C s&./s ■ /uess -tfo NOAA Technical Report NESS 80 ^ OF ^ 8 C -^XLl£ n ^r ES o* \ / Calculation of Atmospheric Radiances and Brightness Temperatures in Infrared Window Channels of Satellite Radiometers Washington, D.C. March 1980 U.S. DEPARTMENT OF COMMERCE National Oceanic and Atmospheric Administration National Environmental Satellite Service NOAA TECHNICAL REPORTS National Environmental Satellite Service Series Environmental Satellite Service (NESS) is responsible for the establishment and operation environmental satellite systems of NOAA. tion of a report in NOAA Technical Report NESS series will not preclude later publication in an i or modified form in scientific journals. NESS series of NOAA Technical Reports is a continua- and retains the consecutive numbering sequence of, the former series, ESSA Technical Report Environmental Satellite Center (NESC) , and of the earlier series, Weather Bureau Meteorological (MSL) Report. Reports 1 through 39 are listed in publication NESC 56 of this ser- ies. i the series are available from the National Technical Information Service (NTIS), U.S. tment of Commerce, Sills Bldg. , 5285 Port Royal Road, Springfield, VA 22161, in paper copy or mi- :he form. Order by accession number, when given, in parentheses. Beginning with 64, printed is of the reports, if available, can be ordered through the Superintendent of Documents, U.S. Gov- Lnting Office, Washington, DC 20402. Prices given on request from the Superintendent of Docu- ments or NTIS. ESSA Technical Reports 46 Monthly and Seasonal Mean Global Charts of Brightness From ESSA 3 and ESSA 5 Digitized Pic- tures, February 1967-February 1968. V. Ray Taylor and Jay S. Winston, November 1968, 9 pp. plus 17 charts. (PB-180-717) C 47 A Polynomial Representation of Carbon Dioxide and Water Vapor Transmission. William L. Smith, February 1969 (reprinted April 1971), 20 pp. (PB-183-296) Statistical Estimation of the Atmosphere's Geopotential Height Distribution From Satellite Radiation Measurements. William L. Smith, February 1969, 29 pp. (PB-183-297) Synoptic/Dynamic Diagnosis of a Developing Low-Level Cyclone and Its Satellite-Viewed Cloud Patterns. Harold J. Brodrick and E. Paul McClain, May 1969, 26 pp. (PB-184-612) Estimating Maximum Wind Speed of Tropical Storms From High Resolution Infrared Data. L. F. Hubert, A. Timchalk, and S. Fritz, May 1969, 33 pp. (PB-184-611) 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, Arthur H. Smith, James F. W. Purdom, and Ranee W. Skidmore, March 1974 (reprint and revision of NESC 51, September 1969, and inclusion of Supplement, November 1971, and Supplement 2, March 1973), pp. 1 — 6C-18 plus references. : 52 Data Reduction Processes for Spinning Flat-Plate Satellite-Borne Radiometers. Torrence H. MacDonald, July 1970, 37 pp. (COM-71-00132) Archiving and Climatological Applications of Meteorological Satellite Data. John A. Leese, Arthur L. Booth, and Frederick A. Godshall, July 1970, pp. 1-1 — 5-8 plus references and appen- dixes A through D. (COM-71-00076) 54 Estimating Cloud Amount and Height From Satellite Infrared Radiation Data. P. Krishna Rao, July 1970, 11 pp. (PB-194-685) ;SC 56 Time-Longitude Sections of Tropical Cloudiness (December 1966-November 1967). J. M. Wallace, July 1970, 37 pp. (COM-71-00131) 55 The Use of Satellite-Observed Cloud Patterns in Northern Hemisphere 500-mb Numerical Analysis. Roland E. Nagle and Christopher M. Hayden, April 1971, 25 pp. plus appendixes A, B, and C. (COM-73-50262) 57 Table of Scattering Function of Infrared Radiation for Water Clouds. Giichi Yamamoto, Masayuki Tanaka, and Shoji Asano, April 1971, 8 pp. plus tables. (COM-71-50312) 'he Airborne ITPR Brassboard Experiment. W. L. Smith, D. T. Hilleary, E. C. Baldwin, W. Jacob, H. Jacobowitz, G. Nelson, S. Soules, and D. Q. Wark, March 1972, 74 pp. (COM-72-10557) (Continued on inside back cover) -II HMOS'-, ""IUe"*^ NOAA Technical Report NESS 80 Calculation of Atmospheric Radiances and Brightness Temperatures in Infrared Window Channels of Satellite Radiometers Michael P. Weinreb and Michael L. Hill Washington, D.C. March 1980 U.S. DEPARTMENT OF COMMERCE Philip M. Klutznick, Secretary National Oceanic and Atmospheric Administration i o Q. g Richard A. Frank., Administrator P National Environmental Satellite Service David S. Johnson, Director CONTENTS Abstract 1 1. Introduction 1 2. Application of the radiative transfer equation in vide spectral intervals 2 2.1 The radiative transfer equation 2 2.2 Radiance calculations in wide spectral intervals 3 2.3 Conversion of atmospheric radiances to brightness temperatures. 5 3. Calculations of transmittances 6 3.1 General 6 3.2 Atmospheric Absorption Spectra 6 3.3 Transmittance of Atmospheric Constituents 7 3.3-1 Water vapor 7 3.3.2 Molecular nitrogen 9 3.3.3 Uniformly mixed gases 10 h. Computational details and results 12 U.l Quadrature 12 k.2 Examples 13 U.2.1 Atmospheres 13 h . 2. 2 Transmittances 13 i+.2.3 Atmospheric Attenuation 13 5. Conclusion 1^ Acknowledgments 15 References 1° 11 TABLES 1. Polynomial coefficients for spectral lines of water vapor ... 20 2. Coefficients for absorption by continua, nitrogen, and uniformly-mixed gases 22 3. The 100 quadrature points 23 FIGURES 1. Spectral response function for channel k of AVHRR on TIROS-N . 2U 2. As in figure 1, for channel 3 25 3. Rectangular subintervals in region of channel h of AVHRR ... 26 k. As in figure 3, for channel 3 27 5. Errors in calculated blackbody radiances 28 6. Look-up table relating blackbody radiance to temperature ... 29 7- Spectrum of atmospheric transmittance in 11-um region 30 8. As in figure 7, for 2.5- ^.5-ym region 31 9. Look-up table from LOWTRAN h 32 10. Temperature profiles for three atmospheres 33 11. Profiles of water-vapor mixing ratio 3^ 12. Calculated transmittances in 11-um region 35 13. As in figure 12, for 3.7-ym region 36 lk. Attenuation vs precipitable water for AVHRR 37 15. Attenuation vs precipitable water for channel h of AVHRR ... 38 16. Attenuation vs surface temperature for channel k of AVHRR . . 39 17. As in figure l6, for channel 3 ^+0 iii Digitized by the Internet Archive in 2013 http://archive.org/details/calculationofatmOOwein CALCULATION OF ATMOSPHERIC RADIANCES AND BRIGHTNESS TEMPERATURES IN INFRARED WINDOW CHANNELS OF SATELLITE RADIOMETERS Michael P. Weinreb and Michael L. Hill Office of Research, National Environmental Satellite Service, NOAA, Washington, D.C. 20233 ABSTRACT. We describe a method of simulating measurements of atmospheric radiances and brightness temperatures in wide-band window channels (at 11 and 3-7 ym) of satellite radiometers. As input the simulation takes vertical profiles of atmospheric temperature and water-vapor mixing ratio, as well as the spectral response functions of the window channels. It models the atmospheric transmittances and integrates the equation of radiative transfer. We demonstrate the use of the method with applications to the Advanced Very High Resolution Radiometer on the TIROS-N satellite. 