuals are quite reasonable. The average solution by Twomey's method is not greatly different from the previous one, but the differences are characteristic and persist through all solutions carried out by the two methodso TEVE method always gives less tropospheric ozone, more in the stratosphere (layers 2 and 3), less in layers 6 and 7, and more in layer 8 than Twomey's method. Thus the tendency toward solution instability in the TEVE method (because of the introduction of the fourth eigenvector with full weight) is eliminated by using Twomey's method, with a sufficiently large 7, at some expense in increased residual variance. Looking back over Tables 12-15, inclusive, we note evidence of a correspondence between the column weighting vector and the variability of the solution amounts in each layero This suggests an additional constraint which we may impose on the solution, vizo, we may choose a column weighting vector such that the variability of our solution layer-mean partial pressures approximates most closely that found in the atmosphere in

99 TABLE 15 SOLUTION STATISTICS FOR AROSA DATA SAMPLE FOR SOLUTIONS BY TEVE AND TWOMEY METHODS WITH COLUMN SCALING VECTOR DETERMINED FROM DERIVATIVE MATRIX, WQ = 0.1O, AND y = 0,1 Layer-Mean Standard Deviations Column Partial of Layer-Mean Layer Weights Pressures Partial Pressures (Twmb) (omb) TEVE Twomey TEVE Twomey 1 7.52 29 33 16o 4 13o 6 2 8.90 62 57 26.3 25,9 3 7.o 34 96 90 27.2 24o 6 4 6o 05 130 126 21o4 18.6 5 5.19 117 116 15.9 14.3 6 4.32 65 70 11o7 11.2 7 2.60 44 46 5.7 5.8 8 1.10 26 25 3.2 2,7 9 o.48 10 10 1.7 1.3 Mean Residual Variance (N-units)2 2.3 4.7 RMS Residual (N-units) O,44 o.63 Mean Total Ozone Residual (m atm-cm) -0.7 1.9 RMS Total Ozone Residual (m atm-cm) 1,7 3.4

100 actual balloon soundings~ The statistical parameters for the balloon sounding data available to the writer are given in Table 16. The method of processing the data is described in Appendix D along with a listing of the sources of the data~ We find that two distinct patterns of variability are in evidence in the lower layers of the atmosphere~ The first pattern, applicable to moderate and large ozone amounts, indicates a maximum of variability in layers 2 or 3, while the second pattern, based, however, on relatively few soundings for low ozone, shows a maximum variability in layer 4~ Column weighting vectors actually selected for use in the solution procedures used for further work are given in Table 17o The solutions described in Chapter 4 were all calculated from DUtsch's Standard Distribution I (hereinafter SI) using column weighting vector CI with W2 = 0ol Statistics for solutions with column weighting vector CI, with Wn = 0ol, for both TEVE and Twomey methods are listed in Table 180 Solutions by the Twomey method have been carried out using Z = 0~25, 0o5, and lo.0 The differences between the two methods, as described earlier, are readily apparento In addition, as we decrease 7', we note the two solutions approach each other~ When we increase 7y the persistent differences increase in magnitude, the solution variability decreases, and the unexplained variance increaseso In Table 19, solution statistics are listed to demonstrate the effect of changing the weight WS- of the ozone conservation equation in the

101 TABLE 16 AVERAGES AND STANDARD DEVIATIONS OF LAYER-MEAN PARTIAL PRESSURES FOR BALLOON SOUNDINGS AND UMKEIHR DATA Standard Deviations of LayersMean Partial Pressures Layer-Mean Partial Pressures LayerMean Partial Pressures Layer, (cub),, ( Lmb) I II III IV V I II III IV V 1 14 12 6 18 15 23 22 13 32 23 2 35 26 13 26 36 57 49 22 97 46 3 31 22 19 30 14 104 102 68 130 91 4 28 22 27 23 17 149 143 128 174 141 5 (17) (17) (12) (20).(14) (116) (114) (111) (122) (116) 6 (11) (12) (8) (10) (10) (78) (79) (74) (80) (82) 7 (6) (6) (5) (6) (7) (48) (47) (45) (51) (5i) 8 (2.3) (2,.2) (2,3) (2.2) (3.2) (22) (22) (22) (22) (23).9 (1o?) (:o8) (1o?) (1.3) (2,3) (11) (11) (10) (11) (11) Average Total Ozone (m atm-cm) 345 334 279 408 333 Io 121 soundings II + III + IV II 71 soundings 300 ~ 2 < 375 m atm-cm III0 18 soundings K < 300 IV~ 32 soundings a > 375 V~ 29 soundings simultaneous with Umkehr observations

102 TABLE 17 COLUMN WEIGHTING VECTORS USED WITH DERIVATIVE MATRICES IN SOLUTIONS Column Weighting Vectors (Layer) CI CII CIII 1 12 10 5 2 30 20 10 3 24 24 15 4 18 18 20 5 14 14 15 6 9 9 9 7 6 6 6 8 3 3 3 9 1.5 1.5 1.5 two methods. Except for an increase in the variability of the solution in the troposphere, a decrease in the variability of the total ozone residual, and in its average, the results are virtually the same, on the average, as W2 is increased. In the case of the TEVE solutions, the stronger forcing on ozone conservation gives ozone partial pressures in the troposphere which are just negative in two cases. Next, we shall illustrate two effects. First, we have used column vectors CII and CIII, with Wo = 0.1 and y = 0.5, to obtain solutions by Twomey's method. Second, we have computed solutions with the combined method referred to in the last chapter, where only the first four eigenvectors are used but the fourth one is weighted by using y = 0.5. Column weighting vector CI has been used in this case. These results are presented in Table 20. The solutions statistics for CII and CIII illustrate rather clearly the effect of the column weighting vector on solution vari

10o3 TABLE 18 STATISTICS FOR AROSA DATA SAMPLE SOLUTIONS CARRIED OUT BY TEVE AND TWOMEY METHODS USING COLUMN WEIGHTING VECTOR CI, WQ = 0.1, AND SI Standard Derivations of Layer-Mean Partial Pressures an Layer-Mean Partial Pressures Layer (Lmb) (p mb) TEVE '......Twomey TEVE Twomey TEVE 7=.25 7=.5 7=10 TEVE...0, _ r=,2~ _~Y=. 5 Y=,L.,,y=4, _ _,..,,, 1 26 27 28 29 11.8 11,0 8.9 7.1 2 73 73 70 66 33. 3 36.0 33.3 30.7 3 96 94 92 88 29.3 30..5 28.7 26.9 4 124 123 122 121 20.8 19.2 19.0 18.5 5 112 110 111 112 15.5 14.6 14.5 14.3 6 73 75 76 77 9.7 9.4 9.1 8.7 7 43 44 45 46 6.6 6.7 6.1 5.6 8 24 24 24 23 2.8 2.8 2.6 2.4 9 11 11 10 10 1.9 1.6 1.5 1.4 Mean Residual Variance (N-units)2 2.1 2.2 3.4 6.4 RMS Residual (N-units) 0.42 0.43 0. 53 0.73 Mean Total Ozone Residual (m atm-cm) -0.5 0.1 0.9 3.0 RMS Total Ozone Residual (m atm-cm) 2.4 2.5 3.8 6.3

TABLE 19 STATISTICS FOR AROSA DATA SAMPLE SOLUTIONS TO ILLUSTRATE THE EFFECT OF VARIATIONS IN' THE WEIGHT ON THE OZONE CONSERVATION EQUATION, WITH 7 = 0.5 Standard Deviation of Layer-Mean Partial Pressures Layer-Mean Partial Pressures Layer ( ~mb) (minb) TEVE Twomey TEVE Twomey W = 0.2 wa= l.O wa = o.2 wa = lo = 0.2 w= 1O. w= o.2 Wa = 1.o 1 26 25 28 29 12.4 12.6 10.o7 11.4 2 72 72 71 71 32.3 32.1 3359 34.2 3 97 97 92 92 29,4 29.4 28.7 28.7 4 125 125 122 122 21.0 21.1 19.0 19.0 5 112 112 111 111 15.7 15.7 14,4 14.4 6 73 73 76 76 9.7 9~7 9.1 9.2 7 43 43 45 45 6.4 6.4 6.2 6.2 8 24 24 24 24 2.7 2.7 2.6 2.6 9 11 11 10 10 1.9 1.9 1.5 1.5 Mean Residual Variance (N-units)2 2,2 2~2 3.4 354 RMS Residual (N-units) O.43 O.43 0.53 0.53 Mean Total Ozone Residual (m atm-cm) -0.1 0.0 0.3 0.01 RMS Total Ozone Residual (m atm-cm) 0~5 0.02 0.5 0.05

105 TABLE 20 STATISTICS FOR AROSA DATA SAMPLE SOLUTIONS USING CII AND CIII WITH TWOMEY'S METHOD AND USING CI WITH THE "COMBINED" TEVE-TWOMEY METHOD (y = 0.5 and WQ = 0.1) Standard Deviations of Layer-Mean Partial Pressures Layer-Mean Partial Pressures Layer Layer (mb) (Gtmb) Twomey Twomey Combined CII CIII Combined CII CIII CII CIII CII CIII 1 30 29 27 8.8 8.7 5.0 2 76 60 51 3355.2 20.9 11o 3 3 94 97 98 28.4 34.5 23.2 4 122 123 131 19.8 20.6 34.3 5 110 111 110 15 3 14.4 16.3 6 74 75 74 9.4 8 9 8,2 7 44 45 44 6.0 6.0 5.6 8 24 24 24 2.8 2.6 2.6 9 11 10 10 1.8 1.5 1.6 Mean Residual Variance (N-units)2 27 3.7 5.4 RMS Residual (N-units) 0.47 0.55 0.67 Mean Total Ozone Residual (m atm-cm) -3.7 2.1 6.2 RMS Total Ozone Residual (m atm-cm) 5.7 5.1 11.6

106 ability~ We note also some tendency, with CIII, which has a large weight in the fourth column, to produce a more pronounced maximum in layer 4 in the mean solution. It is also evident that a larger value of WQ needs to be used with CIII and possibly also with CII to improve the fit with total ozoneo The "combined" method results are not much different from those with CI and TEVE or Twomey's methodso The explained variance is somewhere between the other twoo The main difference is that ozone has been added in the troposphere and lower stratosphere in the mean and the resulting fit with total ozone is not as goodo Statistics for solutions with respect to DUtsch's Standard Distribution SII, for 42 Arosa curves when total ozone was less than 300 m atm-cm, are listed in Table 210 In view of the persistent secondary maximum obtained when CI was used, this column weighting vector was considered unsuitable for use with SIIo Although there is not much to choose between CII and CIII, it was decided to use CII for future solutions with SII, largely because of the somewhat better fit obtainedo The CI column weighting vector has been used for solutions with Standard Distribution III with 7 = 0.5 and Wo = 0.1 and the solution statistics are listed in Table 220 These quantities have been used in future solutions with Standard Distribution IIIo However, in view of the moderately large mean residual for total ozone, it is evident that a larger value of WO should have been usedo Since these residuals are negative, we know from the earlier results that the effect of increasing WQ- would be to decrease the solution values in the lowest layers of the atmosphere.

107 TABLE 21 STATISTICS FOR AROSA DATA SAMPLE SOLUTIONS WITH RESPECT TO STANDARD DISTRIBUTION II WHEN TOTAL OZONE IS LESS THAN 500 M ATM-CM,WITH WQ = 0.5 Standard Deviations of Layer-Mean 'Partial Pressures Layr-a Layer-Mean Partial Pressures Layer e(pmb) (pymb) TEVE Twomey_ (y=5) TEVE Twomey ('=.5) CII CI CII CIII CII CI CII CIII 1 24 25 27 25 9.7 9.6 9.0 6,0 2 47 54 43 37 6,7 7.8 5.8 5.8 3 63 49 56 60 8.5 7.9 8.1 4.9 4 88 85 86 96 8.0 6.8 7.0 9.1 114 117 116 115 7.5.5 6. 7.9 6 71 75 74 72 6.5 5.6 5.6 5,4 7 40 41 41 41 4.0 5.5 5.5 5.5 8 24 24 24 24 1.2 1.1 1.1 1.1 9 10 10 10 10 1,1 o.8 0.8 o.8 Mean Residual Variance (N-units)2 1o8 2.4 2.5 3.4 RMS Residual (N-units) 0,59 o.45 o.46 0.53 Mean Total Ozone Residual (m-atm-cm) 0o0o4 0.2 0.2 1.1 RMS Total Ozone Residual (m atm-cm) 0.07 0~2 0.2 0.5

108 TABLE 22 STATISTICS FOR AROSA DATA SAMPLE SOLUTIONS WITH RESPECT TO STANDARD DISTRIBUTION III WHEN TOTAL OZONE EXCEEDS 375 M ATM-CM (CI, 7Y = 005 and WQ = 0.1) Standard Deviations of Layer-Mean Partial Pressures Layer-Mean Partial Pressures Layer Layer (mb) ( mb) TEVE Twomey TEVE' Twomey 1 34 39 11.7 9.3 2 82 91 21.1 19o5 3 152 146 17.0 16.2 4 158 149 13.5 10.0 5 132 127 8,5 6.6 6 81 84 5~2 4.6 7 46 51 5.3 4.3 8 28 27 2.2 21ol 9 13 12 1.5 1.3 Mean Residual Variance (N-units)2 3.0 4.8 RMS Residual (N-units) O.50 063 Mean Total Ozone Residual (m atm-cm) -2.3 -4.0 RMS Total Ozone Residual (m atm-cm) 303 5,7

io09 5o2 THE EFFECT OF ADDING RANDOM NOISE TO THE OBSERVATIONS In order to be sure that our solution procedure is a computationally stable one, noise has been added to the observations using a random number generation subroutine directly available as a system subroutine at the Computing Center, The University of Michigan~ The random numbers generated had a normal distribution with zero mean and a standard deviation of 0.5 N-units. Solutions have been obtained for these curves by the TEVE and Twomey methods, with respect to SI, using CI, W2 = 0ol, and 7 = 0050 The solution statistics are given in Table 230 We find that the average solutions are virtually identical to their counterparts in Table 180 The solutions are, however, slightly more variable. The total ozone fit is about the same~ The slight increase in tropospheric solution variability has given rise to a few very small negative ozone concentrations in the troposphere in the solutions by the TEVE method. The fact that the solution variability has increased by only a small amount indicates that the above procedures are essentially stable ones for evaluation of Umkehr observationso 503 THE NEED FOR MORE THAN ONE STANDARD DISTRIBUTION 5o301 Convergence of the Iterative Procedure Using the Second Derivatives One way of examining the need for more than one standard distribution is to look at the number of iterations required to achieve convergence when the second order partial derivatives are used as indicated in Eqs. (21), (22), and (23)o To carry out such a test, the Arosa data sample

110 TABLE 23 STATISTICS FOR AROSA DATA SAMPLE SOLUTIONS WHEN RANDOM NOISE HAS BEEN ADDED TO THE UMKEHR CURVES (Solutions are relative to SI, with CI, WQ = 0.1, and y = 0~5) Standard Deviations of Layer-Mean Partial Pressures Layer-Mean Partial Pressures Layer (4mb) (4mb) TEVE. Twomey TEVE Twomey 1 25 27 12.2 9.1 2 72 71 533.1 33.1 3 97 92 29.8 29.0 4 12122 21.4 19.5 5 112 111 15.9 14.9 6 73 76 9.8 9o 3 7 43 45 6.7 6.3 8 24 24 2.8 202 9 11 10 1.9 1 6 Mean Residual Variance (N-units)2 4.5 5.8 RMS Residual (N-units) 0 61 o 69 Mean Total Ozone Residual (m atm-cm) -0.8 0o.6 RMS Total Ozone Residual (m atm-cm) 2.7 3.8

111 was divided up into the following subsamples, according to the total amount of ozone: BI(Q < 275); BII(275 < 2 < 300); BIII(300 < Q < 375); BIV(375 < Q < 410), BV(S > 410)o Subsamples BI and BII were evaluated with respect to SI and SII; B:III with respect to SI; and BIV and BV with respect to SI and SIIIo The evaluation was carried out using both the TEVE and Twomey methods and the final solution was that obtained on the mth iteration, such that9 in the notation of (23), 12 Sk(m)- Sk < 1o0 N-units o (70) k=l The frequency distribution of the number of iterations required is given in Table 24~ Looking at the frequency distributions, we note an increase in the number of iterations required as the total ozone deviation from the standard distribution value becomes greater (336 m atm-cm for SI)o There is a marked improvement when three standard distributions are used, with 90 out of 100 cases converging in two iterations, whereas only 56 cases converge in two iterations when SI is used aloneo The above is for Twomey's method; for the TEVE method the corresponding numbers are 84 and 39. Thus we note that the additional smoothing imposed by Twomey's solution method appears to be benificial in terms of reducing the number of iterations required. 5~3o2 Average Solutions First, we shall consider those cases where total ozone is less than

