SSS<7 NOAA TR NESS 59 A UNITED STATES DEPARTMENT OF COMMERCE PUBLICATION •X NOAA Technical Report NESS 59 U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration National Environmental Satellite Service Temperature Soundin From Satellites S. Fritz D. Q. Wark H. E. Fleming W. L. Smith H. Jacobowitz D. T. Hilleary J. C. Alishouse WASHINGTON, D.C. July 1972 NOAA TECHNICAL REPORTS National Environmental Satellite Service Series The National Environmental Satellite Service (NESS) is responsible for the estab- lishment and operation of the National Operational Meteorological Satellite System and of the environmental satellite systems of NOAA. The three principal Offices of NESS are Operations, Systems Engineering, and Research. The NOAA Technical Report NESS series is used by these Offices to facilitate early distribution of research results, data handling procedures, systems analyses, and other information of interest to NOAA organizations. Publication of a Report in NOAA Technical Report NESS series will not preclude later publication in an expanded or modified form in scientific journals. NESS series of NOAA Technical Reports is a continuation of, and retains the consecutive numbering sequence of, the former series, ESSA Technical Report National Environmental Satellite Center (NESC) , and of the earlier series, Weather Bureau Meteorological Satellite Laboratory (MSL) Report. Reports 1 to 37 are listed in publication NESC 56 of this series. Reports 1 to 50 in the series are available from the National Technical Information Service, U.S. Department of Commerce, Sills Bldg., 5285 Port Royal Road, Springfield, Va. 22151. Price: $3.00 paper copy; $0.95 microfiche. Order by accession number, when given, at end of each entry. Beginning with 51, Reports are available through the Super- intendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. ESSA Technical Reports NESC 38. Angular Distribution of Solar Radiation Reflected from Clouds as Determined from TIROS IV Radiometer Measurements, I. Ruff, R. Koffler, S. Fritz, J. S. Winston, and P. K. Rao, March 1967. (PB 174 729) NESC 39. Motions in the Upper Troposphere as Revealed by Satellite Observed Cirrus Formation, H. McClure Johnson, October 1966. (PB 173 996) NESC 40. Cloud Measurements Using Aircraft Time-Lapse Photography, L. F. Whitney, Jr., and E. Paul McClain, April 1967. (PB 174 728) NESC 41. The SINAP Problem: Present Status and Future Prospects. Proceedings of a Conference held at the National Environmental Satellite Center, Suitland, Md., January 18-20, 1967, E. Paul McClain, Reporter, October 1967. (PB 176 570) NESC 42. Operational Processing of Low Resolution Infrared (LRIR) Data from ESSA Satellites, Louis Rubin, February 1968. (PB 178 123) NESC 43. Atlas of World Maps of Long-Wave Radiation and Albedo -- For Seasons and Months Based on Measurements from TIROS IV and TIROS VII, J. S. Winston and V. Ray Taylor, September 1967. (PB 176 569) NESC 44. Processing and Display Experiments Using Digitized ATS-1 Spin Scan Camera Data, M. B. Whitney, R. C. Doolittle, and B. Goddard, April 1968. (PB 178 424) NESC 45. The Nature of Intermediate-Scale Cloud Spirals, Linwood F. Whitney, Jr., and Leroy D. Herman, May 1968. (AD-673 681) NESC 46. Monthly and Seasonal Mean Global Charts of Brightness From ESSA 3 and ESSA 5 Digitized Pictures, February 1967-February 1968, V. Ray Taylor and Jay S. Winston, November 1968, (PB 180 717) NESC 47. A Polynomial Representation of Carbon Dioxide and Water Vapor Transmission, William L. Smith, February 1969. (PB-183 296) NESC 48. Statistical Estimation of the Atmosphere's Geopotential Height Distribution From Satellite Radiation Measurements, William L. Smith, February 1969. (PB 183 297) NESC 49. Synoptic/Dynamic Diagnosis of a Developing Low-Level Cyclone and Its Satellite- Viewed Cloud Patterns, Harold J. Brodrick and E. Paul McClain, May 1969. (PB 184 612) NESC 50. Estimating Maximum Wind Speed of Tropical Storms from High Resolution Infrared Data, L. F. Hubert, A. Timchalk, and S. Fritz, May 1969. (PB 184 611) NESC 51. Application of Meteorological Satellite Data in Analysis and Forecasting, R. K. Anderson, J. P. Ashman, F. Bittner, G. R. Farr, E. W. Ferguson, V. J. Oliver, and A. H. Smith, September 1969. (AD-697 033) NESC 52. Data Reduction Processes for Spinning Flat-Plate Satellite-Borne Radiometers, Torrence H. MacDonald, July 1970. NESC 53. Archiving and Climatological Applications of Meteorological Satellite Data, John A. Leese, Arthur L. Booth, and Frederick A. Godshall, July 1970. (COM-71-00076) NESC 54. Estimating Cloud Amount and Height From Satellite Infrared Radiation Data, P. Krishna Rao, July 1970. (PB-194 685) NESC 56. Time Longitude Sections of Tropical Cloudiness (December 1966-November 1967), J. M. Wallace, July 1970. NOAA Technical Reports NESS 55. The Use of Satellite-Observed Cloud Patterns in Northern Hemisphere 500-mb Numerical Analysis, Roland E. Nagle and Christopher M. Hayden. April 1971. (Continued inside back cover) ^p ATMOSfl^ 'Ment of U.S. DEPARTMENT OF COMMERCE Peter G. Peterson, Secretary NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION Robert M. White, Administrator NATIONAL ENVIRONMENTAL SATELLITE SERVICE David S. Johnson, Director 1 NOAA Technical Report NESS 59 Temperature Sounding From Satellites S. Fritz D. Q. Wark H. E. Fleming W. L. Smith H. Jacobowitz D. T. Hilleary J. C. Alishouse •a o • o i/5 WASHINGTON, D.C. JULY 1972 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C, 20402. Price 55 cents. UDC 551.501.724:551.507.362.2 551.5 Meteorology .501 Methods of observation and computation .724 Upper air temperature computation .507.362.2 Satellites li CONTENTS List of acronyms iv Abstract 1 1. Introduction 1 2. Theoretical background for temperature sensing through a cloudless atmosphere 3 A. General considerations 3 B. Spectral regions 7 C. Gas transmittances 9 3. Inversion procedures - cloudless conditions n A. Modifications to equation (16) 13 B. Regression method 15 C. Inverse matrix methods 17 Full statistics 17 Minimum information 13 D. Direct iteration retrieval methods 20 E. Conclusions 23 4. Clouds and aerosols 24 A. The cloud problem 25 Single field of view approaches 27 Multiple field of view approach 28 B. The effects of aerosols . 3Q 5. Operational experience - error analyses 30 A. SIRS A experience - regression method 30 B. SIRS B experience - "minimum information" method .... 