THE UNIVERSITY OF MI CHIGAN COLLEGE OF ENGINEERING Department of Aeronautical and Astronautical Engineering High Altitude Engineering Laboratory Technical Report ERRORS IN ATMOSPHERIC TEMPERATURE STRUCTURE SOLUTIONS FROM REMOTE RADIOMETRIC MEASUREMENTS S: Roland D3rasorn ORA Project 05863 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION CONTRACT NO NASr-54(03 ) WASHINGTON, D C administered through~ OFFICE OF RESEARCH ADMINISTRATION AE ARBOR December 1963

r\ -1 1~P (\\ v

TABLE OF CONTENTS Page LIST OF TABLES v ABSTRACT vii INTRODUCTION 1 METHOD OF CALCULATION 2 LIMITATIONS 8 EIGENVECTOR SOLUTIONS OF TE: MATRIX EQUATION 10 CONCLUSIONS 20 ACKNOWLEDGMENTS 21 REFERENCES 22 APPENDIX~ LEAST SQUARES SOLUTIONS 23 iii

I

LIST OF TABLES Table Page 1. Ci = 6(logeI)/cTi(~C) for Middle Route 3 2. Comparison of Computed with Actual Temperatures (~K) Without and with Systematic Errors. Basic Signal ~3x10-5 Watts/f Steradian 3 3. Inverse of Matrix Formed by First 7 Columns of Table 1 Determinant of Coefficient Matrix = -4o05x10-20 4 4. Summary of Effect of Errors Using Selected Channels 6 5a. Eigenvalues of Matrix A'A 13 5b. Eigenvectors of Matrix A'A 13 6. Maximum Value of IATiI for p = 2,..,7 with I AIi = % 14 7a. Solution of Eq. (6) Using Value of AI Calculated from Eq. (3) with ATi = +1~K, all i 15 7b. AI2-AI1 15 8a. Same as Table 7a with AT300 = +50K, all other ATi = 0 17 8b. AI 2-II 1 17 9ao Solution with AI = +1* for Each Channel 18 9b, Values of AI Obtained by Substituting Values of ATi in Table 9a 18 v

ABSTRACT Calculations have been made which show that the library method of solution of the'Kaplan' experiment yields temperature structure solutions which are dominated by the random errors in measurement and computation of radiation~ A least squares method of solution is developed and modified by consideration of the eigenvalues and eigenvecters of the matrix equation. It exhibits some of the limitations of the experiment and shows that, using radiometric measurements, certain quite different profiles are essentially indistinguishable from one another. vii

INTRODUCTION An earlier report1 stressed the importance of determining errors in the calculation and measurement of the intensity of infrared radiation in the Kaplan experiment. Workers in the field well know that the problem of inverting the radiation equation to deduce the temperature structure of the atmosphere is complicated by the introduction of quite small errors arising from either physical measurements or inaccuracies in calculations. Kaplan2 -showed that the effect of systematic errors was small but stated that random errors introduced much more serious discrepencieso Unfortunately, his paper gave no details, In order to investigate such solutions, the coefficients given by Kaplan were used to make a preliminary analysis on which to base more accurate and detailed calculations.

IMETHOD OF CALCULATION Kaplan's method follows, briefly: The intensity of radiation at frequency v, I(v) was calculated for a number of atmospheric models, which were determined by the temperature at a number of pressure levels in the atmosphere (50, 100, 200, 300, 40o, 700, and 1000 mb). The intensity of radiation Io(v) was determined for some fixed model, and the value of I(v) for a slightly different model is assumed to be expanded in the form I(v) -1(v) = X CiATi + CijATiATj +.(1) whIere i( v)i i i,j where ATi is the difference in temperature from the fixed model at the level i. The coefficients Ci and Cij were determined from calculations of I(v) for various models. In the following analysis we shall consider only the first order coefficients; second order terms should be included for an accurate solution, but the essential characteristics and behavior of solutions can be demonstrated without them. Table 1 (reproduced from Kaplan's paper) gives the coefficients Ci for the "middle route" for each of the nine channels calculated. Table 2 (also reproduced) gives details of the "middle route." Kaplan chose the first seven frequencies (675, 685, 695, 700, 705, 710, and 730 cm-') to make his calculations, and neglected the two remaining channels (745 and 760 cm-l). To find the value of ATi one must invert the matrix formed by the first seven columns of Table 1. The inverted matrix is sh-wn in Table 3. The temperature differences induced by a 1/35 systematic error may be calculated by putting I(v)-Io(v) = 1/300 for each of the 2