1 . INTRODUCTION The environmental satellites operated by NOAA's National Environmental Satellite Service (NESS) carry instruments that measure the intensity of radiation upwelling from the earth's surface in the infrared "windows" at wavelengths near 11 ym and 3.7 ym. Among these instruments are the Scanning Radiometers (SR) on the ITOS 1 and NOAA 2-5 satellites, the Very High Resolution Radiometers (VHRR) on the NOAA 2-5 satellites, the Visible and Infrared Spin-Scan Radiometers (VISSR) on the current SMS/GOES satellites, and the Advanced VHRR's (AVHRR) on the current TIROS-N series of satellites. Windows are spectral intervals in which the atmosphere is nearly transparent to the radiation emitted by the earth's surface. Although the atmosphere's effect on the radiation is small, it is, nevertheless, not negligible. For example, in the 11-ym region a moist atmosphere may attenuate the radiation emitted by the earth's surface to space by 5-10%. Therefore, to infer properties of the surface from measurements in windows, we need to model theoretically the radiative transfer in the atmosphere. This report documents a procedure for computing, in the 11- and 3.7-ym windows, the radiances and brightness temperatures that would be measured by an orbiting radiometer, given the temperature of the earth's surface and the vertical temperature and water-vapor profiles of the atmosphere. The model also takes into account other gases that absorb and emit radiation at these wavelengths. For the instruments considered here, which view ih the nadir or near-nadir, these gases are carbon dioxide, nitrogen, nitrous oxide, and methane. The computations apply to spectral intervals whose widths range from several tens to several hundreds of cm" . A computer program that incorporates the method described in this paper is currently in use at NESS. Copies of this program are available on request. -1- The literature describes a number of earlier transmittance models that have been applied in the 11-um window (Wark et al. 1962, Davis and Viezee 196*1, Saiedy and Hilleary 1967 , Anding and Kauth 1969, Smith et al. 1970, and Maul and Sidran 1973). These works were published before the importance of the self-broadened water-vapor continuum (Bignell 1970) was recognized. More recently the computer code LOWTRAN (McClatchey et al. 1972, , S.elby et al. 1978) was developed for modelling radiative transfer in the atmosphere in 20-cm -1 intervals throughout the infrared spectrum, including both the 11-um and 3-7-um window regions. The method described in this report evolved from the algorithm (Wark et al. 197^, Weinreb and Neuendorf fer 1973) developed at NESS to calculate radiances in the 11-um window channel of the Vertical Temperature Profile Radiometers (VTPR) (McMillin et al. 1973) on the NOAA 2-5 satellites. The present method has the following new features : 1. It can be applied to spectral intervals that are several hundred cm in width. (The VTPR interval was about 8 cm" 1 in width.) 2. It applies in both the 3.7- and the 11-um windows. 3. It incorporates recent advances in calculating transmittances , particularly those in modelling the nitrogen absorption near k urn and the water-vapor continua in the 11- and 3. 7- urn regions. Section 2 of this report introduces the radiative transfer equation and describes our techniques of calculating radiances and brightness temperatures while coping with the variation of the Planck function over the considerable width (in wavenumber) of the spectral intervals. Section 3 describes the transmittance calculations , which include the effects of H2O lines and continua, the collision-induced Ng band near k urn, and the "uniformly mixed" gases, particularly C02» CH1±, and N2O. Section k describes the numerical procedures for calculating transmittances and integrating the radiative transfer equation. The report concludes with a few applications of the calculations to the AVHRR on TIROS-N. 2. APPLICATION OF THE RADIATIVE TRANSFER EQUATION IN WIDE SPECTRAL INTERVALS 2.1 The Radiative Transfer Equation The upwelling radiance R(v) at wavenumber v can be calculated from knowledge of the temperature of the earth's surface, the atmospheric vertical temperature profile, and the vertical profiles of concentrations of the gases that absorb radiation at v. We accomplish this by numerically integrating the equation of radiative transfer in its integral form (see, e.g., Wark and Fleming 1966), -2- t(p s ,v) R(v) = B(T g ,v) x(p s ,v) -J B(T(p),v) dx(p,v), (l) 1 where B = Planck radiance, T = atmospheric temperature, s = subscript indicating that a quantity is to be evaluated at the earth's surface, p = atmospheric pressure, and r(p,v) = transmittance between the satellite and the level of the atmosphere with pressure p. The Planck radiance is given by 2 3 , v _ 2hc v J BlT ' Vj '" exp[hcv/kT]- 1 ' where h, c, and k are, respectively, Planck's constant, the speed of light, and Boltzmann's constant. Equation (l) holds under cloudless conditions for nonscattering, plane- parallel atmospheres in local thermodynamic equilibrium. 2.2 Radiance Calculations in Wide Spectral Intervals Equation (l) holds only for monochromatic radiation. For it to be applied to a spectral interval of a broad-band instrument , it must be convoluted with the spectral response function (v) of the interval. Figures 1 and 2 show such functions for channels k and 3 of the AVHRR on TIROS-N. The radiance R, that would be measured in any of these intervals is then given by * % 00 I 00 = J R(v) <|>(v)dv / I (v)dv , (2) where R(v) is computed from equation (l). To integrate equation (2) numerically, one must first integrate equation (l) numerically for a large number of closely spaced values of v. This is too cumbersome for our purposes. However, if the function (v) is narrow enough (say, 30 cm" or less in half-width), we can find a wavenumber v Q such that R, can be approximated adequately (Wark and Fleming 1966) by -3- -I l(Ps* v J S» "o R * = B(T s' V o } T(p s' V o } " J B(T(p),v o ) dx(p,v o ), (3a) where x(p,v Q ) = t( P ,v) (v)dv / I (v)dv . (3b) o Unfortunately, we are working with spectral intervals having widths of 100 cm -1 or greater, for which eq. (3) produces errors comparable to or exceeding the noise in the measurements of Ra. We resort, then, to a more accurate procedure. We first subdivide each spectral interval into rectangular subintervals, 30 or 20 cm" 1 in width, as shown in figures 3 and h. Within these subintervals eq. (3) is an adequate approximation (see below); i.e., if the index i labels each subinterval, we can apply eq. (3) to compute a value of radiance Rj_ in the subinterval. Then to estimate the radiance for the full interval, we compute the weighted mean of the R-^'s, where the weights are the heights h^ shown in figures 3 and k. That is, ■ E Vj/e h i w In our calculations we chose the v Q 's to be at the center of each subinterval, and we chose the h-j/s so that in each subinterval the area under the spectral response function equals the area of the rectangle. As shown in figures 3 and h, the widths of the subintervals are 30 cm" 1 near 11 