VOL. 11, NO. 5 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY OCTOBER 1994Evaluating the Design of an Earth Radiation Budget Instrument with System Simulations. Part II: Minimization of Instantaneous Sampling Errors for CERES-][ LARRY STOWENOAA/National Environmental Satellite Data and Information Service, Washington, D.C.RICHARD HUCEK, PHILIP ARDANUY, AND ROBERT $OYCEResearch and Data Systems Corporation, Greenbelt, Maryland(Manuscript received 19 May 1993, in final form 26 January 1994) ABSTRACT Much of the new record of broadband earth radiation budget satellite measurements to be obtained duringthe late 1990s and early twenty-first century will come from the dual-radiometer Clouds and Earth's RadiantEnergy System Instrument (CERES-I) flown aboard sun-synchronous polar orbiters. Simulation studies conductedin this work for an early afternoon satellite orbit indicate that spatial rms sampling errors of instantaneousCERES-I shortwave flux estimates will range from about 8.5 to 14.0 W m-2 on a 2.5- latitude and longitudegrid resolution. Root-mean-square errors in longwave flux estimates are only about 20% as large and range from1.5 to 3.5 W m-2. These results are based on an optimal cross-track scanner design that includes 50% footprintoverlap to eliminate gaps in the top-of-the-atmosphere coverage, and a "smallest" footprint size to increase theratio in the number of observations lying within to the number of observations lying on grid area boundaries. Total instantaneous measurement error depends additionally on the variability of anisotropic reflectance andemission patterns and on the retrieval methods used to generate target area fluxes. Three retrieval proceduresare investigated, all relying on a maximum-likelihood estimation technique for scene identification. Observationsfrom both CERES-I scanners (cross-track and rotating azimuth plane) are used. One method is the baselineEarth Radiation Budget Experiment (ERBE) procedure, which assumes that errors due to the use of meanangular dependence models (ADMs) in the radiance4o-flux inversion process nearly cancel when averaged overgrid areas. In a second (estimation of N) method, instantaneous ADMs are estimated from the multiangular,collocated observations of the two scanners. These observed models replace the mean models in the computationof the satellite fiux estimates. In the third (scene flux) approach, separate target-area retrievals are conductedfor each ERBE scene category and their results are combined using area weighting by scene type. The ERBEretrieval performs best when the simulated radiance field departs from the ERBE mean models by less than10%. For larger perturbations, both the scene flux and collocation methods produce less error than the ERBEretrieval. The scene flux technique is preferable, however, because it involves fewer restrictive assumptions.I. Introduction Measurement of the earth's radiation budget (ERB)from orbiting satellites has been the objective of scientific research and development for over three decades. The reasons for this effort stem from the fundamental role that radiation plays in determining longterm variations in the earth's weather and climate. Oneof the major achievements of ERB measurements, sincethe deployment of broadband instruments in 1978, isthe decisive conclusion that global cloud cover lowersthe net radiation at the top of the atmosphere, that is,has a cooling effect (Ramanathan et al. 1989; Ardanuyet al. 1991 ). It is also expected that broadband measurements of ERB parameters at the top, bottom, and Corresponding author address: Larry Stowe, NOAA/NESDIS/ASB, E/Ral I:LS, World Weather Bldg., #711, Washington, DC20233.within the atmosphere will improve the forecasts ofmedium- and long-range weather. These measurementsmay also assist in understanding and predicting theeffects on climate of natural and anthropogenically induced changes in the "earth-atmosphere system"(NOAA 1988). Currently, there are no broadband scanners in space,but a series of future satellite missions plan to havethem aboard. In the mid-1990s, the French SCARABis scheduled for launch aboard a Russian Meteor satellite, and the joint Japanese-American TropicalRainfall Measuring Mission (TRMM) will includethe National Aeronautics and Space Administration(NASA)-de~eloped CERES-I (the Clouds and Earth'sRadiant Energy System Instrument). Late in the 1990s,NASA will also employ CERES-I on its sun-synchronous polar platforms, which will be part of the EarthObserving System (EOS). Thus, the baseline ofERB broadband scanner measurements begun byc 1994 American Meteorological Society11691170 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11Nimbus-7 (Jacobowitz et al. 1984) and continued by usERBE (Barkstrom 1984) will be extended into thetwenty-first century, with CERES-I providing much of ssthe new data record.The measurement.accuracy of CERES-I was con- assidered in Part I of this series (Stowe et al. 1993 ) whereerrors in instantaneous, top-of-the-atmosphere (TOA) ~ssimulated satellite fluxes were compared for five uniqueinstrument designs. The designs covered a range of s"scanner" concepts including a fixed array of radiometers, and biaxial, conical, and cross-track scanning -smotions. CERES-I (Fig. 1 ) consists of two componentradiometers, each very similar to the successful ERBE - ~ sscanner. One operates continuously in the ERBE crosstrack mode, while the other rotates the scan planethrough 180- every 36 s. Although spatial samplingerrors were least for the CERES-1 design, its total sampling errors were at or above the 10 W m-2 limit forshortwave flux accuracy required by the user.community (NOAA 1988). Additional uncertainties dueto errors in calibration and earth location, for example,would increase the total measurement error and raisethe likelihood that actual errors will exceed the userrequirement. Refinements to the basic CERES scandesign and to retrieval procedures are investigated inthe current study, therefore, as a means of reducingthe-total sampling error (spatial and angular). Diurnally varying errors will be.the subject of our next publication in this series.Spatial sampling errors are taken up first by examining the effects of different field-of-view (FOV) sizes,scan patterns, and footprint overlap ratios for the .CERES-I cross-track plane scanner (CTPS). Optimalparameter values are sought that yield minimum spatialsampling errors in shortwave reflected flux estimates.When angular sampling errors are added, the totalsampling error rises above the minimum spatial-onlycomponent. Alternative procedures for combining theradiances of the CTPS and the rotating-azimuth planescanner (RAPS) are explored, all based on scene identification using the maximum-likelihood estimation(MLE) procedure of Wielicki and Green ( 1989). During this latter phase, the design parameters of both radiometers are set to the optimal values found in thespatial error analysis for the CTPS, and only modifications to the MLE retrieval procedure are investigated.A description of the TOA reference fields used tosimulate the earth and atmosphere is presented in section 2, and the time and space domains of the studyare established. Section 3 treats the spatial samplingproblem by considering six possible designs for theCERES CTPS, each with a different com~bination ofFOV size and overlap configuration (both along- andacross-track overlap are considered), and scan pattern.Angular modeling errors are investigated in section 4by introducing reasonable variations of TOA anisotropy within ERBE scene categories. When satellite radiances are inverted using the ERBE reference models,-12o -11, o -too -9o -8o -70 -6o -.50F1G. 1. CERES-I scan pattern for one complete rotation cycleof the RAPS. Dots indicate TOA locations of FOV midpoints.spatial and angular errors in flux estimates are made.These component errors as well as their total are examined for four