TABLE 1 Ci = 6(logeI)/aTi( C) FOR MIDDLE ROUTE v( cm1) 675 685 695 700 705 710 730 745 76o (cgs Units Per Steradian) 1853o 182.4 199.8 227o8 264,7 301.9 356o8 391o6 444.5 Pressure (mb) 50 o0145.0132 00oo68 o0038 o0021.0011.o0005.0002 0 100.0057.0062 o0049 o0030 o 0016.o0009.0004.0001 200 o 0018.0026.oo48 00037.0022.0013.ooo6.0002 o 300.0001 o0003 o0024.0029.0024.0016 0009.o0004 o 400 0 0 o 0010.0030.0042 o0040.0027.0017 ooo0004 700 0 0.0001.0009.0025.0037 o0040.0031.0010 1000 0 0 0 0 0003 0015.oo00.0086.o 0135 TABLE 2 COMPARISON OF COMPUTED WITH ACTUAL TEMPERATURES ( K) WITHOUT AND WITH SYSTEMATIC ERRORS, BASIC SIGNAL -3xlO WATTS/STERADIAN Pressure (mb) 50 100 200 300 400 700 1000 Middle Route 210.0 210,0 220,00 2300 250.0 260~0 270.0 +10-7 wat'ts/ster 210.2 210,5 220.0 230.5 250.2 26003 27001 +10-6 watts/ster 211.7 214o8 219,8 235.1 251 5 26209 270.8 5

TABLE 3 INVERSE OF MATRIX FORMED BY FIRST 7 COLUMNS OF TABLE 1 DETERMINANT OF COEFFICIENT MATRIX = -4.05x10-20 4.91X102 -1.05x103 -1.61xlO2 1l45x103 -9o84xl02 683x1l02 -20 02xl02 2.24X102 -1095x103 4o81x103 -7o91x103 L420x103 -2.73x103 817X102 -4024X103 1.89x104 -2078x104 3o63x104 -1.78x1.04 o 13x104 -3o40x103 1o02x104 -4o54x104 6o60x104 -8.40x104 4o02xi 04 -2o54x104 7o68x103 -1o29x1.04 5072x104 -8o29x104 lox05105 -4o90x104 5 305x104 -9o18x103 7o92x10 -3.51xlO4 5o08x104 -6o41x104 2497x0lo -1 479x104 5o23x103 -1,46xl03 6o45xi03 -9o34x103 1o18x1o4 -5[43xl03 3o23x103 -7o43x102 seven frequencies v in question, This yields AT50 = +0 90K, AT00 = -0K, AT200 = +17K, A = 4~K AT400 = +2.9~K, AT700 = -1o3~K, 100G = +0 7~K The most surprising aspect of the matrix inversion was the magnitude of the ATi for non-systematic errors in radiation. For +1, -1, +1, -1, +1, -1, +1% error respectively for v = 675, 685, 695, 700, 705, 710, 730 cm-1 the ATi were -3650K, +1640~K, -24200K, +31000K, -1470~K, +918~K, -2730~K It should be emphasized that, with ATi of this magnitude, Eqc (1) is not applicable. Nevertheless, it does give some indication that -the temperature errors will be inconveniently large. The reason for the large ATi is easy to see by examining the inverse matrix. The coefficients are large in absolute value, but their algebraic sums, taken by columns, are smallo It is not slurprising that Kaplan failed to get convergent solutions in a number of cases. 4