urn and 20 cm~l in the 3.7-um region. The accuracy of eq. (k) depends on the behavior of not only the Planck radiance, but also the transmittance, as functions of wavenumber in the spectral interval of interest. A simple, rough way to estimate the accuracy of eq. (h) is to ignore the transmittances , i.e., work with blackbody radiances instead of atmospheric radiances. Following this approach, for fixed temperature T we computed the blackbody radiances in three separate ways, the "exact" calculation and two approximations. For the exact calculation, we convoluted the Planck function B(T,v) with the spectral response function, i.e., we applied eq. (2) with R(v) replaced by B(T,v). In numerically evaluating the integrals we applied the trapezoidal rule on points spaced every 0.1 cm -1 . The first approximation was simply the Planck function evaluated at the centroid of the spectral response function. (This is the form of eq. (3) for blackbody radiances.) For the spectral response functions shown in figures 1 and 2, the centroids are at 913.3 cm -1 and 2656.3 cm _ l, respectively. The second approximation was the one of eq. (k) , -4- with Rj_ replaced by B(T,v). The errors in each approximation were computed as the differences between the results of the approximations and those of the "exact" calculation. The absolute values of these errors are shown in figure 5 as functions of temperature. The upper panel applies to the AVHRR's 11-um channel, and the lower to the AVHRR's 3.7-um channel. The horizontal dashed lines are the nominal values of the NEAN's (instrument noise, in radiance units) in the two channels. At 11 um, the first approximation (centroid) produces errors comparable in magnitude to the NEAN, while the second approximation (eq. (h)) holds the errors to values less than half of the NEAN. At 3.7 um, the first approximation (centroid) produces errors many times larger than the NEAN, while eq. (k) reduces these errors to values approximately equal to the NEAN. However, since we have ignored the transmittances in this analysis, the results provide only an estimate of the errors. In the remainder of this report, eq. (k) is used in computations of atmospheric radiances. 2.3 Conversion of Atmospheric Radiances to Brightness Temperatures In many applications, users of satellite data prefer to work with equivalent brightness temperature rather than radiance. We convert our calculated atmospheric radiances to brightness temperatures through look-up tables, one for each channel. Each table consists of 1501 pairs of blackbody radiances and their corresponding temperatures. The pairs are specified every 0.1°K between l80°K and 330°K. Each value of blackbody radiance in a table depends upon the spectral response function and is computed from eq. (U), with R^ replaced by B(T,v ). In other words, it is the weighted average of Planck radiances evaluated at the centers of the rectangular subintervals of figures 3 or it, with the weights given by the heights h± . The data in the look-up tables for channels 3 and k of the AVHRR are graphed in figure 6. It is important to realize that because they are computed from eq. (h) , the blackbody radiances in the look-up tables are not error-free, but carry with them the errors shown in figure 5. However, by computing the look-up table this way, we tend to minimize the errors in the inferred equivalent bright- ness temperatures, for the following reason: Recall that the first step in deriving an equivalent brightness temperature is the application of eqs . (3) and (h) to the atmospheric temperature profile to produce an atmospheric radiance. As previously described, this radiance carries with it an error than can be estimated from figure 5. The second step is to refer to this value of radiance in the look-up table and extract the corresponding equivalent brightness temperature. If the blackbody radiances in the look-up table are subject to exactly the same errors as are the atmospheric radiances, the errors in the two steps will compensate, and the derived equivalent brightness temperatures will be error-free. As described earlier, however, the errors in the blackbody radiances will not coincide exactly with the errors in the atmospheric radiances. Hence, using the look-up table generated from eq. (k) 9 we will minimize the errors in the inferred equivalent brightness temperatures but not eliminate them. -5- 3. CALCULATIONS OF TRANSMITTANCES 3.1 General The first step in calculating radiances is to generate transmittances in each of the subintervals shown in figures 3 and k. In the 11-um window, eight intervals, each 30 cm~l wide, span the region from 760 to 1000 cm - -'-. In the 3-7-ym window, 23 intervals, each 20 cm -1 wide, span the region from 2UU0 to 2900 cm~l . In each subinterval, the transmittance of the atmosphere is treated as a product of the transmittances of the atmospheric constituents that absorb radiation. In the 11-um region, the constituents are water vapor and the "uniformly mixed gases" (McClatchey et al . , 1972), principally carbon dioxide. In the 3.7-ym region, the constituents are water vapor, molecular nitrogen, and the uniformly mixed gases, chiefly carbon dioxide, nitrous oxide, and methane. We have intentionally neglected ozone. It is important only between 980 and 1000 cm - !, whereas the responses of our satellite instruments, as measured by the (v) functions, are small, if not zero, in this subinterval. The effects of aerosols and clouds are also ignored. 3.2 Atmospheric Absorption Spectra The purpose of this section is to describe generally the nature of atmospheric absorption in the window regions. Figures 7 and 8 are measured absorption spectra of the atmosphere in the 11- and 3.7-ym regions, respectively (Weinreb, Planet, and Jones 1977). (Figure 7 does not cover the "entire 76O-IOOO cm - -*- range, but it is useful, nonetheless, for the qualitative discussion here.) The spectra were taken with a spectrometer receiving solar radiation through the McMath solar telescope at the Kitt Peak National Observatory. The spectral resolution is about 0.7 cm near 11 pm, and 7.0 cm~l near 3-7 ym. In the 11-um window, water vapor dominates the absorption, contributing spectral lines and a continuum. The continuum (Bignell 1970) is absorption that has little dependence on wavenumber. In figure 7 its effect is most noticeable between the spectral lines , where the envelope of the spectrum has a value of transmittance less than one. In this case it is about O.98, so the continuum absorption is about 2%. Incident ially, this spectrum was taken under very dry conditions (precipitable water = 0.6 cm). Under typical mid- latitude or tropical conditions (precipitable water ~ 2 or 6 cm, respectively), the absorption is considerably stronger. For wavenumbers lower than 820 cm~l , carbon dioxide makes some contribution to the absorption. Its effect is seen in the strong line near 792 cm - -'- and in the background absorption that increases with decreasing wavenumber. Carbon dioxide also has a small effect between 930 and 1000 cm" (not shown in figure 7). As mentioned in the preceding section, our calculations ignore the absorption by ozone, which is measurable only for v > 980 cm . -6- In the 3.7-ym window, the principal absorbers are nitrous oxide, carbon dioxide, methane, water vapor (mostly as HDO), and molecular nitrogen. Nitrous oxide contributes the band near 2570 cm" 1 and the high -wave number part of the sharp fall-off between 21+00 and 2500 cm"" 1 . Near 21+00 cm" 1 carbon dioxide dominates, but it rapidly loses strength toward higher wavenumbers. The region between 21+00 and 2500 cm -1 is also affected by the collision-induced nitrogen absorption (Shapiro and Gush 1966, and Farmer and Houghton 1966). Between 2700 and 2900 cm -1 , water vapor and methane are the principal absorbers. Throughout the 3.7-pm window there is also a small contribution from the water-vapor continuum (White et al . 