GREENWALD-2005-1

NOTES AND CORRESPONDENCEFast Computation of Microwave Radiances for Data Assimilation Using the “Successive Order of Scattering”MethodT HOMAS G REENWALDCooperative Institute for Meteorological Satellite Studies,University of Wisconsin—Madison,Madison,WisconsinR ALF B ENNARTZ AND C HRISTOPHER O’D ELLDepartment of Oceanic and Atmospheric Sciences,University of Wisconsin—Madison,Madison,WisconsinA NDREW H EIDINGERNational Environmental Satellite,Data,and Information Service,National Oceanic and Atmospheric Administration,Madison,Wisconsin(Manuscript received30April2004,in final form6December2004)ABSTRACTFast and accurate radiative transfer(RT)models are crucial in making use of microwave satellite data feasible under all weather conditions in numerical weather prediction(NWP)data assimilation.A multi-stream“successive order of scattering”(SOS)RT model has been developed to determine its suitability in NWP for computing microwave radiances in precipitating clouds.Results show that the two-stream SOS model is up to10times as fast as and is as accurate as the commonly used delta-Eddington model for weaker scattering[column scattering optical depth(CSOD)Ͻ0.01],but it is less accurate and is slower for higher frequencies(Ͼ30GHz)in cases of moderately strong to strong scattering(CSODϾ5).If two-and four-stream SOS models are used in combination,however,it was found that85.5-GHz brightness tem-peratures computed for1°ϫ1°global forecast fields were more accurate(Ͻ0.5K vs1.5K for CSODϾ0.1) and were executed4times as fast as the delta-Eddington model.The SOS method has been demonstrated as an alternative to other fast RT models for providing accurate and very rapid multiple-scattering calcu-lations at thermal wavelengths for remote sensing studies and demanding applications such as operational NWP data assimilation.1.IntroductionNumerical weather prediction(NWP)models are re-lying increasingly on satellite data,especially passive microwave data,for enhancing forecasts(English et al. 2000;Hou et al.2001).Perhaps the most promising assimilation methods are those that utilize radiance data directly(Eyre et al.1993).These methods have the principal advantage of allowing for better control of errors in the assimilation environment.The downside is that the radiative transfer models and their associated adjoints must be extremely fast,especially for opera-tional NWP,and must treat multiple scattering in cloudy conditions(Greenwald et al.2002).The preferred methods for rapidly computing ther-mal radiances in multiple-scattering atmospheres are the Eddington approximation(e.g.,Kummerow1993) and discrete-ordinate techniques(e.g.,Liu1998;Liu and Weng2002)that attain accuracies to within1–3K, depending on the conditions(Kummerow1993;Smith et al.2002;Liu1998).Another approach that has re-ceived somewhat less attention is the“successive order of scattering”(SOS)method(e.g.,Weinman and Guetter1977;Wendisch and von Hoyningen-Huene 1991).It has the advantage of allowing greater physical insight into the radiative transfer problem and is com-putationally practical for microwave radiative transferCorresponding author address:Dr.Thomas Greenwald,CIMSS/ SSEC,University of Wisconsin—Madison,1225W.Dayton St., Madison,WI53706.E-mail:tomg@©2005American Meteorological Societyproblems because scattering is generally relatively weak.The aim of this note is to evaluate the potential of the SOS method for computing microwave brightness tem-peratures in precipitating clouds for use in global data assimilation through comparisons with a fast radiative transfer model(delta Eddington)and more“exact”Monte Carlo calculations.The delta-Eddington model is the standard in terms of speed and