£ '3;/fe\ NOAA TR NESS 67 A UNITED STATES DEPARTMENT OF COMMERCE PUBLICATION NOAA Technical Report NESS 67 #* ° f ^ ^r E s o< ' U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration National Environmental Satellite Service Vertical Resolution of Temperature Profiles for High Resolution Infrared Radiation Sounder (HIRS) Y. M. CHEN H. M. WOOLF W. L. SMITH WASHINGTON, D.C. January 1974 *♦ *% ■0 *s* ^ $l ** tfA o. 'Or %ENTS cS>> # V NOAA TECHNICAL REPORTS National Environmental Satellite Service Series The National Environmental Satellite Service (NESS) is responsible for the establishment and operation of the National Operational Meteorological Satellite System and of the environmental satellite systems The three principal offices of NESS are Operations, Systems Engineering, and Research. 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Johnson, Director NOAA Technical Report NESS 67 Vertical Resolution of Temperature Profiles for High Resolution Infrared Radiation Sounder (HIRS) Y. M. Chen H. M. Woolf W. L. Smith O u / ^6-A9l' b o a. 4) a WASHINGTON, D.C. JANUARY 1974 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402. Price $0.55 CONTENTS Abstract 1 1. Introduction 1 2. The Method of Backus and Gilbert 1 3. Application to HIRS 4 4. Discussion 12 Acknowledgements 12 References 14 111 VERTICAL RESOLUTION OF TEMPERATURE PROFILES FOR HIGH RESOLUTION INFRARED RADIATION SOUNDER (HIRS) Y. M. Chen", H. M. Woolf and W. L. Smith National Environmental Satellite Service National Oceanic and Atmospheric Administration Washington, D. C. ABSTRACT. By using the method of Backus and Gilbert, the intrinsic vertical resolution of temperature pro- files obtained from the High Resolution Infrared Radiation Sounder (HIRS) on the Nimbus-F satellite, which contains channels in both 4.3-ym and 15-ym CO2 absorption bands, is evaluated and compared with those of channels in the M-.3-ym band and the 15-um band alone, It is found that the combination of 4.3-ym and 15-ym bands is superior to the other two cases for all levels of the pressure. 1. INTRODUCTION One of the basic problems in atmospheric temperature profile inversion techniques is the intrinsic vertical resolution of the measurement system. Recently, the fundamental technique for obtaining a direct quantitative value in answer to the problem was developed by Backus and Gilbert (1968, 1970) for the analysis of the solid earth. Conrath (1971) has applied it to the problem of the vertical sounding of the atmosphere by means of remote radiation measurements (with random noise) in the 15-ym CO2 absorp- tion band. The purposes of this paper are to evaluate the intrinsic vertical resolu- tion of temperature profiles obtained from the High Resolution Infrared Sounder (HIRS) on the Nimbus-F satellite (which contains channels in both 4.3-ym and 15-ym CO2 absorption bands) and to compare it with those obtained separately in the 4-,3-um and 15-ym bands. First, a brief review of the formulation of Backus and Gilbert is presented. Then for various levels of atmospheric pressure p, the averaging kernel A(x,x'), the center of A(x,x'), and the resolving length of A(x,x T ) are computed and plotted as functions of p for three individual cases, the 4.3-ym band, the 15-ym band and the 4.3-ym + 15-ym bands (HIRS). Finally, we discuss and compare these three cases. 2. THE METHOD OF BACKUS AND GILBERT In the remote sensing of atmospheric temperature profiles, measurements are made of radiances in a finite number of spectral intervals within "Permanent address: Department of Applied Mathematics and Statistics State University of New York, Stony Brook, New York 11790 atmospheric absorption bands. For a nonscattering atmosphere in local thermodynamic equilibrium, the expression for the radiances has the form r i r x( Po) r I(v i ) = B T( Po ),v i T( Po ,v i ) -\ B T(p),v i "| d^^v^ dx(p) , (l) J Jx(o) L -1 dx ^P) i = 1j 2, . . . , m , where B[T(p),v] is the Planck function for temperature T(p) at pressure p; the independent variable x can be any monotonic function of p (x = In p being used here); p Q is the surface pressure; and x(p 9 uj_) is