1978). 3.3 Transmittance of Atmospheric Constituents 3.3.1 Water Vapor The transmittance of water vapor in the rectangular subintervals is treated as a product of the transmittances of spectral lines and of continua. For calculating transmittances of spectral lines, we use the method of Weinreb and Neuendorffer (1973). This method demands far less time and memory on the computer than does the line-by-line technique, yet it is nearly as accurate. The method treats the atmosphere as a succession of homogeneous layers, in each of which the pressure, temperature, and mixing ratio are constant. Over the path between the satellite and the bottom of a given layer, the transmittance t^ is computed from a function of the layer's total pressure (P) and temperature (T), and a scaled value of the water vapor amount (U) . For the function of P, T, and U we chose a polynomial representation similar to that suggested by Smith (1969). In our calculation we used the following polynomial expression : lU ln(-lnx £ ) = S ^(v) X i , i~l where t^ is transmittance averaged over the rectangular subinterval, X ± = 1, X 2 = 0.1 In (UT/273), X = ln(P/l000), x k = md/273), x 5 = x 2 x 3 , x 6 = x 2 x h , 2 X 7 - X 2 ' x 8 " X l+ X 7 ' X 9 " X 3 X 1+' 2 X 10 " X 2 X 7' X ll " X U X 6 ' X 12 " X l+ ' X 13 = X 3 X 6 , and X lk = X^. -7- The polynomial coefficients Cj_ were derived by a least -squares fitting of the polynomial to transmittances calculated line by line ( Neuendorf fer 1977) and averaged over the rectangular subintervals for a large dependent sample of homogeneous paths. Table 1 lists these coefficients. The heart of the approximation is the procedure for calculating the scaled values of U in each layer. This is described in detail by Weinreb and Neuendorf fer (1973). The two water vapor continua are usually termed the self -broadened and the foreign-broadened continua. In the former, the absorption coefficient is proportional to the partial pressure of water vapor, while in the latter it is proportional to the partial pressure of the dry atmosphere. Following Roberts et al. (1976), the atmospheric transmittance in the self-broadened continuum is given by the equation, L -in x sb (L,v) = C°(v) f W^ P exp[T o (| - ^)]d£ , ( 5 ) o where t , (L,v) = transmittance between the satellite and a level in the S D atmosphere at a distance L cm from the satellite, W„ = density of water vapor, in molecules cm~3, P = partial pressure of water vapor, in atmospheres, T = reference temperature described below, and C°(v) = coefficient described below. For convenience in computation we have applied the hydrostatic equation to eq. (5) and made some changes in units to obtain -In x . sb P (P,v) = 5.^1xl0 13 C°(v) sec9 p r 2 exp[T Q (| - g^ldp, (6) where 6 = angle between line of sight and the local vertical, P = atmospheric pressure in mb, r = mass mixing ratio of water vapor in g/kg, and t , (P,v) = transmittance between the satellite and a point in the S D atmosphere with pressure P. The values of C° in eq. (6) are listed in table 2. For the 11 ~um region, these values were derived from Roberts et al. (1976). For the 3-7-um region they came from Burch et al . (l97l). For T we use l800°K in the 11-um region, (Roberts et al . 1976), and in the 3-7-ym region we use the value of 1300°K, which was derived from the data of Burch et al. (l97l). -8- The foreign-broadened continuum was ignored in the 11 yra region, because its effect is reported to be negligible (Roberts et al. 1976). In the 3.7-ym region, however, it cannot be ignored, The transmittance Xf for this continuum is given by (Burch 1971 ), L x f (L,v) = y C°(v) j W H Q P D Al . (7) -In The notation in eq. (7) is the same as in eq. (5). Also, P D is the partial pressure of the dry atmosphere, and y is the ratio of foreign broaden- ing to self-broadening. Equation (7) contains no temperature dependence, in part because it is poorly known, and in part because it is small. As suggested by Burch et al. (l97l) we adopted the value 0.12 for y. Inserting this into eq. (7), using the approximation P D = total pressure, and manipulating eq. (7) as we did eq. (5), we obtain -in t (P,v) = U.04xl(T' (T(v) secG I T_ (P,v) = U.Oi+xlO 15 C°(v) sec6 I p r dp. (8) Note that the coefficients C (v) have been selected at the centers of the subintervals of figures 3 and h. Since these coefficients are slowly varying functions of wavenumber, the transmittances calculated from them are representative of averages over the subintervals of figures 3 and k. We have discussed separately the methods of calculation of transmittances for water vapor in spectral lines and the two continua. To obtain the over- all transmittance of water vapor, we take the product of the transmittances of these three components. 3.3-2 Molecular Nitrogen Molecular nitrogen has a collision-induced absorption band centered at 2330 cm - -*-. Following Burch et al. (1971), the transmittance tjj in this band for homogeneous paths (paths where pressure, temperature, and mixing ratios are all constant), is given by, ?6 P A L -In x N (L,v) = 5.67x10 ° C N (T,v) -=— , (9) where P^ = atmospheric pressure in atmospheres, and Cjj(T,v) = Burch's "self- induced" coefficient for nitrogen absorption, in units of molecules" cm^ atm~l. The remaining notation is as in eq. (5). -9- The coefficient Cjj(T,v) has a pronounced dependence on temperature, which is formulated as follows: Susskind and Searl (l9TT) demonstrate that the . ln T N . quantity — = — is very nearly independent of T, where p is numher density of P L _ 3 N2 in units of molecules cm . By combining eq. (9) with the hydrostatic equation, we get ln t n (L,v) C N (T,v) p L T Therefore, Cjj(T,v) has a temperature dependence of — , and we can define a temperature-independent coefficient C (296,v) by C N (296 ' V) = 2^ C N (T ' V) * (10) Combining eqs. (9) and (10) and applying the result to an atmospheric slant path, we obtain 2 t n (L,v) =: 5.67X10 26 C N (2 9 6,v) f (52£) -| d£. (11) 1 Manipulating eq. (ll) as we did eq. (5), we obtain the form used in our computations, -ln x N (P,v) = M7xl0 21 C N (296,v) secB jj p/T d V -, (12) The coefficients Cjj(296,v) were derived from Shapiro