distinct retrieval procedures for bothLW and SW radiation. Section 5 concludes by summarizing the likely range of instantaneous total sampling errors that are to be obtained for the CERES-Iand offers guidelines for the instrument design and inversion procedure.2. TOA reference fields A geographic area, extending from 15-S to 45-Nand from 50- to 120-W, is chosen for the study domain(see Fig. 1 ). This region, which occupies roughly 10%of the globe and encompasses extended portions ofocean and land surfaces in both the Tropics and mid~latitudes, is subdivided to a scale of 0.1 - in latitudeand longitude that defines a grid system of approximately 10-km resolution. Reference field ERB data arederived at each grid element from 1800 UTC Geosynchronous Operational Environmental Satellite (GOES)observations and consist of instantaneous TOA longwave (LW) and shortwave (SW) fluxes, and sceneidentification (ID). Six unique GOES images are utilized, one for each day of the study period 25-30 January 1984. The process begins by mapping 8-km (atnadir) International Satellite Cloud Climatology Project (ISCCP) GOES-5 B 1 visible (VIS) and infrared(IR) narrowband counts (Schiffer and Rossow 1985 )to their earth locations. A series of calibration, classi..fication, interpolation, and regression steps transform.the narrowband counts to broadband radiances. Theseprocedures are documented in detail in Stowe et al.(1991, 1993). Scene classification is next performed based .on theGOES-derived broadband radiances and MLE cloudretrieval (Wielicki and Green 1989). Although ERBEscenes (Suttles et al. 1988, 1989) are defined for fivesurface types (ocean, land, snow, desert, and coastal)OCTOBER 1994 STOWE ET AL. 1171and four cloud categories (dear, partly cloudy, mostlycloudy, and ovemast), snow and coastal surface classifications are not employed. A high-resolution landocean map is used to define continental boundaries,and grid elements straddling the coastlines are takenas land. Land-desert boundaries are defined at 1 o resolution using the land-use dataset given by Matthews(1985). The distribution of snow is variable and notalways known; thus, snow-covered grid areas are treatedsimply as land or desert. This misrepresentation of thesurface should have little impact on the relative or absolute accuracy of different CERES design estimatesbecause the low IR and large VIS radiances obtainedfrom these areas are interpreted as arising from clouds.Because snow and clouds behave nearly the same radiatively, the magnitude of reference fluxes and absolute errors in their satellite estimates should be modifiedonly slightly by the change from snow to cloudy scenes.Relative errors should be even less sensitive to thesechanges because all instrument designs are flown overthe same TOA flux fields and each is influenced in asimilar way. Reference fluxes for each grid element, Fref, arecomputed from the GOES-derived broadband radiances, RGOES, and the scene classification using the relationship ~rRGoES F~e~ = ~, (1) Prefwhere 0~r is the reflectance or emission anisotropicfactor for the ERBE scene category. Regional valuesof the reference field, FREF, are formed by averaging0.1 o grid values over the coarser resolution of targetareas. Both 1.0- and 2.5- target areas, containing 100and 625 grid elements, respectively, are utilized duringanalysis of regional satellite flux errors. Simulations of the proposed EOS-A sun-synchronous polar platform are flown in an orbit of altitude1.153525155-5 -15 -t20 x~- t C5:~ -110 -100 -90 -80 -?0 -60 -50 FIG. 2. Mean 1800 UTC SW flux reference field for the 2.5- targetareas for the period 25-30 January 1984. Contour interval is 60W m-2. sun .... PRINCIPAL PLANE OF SUN I~G. 3. Bidirectional reflectance models for clear ocean in the principal plane of the sun. Isotropic (solid), ERBE (dashed), and perturbed ERBE (N = 1.3, dotted) model curves are shown for a solarzenith angle of 30-.705 km and ascending node (AN) local equator crossing time of 1330. The earth-sun geometry is frozen atthe instantaneous time of 1800 UTC (1830 UTC forthe total sampling error studies in section 4), but asatellite ground track on a "rotating earth" is used toobtain a realistic spatial distribution of observations atthe TOA. Although solar insolation and cloud amountare held fixed during any one orbit, daily variations incloud properties and distribution are included by extending the study over six days (25-30 January 1984).Figure 2 is a contour map of the 6-day mean 1800UTC SW flux reference field over the study domain.Fluxes range from a maximum over the Andes mountains in excess of 600 W m-2 to a minimum over theAtlantic and Pacific Oceans of less than 120 W m-2.The maximum flux values are associated with areas ofextensive cloud cover (northern United States, centralMexico, and the Andes). The minima occur over oceanareas with little cloud cover. The distribution of the reference fluxes into upwelling radiances is performed using perturbed anisotropic models p' (Green 1980) that are related to thereference models in the form p'- 1 -- - N, (2) Pref- 1where the isotropic value of I is removed from bothp' and prcr in forming their ratio, and Nis a scale factorthat may increase (N > 1 ) or decrease (N < 1 ) theanisotropic component of the postulated models telative to the reference models. Notice that by rearranging the form of Eq. (2), it is easy to show that p' isnormalized to unity (when integrated over the upwelling hemisphere) if Pref is normalized. Three patterns of TOA emission and reflection areexamined and illustrated in Fig. 3 where scattering forclear ocean in the principal plane of the sun is depicted.Notice in the ERBE models (dashed curve) the lobeof greater-than-isotropic scattering that takes place inthe forward direction near 30- of viewing zenith angle.1172 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME ll TABLE 1. Scan parameters for four CERES-I study designs. Scan characteristics Along-scan Nadir Sampling rate Scan velocity FOV size Along-track overlap repeat On-earth (observations (degrees of nadir angleRadiometer (percent of baseline) overlap (%) (%) cycle (s) time (s) per second) per second)Baseline* 100 50 50 4 1.94 30 66.67HOTHOS 50 50 50 2 0.97 120 133.34COTCOS 50 0 0 4 1.94** 302 66.67**Triangular 50 variable 50 2 1.6 ~ 732 80.96 * Baseline: Diamond-shaped footprint corresponding in area to a full-cone angle of 3.6-.** Effective value (see Lext).This scattering peak is due to specular reflection fromthe ocean surface and, in the ERBE models, occurs atan angle of reflection that is nearly the same as theangle of solar incidence. An isotropic model (N = 0,solid curve) is used to study spatial sampling error,defined as the discrepancy between satellite-derivedTOA flux estimates and the reference field when nouncertainty in the application of ADMs exists. Underknown conditions ofisotropic emission and reflection,satellite radiances can be inverted using Pref = I and,thus, errors arising from scene identification and ADMsvanish. ERBE model reflection' and emission (N = 1,dotted curve) are employed to obtain minimum totalsatellite sampling errors because, in this case, the basicanisotropic behavior of the atmosphere is properlyrepresented by the inversion models (i.e., the ERBEmodels). Angular errors are not eliminated entirely,however, because while CERES observations are inverted using the discrete ERBE scene models, actualfootprints are best described by composite models thataccount for the fractions of different scene types existingwithin an FOV. Anisotropic reflection and emission30% greater than in the ERBE models'(N = 1.3, solidcurve) are used to investigate angular sampling errorswhen the patterns ofTOA anisotropy are different fromthe inversion models. Thirty percent is the range ofanisotropic variability expected for the earth/atmosphere (Stowe et al. 1991 ) and this should also yieldthe range of angular sampling errors produced by theCERES-I. This curve crosses the ERBE model at isotropic values where Pre = 1 [ cf. Eq. (2)]. Near nadir,the ERBE model is less than isotropic and p' < p~f; atlarger zenith angles, the ERBE model is greater thanisotropic and 0' >Prcf.3. Minimization of spatial sampling errora. Simulation of scanner operation Six CERES scanning designs have been tested fortheir ability to measure the reference flux from isotropically reflecting 10-km surfaces within 1- and 2.5-target areas (Stowe et al. 1990). Table I contains asummary of four of these six designs, which yieldeduniquely different spatial sampling error characteristic:;.This includes the ERBE baseline design (also simplyERBE) for which a TOA footprint is modeled as adiamond shape corresponding in area at nadir to a fullcone angle of 3.6-. Figure 4 illustrates the samplingcoverage of ERBE from a satellite position near th,:ascending node equator crossing at 67.5 -W and viewing to the east at approximately 50- of satellite zenithangle. The pattern of half-overlapping footprints bothalong-track and along-scan leaves no gaps in the TOAcoverage and no sampling errors arise due to this effec,~.