31 C. Error analyses 32 SIRS A 32 SIRS B 33 6. Sounding instruments 33 A. Infrared 33 B. Microwave radiometers 33 C. Further discussion 39 7. Some remaining problems 39 A. Atmospheric transmittance 39 B. Clouds and particles 40 C. Retrieval methods 41 iii D. Use of other auxiliary information 42 E. Error analysis 42 F. Sounding instruments 43 References 44 LIST OF ACRONYMS IRIS - Infrared Interferometer Spectrometer ITOS - Improved TIROS Operation Satellite ITPR - Infrared Temperature Profile Radiometer NASA - National Aeronautics and Space Administration NESS - National Environmental Satellite Service, NOAA NMC - National Meteorological Center NOAA - National Oceanic and Atmospheric Administration SCR - Selective Chopper Radiometer SIRS - Satellite Infrared Spectrometer VTPR - Vertical Temperature Profile Radiometer IV TEMPERATURE SOUNDING FROM SATELLITES Compiled by Sigmund Fritz National Environmental Satellite Service National Oceanic and Atmospheric Administration Washington, D.C. ABSTRACT. Vertical temperature sounding of the atmosphere can be calculated from satellite radi- ance measurements. The theory relating satellite radiance measurements to temperatures in the atmosphere for both cloudless and cloudy condi- tions is presented. The accuracy of vertical temperature soundings from satellites in compari- son with radiosonde measurements is judged on the basis of two years of operational experience. RMS temperature "errors" varied from 1.1K to 3K for "clear" conditions, and from 1.6K to 3.5K for conditions during which high clouds were judged to be present. Sounding instruments already flown on Nimbus satellites and instruments to be flown in the near future are reviewed. Remaining problems which require further work to improve the accuracy of remote vertical temperature soundings are discussed. 1 . INTRODUCTION Quantitative measurements of temperature at various pressure levels in the atmosphere are among the fundamental observations required in present- day weather forecasting. Until recently, such free-air temperature measurements were obtained almost entirely from balloons and rockets. Since these were available mainly from populated land areas and a few ships, large gaps existed in the observational network. The advent of satellites offered the possibility of obtaining a truly world-wide distribution of temperature soundings. A method for doing that was suggested by Kaplan (1959). He showed that vertical temperature sound- ings of the atmosphere could be inferred from satellite spectral radiance measurements in the 15|jm band of CO2. On the basis of Kaplan's suggestion, two instruments were developed to sound the atmosphere from satellites. These were the SIRS (Satellite Infrared Spectrometer) and the IRIS (Infrared Interferometer Spectrometer). However, it was not until these instruments were launched on Nimbus 3 in April 1969, that the accuracy of vertical temperature profiles retrieved in operational practice could be determined. Operational experience with SIRS indicates that much can be done to retrieve vertical temperature profiles for use in operational weather fore- casting. However, many problems remain. The transmit tances of the atmosphere, not only in the CO2 band but also in the water vapor and ozone bands, were brought into question. The absolute value of the calibration of the satellite instruments has some uncertainty amounting to perhaps 1 or 2 percent. The influence of clouds varies, rang- ing from annoyance to complete obstacle depending on the amount, opacity, and height of the clouds. It became apparent also that the departure of the initial estimate of the temperature profile, or of the so-called "first guess", from the "true" temperatures affects the final result; and this influence depends somewhat on the method of temperature retrieval. This report summarizes operational experience in the United States, although occasional reference is made to work in other countries. As shown in the list of contents, the report begins with the theoretical and techni- cal basis for temperature soundings from satellites. The theoretical basis for temperature soundings in a cloudless atmosphere is presented in Sections 2 and 3. The troublesome problem of removing the influence of clouds and an assessment of the role of aerosols is given in Section 4. The theoretical basis is followed by a summary of existing experience. Operational experience with SIRS, including an error analysis, is summar- ized in Section 5. However this does not contain summaries of all the research experience obtained from SIRS and IRIS, some of which had its impact on the operational procedures. Discussions about the instruments used so far, and a word about the instruments which may be used in the near future, are included in Section 6. Finally, in Section 7 some problems which remain before further improvement in the accuracy of temperature retrieval can be achieved are discussed. The main part of each section was prepared by the following people of the National Environmental Satellite Service, NOAA, although there was inter- action with many individuals. S. Fritz was the convener, or editor. Section 1. - Introduction (S. Fritz) Section 2. - Theoretical Background (D.Q. Wark) Section 3. - Inversion Procedures (H.E. Fleming) Section 4. - Clouds (W.L. Smith) and Aerosols (H. Jacobowitz) Section 5. - Operational Experience (S. Fritz) Section 6. - Sounding Instruments (D.T. Hilleary-IR; J.C. Alishouse -Microwave) Section 7. - Some Remaining Problems (mainly Fritz, Fleming, and Smith) Additional suggestions were received by mail from several scientists; some of these suggestions have been incorporated into the text. Mailed suggestions were received from M.T. Chahine, formerly of the Jet Propulsion Laboratory, Pasadena, California, now at American University of Beirut; R. Fraser, and V.G. Kunde, both of Goddard Space Flight Center, NASA; L.L. Stowe, Jr., University of California, Los Angeles. Useful comments were also supplied by C.M. Hayden, A.W. Johnson, D.S. Johnson, and H.W. Yates, all of NESS, who between them reviewed the whole manuscript. 2. THEORETICAL BACKGROUND FOR TEMPERATURE SENSING THROUGH A CLOUDLESS ATMOSPHERE The simplest case to consider is the temperature retrieval in a cloudless atmosphere resting on a blackbody surface. But even in this case atmos- pheric composition and transmittance must be known accurately for certain methods of temperature retrieval from satellite measured radiances. We generally assume the C02 is mixed well enough in the atmosphere so that its composition can be taken as invariant with time and place; this is obvi- ously an approximation, but probably does not introduce serious errors on a global scale. However amounts of other gases, such as water vapor, are highly variable and modify the atmospheric transmittance, expecially through the lower atmosphere, even in the CO2 emission bands. The atmosphere is a mixture of gases; the transmittance of its varying combinations must be approximated. The satellite instrument spectrally measures the emission from this com- bination of gases. However, no instrument has a pure, monochromatic trans- mission, so the spectral characteristics of each instrument must be measured in the laboratory. Since the measured radiances depend on the air temperature, on the spectral transmittance of the atmosphere, and on the instrumental characteristics, it is necessary to consider the