and Gush (1966) and are given in the fourth column of table 2. Since they are slowly varying functions of wavenumber, the transmittances calculated from them are representative of averages over the subintervals of figures 3 and h. 3.3.3 Uniformly mixed gases The uniformly-mixed gases comprise CO2, N2O, CO, CH1|, and 02- As mentioned in section 3.2, C02 absorbs weakly in the 11-um window, whereas CO2, N 2 and CHI; absorb weakly in the 3.7-um window. The method of calculating transmittances is taken from LOWTRAN (McClatchey et al. 1972 and Selby et al. 1978) and is summarized in the following paragraphs. -10- The transmittances from LOWTRAN have a spectral resolution of 20 cm and are specified at every 5 cm"-*- throughout the visible and infrared spectrum. Transmittances are computed separately for each of the processes in the atmosphere, e.g., absorption by uniformly mixed gases, absorption by water vapor continua, molecular scattering, etc. The uniformly mixed gases are treated as a unit, with relative concentrations given in McClatchey et al . (1972). These concentrations are built into the model, so that the user does not have to specify them. The basic idea of LOWTRAN is that transmittances through any slant path can be calculated rapidly in a single operation by the equation, x u (v, P,T) = F C u (v) + log 10 u(P,T). , (13) where t u = transmittance of the uniformly mixed gases , C u (v) = wavenumber-dependent coefficient, related to the absorption coefficient representing a 20 cm - -'- interval, a) = "equivalent absorber amount" in the slant path, and F = a known function, specified in a look-up table, as described below. LOWTRAN applies eq. (13) in calculations of transmittances for all molecular species. However, the following material, which describes the variable w and the coefficient C u (v), applies only to the uniformly mixed gases. The "equivalent absorber amount" to, between an atmospheric layer at altitude Z (km) and the satellite, is given by 7 A / \ U /8 . = sece J^ j (Jfj dz, (1U) where T Q and Po are standard temperature and pressure (273.15 K and 1013 mb in LOWTRAN). Manipulating eq. (l^) as we did eq. (5), we obtain the form used in our computations, P , 3. u> = 7.89xl0" 3 sec9 pH^") ! ' dp ' (l5) where the notation is the same as in eq. (6). The wavenumber-dependent coefficients C u (v), taken from Selby et al. (1978), are listed in table 2. -11- For convenience in describing the function F, we define its argument to "be 6, i.e. , 8 = C u (v) + log 10 io(P,T). The argument 6 is evaluated by the procedures already described. The final step in determining t u is to use the look-up table relating 3 to t '. In figure 9 this relation is graphed over the part of its domain that corresponds to .900 < x <_ .999. This range covers the values of transmittance that are encountered in the 3-7- and 11-um windows for the vertical or nearly vertical paths considered here. As an example of the use of the look-up table, figure 9 shows that if one enters a value of -0.5 for 8, he finds a value of approximately 0.97 for x u . Since the values of transmittance derived in this way are averages over 20-cnr" 1 intervals, they can be applied directly in the 20-cm rectangular subintervals in the 3.7-ym window. In the 11-um window, we used the 20-cm - -'- averages for LOWTRAN to represent the transmittances in the 30-cm~l wide rectangular subintervals. Since the absorption is small and varying slowly with wavenumber, we are confident that this additional approximation is permissible. k. COMPUTATIONAL DETAILS AND RESULTS k.l Quadrature The computations of transmittances employ eqs. (6), (8), ( 12), and (15), which involve integrals with respect to atmospheric pressure. In numerically evaluating these integrals, we apply the trapezoidal rule on the 100 quadrature points listed in table 3. These points represent equal increments on the scale of Pw7. In computing radiances we evaluate eq. (3a) by the trapezoidal rule, specifying B and x at the 100 quadrature points listed in table 3. Because the transmittances are already averages over the rectangular subintervals of figures 3 and U, as discussed in the previous section, we do not use eq. (3b). As also discussed previously, the wide-band radiances are computed from eq. (h) , and brightness temperatures are inferred from the look-up table as illustrated in figure 6. With the 100 quadrature points of table 3, we can most conveniently compute radiances for those atmospheres that have their surface pressure equal to 1000 mb. However, the computer program that performs these calculations can also handle atmospheres with surface pressures different from 1000 mb. If the surface has a pressure greater than 1000 mb, we simply include that level in all the integrations as a 101st quadrature point. On the other hand, if the surface has a pressure less than 1000 mb , we retain all 100 quadrature levels in the integrations. However, to all levels whose pressures exceed the surface pressure, we assign a value of temperature equal to the surface temperature and a value of mixing ratio equal to the mixing ratio at the surface. Also, in eq. (3a) the surface term is placed at the 1000 mb level. The radiances that are computed by this procedure are identical to the -12- radiances that would have been computed if we had integrated from the top of the atmosphere down only as far as the actual surface. For completeness we should also mention that the computer program is designed to use two values of temperature at the surface; one to represent the radiative temperature of the surface itself (T s ), and the second to represent the temperature of the atmosphere at the surface (the shelter temperature), e.g. the value of T at P = 1000 mb. Often, however, atmospheres are provided without a value of T s being specified. In that case the program automatically picks a value of T s equal to the shelter temperature. U.2 Examples U.2.1 Atmospheres All calculations were done for the three atmospheres whose temperature profiles are shown in figure 10, and whose profiles of water-vapor mixing ratio are shown in figure 11. They represent a diversity of conditions. All three of these examples have P = 1000 mb at the earth's surface. U.2.2. Transmittances Figures 12 and 13 show transmittances calculated in each of the sub- intervals in the 11- and 3.7-ym windows, respectively, for water-vapor, nitrogen, and the uniformly-mixed gases. The AVHRR spectral response functions also appear in these figures. In figure 12 the water-vapor transmittances were calculated for two atmospheres — 6H°N, labelled "dry" (total precipitable water = 0.75 cm), and 9°N, labelled "moist" (total precipitable water = h.Q6 cm). The absorption by the uniformly mixed gases is unchanged for the two atmospheres. Not only is water vapor the strongest absorber in this spectral region, but it is also the most variable. In figure 13 the transmittances were calculated for only the moist (9°N) atmosphere. Since the absorption is weak in this region of the spectrum, its variability from wet to