112 TABLE 24 FREQUENCY DISTRIBUTIONS OF THE NUMBER OF ITERATIONS REQUIRED FOR CONVERGENCE WHEN THE SECOND ORDER PARTIAL DERIVATIVES ARE USED Standard,, ' Frequency Distributions Sample Distribution Number of Iterations 1 2 3 4 5 6 7 Twomey's Method BI SI 0 0 10 10 1 0 0 BII 0 12 7 2 0 0 0 BIII 2 22 4 0 1 0 0 BIV 0 14 2 0 0 0 0 BV 0 6 7 0 0 0 0 BI SII 0 19 2 0 0 0 0 BII 0 21 0 0 0 0 0 BIV SIII 0 16 0 0 0 0 0 BV 0 10 3 0 0 0 0 TEVE Method BI SI 0 0 2 10 9 0 0 BII 0 9 10 3 1 0 0 BIII 1 20 5 2 0 0 1 BIV 0 9 7 0 0 0 0 BV 0 0 13 0 0 0 0 BI SII 0 19 2 0 0 0 0 BII 1 20 0 0 0 0 0 BIV SIII 0 13 3 0 0 0 0 BV 0 10 3 0 0 0 0

113 300 m atm-cm, for which average solutions are listed in Table 25. The usual characteristic differences between the TEVE and Twomey methods are apparent. However, the main difference is between solutions carried out with respect to SI and SILo In every case, the ozone maximum is shifted from layer 4 with SI to layer 5 with SIIo The difference cannot be attributed to the choice of the column weighting vector (CI with SI and CII with SII) since the same pattern is evident when we use CI with SII (Table 21). The reason for the difference lies in the fact that we are obtaining a solution which represents a minimum deviation from a trial solution (Twomey), the trial solution being the standard distribution. Examining the standard vertical distributions in Table B-1, we find that SII does have a maximum in layer 5, whereas SI has a rather broad maximum with layers 4 and 5 having approximately equal concentrations. We must conclude' that, if indeed there is a significant difference between the mean vertical distributions at low and moderate ozone values, then a separate standard distribution should be usedo The evidence of Table 16 is that the solutions with respect to SI are more nearly correct. However, the sample, on which the low-ozone means of Table 16 are based, is much too small and, in addition, contains more soundings for Liverpool (11) than for Arosa (5). Moreover, the concentration in layer 5 is based in part on average Umkehr resultso Although Twomey's method as presented here, can be modified to obtain a solution which is a minimum deviation from a trial distribution other than the standard, we are in favor, in view of the iteration results, of using an additional standard distribu

114 TABLE 25 AVERAGE SOLUTIONS FOR AROSA DATA SAMPLE WITH SECOND DERIVATIVE CORRECTIONS INCLUDED WHEN TOTAL OZONE IS LESS THAN 300 M ATM-CM Layer-Mean Partial Pressures Solution Method: TEVE Twomey Standard Distribution: SI I S SI S SII SI SI S SII SI SII Sub-Sample~ BI BI BII BII BI BI BII BII Layer I 1 22 21 26 27 24 24 28 29 2 38 45 43 49 36 42 37 46 3 64 6o 67 65 61 54 64 60 4 101 84 107 91 100 82 107 90 5 95 109og 104 117 95 111 106 120 6 65 66 74 75 68 70 77 78 7 4o 38 45 42 41 40 45 43 8 24 25 23 24 23 24 23 24 9 11 11 10 10 10 11 10 10 Solution Total Ozone 264 264 287 288 264 263 286 288

115 tion for low total ozoneo Next, we shall consider cases where total ozone exceeds 375 m atm-cm, for which average results are listed in Table 26. In this case, statistics were computed for subsample BIV and for BIV + BV combinedo Comparing the differences between the SI and SIII solutions, we find them less pronounced~ The major difference, in all cases, is that ozone is removed from the lower stratosphere (layer 2) and added to the troposphere (layer 1) and to layer 3 in large amounts and to layers 4, 5, 6, and 7 in small amounts when solutions are with respect to SIII instead of SIL This difference is probably due to the nature of the CI column weighting vector used with both standard distributions. When much ozone has to be added to the standard distribution, as is the case when SI is used with high total ozone, there is a tendency to add it in layer 2 because of its larger weight in the solution procedure~ Comparing these results with those of the balloon soundings as given in Table 16, we find that we should probably have a more pronounced peak in layer 4 in our standard distribution SIIIo 5~4 THE NEED FOR SECOND DERIVATIVE CORRECTIONS What is the effect of applying the corrections for the second order derivatives in the Taylor expansion? Since we are already well aware of the differences between the two solution methods, TEVE and Twomey, aver-/ age solutions with and without these corrections are given in Table 27 for the Twomey method onlyO Moreover, since the differences will be

116 TABLE 26 AVERAGE SOLUTIONS FOR AROSA DATA SAMPLE WITH SECOND DERIVATIVE CORRECTIONS INCLUDED WHEN TOTAL OZONE EXCEEDS 375 M ATM-CM Layer-Mean Partial Pressures Solution Methodo TEVE Twomey Standard Distribution: SI SIII SI SIII SI SIII SI SIII Sub-Sample: BIV BIV BIV+V BIV+V BIV BIV BIV+V BIV+V Layer l 1 24 33 27 36 25 37 27 39 2 99 70 113 83 100 79 116 92 3 131 142 140 151 126 138 136 147 4 -150 152 153 156 145 145 148 149 5 129 129 129 130 136 125 125 125 6 80 82 78 81 82 84 80 83 7 47 48 47 47 51 52 51 52 8 28 28 29 29 27 27 28 28 9 14 13 14 14 12 12 13 13 Solution Total Ozone 398 398 415 414 395 400 411 415

117 TABLE 27 AVERAGE SOLUTIONS FOR AROSA DATA SAMPLE TO ILLUSTRATE DIFFERENCES BETWEEN LINEAR (L) AND NONLINEAP (NL) SOLUTIONS Layer-Mean Partial Pressures Layer Q < 300 All Q Q > 375 L L NL L NL 1 26 26 28 27 31 27 2 44 37 70 67 115 116 3 66 62 92 91 131 136 4 104 104 122 122 145 148 5 98 101 111 112 125 125 6 69 72 76 77 82 80 7 40 43 45 47 50 51 8 22 23 24 24 27 28 9 10 10 10 11 12 13 greatest when we use SI for low and high ozone, only solutions with respect to SI are given. For low ozone, the effect of the second order derivatives is to transfer ozone from layers 2 and 3 to layers 5, 6, and 7. A similar comment applies, but to a lesser extent, when all values of total ozone are lumped together. The effect for high total ozone is less clear-cut. The influence of the corrections is to remove ozone from layers 1 and 6 and to add it in layers 3 and 4. It is somewhat more instructive to look at individual solutions since, on the average, we might expect the effect of the second derivative corrections to cancel out. Solution instability first manifests itself, in the linear solution system, in the form of low ozone values in layers 6 and/or 7 and in the troposphere. Since these effects show up in a more pronounced manner in the TEVE solutions, examples of each

118 are shown in Table 28. The solution which required 7 iterations, which also suffers from low ozone in layers 6 and 7, is one of those listed. We see that the effect of introducing the second order corrections is to improve the smoothness of the solutions. Thus, although we might be inclined to discard the second order corrections as unnecessary in viewing only the average solutions, they do serve a useful purpose in smoothing out unrealistic irregularities in some solutions. TABLE 28 INDIVIDUAL SOLUTIONS CHOSEN TO ILLUSTRATE THE DIFFERENCES BETWEEN LINEAR (L) AND NONLINEAR (NL) SOLUTIONS Layer-Mean Partial Pressures Layer 4/4/61 10/20/61 10/31/61 L NL L NL L NL 1 0 4 22 32 15 18 2 61 62 100 99 43 34 3 122 118 115 102 72 64 4 150 146 126 116 104 101 5 128 125 97 94 91 93 6 71 72 52 57 57 63 7 36 38 27 34 32 38 8 24 25 24 25 23 24 9 12 12 12 12 11 11 Observed Q 334 334 327 327 254 254 Solution Q 335 335 327 329 255 255 No. of iterations 3 7 5 5.5 COMPARISON WITH DUTSCH'S SOLUTIONS Dr. DUtsch has kindly provided the writer with a duplicate card deck of solutions for the cases given in his 1963 report. However, the card

119 deck solutions had undergone a further processing to achieve a closer fit between observed total ozone and that implied by the solutiono In selecting those cards which corresponded to the sample used in this study, it was found that our sample contained five duplicates, all for clear sky Umkehrs, where Umkehr curves had been included for the cases with and without luxmeter correctionso In addition there were two curves in our sample for which no solution card was available in DUitsch's solution deck~ Thus the sample for which an exact correspondence existed was reduced to 93 cases~ In setting up duplicate decks, the appropriate solution from SI, SII or SIII was chosen from the present work~ Average solutions and solution variabilities are listed in Table 29 for this sample, for the three methods: DUtsch, TEVES and Twomeyo The' individual solutions are listed in Appendix E, which also includes the duplicates and the two missing solutions~ The data are listed in order of increasing total ozone and, in the case of the duplicates, the first solution listed corresponds to Dutsch'so Also listed in Table 29 are the fractional eigenvalues of the correlation matrix of the solution partial pressures. On the whole, we find only rather small differences between the solution statistics, indicating that DUtsch's technique of averaging solutions from several systems of overlapping layers has been successful in eliminating the instabilities from the systemo The correlation matrix eigenvalues are an indication of the number of independent linear combinations that are present in the normalized solutions~ As it should, from its basic formulation, the TEVE solution method explains more of the normalized

TABLE 29 SOLUTION STATISTICS FOR AROSA DATA SAMPLE TO COMPARE THE DUTSCH, TEVE, AND TWOMEY METHODS OF SOLUTION WITH SECOND DERIVATIVE CORRECTIONS APPLIED Standard Deviation of Layer-Mean Partial Fractional-. Eigerialues Layer-Mean Partial Pressures of Partial Pressure Layer Pressures Correlation Matrix (Cimb) _ _ (mb) __ _ DUtsch TEVE Twomey Duitsch TEVE Twomey Duitsch TEVE Twomey 1 32 28 30 11.9 11.8 10.6.424.536.578 2 57 63 63 25 e4 23.2 26.6.o 339.230.176 3 97 96 92 41.1 38.6 39.2.121.144.135 4 122 120 117 28.6 28.9 26.3.081.064.061 5 113 117 116 11.4 11.4 8.8.032.016.035 o 6 78 75 78 7.7 7.2 6.5.002.009.012 7 47 44 46 5.9 5.1 5.4 -.003 8 23 25 25 2.4 3.0 2.8.001 9 11 11 11 1.8 2.2 1.7

121 solution variance with the first four eigenvectors, but the differences do not appear to be significant. 5.6 ANOTHER ORTHOGONAL VECTOR EXPANSION FOR SOLUTIONS Another possibility is to expand the vertical distribution in terms of its own characteristic patterns, or empirical orthogonal functions, truncating the expansion after the first three or four pattern vectors have been used~ One of the problems here is to secure empirical data giving the vertical distribution in all of the nine layers of the atmosphere that are used in the current studyo The data available for ozonesonde intercomparisons carried out at Arosa in July-August, 1958 (Brewer et al., 1960), in the summer of 1961, and again during the spring of 1962, with simultaneous Umkehr data on most occasions, provide the closest approach to the required informationo A total of 29 such vertical distributions were synthesized by matching up the balloon results for the lower atmosphere and the Umkehr results for higher levelso The correlation matrix, and its eigenvalues and vectors were calculated for this small sample~ Solutions have been carried out using the first four eigenvectors of the correlation matrix. In this casey we set or P~~~P bV S_~~i~~9P(71)

122 where Z and p are the standard deviations and means, respectively, of the ozone partial pressures in the various layers for the sample of 290 These are given in the columns designated V in Table 16o The linear equation system (42) now becomes DZy =u D - (72) Setting = W (73) we have DZW*b = AW*b = u Dp (74) with the least squares solution b (WA*AW*) WA*(u-Dp) (75) In this case, the coefficients bj do not explain independent segments of Umkehr curve variance, nor can they be computed independently, since (WA*AW*) is no longer a completely diagonal matrix. It should be noted that the matrix W, as used in this context, comprises only the first four eigenvectors of the correlation matrix~ Solutions have been carried out according to this scheme using the Arosa data sample for low ozone (~ < 300) re SII, for all ozone re SI, and for high ozone (s > 375) re SIIIo In each case, W- = 0ol was used.

125 In this section and the next one, average solution results are presented in the form of smooth curves on ozonagrams. In drawing these smooth curves through the block distributions implied by the actual solutions, the smoothest curve which keeps the same amount of ozone in each layer is drawn subjectively. This is illustrated in Fig. 24 which shows the block distribution and the smooth curve for the case SI (TEVE) of Fig. 25. In drawing this curve, use has been made of our a priori knowledge from balloon soundings that the tropospheric distribution tends to follow a constant mixing ratio line. The usual sharp increase in partial pressure atthe tropopause has been smoothed out since we are dealing with mean solutions. A discussion of objective criteria for drawing these curves has been given by Godson (1962). It has to be emphasized that all of the smoothed curves presented here are for average solutions and that the subtle changes in curvature of the curves from one layer to the next are properties of the solution system used. Inferences about differences in mean solutions (i.e., in solution systems) and about differences between individual solutions computed from the same system should be made only for rather broad atmospheric layers. Average solutions for 42 low ozone cases are shown in Fig. 25, where p has been taken as zero (reference CP, 0 in the figure). In addition, curves for average linear solutions from the previous work are shown.

124 45 5 10 2o ~ ) E 20 s 5 10025 H $ E R _ _ _ _ _ _ G 50 E I \ I /I I I I 1 -20 (mb)' 100 \ (kin) 200 distribution. 500 -5 0 50 100 150 200 250 300 distribution.

125 Re S E (CP,O ) 45 ___ Re S I (CP,O) ~2 4\\ \ ~,.............. ReSI (TEVE).Re S XU (TEVE) ]0 10 P R E 20 E R E 20 (mb) zH 100 T _15 (km) 5 0 50 100 150 200 250 300 PARTIAL PRESSURE OF OZONE (,umb) Fig. 25. Average solutions for 42 low-ozone cases at Arosa using the Characteristic Pattern method, with TEVE: solutions for comparison.

126 The result using the characteristic' patterns with respect to SI is quite ridiculous and indicates that the linear combinations implied by the characteristic patterns are not necessarily ones whose coefficients can be inferred from the Umkehr observations, at least when the Co Po's are determined from such a small sampleo Average solutions for 29 high ozone cases are shown in Fig. 26. In one case (designated CP,V) ~ has been set equal to the appropriate vector in Table 16. The characteristic pattern solutions, except perhaps for case CP, 0 re SIII, are considerably less smooth than those obtained by the linear TEVE method. The interesting feature of these solutions is the ozone maximum in the lower stratosphere in cases CP, 0 re SI and CP, V re SIIIo There is, of course, also a suggestion of such a maximum in the TEVE solution. The reasons for a high ozone content in layer 2 in this case have already been discussed. The reason for the high ozone content in layer 2 in the other two cases is that the p vector corresponds to about the same amount of ozone as is present in SI. Moreover, the column weighting vector used here assigns a high weight to layers 2 and 4, but a low weight to layer 30 Thus, there is a tendency to add ozone in layers 2 and 4 in preference to layer 3. Since much ozone has to be added for these high ozone cases, we end up with the double maximum structure. Thus we do not get the lower stratospheric maximum from the Umkehr curve, but rather because we said in advance that it would be there whenever total ozone is higho In Fig. 27, average linear solutions, all with respect to SI, for

127..... ___ Re si. (CP,O) Re S I (CP, O) x- x — Re Sm' (CP,V)........... Re S m (TEVE)....Re SI (TEVE) a40 6 Avrg5eoutosf 29hg-ooe ae a uinh 10 E 20 S s H R 5E E 20 (mb) ~s ~~T 100 15 ( km) 200 500 1000 NI0 50 100 150 200 250 300 PARTIAL PRESSURE OF OZONE (,,umb) Fig. 26. Average solutions for 29 high-ozone cases at Arosa using the Characteristic Pattern method, with TEVE solutions Lor comparison.

128 45 Re SI (CP,O)..........Re SI (CP,V) ~~~~~2 ~..-...Re SI (TEVE) 40 5 35 10 030 P R E 20 20 __25 "91g S c~~ ~~H U E 100~~~~~~~~~~~~~~~~~~~1 50 ' —.I E ~ 20 t 1 /~ G (m b) H,.~ ~ ~ ~~~~~~~~~~~~~~~~~ —10,00 -15 (km) 1000 0 50 100 150 200 250 300 PARTIAL PRESSURE OF OZONE (,umb) Fig. 27. Average solutions ~or 100 Arosa Umkehr curves using the Characteristic Pattern method, with TEVE solutions for comparison.

129 the entire Arosa data sample are plotted. The tendency toward a small hump in the lower stratosphere on the TEVE curve largely disappears when the curves for high ozone values entered in the average are replaced by those taken with respect to SIII. Although the use of the empirical orthogonal functions or characteristic patterns of the vertical distribution is an attractive way of imposing a priori knowledge on the solution system, it clearly requires a reasonably sized sample of complete vertical distributions for calculation of the Characteristic Patternso In addition, the p vector should be used in the selection of the standard vertical distribution. Thus, with the column weighting vector also smoothed out by a larger sample, we should tend to have smooth solutions similar to those obtained by the previously discussed methodso The advantage of the previous methods is that the solution is expanded in terms of those linear combinations of the unknowns on which the measurements can provide information9 within the restrictions of the physical-mathematical model used. 507 VERTICAL DISTRIBUTIONS USING OTHER WAVELENGTH PAIRS Heretofore, only Umkehr observations on the C wavelength pair have been much used to estimate the vertical distribution of ozoneo It is well known that discrepancies exist between observations of total ozone amounts on the different wavelength pairs, when the ozone absorption coefficients determined in the laboratory are used with the spectrophotometer (see Appendix A for a brief discussion of this point).