the trans- mittance of the atmosphere above pressure p at u:. The inversion problem then is to estimate the temperature profile T(p), given the atmospheric transmittances x(p,v. ) and measurements of the radiances (l(vi)}, i = 1, 2,..., m. If the contribution from the non- linearity of eq (1) is small, then only the first iteration of the iterative algorithm of Chen (1973) or the procedure of Backus and Gilbert will be sufficient for our purpose. Hence, the contribution from nonlinearity is assumed to be small throughout. Let T (p) = T (p) + 6T(p) , (2) where T (p) is the reference profile and is taken to be the U.S. Standard Atmosphere. Then from Chen (1973), 6T(p) should be the best approximate solution of the linearized equation f x o I kIx.v^ 6T(x) dx = 6I(v 1 ) , i = 1, 2,..., m. (3) where K [x,v. | = d B [t(x), v-] i dx(x,v 1 - ) L 1 -l d T T dx I o $x fi|T (x) > v i l dx(x,v n -) dx - I(vi) (4) dx (5) and dB/dT is the Frechet derivative of b[t], The weighted average of 61 at x which gives heavy emphasis to points close to x and very little to distant points is defined by S x o A(x,x') 6T(x') dx' , (6) _00 where the averaging kernel A is normalized according to \ x o A(x.x') dx 1 = 1 . (7) <6T> is the most localized weighted average if and only if the averaging kernel A closely resembles the Dirac delta function 5(x'-x). The spread of h. from x is defined by Q( x, A ) = a T \ ° J(x,x') A 2 (x,x') dx' , (8) where J(x,x ? ) is a chosen infinitely differentiable function of x' such that J(x,x) = and increases monotonically as x' increases or decreases away from x with dimension of x'^, and a J i: 2 (9) J(x,x') Aj(x,x') dx' with A£(x,x') = l/l , x - */2 and {6I(v i )} J i = l,2,...,m, all de- pend linearly on the function 6T, it follows that <6T> X must depend line- arly on {Sl(v^)}. Therefore, there should exist constants {aj'(x)}, i = 1,2, . . . ,m, depending on the fixed point x such that < 6T >x = £ a i< x > 6i < v i> • ( i3 > 1=1 Hence from (3) and (6), A(x,x') = J ai (x) K|x',Vil . (14) It follows from (8) and (14) that Q(x, A) is a positive-definite quadratic function of {a^(x)}. Since the determination of the most localized weighted average <5T> is equivalent to minimizing the spread of A from x, subject to the constraint (7), by using the method of Lagrange multipliers, the proper set of {a^(x)} satisfies the following set of m+1 linear algebraic equations : in 5=1 1 12 I " (x-xM 2 K|~x f ,v IkTx'.v. dx'l a.(x) + Al ^fx'^ dx'=0, ki!-l*^] i = 1,2,. . . ,m. (15) dx'J> a.(x) = 1 , where A is a Lagrange multiplier. The vertical resolution of the temperature profile at the level x obtain- able from a given set of radiance measurements is determined by the close- ness of A(x,x') to 6(x' - x). Visually, the plot of A (x,x T ) vs. x' gives a qualitative estimate of the vertical resolution at x. To characterize the behavior of A(x,x') more precisely, the "resolving length" of A(x,x') is introduced and defined as the spread about its center, w(x) = 12 \! [ c(x) - x ] c(x) - x'l 2 A 2 (x,x') dx' , (16) where f X o 2 /f X o O c(x) = I x 1 A^(x,x'> dx'/ 1 A 2 (x,x') dx' (17) is the "center" of A(x,x'). 3. APPLICATION TO, HIRS The method of Backus and Gilbert has been applied to an analysis of the vertical resolution of temperature profiles inferred from radiance measure- ments in two CO2 absorption bands observed by the High Resolution Infrared Radiation Sounder (HIRS) , The 7 channels in the 15-um C0 9 band and the 5 channels in the 4.3-um CO2 band in the study are listed in table 1. The dx^/dx(p) vs. p curves for 15-um and 4.3-um C0 ? bands are shown in figures 1 and 2 respectively. Similarly, the dB^/dT |T_.dT£/dx(p) radiative transfer kernels vs. p for 15-um and 4.3-um CO2 bands are shown in figure 3. The averaging kernel A for each of various pressure levels, ranging from 1 mb to 1000 mb, is computed for 3 separate cases: the 5 channels in the 4-,3-um band alone, the 7 channels in the 15-um band alone, and the 12 channels in the combination of 4.3-um and 15-um bands. The corresponding set of three averaging kernels at various levels of p are plotted as functions