dry is less than that observed at 11 urn. In any case, water vapor is the principal absorber for wavenumbers higher than about 2600 cm - -'-. Below 2600 cm~l, the nitrogen collision-induced absorption is of significance, and it becomes dominant below 2500 cm"". U.2.3 Atmospheric Attenuation Atmospheric attenuation is used here to mean the difference between the earth's surface temperature and the brightness temperature measured at the satellite. This quantity is of interest because one of the major applications of data from satellite-borne radiometers is in determining surface temperature. In operational sounding we seldom have available the atmospheric temperature and humidity profiles necessary to calculate attenuation by the method of this report. However, these calculations are useful for simulations and for case studies, where data from both the satellite radiometer and concurrent radiosondes are available. -13- Figure ik shows the variation of attenuation with total precipitable water in the atmosphere for a vertical path (sec0 = l). The calculations were done for the AVHRR's channel k (lower panel) and channel 3 (upper panel). In all calculations the surface temperatures were set equal to the temperatures at 1000 mb. Calculations were done separately for each of the three atmospheres of figures 10 and 11. We varied the total precipitable water by multiplying the mixing ratio profiles from figure 10 by 0.1, 0.2, 0.5, 1, 2, 3, and k. However, the mixing-ratios were never allowed to exceed values at saturation. As expected, the attenuation at 11 ym is a stronger function of precipitable water than is the attenuation at 3-7 ym. One interesting result is that for precipitable water less than about 2 cm, the attenuation is greater at 3.7 ym than it is at 11 ym, while for the higher water vapor amounts the reverse is true. This is a consequence of the fact that water vapor is the major absorber in one interval, while the uniformly mixed gases and nitrogen predominate in the other. Figure 15 illustrates a second, perhaps more realistic, approach to demonstrate the dependence of attenuation on total precipitable water in the 11-ym window. The data in this figure were compiled by E.P. McClain (1979) from a set of 60 atmospheres representing a range of typical maritime conditions around the globe. From our computer program, he obtained the attenuation in a vertical path for each atmosphere, and these are plotted as the +'s in the figure. Incident ially, in all calculations, he set the surface temperatures equal to the temperatures at 1000 mb. The solid lines in the figure are the data from figure lk, which are included for the purpose of comparison. Figures l6 and 17 show the variation of attenuation with surface temperature for channels h and 3, respectively, for vertical paths. These calculations were also done separately for each of the three atmospheres. In each, we varied the surface temperature about the 1000 mb value given in figure 10 by 0°K, +3°K, +6°K, +9°K, and +12°K. The arrows indicate the temperature at 1000 mb for each atmosphere. 5 . CONCLUSION The method described here enables us to calculate atmospheric radiances , brightness temperatures, and attenuations in the wide spectral intervals of satellite radiometers in the 3.7- and 11-ym windows. The method involves integrating the radiative transfer equation. Because transmittances are modelled rather than computed line by line, the calculations are rapid. As input, the method requires vertical profiles of atmospheric temperature and water-vapor mixing ratio. Hence its main utility is in case studies and simulations, not in real-time retrievals of surface temperatures from satellite data. -lU- We are better able to estimate the accuracy of the calculations at 11 um than at 3.7 ym. Since we have considerable experience (dating back to the SIRS instruments on the Nimbus 3 and h satellites in the late 1960s) with calculations and measurements at 11 ym, we feel that these calculations are not grossly in error. The error is probably less than 1 mW/(m2 sr cm~l). However, at 3. 7 ym we have little experience, and we are not in a position to offer an estimate of accuracy. There is a need for work on the absorption properties of atmospheric gases in this spectral region. In addition, studies comparing measurements with the calculations have yet to be carried out. The method described here is incorporated in a computer program that is available by writing to the authors. ACKNOWLEDGMENTS The authors thank A.C. Neuendorf fer of NESS for computing the polynomial coefficients in table 1, E. P. McClain of NESS for supplying data in figure l6, and Kay Collins for preparing the manuscript. -15- REFERENCES Anding, D. and R. Kauth, 1969: Atmospheric modelling in the infrared spectral region: atmospheric effects on miltispectral sensing of sea-surface temperatures from space. Report 2676-l-P, Willow Run Laboratories, Institute of Science and Technology, The University of Michigan, Ann Arbor, MI. Bignell, K.J. , 1970: The water vapour infra-red continuum. Quant . J . R . Met. Soc , 96, 390-403. Burch, D.E. , D.A. Gryvnak, and J.D. Pembrook, 1971: Investigation of the absorption of infrared radiation by atmospheric gases: water, nitrogen, nitrous oxide. Semi -Annual Technical Report . No. 2_, AFCRL-71-0124 , Philco-Ford Corporation, Aeronutronic Division, Newport Beach, CA, 25 pp. Davis, P. A. and W. Viezee, 1964: A model for computing infrared transmission through atmospheric water vapor and carbon dioxide. J . Geophys . Res . , 69, 3785-3794. Farmer, C.B. and J.T. Houghton, 1966: Collison-induced absorption in the Earth's atmosphere. Nature , 209, 1341-1342. Lauritson, Levin (National Environmental Satellite Service, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Washington, D.C.), 1979 (personal communication). Maul, George A. and M. Sidran, 1973: Atmospheric effects on ocean surface temperature sensing from the NOAA satellite scanning radiometer. J. Geophys. Res. , 78, 1909-1916. McClain, E.P. (National Environmental Satellite Service, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Washington, D.C.), 1979 (personal communication). McClatchey, Robert A., R.W. Fenn, J.E.A. Selby, F.E. Volz, and J.S. Garing, 1972: Optical properties of the atmosphere (third edition). AFCRL-72-0497, Air Force Cambridge Research Laboratories, L.G. Hanscom Field, Bedford, MA, 108 pp. McClatchey, Robert A., W.S. Benedict, S.A. Clough, D.E. Burch, R.F. Calfee, K. Fox, L.S. Rothman, and J.S. Garing, 1973: AFCRL atmospheric absorption line parameters compilation. AFCRL-TR -7 3-0096, Air Force Cambridge Research Laboratories, L.G. Hanscom Field, Bedford, MA, 78 pp. -16- McMillin, L.M. , D.Q. Wark, J.M. Siomkajlo, P.G. Abel, A. Werbowetski, L.A. Lauritson, J. A. Pritchard, D.S. Crosby, H.M. Woolf, R.C. Luebbe , M.P. Weinreb, H.E. Fleming, F.E. Bittner, and CM. Hayden, 1973 : Satellite infrared soundings from NOAA spacecraft . NOAA Technical Report NESS 65, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Washington, D.C. 112 pp. Neuendorf