accuracy that is currently used or proposed for use at operational NWP centers.The next section describes the development of a multistream SOS model that includes an algorithm for accelerating convergence of the scattering series.Re-sults are then presented at selected microwave window frequencies(10.7,19.4,37,and85.5GHz)using several case studies,representing a wide range of conditions, and National Centers for Environmental Prediction (NCEP)Global Forecast System(GFS)model fields.2.Model descriptionConsider a plane-parallel atmosphere in which the radiance field is independent of azimuth and obeys the Rayleigh–Jeans law.The governing equations for the upwelling(ϩ)and downwelling(Ϫ)brightness tem-peratures at cosine of the zenith angle␮with respect to optical depth␶are thusdTb͑␶,Ϯ␮͒d␶ϭTb͑␶,Ϯ␮͒ϪJ͑␶,␮͒,͑1͒where the scattering source plus thermal emission source(assumed as isotropic)isJ͑␶,␮͒ϭ␻o2͵Ϫ11P͑␮,␮Ј͒T b͑␶,␮Ј͒d␮Јϩ͑1Ϫ␻o͒T͑␶͒,͑2͒and where␻o is the single-scatter albedo,P(␮,␮Ј)is the scattering phase function,and T(␶)is the atmospheric temperature at␶.The only source of polarization con-sidered here is the ocean or ground surface.Observa-tional and modeling studies both have shown that po-larization originating from other sources such as pre-cipitation particles is typically only a few kelvin(Liu and Simmer1996;Weinman and Guetter1977),though polarization signatures can reach10–20K in rare cases for very intense storms(Spencer et al.1983;Czekala et al.2001).Atmospheric polarization sources are of less concern in the context of data assimilation mainly be-cause of the primitive state of NWP microphysical pa-rameterizations,which provide little or no information on the type and shape of particles responsible for gen-erating polarization signatures observed in clouds.Moreover,there are additional greater sources of error from uncertainties in the dielectric properties of ice particles(Bauer et al.2000)and from3D effects,which have a prominent impact on polarization(Liu and Sim-mer1996).In the SOS method,photons that experience a dif-ferent number of scattering events contributing to the total brightness temperature can be expressed simply as a sum over brightness temperatures associated with each scattering event n asT b͑␶,␮͒ϭT b,0͑␶,␮͒ϩ͚nϭ1ϱT b,n͑␶,␮͒,where T b,0(␶,␮)is the brightness temperature due to thermal emission only(i.e.,zeroth-order scatter): T b,0͑␶,Ϯ␮͒ϭ͑1Ϫ␻o͒͵␶␶1T͑␶Ј͒exp͓Ϫ͑␶ЈϪ␶͒ր␮͔d␶Јր␮,͑3͒subject to the boundary conditions T b(␶ϭ␶*;␮)ϭ␧T s and T b(␶ϭ0;Ϫ␮)ϭ2.725K,where␶*is the total optical depth of the atmosphere,T s is the surface tem-perature,and␧is the surface emissivity.If one assumes that the thermal source varies linearly with height,an analytic expression may be derived for(3)across a thin layer(see the appendix).The scattering source function and upwelling brightness temperatures for each order of scatter n may then be computed recursively through the relationshipsJ nϩ1͑␶,␮͒ϭ␻02͵Ϫ11P͑␮,␮Ј͒T b,n͑␶,␮Ј͒d␮ЈfornՆ0,and͑4͒T b,n͑␶,␮͒ϭ͵␶␶1J n͑␶Ј,␮͒exp͓Ϫ͑␶ЈϪ␶͒ր␮͔d␶Јր␮for nՆ0,͑5͒with the boundary conditions T b(␶ϭ␶*;␮)ϭ(1Ϫ␧)T s and T b(␶ϭ0;Ϫ␮)ϭ0.To solve(3)–(5)numerically,we first employ the standard practices of discretizing the radiance field in␮using quadrature and expanding the phase function as a sum of Legendre polynomials.At microwave wave-lengths,the Henyey–Greenstein function is a very good approximation to the phase function derived by more rigorous means(such as Lorenz–Mie theory),where the expansion yieldsP͑␮,␮Ј͒ϭ͚lϭ02NϪ1͑2lϩ1͒g l P l͑␮͒P l͑␮Ј͒,where N is the number of quadrature points in each hemisphere,g is the asymmetry factor,and P l(␮)and P l(␮Ј)are the l th-order Legendre polynomials.The method of Joseph et al.