relationships between these quantities. From these relationships, we may be able to derive the atmospheric temperature profile from the measured radiances and a knowledge of other quantities by methods which are not based entirely on empirical regression. A. General Considerations The response of a satellite -borne instrument will depend on the radiance of the earth within the field of view. The response, whether linear or not, can be determined from a calibration procedure, using a uniform source of known radiance. The earth's radiance is generally not uniform (mainly be- cause of variable cloudiness), but the response of the instrument will nevertheless be a measure of the radiative flux at the detector. The equiv- alent radiance of the calibration source is the quantity frequently referred to as the earth's radiance. The physical basis for sounding the cloudless atmosphere is discussed in this section. The problems introduced by variable cloudiness will be treated in Section 4. The observed "radiance" in a narrow spectral interval centered at v can be expressed as J tt/2 2 fT KvJ - J J [Kv,,cp,co') R(v 4 ,cd,cp')] dcpdcp' + E( v .), (1) where I « radiance, cp, ©' m spherical space coordinates (azimuth and elevation, respectively), R ■ instantaneous and normalized field response of the instrument, and E ■> error of measurement. The radiance is an ever-changing quantity as the satellite progresses and scans from side to side. Variations arise from clouds, different atmos- pheric and surface temperatures, random errors of measurement, and changes in concentrations of gaseous and particulate constituents. Consider an idealized situation in which there are no clouds in the field- of-view and temperatures are uniform over the horizontal field. Then the measured radiance is, aside from errors of measurement, proportional to I(v>cpjCp'). In this degenerate condition the equation of radiative transfer may be approached. In an idealized atmosphere which is plane-parallel, is free of scattering agents, and is in local thermodynamic equilibrium, dKv.coV) = [-I(v,co,cc') +B[ v ,T(s)] j k( v ,s) p (s)ds , (2) where B = Planck radiance, s = geometrical path, k = mass absorption coefficient, p = density of absorbing gas, v = spectral frequency, and T = temperature. By means of the hydrostatic equation, the variable "s" can be transformed to pressure so that pds= . £ sece dp . the path of the beam is at angle e> g from the local vertical, "p" is atmospheric pressure, "q n is the mass frac- tion of the absorbing gas, and "g" is the acceleration of gravity. Eq. (2) now becomes dl(v,e) = [+Kv, 9) - B[ v ,T(p)] j k( v ,p) iM sace dp. (3) o If the surface of the earth is assumed to be black, (3) can be written in the integral form .A Kv>e) = B[ v ,T(p o )] exp[- |^ kCv.p^qCp 1 ) sece dp'] o Po E + I C B[ v ,T(p)] exp[- |^ kCv.p^qCp 1 ) sece dp'] (A) x k( v ,p)q(p) sece dp , where the subscript refers to the lower boundary (surface). This form of the equation shows the two components of the radiances: that arising from the surface and attenuated by the atmosphere, and that arising from the atmosphere. It may be noted that a "black" cloud in the field affects only the pressure of the boundary level. At this point one must consider the instrument's field-of-view. If R(cd, cp 1 ) in (1) is zero outside some small angle from the optical axis, then sec0 varies only slightly over the field and its value at the center of the field may be used to represent the entire field. That is, integration over 6 is not, in practice, necessary. It is convenient to simplify the notation by introducing the fractional transmittance of the beam between level p and the effective top of the atmosphere P t(v>P> ft) = exp[- I V k( v ,p')q(p') sece dp']. (5) o Then (4) becomes Kv,e) = B[ v ,T(pJ] T(v,P n , 9) s t(v,p , e) B[v,T(p)] d T (v,P,e). (6) 1 Equation (6) expresses the radiance for a single spectral frequency. The finite width of the spectral interval being observed requires an appropriate integration over the response of the instrument. Considering the case of a filter-type radiometer, the measurable radiance for the j tn filter is r«- Kvj,e) = J Kv, 9) tjCv) d v B[v,T(p o )] t( v ,P , G) t ( v ) d v r°^ T( v ,P rt ,e) JJ B[ v ,T(p)] t ( v ) d T (v,p, 0) d v 1 (7) ° 1 J ) -f-J t.( v ) d v o In eq (7), t(v) is the filter transmittance . A similar integral can be written for a spectrometer. Equation (7) is hopelessly complicated if one wants to retrieve B(v,T). However, as pointed out by Elsasser (1938) if the spectral interval over which tj(v) is nonzero is small, then B( V ,T) varies little and is nearly linear in the interval. It can, therefore, be replaced by its value B(v4,T) at a properly defined mean frequency v<» and factored out of the integral with respect to v after the order of integration is interchanged. It becomes necessary to define the mean transmittance and its derivative by \ T(v,p, 0) t,(v) d v t(vj,P, 6) = ^ tjCv) d v (8a) and o oO 5 ( dT /dp)(»»p, e) tj( v ) d v tr (v i' p ' e) = c^ dp \ t ( v ) d v (8b) In the instruments discussed in this technical note, the spectral inter- vals are narrow (about 10 cm"*) and the filter functions ti( v ) are nearly symmetrical about a vertical line through the centroid. Therefore, one may set vi equal to the mean frequency at the centroid of ti(v) and use (8a) and (8b) to rewrite (6) in the form KvjS) = B[ Vj ,T(p o )] T( Vj ,P o ,e) -S >t(vj,p > e) B[ Vj ,T(p)] dT( Vj ,P,0) (6 r ) Equations (6) and (6') are identical in form, but the latter contains the approximation B(v*,T), introduces a mean transmittance, and assumes that v- ■ v-. The step in going from one equation to the other is not trivial, but yields physically realizable functions. B. Spectral Regions In the design of an indirect sounding system it is necessary to review the absorption bands which occur in that part of the spectrum where measur- able self -emission occurs in the earth's atmosphere. Measurements in selected "windows" of low absorption are also required to examine surface temperatures and the influence of clouds upon the outgoing radiation. Prac- tical considerations limit measurements of the spectrum to a range of about 3.5pm to about 20 mm. Absorptions significant for indirect temperature soundings in the atmos- phere result from oxygen, carbon dioxide, water vapor, ozone, nitrous oxide, and methane. Table 1 lists the various constituents, their mixtures (by volume), and their absorption bands; the locations of several important "windows" also are indicated. Radiances in bands of gases with fixed mix- tures can be used to deduce temperature profiles; for gases with variable mixtures, satellite measurements may be used to estimate the mixtures. Figure 1 (Valley 1965) displays the approximate strengths and shapes of the bands, but not the fine structure of the individual lines. Table 1 shows that temperature profiles must be deduced from measurements in one or more of three spectral regions: The combined CO2 and N 2 bands in the 2160-2393 cm" 1 region; the CO2 band between 550 cm"l and 760 cm" 1 ; and the oxygen complex in the 1.7-2.3 cm -1 region. Determination of the mixtures of ozone and water vapor may result from measurements (estimates of these may be needed for temperature determina- tions) in the following ranges: For water vapor, 1200-2000 cm" 1 , near 800 cm" 1 , or 0.74-550 cm" 1 , for ozone, 1000-1100 cm" 1 . Table 1. --Absorption bands and windows important to indirect soundings v Mixture X by volume (cm" 1 ) (urn) Species (percent) Application 0.00005 fair 033 | temperature 2,160-2,270 4.4-4.6 N 2 2,240-2,393 4.18-4.5 C0 2 560- 770 13 - 18 co 2 1.7-2.3 4300-5900 (51-69 GHz) °2 1,200-2,000 5.0-8.3 H 2 0.74- 800 12.5-13,500 (22.2 GHz) H 2 1,000-1,100 9-10 o 3 650- 750 13.3-15.4 °3 2,393-2,870 3.5-4.18 window 770-1,000 10.0-13.0 window 0.9-1.4 7,000-11,000 (27-42 GHz) window < 0.65 > 15, 000 (19.5 GHz) window f0. 