fer , A.C., 1911' Rapid atmospheric transmittance through fast Fourier convolution. J. Opt. Soc . Am. , 67, 1376. Roberts, Robert E. , J.E.A. Selby, and L.M. Biberman, 1976: Infrared continuum absorption by atmospheric water vapor in the 8-12 urn window. Appl. Opt. , 15, 2085-2090. Saiedy F. and D.T. Hilleary, 1967: Remote sensing of surface and cloud temperatures using the 899 cm - interval. Appl . Opt . , 6, 911-917- Selby, J.E.A. , F.X. Kneizys, J.H. Chetwynd, Jr., and R.A. McClatchey, 1978: Atmospheric transmittance/radiance: Computer code LOWTRAN h. AFGL-TR-78-0053, Air Force Geophysics Laboratory, Hanscom AFB, Bedford, MA, 100 pp. Shapiro, M.M. and H.P. Gush, 1966: The collision-induced fundamental and first overtone bands of oxygen and nitrogen. Can . J . Phys . , kh , 949-963. Smith, W.L. 1969: A polynomial representation of carbon dioxide and water vapor transmission. ESSA Technical Report NESC 47, Environmental Science Services Administration, U.S. Department of Commerce, Washington, D.C, 20 pp. Smith, W.L., P.K. Rao, R. Koffler, and W.R. Curtis, 1970: The determination of sea-surface temperature from satellite high-resolution infrared window radiation measurements. Mon. Wea. Rev. , 8, 6o4-6ll. Susskind, J. and J.E. Searl , 1977: Atmospheric absorption near 2400 cm J. Quant. Spectrosc. Radiat . Transfer , 18 , 581-587 . Wark, David Q. , Yamamoto, G. , and Lienesch, J.H., 1962: Methods of estimat- ing infrared flux and surface temperature from meteorological satellites. J. Atmos. Sci. , 19, 369-384. Wark, David Q. and H.E. Fleming, 1966: Indirect measurements of atmospheric temperature profiles from satellites: 1. Introduction. Mon. Wea. Rev. , 94, 351-362. Wark, David Q. , J.H. Lienesch, and M.P. Weinreb, 1974: Satellite observations of atmospheric water vapor. Appl . Opt . , 13, 507-511. -17- Weinreb, Michael P. and A.C. Neuendorf fer , 1973: Method to apply homogeneous-path transmittance models to inhomogeneous atmospheres. J. Atmos. Sci . , 30, 662-666. Weinreb, M.P., W.G. Planet, and G.D. Jones, 1977: Transmittance of the atmosphere to infrared solar radiation. J. Opt . Soc . Am. , 67, 1377. White, Kenneth 0., W.R. Watkins, C.W. Bruce, R.E. Meredith, and F.G. Smith, 1978: Water vapor continuum absorption in the 3.5 - U.O-um region. Appl . Opt . , 17, 2711-2720. -18- Table 1. — Polynomial coefficients for spectral lines of water vapor Subinterval v Q (cm 1 )* C l C 2 C 3 C l+ C 5 C 6 1 775. -2.23135 6.123U6 0.1+5051 2.71^98 1.20255 -1.96786 2 805.0 -1.85030 5.82891+ O.U1698 2.97656 l.lll+0l+ -2.671*01 3 835.0 -3.09180 7.0259^ 0.3299*+ 3.39958 1.23001+ -3.1+5263 J» 865.0 -2.81U3O 6.57833 0.39086 3.97101 1.26900 -1+ . 1+6690 5 895.0 -3.271OU 7. 081+77 O.36U51 3.99316 1.13821+ -2.2901+8 6 925.0 -3.87608 7. 81+570 0.28821+ 5.01+81+3 1.23165 -2.31676 7 955.0 -3.96672 7.95368 0.26595 1+. 60590 1.021+71 -1.36628 8 985.0 -1+. 05978 8.80581+ O.187U6 5. 101+ 1+9 1.06l5l+ -2.1+6299 9 2U50.0 -8.1+3013 15.21933 0.60791 0.0 0.0 15.86929 10 21+70.0 -8.72307 23.63057 0.0 0.0 0.0 0.0 11 21+90.0 -6.99218 8.2l+59 I + 0.0 0.0 0.0 38.6951*5 12 2510.0 -6.3853U 8.55110 0.0 0.0 0.91916 35.69U87 13 2530.0 -5.9869I+ 9.68929 -0.17617 3.36765 0.0 2.67673 ik 2550.0 -5.33819 9.31851 0.0 2.95701 0.0 0.0 15 2570.0 _U .2UU0T 9.72573 0.0 1.91971 0.0 -0.9^502 16 2590.0 -3.75231 9.61935 0.0523!+ 1.01+1+05 0.1+61+35 -0.50525 IT 2610.0 -3.21+81+8 9.12599 0.06981+ 0.275^9 0.57222 -0.1+8066 18 2630.0 -2.55092 8.73256 0.16667 -0.1+1303 1.21+931* 1.01+95U 19 2650.0 -2.62371 8.72I+98 0.18783 -0.92050 1.16865 1.31301 20 2670.0 -2.5731+2 8.50253 0.21895 -1.3567^ 1.28026 1.95386 21 2690.0 -3.01532 8.69507 0.13112 -1.1+9397 0.79516 1.757U 22 2710.0 -2.63606 9. 21+895 0.15821+ -0.2769!+ D. 80061 1.56005 23 2730.0 -2.07502 8.203^ 0.12908 -1.29617 1.31776 1.96238 2U 2750.0 -3.17113 8.81213 0.12185 -1.67030 0.65181 1.1+6866 25 2770.0 -2.57576 8.3901+1 0.19882 -1.78166 1.2691+7 1.61+839 26 2790.0 -2.31882 8.7521+9 0.20187 -1.251+1+9 1.13656 1.53516 27 2810.0 -2.292I+2 8.585I+I 0.17273 -0. 7^559 1.09597 1.18283 28 2830.0 -2.58936 9.27357 0.10902 -0.05086 0.83697 0.96999 29 2850.0 -3.20658 9-73757 0.02928 0.23195 0. 325^9 -0.1+7011 30 2870.0 -3.381+21 9.50275 0.01+329 0.71+368 0.31+1+26 -0.78017 31 2890.0 -3.3621+9 8.3211+1 0.18909 1.1+0180 0.59159 -2.23602 *Midpoint of subinterval -20- Table 1. — Polynomial coefficients for spectral lines of water vapor (continued) Sub interval C 7 C 8 C 9 c io c ll C 12 C 13 c ll+ 1 -3.76620 0.16593 0.1+3179 3.83257 2.18960 -1.57369 3.31522 -1.31+526 2 -3.926U1 -2.771+98 0.31976 2.79153 0.3291+5 -1.521U7 3.1+2676 -1.05728 3 -6.1+0380 -1.00006 0.61283 5.99156 0.66930 -0.591+1+3 7. 081+75 -1+. 82659 k -5.7^839 -0.50223 0.78782 6.00689 0.1+8105 -2.76763 7.00733 -1+.611+33 5 -5.69290 -O.6U269 0.30803 7. 71+251 -2.16183 -0.78133 3.28952 -5.121+07 6 -5.89552 -3.71300 0.361+73 6.08009 -3.61897 -0.2180U 6.03967 -1+. 22350 7 -U.T28U6 -3.681+52 0.30339 5.13370 -1.26021 -0.67621 5.1+2607 -1+. 195^6 8 -5.15066 -9.3811+8 0.1+31+82 2.28616 0.52201+ -2.57286 6.1H268 -3.1+9313 9 -20.0U587 0.0 -8.8937^ 20.6359^-116.71+71+7 33.37996 0.0 0.0 10 -50.0381+8 36.53720 -10.09H01 55.29951-13^.7^1+05 1+0.28809 0.0 16.6523!+ 11 2. 2U573 7l+. 765^6 0.0 0.0 0.0 0.0 0.0 5.02699 12 0.0 71.!+ 1+127 0.0 5.61771- 35.201+01 8.97379 -7.0511 k 0.0 13 k. 28756 10.06262 0.0 -13.5551+9 -9.7501+2 0.0 0.0 9.83851 lit 5.W571 -3.73073 0.0 -17.92360 0.0 -2.39925 0.0 1+.1+921+3 15 0.71+369 -2.61+936 0.0 -1321881+ 0.0 -1.99500 0.0 6. 21+828 16 -1.28595 -2.82817 0.0 10.27631+ 0.0 -1 . 52861 0.0 3.20701+ 17 -3.1+9510 -0.911+3!+ 0.0 -7. 71+922 0.0 -0.96192 0.0 1.88801+ 18 -6.33610 -3.22005 -0.00891 0.95721+ 0.0 -0.57106 3. 56l 1+9 -3.29910 19 -5.872U6 -3.23281+ 0.0 3.3281+0 0.0 -0.1+8695 3.61152 -3.901+07 20 -6.16732 -2.3521+1 -0.09758 1+. 72198 0.0 0.0 3.52096 -k. 33986 21 -3.80712 1.72189 -0.11658 -1 . 68101+ 0.0 0.0 0.0 1.11738 22 -1+ . 91668 -3.52099 0.01092 0.1+1617 0.0 0.0 5.7521+6 -1+.6001+7 23 -6.621+88 0.0 -0.16732 1.01783 -2.1+3217 0.0 u . 3U9145 -2.02269 2k -3.1+1+052 1.19811+ 0.0 -1.701614 0.0 0.0 0.0 0.98203 25 -6.19237 0.0 -0.1+1+376 1+. 1+1116 0.0 0.0 0.0 -3.59585 26 -5.86578 -2.1+0387 -0.15961 3.21331+ 0.0 0.0 2.97682 -U. 05581 27 -5.801+59 -3.1+1682 0.0 1+. 01662 0.0 -0.1+5678 3.1+3357 -3.93663 28 -1+. 70158 -6.29812 0.0 1.211+37 0.0 0.0 5.65171+ -3.1+61+06 29 -1.50509 -0.80652 0.0 -6.01+562 0.0 0.0 0.0 1.16613 30 -1.87255 0.0 0.0 -3. 551+67 0.0 0.85632 0.0 0.0 31 -2.62032 2.13675 0.22333 2.59861 2.1U038 0.381+28 0.0 -0.1+1753 -21- Table 2. — Coefficients for water-vapor continuum, nitrogen absorption, and uniformly mixed gases C°(v)xl0 2U C N (296,v)xl0 28 Sub interval v o (cm ) (molec cm atm ) (molec cm atm ) C u (v) 1 775 500 -0.53 2 805 1+21 -1.18 3 835 359 -2.51 1+ 865 310 -5.00 5 895 271 -5.00 6 925 2l+0 -1.71 7 955 216 -1.11 8 985 197 -1.33 9 21+50 1+.30 323 -1.12 10 21+70 3.95 21+0 -1.20 11 21+90 3.65 161+ -1.51+ 12 2510 3.1+0 99.9 -2.26 13 2530 3.15 61+.6 -1.06 lit 2550 2.90 1+7.0 -0.1+5 15 2570 2.75 29. !