OCTOBER 1996 NOTES AND CORRESPONDENCE 1101NOTES AND CORRESPONDENCESpatial Correlation of Beam-Filling Error in Microwave ]gain-Rate RetrievalsGERALD R. NORTH AND ILYA POLYAKClimate System Research Program, College of Geosciences and Maritime Studies, Texas A &M University, College Station, Texas7 November 1995 and 20 April 1996ABSTRACT In this paper the authors consider the possibility of correlations between the random part of the so-calledbeam-filling error between neighboring fields of view in the microwave retrieval of rain rate over oceans. Thestudy is based upon the GARP (Global Atmospheric Reseamh Program) Atlantic Tropical Experiment (GATE)rain-rate dataset, and it is found that there is a correlation of between 0.35 and 0.50 between the errors in adjacentrainy fields of view. The net effect of this correlation is reducing the number of statistically independent termsaccumulated in forming area and time averages of rain-rate estimates. In GATE-like rain areas, this reductioncan be of the order of a factor of 3, making accumulated standard error percentages increase by a factor of theorder of ,/~. For the Tropical Rainfall Measuring Mission using the microwave radiometer alone, this couldincrease the accumulated random part of the beam-filling error for month-long 5- x 5- boxes from about 1.2%to 2%. The effect is larger for less rainy areas away from the equatorial zone.1. Introduction Understanding the error budget for the retrieval ofmonth-long 5- x 5- box averaged rain rates forthe upcoming Tropical Rainfall Measuring Mission(TRMM; Simpson et al. 1990) is clearly a high priorityfor prelaunch research. One of the largest terms in theerror budget is the sampling error variance due to thetemporal gappiness associated with low earth orbitingsatellites. The main sensor for estimating rain rates overthe oceans is the TRMM Microwave Imager (TMI),which is a dual-polarization, multichannel microwaveradiometer. The individual channels have differentfootprint sizes, but when combined they might bethought of as having a nominal 25-km resolution. Thismeans that features in the rain-rate field smaller thanthis cannot be resolved. When such an instrument isused for the retrieval of rain rates over the ocean, thereis inevitably the so-called beam-filling error (BFE). The BFE comes about because the radiometer measures a field-of-view (FOV) area average of the apparent microwave emission temperature, while the observer desires an estimate of the FOV area-average rainrate. The problem is that the formula relating the pointvalue of the microwave temperature and the point valueof the rain rate is nonlinear (approximately a rising Corresponding author address: Dr. Ilya Polyak, Climate SystemsResearch Program, College of Geosciences and Maritime Studies,Texas A&M University, College Station, TX 77843-3150.E-mail: polyak@csrp.tamu.edusaturating exponential for low rain rates). Hence,straightforward insertion of the measured FOV microwave temperature into the formula does not lead to theFOV average rain rate because of the heterogeneity ofrain rates within the FOV. In fact, the individual retrieval is not unique--infinitely many FOV averagerain rates can be responsible for a single microwavetemperature reading. In this paper we have simplifiedthe problem of the BFE to that of an ideal single-channel instrument (we think of the 19.6-GHz channel, butthat is not necessary). While use of data from morethan one channel can improve accuracy, it can neverremove the ambiguity associated with the BFE. Thissimplification to a single channel allows us to applysimple analytical techniques, bringing out the essentialfeatures of the problem. The BFE has two components, a bias or offset partand a random part with ensemble mean zero. Chiu etal. (1990) used GATE [GARP (Global AtmosphericResearch Program) Atlantic Tropical Experiment]rain-rate data to show that these are each large: thebias is of the order of 40% of the actual rain rate, andthe standard deviation of the