031-0 21 variable j H 2 abundance (0-2) J and cloud (variable 0-0.0003 emissivity O3 abundance or transmittance Surface and cloud temperature j Surface j temperature There are two practical ranges for estimating water vapor concentration. These ranges are determined partly by instrumental capabilities; they are: near the 0.74 cm" 1 line with microwave radiometers, and on the low wavenum- ber side of 550 cm" 1 using infrared techniques. The lower limit for most purposes is about 300 cm" 1 . Below that, absorption by water vapor is so great that the earth's radiance arises from near the tropopause. This causes insensitivity to the radiance of the water vapor mixture; that quan- tity, then, becomes largely indeterminate. Two minor absorption bands deserve mention because they coincide with more important bands. In the region 650-750 cm" 1 , the 701 cm~l ozone band imposes a variable absorption upon the nearly constant absorption by carbon dioxide. Strictly speaking the ozone mixture should be known if one wishes to infer the temperature profile. From experience with the SIRS instrument it has been found that a rough estimate of the ozone usually suffices for this purpose. In a similar way, the methane band (not shown in table 1) between 1200 cm" 1 and 1400 cm" 1 has been found by Hanel, et al. (1971b) to render that part of the v 2 band of water vapor to be useless in retrieving the mixture of that constituent (the reverse is also true for the methane mixture) . C. Gas Transmittances Equation (6 1 ) shows that the measured radiance depends on the atmospheric transmittance and on the Planck function. To calculate the Planck function, it is necessary to know the transmittance in most temperature retrieval methods. The spectral transmittance of an atmospheric gas, i, between pressure level p and the satellite is defined, as in (5), by 1 r P TiUp.O) - exp[- I V kiCv'.p') q ± (p 1 ) sece dp'] . (5«) o It follows that if N constituents absorb at the same frequency, the total spectral transmittance is N t(v,p,6) - J]~ TiUp.e) . (9) i=l For a satellite instrument, with its finite spectral bandpass, a new trans- mittance must be defined. This incorporates both the optical properties of the atmosphere and the characteristic of the instrument's spectral response, as in (7) and (8). In the case of a dispersed spectrum with the detector at the focal plane, the transmittances must be defined by ©o o° T i ( Vj ,p,e) = ^ T.(v,P,9) f(vj-v) dvA f j (v)d V (10) o o where v . is the abscissa of the centroid of the instrument's "slit function" f, for the j t " spectral interval. Similarly, the transmittances for a "straight through" filter type of instrument are given, from Eq (8a), by •^(vyP, e) - \ T^p, e) tj ( v ) d v / \ t.( v ) d v , (li) o o where v . i s the abscissa of the centroid of the filter transmittance, t. J In applying (10) and (11) to the atmosphere, it is usually assumed that (9) may be applied in the form N t( Vj ,p, s) = 77 ^(v , P ,e) . (12) i=l 10 This may be justified only on practical grounds; it assumes that there is no systematic relation between the absorptions by any two gases. It is required wherever gases with variable mixtures are involved. Tests of the validity of this assumption must be made, and, if necessary, modifications made to suit each spectral interval. Transmittances for the fixed constituents can best be made by point-by- point calculations through spectral lines for a series of pressure levels. Drayson (1966) has performed these calculations for CO2 using line positions and parameters taken from laboratory experiments and from theory, for a spectral mesh which is much smaller than the line widths. When the results were used with SIRS, a slight modification in some spectral intervals was required to bring the observed temperature profiles from radiosondes into agreement with the measured radiances through eq (6 1 ). Similar calculations have been made in the 2000-2500 cm -1 region for CO2 (McClatchey 1965) and in the 4-6 mm region for O2 (Meeks and Lilley 1963). Gases whose properties vary require a different approach. Because the point-by-point calculations consume so much time, simpler approaches are required. One method is to compute t from eq (10) or (11) using point-by- point calculations for a series of homogeneous path lengths, varying pressure and temperature. The transmittances may then be fit to a suitable expansion of a function, W, in three variables T( Vj ,T£, p m , u n ) = expt-WjO^, p m , u n )] , (13) where u = optical path length, and ^,m, n = indices for temperature, pressure, and optical path length. Then any atmosphere may be fit incrementally for quadrature purposes by 6T( Vj ,T,p,u) « -exp[-W(T,p,u)] 6W(T,p,u) . (14) For water vapor, Smith (1969a) found that computed transmittances could be fit well to a series expansion of the form x i y i z i W(T,p,u) = exp[rC X Y Z x ) , i where X, Y, and Z involve logarithms of u, p, and T, and the coefficients, C-, were determined from a least-square solution. The quantities, x^, y., z-, take on values of 0, 1, and 2 in several combinations. Ozone is not so well defined in terms of the line parameters, so no point- by-point calculations have been performed in the infrared. However, the 701 cm" 1- band is rather weak, absorbing only about 5 percent for an average ozone amount in the atmosphere. From the laboratory measurements of McCaa and Shaw (1968), a good estimate of atmospheric ozone transmittances can be made. No temperature adjustments are justified. 