(1976),which approximates the phase function in the forward direction with a delta function,is often used to improve solution accuracy in cases of strong forward scattering.However,this study found that,overall,delta scaling slightly worsened the results for oblique angles but did provide some im-provement in certain cases at a zenith angle of0°.Delta scaling was therefore not used in the results shown in the next section.Because we want to know the upwelling brightness temperature at the satellite sensor’s zenith angle,which almost never corresponds to the quadrature points,a method must be devised to compute the brightness temperature at any angle.Our solution was to solve simultaneously another set of(3)–(5)at an arbitrary angle␮o.Thus,for example,(4)and(5)becomeJ nϩ1͑␶,␮o͒ϭ␻02͵Ϫ11P͑␮o,␮Ј͒T b,n͑␶,␮Ј͒d␮ЈfornՆ0and͑6͒T b,n͑␶,␮o͒ϭ͵␶␶1J n͑␶Ј,␮o͒exp͓Ϫ͑␶ЈϪ␶͒ր␮͔d␶Јր␮for nՆ0,͑7͒respectively.Both sets of equations(those at the quadrature points and those at␮o)are coupled through T b,n(␶,␮Ј)in(6).To evaluate(5)and(7)numerically in optical depth, a given atmospheric layer was divided into sublayers (referred to here as“layer splitting”)that are thin enough so that essentially a single-scatter approxima-tion can be applied during integration.The key assump-tion is that for a thin layer(5)may be approximated asT b,n͑⌬␶,␮͒ϷJ n͑⌬␶,␮͒͵0⌬␶eϪ␶Јր␮d␶Јր␮ϭJ n͑⌬␶,␮͒͑1ϪeϪ⌬␶ր␮͒,where⌬␶is the sublayer optical depth and J n(⌬␶,␮)is the local scattering source function.Each sublayer was assigned identical␻o and g values.As will be shown later,layer splitting becomes inefficient when scattering becomes more pronounced.The SOS model was validated in32-stream mode by comparing it with a backward Monte Carlo model (Petty1994)for the same test cases and global model fields that are described in the following section.A thin layer optical depth(⌬␶)of0.005was used.Results in-dicated that the SOS model was within0.15K of the Monte Carlo model over the full range of scattering conditions.For moderate scattering,this value repre-sents an error of roughly0.06%.Additional tests of the SOS model to determine the best choice of⌬␶to use(in terms of speed and accu-racy)for a given situation revealed that it depends somewhat on the number of streams in certain scatter-ing regimes.In general,smaller values of⌬␶were re-quired as the number of streams increased.For two, four,and six streams,values of⌬␶of0.035,0.02,and 0.015,respectively,were assumed;all higher streams were set as0.01.Further improvements in speed were obtained by ap-plying a series accelerator algorithm.The algorithm in-volves the addition of a term to the summation of brightness temperatures that is inversely proportional to the gradient of the brightness temperature in the order-of-scatter series.For iteration k it can be writ-ten asQ kϭ͚iϭ1kϪ1T b,iϩT b,k2T b,kϪ1ϪT b,k for kϾ1.The algorithm has the correct behavior in the limit as k→ϱ:because T b,k→0as k→ϱ,the second term vanishes and Q k reduces to the sum of the brightness temperatures.Figure1shows an example of the accel-eration for a highly scattering situation at85.5GHz.In this case computational time was reduced byabout F IG.1.Convergence characteristics of the accelerated SOS model in two-stream mode and Monte Carlo calculations for case 1(see Table1).Zenith angle is53.1°.30%.The benefit of the acceleration,however,is lost when scattering is weak;therefore it was only applied if the column scattering optical depth exceeded0.1.It is common to terminate the series at roughly the precision of the measurements,which was set as0.1K.3.ResultsThe first test of the multistream SOS model uses the same four cases from Smith et al.