(i.e., holes in TOA coverage). Yet the footprints andtheir spacing are relatively large at this angle, compared.to a 1- or 2.5- target area, and increase even furtherat larger viewing angles. As a result, a significant fraction of all observations falling within a target also extends over its boundary. This is a second source ofspatial sampling error because information from outside the target is brought into the area average. The scan parameters for the remaining three designs(Table 1 ) are variants of the ERBE design. Each usesa reduced footprint size corresponding in angle to halfthe baseline FOV size, or a 1.8- full-cone angle. Atthis reduced size (one-fourth area of ERBE footprint)50% overlap along-track and along-scan is achieved byhalving the nadir repeat cycle of the ERBE design toer 2 o -63 -62 -61 -60 -59 -58 LONGITUDE (DEGREE)FIG. 4. Spatial sampling coverage of ERBE baseline scanner at 50- of satellite zenith angle.OCTOBER 1994 STOWE ET AL. 11732 s and sampling along-scan at twice the ERBE nadirangle rate, or about 1.11 o per observation. This designwe refer to as HOTHOS (half overlap along-track andhalf overlap along-scan). Exact scanning and samplingrates of 133.34- of nadir angle per second and 120observations per second, respectively, are specified bymaintaining the ERBE ratio of on-earth time to totalscan period constant. Note that this sampling rate isfour times the ERBE value. Footprint overlap is nearly eliminated along-trackand along-scan by subsampling alternate HOTHOSscan lines and observations. This is the COTCOS(contiguous overlap along-track, contiguous overlapalong-scan) design. It effectively represents the ERBE4-s scan repeat cycle and 2.22- per observation nadirangle sampling rate, but for a radiometer with half theERBE FOV. The COTCOS sampling pattern is shownin Fig. 5 and in it, sampling gaps are evident due tothe reduction in footprint size. Near nadir (not shown),adjacent pixels adjoin but they do not overlap. Awayfrom nadir, larger footprints cause some overlap in thealong-track direction. However, no overlap takes placein the along-scan direction because contiguous angularFOVs are converted to abutting footprints when projected to the TOA. Thus, complete spatial coverage isnot achieved by COTCOS. A final reduced-FOV design is the bidirectional/triangular scan (triangular because this is roughly thepattern made on the earth by consecutive scan lines).It uses the 4-s full-scan period of ERBE, but allows forearth sampling on both the "across" and "return"phases of a full-scan cycle. Sampling along-scan is doneat a nadir angle rate of 1.11 o per observation so thathalf overlap along-scan is achieved. Along-track, overlap is variable, but becomes half overlap near satellitenadir. Scanning and sampling rates of 80.96- s-t and73 observations per second, respectively, are based onan off-earth time of 0.8 s at each horizon. Errors forthe other two designs not reported here, the COTHOSdesign and the HOTCOS design, were very similar toHOTHOS and COTCOS, respectively.b. Satellite data inversion method Satellite radiances Rsat are computed during orbitalsimulations by numerical integration of the TOA radiance field over the scanner FOVs, the instrumentpoint spread function set to unity. Measurements aresorted into target areas ( 1 o or 2.5- regions) by the location of their FOV midpoints and individually converted to TOA fluxes, F~t, given by Eq. (3), assumingisotropy: Fsat = lrRsat. (3)This procedure is similar to the ERBE method exceptthat scene identification and the ERBE anisotropicmodels, p~r, are not required when isotropy is assumedas we have done for the spatial sampling problem (seeEQ ' 62oW 61-W 60-WLONGITUDEFIG. 5. Same as Fig. 4 but for COTCOS scanner.section 2). Satellite regional mean values are computedusing 1 n F~E^S = -- Z Fsat.i , (4) /'/ i=1where the summation extends over the number of observations n falling within the target area. Note that allsatellite flux estimates enter the regional averageequally, without weighting. Spatial sampling errors in satellite-derived regionalflux estimates are defined by E = FMEAS -- FREF, (5)where FREF is given by Eq. ( 1 ) for 0.1 - grid elementsaveraged over the target area. Thus, the error dependsonly on the uniformity of sampling within target areasand the amount of extraneous radiation brought intothe satellite value due to footprints extending beyondthe target-area boundaries.c. Error analysis and results The relative accuracies of the different CERES scanner designs are evaluated by computing the bias, rootmean-square (rms), and standard deviation (std) oftheir target area flux errors over the orbital swath anddays used in the simulation. Figure 6 illustrates thesensitivity of the rms error of the ERBE baseline designto a change in satellite zenith angle ~~at processingthreshold for 2.5- target-area resolution. As samplingis restricted to progressively narrower swaths about thesubsatellite track, fewer targets are processed and theerror decreases, at first rapidly, and then more slowlyas a minimum error plateau is approached (solidsquares). Error profiles similar to this are also applicable to the other instrument designs and to 1.0- targetarea resolution as well. The second curve on the figure(open squares) shows the percentage of target areasexcluded from processing as the cutoffangle decreases.To be near the minimum error plateau, a ~'~t processingthreshold of 70- is adopted for this study. This wasalso the maximum satellite zenith angle allowed duringthe processing of ERBE observations. Variations in1174 JOURNAL OF ATMOSPHERIC 25 100 20- -80OO 10- -40n 5- ~ -20 0 0 45 50 55 60 65 70 75 80 85 MAXIMUM SATELLUTE ZENITH ANGLE --=- ERROR -~- REJECTION RATIO FIG. 6. Rms error of ERBE baseline, 2.5- resolution, isotropic SWflux estimates as a function of satellite zenith angle processing threshold (filled squares). The percentage of observations excluded fromprocessing is also shown (empty squares).AND OCEANIC TECHNOLOGYVOLUMELt53525155 _-5 _-15 ~ -120,,, -110 -100 -90 -80 -70 -60 -50l~o. 8. Same as Fig. 7 but for rms daily error.Contour interval is 4 W m-2.cloud meteorology are included by computing "ensemble'' statistics that combine the observations of alldays of the simulation period. Each day and target areavalue enter the ensemble as independent measure. ments. The 6-day mean error (bias) for the ERBE design instrument on board the EOS-A satellite (remember orbit longitude has been fixed at 1800 UTC for the 6 day statistics) is shown as a contour plot in Fig. 7. Maximum bias errors occur along the edges of the sampling.swath and both positive (solid) and negative (dashed) values are found. They range from about + 18 to -30 W m-2 (contour spacing is 