11 Transmittances by water vapor in the infrared have recently been found to contain an element which is not explained by classical approaches. In the •"windows", for example, Varanasi et al. (1968) found that there is a con- tinuum whose absorption coefficient is proportional to the partial pressure of water vapor, and which has a large negative exponential dependence upon temperature. Bignell (1970) found the same phenomenon in the windows and in the rotation band above 200 cm" 1 . Thus, to explain the observed atmospheric transmittances of water vapor in the continuum it is necessary to multiply the right-hand side of eq (13) by T(v j ,P,T) = exp[-Kj V p« q 2 FoP sece dp'] (15) where Ki includes the constant part of the absorption coefficient and other constants. It has been found that n=5 yields reasonably good results. The explanation by Varanasi, et al. (1968) is that this absorption is caused by the water vapor dimer, (H20)2. Observations fit the chemical equilibrium theory. Eq (13) applies for the lines in the spectrum;eq (15) applies to the con- tinuum. Therefore, atmospheric transmittance of water vapor calculated from considerations in this section now becomes the product of eq (13) and (15), in order to include the effects of both the line and continuum components. Eq (6') defines the relationship between the satellite measured radiance, I, and the atmospheric temperature profile, expressed through B[T(p)]. How- ever, the radiances, I, also depend on the atmospheric transmittances, t. It is therefore evident, from the above discussion of transmittance, that difficult problems can arise in the calculation of T from measured I's with methods based on eq (6 1 ). 3. INVERSION PROCEDURES - CLOUDLESS CONDITIONS The integral form of the radiative transfer equation for a cloudless atmos- phere, eq (6'), is given in Section 2. That equation can be modified to yield the relation x o I( v ) = B[ v ,T(x )]t(v,x ) - V B[ v ,T(x)] d ^ x > dx (16) " u «J dx o An "inversion procedure" or more precisely the "inverse problem" of radia- tive transfer, is a method for finding the temperature T in eq (16), as a function of the independent variable x, from satellite observations of the radiance, I, at select mean (or central) frequencies v . The variable x, can be any single valued function of pressure. The transmission functions, t, and the weighting functions, dr/dx, are assumed known in some inversion methods. However, even if I is completely known as a function of v. ec l (16) 12 100 100 100 Y J I I I I I 1 I 1 I 1 I I L CO CH. i i : i_ i i _i i l i i j i i l N?0 j i ' " i i i i i i i i i i S. 100 § o V) GO < ,, 100 j i i i i i i i !l! i i I I L V" ^v W^~ CO, i \J i i_LJ i i i i I I i I L -~v HDO 8000 5000 3000 2000 1600 1400 1200 1000 900 800 700 (cm -1 ) J — |— ' H 1 — ' 1 1 r 1 1 ' 1 r— ' 1 I 2 3 4 5 6 7 8 9 10 II 12 13 14 (I5 M ) Figure 1 . --Comparison of the near-infrared solar spectrum with laboratory spectra of various atmospheric gases (from Valley 1965). 13 generally does not have a unique solution (Courant and Hilbert 1953, pp. 159-160). In practice I( v ) is measured at only a finite number of frequen- cies, v» an d when errors in the observations are considered, the solution to the inverse problem may be unstable in the sense that small errors in I( v ) yield disproportionately large errors in T(x) (see Phillips (1962) for a dis- cussion of this) . In view of these difficulties it is apparent that temperature accuracies will vary from one solution algorithm to another. For this reason three classes of inversion procedures will be discussed; they include regression, inverse matrix, and direct methods. Of these, in a cloudless atmosphere, the regression method need not deal with the physical eq (16); therefore the physical parameters are not required. However, in the cloudy atmosphere, the physical parameters still may be required even with some versions of the regression method. A. Modifications to Equation (16) Before considering the various solutions of the inverse problem, one can do three things to make eq (16) more tractable. First, a climatological mean T(x) (or a forecast profile, when available) can be subtracted from T(x) to reduce the problem to that of solving for the deviation "h" of T from T. With a forecast profile, "h" will usually be smaller than with a climatological-mean profile. Second, since generally more than 85 percent of the energy in the ll^m "window" comes from the boundary term in (16), this measurement and the first approximation, T, can be used in (16) to determine a good first approx- imation of the surface temperature T s (x ), for cloudless conditions. Conse- quently, the boundary terms for all the spectral frequencies can be calcula- ted and subtracted from eq (16). This puts (16) into the more classical form of an integral equation of the first kind. Separation of the surface boundary from the atmosphere also takes care of the situation in which a discontinuity exists between the ground temperature and the shelter-air- temperature. Third, eq (16) is nonlinear; it is necessary to remove the simultaneous dependence of B on v anc * T under the integral. Also, B depends nonlinear ly on T. It is possible to form a linear equation in which the Planck function has been approximated by the Taylor expansion, B( V ,T) - BUT) = h dB (17) dT where h(x) = T(x) - f(x) . (18) 14 The quantity dB(v,T)/dT can be calculated at every pressure level in ad- vance for each frequency once T has been chosen. Other approximations are given in Wark and Fleming (1966), Fleming and Smith (1972), Smith, Woolf, and Jacob (1970), and Smith, Woolf, and Fleming (1972). If these three steps are incorporated into (16), it can be written as a linear Fredholm equation of the first kind: x o (v) = ^ K(v ' x) h < x > dx ' (19) where x r(v) = Kv) - B[ v ,T s (x )]t( v ,x ) + J B[ v ,T(x) ] d ] ( v x > dx (20) " o dx v( v \ - dT x > . dB (v,T) MV, X) = -— y dx dT and h(x) is defined by (18). Since I(v)> a nd hence r(v), is measured at a finite number, M, of frequen- cies, and the kernel, K(v, x), is generally known only in tabular form, eq (19) is approximated and solved numerically. Partition the interval [0, x ] at N + 1 points and let w:, j = 1,2, ...,N be the quadrature weights associ- ated with Xj, then (19) can be approximated by N r i = E a ii h i > i = 1>2,...,M , j = l J or in matrix form by r = A h, (21) where a.j = WjK( Vi ,Xj)AXj The quantities, a.., are kernel functions, that is, they are values by which the temperature deviations must be multiplied, or weighted, to get the radiance deviations in eq (21). The shape of a^ is mainly determined by dT/dx. An illustration of dT/dx as a function of pressure is shown in fig- ure 2. Generally the weighting functions arising in temperature sounding problems with downward pointing instruments are very smooth and overlapping (fig. 2), with the result that the matrix A in (21) is usually ill-conditioned with respect to matrix inversion. Hence, a direct solution of the linear 15 equation (21) is impractical (see Phillips 1962, and Wark and Fleming 1966). Other methods of solution, such as those discussed next, must be used. 01 -L 1 I 1 ' i 1 ■ 1 1 1 SIRS - B - 5 668 7cm' - 10 2 5 ^V^/ 679.8crr 50 ~ 100 "">>692.0cm ' - 200 \~ 7010cm' ■ 300 ^^709.0cm' ^ 400 500 734.0cm' ' 700 850 nnn f |~^<^99.0 f m ' ^"»73O.0cit|-' - 