130 To gain some insight into this problem, solutions have been carried out, using the Twomey technique (with y = 0O5) on wavelength pairs A, C, D, AD, AC, and CD, and on the combined pairs A-C-D, AC-CD, and AD-ACo The sample of 98 Umkehr curves from the North American network and Dutsch's first derivative matrices for Arosa have been usedo Although this procedure is not strictly correct because Arosa has a mean surface pressure of 814 mb, the differences are not large (see DUtsch, 1957, and Mateer, 1960), and the results will serve to indicate the general nature of the problem. Unfortunately, DUtsch's second derivatives were not computed for the full range of zenith angle used here, so that only linear solutions are possible and these are all with respect to SI. Column weighting vector CIII was used for these solutions since it was found that CI gave negative partial pressures in layer 2 for low total ozone due to the large weight assigned to that layer by CIo In addition, a value of WQ = 1.0 was used throughouto Total ozone amounts in the North American network are based on the double-pair AD measure,mentso Corrections, indicated in Table 30 (after Dobson, 1963) were applied to the total ozone amounts to render them "compatible" with the ozone absorption coefficients actually used in the construction of the derivative matrices~ The solution amounts were then "recorrected" back to the AD ozone scale~ When multiple pairs were used, vizo, A-C-D, AC-CD, and AD-AC, no corrections were applied~ The average solutions are shown in Figs~ 28, 29, and 30~ In addition, solutions have been carried without corrections for total ozone, and the result for the C wavelengths is given

131 TABLE 30 CORRECTION FACTORS FOR AD TOTAL OZONE MEASUREMENTS USED IN UMKEHR EVALUATIONS Wavelength Correction Factor Re-Correction Factor Pair for AD Total Ozone for Solution Ozone AD 1.00 1.000 A 0.99 1.011 C 0.93 1.076 D 0.95 1.056 AC 1.05 0.956 CD 0.92 1.091 in Fig. 31, for comparison with the corrected solution. For the C wavelengths, solutions were also carried out with the CI column weighting vector, using SI, WQ = 0.1, y = 0.5, Twomey's method, and applying the second derivative corrections. The average of these latter solutions is also shown in Fig. 31, as Case III, and the solutions are listed in full detail in Appendix E. Looking first at the individual wavelength solutions in Fig. 28, we note a progressive decrease in the amount of ozone at high levels as we go from A to C to D. The reverse is true at low levels. The main maximum on the A and C curves is at about the same level, but that for the D curve is a little lower. Referring to Fig. 31, where Case I is the curve of Fig. 28 and Case II is the mean curve when total ozone is uncorrected, we note a downward shift of the C curve maximum in the latter case, and an increase in the low-level concentration. Similar remarks apply to the uncorrected D curve (not shown). We note, however, that the application

132 5..... A Wavelengths 45...........C Wavelengths D Wavelengths 40 5\ 5 10< _ R. E 20 $. 20H U E 1000 10' 2~050101020 2 0 3 50 \0 H PARTIAL PRESSURE OF OZONE ((mb) Fig. 28. Average solutions for 98 North American Umkehr curves for the individual wavelength pairs.

133.... __ _AD Wavelengths — 45.. AC Wavelengths 2 p; Xt\.......... CD Wavelengths 10 ~40 P ~ \P R E 20 20 -S H 100 205 PARTIAL PRESSURE OF OZONE (jimb) doubl~e wave~length pairs.

134 A-C-D Wavelengths N \45 -__ AC-CD Wavelengths........ AD-AC Wavelengths 10 R ol E U 50 G E 20 200[ \ __,Jay l 0 50 100 150 200 250 300 PARTIA RSUEO ZN ~b F~ig.3.Aeaesltosfr9 Nort Aeica gmeh cuvsfoo-1 bined\ waeeghpiraddobepis

135 WAVELENGTHS C 45..........Case I Case II 2 ______ Case I]f 40 ( See Text) 5 10 R. E 20 I\\\ -22d 25 H S E R 0 G:. 50 E H (mb) T 100 (km) 15 500 00 250 300 50 100 150 200 250 300 PARTIAL PRESSURE OF OZONE (/ mb) Fig. 31. Average solutions for 98 North American Umkehr curves for the C wavelength pair by different methods.

136 of the total ozone correction does not appear to change high level ozone. Actually there is a slight increase in high level ozone when the correction is applied, but it is too small to show up on the scale of the ozonogram. We may conclude that the application of the total ozone correction in the solution acts to decrease the discrepancies that would otherwise be observed when we compare solutions for the individual wavelength pairs. Referring next to Figo 29, we again find differences in solutions with the double pairs. Comparing the AD and AC curves, we find a somewhat more flattened maximum and more ozone at high and low levels with the latter. Comparing AD and CD curves, we find that the latter curve has a sharper maximum at a slightly lower level, and slightly less ozone at high and low levels. The curves shown all embody the total ozone corrections. If we do not include these corrections, the uncorrected CD curve (not shown), has a somewhat more flattened maximum at a lower level, with less ozone at high levels and more at low levels, than the corrected curve. In the case of the uncorrected AC curve (not shown), the maximum is somewhat more intense and there is a little less ozone at low levels. The effect of not applying the correction is to transfer ozone from levels below about 16 km to the 16-33 km layer. From Figo 29 and the above description, we may conclude that the agreement is somewhat better perhaps when the corrections are not applied. However, it is quite clear that neither of the methods used is the "correct" oneo In Fig. 30, we again find differences between the various solutions. The AD-AC combination suggests more ozone above about 20 km and less be

137 low than either of the other two combinations and also has the sharpest maximum. The A-C-D solutions provide the other extreme with more ozone below 20 km and less above- this levelo All three curves show maxima at about the same levelo In Figo 31, comparing the solutions with total ozone uncorrected but with the second derivative corrections applied and using CI as a column weighting vector, we find a tendency toward a maximum in the low stratosphere, characteristic of solutions using CI which also cover a wide range of total ozoneo These characteristic differences should be borne in mind in examining the individual solutions of Appendix E. We may inquire into the information that may be detected by the systems of linear equations with which the above solutions have been carried out and into the goodness of fit of the solutions. The first four eigenvalues of A*A for each of the systems, together with the mean and rms residuals for the linear solutions, are given in Table 31o Statistics on the solution residuals were not computed by the program used for the combined pairs, which also computes the solutions when the second order corrections are applied~ We note that the first eigenvalue dominates the trace of A when we choose WS = l.0o However, the third and fourth eigenvalues are about the same as those given for the C wavelength pairs in Table 7~ Thus, although there is strong forcing on total ozone to be the same in the solution as observed, the predictability of the third and fourth eigenvector coefficients is not impaired~ We note also that there is somewhat less information i.n the D measurements than in the

138 TABLE 31 FIRST FOUR EIGENVALUES OF A*A WITH MEAN AND RMS RESIDUALS FOR THE VARIOUS WAVELENGTH COMBINATIONS Eigenvalues Mean RMS Wavelengths Residual Residual 1 2 3 4 (N-units) (N-units) A 423.0 42.7 8.7 1.5 0.19 0.64 C 365.3 24.2 4.9 0.8 0.07 o.60 D 360.6 11.1 1.8 0.2 -o.o6 0.65 AD 475.0 32.4 10.0 2.1 o.46 0.88 AC 399.3 13.4 4.1 1.2 0.82 1.19 CD 368.5 8.5 3.1 0.7 0.28 0.60 A-C-D 427.2 42.2 14.9 1.8 - - AD-AC 366.4 57.3 6.4 o.8 - - AC-CD 379.2 22.9 7.9 0.9 - - C, and less in C than in A, as evidenced by the magnitude of the fourth eigenvalue. Insofar as the goodness of fit is concerned, we have about the same rms residual in each case for the single wavelength pairs, but the mean residual is considerably larger for the A pair, largely because of persistent positive residuals for zenith angles of 65~, 700, and 74o. All of the double pair solutions show fairly large mean residuals and also rms residuals. We may attribute this, at least in part, to the use of incorrect tables for the standard distributions. As one might expect, the fit becomes quite poor in the case of the multiple pairs and double pairs, a visual examination suggesting that rms residuals will be of the order of 2 N-units. This may be attributed in part to the use of the Arosa tables, but is more likely due to absorption coefficient uncertainties and modeling errors.

139 Since we would expect the effects of multiple scattering and aerosol scattering to be somewhat less for the double wavelength pairs, the above difficulties should be further exploredo The use of multiple pairs is particularly attractive for Arctic and Antarctic regions, where the range of solar elevation is small, because the A wavelength measurements are always sampling the atmosphere at a higher level, and the D at a lower level, than those of the C wavelength pairo

60 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK We have shown from a statistical examination of the Umkehr curves themselves, from a mathematical examination of the linearized solution system, and from studies of the curves plus the system, that there are at most four pieces of information about the vertical distribution of ozone to be obtained from Umkehr observations. Moreover, even when we solve for four pieces of information about the vertical distribution, the solution depends on the way in which the problem is set up, Vizo, on the standard distribution from which the solution is computed, and on the scaling of the equations and the variables. The main information contained in the Umkehr curve concerns the total amount of ozone in the atmosphere and the fact that there is much more ozone between the tropopause and 30 km than there is above and below this layer. Any intercomparisons between day-to-day changes in the vertical distribution and inferences therefrom on atmospheric motions, as computed from Umkehr observations, are meaningful only when the solutions are computed by the same objective technique. In addition, it appears that little or no additional information is obtained by observations on more than a single wavelength pair, except in the rather special circumstances of Arctic or Antarctic observations where the range of solar zenith angle is somewhat restricted. Even here, the additional information that may be gained is jeopardized by uncer140

141 tainties in the ozone absorption coefficients that are used with the ozone spectrophotometer o In setting up an objective solution technique, we should use more than one standard distributiono These standard distributions should be based upon information about the mean vertical distribution obtained from balloon soundings. If the vertical distributions of ozone obtained from Umkehr observations are to be compared with each other, then the same set of standard vertical distributions should be used for data from all stations~ However (see below), if vertical distributions are to be compared with those obtained from balloon soundings, different criteria will have to be establishedo We may well inquire into the present-day usefulness of Umkehr observations, now that the gross features of the vertical distribution have been known for some time and information on the finer structure must await the more wide-spread use of ozone sondeso First, the Umkehr evaluations do give fairly consistent results in the uppermost layers of the atmosphere and the relative seasonal variation in these layers may be inferred with some degree of confidenceo These are also the layers which are most nearly in photochemical equilibrium so that comparisons may be made with photochemical calculations for the purpose of checking the lattero Second, the absolute calibrations of the various ozone sondes still leave something to be desired. Thus, when Umkehr observations are combined with sonde measurements, it should be possible to have the ozone

142 sonde measurements specify the fine structure and to use the Umkehr observations to infer an adjustment factor for this fine structure plus the distribution picture at higher levels. Third, since ozone sondes are still expensive and are not used every day, Umkehr observations should provide a useful means of interpolating between soundings so that some continuity may be maintained. Finally, within the context of an objective evaluation system, the vast number of Umkehr observations now available may be used to determine the main features of the differences in the vertical distributions in different geographical areas. All of these are worthy of further work and investigation at the present time. Inevitably, in the future, as ozone sondes improve in reliability and become less expensive, as balloon performance improves, and as rocket techniques are developed, the Umkehr technique will gradually be replaced completely by these direct methods.

APPENDIX A TERMINOLOGY AND UNITS USED IN OZONE-METEOROLOGICAL RESEARCH 143

144 The instrument commonly used for the measurements discussed in thisreport is the Dobson (1931) ozone spectrophotometer, manufactured commercially by R & J Beck, Londono This is a double monochromator which compares the intensities of two wavelengths in the solar ultraviolet. The shorter and less intense of these wavelengths is strongly absorbed by ozone, the longer and more intense weakly absorbed. The instrument, which employs a photo-multiplier as detector, is very sensitive and observations may be taken on the light scattered downwards from the zenith sky as well as on the direct solar beam. The use of the instrument and its adjustment and calibration have been described by Dobson (1957 a, b)o The measurements of the instrument, when taken on direct sunlight, are adjusted to read directly in units of 100 log It -log Io' I Io where the logarithm is to the base 10, I Io, are the extra-terrestrial intensities of the long and short wavelengths, respectively, and I', I, are the intensities at the point of measurement. The measured values are commonly referred to as N-values and the units as N-units and this procedure is followed throughout this report. Since I' > I and I'/I > Io'/Io, because of the greater absorption of the shorter wavelength, the N-values are always positive. Measurements on direct sunlight are used for determination of the total amount of ozone in the atmosphere. When the measurements are taken on zenith skylight, an additional unknown

145 constant enters into the picture and we may consider the measured quantity, quite simply, to be 100 log -- + C where C is the unknown constant, now including the extra-terrestrial logintensity ratio. These measured values are also referred to as N-valueso Four pairs of wavelengths are used with the spectrophotometer, and these have been designated A (3055/3254 X)9 B (3088/3291), C (3114/ 3324), and D (3176/3398)o Most observations of the Umkehr effect have been taken (on zenith skylight) with the C wavelength pair, although a large body of data including the A and D pairs is being built up, particularly in North Americao The B wavelength pairs are used rather infrequently~ In most measurements of total ozone, the difference between two pairs of wavelengths is used (AD is standard) because the effects of scattering by air molecules, aerosols, and dust are largely eliminated. Unfortunately, the laboratory measurements of the ozone absorption coefficients (Vigroux, 1953), when used with the Dobson spectrophotometer, do not lead to consistent resultso These inconsistencies have been summarized recently by Dobson (1963). In the case of total ozone measurements, the problem of correcting the measurements to a uniform or standard scale is a relatively simple one, since one is concerned only with differences ( v-ac) in the absorption coefficients for the short and long wavelengthso However, in the case of Umkehr evaluations, the two coefficients aav enter individually into the calculations and, since the

corrections for the individual coefficients are almost certainly not the same, these corrections will also differ from that for the difference. This problem is still unresolved and is one of the reasons why the additional wavelength pairs are not much used in Umkehr evaluations. The unit used to express ozone amount is the reduced thickness. This is the thickness of the layer of pure ozone, at standard temperature (O~C) and pressure (1013.250 mb), which would result if all the ozone in a vertical column (encompassing that layer of the atmosphere in which we are interested) were collected into such a layer. The reduced thickness is expressed in atmosphere-centimeters, abbreviated atm-cm, or milli atmosphere-centimeters, abbreviated m atm-cm. If all the ozone in the entire atmosphere were collected into a layer of pure gas, it would occupy a layer between 2 and 6 mm thick. Consequently, total ozone amounts are between 200 and 600 m atm-cm. These units are occasionally referred to in the literature as Dobsons or Dobson units, as 10 3 cm STP, or just as 10- cm. Ozone concentrations are referred to in terms of ozone density (micrograms per cubic meter or Lg/m3), ozone partial pressure (micromillibars or jmb), or ozone mixing ratio (by mass: micrograms per gram or 4g/g; by volume: parts per million or per hundred million)O The only quantities used in this report are the partial pressure (p in 4mb), the reduced thickness for a layer (x in m atm-cm), or total amount (Q in m atm-cm), and the mixing ratio (r3). In order that vertical distributions of ozone might be displayed in

147 a common format by all workers in the field, the International Ozone Commission asked Godson to prepare a suitable diagram for this purpose. This diagram, known as the ozonogram, was described by Godson (1962) and is gradually being adopted by most workers in published papers and in routine worko The ozonogram, or at least that portion of it used to plot the vertical distribution of ozone, is used exclusively in the present report. It first appears in Figo 20 The abscissa is ozone partial pressure in ~tmbo The ordinate is the logarithm of atmospheric pressure in millibars (mb)o Along the right-hand side of the diagram is a height scale (km) corresponding to the pressure-height relationship in the standard atmosphere~ The curved lines which slope down from upper left to lower right are lines of constant ozone mixing ratio, r3, in [Ag/go The relationship between partial pressure, p, atmospheric pressure, P, and ozone mixing ratio is r3 ( g/g) 1 657 p(=mb) P (mb)

APPENDIX B FURTHER DETAILS OF DUTSCH'S EVALUATION- METHOD 148

149 The material presented here is intended to supplement the description of the method as presented in Section 204 of the study. First of all, the standard distributions used are listed in Table B-lo The original layers used were chosen such that the pressure at the bottom of each layer was N times that at the top, with 500 mb as a reference~ In actual practice, these smaller layers have been combined into broader layers with bottom pressure twice that at the top. The standard distribution Umkehr curves, which include the effects of secondary scattering, are listed in Table B-2. The first order partial derivatives for the various standard distributions and wavelength pairs are listed in Tables B-3 through B-7, inclusive. As noted in the text, the derivatives have units: (N-units)/(unit fractional change in layer ozone content), and also have the effects of secondary scattering incorporated in them~ It has to be emphasized that these tables contain data appropriate to stations at a mean surface pressure of 814 mbo The formulation used by DUtsch to evaluate secondary scattering is summarized in the following equationm n n 4 10 S -Iok ) (0.025) A AP G F PO i.j ~ kF 1k i=l j=l k=l A=1 0 X exp {bi P P j Po +eo Po j n X exp -a Xm+ C mXm+ j aj m, kQGxmI m=l m=in m=j+l

150 where quantities not previously defined are: S = intensity of secondary scattered light at the instrument n = number of layers in scattering atmosphere. The scattering atmosphere extends well above the ozone absorbing atmosphere. i = index referring to zenith layer in which secondary scattering occurs j = index referring to layer in which primary scattering occurs k = index referring to azimuth angle. Four azimuth angles: 00 0, 180, and 2700 were used. ~ = index referring to angle of incidence of primary scattered beam arriving at secondary scattering layer. Ten angles were chosen such that each was at the midpoint of equal solid angleso Gi = scattering volume determined by geometry Fk,e G = scattering function (cf., l+cos2G for primary scattering in the zenith) bi,j, ~ = Bemporad's function for the path between primary and secondary scattering layers ej,, 2 = Bemporad's function for the path from outer space to primary scattering layer

151 Cm = relative ozone absorption slant path through mth layer between primary and secondary scattering layers. aj m k ~ g = relative ozone absorption slant path through the mth layer between the primary scattering layer and outer space xm - ozone content of mth layero The numerical constant 0~025 (= 1/40) takes account of the fact that the contributions from 40 equal solid angles are being summed. This constant could equally well be incorporated in the quantity Gi,j,. Ground reflection is not considered in this model for secondary scattering. Also of interest is the elaborate system of overlapping layers which DUtsch has used to suppress the instabilities and obtain a smooth solution. One such system is shown in Table B-8. This system is used whenever all zenith angles are available. Each subset is solved as an even-determined system using the zenith angles shown in the individual columns. The final solution for the fractional change in layer 1, for example, is obtained by averaging the solutions over each set and combining these into a final averageo Thus f =fl(l1)+fl(1(2)i + fl(21)+fl(22 + fl(3.1)+fl(3o2) + fl(41)+fl(4o2)+fl(43} According to Dlitsch, these smoothed solutions generally converge on the second iteration.