4 W m-2) with the largest errors associated with regions of maximum cloudiness. Over the less cloudy oceans, errors at theq53525155-5-15 -120 , , .I '1 I I I I,',';".1 ','~ . -' 't'=75~ I -110 -100 -90 -80 -70 -60 -50 FIG. 7. Distribution of mean bias error of ERBE baseline, 2.5-resolution, isotropic SW flux estimates for the ERBE baseline scannerdesign for the period 25-30 January 1984. Error estimates are basedon one pass per day of a sun-synchronous satellite crossing the equatorat 1330 LT with orbit trajectory repeated each day of the samplingperiod. Contour interval of 4 W m-2: solid--positive, dashed negative.swath edges are less than 6 W m-2. These errors arecaused primarily by the elongation of TOA footprintsas satellite zenith angle increases, allowing radiancesfrom outside the target area to influence the area average within. Near the center of the swath, the biaserrors are everywhere less than 6 W m-2. To examine the variability of error with changes inmeteorology (clouds), Fig. 8 shows the rms error of2.5 - regional daily flux estimates computed for 6 days..Generally, the largest rms errors are in regions havingthe largest bias errors, with the exception of the 40W m-2 error at 3.75-S, 56.25-W. The greatest day-today variability in cloud cover is apparently at this location. Root-mean-square errors are least near thecenter of the swath with values less than 4 W m-2. Table 2 compares the performance of the fourCERES designs at 2.5- target resolution as well as 1.0-in terms of the 6-day ensemble mean (bias), rms, andstd (standard deviation) of the error in satellite fluxestimates over the orbital swath. All designs have smalitbias errors (less than 0.5 W m-2), so the rms error isthe only meaningful statistic upon which to evaluate.TABLE 2. Shortwave flux spatial sampling errors (W m-2) at 2.5-and 1.0- target-area resolution for four CERES-I study designs. Standard Root-meanRadiometer Bias deviation square2.5- Target-area resolutionBaseline -0.43 9.76 9.76HOTHOS -0.16 6.06 6.06COTCOS -0.22 10.72 10.72Triangular -0.24 5.92 5.921.0- Target-area resolutionBaseline -0.67 25.23 25.24HOTHOS -0.25 10.87 10.88COTCOS -0.30 .27.82 27.82Triangular 0.00 11.47 11.47OCTOBER 1994 STOWE ET AL. 1175Errors for the ERBE and HOTHOS designs bring outthe effects of radiometer FOV size because they employthe same scan type (unidirectional) and FOV overlapconfigurations (half along-track and along-scan). Thehalf-overlapping FOVs lead to relatively uniform TOAsampling by each radiometer and no gaps in coverageexist. Yet HOTHOS yields 40% and 60% less rms errorat 2.5- and 1.0- resolution, respectively, than ERBE.The differences are due mainly to a greater "sampling"ratio (i.e., the ratio of the number of observations lyingwithin to the number of observations lying along theboundaries of a target area) obtained when smallerfootprints are used. For the same FOV overlap configuration, the sampling ratio is directly proportional tothe linear dimension of the target area and inverselyproportional to the radiometer footprint cone angle.Thus, halving the FOV size will double the samplingratio and thus reduce the influence of observationstaken along the target perimeter where extraneous information is acquired. Because the TOA footprint sizesof all the CERES designs increase with ~'~t, their sampling ratios depend on ~'~t and also, therefore, theirrms errors, as illustrated for ERBE in Fig. 6. A marked error change occurs between the halfoverlap HOTHOS and contiguous overlap COTCOSscan designs even though both configurations employthe same FOV size and unidirectional sampling sweep.The lower errors for HOTHOS are primarily due tothe elimination of along-scan sampling gaps. Thesegaps are not removed by COTCOS, even at large satellite zenith angles, because no footprint overlap occursin the along-scan direction. We have also considered scanners that eliminateoverlap in only the along-track (alternate HOTHOSscan lines) or along-scan (alternate HOTHOS observations within scan lines) directions. Sampling errorsfor these designs are nearly equal to HOTHOS andCOTCOS errors, respectively. Total design error isdominated not by along-track overlap configuration,but by along-scan overlap. Additional details are givenin Stowe et al. (1990). Examination of Table 2, for 1 o regions, shows asimilar separation of designs by rms error. The onlydifferences are in magnitude, with the minor exceptionthat the triangular design produces an error that isslightly larger than for HOTHOS. The magnitudes ofthe rms errors are two to three times larger than thevalues for 2.5- regions. This is the result of havingamplified the two sources of spatial sampling error (gapsand out-of-target radiances) when the same FOV sizesare mapped into smaller target areas.d. Selection of scanner design To minimize spatial sampling error in the measurement of instantaneous regional fluxes, the CERESscanner should be designed with the smallest possibleFOV size (to eliminate sampling outside the targetarea), and have scanning and sampling rates that avoidgaps in spatial coverage. This may be accomplished byemploying a design where successive along-track andalong-scan footprints overlap by 50%. Not only aregaps in spatial coverage eliminated, but also approximately uniform sampling at the TOA is achieved. Thisimproves the accuracy of retrieval systems such as theERBE method used here where an underlying assumption is that all area elements of the TOA are represented an equal number of times in the arithmeticaverage of Eq. (4). To have a margin in meeting the10 W m-2 user requirements (NOAA 1988) whenother sources of error, such as angular modeling, areincluded, it is also recommended that 1 ) 2.5- targetareas be used for CERES ERB products; and 2) that abidirectional scan pattern be implemented. All of theserecommendations will keep spatial sampling errors ator below the 6 W m-2 level. The bidirectional scan ispreferable because, for the same spatial sampling erroras the HOTHOS design, it has no rapid retrace, a slowerscan rate, and a lower sampling rate, all contributorsto longer life in orbit.4. Minimization of total sampling error A triangular scan pattern for the CERES-I radiometers is adopted and, in accordance with the findings ofsection 3, a 50% footprint overlap at nadir is demandedin both the along-track and along-scan directions. At64.67- nadir angle per second scan velocity (2-s earthsweep time) and a 6-s full scan period ( 1 s offthe earthat each horizon) are chosen. The scan velocity corresponds to approximate ERBE technology (Table 1 ) toobtain 50% overlap every 3 s at nadir, but a diamondcenter-to-apex angle of 1.65- and an increased sampling rate (39.2 observations per second) are required.The two radiometers making up CERES-I are identicalexcept that the RAPS slews in azimuth at a fixed rateof 5- s-t, or equivalently 180- every 36 s. From engineering constraints, after turning 180- in one direction, it reverses the direction of rotation and returnsto 0-. A complete rotation cycle consumes 72 s.a. ERBE retrieval Individual satellite radiance observations are processed using the conversion equation F~t = --, (6) Prefwhere as before F~t is the satellite flux estimate, andp~-f is the ERBE reference model anisotropic factor forthe scene type determined using MLE. Note that whileTOA radiances, and thus R~at, are generated using theanisotropic models of Eq. (2), the radiance-to-fluxconversion is carried out using the reference models.Thus, although anisotropic departures from the meanmodels may exist at the TOA, they are unaccounted1176 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11for in the ERBE retrieval. Regional mean satellite fluxesare computed as in Eq. (4) with a ~'~t threshold of 70 oagain used to reject data at large viewing zenith angles.The number of observations n is variable and dependson whether one or both CERES radiometers are utilizedfor