(2002),based on me-soscale simulations of continental convection during the1986Cooperative Huntsville Meteorological Ex-periment(COHMEX).These cases represent a wide range of conditions(see Table1).Details concerning microphysical assumptions and single-scattering calcu-lations are given by Smith et al.(2002).For all cases and frequencies,the surface emissivity was fixed to0.4and the surface temperature was300K.Polarization was not considered in these calculations.Window frequen-cies at10.7,19.4,37,and85.5GHz were chosen to highlight scattering by precipitation particles.Back-ward Monte Carlo calculations(Petty1994)were also performed to serve as an independent benchmark. Figure2summarizes the results for the two-and four-stream SOS models for an oblique angle often used by conical scanning instruments.For reference, Table2provides brightness temperatures computed from the Monte Carlo model for these cases and the contributions to the brightness temperatures from scat-tering only.It was found that less than10orders of scatter were usually needed to achieve convergence; however,for case4at85.5GHz,18orders of scatter were needed.In weakly scattering situations(i.e.,col-umn scattering optical depthsϽ0.01),the two-and four-stream SOS model errors are under0.3and0.1K, respectively,which are both less than the scattering sig-nals themselves(see Table2).These errors are compa-rable to the delta-Eddington model(results not shown). Errors increase for moderate and strong scattering(i.e., column scattering optical depthsϾ0.01),typically rang-ing between0.5and3K but sometimes as large as4.6K for the two-stream SOS model.The delta-Eddington model outperforms the two-stream model in these situ-ations.We suspect this result may be due to the Ed-dington approximation’s better representation of the radiance field by virtue of the spherical harmonics ex-pansion.The4-stream SOS model,on the other hand, performs at least as well as the delta-Eddington model except for one or two of the more intense precipitation cases(results not shown).As was generally expected, the errors are reduced as the number of streams in the SOS model is increased.There was an exception,how-ever,for case3at85.5GHz,in which the four-stream error was slightly worse than that of the two stream. Recall,the accuracy of the SOS model is not only de-termined by the number of streams,but also by the thin layer thickness⌬␶,which,as discussed earlier,must be reduced as the number of streams increases.Therefore, for this particular case,a smaller value of⌬␶was most likely needed.A more adaptive approach todetermin-F IG.2.SOS model errors as a function of scattering optical depth(defined as the column optical depth weighted by the single-scatter albedo)for the Smith et al.(2002)cases.Zenith angle is53.1°.Also shown are weak,moderate,and strong scat-tering regimes as defined by the relative contribution of scattering to the total brightness temperature(see Table2).T ABLE1.Characteristics of the four cases described by Smith et al.(2002)for testing the accuracy and speed of the radiativetransfer models.Column scattering optical depth*Case Surface rain rate Liquid water Ice water10.6GHz19.35GHz37GHz85.5GHz 10.94Low Low0.00100.00880.0620.35 20.34Low High0.000990.00780.0450.29 39.46High Low0.0180.20 1.6 6.0 490.2High High0.20 1.91235*Defined as the vertically integrated optical depth weighted by the single-scatter albedo.ing the optimal value of⌬␶could be implemented in the future.The four Smith et al.