of atmos- pheric pressure in figures M- to 12. These curves give a qualitative esti- mate of the vertical resolution at the indicated levels and offer a visual comparison for the resolving length and the center of A(x,x') among the Table 1. — HIRS channel spectral characteristics Channel Center C0 2 band no< frequency (cm x ) 15 ym 1 668.5 680.0 690.0 703.0 716.0 733.0 749.0 4.3 ym 11 2190.0 12 2210.0 13 2240.0 14 2270.0 15 2360.0 : 1 \ ' ' ■ 1 1 1 I 15MmC0 2 i 5 - - 10 - 668 5 cm-1 - £ 20 as => - s^ ^v 680 cm-1 - £ 50 - 100 \ ^ - 690 cm ~ 1 200 - V^^V 703 cm-1 N. \ J 716 cm-i - 500 - f 749 cm-'^ ~~^X^^^/^\ 733 cm _1 - 1 .3 .4 _dr dx Figure 1. — The derivative of transmittance with respect to the logarithm of pressure for the 15-\xm CO absorption band. 1 '2 -T— 1 1 1 1 1 1 \ 4.3^i.m CO2 5 10 \\ \ 2360 cm-1 5" 1 20 3£ - D n £ 50 - \ \ \S \. 2270 cm-1 100 - 200 [ \ >v ^v/ 2240 cm-1 500 \yS ^^^^"^ ^\J 2190 cm-i 1 3 4 dT dx Figure 2. — The derivative of transmittance with respect to the logarithm of pressure for the 4.3-\im CO absorption band. TYT '.'J •r. t- -5 20 to 100 200 500 1000 \ J . • \ 2270 n • ■. \ •-v V-y !• A. 733 ' / V- \O 240< 716 •15^mC02 ■43^mC02 668 5 T\ r \ \ \ / 2360 cm-i / / fj8C 690' 703 \ " • • . .>v # . » I .-.** *•". 2210 749 2190- ■■ t r~ — r 7$ •-^^>l - t 1 i i i V I - — - i I i 1 (avg kernel) Figure 5. — Averaging kernels at 10 mb for the 4.3-V-m band, 15-]im band, and 4.3-um + 15-ym bands plotted as functions of atmospheric pressure. 5 - 10 E 20- 50- 100 - 200 - 500 1000 \ '. * T 1 1 1 i - • \ ' ' co 2 - 4 3// m / • N I i v 20 mb - i i i i I ': ) - ■ / - - r •V ' i i i i i (avg kernel Figure 6. — Averaging kernels at 20 mb for the 4.3-um band, 15-\im band, and 4.3-]im + 15-\xm bands plotted as functions of atmospheric pressure. 10 J 20 LLI r> £ 50 O- 100 200 500 1000 Figure 7. — Averaging kernels at 50 mb for the 4.3-\xm band, 15-\xm band, and 4.3-\im + 15-\im bands plotted as functions of atmospheric pressure. 10 E 20 50 100 - 200 - 1 i i i i i i \ i _ ; \ co 2 / '. ' 4.3iifn - ; * 70 mb - v- \ ^k \ • J \ •' ' s-<^ r ^' < At' — /• \ A > . V _^^ _ f7_j ~~ C__ \ ~~} \i--r J i i i i i i 500- _.U 1000 .2 .3 A |av 4 5 kernel) Figure 8. —Averaging kernels at 70 mb for the 4.3-\im band, 15-\ym band, and 4.3-\im + 15-\im bands plotted as functions of atmospheric pressure. 10 1 1 1 1 i i i I 1 2 i CO 2 \ 15/* m 5 \ 4 3/z m \ \ 100 mb 10 — I \ '• \ \\ \ L 20 l ) ) \ v. ! 50 100 ■*> ™ — --— \ '-. . \ 200 - \ : ; .•• . - 500 - .•••• — / nnn %---T } 1 1 1 1 1 1 -.1 .1 2 3 4 5 6 A (avg kernel) Figure 9. — Averaging kernels at 100 mb for the 4.3-]xm band, 15-]sm band, and 4.3-\xm + 15-\im bands plotted as functions of atmospheric pressure. i 1 1 r 2 E 20 50 IOC 200 500 1000 12 3 4 5-6 A lava kernel Figure 10. — Averaging kernels at 200 mb for the 4.3-um band, I5-um band, and 4.3-u/n + 15-\xm bands plotted as functions of atmospheric pressure. 11 1 1 i i i i i i co 2 i 2 4. 3/i m + 1 5 /i m 4 3/i m 5 — 500 mb 10 -' £ 20 3i - -O a so - 100 - \ \ 200 ■**^Sxi 500 ss i 1 " •< • • •_ i *r*~ i i i i i -.2 6 .8 1.0 12 1.4 16 I.I A (avg. kernel) Figure 11. — Averaging kernels at 500 mb for the 4.3-\im band, 15-\im band, and 4.3-\xm + 15-\im bands plotted as functions of atmospheric pressure. 2- E 20 50 - 100 200 500 1000 ' - 1 1 1 1 1 1 1 1 1 1 co 2 1 1 — 4.3/i m 700 mb _ — - 1 f. - - 1 u " ' ,, --*-r..^ t _ ■ 1 1 10 1.2 1.4 1.6 A (avg, kernel) 8 2.0 2.2 2.4 Figure 12. — Averaging kernels at 700 mb for the 4.3-\xm band, 15-\im band, and 4.3-uzn + 15-\im bands plotted as functions of atmospheric pressure. 12 three cases. Finally, the center and the resolving of A(x,x') for the above- mentioned three cases are computed and plotted in figures 13 and 14, respec- tively. 