mapping the ERB. Two choices are considered: usethe observations of 1 ) only the CTPS, referred to asERBE (CTPS); or 2) both radiometers, ERBE (CTPSand RAPS).b. Estimation of N retrieval When TOA bidirectional reflectance or limb-darkening functions are related to the ERBE modelsthrough the anisotropic scale factor N of Eq. (2), it ispossible to obtain estimates of N by extracting information common to collocated observations of theCTPS and RAPS. Although differing in size, shape,and orientation, observations are considered collocatedwhen their FOV midpoints fall within the same 0.1 olatitude-longitude grid element. Only a relatively smallnumber of all CERES observations are collocated and,thus, their estimates of N must be assumed applicableto nearby uncollocated pixels as well. The region overwhich N is transferable is a key issue that is not addressed in this study. It is simply taken to be the 2.5-target-area resolution used for the generation of regional flux retrievals. Estimation of N is most simplyperformed for FOVs that contain only one ERBE scenetype and only this case is examined. Fields of viewcontaining more than one ERBE scene type are assumed to have the same N as homogeneous sceneFOVs. Retrieval of regional satellite flux estimates is againconducted using Eqs. (4) and (6), but the referencemodels in Eq. (6) are now replaced by in situ models,p', related to N and ~Oref in the form of Eq. (2). HereN is the regional anisotropy scale factor determinedthrough analyses of collocated CTPS and RAPS observations. A detailed derivation of N is given in theappendix. Summation over n in Eq. (4) uses only theuniformly spaced array of CTPS observations. Thiseliminates spatial oversampling that would occur byincluding the scattered distribution of RAPS footprints.c. Scene fiux retrieval Like estimation of N, the scene flux retrieval makesuse of the observations from both CERES-I radiometersin seeking to minimize angular sampling errors. Butunlike estimation of N, it is unnecessary to assumethat perturbations in TOA anisotropy from the ERBEmodels scale by N. All that is supposed is that the patterns of anisotropy at the TOA cross or nearly crossunity at the same angles as occur in the ERBE models.Then, if the radiance distribution of the combinedscanner dataset is centered closer to isotropic anglesthan for the CTPS alone, ADM error (using the ERBEretrieval method) will be less for the combined CTPSand RAPS dataset. This is illustrated in Fig. 9 for clearocean in the principal plane of the sun along the backscatter direction at a solar zenith angle ~'s,n'Of 50 o. TWOTOA bidirectional reflectance models are represented:the ERBE model (asterisks) and an N = 1.3 perturbation of the ERBE model (solid line). Three observations are depicted on the diagram corresponding 'toa CTPS observation at 20- and two RAPS observationsnear 35- and 60- of viewing zenith. During inversioa,the true (perturbed) atmospheric BDR model (solidline no symbols) should be used. However, as this isunknown, we rely on the ERBE mean models and assume that the resulting error for any one observationcancels when averaged with many other observationsof the same scene type. The vertical distance separatingthe curves gives a measure of this error for individualobservations: the larger the separation between thecurves, the greater the error. In the situation shown,the RAPS observations contain less angular error, andthe errors are of opposite sign, so these errors will tendto cancel. While averaging the observations of the RAPS andthe CTPS should reduce angular modeling errors, anincrease in spatial sampling error should also occur.The multiangular views of the RAPS are distributedirregularly over a region and, because of this, they dis-rupt the uniform array of scan spots produced by theCTPS. The result is a trade-off in which angular sampling error may decrease, but at the expense of increasing spatial sampling error. One way of reducing theeffect of spatial oversampling, having averaged CTPSand RAPS observations by scene type, is to combinethese fluxes weighed by the area occupied by each scenetype.2.02O 1.5122C)O~ 0.5ERBE ~ .....0'O0 1'0 2'0' 3'0' d0 5'0. ' 6'0 ' 7'0 8'0 90 VIEWING ZENITH ANGLE FIG. 9. Bidirectional reflectance factors for clear ocean in the principal plane of the sun for backscattered radiation at a solar zenithangle of 50-. The ERBE mean (asterisks) and ERBE perturl~d (solidline, no symbols) models are shown. The ERBE mean-ERBE perturbed difference is indicated for observations at 20- (CTPS), 35-(RAPS), and 60- (RAPS) of satellite zenith angle.OCTOBER 1994 STOWE ET AL. 1177 If ffj represents the arithmetic average of all CTPSand RAPS observations of scene type j, then the targetarea average flux is given by 1 J FMEAS = ~ ~, nCTPS,i/~. (7) nCTPS j= 1Scene weights in Eq. (7) are based on the uniformspatial array of CTPS observations and given by ratioof the number of observations for a specified scenetype to the total number of CTPS observations.d. Results In Table 3, bias and rms errors in regional satelliteSW flux estimates are given for four retrieval procedures: 1 ) ERBE (CTPS), 2) ERBE (CTPS and RAPS),3) estimation of N, and 4) scene flux. In each case,regional errors are computed separately for each of thesix simulation days and treated as independent observations in 6-day ensemble statistics. Although the samesuborbital track is used each day, the daily variationof error due to changes in satellite viewing angle atfixed earth locations is accounted for by reporting rmserrors across a scan swath, equivalent to rms error overan orbit repeat cycle. The ERBE (CTPS) retrieval results correspond closely to errors obtained by Stowe etal. (1993) for the ERBE instrument. Some differencesare expected because the design parameters of the CTPSand the processing criteria used here depart slightlyfrom those of ERBE. A calibration error in Stowe etal. (1993) resulted in underestimates of instrument errors in that study. The same calibration error has causedunderestimates in the spatial sampling errors reportedin section 3. However, these results accurately representinstrument errors relative to one another and, consequently, the choice of an optimal scan design for thestudy of total sampling error is unaffected by this calibration error (Stowe et al. 1993). We have removed this calibration error in the totalsampling error study reported in this section. Errorsfor the CTPS provide benchmarks for an ERBE-typeinstrument and may be used as a guide in assessing theefficiency of other retrieval procedures. Three cases ofearth-atmosphere anisotropy have been examined andinclude isotropic (N = 0), ERBE mean (N = 1 ), andERBE enhanced anisotropic (N = 1.3) reflectance patterns. The N = 0 experiments provide only spatialsampling errors for the different retrievals since theirADM errors are eliminated by the use of isotropicmodels both at the TOA and in the satellite retrievalsystem. These results are reported in the spatial rmscolumn of Tables 3 and 4. This is done since total rmssampling error has been decomposed into spatial andangular components, given that regional spatial andangular errors are uncorrelated (mean-square errorcomponents are additive) over the study domain asshown in Stowe et al. (1993). For the ERBE (CTPS), spatial sampling error is 8.5W m-2. Note that this is significantly different fromthe value of 5.9 W m-2 reported in Table 2 for thesame 2.5- target resolution, bidirectional (triangular)scan mode, and 70- ~t processing threshold. The discrepancy can be attributed to three factors, two of whichare known to reduced errors in Table 2 relative to Table3. First, calibration error results in about a 20% underestimate in values given in Table 2 (Stowe et al.1993). Second, although overlap configurations are thesame, the radiometer FOV is smaller, and the scanningand sampling rates are faster in section 3 than in section4. These