From a purely physical viewpoint one should not expect the first seven channels to give the best results. Table 1 shows that nearly all the radiation comes from ground level at 760 cm-1, so this channel should certainly be included for an accurate determination of Tl1000 A number of coefficient matrices were inverted, and the systematic and maximum "random" errors were calculated. The results are summarized in Table 4, The determinant of the coefficient matrix provides a rough guide to the best choice of channels. The effect of 1l random noise is still very severe and quite unacceptable, even in the most favorable cases. With values of ATi so large, it is clear that the use of Eqo (1) is not justified, and that any physical result obtained from it will be meaningless. A least squares method can be used for cases in which the number of observations exceeds the number of unknowns, The theory of this method is developed in the Appendix, which should be consulted for details. The method was applied to Eqo (1), using seven temperature levels and observations on nine channels. The maximum values of the IATii were very close to the best result using seven channels and offer very little improvement. In addition, the systematic errors were not markedly different. The number of unknown temperatures was reduced to four, by two different sets of assumptionso Case I T50 T100 T200 = 1/2(T100 + T300) T70 = 1/2(T40oo + T1000) Case II T200 = 1/2(Too0 + T300) T400 = T300 + 20OK T700 = T100O0 - 100K 5

TABLE 4 SUMMARY OF EFFECT OF ERRORS USING SELECTED CHANNELS Omitting Channels Maximum Value of | ATi From Linear Approximation Determinant Centered at With Absolute Value of Error 1% O~K), of Coeffi- Remarks cm-1 cm 1 50 100 200 300 400 700 1000 cient Matrix 745 760 365 1640 2420 3100 1470 918 273 4 05x10-20 Used by Kaplan 705 745 39 147 171 167 50 15 3 4,57xl0-18 700 745 40 135 170 150 40 17 3 3.82xlO8 695 745 55 2620 220 220 55 18 3 1o69x10-18 None 35 133 151 175 56 20 2 --- Least Squares

In Case I, the ATi were also calculated from the four channels 675, 700, 710, 760 cm-1 subject to errors of +1, -1, +1, -1% respectively, giving T100oo = +.90K, AT300 = -8.50K AT400 = +5o2~K, ATloo0 = -1.1~K The matrix (A'A)-lA' was found for both cases. The maximum temperature differences subject to 1% errors: Case I ATlooi = -o 9~KIAT3001 + 60 4K, IAT4001 + 4.80K, ATlo0001 = 1.40K Case II IAT501 = 39~KIATlo001 + 8.20K,/IAT3oo = 3.8oKIAT10001 = 1.20K In Case I the least squares method yields a rather better solution, In fact it is slightly better that the figures indicate, since the IATi will not all achieve their maximum for the same error distribution, In addition, one is much less likely to have all the errors go'the wrong way' over nine channels as compared with a four channel system. One can therefore conclude that the least squares method will yield a useful but limited improvement over a plain solution of linear equations. 7

LIMITATIONS This section will attempt to give a mathematical explanation for the large errors. Physically they arise from the fact that two quite different atmospheric structures may produce an almost equal radiation intensity over the whole of the 151 C02 band under consideration. Consider an integral. equation of the first kind, b h(x) / f(xy)g(y)dy (2) The problem is to invert this equation, solving for g(y) assuming that f and h are given functions. In practice, sets of numerical values will be given for f and h, solving Eq. (2) by reducing it to a set of simultaneous linear equations. Now replace the left-hand side of Eq. (2) by h(x) + c(x), where e(x) is a small error function. We are interested in knowing its effect on the solution g(y). Unfortunately9 a small. E(x) may produce a large change in g(y). Phillips3 gives an examples showing that f f(x,y)sin(my)dy -+ 0 as m + c for any integrable kernel, fo Thus there is a basic instability in Eq. (2); any solution obtained from it may be correct from a mathematic standpoint, but physically meaningless. For this reason a least squares solution will, in general, provide only limited aid. Phillips goes on to develop a method whereby a smoothness constraint placed on the solution eliminates these unwanted oscillations~ The technique has been generalized and modifbed by Iwomey to a form more convenient for numerical solution0 8

A basic difficulty is introduced: how smooth shall the solution be? If an insufficient degree of smoothness is introduced, oscillations will still predominate. On the other hand, too much smoothing will destroy the essential physical characteristics in which we are most interested. The reduction of the number of unknown temperatures in the previous sections operates as a smoothing constraint in that it prevents fluctuation of intermediate temperatures. Approximation by a polynomial of low degree acts in the same way. So far we have made little or no use of known properties of the atmospheric temperature structure. Because we have some idea of its form, it is possible to seek a solution which best approximates (in some sense to be defined) a standard structure. In the next section, we shall again return to the least squares method of solving the matrix equations, examining closely which components contribute most to the large errors.