random part is of aboutthe same size for an individual retrieval. Theoreticalstudies with a variety of random rain-rate fields haveshown essentially the same results (Ha and North1995 ). Short and North (1990) showed that the BFEwas responsible for most of the error in retrievingrain rates from ESMR-5 using data taken from theNimbus-5 satellite in coincidence with the GATE experiment in 1974.c 1996 American Meteorological Society1102 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VO~zUME 13 The philosophy in compiling month-long averagesover grid boxes has been to correct for the bias basedupon climatology and assume the random part wouldunder accumulation essentially cancel to a negligibleresidual. A back-of-the-envelope calculation of thecancelation effect consists of noting that over themonth there will be about 12 000 readings for an equatorial 5- x 5- grid box. In this estimate, we took anominal footprint size of 25 km (about 400 contiguousFOVs cover a grid box) and assume 30 flush (flushmeans intersection of satellite swath on an overpass andgrid box = "grid box" ) visits (or more realistically 60half-flush visits) to the grid box during the month. Ofthe 12 000 FOVs about 10% will be raining (at leastin GATE-like areas). This gives 1200 raining FOVs.Under accumulation of the data in the box over amonth, cancelation of the random error leads to a reduction in aggregated error at the end of the month oforder or/ 12x/-~0~ ~ 1.15% of the mean rain rate, wherewe have estimated cr m 40% x (average rain rate).This is considered to be a tolerable level of error considering the many other difficulties associated with themeasurements. It should be noted, however, that thepercentage error rises as one leaves the heavy rain-ratezones near the equator. The question we pose in this paper is what if theerrors from one FOV to another are correlated? Thetotal number of independent random error contributions will be effectively less than 1200. We examinethe correlations of random BFE between neighboringFOVs and ask how the effective number of independentFOVs is reduced by this correlation. There are manypossible approaches to the problem. We have chosen aconceptually simple one. The steps in our procedureare to first find the spatial correlation of the BFE ofadjacent FOVs. Having estimated this correlation forthe GATE data, we can estimate the effective numberof independent samples across a GATE scene. This partis subtle since the raining areas are not distributed randomly (independently) across a scene. Actually thereis considerable clustering of the rainy areas.2~ Definitions Consider measurements of the rain rate by a singlechannel microwave radiometer, the FOV of whichcovers a square composed of 7 x 7 GATE files (theseare each 4 km x 4 km) and r(i,j) is a value of instantaneous rain rate observed for the tile (i, j). We havetaken our nominal FOV to have this 28 km x 28 kmsize. In fact, we recall that TRMM will make use ofseveral microwave channels, but we are using only aone-channel analog here. Next we label the tiles within a FOV from 1 to 7 x 7= 49, and let Rk be the rain rate corresponding to thetile number k. Then the area-average rain rate JR]0 isdefined as 49 [R]0 = ~ Rk, (1)and the BFE is 6BF = R([T]0) - JR]0 = calculated - true, (2)where R(T) is the formula for the point-by-point relationship between microwave temperature and rainrate. A first-order formula for the BFE can be derived(Chiu et al. 1990; Ha and North 1995): R 2 (R) 6BF = [(Rk- [ ]o) ]0',,r,,-------77--~,~, ! ' (3)where the derivatives are to be evaluated at a typicalrain rate Ro. For a saturating exponential folTnula[T(R) = A - Be-CR; A, B, C constants depending onmicrowave frequency ], the factor involving derivativesis simply a constant (-C/2) independent of R0. Wesee that the BFE to first order is just proportional ~:o thevariability (sample variance) of the rain rate within theFOV (Chiu et al. 1990; Ha and North 1995): C1 496~F = -- ~ 4-~ ~ (R~ - [R]0)2 k=l C [(Rk- [R]0)2]o. (4) 