4. DISCUSSION Qualitatively, the curves of figures 4 to 12 indicate that the 12 channels of the combined 4.3-um and 15-um CC>2 bands has better vertical resolution of temperature profiles than that of the 7 channels of the 15-um band, uniformly for all p, which in turn has better vertical resolution than that of the 5 channels of the 4.3-um band. This is substantiated by the center vs. p curves in figure 13 and the resolving length w(p) vs. p curves in figure 14. It is found that the addition of the 5 channels of the 4.3-um band to the 15-um band improves the resolving length of the averaging kernel for the 15-utii band alone about 1 to 15% in the troposphere, 15 to 43% in the stratosphere except in the range of 3.5 to 8.5 mb, and 15% at the tropo- pause. Note that the w(p) vs. p curve for the 15-um band is comparable to that of Conrath (1971). Furthermore, figure 13 shows that in general the cen- ter of the averaging kernel for the combination of 4.3-um and 15-um bands is closer to the level to which the kernel pertained than the other two cases. This is particularly true in the upper stratosphere, for example in the layer of p < 10 mb. Although the resolving length of the 15-um band is shorter than that of the combined 4.3-um and 15-um bands in the range of 3.5 to 8.5 mb , the large deviation of the center of the averaging kernel for the 15-um band from the level to which the kernel pertains does away with any advantage of shorter resolving length. In our opinion, these facts indicate the uniform superiority of the combination of 4.3-um and 15-um bands over the other two cases for all p in the vertical resolution of temperature profiles. Figure 14 indicates that the resolving length for the 4.3-um C0~ channels is poor in the range of 8 to 60 mb. This is mainly because of the lack of weight concentration of K[x,v^]'s in that particular range (figure 4). Unfortunately, the method of Backus and Gilbert will not yield accurate results under this circumstance. To obtain more accurate results, one has to experiment with various J-f unctions other than that of eq (12). It is known that the instrument noise will contaminate the above results (Backus and Gilbert, 1970). The general effect of the noise here is be- lieved to be the same as that in the analysis of Conrath (1971). ACKNOWLEDGEMENTS The authors wish to thank Messrs. C. M. Jacobson, L. P. Mannello and L. D. Hatton for their assistance in the preparation of this paper. One of the authors (YMC) also wishes to express his appreciation to Dr. J. S. Winston for a very pleasant stay (summer, 1972) at the Meteorological Satellite Laboratory, National Environmental Satellite Service, NOAA, where this research was carried out. 13 E 20 - 1000 — 43p. m + 15/im • . 15iim • — 4.3^ m 100 50 20 10 CENTER (mb) Figure 13. — Averaging kernel center for the 4.3-\xm band, 15-)im band, and 4.3-\im + 15-\im bands plotted as functions of atmospheric pressure. 5 10 20 50 100 200 RESOLVING LENGTH (mb) 500 1000 Figure 14. — Resolving lengths for the 4,3-\xm band, 15-\im band, and 4.3-\im + 15-]im bands plotted as functions of atmospheric pressure. 14 REFERENCES Backus, George, E. and Gilbert, J. Freeman, "The Resolving Power of Gross Earth Data," Geophysics Journal Royal Astronomical Society , Vol. 16, pp. 169-205, 1968. Backus, George, E. and Gilbert, J. Freeman, "Uniqueness in the Inversion of Inaccurate Gross Earth Data," Philosophical Transactions Royal Society of London , Vol. A266, pp. 123-192, 1970. Chen, Yung Ming, "Iterative Algorithms for Constructing Approximate Solu- tions of Nonlinear Problems from Inadequate Data," Unpublished Manuscript, 1973. Conrath, Barney J., "Vertical Resolution of Temperature Profiles Obtained from Remote Radiation Measurements," NASA (Goddard Space Flight Center) preprint X-622-71-519, 1971. ■& V. S. GOVERNMENT PRINTING OFFICE : 1 97 1 *--5'»26S3/' 23 (Continued from inside front cover) NESC 51 Application of Meteorological Satellite Data in Analysis and Forecasting. Ralph K. Anderson, Jerome P. Ashman, Fred Bittner, Golden R. Farr, Edward W. Ferguson, Vincent J. Oliver, and Arthur H. Smith, September 1969. Price $1.75 (AD-697-033) Supplement price $0.65 (AD-740- 017) NESC 52 Data Reduction Processes for Spinning Flat-Plate Satellite-Borne Radiometers. Torrence H. MacDonald, July 1970. Price $0.50 (COM-71-00132) NESC 53 Archiving and Climatological Applications of Meteorological Satellite Data. John A. Leese, Arthur L. Booth, and Frederick A. 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