TABLE B- 1 STANDARD VERTICAL DISTRIBUTIONS OF OZONE USED BY DUTSCH Standard Distribution Pressure Range SI SII SIII Layer (mb) x P x P x P (m atm-cm) ( mb) ( ( atm-cm) ( 4mb) (m atm-cm) ( Lmb) 814 - 500 9.65 25.1 4.92 12.8 18.60 48.3 1 500 - 250 12.86 23.5 6.38 11.7 28.60 52.3 2 250 - 125 23.06 42.1 9.60 17.5 47.90 87.5 3 125 - 62.5 46.14 84.3 20.65 37.7 68.50 125 1 4 62.5 - 31.2 72.60 132.6 49.20 89.9 78.40 14-3.2 5 31.2 - 1 5.6 73.30.133.9 70.40 128.6 73530 133.9 6 15.6 - 7.8 52.10 95.2 47.90 87.5 52.10 95.2 7 7.8 - 3.9 29.22 53.4 25.30 46.2 29.22 53.4 8 3.9 - 1.95 11.00 20.1 11.23 20.5 11.00 20.1 9 1.95 -.98 3-82 7.0 3.82 7.0 3.82 7.0 -.98 -.03 3.06 3.o6 - 3.06 Total ozone (m atm-cm) 335.8 251.5 413.5

153 TABLE B-2 UMKEHR CURVE POINTS FOR THE VARIOUS STANDARD) DISTRIBUTIONS AND WAVELENGTH PAIRS, WITH SECONDARY SCATTERING EFFECTS INCLUDED Zenith N-Values for the Various Curve Points Angle Distributions: I I I II III (degrees) Wavelengths: A C D C C 90 151o9 124.8 94~9 118.4 131.4 89 161,8 132o8 97~7 125o7 139o 1 88 169o5 137.3 96.7 129,4 143.5 86,5 179.4 141.6 91.3 131.6 147.6 85 186.8 141.8 83.8 128.7 148.9 83 192.5 136.8 73.3 119.2 145.9 80 19353 121,8 59.6 100,7 134.2 77 185.0 105o.7 49.0 84.7 119.7 74 170o8 91o7 411! 72.2 106.1 70 150.4 77~1 3355 59~ 5 90.9 65 128.7 64,1 26~8 48.7 76.5 60 111.9 54~6 22,1 41.0 66.3

154 TABLE B-3 FIRST ORDER PARTIAL DERIVATIVES FOR STANDARD DISTRIBUTION I, A WAVELENGTH PAIR, WITH SECONDARY SCATTERING EFFECTS INCLUDED Zenith Derivatives* Angle (egrees Layers: 1 2 3 4 5 6 7 8 9 90 4o5 415 748 916 526 61 1 811 1849 89 403 402 666 768 470 156 313 1080 1497 88 400 388 634 738 534 326 673 1156 1241 86.5 400 369 613 773 565 633 1126 1120 981 85 394 360 621 869 878 1005 1414 1058 808 83 391 360 679 1051 1240 1488 1614 960 652 80 4oo 425 90o8 1584 2024 2100 1701 821 505 77 436 559 1270 2290 2750 2470 1688 717 414 74 486 680 1547 2740 3133 2530 1570 630 349 70 534 750 1690 2850 3090 2350 1379 533 288 65 553 746 1600 2641 2779 2044 1171 445 239 60 553 704 1450o 2360 2445 1771 1003 380 203 *Derivatives above have to be multiplied by 10-2 to get units specified in text.

155 TABLE B-4 FIRST ORDER PARTIAL DERIVATIVES FOR STANDARD DISTRIBUTION I, C WAVELENGTH PAIR, WITH SECONDARY SCATTERING EFFECTS INCLUDED Zenith Derivatives* Angle (degrees) Layers: 1 2 3 4 5 6 7 8 9 90 201 214 361 374 192 128 602 1158 1047 89 201 204 332 552 287 387 1018 1082 822 88 201 197 326 412 435 755 1265 961 670 86.5 200 194 364 578 809 12530 1405 814 524 85 201 218 477 900 1322 1596 1417 701 422 83 212 290 718 14351 1921 1870 1344 587 334 80 246 405 1021 1922 2290 1882 1175 468 255 77 276 462 1106 1963 2190 1682 993 383 206 74 292 467 1070 1833 1972 1469 849 323 173 70 300 446 975 1619 1701 1240 707 268 143 65 297 404 856 1386 1428 1031 586 220 118 60 289 372 758 1205 1230 882 498 187 101 *Derivatives above have to be multiplied by 10-2 to get units specified in text.

TABLE B-5 FIRST ORDER PARTIAL DERIVATIVES FOR STANDARD DISTRIBUTION I, D WAVELENGTH PAIR, WITH SECONDARY SCATTERING EFFECTS INCLUDED Zenith Derivatives* Angle (degrees) Layers: 1 2 3 4 5 6 7 8 9 90 89 98 180 241 339 766 1205 800 441 89 90 100 212 402 717 1157 1232 642 345 88 90 111 291 658 1105 1370 1150 525 281 86.5 93 147 451 1034 1475 1438 997 417 219 85 101 194 587 1249 1600 1370 863 340 179 83 114 236 663 1300 1520 1200 712 273 143 80 130 255 647 1170 1284 960 553 209 110 77 138 248 587 1010 1073 789 448 169 90 74 141 232 522 875 915 663 375 141 75 70 139 209 452 737 760 545 308 116 61 65 135 186 387 617 630 449 253 95 51 60 130 168 338 532 540 384 215 81 43 *Derivatives above have to be multiplied by 10-2 to get units specified in text.

157 TABLE B-6 FIRST ORDER PARTIAL DERIVATIVES FOR STANDARD DISTRIBUTION II, C WAVELENGTH PAIR, WITH SECONDARY SCATTERING EFFECTS INCLUDED Zenith Derivatives* Angle (degrees) Layers: 1 2 3 4 5 6 7 8 9 90 102 90 16o 244 199 167 717 1195 1042 89 101 87 152 257 291 469 1072 1134 819 88 101 86 161 331 517 842 1257 1013 669 86.5 102 95 220 578 1082 1380 1352 852 520 85 106 121 331 960 1761 1776 1351 730 419 83 117 168 483 1391 2370 1990 1250 612 3355 80 138 212 576 1573 2490 1864 1054.485 255 77 150 220 563 1465 2240 1604 876 395 206 74 152 213 519 1315 1964 1383 743 333 173 70 152 196 458 1130 1652 1150 614 274 143 65 151 178 395 956 1381 952 506 225 117 60 148 160 346 827 1187 811 430 191 100 *Derivatives above have to be multiplied by 10-2 to get units specified in text.

158 TABLE B-7 FIRST ORDER PARTIAL DERIVATIVES FOR STANDARD DISTRIBUTION III, C WAVELENGTH PAIR, WITH SECONDARY SCATTERING EFFECTS INCLUDED Zenith Derivatives* Angle (degrees) Layers: 1 2 3 4 5 6 7 8 9 90 420 435 544 408 187 119 595 1160 1048 89 421 419 494 391 274 378 1014 1082 821 88 419 410 480 435 423 715 1256 963 669 86.5 416 395 504 576 770 1193 1391 805 520 85 413 409 597 836 1211 1542 1400 697 421 83 420 490 845 1295 1734 1780 1320 584 335 80 473 682 1254 1818 2110 1810 1151 468 256 77 532 817 1451 1963 2090 1643 982 382 206 74 575 871 1474 1884 1929 1456 846 324 173 70 602 862 1386 1704 1680 1235 705 267 144 65 604 803 1254 1476 1421 1031 582 220 117 60 596 738 1095 1288 1222 880 496 187 100 *Derivatives above have to be multiplied by 10-2 to get units specified in text.

1~59 TABLE B-8 THE OVERLAPPING-LAYER ZENITH-ANGLE SYSTEM USED BY DUTSCH TO OBTAIN SMOOTH SOLUTIONS...... __Zenith Angles Used...... Set 1 2 3 4 Sub-Set 1.1 1.2 2.1 2.2 3.1 3.2 4.1 4.2 4.3 Layer 1 6o 60 70 65 70 65 2 74 74 77 77 3 77 8o 80.. 4 80 80 83 83 5 I 83 85 85 85 6 i 89 96586.5 86.5 7 86.5 88 88 88 8 88 89 89 89 89 9 90 90/ 90 90

APPENDIX C THE COMPUTATION OF THE GENERATING FUNCTION CURVES FOR PRIMARY SCATTERING 160

161 The basic equations used in this computation are (6) and (7) of Section 2.1. Pressure and density data were taken from UOS. Standard Atmosphere, 1962, for 1 km intervals from the surface to 80 km. Ozone mixing ratio values were used for the same height interval (Fig. 18). Values of the total atmospheric refraction were inferred from Table 137, Smithsonian Meteorological Tables (List, 1958), for each of the 12 zenith angles used by DUtsch. It was assumed that atmospheric refractive index could be represented by the simple formula: i(z) = 1 + Clp(z) (76) where the value C1 = 2.357x10O-4 m3/kg was taken from work by Komhyr- (1956) on refraction of air in the ultraviolet. In a spherical atmosphere, we use the modified index of refraction M(z), M(z) = N(z)(l+z/R) (77) where R is the radius of the eartho Snell's law now takes the form (sin Szh)M(h) Cz, a constant (78) where ~zh is the angle of incidence, at height h, of the direct solar beam which is incident in the zenith direction at height z. The ray constant Cz was calculated from (78), based on the assumed value of atmospheric refraction for the appropriate zenith angle. It was further assumed that the amount of refraction at level z was given by

162 6(z) = 6(0) (79) The angle zh could then be computed from Eq. (78). Although the above model for atmospheric refraction is somewhat crude, it will suffice for the present purposes since the sphericity of the atmosphere is the dominating influenceo The integral of Eq. (7) was evaluated for z = 0 (2) 80 km. Above 80 km, only scattering is important and atmospheric attenuation of the incoming solar beam was assumed to be negligibly small. The integration was carried out using 4-point Gaussian quadrature in each 2 km interval above the scattering point z. Quadratic interpolation was used, within each 2 km interval, to obtain values of p(h), r3(h) at the appropriate abscissas. Finally, X(O,z) was calculated and the integration of Eq. (6) 00 Q(0) = X(G,z)dz (80) was performed using the 2-point Newton-Coates quadrature formula in each 2 km interval (i.e., the trapezoidal rule). As z+80 km, it was found that X(Q,z)+p(z). Hence, based on the densities at 70 and 80 km, an exponential density decrease with height was assumed to obtain a small correction for the downward scattering from the layer above 80 km.

APPENDIX D THE PROCEDURE USED WITH OZONE SONDE DATA 163

164 The ozone sounding data used here were mostly for the ozone sondes developed at Oxford by Brewer and Milford (1960) and by Griggs (unpublished PhoDo thesis). Most of the flights were made at Liverpool. In addition, data for 13 flights were kindly provided by Hering (personal communication) for Fort Collins during the winter 1962-63o Data for an additional 29 flights at Arosa during ozone-sonde intercomparisons were provided by DUtsch. The Arosa data were matched up subjectively with the corresponding results of the evaluation of simultaneous Umkehr observationso The remaining data were treated as follows. No sounding was used unless it extended above the middle of layer 4, ioe., about 45 mb. An estimate of the ozone content for the layer in which the ascent terminated was made whenever the ascent went above the middle of the layer. The ozone content, or mean concentration, for each of the layers was read off and converted to layer-mean partial pressures, if not already in these units. When a total ozone measurement was not available, as was the case for seven soundings, an estimate was made based on the sum of the partial pressures in layers 1 through 4, inclusiveo This estimate was based on the following equation which was derived from those cases where total ozone was available Q = + b.(Ep - Ep + Nd(O,.) (81)

where = is the estimate = average total ozone for the sample = 347 m atm-cm b = 0o307 (m atm-cm/[mb) ZPi = sum of the partial pressures in layers 1 to 4 (Imb) EPi = mean sum for the sample (etmb) Nd(O,a) = normally distributed random variable with zero mean and standard deviation a generated by random number generator subroutine = standard error of estimate of the regression equation 31o4 m atm-cmo The partial pressures in the layers above the top of the balloon sounding were estimated from a regression based on the Arosa solutions given by DItsch (1963), using only those cases where cloudiness (on his scale) was 0 or 1o The regression equation used was based on the total amount of ozone and is pj = pj + bj(i-i) +N(Oj) (82) where pj = the estimate of layer-mean partial pressure in layer j (tmb) pj = the average layer-mean partial pressure in layer j for the sample (pmb), tabulated in the second column of Table D-l bj = the regression coefficients, tabulated in the third column of Table D-lo

166 = total ozone for the case being estimate (m atm-cm) = mean total ozone for the sample = 314 (m atm-cm) aj = standard error of estimate of the regression equation (limb), listed in the fourth column of Table D-l. TABLE D-l STATISTICAL PARAMETERS USED IN ESTIMATING LAYER-MEAN PARTIAL PRESSURES FROM TOTAL OZONE Average Layer- Regression Standard Error Mean Partial Layer Coefficient of Estimate Pressure (Gtmb/m atm-cm) (pmb) (pmb) 5 a110.1.0926 9.1 6 76.4.0678 8.3 7 45.9.0567 4.3 8 21.7.0057 1.8 9 10. 5.0031 1.5 The unusual procedure of adding random noise to the regression estimates was used so that the estimates would have the same variability as in the original sample. It was originally intended to derive characteristic patterns of the vertical distribution from these data and to use these patterns in the solution procedure outlined in Section 5.6. However, these solutions proved rather unstable and the method was discarded. In the present context, it is simply an elaborate way of ensuring that the "processed" balloon sounding data will have the same variability in the upper layers as the Umkehr solutions. Finally, the synthesized vertical distribution was summed to obtain

167 a total amount of ozone and an adjustment factor for the balloon sounding portion was calculated to ensure agreement between observed total ozone and that in the synthetic distribution.