effects combine to again decrease the spatialsampling error (cf. baseline and HOTHOS in Table 2)and contribute to the lower estimates of Table 2. Thethird factor is that slightly different equator crossingtimes ( 1800 and 1830 UTC)--that is, orbital swaths-are analyzed. The effect of this difference is unknown. For N = 1.0 experiments, only relatively small angular errors are found, so that the total rms error isvery nearly equal to the spatial error component. Whenperturbations from the ERBE mean atmosphere exist(N = 1.3), relatively large angular sampling errors of TABLE 3. Instantaneous shortwave measurement errors (W m-2) for the CERES instrument for four retrieval methods: ERBE (CTPS),ERBE (CTPS and RAPS), estimation of N, and scene flux. In parentheses, bias and total rms errors are also given in terms of the percentageof the average reflected flux that they represent. In addition, bias error gradients (south to north) are provided in units of watts per squaremeter per degree of latitude. Two anisotropic experiments are reported, N = 1 (no departure from ERBE mean models) and N = 1.3(approximately 30% more anisotropic than the ERBE models) for the period 25-30 January 1984. Anisotropic Spatial Angular TotalProcessing procedure experiment Bias Bias gradient rms rms rmsERBE N = 1.0 0.2 (0.1) 0.0 8.5 1.9 8.7 (2.6)(CTPS) N = 1.3 -3.6 (-1.0) -0.5 8.5 10.7 13.7 (4.1)ERBE N = 1.0 0.4 (0.1) 0.0 13.8 2.4 14.0 (4.2)(CTPS and RAPS) N = 1.3 -2.2 (-0.7) -0.4 13.8 8.9 16.4 (4.9)Estimation of N N = 1.0 0.4 (0.1) -0.1 8.5 6.5 10.7 (3.2) N = 1.3 0.6 (0.2) -0.3 8.5 8.3 11.9 (3.5)Scene flux N = 1.0 0.1 (0.0) 0.0 9.8 2.0 10.0 (3.0) N = 1.3 -3.1 (-0.9) -0.4 9.8 9.0 13.3 (4.0)1178 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME II TABLE 4. Instantaneous longwave measurement errors (W m-2) for the CERES instrument for four retrieval methods: ERBE (CTPS),ERBE (CTPS and RAPS), estimation of N, and scene flux. In parentheses, bias and total rms errors are also given in terms of the percentageof the average emitted flux that they represent. In addition, bias error gradients (south to north) are provided in units of watts per squaremeter per degree of latitude. The period of coverage is 25-30 January 1984. Anisotropic Spatial Angular TotalProcessing procedure experiment Bias Bias gradient rms rms rmsERBE N = 1.0 0.1 (0.0) 0.0041 1.4 0.2 1.4 (0.6)(CTPS) N = 1.3 0.5 (0.2) -0.0022 1.4 2.3 2.7 (1.1)ERBE N = 1,0 0.0 (0.0) 0.0094 2.4 0.2 2.4 (1.0)(CTPS and RAPS) N = 1.3 0.2 (0. l) 0.0051 2.4 2. l 3.2 (1.3)Estimation of N N = 1.0 0.0 (0.0) 0.0048 1.4 1.3 1.9 (0.8) N = 1.3 0.0 (0.0) -0.0171 1.4 2.1 2.5 (1.0)Scene flux N = 1.0, 0.0 (0.0) 0.00094 1.6 0.2 1.6 (0.7) N = 1.3 0.3 (0.1) 0.00021 1.6 2.2 2.7 (1.;.)10.7 W m-2 arise, and the total sampling error increasesto 13.7 W m-2. By adding the observations of theRAPS, the ERBE retrieval spatial sampling error goesup from 8.5 to 13.8 W m-2. So much oversamplingoccurs, in fact, that the spatial component alone of theERBE (CTPS and RAPS) is greater than the total sampling error for ERBE (CTPS) in a perturbed anisotropicatmosphere (N = 1.3 ). It is noteworthy, nevertheless,that the ERBE (CTPS and RAPS) retrieval produces,as expected, a decrease in angular error componentwhen compared to the ERBE (CTPS), for an N = 1.3experiment. This is a consequence of the better angularcoverage provided by both CERES-I radiometers thanby either scanner alone. Spatial sampling errors for the estimation of N retrieval are identical to those of the ERBE (CTPS) because regional satellite fluxes are, in the reprocessingstep, derived only from the distribution of CTPS footprints. For an N = 1.0 experiment, 6.5 W m-2 of angular sampling error is incurred, larger than either ofthe ERBE retrievals and due to random errors in theregional estimates of N. When the atmosphere becomesperturbed, however, only 8.3 W m-2 of angular sampiing error are found. This is lower than for either ofthe ERBE retrievals and leads to a total rms error of11.9 W m-2, about 2 W m-2 greater than the 10 W m-2instantaneous error level desired by the climate researchcommunity. In the scene flux method, spatial sampling error isdetermined by eliminating both scene misclassificationsand the application of incorrect ADMs during radiance-to-flux inversion..Here, N = 0 experiments aloneare not suitable for this exercise because an accurateseparation of observations into scene types, as requiredby the method, is not possible under isotropic reflectionand emission (using MLE and the reference models).An N = 1.0 experiment is run first to identify the scenesfor all pixels. Although random scene misclassificati0noccurs (with N = 1.0), systematic errors vanish andthe random sampling errors that result cancel in themean. With scene ID supplied, sorting and averagingof observations into scene categories can be performedfor N = 0 and, as always, isotropic models are usedduring inversion to eliminate ADM errors. In the table,spatial sampling error is reported as 9.8 W m-2. Thisis considerably less than the 13.8 W m-2 error of theERBE (CTPS and RAPS) and underscores the correctness of the assumption that the effects of oversampling can be reduced by averaging within ERBE scenecategories. Furthermore, the other primary assumptionof this method is also justified by comparing the angularsampling error components of the scene flux and ERBE(CTPS) retrievals for a perturbed atmosphere. The 9.0W m-2 value obtained with the scene flux method rel:,resents a 17% reduction in the angular sampling errorof the ERBE (CTPS) retrieval and demonstrates thatthe addition of the RAPS observations moves the combined observation dataset closer to a zero-error isotropic sampling angle. Overall, total sampling errors forthe scene flux method are greater than the baselineERBE (CTPS) for N = 1.0 by 1.3 W m-2, but lowerby 0.4 W m-2 for an N = 1.3 experiment. Latitudinal gradients in bias error are found for allretrieval methods, but are greatest for the ERBE(CTPS) (-0.5 W m-2 per degree latitude) and leastfor the estimation of N (-0.3 W m-2 per degree latitude) methods. When converted to flux errors frorn15-S to 45-N, these become -30 and -18 W m-2,respectively. A latitudinal gradient exists because erron;in the retrieval are related to the anisotropic departureof the atmospheric ADMs from the ERBE models (cf.Stowe et al. 1993). The magnitude of latitudinal gra..dients depends on several factors including the retrieval.procedure, the declination of the sun, and the equatorcrossing time of the satellite orbit. Errors in LW fluxes (Table 4) for the different re-.trievals generally follow the trends already seen in the:SW flux analysis, but they are reduced in magnitudeby a factor of 5 or 6 in absolute error and by a factor'of 4 in percent error. ERBE (CTPS) and estimationof Nhave the lowest spatial sampling error componentsat about 1.4 W m-2, and ERBE (CTPS and RAPS)OCTOBER 1994 STOWE ET AL. 1179the largest at 2.4 W m-2. Angular sampling errors forN = 1.0 are 0.2 W m-a for all retrievals except estimation of N, which has a relatively large value of 1.3W m-a. This was also the case for SW fluxes. In addition to the effect of random errors in N, this discrepancy may be due, in part, to the application of thesame regional N factors found for SW to the LW problem. Because LW anisotropic emission factors varyonly slightly with variations in cloud amount [cf., e.g.,ERBE models (Suttles et al. 1989)], scene identification is less important, and mixed as well as homogeneous scenes become suitable to the method. Not onlydoes the statistical number of collocation events increase, but N estimates from individual collocationspots may be more accurate due to reduced scene IDerrors. As a result, LW (not SW) radiation may bemore amenable to the estimation of N method, andthe angular errors given in Table 4 are probably overestimates for this retrieval procedure. When perturbations to the