(2002)cases were also used to provide some assessment of the SOS model’s speed relative to the delta-Eddington model.In each case the models were iterated100times on a3-GHz Intel,Inc., Xeon personal computer using the STMicroelectronics, Inc.,Portland Group FORTRAN90compiler with no optimization.It is recognized that the results of this kind of test depend to a certain extent on how effi-ciently the code is written;the large differences seen are unlikely to be overcome by changes in coding alone, however.Results show that for all cases and all fre-quencies,except the two highest frequencies for cases3 and4,the SOS model from two to eight streams is considerably faster than the delta-Eddington model (see Fig.3).These differences can be as large as a factor of10.The clear disadvantage of the SOS model,how-ever,is computing brightness temperatures at higher frequencies in the presence of strong scattering.This slowdown is mainly due to the layer-splitting integra-tion technique that can result in the use of hundreds or even thousands of sublayers.Note that case4is an extreme situation and thus would be a rare event in the context of regional or global data assimilation.It should be emphasized that the previous compari-sons provided limited tests of the SOS model’s perfor-mance.A more complete assessment of the SOS mod-el’s accuracy and speed was undertaken using NCEP GFS fields.These data are made available online and are degraded products(1°ϫ1°horizontal grid and26 vertical levels).The following quantities were taken from the forecast model fields:temperature,relative humidity,and liquid water mixing ratio at all levels and precipitation rate at the ground.Neither cloud ice mix-ing ratio nor instantaneous cloud fraction was available. Because the GFS data products included precipitation rate at the surface only,several crude assumptions were made to distribute vertically the total rate into liquid and ice species.A constant rain rate was assumed from the surface up to cloud base.From there,the fraction of ice(liquid water)precipitation rates was linearly in-creased(decreased)from0to1(from1to0)until a temperature ofϪ20°C was reached,above which only ice existed.Cloud liquid water and ice were similarly partitioned from the freezing level toϪ20°C.Precipi-tation rates were converted to water content using an exponential size distribution.Single-scattering proper-ties for rain,snow,graupel,and hail were computed from Lorenz–Mie theory at discrete frequencies,tem-peratures,and water contents,assuming an exponential size distribution.These calculations were organized in tabular form for interpolation purposes.Extinction co-efficients for gas(water vapor and oxygen)were ob-tained from the optical path transmittance(OPTRAN) model(McMillin et al.1995),and absorption due to cloud liquid water was computed from Liebe et al. (1992).For simplicity,the surface emissivity was fixed to0.4and surface temperature was set to300K,re-gardless of region.Figure4shows differences between the SOS/delta-Eddington models and Monte Carlo calculations at85.5 GHz using GFS12-h forecast fields at0000UTC3 February2004.Both of the models exhibit smallerrors F IG.3.Execution time for100iterations of the SOS model as a function of the number of streams for the four cases of Smith et al. (2002)at three frequencies.Zenith angle is53.1°.The range of results from the delta-Eddington model(dual horizontal lines)is also shown for comparison.T ABLE2.Total brightness temperatures T b computed by the Monte Carlo model for the Smith et al.