APPENDIX E TABULATIONS OF INDIVIDUAL SOLUTIONS 168

TABLE E-1 INDIVIDUAL SOLUTIONS FOR AROSA DATA SAMPLE., BY TWOMEY'S METHOD,9 WITH RESPECT TO SI,9 AND WITH SECOND DERIVATIVE CORRECTIONS APPLIED Vertical. Distribution of Ozone -Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution ARCSA 2 11 61 P 21 23 52 95 92 66 4C 24 11 243 244 0 ARCSA I 11 61 A 23 25 51 94 92 66 41 24 11 247 247 0 ARCS/ 31 IC 61 P 20 34 61 99 92 64 41 24 11 254 256 0 ARCS/ 5 10 61 A 12 3C 68 106 98 69 43 24 11 256 261 0 ARCS/A 31 10 61 /i 27 41 58 95 89 62 38 23 10 257 257 0 ARCS, 15 10 61 P 19 36 65 102 95 66 41 24 11 259 262 0 ARCS/A 11 1C 61 A 19 37 65 101 94 66 43 24 11 260 263 0 A RCSA 15 10 61 A 18 36 67 104 97 66 41 24 11 262 265 0 ARCSA 10 2 62 A 20 35 67 11C 105 69 37 20 8 266 269 0 ARCSA 13 IC 61 P 20 36 65 103 98 69 41 24 11 266 268 0 ARCSA 13 1C 61 A 23 36 62 101 96 68 43 24 11 266 266 0 ARCSA 23 9 61 A 32 4C 53 93 91 68 43 23 1C 267 264 C ARCS/A 23 9 61 A 33 46 55 92 88 66 41 23 10 267 265 0 ARCSA 24 9 61 A 26 3E 61 100 95 68 42 24 11 269 268 0 ARCSA 25 q 61 P 33 35 51 94 95 70 42 23 10 269 265 0 ARCSA 25 9 61 A 26 35 59 10C g9 71 43 23 10 270 269 0 ARCSB' 1C 2 62 P 17 43 73 108 99 68 40 23 10 270 274 0 ARCSA 22 s 61 a 30 4F 60 95 90 67 42 24 11 273 271 0 ARCS/ 21 9 61 A 29 3S; 59 100 96 69 42 23 10 273 271 1 ARCS/ 21 9 61 A 24 34 62 104 100 72 44 24 10 273 272 1 ARCSA 20 9 61 A 23 29 63 107 105 74 43 24 11 274 274 0 ARCSa 22 9 61 P 38 54 56 91 87 64 41 22 10 276 272 0 ARCSA 22 S 61 P 36 44 54 94 93 68 41 23 10 276 271 0 ARCSA 1S 9 61 a 25 35 62 105 103 74 45 23 10 278 277 0 ARCSA 16 9 61 P 34 29 54 103 105 74 41 21 9 279 274 0 A refers to AM observations P refers to PM observations

TABLE E-1 ( Continued) Vertical Distribution of OzoneToaOzn Station Date Mean Ozone Partial Pressures for the Various Layers --- Codns..~~~~~~~~~~~~~~~Codns 1 2 3 4 5 6 7 8 9 Observed Solution ~RCSA 16 9 61 ~ 24 31 63 106 105 75 46 24 1C 279 27& 0 ~RCS~ 19 2 62 P 13 26 73 119 116 82 46 22 9 281 286 0 ~Rcsa 26 7 62 ~ 29 28 55 104 108 83 48 22 9 283 281 0 ~RCSA 1 5 61 P 30 39 60 102 103 77 46 22 10 285 283 1 ~RCS~ 27 8 61 P 32 31 57 105 1C8 79 46 22 10 288 284 0 ~RCS~ 2S 8 61 P 30 34 61 106 1C7 79 45 22 10 289 286 0 PRCSfi 4 5 61 ~ 33 52 67 103 97 69 41 23 1C 290 287 3 ~RCSA 25 7 62 P 35 43 61 104 105 75 41 21 9 292 288 3 ~RCSfi 11 9 61 P 1.9 33 73 116 113 79 48 25 11 293 294 0 ~RCS~ 2C 7 62 P 29 4 1 64 107 107 8 1 49 22 9 294 292 0 fiRCSA 2C 2 62 a 13 27 79 125 121 83 47 24 11 295 298 0 ~RCS~ lO 9 61 A 20 42 76 114 log 78 49 25 11 296 298 0 ~R~Sfi 3C 8 61 P 32 44 64 105 104 78 48 23 9 296 293 0 ~~RESA 15 9 61 P 32 35 66 112 111 74 38 23 11 296 291 20 ~RESA 21 7 62 ~ 43 3~ 51 102 107 81 43 18 7 297 289 1 ~RCSfi 2C 7 62 P 22 29 69 118 119 86 49 21 8 297 297 2 ~RCSB 13 9 61 P 23 47 76 110 I03 75 48 26 12 298 298 2 ~RCSA 20 2 62 P 20 36 75 118 116 81 48 24 10 300 301 1 aRCSA 14 7 62 ~ 32 43 68 11! 110 78 46 21 9 3C2 299 2 PRCSA 26 8 6 1 ~ 2C 38 77 1 19 1 16 83 49 23 10 303 304 0 aRcsfi 14 7 61 ~ 6 37 91 131 122 85 5C 24 11 304 311 0 ARCSfi 24 6 62 P 33 49 6~ lo9 1C7 76 47 23 10 306 302 3 ~RCSA 9 7 62 a 40 5'7 65 105 105 80 45 19 7 309 304 2 ~RCSA 5 5 6 1P 26 58 8 1 11t4 106 77 51 23 9 312 312 2 ~RcSa 5 5 61 a 17 5C 89 124 116 80 49 24 10 313 316 2 A refers to AMI observations P refers to PM observations

TABLE E-1 (Continued) Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 ~ 6 7 8 9 Observed Solution AR~SA 26 7 61 P 35 57 76 li5 ll2 78 38 19 8 314 311 ARCS~ 31 5 62 ~ 19 51 89 128 121 84 49 24 11 325 326 AR~SA 20 10 61 A 26 91 101 118 97 64 41 25 12 327 329 ARCSA 20 10 61 P 31 97 101 116 93 58 38 24 11 327 327 ARCSA 24 4 62 P 23 68 96 128 117 77 4.3 21 9 329 331' ARCSA 23 7 61 P 32 54 80 122 120 84 46 22 9 331 328 1 ARCSA 21 2 62 A 20. 47 90 131 126 86 51 25 11 331 332 0 ARCSA 4 4 61 A 13 65 109 139 123 77 42 24 11 334 339 2 ARCSA 24 7 61 ~ 22 61 94 129 122 84 49 24 10 337 338.0 ARCSA 28 6 62 A 31 65 87 122 117 83 49 22 8 337' 335 0 6RCSA 22 4 62 P 32 61 86 125 120 80 44 23 10 338 335 ARCSA 19 10 61 ~ 31 101 104 118 98 65 42 25 12 343 342 ARCSA 30 5 61 ~ 25 56 90 130 127 91 51 23 10 344 343 I —' ARCSA 9 6 62 ~ 25 67 98 133 124 84 48 21 8 346 346 — ~ ARCSA' 17 5 62 ~ 29 66 94 131 126 90 53 22 8 355 353 ~-J ARCS~ 2 5 61 A 33 84 lO1 ]27 lI5 76 44 24 I1 356 ']53 ARCSA 6 6 62 ~ 55 71 70 112 115 86 52 21 8 361 348 ARCSA 26 4 62 F 40 77 87 120 118 88 53 24 10 365 358 ~RCS~ 28 2 62 F 19 61 108 lql 130 85 54 32 15 36? 365 ARCSA 28 3 6l P 41 74 93 131 125 82 44 22 9 367 360 /~RCS/~ 7 3 62 P 30 91 110 134 120 79 46 24 11 3?0.368 ~RC:SA 25 5 61 A 23 85 115 141 128 88 53 24 '10 377 378 ARCSA 20 3 62 /~ 16 g3 12g 148 125 79 49 27 12 ]Tg 382 ARCSa 27 2 62 P 29 ll2 124 138 lI6 75 46 25 11 386 386 ARCS~ 12 4 62 ~ 31 10C 116 135 118 80 52 27.12 386 383 A refers to AM observations P refers to PM observations

TABLE E-1 (Concluded) Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 ' ~ 6 7 8 9 Observed Solution ~ ~R[S~ 12 4 ~2 ~ 38 84 102 13~ 125 86 53 26 11 386 379 1 aR[Sa 21 3 62 ~ 23 101 128 146 126 80 50 28 13 394 394 0 ~CS~ 12 4 62 P 34 9S 118 137 119 77 50 29 14 394 388 3 ~RCSa 19 3 62 ~ 17 109 lq3 153 126 76 46 29 14 398 40'1 0 ~RESfi 19 3 62 P 31 113 128 143 122 78 45 26 13 4C0 398 0 ~R~S~ 24 3 ~2 ~ 20 89 125 148 1.33' 91 62 30 13 401 401 0 ~RES~ 24 3 62 ~ 13 84 133 156 139 90 57 32 15 401 403 0 ~RCSA 21 3 ~2 P 22 98 132 152 130 83 52 29 13 403 40'3 1 ~RCS~ 21 3 62 P 17 92 133 154 133 85 57 31 15 40.3 404 1 ~RCSA 2 q 62 P 39 122 122 135 116 77 4g 25 11 404 400 1 ~RCS~ 25 3 62 ~ 28 101 125 145 129 87 55 27 12 405 403 S ~R/Sfi 2C 3 61 P 20 125 148 156 127 78 44 24 11 408 413 0 ARCS~ 25 2 62 ~ 9 125 159 161 128 79 5C 28 13 410 418 0 F'J ~RCS~ 1 4 62 F 25 127 142 148 122 79 51 27 12 413 41YJ 2 ~RCS~ ll 4 62 fi 42 12C 123 137 llg 83 52 26 II 417 410 2 ~RCSA 11 4 62 P 48 101 109 137 128 86 51 27 12 418 406 4 ~RCSA 18 3 62 P 22 123 148 155 127 78 50 30 14 422 423 3 ~RCSA 24 2 62 ~ 21 155 162 154 117 71 48 27 12 427 432 0,~RES,~ 24 2 62 P 27 152 155 151 117 73 47 26 12 428 430 0 ~RCS~ 22 3 61 P 27 142 155 157 127 79 47 25 11 434 436 1 6RCSfi 17 3 62 ~ 2C 135 16C 161 128 77 51 32 15 438 440 3 ~RESfi 18 3 62 ~ 28 134 150 153 125 78 53 33 16 441 438 3 ~gCS~ 26 3 62 ~ 26 148 160 159 128 82 54 30 14 452 453 B ~RCSA 16 4 62 ~ 49 164 142 138 111 76 53 26 11 453 ~46 1 ~RCS~ 26 ] 62 P 27 132 156 163 134 83 55 32 15 455 452 3 A refers to AM observations P refers to PM observations

TABLE E-2 INDIVIDUAL SOLUTIONS FOR 42 LOW-OZONE AROSA UIMKEHRS, BY TWOMEY'S METHIOD, WITH RESPECT TO SII, AND WITH SECOND DERIVATIVE CORRECTIONS APPLIED Vertical Distribution of OzoneToaOzn Station Date Mean Ozone Partial Pressures for the Various Layers -- -- -- Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution aRCS~ 2 11 61 P 20 31 42 78 109 69 39 25 11 243 24'3, 0 ~RCS~ 1 11 61 a 23 32 4i 77 109 69 40 25 11 247 247 0 fiRCSA 31 10 61 P 19 40 54 81 107 67 40 25 ' 11 254 254 0 ~RCS~ 9 1C 61 ~ 10 37 59 87 113 71 41 25 11 256 256 0 ~RCS,~ 31 1C 61 a 28 46 52 77 104 64 36 24 11 257 257 0 ARC5SA 15 10 61 P 18 42 58 84 110 68 359 25 11 259 259 0 ARI]SA 11 1C 61 P 19 42 56 82 1C9 69 41 25 11 260 260 0 PRI]SA 15 1C 61 A 17 42 61 86 111 68 39 25 11 262 262 0 ~R(S~ '10 2 62 Pa 19 43 65 93 117 68 34 21 9 266 266 0 ARCS~ 13 10 61 P 20 42 57 85 1'13 71 40 25 11 266 266 0 ~RISS, 13 1C 61 ~ 23 43 54 83 110 71 42 25 11 266 266 0 ~RCSA 23 9 61,a 33 45 46 76 1C]8 71 41 25 11 267 267 0 aRt]S, 2 3 5 61 ~ 35 5C 48 74 105 69 39 24 11 267 266 0 ARCSA 24 9 61 a 27 44 53 81 110 71 41 25 11 269 269 0 PR~SA 25 9 61 P 35 42 43 77 111 74 42 24 10 269 268 0 ~RCS~ 25 9 61 Pa 27 42 51 82 114 74 41 24 11 270 270 0 ~RCSA 1C 2 ~2 P 16 48 69 sa 112 68 38 24 10 270 270 0 ~RCS~ 22 9 61 P 32 52 54 77 106 69 41 25 11 273 273 0 ARCS,~ 21 9 61 ~ 31 46 53 81 111 72 41 24 10 273 273 1 ~RCSA 21 9 61 ~ 25 42 55 86 114 74 43 25 1C 2.73 273 1 ~RCS~ 20 9 61 ~ 22 38 56 89 119 76 42 25 11 274 274 0 ~RCSA 22 5 61 P 41 58 51 73 1C2 67 39 24 1C 276 275 0 ~RCSa 22 9 61 P 39 5C 49 77 108 '70 3'g 24 11 276 ' 275 0 ~RCS~ 19~ 5 61 P 26 43 56 87 117 76 43 24.1C 278 27'8 0 PRCSP 16 S 61 P 36 41 48 85 118 76 40 23 9 27280.~~~~ ~ ~ ~~~~~~~~~ 2. 2 _ 0 A refers to AM observations P refers to PM observations

TABLE E-2 (Concluced) Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution ARCSA 16 9 61 A 24 40 57 89 119 76 43 25 11 279 279 0 ARCSA 19 2 62 P 10 37 69 101 128 81 43 23 9 281 281 0 ARCSA 26 7 62 A 29 38 49 87 124 84 46 23 9 283 283 0 ARCSA 1 9 61 A 32 45 54 85 118 79 44 24 10 285 285 1 ARCSA 27 8 61 P 33 42 53 88 122 80 44 24 10 288 287 0 ARCSA 29 8 61 P 31 44 57 89 121 79 43 24 10 289 289 0 ARCSA 4 9 61 A 36 58 63 84 110 70 4C 24 11 290 289 3 ARCSA 25 7 62 P 38 51 57 87 118 75 39 23 10 292 291 3 ARCSA 11 9 61 A 18 43 70 98 124 79 46 26 11 293 293 0 ARCSA 20 7 62 P 30 49 60 89 120 82 47 23 9 294 294 0 ARCSA 20 2 62 A 11 39 77 107 132 81 44 25 11 295 295 0 ARCSA 10 9 61 A 19 49 73 96 121 79 46 25 11 296 296 0 ARCSA 30 8 61 P 34 51 60 87 118 80 45 24 10 296 296 0 ARCSA 15 9 61 P 34 47 65. 95 122 72 37 25 11 296 295 2 ARCSA 21 7 62 A 48 49 47 85 122 82 41 20 8 297 296 1 AROSA 20 7 62 A 21 40 66 101 131 86 46 22 8 297 297 2 ARCSA 13 9 61 P 25 54 72 91 116 76 46 27 12 298 298 2 A refers to AM observations P refers to PM observations

TABLE E-3 INDIVIDUAL SOLUTIONS FOR 29 HIGH-OZONE AROSA UMKEHRS, BY TWOMEYIS METHOD, WITH RESPECT TO LE11, AND WITH SECOND DERIVATIVE CORRECTIONS APPLIED Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressurea for the Varioua Layera Cloudineac 1 2 3 4 5 6 7 8 9 Observed Solution ARCSA 25 5 61 A 34 66 127 14C 126 90 54 24 IC 377 383 0 ARCSA 20 3 62 A 29 67 141 150 125 81 49 27 12 379 387 0 ARCSA 27 2 62 P 40 91 138 138 115 77 46 25 11 386 391 0 ARCSA 12 4 62 A 43 81 128 135 116 81 53 27 12 386 389 1 ARCSA 12 4 62 A 49 71 114 131 122 88 54 26 11 386 385 1 ARCSA 21 3 62 A 35 7i 140 147 125 82 51 28 13 394 399 0 ARCSA 12 4 62 P 46 80 129 138 117 78 51 29 14 394 393 3 ARCSA 19 3 62 A 29 80 154 155 127 79 47 29 14 398 406 0 ARCSA 19 3 62 P 42 92 141 142 121 80 46 26 13 400 404 0 ARCSA 24 3 62 A 32 69 136 147 131 93 63 30 13 401 406 0 ARCSA 24 3 62 A 25 60 145 156 138 93 58 31 15 401 408 0 ARCSA 21 3 62 P 35 75 143 152 130 85 53 29 13 403 407 1 ARCSA 21 3 62 P 30 68 143 153 133 88 58 31 14 403 408 1 ARCSA 2 4 62 P 49 104 135 135 114 79 Sc 25 11 404 405 1 ARCSA 25 3 62 A 4C 81 137 145 127 89 56 27 12 405 408 3 ARCSA 20 3 61 P 32 96 161 157 129 81 44 24 11 408 417 0 ARCSA 25 2 62 A 22 92 170 163 130 82 SC 28 13 410 422 0 ARCSA 1 4 62 P 36 101 155 149 122 81 52 27 12 413 419 2 ARCSA 11 4 62 A 53 102 134 137 118 85 53 26 11 417 416 2 ARCSA 11 4 62 P 60 87 120 135 125 88 53 27 12 418 412 4 ARCSA 18 3 62 P 35 95 158 157 129 81 51 30 14 422 427 3 ARCSA 24 2 62 A 33 123 175 1S6 119 75 49 27 12 427 436 0 ARCSA 24 2 62 P 39 122 167 154 119 76 47 26 12 428 434 0 ARCSSA 22 3 61 P 40 112 165 160 130 83 48 25 11 434 440 1 ARCSA 17 3 62 A 34 104 170 162 130 81 51 32 1S 438 443 3 A refers to AM observations P refers to PM observations