01 02 .03 .04 dr/dx Figure 2. --Derivative of transmittance with respect to xCp^p 2 / 7 (from Smith, Woolf, and Fleming 1972). B. Regression Method If the transmittance properties of the atmosphere are not known, or if there is a large uncertainty in the accuracy of the transmittance functions, the regression method should be used. It can however, also be used even when the transmittance functions are known fairly well. This method also has applicability when calibration difficulties or measurement biases arise. It was used operationally with SIRS A on Nimbus 3. To implement the method, one needs radiosonde measurements that are coin- cident in space and time with satellite measurements. Since the radiosonde measurements do not extend sufficiently high into the atmosphere they must be extended by the use of rocketsonde or grenadesonde measurements; in some applications the use of climatological data is adequate. Suppose L pairs of simultaneous measurements of radiosonde temperature and satellite radiances are available. Let the matrix H = [h jk ] , j = l,2, ...,N; k=l,2,...,L, (22) represent the array of L temperature profiles with values given at N pre- scribed pressure levels, and suppose the corresponding L sets of satellite measurements are given by the matrix 16 R [r ik ] , i=l,2,...,M; k=l,2,...,L. (23) The quantity "h" is defined by (18) and r(v) is the_deviation of I( v ) from the mean of all L satellite radiance measurements, I(v). The principle of the regression method now depends on the following rea- soning. Eq (16) shows that each radiance, I(v)> depends on the temperatures at many levels in the atmosphere. Therefore we can assume that the tempera- ture at any one level influences a few of the radiances significantly. More- over the temperature at one level in the atmosphere is often correlated with the temperature at other, distant levels. Therefore, the temperature at a particular level might be statistically related to all the radiances. With this expectation, it is then required to find a set of coefficients by which the radiances can be multiplied to give the "best" temperature estimate in a least-squares sense. The regression problem is that of finding the N X M matrix C, which best fits the relation H = CR in the sense of least squares. In other words, find the matrix C that minimizes the Euclidean distance N,L M 2 5(C) = E (h. k - EC r ) j,k=l Jk i=l ^ lk = tr {(H-CR) (H-CR) T j, (24 ) where "tr" and the superscript "T" represent the trace and the transpose, respectively. The solution is found by differentiating (24) with respect to the elements of the matrix C, which results in the normal equation dF T T jg = 2HR - 2CRR = , and implies that the matrix of regression coefficients is C = HrVrV 1 , (25) where the superscript -1 indicates the matrix inverse. To avoid the possi- bility of the matrix RR T being singular, one should choose L much larger than M. For any particular set of M radiance values £, the N temperature deviations ^h, comprising the solution, now can be obtained from the equation 17 h ■ Cr , (26) which when added to T yields the desired temperature profile. An advantage of the linear regression method, especially in the cloudless case, is that it does not require explicit knowledge of the weighting func- tions. However a linear relation is often assumed to exist between h and r, when in fact this is not true. The regression method is useful, especially when certain physical quanti- ties, such as atmosphere transmittance are unknown. However, if the weight- ing functions and calibrations are known, the retrieval methods described in the next two sections are appropriate. C. Inverse Matrix Methods Full Statistics These methods have the advantage that simultaneous measurements of r and h are not required. Instead climatological measurements of h and the errors e( v ) of measurement in the corresponding I(v) can be used to estimate the required statistical quantities. This will be discussed after eq (30). Write eq (21) as r = Ah + / e (27) where e are the errors of measurement. Now apply eq (27) to all of the corresponding pairs of the L columns of data in eq (22) and (23), put them into arrays in the appropriate manner (Smith, et al. 1970) to obtain the matrix equation R = AH + E , (28) where E is a known matrix of errors corresponding to R. Eq (28) essentially defines the error matrix, since R, A and H are assumed given. Substitute (28) into (25) and assume that: (a) The errors £ are uncorrelated with Jh; (b) the errors e-^ are random variables with zero mean; and (c) the quadrature errors are negligible with respect to je. It follows that the matrix products HE T and EH* are small relative to any of the other products, and that in (26) C = (HH T ) A T [A(HH T )A T + EE T ] . (29) 18 Compare this C with the one in (25) and note that this one incorporates the correlations of temperature between pressure levels through the quantity (HH T ), while the one in (25) does not. When the right side of (29) is appropriately divided and multiplied by the scalar (L-l), and it is applied to (26), then (26) has the form £ - S h A* [AS h A T + S e ]"\ , (30) where S^ and S fi are, by (18) and assumption (b) above, the covariance matrices of h and je, respectively. Other derivations of equation (30) are given in Foster (1961), Strand and Westwater (1968a, 1968b), and Rodgers (1970). It was noted after eq (28), in connection with the derivation of (30), that knowledge of both R and H are necessary to obtain E. However, in prac- tice S^ can be estimated from climatological collections of radiosonde meas- urements and Secan be estimated from the satellite onboard calibrations; this eliminates the need for simultaneous cloudless measurements of radiances and temperatures which are difficult to obtain in large numbers. But, of course, eq (30) now assumes a good knowledge of the transmittances which are con- tained in A. Minimum Information By making various simplifying assumptions about the covariance matrices in (30), it is possible to consider many special cases. Perhaps the most impor- tant of these is the "minimum information" case of Foster (1961). It assumes that the elements of Jh are uncorrelated, the elements of e^ are uncorrelated, and their corresponding variances o*fi and <7|, respectively, are constants, that is, S h = a h X N ' S e = a e X M ' where I, is the J X J identity matrix. Under these assumptions, equation (30) reduces to h = A T [AA T + YI ]" r , (31) where Y = o*/o u . 