ERBE models are introduced (N = 1.3 ),angular sampling errors for ERBE (CTPS and RAPS),estimation of N, and scene flux are all slightly reducedcompared to ERBE (CTPS), as found previously forSW. The largest total error of 3.2 W m-2 occurs for N= 1.3 and ERBE (CTPS and RAPS); the smallest, 1.4W m-a, is for N = 1.0 and ERBE (CTPS).e. Alternative procedures While the scene flux retrieval reduces angular sampiing error, it also increases spatial sampling error and,thus, the present method may not be an optimal procedure for the reduction of total sampling error. Oneway of decreasing spatial sampling error is through division of the scenes into higher-resolution subdomains,each of which provides a separate scene flux average.The mean scene flux formed by area weighting andaveraging subdomain scene fluxes will usually containlower spatial sampling error than the populationweighted mean scene flux. This is because area weighting reduces the influence on the mean, relative to population weighting, of oversampled, high-error local areafluxes. A limiting spatial scale for this type of approach isdefined by a grid system with area elements centeredon each CTPS observation. Subregions are either sampled by the CTPS alone, in which case no subregionalprocessing or averaging is performed, or they are oversampled by the RAPS. The observation, or weightedaverage of the observations, whose ERBE anisotropicfactor is closest to unity may be retained. Other angularerror reduction strategies are also possible, includingpopulation weighting used in the current scene fluxmethod. Spatial errors should be near the minimumvalues obtainable for a single cross-track scanner. Someangular error reduction, relative to ERBE (CTPS), isalso expected because RAPS and CTPS observationsare still combined to reduce ADM errors. In general,the scene flux method should be optimized for minimum retrieval errors by considering a range of spatialscales and weighted averaging schemes.5. Conclusions Root-mean-square sampling errors of instantaneousCERES SW flux estimates for January will range from8.5 to 14.0 W m-a when based on observations froma 1330 LT sun-synchronous polar platform. Theselimits account for variability in TOA ADMs and differences in retrieval methods, and they apply to a crosstrack scanning design that minimizes the spatial component of the total sampling error. The optimum designrequires the smallest FOV size possible within the constraints of scanning and sampling rates that lead toapproximately half-overlapping footprints, most importantly in the along-scan direction. Small FOV sizedecreases the influence of observations lying along theedges of a target area, where contaminating informationfrom outside the target enters the regional flux estimate.Fifty percent overlap eliminates gaps in spatial coverageand produces nearly uniform sampling at the TOA. Abidirectional scan pattern, while having variable overlap along-track, is preferred over a unidirectional scanbecause it produces near-minimum spatial samplingerrors (6 W m-a at 2.5- and 11.5 W m-2 at 1.0- resolution) at half the scanning and sampling rates. Thus,instruments of this design are likely to survive longerin space. Three methods for the estimation of instantaneousTOA fluxes from satellite radiances were presented(ERBE, estimation of N, and scene flux) and their retrieval accuracies compared. For meteorological conditions in which atmospheric radiances are correctlydescribed by the ERBE angular dependence models,the ERBE retrieval produces the least total rms SWsampling error (8.7 W m-2), although not much lowerthan the error produced by either the scene flux ( 10.0W m-a) or estimation of N ( 10.7 W m-2) methods.All but the estimation of Nmethod meets the 10 W m-2absolute error accuracy requirement identified by theclimate community as necessary for climate modelvalidation. For conditions in which atmospheric emission and reflectance patterns differ significantly (thatis, N = 1.3) from the ERBE model description, theestimation of N method produces the smallest totalerror of 11.9 W m-2. Errors for the scene flux andERBE methods are about 12% and 15% greater, havingvalues of 13.3 and 13.7 W m-2, respectively. Differences in their overall performance is mostly attributableto each method's efficiency in reducing angular sampling error. When atmospheric anisotropic departuresfrom the ERBE models are likely to be large, the estimation of N method does best. This method, however,assumes that perturbations in angular dependencemodels scale by a constant factor N, a condition thatcannot be guaranteed in practice. Thus, the scene flux1180 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME llretrieval is recommended for scenes characterized bylarge anisotropic variations. For other less variablescenes (i.e., IN- 1.01 < 0.1 ), the ERBE inversion isrecommended. There are several aspects of our current simulationthat may cause the errors derived not to be totally realistic. One is the lack of three-dimensional cloud fields,which introduces a systematic error in the observedflux as satellite zenith angle increases, due to the foreshortening of holes in clouds when viewed at large angles. A second is the use of a diamond FOV, ratherthan a more exact replica of the point spread functionfor the truncated diamond FOV of ERBE. A third isthe use of a fixed orbit, rather than a precessing one,which might bias the errors to a particular kind of cloudcover present just over the Andes, for example. Thelast is the assumption that the arithmetic average ofthe 10-kin-element fluxes represents the "true" flux ofthe target area, whereas the correct "true" flux mightbe more accurately computed at each 10-kin elementfrom an integration of the surrounding elements usingprojected solid angle weighting. These sources of errorin the reference field may be explored in the future.However, the results of this study remain valid sincethe various designs are evaluated against a common,and probably quite reasonable, representation of the"truth." Consequently, the relative merits of one designover another should not change significantly, even ifsome of these possible simulation errors significantlyalter the measurement error magnitudes in certain situations. Acknowledgments. Many thanks go to Brenda Vallette of Research and Data Systems Corporation foran outstanding effort in editing and preparing themanuscript for publication. This research was partiallysupported by NASA Contract L90987C. APPENDIX Derivation of Estimation of N Retrievala. Derivation of N for collocated pixel pairs Satellite radiance observations R~, are related to theTOA radiance field RTOA through the measurementequation written as fFOV RToAGd~ Rsat = , (AI) fvGdft ovwhere G is the radiometer point spread function (incorporating the time and angular responses of the detector) and dfi is an element of solid angle at the satellite. The time response is assumed to be instantaneousand the angular response to vary as the cosine of theangle of incident radiation to the sensor normal. Asthis angle is less than 1.65- for the FOV size studied,G is approximated as everywhere equal to unity. TOAradiance Rto^ is expressed in terms of the TOAFret and anisotropy factor O' by P 'Fret RTOA = -- (A2) ~rUsing Eq. (A2) in Eq. (AI), Rsat can be rewritten as Rsat - , (A3)where ~f is the solid-an~e weighted average ofover the measurement footprint and ~' is the averageanisotropic reflectance or emission factor ~ven by fvov o'F~td~ h' = (A4) fvF~d~ ov Within an FOV, ~' is assumed to ~ related to theERBE anisotropic models, pet, throu~ the anisotroificscale factor N of Eq. (2). In general, N varies in space,x, so thin when Eq. (2) is used in Eq. (A4), we obtainfor homogeneous FOVs ~'= ~(h~t - ~) + ~, (AS)where ~ is the desired unknown average anisotropyscale factor for a sin~e FOV. Fomally, ~ is ~ven by fvOV N(p~er - 1 )Fretd~ ~ = , (A6) fFOV (~)F~erd~ 1and is