TABLE E-3 (Concluded) Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution ARCSA 18 3 62 A 41 108 161 154 125 81 54 33 16 441 442 3 ARCSA 26 3 62 A 39 119 171 160 130 85 55 29 13 452 456 3 ARCSA 16 4 62 A 61 144 153 138 111 79 54 26 11 453 450 1 ARCSA 26 3 6.2 P 42 103 163 164 136 86 55 32 15 455 455 3 A refers to AM observations P refers to PM observations Oh

TABLE E-4 INDIVIDUAL SOLUTIONS FOR AROSA DATA SAMPLE, BY TEVE METHOD, WITH RESPECT TO SI, AND WITH SECOND DERIVATIVE CORRECTIONS APPLIED Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution ARCSA 2 11 61 P 21 23 52 95 92 66 40 24 11 243 244 0 ARCSA 1 1 61 A 23 25 51 94 92 66 41 24 11 247 247 0 ARCSA 31 10 61 P 20 34 61 99 92 64 41 24 11 254 256 0 ARCSA 9 10 61 A 12 3C 68 106 98 69 43 24 11 256 261 0 ARCSA 31 10 61 A 27 41 58 95 89 62 38 23 10 257 257 0 APCSA 15 10 61 P 19 36 65 102 95 66 41 24 11 259 262 0 ARCSA 11 1C 61 A 19 37 65 101 94 66 43 24 11 260 263 0 ARCSA 15 1C 61 A 18 36 67 104 97 66 41 24 11 262 265 0 ARCSA 10 2 62 A 20 35 67 110 105 69 37 20 8 266 269 0 ARCSA 13 1C 61 P 20 36 65 103 98 69 41 24 11 266 268 0 ARCSA 13 1C 61 A 23 36 62 101 96 68 43 24 11 266 266 0 ARCSA 23 9 61 A 32 4C 53 93 91 68 43 23 10 267 264 0 ARCSA 23 9 61 A 33 46 55 92 88 66 41 23 10 267 265 0 ARCSA 24 9 61 P 26 3e 61 100 95 68 42 24 11 269 268 0 ARCSA 25 S 61 P 33 35 51 94 95 70 42 23 10 269 265 0 ARCSP 25 9 61 A 26 35 59 100 99 71 43 23 10 270 269 0 ARCSA 1C 2 62 P 17 43 73 108 99 68 40 23 10 270 274 0 ARCSA 22 9 61 A 30 48 60 95 90 67 42 24 11 273 271 0 ARCSA 21 9 61 A 29 39 59 100 96 69 42 23 10 273 271 1 ARC S 21 9 61 A 24 34 62 104 100 72 44 24 10 273 272 1 ARCSA 2C 9 61 A 23 29 63 107 105 74 43 24 11 274 274 0 ARCSA 22 9 61 P 38 54 56 91 87 64 41 22 10 276 272 0 ARCSA 22 9 61 P 36 44 54 94 93 68 41 23 10 276 271 0 ARCSA 19 9 61 A 25 35 62 105 103 74 45 23 10 278 277 0 ARCSA 16 9 61 P 34 29 54 103 105 74 41 21 9 279 274 0 A refers to AM observations P refers to PM observations

TABLE E-4, (Con-tinued) 'Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers — Cloudines s 1 2 3 4 ~ 6 7 $ 9 ~~~~Observed Solution ARCSA 16 9 61 P 20 36 68 108 10.3 72 44 24 11 2(~ 278,~!CSA 19 2 62 P 9 23 77 123 118 79 44 23 10 281 282 6RCS~ 26 7 62 A 29 3] 57 '103 106 80 49 22 9 283 8 ~I~CSP 1 9 61.a BO 43 62 102 101 74 46 23 10 285 285 ARESA 27 8 61 P 29 40 63 104 104 76 46 23 10 288 8 ARCS,6 29 8 61 P 27 42 67 107 104 75 45 23 10 289 287 ARCSA 4 9 61 P 33 59 69 100 94 67 42 24 11 290 8,~RCSb 25 7 62 P 31.53 67 104 99 71 42 22 9 292 8 ARCSP ll 9 61 P 16 35 77 118 112 77 46 25 11 293 293 bRCS,6 20 7 62 P 32 43 62 105 107 80 49 22 9 294 9,~RCSA 20 2 62 4 6 26 85 129 121 79 45 24 11 29.5 9 6RCSb 1C 9 61 P 20 41 77 116 111 76 46 25 11 9 8 ARGSA 30 8 61 P 34 49 64 103 103 77 48 23 1O 26260 - ARCS6 15 9 61 P 22 51 78 111 102 68 41 24 11 262020 ARCSA 21 7 62 P 43 52 54 96 lO1 79 47 19 7 9 2 ARCS/~ 2C 7 62;~ 21 BC 70 118 119 85 49 21 8 9 6 AIRCSA 13 9 61 P 24 49 77 111 104 Y3 46 '27 12 298 0 BRCSB 20 2 62 P 17 38 79 12C 115 78 47 25 11 0 0 ARCSA lq 7 62 A 31 50 70 109 107 77 46 22 9 B02 0 /~R(S~A 26 8 61 A 18 38 79 121 117 81 47 24 10 B0 4OB ARCSA 14 7 61;~ 3 28 94 137 128 82 46 25 1 0 0 bRCS.6 24 6 62 P 33 56 71 107 104 75 47 2]3 10 0 5 ARC.SP 7 62 P 44 64 64 102 10B 78 46 19 7 0 8 LrRC~SA 5 5 61 P 31 57 77 113 109 78 48 24 10 1 5 bRCS6 5 5 61 4 16 47 90 127 119 78 45 24 i 1 1 A refers to ALM observations P refers to PM observations

TABLE E-4 (Continued) Vertical, Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the. Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution ARCSA 26 7 61 F 30 65 83 116 108 73 40 20 8 3.14 311.3 ARCS~ 31 5 62 ~ 16 5C 94 131 122 81 47 25 L1 325 326 3 ~RCSA 20 10 61 B 29 8S 101 119 1GO 62 38 26 13 327 331 l ~RCS~ 20 1C 61 P 32 q9 102 116 94 57 34 25 12 327 329 2 ARESA 24 4 62 P.20 67 100 132 118 75 40 22 9 329 330 1 ARCSA 23 7 61 P 28 61 86 122 116 81 47 23 9 331 329 1 ARCSA 21 2 62 ~ 15 48 95 134 126 84 4c~ 26 11 331 331 0 ARCSA 4 4 61 ~ 4 62 118.146 125 72 38 25 12 334 335 2 ~RCSA 24 7 61 ~ 21 5~ 97 132 123 82 48 24 10 337 338 0 ~RCSA 28 6 62 ~ 33 67 86 122 117 83 48 22 9 337 338 0 ARCS~ 22 q 62 P 27 76 95 126 115 76 44 23 1C 338 335 2 ~RCSA lq1C61 ~ 35 101 105 12C 100 63 39 27 13 343 347 0 tJ ARCS~ 30 5 61 /~ 24 57 94 132 127 87 51 24 10 344 345 0 — 4 /~RCSB 9 6 62 A 24 67 99 134 125 83 46 22 9 346 347 N~). 0 ARISSB 17 5 62 ~ 32 67 92 131 127 90 53 22 8 355 '356 2 ~RCSA 2 5 61 A 3l 9C 106 129 113 73 43 25 11 356 356 3 ~RCSA 6 6 62 ~ 61 89 70 104 108 88 56 21 7 361 357 2 ARCS~ 26 4 62 P 45 B4 88 11g 116 86 55 25 lO 365. 365 3 ~RCSB 28 2 62 P lO 68 120 147 128 80 51 B3 16 367 38b 3 ARCSA 28 3 61 P 35 8S 103 129 lib 79 46 23 9 367 362 3.aRCS.a 7 3 62 P 29 94 114 1'3,6 120 76 44 25 1l 370 371 0 /~RCS~ 25 5 61 A 25 81 115 145 132 87 50 25 11 3?7 380 0.aRCS.a 2C 3 62 /~ 13 88 133 154 131 77 43 28 14 379 ]382 0 ARCS~ 27 2 62 P 30 111 127 142 119 73 42 26 12 386 390 0 BRCSA 12 4 62 B 34 101 118 137 121 79 48 28 13 386 388 1 A re~ers to AM observations P refers to PM observations

TABLE E-4 (Concluded) Vertical. Distribution of Ozone Station Date Mean Ozone Partial Pressurea for the Varioua Layera CloudiTness 1 2 3 4 5 6 7 8 9 Obaerved Solution ARCSA 12 4 62 A 37 93 10C 133 122 84 53 27 12 386 385 1 ARCSA 21 3 62 A 20 ICC 135 152 128 77 46 29 14 394 396 0 ARCSA 12 4 62 P 31 108 126 139 118 74 47 30 15 394 394 3 ARCSA 19 3 62 A 11 104 150 161 130 72 41 30 16 398 401 0 ARCSA 19 3 62 P 28 115 136 148 122 73 42 28 13 4CC 402 0 ARCSA 24 3 62 A 24 83 124 152 139 92 57 30 14 401 405 0 ARCSA 24 3 62 A 7 78 141 166 144 87 53 3 3 16 401 404 0 APCSA 21 3 62 P 18 98 139 157 133 80 48 30 15 403 405 1 ARCSA 21 3 62 P 14 89 139 160 137 84 52 32- 16 403 406 1 ARCSA 2 4 62 P 43 124 124 137 118 76 46 26 12 404 407 1 APSA 25 3 62 A 29 1CC 127 149 132 85 52 28 13 405 408 3 ARCSA 20 3 61 P 19 117 151 163 133 75 39 25 12 408 413 0 ARCSA 25 2 62 A I 108 159 171 139 76 42 30 15 410 418 0H APRSA 1 4 62 P 28 119 143 154 129 77 45 28 14 413 419 2 ARCSA 11 4 62 A 46 125 123 138 121 81 51 27 12 417 419 2 ARCSA 11 4 62 F 45 118 120 136 121 83 53 28 13 418 414 4 ARCSA 1 3 62 P 19 119 155 163 132 75 45 31 16 422 426 3 ARCSA 24 2 62 A 28 14C 159 161 128 71 40 28 14 427 437 0 ARCSA 24 2 62 P 33 142 154 157 126 72 41 27 13 428 435 0 RCSA 22 3 61 P 29 134 155 163 135 78 42 26 12 434 439 1 ARCSA 17 3 62 A 17 129 167 170 135 74 45 33 17 438 443 3 A RSA 18 3 62 A 27 133 158 161 129 75 48 35 18 441 445 3 ARCSA 26 3 62 A 30 139 161 166 136 80 48 31 15 452 459 3 ARCSA 16 4 62 A 62 165 138 138 116 77 50 27 12 453 459 1 ARCSA 26 3 62 P 23 131 165 170 139 80 50 34 17 455 457 3 A refers to AM observations P refers to PM observations

TABLE E-5 AND WITH SECOND DERIVATIVE CORRECTIONS APPLIED Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution ARESA 2 11 61 P 16 36 50 79 106 65 38 25 12 243 24.3' 0 ~RCSA 1 11 61 ~ 18 38 50 78 105 65 39 26 12 247 247 0 ARCSA 31 10 61 P 17 42 58 83 107 64 37 25 12 254 254 0 ~RCSA g 10 61 ~ 7 35 64 gl 114 67 38 26 12 256 256 0 ARCS~ 31 lO 61 A 25 51 59 78 10I 60 34 25 II 257 25? 0 ARCSA 15 1C 61 F 15 43 65 87 109 63 36 26 12 25g 259 0 ARCS~ 11 1C 61,a 17 42 60 85 110 66 38 26 12 260 260 0 ARCSA 15 1(] 61 A 14 44 68 8g 110 63 36 26 12 262 262 0 ARCSA 10 2 62 ~ 15 47 72 95 115 65 32 21 9 266 2'66 0 ARCSA 13 1C 61 P 17 44 65 88 1ll 66 38 26 12 266 266 0 t —' 0o ~RCSA 13 lO 61 ~ 21 45 60 84 109 67 39 26 12 266 266 0 ~J ARCSA 23 g 6i A 32 49 51 76 105 68 4I 25 lI 267 261 0 ~RCS~ 23 9 61 ~ 33 53 54 75 102 65 38 25 II 267 267 0 ~RCSA 24 g 61 ~ 25 48 59 83 108 67 40 25 11 269 269 0 ARC;SA 25 g61P 33 4cj 49 76 106 70 42 25 10 2'6g 269 0 ARCSA 25 9 61,a 24 46 58 84 111 70 41 25 11 270 270 0 ARCSA 1C 2 62 P 15 47 73 93 113 64 35 25 11 270 270 0 ~R~SA 22 g 61 B 31 54 59 7g i04 65 3g 25 II 273 213 0 ARCSA 21 g 61 ~ 2g 50 58 82 109 6g 40 25 lI 273 27'3 'l ARCSA 21 9 61 A 23 45 60 87 I13 71 41 25 II '273 273 I /~RCSA 2C q 61 /1 18 43 65 91 116 71 41 25 II 274 274 0 ARCSA 22 q 61 P 41 61 55 73 lO0 64 38.... 24 11 276 276 0 ARCSA 22 9 61 P 36 57 57 77 103 66 39 25 11 276 2.76 0 ARCS~ 19 g 61 A 24 45 61 88 116 74 42 25 10 278 278 0 ARCSA 16 9 61 P 32 5C 56 83 112 73 41 23 g 279 279 0 A refers to AM observations P refers to PM observations

TABLE E-5 (Concluded) Vertical. Distribution of OzoneToaOzn Station Date Mean Ozone Partial Pressures for the Various Layers --- Clouadiness 1 2 3 4 5 6 7 8 9 Observed Solution AR~SA 1~ 9 ~l a 21 44 64 91 117 73 42 25 11 279 279 0 aRCSA 19 2 62 P 7 35 74 106 129 77 40 24 10 281 281 0 ARCS~ 26 7 62 a 29 41 51 87 121 82 47 23 9 283 283 0 ARCSA 1 9 61 P 3C 48 58 86 116 76 44 24 10 285 285 1 PRCSA 27 8 61 P 30 48 S9 88 117 77 44 24 10 288 288 0 ARCSA 29 8 61 P 28 4S 64 9C 118 76 43 24 10 289 289 0 ARCSA 4 9 6 1 P 34 61 69 85 108 66 39 25 11 290 290 3 6RCSA 25 7 62 P 35 58 66 87 113 71 4C 23 10 292 2 92 3 AR~SA 11 9 61 ~ 15 44 76 101 124 76 43 26 12 293 293 0 AR~S~ 2C 7 62 P 32 48 59 9C 121 81 46 23 9 294 294 0 ~R~S~ 2C 2 d2 P 6 4C 85 111 131 '77 41 25 11 295 295 G ARCSA 10 9 61 P 18 47 7h 100z 123 75 43 26 12 296 296 0 ARCSA 30 8 61 P 34 53 62 88 117 78 45 24 10 296 296 0 I — ~RCSA 15 9 61 P 26 5S 81 95 113 66 37 25 12 296 296 2 CO0 ~RCSA 21 7 62 a 46 57 51 8! 115 81 44 20 7 297 297 1 ~RCSA 2C 7 62 P 21 4C 67 102 132 85 46 22 8 297 297 2 ~RCSA 13 5 61 P 23 52 76 95 117 72 43 28 13 298 2 98 2 A~~ reest Mosrain P refers to PM observations

Vertical Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution ARCSA 25 5 61 fi 34 55 126 147 132 88 50 25 11 377 381 0 ~RCSA 2C 3 62 ~ 23 55 145 158 132 79 43 28 14 379 382 0 ~R~SA 27 2 62 P 38 83 140 144 119 74 43 26 12 386 38<) C ~R'CS~ 12 4 62 ~ 42 77 130 139 120 80 49 27 13 386 388 1 ARCS~ 12 4 62 ~ 46 72 119 134 121 86 53 27 12 386 386 1 ~RCSA 21 3 C2 ~ 30 68 146 155 129 79 46 29 14 394 396 0 ARCS~ 12 4 62 P 4C 81 137 141 118 76 48 30 15 394 393 3 ~RCSA 19 3 62 A 21 67 161 166 133 75 41 30 16 398 401 0 ~RCSfi 19 3 62 P 36 85 149 151 123 75 43 28 13 400 401 0 ARCS~ 24 3 62 A 33 57 134 154 139 94 57 30 14 401 406 0 ~RCSA O0 24 3 62 ~ 18 44 151 169 146 89 53 33 16 401 404 0 ~RCSA 21 3 62 P 28 66 150 160 135 83 48 30 15 403 404 1 ARESA 21 3 62 P 24 56 149 163 139 86 52 32 16 403 406 1 ~RCSA 2 4 62 P 50 101 136 138 117 78 47 26 12 404 407 1 ARESfi 25 3 62 ~ 38 74 138 151 133 87 52 28 13 405 408 3 ARI]SA 20 3 61 P 26 81 164 169 136 77 39 25' 12 408 412 0 ARCSA 25 2 62 ~ 19 69 170 176 142 80 42 29 15 410 417 0 ~RCSA 1 4 62 P 36 88 154 157 130 80 46 28 14 413 418 g AROSA 11 4 62 ~ 54 103 135 139 120 83 51 26 12 417 418 2 ARCSA ll 4 62 P 53 97 130 136 120 85 54 28 12 418 414 4 ~RCSA 18 3 62 P 29 84 165 166 134 78 45 31 16 422 425 3 ~RCSA 24 2 62 ~ 36 106 170 165 130 74 41 28 14 427 435 0 ARCSA 24 2 62 P 41 llC 165 160 127 75 42 27 13 428 434 0 BRCSA 22 3 61 P 37 102 166 167 136 81 4'3 25 12 434 438 1 ~R~SA 17 3 62 ~ 27 91 178 174 137 77 45 33 17 438 441 3 A refers to AM observations A refers to PM observations