2 From one FOV to another in a scene and throughtime, 6~F is a. random variable. Its ensemble mean(when raining) is (~5~F), which is the bias. The randompart by definition has zero mean but may exhibit significant correlation from one FOV to another. This lastis our next line of investigation.3. Statistical dependence of the BFE First, let us ask if there should be a correlation of theBFE between neighboring FOVs. Since the BFE is approximately linearly related to the variability wi~thin,we are asking whether variability is correlated with itself from. one place to another. In effect, is the graininess of rain at finescale persistent over distances of order 25 km? It is possible to answer this question theoretically and we present such an argument in. theappendix of this paper for the case of normal statisticsof the rain rate when raining. The answer is that if thereis correlation in the rain field, there will be correlationin the variability field. Next we turn to the real problem using the GATE1data. We have calculated 6UF(i, j); i = 1, "', .10; j= 1, "', 10 for each FOV in the 10 x 10 array coveting the GATE1 scene. These form 100 time series( 1716 terms each for the GATE1 phase). Having thesetime series one can estimate the spatial correlations foreach pair of adjacent [ 6I~F (i, j), 6BF (i ', j' ) ] FOVs. One additional parameter has to be established here.This parameter is the minimum number of the simulOCTOBER 1996 NOTES AND CORRESPONDENCE 1103Series ASeries B Only these points are used in correlation calculationFIG. 1. Scheme of estimation of the correlationcoefficient of the adjacent FOVs.TABLE 2. Number of observation pairs usedto estimate correlations in Table 1. Ji 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10I 15 39 45 6l 58 57 78 147 2072 20 49 36 75 40 95 93 157 1643 13 54 7t 77 58 110 107 154 t484 0 39 63 88 113 147 127 145 1675 42 59 89 81 171 165 138 16l 1836 43 99 129 123 183 170 127 162 1947 40 97 160 184 181 141 131 157 2228 51 103 158 192 173 153 144 165 2249 37 61 136 189 178 154 106 117 4010 0 47 112 134 108 93 49 0 0taneously nonzero terms (see Fig. 1 ) of each pair ofthe adjacent series for which the spatial correlation coefficient is estimated. This number was taken to beequal to 10. If i is the row number and j is the column number(of the 10 x 10 array) then, fixing the i, one can find9 correlation coefficients between the adjacent time seriesj = 1 andj = 2;j = 2 andj = 3; ...;j = 9 andj= 10. Repeating this procedure for each i, one obtains90 estimates of the correlation coefficient that characterize the statistical dependence of the BFE along aneast-west line. The results in Table 1 give the correlation coefficientestimates when the threshold value (for the FOV areaaveraged rain rate) is taken to be 1 mm h-~. The estimates vary significantly [ as well as the number of thecorresponding nonzero pairs of FOVs (Table 2) usedfor their estimation]. The t-statistic values (Devore 1994) given in Table3 show that there are 41, about 45%, statistically significant (at 95% level) and 43 = 84 - 41 statisticallyinsignificant estimates in this case. The number of thestatistically significant estimates is too large not to reject the hypothesis about a zero value of the spatialcorrelation coefficient analyzed. [Our anonymous reviewer showed that, under some assumptions, the likelihood to get 41 (or more) successes out of 83 bychance is less than 10-4.] Analogous estimates (which we do not provide here)for each pair of the adjacent FOVs along the northsouth line and for other threshold values (2 and 3mm h-l) lead to approximately the same conclusion.We repeat the calculation for different thresholds because it may be that the microwave retrieval algorithmsin actual practice may need some flexibility in thechoice of threshold. The summary results (Table 4) obtained by averaging the correlation coefficients over the entire number of cases (90), over the nonzero estimates and overthe statistically significant