'e '"h Foster (1961) calls Y the ratio of noise to signal power. A direct deriva- 19 tion of (31), given by Smith, Woolf, and Fleming (1972), shows that it can be interpreted as a perturbation technique which minimizes the deviation of the retrieved profile from a profile forecasted from dynamical considerations. This is the method used operationally with SIRS B, Nimbus 4 data. If the matrix identity T T -IT -It A 1 [AA X + YE m ] = [ A X A + YI N ] A 1 is applied to (31), then the solution is that of Twomey (1963, 1965). How- ever, all the methods following from eq (31) can be categorized as the Tikhonov (1963) method of regularization. The minimum information solution is the first of those examined thus far that is essentially independent of a priori statistical data, except for some estimate of cTh» * c no longer contains the covariance matrices; there- fore, T in equations (18) and (20) now can be any initial profile, not nece- ssarily the mean profile. This may be advantageous because one can use a forecast profile which is generally more accurate than a statistical mean for the initial profile. Experience suggests that the accuracy of solutions from the algorithms discussed in this report are dependent upon temperatures selected for the initial profile. Iteration On the other hand, the failure to use statistical information can have an adverse affect on the solution, but some compensation for an inappropriate Y can be made through iteration. This is accomplished by making the following replacements in equations (17), (18), (20), and (21): a. T^ n+1) for T, b. T< n > for T, c. h} * 1 ' for h, d. r< n > for r, e. A (n) for A. When these changes are applied to equation (31) it takes the form ^ =a) AT r (n) . (35 ) Where the superscripted quantities are those described in items (a) through (d) in the sub-section on Iteration, above and the scalar to is a relaxation factor which assures convergence, provided that Tj n) )/dT ij dB( Vo ,TJ n ' , )/dT The definition , v (n) differs from that given for (21) and does not depend upon T . Convergence of (37), as defined by n Q in (33), generally is more rapid than for (35) or (36). All of the direct methods discussed thus far solve the nonlinear eq (16) as the limit of a sequence of linear equations. However, there are methods that are explicitly nonlinear. Perhaps the most familiar of these is that by Chahine (1968, 1970, 1972). Chahine assumes that the Planck function B.(p.) at some pre-selected pres- sure levels, p., can be related to the upward radiances by * n+1 rn.^ m = » n rT/« m I(v i ) (38) B. [T< Pj >] = B i [T( Pj )] i n U) 23 con- I(v) is the measured radiance and I n (v>i) is the computed value of I(vi) after the n th iteration of the vertical temperature profile. When I n (vi) verges sufficiently close to the observed values of I(v^) the value of B n [T(pO] is taken as the solution; from that T(p.) is calculated as the value of T at the pressure level p.. In the process, the temperatures at levels between the selected pressure levels are obtained by interpolation. This method is nonlinear and does not depend on a^ priori statistics, nor on a set of pre -determined coefficients; the quantities by which the observed I's are multiplied changes in every iteration. Some comparative studies with this method (Conrath and Revah 1972) have shown that its solution tends to become somewhat unstable when the number of measured radiances increases from 7 to 16. Smith (1970) has modified Chahine's method. Smith assumes that .n+1 „ . n B (v^p) - B ( V .,P)+ [I( Vi ) - I (v^l , (39) where again I(v) i s the measured radiance. But now Smith allows the B's to be computed at all pressure levels. A separate set of B's will be generated for each v. From the B's, corresponding temperatures are computed. Smith then proposes that the temperature at any pressure level, p, be given by (n+1) N (n+1) / N T (p) = E T (v.,P)AT( Vi ,p) / S AT(v,,P) i=l x I i=l At the surface AT(v.>p) 1S replaced by t(v,P-)> here, p is the surface X s s pressure. A final approach that should be mentioned is derived by Backus and Gilbert (1967, 1968, 1970). Details will not be given here because it appears that no one has applied the method to the temperature sounding problem. This method also is very useful from a diagnostic point of view in that it yields explicit procedures for determining the vertical resolution inherent in the weighting functions and for determining "trade-off" diagrams between resolv- ing power and measurement errors (Conrath 1971). E. Conclusions In comparing temperature retrieval methods, one is always interested in obtaining the most accurate solution; but this often depends upon the kinds of a priori data available. For example, we have seen that without explicit knowledge of the weighting functions one must resort to regression techniques. However, the accuracy that one can obtain from regression methods depends strongly on the nature of 24 the data used in compiling the regression matrices. Consideration must be given to such things as the degree of correlation that exists between the quantities measured and those to be predicted, and the degree of homogeneity of the data versus the number and the degree of independence of the measure- ments. On the other hand, if one knows the atmospheric transmittance functions and their derivatives well, then he may employ one of the inverse matrix or direct retrieval methods. Of these methods, the full statistical method, eq (30), may be the most accurate (Fleming and Smith 1972) if sufficient a_ priori information is supplied. However, experience suggests that the accuracy of the initial profile is important in assessing the accuracy of the other methods. Indeed, there is a point at which the accuracy of the initial solution is sufficiently good, that is, r is sufficiently small, so that the covariance matrices in eq (30) are superfluous provided the atmospheric transmittances are known. At this point the minimum information case, eq (31) or (32), is adequate. But in this case the direct retrieval methods also are competitive, Another consideration is the question of instrumental noise. All the methods described are stable in the sense mentioned on the first page of Section 3, but each method propagates noise into the solution in a unique manner. The effect of noise on the stability and "smoothness" of the solu- tion may be different for different methods. The full statistical method is probably the one most capable of introducing true fine structure, but noise can introduce fine structure that is not real. In an operational situation, where temperature retrievals must be made in a time period much shorter than the time between successive fields-of-view, computer running time is of paramount importance. Therefore, because of their simplicity, the direct retrieval methods that have fast convergence rates are superior. When the matrix inversion methods are used with itera- tion, the inversions must be made repeatedly, due to the approximation, eq (17), which is incorporated into the matrix A. One cannot make a definitive choice of method without first considering the many trade-offs. Section 5 reviews the accuracy obtained operationally by the regression method with SIRS A data and by the "minimum information" method with SIRS-B data. However, no organized effort has yet been made with real data to determine which methods yield the most accurate results. 