+ -0.37 16 2590 2.70 23.5 -0.75 IT 2610 2.70 17.6 -2.60 18 2630 2.75 5.88 -2.51 19 2650 2.95 -2.1+2 20 2670 3.20 -2.1+3 21 2690 3.1+5 -2.68 22 2710 3.70 -2.83 23 2730 1+.00 -2.66 2k 2750 1+.35 -2.26 25 2770 It. 65 -2.02 26 2790 5.00 -1.86 27 2810 5.35 -1.79 28 2830 5.70 -1.69 29 2850 6.05 -I.78 30 2870 6.1+0 -1.21 31 2890 6.80 -0.53 "Midpoint of subinterval -22- Table 3. — The 100 quadrature points for integrations over atmospheric pressure Level Pressure (mb) Level Pressure (rah) Level Pressure 1 .0100 35 30.2057 68 271.21*51+ 2 .0225 36 33.0936 69 28U.8863 3 .01*35 37 36.1736 70 299.0103 k .0756 38 39-1+530 71 313.6276 5 .1220 39 1*2.9395 72 328.71*82 6 .1858 Uo 1*6.61*07 73 3l*l*. 3825 7 .2701+ kl 50.561*1* 71* 360.51+09 8 • 3795 k2 51*. 7183 75 377.2336 9 .5169 k3 59.1101* 76 39l+. 1*712 10 .6866 kk 63.71+88 77 1*12.261*2 11 .8928 1*5 68.61*11* 78 1*30.6233 12 1.1399 k6 73.7966 79 1*1*9-5590 13 1.1*323 kl 79.2226 80 1*69.0821 Ik 1.77V7 kQ 81*. 9277 81 1*89.2031* 15 2. 1719 k9 90.920I* 82 509.9338 16 2.6289 50 97.2092 83 531.281*1 17 3.1507 51 103.8028 8)* 553.2655 18 3.7^27 52 110.7098 85 575.8890 19 k. 1+101 53 117.9389 86 599.1656 20 5.158U 51* 125.1*991 87 623.1066 21 5.9931+ 55 133.3993 88 61*7.7232 22 6.9207 56 11*1.61*85 89 673.0267 23 7.9^62 57 150.2558 90 699.0285 2k 9.0758 58 159.2303 91 725-7!*01 25 10.3158 59 168.5813 92 753.1729 26 II.672U 60 178.3181 93 781.3385 27 13.1517 6l 188.1*501 91+ 810.21*86 28 1U.780U 62 198.9869 95 839.911*7 29 16.5050 63 209.9378 96 870.3I+87 30 18.3920 61* 221.3126 97 901.5623 31 20.1*281* 65 233.1210 98 933.5671+ 32 22.6209 66 21*5.3727 99 966.3760 33 21+.9766 61 258.0775 100 1000.0000 3k 27.5025 -23- en 00 h- t£> ID W o o K H d o £ O T E !+H o o w -3- a: H O UJ 00 3 S3 O £ -P o O «H c\j 00 to d o ft to O U H 03 5h -P o Ph o CD CD | N H CI) g, •H -2U- o CO 00 CM O O < CD CO £ 1 1 O CM CD lO (L> CO O CO r^ CO O CO "1 w i - ■- i z i 1 — ■ 1 — 1 i ~ i L - ro O Q - ii to sz — - O ii - - — - O ii - , 1 1 1 1 1 1 1 " 00 1^ CD m * c •H o 3 CO CVJ 00 H o cd -p en CJ h- K 1 1 m o i° lO to o o o o o 3SN0dS3U 3AI±\n3U OJ d o oo CO m E 9 E 0.04 AVHRR CH. 4 (ll/im) NEAN I CENTROID (913.3 cm"') AVHRR CH. 3 (3.7/jm) 240 250 260 270 280 290 300 310 240 250 260 270 280 290 300 310 TEMPERATURE (°K) Figure 5. — Errors in calculated blackbody radiances vs temperature, for the "centroid" approximation and for the approximation of eq. k. -28- % o t ; i i i 1 1 t t i 1 r i T 1 1 w V I _ \ \ \ 1 m (£> \ 1 V _E - "v \ "" £ 1 \\ \ \ - ^E \ \ \ \ \ \ - UJ cm \ \ o - \ \ 52 \ \ \ \ \ \ I - 2 \ « - w CD ^ O \ \ X - Q \ \ o O \ \ - CD « \l f - O \ \ 3. 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GOVERNMENT PRINTING OFFICE: 1980 311-046/103 1-3 -1+0- (Continued from inside front cover) NOAA Technical Reports NESS 59 Temperature Sounding From Satellites. S. Fritz, D. Q. Wark, H. E. Fleming, W. L. Smith, H. Jacobowitz, D. T. Hilleary, and J. C. Alishouse, July 1972, 49 pp. (COM-7 2-5096 3) NESS 60 Satellite Measurements of Aerosol Backscattered Radiation From the Nimbus F Earth Radiation Budget Experiment. H. Jacobowitz, W. L. Smith, and A. J. Drummond, August 1972, 9 pp. (COM-72- 51031) NESS 61 The Measurement of Atmospheric Transmittance From Sun and Sky With an Infrared Vertical Sounder. W. L. Smith and H. B. Howell, September 1972, 16 pp. (COM-73-50020) NESS 62 Proposed Calibration Target for the Visible Channel of a Satellite Radiometer. K. L. Coulson and H. Jacobowitz, October 1972, 27 pp. (COM-73-10143) NESS 63 Verification of Operational SIRS B Temperature Retrievals. Harold J. Brodrick and Christopher M. Hayden, December 1972, 26 pp. (COM-73-50279) NESS 64 Radiometric Techniques for Observing the Atmosphere From Aircraft. William L. Smith and Warren J. Jacob, January 1973, 12 pp. (COM-73-50376) NESS 65 Satellite Infrared Soundings From NOAA Spacecraft. L. M. McMillin, D. Q. Wark, J.M. Siomkajlo, P. G. Abel, A. Werbowetzki, L. A. Lauritson, J. A. Pritchard, D. S. Crosby, H. M. Woolf, R. C. Luebbe, M. P. Weinreb, H. E. Fleming, F. E. Bittner, and C. M. Hayden, September 1973, 112 pp. (COM-73-50936/6AS) NESS 66 Effects of Aerosols on the Determination of the Temperature of the Earth's Surface From Radi- ance Measurements at 11.2 m. H. Jacobowitz and K. L. Coulson, September 1973, 18 pp. (COM-74- 50013) NESS 67 Vertical Resolution of Temperature Profiles for High Resolution Infrared Radiation Sounder (HIRS). Y. M. Chen, H. M. Woolf, and W. L. Smith, January 1974, 14 pp. (COM-74-50230) NESS 68 Dependence of Antenna Temperature on the Polarization of Emitted Radiation for a Scanning Mi- crowave Radiometer. Norman C. Grody, January 1974, 11 pp. (COM-74-50431/AS) NESS 69 An Evaluation of May 1971 Satellite-Derived Sea Surface Temperatures for the Southern Hemisphere. P. Krishna Rao, April 1974, 13 pp. (COM-74-50643/AS) NESS 70 Compatibility of Low-Cloud Vectors and Rawins for Synoptic Scale Analysis. L. F. Hubert and L. F. Whitney, Jr., October 1974, 26 pp. (C0M-75-50065/AS) NESS 71 An Intercomparison of Meteorological Parameters Derived From Radiosonde and Satellite Vertical Temperature Cross Sections. W. L. Smith and H. M. Woolf, November 1974, 13 pp. (COM-75-10432) NESS 72 An Intercomparison of Radiosonde and Satellite-Derived Cross Sections During the AMTEX. W. C. Shen, W. L. Smith, and H. M. Woolf, February 1975, 18 pp. (COM-75-10439/AS) NESS 73 Evaluation of a Balanced 300-mb Height Analysis as a Reference Level for Satellite-Derived soundings. Albert Thomasell, Jr., December 1975, 25 pp. (PB-253-058) NESS 74 On the Estimation of Areal Windspeed Distribution in Tropical Cyclones With the Use of Satel- lite Data. Andrew Timchalk, August 1976, 41 pp. (PB-261-971) NESS 75 Guide for Designing RF Ground Receiving Stations for TIROS-N. John R. Schneider, December 1976, 126 pp. (PB-262-931) NESS 76 Determination of the Earth-Atmosphere Radiation Budget from NOAA Satellite Data. Arnold Gruber, November 1977, 31 pp. (PB-279-633) NESS 77 Wind Analysis by Conditional Relaxation. Albert Thomasell, Jr., January 1979. NESS 78 Geostationary Operational Environmental Satellite/Data Collection System. July 1979, 86 pp. (PB-301-276) NESS 79 Error Characteristics of Satellite-Derived Winds. Lester F. Hubert and Albert Thomasell, Jr. June 1979, 44 pp. (PB-300-754) PE N N STATE UNIVERSITY LIBRARIES NOAA SCIENTIFIC AND TECHNICAL PUB AQQ007ZDi fl4 77 The National Oceanic and Atmospheric Administration was established as part of the Department of Commerce on October 3, 1970. The mission responsibilities of NOAA are to assess the socioeconomic impact of natural and technological changes in the environment and to monitor and predict the state of the solid Earth, die oceans and their living resources, the atmosphere, and the space environment of the Earth. The major components of NOAA regularly produce various types of scientific and technical informa- tion in the following kinds of publications: PROFESSIONAL PAPERS — Important definitive research results, major techniques, and special inves- tigations. CONTRACT AND GRANT REPORTS — Reports prepared by contractors or grantees under NOAA sponsorship. 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