TABLE E-6 (Concluded) Vertical Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution ARCSA 18 3 62 A 36 10C 169 164 130 77 49 34 18 441 443 3 ARCSA 26 3 62 A 39 107 172 169 138 83 49 30 15 452 457 3 ARCSA 16 4 62 A 68 145 149 138 115 79 51 27 12 453 457 1 ARCSA 26 3 62 P 33 97 174 173 140 84 5C 33 17 455 455 3 A refers to AM observations P refers to PM observations P refers to PM observations~~~~~~

TABLE E-7 INDIVIDUAL SOLTUTIONS FOR 93 AROSA UMKEHRS BY DUTSCH'S TECHNIQUE Vertical Distribution of Ozone Total Ozone Station Data Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution APCSA 2 11 61 P 20 31 42 78 109 69 39 25 11 243 243 0 ARCSA 1 11 61 A 23 32 41 77 109 69 40 25 11 247 247 0 ARCSA 31 10 61 P 19 40 54 81 107 67 40 25 11 254 254 0 ARS A 9 1C 61 A 10 37 59 87 113 71 41 25 [1 256 256 0 ARCSA 31 10 61 A 28 46 52 77 104 64 36 24 11 257 257 0 ARCSA 15 10 61 P 18a 42 58 84 110 68 39 25 11 259 259 0 ARCSA 11 10 61 A 19 42 56 82 109 69 41 25 11 260 260 0 ARCSA 15 10 61 A 17 42 61 86 LII 68 39 25 11 262 262 0 ARCSA 10 2 62 A 19 43 65 93 117 68 34 21 9 266 266 0 ARCSA 13 10 61 P 20 42 57 85 113 71 40 25 11 266 266 0 ARCSA 13 1C 61 A 23 43 54 83 110 71 42 25 11 266 266 0 H ARCSA 23 9 61 A 33 45 46 76 108 71 41 25 11 267 267 0 AROSA 23 9 61 A 35 SC 48 74 105 69 39 24 11 267 266 0 \J1 ARCSA 24 9 61 A 27 44 53 81 110 71 41 25 11 269 269 0 ARCSA 25 9 61 P 35 42 43 77 111 74 42 24 10 269 268 0 ARCSA 25 9 61 A 27 42 51 82 114 74 41 24 11 270 270 0 ARCSA IC 2 62 P 16 48 69 90 112 68 38 24 10 270 270 0 ARCSA 22 9 61 A 32 52 54 77 106 69 41.25 11 273 273 0 ARCSA 21 9 61 A 31 46 53 81 111 72 41 24 10 273 273 1 ARCSA 21 9 61 A 25 42 55 86 114 74 43 25 10 273 273 1 ARCSA 20 9 61 A 22 38 56 89 119 76 42 25 11 274 274 0 ARCSA 22 9 61 P 41 58 51 73 102 67 39 24 10 276 275 0 ARCSA 22 9 61 P 39 50 49 77 108 70 39 24 11 276 275 0 ARCSA 19 9 61 A 26 43 56 87 117 76 43 24 10 278 278 0 ARCSA 16 9 61 P 36 41 48 85 118 76 40 23 9 279 278 0 A refers to AM observations P refers to PM observations

TABLE E-7 (Continued) Vertical Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution AROSA 26 7 62 A 30 41 54 85 114 ql 49 20 lO 283 2813 AROSA 1 g 61 A 36 52 65 77 103 85 47 22 lO 285 285 AROSA 27 8 61 P 32 38 58 9g 113 83 47 21 lI 28~ 28b AROSA 29 8 61 P 28 40 68 97 10'9 84 47 20 1l 2~9 288 AROSA 4 9 61 A 35 47 75 100 96 71 46 22 11 290 290 AROSA 25? 62 P 40 49 51 90 117 82 39 19 I2 292 292 ~ROSA 1l 9 6l A 20 33 72 116 115 80 48 23 11 293 293 AROSA 20 7 62 P 29 42 76 102 100 82 54 22 8 294 294 AROSA 20 2 62 A 10 25 81 126 123 8B 46 21 11 295 295 AROSA 10 9 61 A 20 45 85 104 104 84 50 22 11 296 296 AROSA 30 g 61 P 28 46 82 96 96 84 53 21 9 296 294 AROSA 15 9 61 P 33 34 58 115 127 75 36 21 14 '296 296 kJ AROSA 21 7 62 A 40 43 69 91 107 87 49 18 9 297 297 CID AROSA 20 7 62 h 17 32 81 115 115 88 52 21 8 297 297 Oh AROSA 13 9 61'P 19 48 97 99 92 84 53 23 12 298 298 AROSA 20 2 62 P' 23 36 71 116 122 82 46 22 ].1 300 300 AROSA 14 7 62 A 30 40 82 115 107 78 49 22 10 302 305 AROSA 26 8 61 A 20 37 83 115 115 85 49 21 10 '303 3C~ AROSA 14 7 61 A 2 3.3 105 127 109 86 52 21 10 304 30l AROSA 24 6 62 P 34 41 78 '118 106 74 50 23 10 '306 306 AROSA 9 7 62 A 41 61 84 90 97 87 50 17? 30'9 30'~ AROSA 5 5 61 P 23 45 107 125 86 7B 59 25 8 BI2 312 AROSA 5 5 61 A 14 42 103 13l 106 77 51 2B 10 313 315 AROSA 26 7 61 P 40 62 77 95 125 86 35 15 11 314 314 AROSA Bi 5 62 A 23 46 90 128 121 84 49 22 11 325 325 A refers to AM observations P refers to PM observations

TABLE E- 7 ( C ont inue d) Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution AROSA 20 lO 61 A 36 66 112 [23 90 65 45 23 12 327 ~2~ 1 ARUSA 24 4 62 P 31 54 96 137 119 72 42 20 '9 329 331 l AROSA 23 7 6l P 38 52 80 124 124 84 47 21 lO 33[ 333 1 AROSA 21 2 62 A 2.4 43 84 137 129 84 49 24 11 '.~3l 331 0 AROSA 4 4 61 A [5 48 113 152 [21 7_3 41 21 12 334 334 2 ARf)SA 24 7 61 A 28 56 96 126 122 86 48 22 10 3.37 337 0 AROSA 28 6 62 A 32 60 104 122 107 86 53 21 9 337 337 O A.ROSA lq 10 61 A 41 81 118 117 91 70 44 22 12 34'3 34'3 0 ARUSA 30 5 61 A 32 61 89 115 130 97 50 20 10 344 344 0 AR[1SA 9 6 62 A 24 55 115 145 112 80 53 21 8 346 346 0 AROSA 17 5 62 A 31 60 111 135 113 89 58 22 8 '355 '355 2 AROSA 2 5 61 A 42 70 104 133 118 76 43 2'2 L2 356 356 3 AROSA 6 6 62 A 61 65 84 125 110 83 59 24 8 361 361 2 p..., AROSA 26 4 62 P 54 80 85 107 122 94 53 23 11 3~5 365 3 O0 AROSA 28 2 62 P 27 44 102 159 137 82 51 29 16 367 36! '$ -"] AROSA 28 3 61 H 49 52 89 157 133 74 46 23 11 367 367 3 ARUSA 7 3 62 P 41 72 116 144 11B 77 46 22 11 310 371 0 ARO.%A 25 5 61 A 29 75 131 142 115 88 55 22 9 377 377 0 AROSA 20 3 62 A 18 61 148 178 112 70 52 26 12 379 379 0 AROSA 27 2 62 P 43 91 137 138 110 77 46 22 11 388 387 0 AROSA 12 4 62 A 3() 82 136 14.1 108 79 55 26 11 386 38~, 1 AROSA 21 3 62 A 33 78 141 155 121 79 50 26 13 394 1395 0 AROSA 12 4 62 P 38 73 143 153 112 76 54 28 14.394 394 3 AROSA 19 3 ~J2 A 25 82 156 158 125 78 43 24 15 398 398 0 AROSA 19 3 62 P 52 93 124 141 128 80 41 22 13 400 400 0 A refers to AM observations P refers to PM observations

TABLE E-7 (Concluded) Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 5 4 5 6 7 8 9 Observed Solution AROSA 24 3 62 A 25 77 134 140 141 95 50 26 15.0l 401 0 AROSA 21 3 62 P 30 62 IS1 174 123 76 54 28 13 40~ 403! AROSA 2 4 62 P 56 99 I32 138 ll4 76 50 24 ll 404 404 1 AROSA 25 3 62 A 38 87 143 144 118 89 56 25 ll 405 405 1 AROSA 20 3 61 P 33 95 164 157 120 BO 42 19 11 408 409 0 AROSA 25 2 82 A 19 98 179 I&O 114 80 48 23 I2 410 411 0 AROSA 1 4 62 P 37 104 160 145 114 81 50 24 12 413 413 2 AROSA ll 4 62 ~, 49 104 155 134 102 ~9 59 24.11 417 417.2 ARO~A ll 4 62 P 62 85 113 143 134 89 52 26 14 418 418 4 AROSA 18? ~2 P ~8 6:~ 156 174 127 74 4-8 27 14 422 422 3 AROSA 24 2 62 A 41 111 180 166 104 67 49 24 10 427 42~ 0 AROSA 24 2 62 P 41 1ll 183 162 102 71 50 23!0 ~28 429 O AROSA 22 3 61 P B8 104 179 16B 117 79 49 22 10 434 4B4 1 O0 AROSA 17 3 62 ~ 37 91 172 17q 126 75 ~9 28 lb 4.~8 430 3 (DO AROSA 18 '3 62 A 48 106 153 158 129 82 49 28 16 441 441 3 AROSA 26 S 62 A 47 114 172 167 120 80 53 26 12 452 453 AROSA 16 4 62 A 70 130 169 [4~ 95 78 59 26 I0 45B 45 AROSA 26.3 62 P 37 88 176 [86 1i3 82 54.~,0 16 455 45~ A refers to AM observations P refers to PM obserYations

TABLE E-8 INDIVIDUAL SOLUTIONS FOR 98 NORTH A~ERICAN UMKEHRS ON C WAVELENGTHS~ BY TWOMEY'S METHOD, WITH RESPECT TO SI, AND WITH SECOND DERIVATIVE CORRECTIONS APPLIED Vertical Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 3 4 5 6 7 8 9 Observed Solution EOMONION 9 3 59 A 22 106 14-9 166 [43 89 5.5 31 lb 440 438' EDMONTON 22 3 59 P 42 126 138 156 137 86 44 23 LC 443 437 EDMONI'ON 24 5 5'9 a I5 132 164 I68 138 87.57 30 1.3 445 450 EOMONION II 4 59 'P 21 96 125 144 /28 87 52 25 It 385 ~8~ EDMONTON 14 5 ~"~.a 32 55 89 13'3 130 88 49 22 9 352 '3'46 [OMONfO,*,! 5 1 oO A 18 64 96 126 115 80 5t 24 i~ 325.'~29 EOMON'fON 9 8 O0 '~ 34 49 70 111 109 77 45 23 [0 309 '505 EDMONfON 10 8 oO a 2') 37 o6 Ill 112 81 45 '22 9 29'/ 295 EDMO!'.~TON 18 5 6[ P I9 7'7 120 149 133 87 52 26 It 378 319 EDMONTON 19 5 61 P 40 76 92 126 122 85 49 24 10 368 361. p~ EDMONTON 22 5 61 P j9 68 84 122 /,,'1 87 50 22 9 354 348 OD EDMONTON.14 6 6.1 P 35 56 74 114 111 78 43 ZO 8 315 kO 311 EDMONTON 16 6 6t P 42 45 54 101 10'3 7'5 40 18 7 291 z85 EDMONT(]N 7 8 61...~ 29 56 74 112 106 74 38 t7 6 294 295 Et)MONfON 14 9 61 A 29.35 5'9 102 99 68 '39 22 10 270 268 EDMO~,~TON 3 4 62 A 26 7~ Ill.141 126 81 48 27 12 370 367 EUMONTON 2'7 6 o2 A 35 90 99 12'3 1t2 79 49 23 tO 360 357 EDMONfON 1 8 62 A 56 52 06. 106 104 75 41 21 9 301 297 MOOSOr.iCE 19 4 60 P ~4 162 146 144 118 78 48 23 tO 448 444 MOOSONEE It 5 60 P 31 98 [t~ 136 122 8'3 51 2b 11 385 ~83 Mr)OS[JNEE 12 5 O0 e 20 102 12 I 1'3'7 1 t5 7~ 43 21.3 356 362 0 ~,UOSO:"IEE 13 5 GO P 41 [[1 rio 128 114 78 46 23 tO 387 382 0 MOOSO~'IE*: 20 5 &O? 27 6{3 9'_~ 131 124 84 bO 24 11 347 345 0 P;OUS(J'xtEE 14 6 oO P.~2 'll 9Z 127 122 88 50 22 9 354 3!51 0 Mf]OSO~.IEE 21 6 6,.., h 25 16 110 143 132 89 51 Z3 10 375 374 '0 A re~ers to AM obserYations P re~ers to PM obserYations

TABLE E-8 (Continued) Vertical Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness i 2 3 4 5 6 7 8 9 Observed Solution MOOSONEE 20 7 60 P ]$5 67 92 130 128 91 52 23 10 367 362 MOOSOlEE 23 7 60 P 41 65 78 117 1!6 85 47 20 8 S42 336 MOOSONEE 4 8 60 P 35 80 97 [29 123 8B 52 24 lO 371 367 MOOSONEE 23 8 60 P 35 59 75 113 111 81 47 22 9 325 321 MOOSONEE 24 8 60 A 21+ 42 7'3, I14 112 81 48 24 IO 302 301 MOOSONEE 27 8 60 A.BO 57 77 1'1'3 109 80 48 23 10 316 314 MOOSONEE 2!1 3 ol P 29 73 107 143 137 96 56 26 1! 390 386 MOOSON~E 3 5 61 P 48 i23 Il6 I33 i19 80 46 2I g 409 402 MOOSONEE 4 5 61 P 35 114 126 144 127 86 50 22 9 410 408 STERLING 28 -3 62 P 32 62 92 13I 128 g2 55 26 II 365 360 STERLING 25 4 62 A 35 49 78 124 127 94 55 23 9 349.$43 STERLING 27 4 62 A 39 60 7g 122 I25 95 5"r 24 9 359. 352 STERLING 7ll6'2 P 28 25 60 110 115 86 52 25 11 299 295 I —' STERLING B B 6.5 P 42 63 91 136 140 101 57 '25 10 B94 384 0 STERLING 24 6 63 P 35 67 94 134 133 97 56 23 9 377 372 STERLING 25 6 6B P 42 65 79 120 lgB 94 57 23 9 B63 ~55 STERLING 26 6 6'3 P 39 58 78 121 125 94 56 2B 9 B55 348 STERLING 4 7 03 P 33 45 74 I20 124 95 52 22 9 B3B 329 TORONTO 22 2 60 P 47 IB9 138 148 126 81 50 29 13 454 444 TORONTO 9 3 60 A 20 104 142 160 136 84 51 29.IS.4'16 417 TORONTO 11 S 60 P 26 133 153 161 lB4 83 49 26 12 439 439 [ORONTO 15 3 60 P 23 121 ]61 177 155 99 59 29 13 47.3 471 TORONTO g 5 oO A 32 I02 I26 150 136 92 59 28 12 425 420 TORONTO 7 6 60 A 40 94 109 140 lB2 88 49 2S IO 400 394 TORONTO 8 6 60 A 30 75 106 139 lJ'3 94 58 28 12 )90 385 A refers to AM observations P refers to PM observations