4. CLOUDS AND AEROSOLS We have so far considered the theory and retrieval methods for the cloud- less atmosphere only. However, the atmosphere is rarely completely cloudless over areas in the fields of view of present-day satellite sounding instru- ments. 25 Clouds are highly variable in amount and characteristics. They drastic- ally modify the transmittance and emittance of the atmosphere. Therefore, when the satellite observes the atmosphere in the presence of clouds, the observed radiances are affected not only by the atmospheric temperatures, but also by the clouds. It is necessary, then, to eliminate or to account for the effect of clouds on the observed radiances to retrieve the atmos- pheric temperature from the sounding instrument observations. Two basic approaches have been suggested for dealing with clouds. One utilizes measurements of a single field-of-view; the other employs high spa- tial resolution measurements of multiple fields-of-view. In one version of the single field-of-view method, an attempt is made to estimate what the observed radiances would have been had there been no clouds in the field-of- view (Smith, Woolf, and Jacob 1970). The other version (Rodgers 1970) treats the satellite observations as an ensemble of cloud and clear areas, and esti- mates the "most probable" temperature profile which exists in the observed, cloudy area. The multiple field-of-view method (Smith 1969c) specifies the clear column radiances from spatially independent radiances observed over a non-overcast regime. A. The Cloud Problem Clouds generally exist within the fields of view of a sounding radiometer. In this case Kv) « T)I cd (v) + (1 - T))I clr ( v ) < 40 > where I(v) is the observation, T\ is the fraction of the field covered by clouds, I C( j(v) is the average radiance from the cloud covered portions of the field of view, and I c i r (v) is the average radiance from the cloud -free portions of the field. Many types of clouds, such as fields of fair weather cumulus, stratocumulus, and cirrus, tend to be horizontally stratified and irregularly distributed in the horizontal direction. Cumulonimbus clouds, on the other hand, have a wide range of vertical dimensions. Depending upon the field of view of the sensing instrument and the weather situation, several different types of clouds at different heights may contribute to a given radiance observation. A complete parameterization of clouds in a radiative transfer model for deducing temperature profiles would introduce more variables than there are independent pieces of information in the observations from present-day instru- ments. For the simplest case of a single layer of clouds I cd (v) = F( v ,p c ,0) + {r c ( v )I d ( v ,p c ,0) + e c (v)B[ v ,T(p c )] + T c ( v )I( v ,p c ,p s) J T(v ,p c>0) > (4i) 26 where p is the pressure of the cloud layer, and r c , g , and t are the re- flectivity, emissivity, and transmissivity of the clouds, respectively. The term F(v, P c ,0) is the radiance emitted at frequency v by the atmosphere above the cloud, i.e. between the cloud pressure p c and zero pressure. Id(v>P c >0) is the downward component of the radiance at the cloud level, B[v, T(p c ) ] is the cloud Planck radiance, and I( v , P C ,P S ) is the upward component of the radi- ance at the cloud base, i.e. between the cloud pressure p c and the surface pressure p s . In writing (41) it is assumed that the cloud pressure is some effective mean value. The second term in eq (41) is the radiance, originat- ing at the cloud, which arrives through the overlying atmosphere at the satellite. At intermediate wavelengths in the infrared (e.g. lS^m), the reflectivity of most clouds is near zero. Assume r c = 0, so that T c = l~e , and eq (41) reduces to the simple form I cd (v) = e c (v) I b (v) + [l-e c (v)l I c i r (v) , (42) where I h (v) i s the radiance associated with the opaque 'black' cloud condi- tion (i.e., e c ■ 1) . Substituion of (42) into (40) gives Kv) = ] I clr (v) (43) where ry(v) is the effective cloud amount, He (v) . After making the appropri- ate substitutions for 1^ and I c ir> one obtains Ps I( v ) = B[ v ,T(p s )] T(v,P s ,0) - ^ B[ v ,T(p)] d T (v,p,0) dp dp a(v) {B[ v,T(p s )] t(v,P s ,0) - B[ v ,T(p )] t( v ,P,,0) Ps c - -c _ C B[ v ,T(p)] d T (v, P> 0) d l (44) P c dP The sum of the first two terms on the right is equal to I clr . Even for this simple case of a non-reflecting single layer of clouds, the transfer equation is nonlinear in a/ a ^d p and cannot be solved for the temperature profile by using the linear inversion methods discussed in Section 3. One approach to the solution of this nonlinear cloud problem is to first estimate the equiva- lent clear column radiances from the measured radiances, and then to solve 27 for the temperature profile from the equivalent clear column radiances. Single Field of View approaches It follows from (44) that the equivalent clear column radiance for any field of view is I clr ( v ) = Kv) + a(v)Q(v,P c ) (45) where Q(v>P c ) i s the term enclosed by braces in eq (44). It is tempting to determine the cloud parameters and p from auxili- ary data. For instance, the fraction of cloudiness, 7], might be specified on the basis of high resolution visible or infrared cloud picture data. How- ever, this fraction will not be equivalent to the effective cloud amount & at the frequency v unless the clouds are opaque at all wavelengths, e.g., thick water droplet clouds. Probably a more severe problem is the difficulty in registering the cloud picture data with respect to the field of view of the sounding radiometer. A small degree of misregistration could lead to significant errors of 71. The determination of the cloud pressure altitude p c from auxiliary data is even more difficult. Even if the cloud top pres- sure could be specified (e.g., from high resolution infrared window data) this may not be sufficient for assigning an effective value to p c . Most severe is the fact that the temperature profile below the cloud must be known in order to use the auxiliary cloud information to determine the correction P c ) . Nevertheless, this has not been tried yet, so it is not known what effect such a procedure would have on the accuracy of temperature re- trievals. The cloud parameters and subsequent cloud corrections to the radiances can be estimated directly from the set of radiance observations if they have a common field of view. (Smith 1969b, Smith, Woolf, and Jacob 1970). Since in practice the "first guess" temperature profile is now estimated before- hand, the correction is evaluated by an iterative procedure as follows. Us- ing measurements in two spectral intervals, v , and vo> whose radiation is sensitive to clouds, one can eliminate #(v) from (45) after assuming the cloud emissivity is constant within the spectral domain of observation. The result is [I c lr