(2002)cases,along with the contributions to the brightness temperatures from scattering alone(⌬T b),that is,the sum of the brightness temperatures for first-order scatter and greater.Tb(K)⌬T b(K)Case10.619.353785.510.619.353785.5 1143.15217.0241.03261.420.26 2.6216.754.9 2134.68196.19205.19255.940.22 2.2613.276.5 3241.61269.68260.32254.01 4.6922.542.060.7 4270.38246.25208.33150.1825.672.3133.7137.5below column scattering optical depths of about 0.1.As scattering increases,however,the two-and four-stream SOS models diverge,with the two-stream model errors being greater (2–2.5K)and the four-stream model er-rors being considerably less (ϳ0.5K)than the delta-Eddington model errors.This result suggests that a col-umn optical depth of 0.1may serve as a threshold to switch from two-stream to multistream to improve ac-curacy.If this strategy is utilized by combining the two-and four-stream SOS models,we find that the execu-tion time for computing 85.5-GHz brightness tempera-tures for the same GFS dataset is a factor of 4faster than the delta-Eddington model.4.ConclusionsThe purpose of this study was to evaluate the perfor-mance of a radiative transfer (RT)model for comput-ing microwave brightness temperatures based on the idea of successive order of scattering to determine its suitability for NWP radiance data assimilation.A mul-tistream SOS model was developed that numerically integrates the RT equation in optical depth by splitting optically thick layers into thinner layers and that also utilizes an algorithm for accelerating the convergence of the order-of-scattering series.The SOS model was found to be potentially fast and accurate enough to meet the demands of an operational environment and to serve as an alternative to the delta-Eddington ing a small sample of case studies from a previous study,the two-stream SOS model was shown to be up to 10times as fast as and as accurate as the delta-Eddington model for weaker scattering.The SOS model ’s ability to adjust easily to a given situationto improve accuracy is one of its important advantages over other fast RT models.For moderate to strong scat-tering,at least four streams were needed to obtain good accuracy.The drawback of the SOS model is that it is prohibitively slow at higher window frequencies (30GHz and higher)in cases of very intense precipitation.This quality is mainly a consequence of the layer-splitting technique rather than increased computation time due to greater orders of scatter.However,in a test using 1°ϫ1°global model fields it was found that using a combination of two-and four-stream SOS models can provide better overall accuracy in 85.5-GHz brightness temperatures and has 4times as much speed as the delta-Eddington model.Fast SOS-type methods should also prove useful in passive microwave instrument simulation studies and in precipitation retrieval methods that utilize forward RT models (e.g.,Evans et al.1995).With only small modi-fications,the current model also may be applied to in-frared wavelengths.Acknowledgments.This work was supported finan-cially by the Joint Center for Satellite Data Assimilation through NOAA Cooperative Agreement NA07EC0676.Thanks are given to our colleague Peter Bauer at ECMWF for providing his delta-Eddington model code.APPENDIXDerivation of Zeroth-Order BrightnessTemperature The zeroth-order upwelling brightness temperature at height z 2for a layer of ⌬z thickness (z 2Ϫz 1)may be expressed asT b ,0͑z 2,␮͒ϭ͑1Ϫ␻o ͒͵⌬zT ͑z Ј͒exp ͓Ϫ͑␤ext z Ј͒ր␮͔␤ext dz Јր␮,͑A1͒where ␤ext is the volume extinction coefficient of the layer.Assuming that the optical properties of the layer are vertically constant and the thermal source (in this case temperature)varies linearly across the layer and substituting into (A1)yields T b ,0͑z 2,␮͒ϭ͑1Ϫ␻o ͒͑1Ϫℑ͒ϫͫT ͑z 1͒ϩ⌬Tͩ12Ϫ␮⌬␶ϩℑ1Ϫℑͪͬ,͑A2͒where ℑϭexp(Ϫ␤ext ⌬z /␮),and ⌬␶and ⌬T are,respec-tively,the optical depth and temperaturedifferencesF IG .4.Errors in the SOS and delta-Eddington models as com-pared with Monte Carlo calculations as a function of column scat-tering optical depth (a measure of the degree of scattering)using a sample GFS model field.Zenith angle is 0°.across the layer.In a similar way,for the downwelling brightness temperature at z1,T b,0͑z2Ϫ␮͒ϭ͑1Ϫ␻o͒͑1Ϫℑ͒ϫͫT͑z1͒Ϫ⌬Tͩ12Ϫ␮⌬␶ϩℑ1Ϫℑͪͬ.