EIGEN VECTOR SOLUTIONS OF THE MATRIX EQUATION In the appendix, a least squares method is developed to replace the system of equations A(AT) =(A) (3) where A is a n x m matrix, n > m, by the system (A'A) (AT) A(AI) (4) If the rank of A is m, the matrix A'A is non-singular, and Eqo (4) may be written (AT) = (AA)-1 A(AI) (5) We have shown that the application of Eqo (5) produces physically unrealistic solutions, arising from the fact that small errors in (AI) may produce large errors in (AT), so much so that these errors dominate the solution. Let us consider how they arise by looking at the eigen values of the matrix AVAo Suppose these are 19tz2,ooo0km, ordered in such a way that %i > %2 > > XM, with a corresponding orthonormal system of eigenvectors Then (A'A)vi =- ivi i = looo m Suppose hi is large. Then putting AT = vi in Eqo (4) we see that kivi = A:(AI) and that such a AT will produce a comparatively large change in radiation 10

intensity, i.e., a large AT(AI). However, when Xi is small, the opposite will occur; putting AT = vi produces a small change in the radiation intensity. Now any (AT) can be written uniquely in the form m (AT) = 7 ivi and m (A'A) (AT) = i i=l Thus it is now clear that the vi corresponding to small Ni are precisely those components of (AT) which have little effect on the radiation intensity, and which cannot be determined from radiometric measurements. Let us consider the effects of small errors on the right side of Eq. (2). 1 If (A'A) (AT) = vi then (AT) = ivi. The roles are now reversed: for small hi, the addition of a small error in the vi component of A'(AI) will produce a large error in AT, while for large hi the effect of such an error in A'(AI) will be small. To summarize, the vi corresponding to small hi have little effect on the radiation intensity, while at the same time their inclusion in Eq. (3) produces large temperature errorso The obvious solution is to ignore these components. Since (AT) can be written m (AT) E= ivi i-l m p instead of 7 Bivi we write 7 Bivi with p < m. i=l 11

The value of p chosen depends on the values of Xp and on the expected errors in radiation intensity. So (AT) = VI' where V is the m x p matrix of the eigenvector columns vi,...,vp, and P is a p x 1 matrix. Eq. (3) now becomes (AV') (P) = (AI), with V'(D) = (AT) Solving this by the least squares method (A) [(AVT),(Av)] l(AV?)'(AI) or (AT) = V'(D) = V'[(AV')'(AVv)]- (AV')'(AI) = C(AI) say~ (6) This method lends itself well to computer evaluation, After finding the eigenvalues and vectors the program may be written in order these in the manner indicated. The solution C may now be evaluated for all values of p from 1 to m and the results compared (p = m reduces to Eq. (15)). It should be noted that although Eqo (6) gives ATi for i = l,.oo, m we do not get m independent pieces of information, but only p. Dependence was introduced by assuming that (AT) could be written as a linear combination of vlo...vp. It has the implication that the finer details of the atmospheric structure are obscuredo But it is precisely these details that we cannot expect ot obtain using radiometric techniqueso Previously the number of variables was reduced by assuming a relation between ATi in adjacent layers; the eigenvector method may be considered as the optimum way of choosing a relationship between the various ATio The eigenvalues of the matrix AA' are given in Table 5ao It can be seen that %l is approximately 105 times greater than X7, showing the relative small influence of the v7 component of (AT) on the intensity of radiationo Table 5b shows the eigenvectors; it is interesting to note that as 12