TABLE E-8 (Continued) Vertical Distribution of Ozone - Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness "~. 6 7, 8 9 Observed Solution TORONTO 8 6 60 P 32 LOS [20 142 129 91 55 23 9 404 403 TORONTO. o 6 60 A 25 64 96 132 129 97 6'3 25 9 365 364 TORONTO lO 6 60A 31 52 83 125 126 92 56 26 1! 350 )45 TORONTO 20 6 60 A 23 8l t17 147 1)8 9B 61 PT 12 )98 )98 TORONTO 28 6 60 A 41 39 62 113 123 97 58 Z) 9 338 329' TORONTO 20 7 60 P 28 87 109 135 I24 87 54 25 IO 375 374 TORONTO 28 7 60 A 28 57 81 I26 122 87 52 26 I1 344 34I TORONTO 16 6 6! A;)3 67 107 14I 130 89 5'3 25.ll 368 3.6.7 fORONlO 16 6 61 P 37 64 88 131 129 90 48 21 8 360 354 TORON10 l? 6 6I A S7 86 I03 133 I25 87 51 24 10 384 379 TORONTO L 7 6I A 37 3I 55 I06 II4 87 49 2I g 302 296 TORONTO 1 7 6I P 47 3.5..52 105.116 90 49 21'- 9 321 3..1....0... ~-J TORONTO 4 7 6I A 37 80 IOI 133 I25 87 54 26 11 384 377 ~,0 TORONTO 5 7 6I A ~0 B3 103 129 118 B3 52 25 II 364 ~-J 362 TORONTO 5 761P 4g 67 80 I23 125 8B 44 20 B 364 353 TORONTO 6 T 61 A 34 70 gO I24 ll6 84 50 25 II 35'3 34g TORONTO 6? 61 P 43 67 84 I26 I25 88 47 21 g 36I 353 TORONTO 17? 6I A 39 64 84 124 122 89 53 25 II 36I 353 TORONTO 27? 6I A 29 40 72 II7 119 87 52 25 iO 320 316 TORONTO 3 8 6I P 37 54 77 I2I I23 9I 51 22 9 343 3.37 TORONTO 5 8 61 P 4I 48 72 119 I23 88 48 22 9 340 331 TORONTO 17 8 6[ A 25 39 77 ~122 122 87 53 25 1! 323 321 TORONTO 13 8 6I A 32 76 103 136 I28 89 51 24 [0 375 371 TORONTO 14 8 61 P SI 46 80 I24 I27 96 57 25 10 347 342 TORONTO 17 8.6I A SO 36 70 II6 iI8 87 52 25 II 3.19 JI4 A refers to AM observations P refers to P~ observations

TABLE E-8 (Concluded) Vertical. Distribution of Ozone Total Ozone Station Date Mean Ozone Partial Pressures for the Various Layers Cloudiness 1 2 ~~~~~3 45 6789Observed Solution IOCI~~,0:,I ' 1.7 8 61 P j9l 41 C., IlS 1 L9 8'7 47 21 9 323 316 0 '[IJRO 0N'TO t8' L g 6 L.% 23 35 fz4t [161 117 85 55 27 12 314 312 0 'T OR 0;' l' 18 8 L:,I P J 3 3<? 6 7 114 119 91 54 23 10 322 317 0 Tl-R.[ 19 R 6[ /1 f a,5IA 33 103 Ii6 93 57 2"5 10C. 326 314 0 [L.)R 0NTOI. 31 8 oi I' P 4 26 " 3 /01 114 8'7 49 21 8 305 293 0 '[O0R 0NqT G 27 ' I Ok 4 5 48 15 10 C,3 10i-~8 84 51 24 10 321 312 0 TCt '[,10'f.1', 3 9 51. P 40 2:,~' 49 101 108 82 48 '22 9 294 286 0 TORUNT!_! 7 V4ol P 3 0 35 58, 104 104 77 48 22 9 283 281 0 'i OkO~',,~ TO -3 9 L1.'I A 2J 32 59 105 103 ' 7 49 25 11 280 279 0 T COR 0",,l! 9 9 6l1 P 33 2~9 53 102 107 82 51 24 10 290 285 0 TORONTOl 1.6 9 61 A 3,3 39 6 8 114 116 85 50 24 tO 517 311 0 1 QR o,"~i'ru 16 9 6lt: 32 33 6(_,- 115 119 88 53 25 I I 319 313 O ' II R0RN'[T.} 17 9 61: 25 -35 73 122 12j 93 57 2~' 11 328 325 0 l(] R 0:." il- 17 9 6 L" 2.q.34 TI 120 125 93 5 5 26 II 327 323 0 P TURO," 14 Tr.I 18 ')61 A 26 25 6,(I 118 123:92 57 26 l'l 317 313 0 T['iR ON~TOi 18 9 6'1 P 31.5 17 55 115 1Z6 98 57 24 tO 319 31l 0 'fI..)RON T 0 8 6 A.3~ 61 9L 1'33 13-~5 96 52 23 lO0 370 365 0 T{.',RU;'.IT{]}; 27 6 62 A 24 59 94- 131 126 91 57 27 It 354 353 0 I'URrjN'O 28 6 62 ~ 23 5 6 95 133 127 88 5 3 26 1 2 350 349 0 1(' R Or-, T 0 2'9) 6,2 A,.',7 66 'rg. 1[21 1 24 92 54 2 3 9 370 359 0 I'fRONI'i T.1 C 7 02:A I7 40 81 130 128 9 2 5 4 26 1 2 330 331 0 l'OR\ '.il,_ 1,9' 7 02 '.. L[ 3 66 1 9 126 92 52 24 10 340 3.30 0 T ORO0',J'T 2 4 'I 2 A 30- 63: 5 134 12 90 53 '25 1l1 364 360 0 A refers to AM observations P refers to PM observations

BIBLIOGRAPHY Brewer, A. W., H. U. Dutsch, Jo R. Milford, M. Migeotte, Ho K. Paetzold, F. Piscalar, and E. Vigroux, 1960: Distribution verticale de 1vozone atmospherique. Comparison de diverses methodes. Ann. Geophys., 16, 196-2220 Brewer, Ao W., and JO R. Milford, 1960: The Oxford-Kew ozone sonde. Proc. Phys. Soc. London, A 256, 470-495. Chapman, S., 1930: A theory of upper atmospheric ozone. Mem. Roy~ Met. Soc., 35 1035 Craig, R. A., 1948: The Observations and Photochemistry of Atmospheric Ozone and Their Meteorological Significance. D.o Sc Thesis, Mass. Insto Tech. 1950: The observations and photochemistry of atmospheric ozone and their meteorological significance. Met. Monogr., 1 (2), 1-50. Dave, J. V., and P. Furukawa, 1964: The effect of Lambert-type ground reflection on Umkehr measurementso Jo Atmoso Scio, 21, in presso Dobson, GoM.Bo, 1930~ Observations of the amount of ozone in the earth's atmosphere and its relation to other geophysical conditions. Part IV. Proco Roy. Soc., London, A 129, 411-433o _ 1931: A photoelectric spectrophotometer for measuring the amount of atmospheric ozone. Proc. Phys. Soc., London, 439 324-339~ _ 1957a: Observers' handbook for the ozone spectrophotometero Ann I.G.Y., 5 (1), 46-89., 1957b: Adjustment and calibration of ozone spectrophotometero Ann I.Go.Y., 5 (1), 90-114. v 1963: Note on the measurement of ozone in the atmosphere. Quart. J. Roy. Met. Soc., 89, 409-411. and D. N. Harrison, 1926: Measurements of the amount of ozone in the earth s atmosphere and its relation to other geophysical conditions. Part I.o Proc. Roy. Soco London, A 110, 660-693. 193

194 BIBLIOGRAPHY (Continued) D. N. Harrison, and J. Lawrence, 1927: Measurements of the amount of ozone in the earth's atmosphere and its relation to other geophysical conditions. Part II. Proc. Roy. Soc. London, A 114, 521-541., D. N. Harrison, and J. Lawrence, 1929: Measurements of the amount of ozone in the earth's atmosphere and its relation to other geophysical conditions. Part III. Proc. Roy. Soc. London, A 122, 456-486. Dutsch, H. U., 1946: Photochemische Theorie des Atmospharischen Ozons unter Berucksichtigung von Nichtgleichgewichtszustanden und Luftbewegungen. Doctoral dissertation, University of Zurich. 1957: Evaluation of the Umkehr effect by means of a digital electronic computer. Scientific Report No. 1, "Ozone and General Circulation in the Stratosphere." Arosa, Lichtklimatisches Observatorium., 1959 a: Vertical ozone distribution over Arosa. Final Report, "Ozone and General Circulation in the Stratosphere." Arosa, Lichtklimatisches Observatorium., 1959 b: Vertical ozone distribution from Umkehr observations. Arch. Met. Geophys. Biokl. A 11, 240-251., 1963: Vertical ozone distributions over Arosa. Tech. Rep. No. 1, Vertical Ozone Distribution and Stratospheric Circulation. Boulder, National Center for Atmospheric Research. Epstein, E. S., C. Osterberg, and A. Adel, 1956: A new method for the determination of the vertical distribution of ozone from a ground station. J. Met., 13, 319-334. Godson, W. L., 1962: The representation and analysis of vertical distributions of ozone. Quart. J. Roy. Met. Soc., 88, 220-232. Goody, R. M., and W. T. Roach, 1956: Determination of the vertical distribution of ozone from emission spectra. Quart. J. Roy. Met. Soc., 82, 217-221. Gotz, F.W.P., 1931: Zum Strahlungsklima des Spitzbergen Sommers. Gerlands Beitrage zur Geophys., 31, 119-154.

195 BIBLIOGRAPHY (Continued), 1951: Ozone in the atmosphere. Compendium of Meteorology, Amero Met. Soco, Boston, 275-291L Ao R. Meetham, and GoMoB. Dobson, 1934: The vertical distribution of ozone in the atmosphere. Proco Royo Soco London, A 145, 416-446, Grimmer, M., 1963: The space-filtering of monthly surface anomaly data in terms of pattern, using empirical orthogonal functions. Quart. J. Roy. Met. Soc., 89, 395-408. Johnson, Fo S., Jo D. Purcell, R. Tousey, and K. Watanabe, 1952: Direct measurements of the vertical distribution of atmospheric ozone to 70 km altitude. J. Geophys. Res., 57, 157-176. Karandikar, R. V., and K. R. Ramanathan, 1949: Vertical distribution of atmospheric ozone in low latitudes. Proc. Indo Acado Sci.o A 29, 330-348. Kay, R. H., A. W. Brewer, and G.M.B. Dobson, 1954: Some measurements of the vertical distribution of atmospheric ozone by a chemical method to heights of 15 km from aircraft. Scio Proco Into Assoco Met., Rome, 189-193o Kendall, M. Go, 1957: A Course in Multivariate Analysiso London, Charles Griffin and Co., Ltd., 185 pp. Komhyr, W. D., 1956: Unpublished M. Sc. Thesis, University of Alberta. Kulcke, Wo, and Ho Ko Paetzold, 1957: Uber ein Radiosonde zur Bestimmung der vertikalen Ozonverteilung. Ann. d. Meto, 8, 47-53~ Lanczos, C., 1956: Applied Analysis. Englewood Cliffs, Prentice-Hall, 539 pp. Larsen, SoHH.o, 1959: On the scattering of ultraviolet radiation in the atmosphere with the ozone absorption consideredo Geofyso Publ., 21 (4), 1-44. Lawley, D. N., and A. E. Maxwell, 1963: Factor Analysis as a Statistical Methodo London, Butterworths, 117 ppo List, Ro J., 1958: Smithsonian Meteorological Tables. Washington, Smithsonian Institution Pub. NoO 4014, 527 pp.

196 BIBLIOGRAPHY (Continued) Lorenz, E. N., 1956: Empirical orthogonal functions and statistical weather prediction. Scio Rep. No. 1, Statistical Forecasting Project, Dept. of Met., Mass. Inst. Tech., 1959: Prospects for statistical weather forecasting. Final Rep., Statistical Forecasting Project, Dept. of Met., Mass. Inst. Tech. Mateer, C. L.o 1960: A rapid technique for estimating the vertical distribution of ozone from Umkehr observations. Met. Branch, Toronto, Circo No. 3291o Mecke, R.o 1931: Zur Deutung des Ozongehalts der Atmosphareo Z. phys. Chem, Bodenstein Festband, 392-404. Muramatsu, H., 1961: Vertical distributions of ozone estimated from Umkehr observations at Marcus Island, Tateno and Sapporo. J. Aerol. Obs. Tateno, 7, 1-8. Paetzold, Ho K., 1952: Erfassung der vertikalen Ozonverteilung in verschiedenen geographischen Breiten bei Mondfinsternissen. J. Atmos. Terr. Phys., 2, 183-188., 1954: New experimental and theoretical investigations on the atmospheric ozone layero Proco Int. Assoc. Met., Rome, 201-212. Phillips, D. L,: 1962: A technique for the numerical solution of certain integral equations of the first kindo Jo Assoco Comp. Mach., 9, 84-97. Pittock, A. B., 1961: A twilight method of determining the vertical distribution of ozone, Nature, 190, 426-427., 1963: Determination of the vertical distribution of ozone by twilight balloon photometry. J. Geophys. Res., 68, 5143-5155o Ramanathan, K. Ro, V. R. Moorthy, and R. N. Kulkarni, 1952: The effect of secondary scattering in the calculation of the vertical distribution of atmospheric ozone from the Gotz inversion-effect. Quarto J. Roy. Met. Soc., 78, 625-626. Ramanathan, Ko R., and J0 V. Dave, 1957: The calculation of the vertical distribution of ozone by the Gotz Umkehr-effect (Method B). Ann. IoGoYo, 5 (1), 25-45.

197 BIBLIOGRAPHY (Continued) Rawcliffe, R. Do, Go Eo Meloy, Ro M. Friedman, and E0 H. Rogers, 1963: Measurement of vertical distribution of ozone from a polar orbiting satellite. Jo Geophys. Res., 68, 6425-6429. Regener, Eo, and V. H. Regener, 1934- Aufnahme des ultravioletten Sonnenspektrums in der Stratosphare und die verticale Ozonverteilung, Physo Zeit., 35, 788-7935 Regener, V. H., 1960: On a sensitive method for the recording of atmospheric ozone. J. Geophys. Res., 65, 3975-3977. Scherhag, Ro, 1952: Die explosionsartigen Stratospharenerwarmungen des Spatwinters 1951-1952. Berichte des Deutschen Wetterdienstes in der U. SO Zone, 6, 51-635 Schr3er, E., 1949: Theorie der Entstehung, Zersetzung, und Verteilung des atmospharischen Ozons. Berichte des Deutschen Wetterdienstes in der U. SO Zone, No. 11, 13-23. Sekera, Z., and J. V. Dave, 1961: Diffuse transmission of solar ultraviolet radiation in the presence of ozone. Astrophys. J.O 133, 210-227. Singer, SO Fo., and Ro Co Wentworth, 1957: A method for the determination of the vertical ozone distribution from a satellite, Jo Geophyso Reso, 62, 299-308. Taba, H., 1961: Ozone observations and their meteorological applications. Geneva, World Met. Org., Tech. Note. No. 36. Teweles, S., and F. G. Finger, 1958: An abrupt change in stratospheric circulation beginning in mid-January 1958. Mono Wea. Rev., 86, 23a-28 T6nsberg, E., and K. Langlo, 1944: Investigations Qn atmospheric ozone at Nordlysobservatoriet Tromso. Geofys. Publo, 13 (12), 1-39. Twomey, S., 1961: On the deduction of the vertical distribution of ozone by ultraviolet spectral measurements from a satellite. J. Geophys. Reso 66, 2153-2162. 1963: On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. JO Assoc. Comp. Macho, 10, 97.1Ol.

198 BIBLIOGRAPHY (Continued) and H. B. Howell, 1963: A discussion of indirect sounding methods with special reference to the deduction of vertical ozone distribution from light scattering measurements. Mono Wea. Rev., 91, 659-664. U. S. Standard Atmosphere, 1962. Washington, U. S. Government Printing Office, 278 pp. Vassy, A., 1958: Radio-sonde speciale pour la mesure de la repartition verticale de lozone atmospheriqueo J. Sci. Meto 10, 63-75. Venkateswaran, S. V., J. G. Moore, and A. J. Kreuger, 1961: Determination of the vertical distribution of ozone by satellite photometry. J. Geophyso Res., 66, 1751-1771. Vigroux, E., 1953: Contribution a l'etude experimentale de l'absorption de I ozone. Ann. de Phys., 8, 709-762. 1959: Distribution verticale de l'ozone atmospherique d'apres les observations de la bande 9.6 i. C. R. Acad. Sci. Paris, 18, 2622-26240 Walton, Go F o, 1953: The vertical distribution of ozone. Bruxelles, IO U. Go Go General Assembly, Proces-Verbaux Io A o M., 316-322, 1957: The calculation of the vertical distribution of ozone by the Gotz Umkehr-effect (Method A). Ann. IoG.Y., 5 (1), 9-22., 1959: Calculation of the vertical distribution of atmospheric ozone. J. Atmos. Terr. Physo. 16, 1-9. Wilkes, M. V., 1954: A table of Chapman's grazing incidence integral Ch(x,X). Proc. Phys. Soc. London, B 67, 304-308o Wulf, O. R., 1932: A theory of the ozone of the lower atmosphere and its relation to the general problem of atmospheric ozoneo Phys. Rev., 41, 375-376., 1934: Steady states produced by radiation with application to the distribution of atmospheric ozone. Phil Mag.o, 17, 251-263.

199 BIBLIOGRAPHY (Concluded ), and L. S. Deming, 1936 a: The theoretical calculation of the distribution of photo-chemically formed ozone in the atmosphere. Terr. Magn. Atmos. Elect., 41, 299-310. and L. S. Deming, 1936 b: The effect of visible solar radiation on the calculated distribution of atmospheric ozone. Terr. Magn. Atmos. Elect., 41, 375-378. and L. So Deming, 1937: The distribution of atmospheric ozone in equilibrium with solar radiation and the rate of maintenance of the distribution. Terr. Magn. Atmos. Elect., 42, 195-202.

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