͑A3͒It can be shown that the last term in parentheses multiplied by⌬T in(A2)and(A3)is highly subject to round-off error for small⌬␶;therefore,for⌬␶Ͻ0.2,the following approximation is used:1 2Ϫ␮⌬␶ϩℑ1ϪℑϷ112⌬␶␮.REFERENCESBauer,P.,A.Khain,A.Pokrovsky,R.Meneghini,C.Kummerow,F.Marzano,and J.P.V.P.Baptista,2000:Combined cloud–microwave radiative transfer modeling of stratiform rainfall.J.Atmos.Sci.,57,1082–1104.Czekala,H.,S.Crewell,C.Simmer,A.Thiele,A.Hornbostel,andA.Schroth,2001:Interpretation of polarization features inground-based microwave observations as caused by horizon-tally aligned oblate raindrops.J.Appl.Meteor.,40,1918–1932.English,S.J.,R.J.Renshaw,P.C.Dibben,A.J.Smith,P.J.Rayer,C.Poulsen,F.W.Saunders,and J.R.Eyre,2000:A comparison of the impact of TOVS and ATOVS satellite sounding data on the accuracy of numerical weather fore-casts.Quart.J.Roy.Meteor.Soc.,126,2911–2931. Evans,K.F.,J.Turk,T.Wong,and G.L.Stephens,1995:A Bayesian approach to microwave precipitation profile re-trieval.J.Appl.Meteor.,34,260–279.Eyre,J.R.,G.A.Kelly,A.P.McNally,E.Andersson,and A.Persson,1993:Assimilation of TOVS radiance information through one-dimensional variational analysis.Quart.J.Roy.Meteor.Soc.,119,1427–1463.Greenwald,T.J.,R.Hertenstein,and T.Vukicevic,2002:An all-weather observational operator for radiance data assimi-lation with mesoscale forecast models.Mon.Wea.Rev.,130, 1882–1897.Hou,A.Y.,S.Q.Zhang,A.M.da Silva,W.S.Olson,C.D.Kummerow,and J.Simpson,2001:Improving global analysis and short-range forecast using rainfall and moisture observa-tions derived from TRMM and SSM/I passive microwave sensors.Bull.Amer.Meteor.Soc.,82,659–679.Joseph,J.H.,W.J.Wiscombe,and J.A.Weinman,1976:The delta-Eddington approximation for radiative flux transfer.J.Atmos.Sci.,33,2452–2459.Kummerow,C.D.,1993:On the accuracy of the Eddington ap-proximation for radiative transfer in the microwave frequen-cies.J.Geophys.Res.,98,2757–2765.Liebe,H.J.,P.Rosenkranz,and G.A.Hufford,1992:Atmo-spheric60GHz oxygen spectrum:New laboratory measure-ments and line parameters.J.Quant.Spectrosc.Radiat.Transfer,48,629–643.Liu,G.,1998:A fast and accurate model for microwave radiance calculations.J.Meteor.Soc.Japan,76,335–343.Liu,Q.,and C.Simmer,1996:Polarization and intensity in micro-wave radiative transfer.Contrib.Atmos.Phys.,69,535–545.——,and F.Weng,2002:A microwave polarimetric two-stream radiative transfer model.J.Atmos.Sci.,59,2396–2402. McMillin,L.M.,L.J.Crone,M.D.Goldberg,and T.J.Kleespies, 1995:Atmospheric transmittance of an absorbing gas,4.OPTRAN:A computationally fast and accurate transmit-tance model for absorbing gases with fixed and variable mix-ing ratios at variable viewing angles.Appl.Opt.,34,6269–6274.Petty,G.W.,1994:Physical retrievals of over-ocean rain rate from multichannel microwave imagery.1.Theoretical characteris-tics of normalized polarization and scattering indexes.Me-teor.Atmos.Phys,54,79–99.Smith,E.A.,P.Bauer,F.S.Marzano,C.D.Kummerow,D.McKague,A.Mugnai,and G.Panegrossi,2002:Intercom-parison of microwave radiative transfer models for precipi-tating clouds.IEEE Trans.Geosci.Remote Sens.,40,541–549.Spencer,R.W.,B.B.Hinton,and W.S.Olson,1983:Nimbus-737 GHz radiances correlated with radar rain rates over the Gulf of Mexico.J.Climate Appl.Meteor.,22,2095–2099. Weinman,J.A.,and P.J.Guetter,1977:Determination of rainfall distributions from microwave radiation measured by the Nimbus6ESMR.J.Appl.Meteor.,16,437–442. Wendisch,M.,and W.von Hoyningen-Huene,1991:High speed version of the method of successive order of scattering and its application to remote sensing.Contrib.Atmos.Phys.,64,83–91.。