TABLE 5a EIGENVALUES OF MATRIX A'A k~ = 5098x10-4 \5 = L 99x10O6 2= 3o 17x10-4 e 1 52x107 \3 = 8091x10-5 7 = 7.61x10-9 4 = 195x10-5 TABLE 5b EIGENVECTORS OF MATRIX A'A v1 v2 V3 v4.. 7 8o576x1o- t =io064x101 i -2555xlO' -2 o 747x1O-1 - 1 773x10-1 2569x10-1 1.224x10-1 4.197xlO-1 -0o 336x1O-1 0o 576x1-o1 2 448xO-1 2 9934x10-1 -6 631x10-1 -4 835x1o-1 2o443xlO1 0o203x10-1 35 431xlO-1 6o049x1o-1 3. 094x1o-1 1137x101'1 5.896xo1o 027x10-1 0o 587xlO-1 3o 961xiO-1 3o 228x10-1 -2~118x10 -O 5o620x10-1 -6.035x10-1 1o 020x10-1 1 881xlo-' 6o258xl0-1 -2.195xlO 01 -5 848x10l -3692x1O1 1.904x1o.6617xlO1'' 2.668x10a 1 4o 161.xLO 1 -5 689x10-L 629x10xO1 171OX10-1 -0.563x10-1 0o627xlO-1 9.365xio-1 -3 034x10- 1o 505x10-1 o -o 648xL0-1 -o 069x10 -1 -o 006x10c~~~~~~5l-,3_0_4xl 1,_5_5., 4

Xi decreases, the number of changes in sign of the vi increases. This is the reason why the "straight" solution gives ATi which are alternatively positive and negative. Calculations of the matrix C were made for all values of p from p = 2 to p = 7 in order to compare the solutions obtained from various sets of assumptions. Table 6 shows the rapid increase with p of the maximum values of ATi with AIit = 1o, obtained by taking the sum of the absolute values of the rows of the matrix CO For p 4 all ATi are less than 40K, while for p = 5 they rise to above 13~Ko TABLE 6 MAXIMUM VALUE OF IATil FOR p = 2,.o,7 WITH:AItl = 1% p= 2 P= 3 p= 4 P= 5 p=6 7 AT50o.8K 1.O0K 19~0K 1080K 160K 35~K AT100 o4 ~5 1l4 5o6 40 133 AT200.2 io0 3o5 5o9 11 151 AT300 o 2 o 1 20O 4o3 35 175 AT400 o3 1o8 200 11O7 29 56 AT700 ~3 1o3 3~7 13o1 18 20 ATlo00 1.o1. I 14 i-o4 1o7 2 2 It is important to know how well a given temperature structure can be reproduced. The values of AI were calculated for each of the nine frequencies on the assumption that ATi = +1~K for every io Equation (6) was solved using these AI, giving the results in Table 7a; p = 2 gives a poor approximation, expecially for the middle of the atmosphereo However, p = 3 is much improved; 1h

TABLE 7a SOLUTION OF EQ. (6) USING VALUES OF AI CALCULATED FROM EQo (3) WITH ATi = +10K ALL i p = 2 p =3 p =4 p =5 p =6 p 7 AT50 1.44K 112K 1.050K 101K 103012K 100KloO~K AT100 o74 o81 o87 93 88 1o 00 AT200 48 o92 1,08 1o 14 1.14 lo O AT300 27 77 o86 o82 o 85 1.o 00 AT400 o44 1o 24 11i8 1O07 10o4 1o00 AT700 o 47 80 o85 =97 o99 1o00 AT1000 1 01 o 97 1 01 oO00 100 1o00 TABLE 7b v p = 2 P = 3 p = 4 P = 5 p = 6 p = 7 I 675,+~ 39t% +o5 +Ol+.00% 0OO% 00% 2o 21% 685 +26 +o01.00 o00 000 00o 2~23 695 -.31 - o8 -o 1 OO - O1 o00 2 00 700 -.o6 +001 000 000 o00 1o73 705 -o 61 -o 03 01.0 O0.00 00 1 53 710 -58 03 00o 00 OC.00.00 1 41 730 -o44 - 06 -o 02 0o 0 o 00.00 1o46 745 -o28 -o6 -o01 o00.00 o00 1043 760 -o6 -.05 +o 0 oOo.00 o00 1o49 *AI1 calculated from ATi = 1 OK all i AI2 calculated from ATi in Table 7a