related to the continuous v~able Nby avera~ngand weighting ~th the anisotropic component of theradiance field, (Pr~r - 1 )F~/~. Because variations in p~r due to chan~ng geometrywithin a footprint.are usually small' (on the order of2%), we approximate ~ ~et by the value of mam theFOV midpoint ~. Equation (A5) becomes ~'= ~r~(~) -- l] + 1. (AT)We recall that there is no restriction on the variationof N(x) over a footprint and that ~is its FOV average.Substituting ~. (A7) in Eq. (A3), the measured ra~ance is related to ~ and the ERBE anisotropic modelfor the measurement by ~r(~)- l] + R~ = (A8) In~du~ represen~tions ofEq. (A8) can be ~ttenfor the obse~ations of the CTPS and ~PS with, ingeneral, different values for each of the parameters andOCTOBER 1994 STOWE ET AL. 1181variables. When this is done and their ratio taken, weobtain RCTPS = ]~CTPS(~OCTPS- 1 )+ 1 Fctvs RR^ps /~R^PS(PRAPS -- 1 ) + 1 ffRAPS' (A9)where the subscripts CTPS and RAPS now identify thesource radiometer for the different terms. Altho_ugh Eq. (A9) applies to collocated measurements, Nc2r_PS may nevertheless differ from ~R^PS andsimilarly FCTPS from PRIPS because the footprints arenot necessarily coangular nor coincident. However,their differences are typically small so new parameterswith values near unity, a and/5, are introduced relatingthe pairs. These are defined by PRAPS = O/~CTPS, (A 10 ) ]~RAPS = /5]~CTPS, (A 1 1 )and are useful for error analysis (next section). Equation (Ag) can now be written ]-CTPs(PcTPS- 1)+ 1 1 -, (AI2) K = /5]~CTPS(PRAPS -- 1 ) + 1 awhere K = RCTPS/RRAPS, lOops, and lORAPS are observable quantities; tz and/5 vary about unity; and ~CTPSiS the desired unknown. A solution for ]~crvs, hereafterreferred to as ]~, is given by 1 - aK /~ = (AI3) a/5K(lORAPS -- I ) -- (lOCTPS -- I ) 'Because a and/5 are unknown, they are set to theirnom'inal values of I and the estimate for 2- is given by 1-K /- = (A14) K(PRAPS- 1)- (PcTPs- 1)'b. Choice of regional values of N The distribution of the combined number of regionalCTPS and RAPS observations for a single orbit isshown in Fig. A1 where the 24 rows and 28 columnsof 2.5 o target areas composing the study domain arearranged from north (row 1 ) to south (row 24) downthe figure and west (column 1 ) to east (column 28)leA-to-fight across the figure. The number densities thatare given apply after the rejection of observations atsatellite zenith angles greater than 70-. In addition,target areas located along the east-west edges of thesampling pattern are rejected when application of the70- satellite zenith angle cutoff interrupts the normalsampling coverage of the CTPS within them. A decrease in number density is noted from nadir (northsouth line through the center of sampling distribution)east or west toward the horizon, and from the equator(located between rows 18 and 19 ) toward the northernextreme of the study area. The former is due to theincreased spacing of satellite observations at the TOAColmma o~ ~ ~ ~~ ~g~ ==~ ~g~e -~S~ ~ ~ --- o ~,~o~EE~M~ ~ ~ g~.~ o ~o~ ~ o ~ ~. ~E~m~o _ o ~ ~~-~ ~ ~M~o~=~~ =~ ~gX~o ~~g~ ~0~ F~G. A1. Total number of CTPS and RAPS obse~ations ~thinthe 2.5- remlution m~et are~ of the study domain ~r one orbit ofa sun-synchronous ~tellite crossing the equator at 1330 LT. Rownumbers 1-24 cover the latitude range ~om 45-N to 15-S, res~ctively; column num~n 1-28 span lon~tudes ~om 120- to 50-W.when viewed in equal nadir angle increments at thesatellite. The latter is caused by the gradual decreasein width of regional target areas as longitude lines converge toward the pole. In Fig. A2, the number of collocated pairs of CTPS and RAPS observations withintarget areas are shown to vary from as many as 95 nearnadir to I or 0 toward the edge of the scan swath. Because many of these FOVs have values of a and/5 thatdiffer significantly from unity or contain mixed scenetypes, they may lead to erroneous values of N. It isimportant, therefore, to either identify these observations and eliminate them from further considerationor to use only observations that are insensitive to variations in a and/5. When the differential of ~ in Eq. (A 13) is takenwith respect to a and/5, we obtain PCTPS -- lORAPS K( 1 -- lORAPS)/h72/k/5' (AI5)1182 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11F1G. A2. Same as Fig. 13 but for the total number of collocated CTPS and RAPS observations.where the derivatives are evaluated at a = 1 and /~= 1, and Aa = a -- 1 and/x/~ =/~ - 1. From Eq. (A15),we see that errors in a and/~ will be amplified to largeerrors in 37, when K is near unity. This may occur, forexample, when the CTPS and RAPS view a scene fromapproximately the same direction. The two observations then provide redundant information and no advantage can be gained from the collocated measurements. Since the first term on the right-hand side .ofEq. (A 1 5 ) depends on ( 1 - K)-2, it is the most sensitiveto the difference, 1 - K. We define, therefore, an errorsensitivity factor S as S = (pc.us - O~^PS) 372. (A16) (1 - K)2 For collocated observations, typical ~aiues for ffcxPsand/~A~S may be 300 and 330 W m-2, respectively,with resultant values for Aa of about 0.1. Using thisvalue for Aa in Eq. (AI5) and dropping the secondterm on the right-hand side, we obtain ~ = 0.1S. (A17) In Eq. (A 1 7 ), errors in 37 will be less than 0.1 if theerror sensitivity factor is less than unity. In practice,observed values of $ are usually greater than 4.0 and,as a result, reliable estimates of 37 are not always obtained from individual observations. A reasonableprocedure to use when there are several estimates of37 available for a region is to form a weighted average37x over the different values. Thus, 2 37~- 2 (^:ts)where wi is given by 1/S/, the inverse of the sensitivityof 37 to changes in a. The square of w~ is used as 'theweight factor because it does not change the sign of'thecontributing terms.c. Rejection criteria In the application ofEq. (A18), we wish to eliminatemixed-scene FOVs and other observations that havelittle probability of producing an accurate estimate., of37. From the ERBE anisotropic models and consid FIG. A3. Same as Fig. 13 but for the remaining number of collocated CTPS and RAPS observations after the application 6f datarejection Criteria.OCTOBER 1994 STOWE ET AL. 1183eration of physical constraints on reflected energy, values ofiqless than 0 or greater than 1.69 are unplausible(Stowe et at. 1991 ) and wi is assigned a value of zero.Such observations may arise from mixed-scene FOVsor because a and t5 differ significantly from unity. Here,wi is also assigned a value of 0 if MLE scene determinations for collocated footprints of the CTPS andRAPS differ. These situations are most likely to occurwhen the FOV contains a mix of scene types. In addition, wi is set to zero for observations whose errorsensitivity factor $ is outside the range I SI ~< 10. Incases where no acceptable estimates of N can be obtained for a target area, the ERBE default value of 1 isused for 37a. Figure A3 shows the number of collocatedobservations that remain on a regional basis after theapplication of all data rejection criteria. The large reduction in the number of suitable collocations is determined primarily by the condition that I S[ ~< 10.REFERENCESArdanuy, P. E., L. L. Stowe, A. Gruber, and M. Weiss, 1991: Short wave, longwave, and net cloud-radiative forcing as determined from Nimbus-7 observations. J. Geophys. Res., 96, 18 537 18 549.Barkstrom, B. R., 1984: The Earth Radiation Budget Experiment (ERBE). Bull. Amer. Meteor. Soc., 65, 1170-1185.Green, R. N., 1980: The effect of directional radiation models on the interpretation of earth radiation budget measurements. J. Atmos. Sci., 37, 2298-2313.Jacobowitz, H., H. V. Soule, H. L. Kyle, F. B. House, and the Nimbus-7 ERB experiment team, 1984: The Earth Radiation Budget (ERB) Experiment: An overview. 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