this improvement continues gradually for p - 4, 5, 6, giving successively better approximationso If we calculate the difference between the AI for these temperatures and the AI for AT - +1~K all i, we find that even for P = 3 this difference is less than 01% in absolute value and does not exceed o0.02% for p 4 (Table 3b) both of these being much less than the expected experimental error. Thus the experiment cannot distinguish between any of the solution profiles for p > 3The limitations of the technique are ill.ustrated by performinrg the same computations with AT3CO =- +5~K and all ot.her ANTi = OKo Even for p = 6 the errors in ATi are almost as h.igh at 2~0K, wbi..l.e for p 4, AT300 is 1.37~K instead of 5~K (Tabl.e 8a)o however. the difference in radiation intensity is less than 0l* for each of the nin-le charnnels for p =4 and rather less for P = 5 and 6o It is precisely these sharp-peakled variations that the technique is not able to resolve, instead we obtain. a smocthed solution, the degree of smoothing depending onr how'low a value of p is selected~ Lastly, Al = +1% over all nine chanrnels was used to obtain values of ATio The result was somewhat surprising~ For values of p from p = 3 to p = 6, for each pres-sre level the,ralal. of'AL differed by less than Ool0K, but differed greatly from the solution wiith p - 7 (Table 9aYo Again, the difference in radiation intensity is less than ~0G5 f:or 3 < p < 7, showing the essentially indistinguishable natu.re cf the various solutions. We should like to be able to cor.clu.de wh i.rCh value o:f p gives the best results. However, this is not a question to wh i.ch an a'bsolute answer can be given for it depends on what is meant by a "''best'v solutiono If the expected error lies arou-.na -~110.o thren Tabl.e 2 sshows th..at p > 5 must be eliminatedo Likewise, p F2 irrmst be discarded on +t; Se grounds that it does not give enough detail in the middle atmrosp~hereo From the l.im.ited calculations made, it

TABLE 8a SAME AS TABLE 7a WITH AT300 = +5~K, ALL OTHER ATi = 0 p = 2 p = 3 p== 4 p 6 7 AT50.41~K -.10~K -.54~K - o350K.370K OOK AT100.20.32.71.40 -1.45 0 AT200.12.81 1o79 1.46 1.78 0 AT300 -.07.85 1.37 1,.60 3518 5.00 AT400.11 1o35.99 161 ~.57 0 AT700.11.82.01 - 65 -.17 0 AT1000.31 -1.11.0,5.02.00 0 TABLE 8b A 2-AI l* v p = 2 P = 3 P= 4 p= 5 p = 6 P = 7 AI1 675 +,68% +.14% -.09% -.05% +,oi%.00%. 685 +.55 +.15 +.o08 +0,o6.oo.00.15 695 -o74 -.38 +,07.00 +o01.00 1.20 700 -1.13 -.37 -.08 -.03 -.01.o00 145 705 -.96 -.05 -o6 +001.00.00 1.20 710 -.62 +.14 +o05 +o05 +.02.00 o80 730 -.17 -.24 +o03 +.02 -.01.00.45 745 +.13 -.62 +,02 -.02.00.00.20 760 + A43 -1i36 -.03 +.03 +,01.00.00 *AI1 calculated from AT300 = +50K, all other ATi = 0 AI2 calculated from ATi in Table 8a 17

TABLE 9a SOLUTION WITH AI = +1i FOR EACH CHANNEL p=2 p= 3 p =4 P = 5 p 6 p= 7 AT50.69 ~c.44 ~ o 45 ~C.47 ~C 47~ c.81~C ATlo0.36 41.41 o 37.o 37 o 95 T200 o 23.59.o 57 o 53 o 53 2 16 AT300.15.55.54.56,56 -1o07 AT400.27.o 90.o 91 98.98 1.50 AT700 0.73.75.67.67.53 AT1000.95.66.66.66.66.66 TABLE 9b VALUES OF AI OBTAINED BY SUBSTITUTING VALUES OF ATi IN TABLE 9a v p=2 p=3 p=4 = p=6 p=7 675 cm-1 1o25%.98%.99%.99%.99% 1.01% 685 1 20 1o00 1o01 loO0 1oO0 1oO1 695.82 1.01 1o03.99.99 1.02 700.61 1.00 1.00 1.00 1.00 1,01 705 51 1.00 o 00 1o 03 1.03 1 02 710 o 52.98.97 o 98.98 1.01 730.79 1.o 02 o 03 1o 02 102 1o 02 745.98.99 1.o 00 o 99.99 o99 760 1o 3 100 10o 0 1.o 00 100 1.00 18

would appear that p - 4 is the most suitable choice, giving a significantly better result than p = 3 in Tables 4a and 4b. The only way to come to a valid decision is to make a statistical analysis based on real atmospheric temperature soundings. The model chosen by Kaplan does not sufficiently approximate any part of the earthts atmosphere to justify such an analysis. The method outlined has a number of advantagesa. It used a library method, making calculation of results comparatively simple. The bulk of the calculations can be made in advance, so that even when large amounts of data are obtained interpretation can be made at once. b. Known physical characteristics of the atmosphere are used in taking the basic model to be that of a standard atmosphere, the choice depending on the latitude and season of the observation. c. The essential limitations of the technique are recognized, and precisely those features which we cannot hope to measure and which contribute so greatly to unwanted oscillations in the solution are eliminated. d. It lends itself to a statistical analysis based on the deviation of temperature structures from the standard atmosphereso 19

CONCLUSIONS The method described appears to be promising enough to justify further development. A series of calculations will shortly be made with the following features: 1. New and more accurate transmission functionso 20 The SIRS channel frequencies and their triangular response functions will be used to calculate radiation intensities. 3. A flexible program to allow radiation from any temperature structure to be readily calculatedo 4. A statistical analysis leading to a choice of p. 5. A detailed analysis of the effect of intensity errorso 6. Influence of cloudy or partly cloudy conditions on solutions. 20

ACKNOWLEDGMENTS I am very grateful to C. L. Mateer, Department of Meteorology and Oceanography, The University of Michigan, who developed the eigenvector technique for atmospheric ozone soundings~ In addition, I should like to thank members of the project for useful discussions and for checking calculations~ 21

REFERENCES 1. Drayson, S. R., A survey of progress and problems in the'Kaplan' Experiment. The University of Michigan, Technical Report No. 05863-2-T, July 1963. 2. Kaplan, L. D., The spectroscope as a tool for atmospheric sounding by satellites, Instr. Soc. Am., Conf. Preprint No. 9-NY 60. Published in J. Quart. Spectry. Radiative Transfer, 1, 98 (1961). 3. Phillips, D. L., A technique for the numerical solution of certain integral equations of the first kind, Jo Assn. for Comput. Mach., 9, 84 (1962). 4. Twomey, S., On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadratureJo Assn. for Comput. Mach. 10, 97 (1963). 22

APPENDIX LEAST SQUARES SOLUTIONS Suppose we have n equations in m unknowns fi((xlx2. oogXm ) - Pi = 0 i = 1,2 oo n (Al) with n > m. In general, these will be mutually inconsistent and have no solution. However, we can look for a solution (xl,x2,ooo,m) for which the sum of the squares of the left-hand side of Eqo (Al) is a minimum, i eo n 2 F(x) = ) [fi(xiooo xrm) - Pi] is a minimumo A necessary, but not normally sufficient condition, is F 0 j = o m xj ioeo [fi(X oooxm) - = i]J = l,0, m (A2) i=l m f (xlJ,00sxm) = ik Xk i l,...,n k=L Then Eqo (A2) reduces to n m Cij (Caih Xh = Bi) = 0 g = l,, m (A3) 23

This may conveniently be written in matrix form A'A(x) = A'(P) (A4) where A = (oi), x = (xl,...,xm)t' () = (ly,. n)', the' denoting matrix transposition. Equation (A4) will yield a unique solution (x) provided that the matrix AtA is nonsingularo' It will be assumed that rank A = m. If this is not m the case, there exist less than m linearly independent forms C ij xj and i=l m may be reduced to m-l1 Because rank A = m, there exist nonsingular matrices P, Q such that P is n x n matrix, Q is m x m, Im is the m x m identity matrix, and 0 is the (n-m)x(n-m) null matrix. Then AA = Q' (ImO)P p (I)Q Since P is nonsingular, P'P belongs to a positive definite quadratic form, which implies that every principal minor of P'P is positive. Hence the m x m matrix B in the top left corner of P'P is nonsingularo But (ImO)(PP)(m) = B, and therefore A'A = Q BQ is nonsingular, being the product of nonsingular matrices. Equation (A4) may now be rewritten 241

(x) = (A'A)-1 A' () (A5) which is in a form convenient for computer evaluation. In the special case where m = n, Eq. (A5) reduces to (x) = A-' (P), which is the original equation.

UNIVERSITY OF MICHIGAN 3 90 15 02653 5776 3II 9015 02653 5776