estimates, show that thenumber of the statistically significant estimates variesfrom about 29% to 48%; the number of the statisticallyinsignificant estimates varies from about 48% to 51%;and the number of cases for which the results could notbe obtained are from 3% to 20% (for the interval ofthe GATE1 observations). The roughly equal number of the statistically significant and insignificant estimates is, possibly, explainedTABLE 1. Spatial correlations (threshold is 1 mm h-~, west-east direction).TABLE 3. The t-statistic values correspondingto the correlations in Table 1.I -0.03 0.56 0.12 0.13 0.44 -0.01 -0.06 0.14 0.36 1 -0.1 4.1 0.8 1.0 3.6 -0.1 -0.6 1.7 5.52 0.31 -0.04 0.03 0.34 0.44 0.04 0.21 0.14 0.05 2 1.4 -0.3 0.2 3.1 3.0 0.4 2.0 1.8 0.63 0.09 0.56 0.09 0.19 0.47 0.17 0.03 0.15 0.00 3 0.3 4.9 0.7 1.7 4.0 1.7 0.3 1.9 0.14 0.00 -0.12 0.01 0.15 0.18 0.25 0.11 0.18 0.26 4 0.0 -0.8 0.1 1.5 2.0 3.1 1.3 2.2 3.55 -0.06 0.15 0.36 -0.11 0.21 0.54 0.13 O.08 0.07 5 -0.4 1.2 3.6 -1.0 2.8 8.1 1.5 1.0 0.96 0.64 0.03 0.34 0.05 0.22 0.19 0.47 0.15 0.38 6 5.3 0.3 4.1 0.6 3.0 2.6 6.0 1.9 5.77 0.56 -0.03 -0.01 0.21 0.09 -0.03 0.20 0.29 0.23 7 4.2 -0.3 -0.1 2.9 1.2 -0.3 2.4 3.8 3.58 0.50 0.21 0.52 0.36 0.03 0.46 0.38 0.12 0.14 8 4.0 2.1 7.6 5.4 0.3 6.4 4.9 1.5 2.09 0.28 0.00 0.32 0.02 0.19 0.80 0.67 0.18 0.22 9 1.7 0.0 3.9 0.3 2.6 16.3 9.3 2.0 1.410 0.00 0.12 0.03 0.37 0.35 0.28 0.18 0.00 0.00 10 0.0 0.8 0.4 4.6 3.8 2.8 1.3 0.0 0.0 Ji 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 i 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-101104 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUX~E 13TABLE 4, Summary results: mean values of the correlationsobtained by averaging over different number of cases. Over all Over nonzero Overestimates estimates statistically significant estimatesN of N of N ofThreshold cases p cases p cases p Along rows (W-E direction)1 90 0.20 84 0.21 41 0.362 90 0.20 81 0.22 36 0.423 90 0.21 72 0.27 26 0.51 Along columns (N-S direction)1 90 0.20 ' 87 0.21 43 0.352 90 0.23 85 0.24 39 0.443 90 0.22 81 0.24 36 0.47by the predominate direction of wind for the GATE1time interval (see Fig. 2). When this direction is alongthe adjacent FOV areas (from one to another), the correlation is maximized; when it is in normal with theadjacent FOV areas, the correlation is minimized. We think that these results confirm the theoreticalconclusion about spatial statistical dependence of theBFE for the adjacent FOVs. The most reliable, statistically significant estimates show that the correlationcoefficient is in the interval of 0.35-0.5. These results characterize only the natural statisticaldependence of the BFE. In the case of satellite observations; some additional component, conditioned bythe possible features of the particular measurement device used can slightly increase the correlations.4. Equivalent number of statistically independent FOVs Next consider the real situation as represented in theGATE data. We will show results for several differentspatial correlation_values between neighboring FOVs.The bottom line is that because rainy areas tend to beclustered, there is considerable reduction in the effective number of independent raining FOVs. Dependingon threshold and correlation coefficient, one can seethat the number is reduced by a factor of roughly 1/3. We assume that the BFE field is approximately isotropic and its spatial correlations can be approximatedby the exponential correlation function p(ij; i'j') = pd,j:,,r, (5)where p(ij; i'j') is the spatial correlation of the (i, j)and (i', j') FOVs, di;;r~, is the distance between FOVs(i,j) and (i',/'). The average BFE over a scene-is 0-2 - ~ 6BF(i, j), (6) Escene N~c (i,j)6~cwhere summations must be conducted over the FOVswith [R]0 ~> threshold, ~ is the set of the FOVs forwhich [R]0 ~> threshold, and Nx is the number of raembers of the set ~f. The ensemble average of E ..... is just the bias referred to earlier. We are interested in the varian.ze ofEs ..... whose square root gives us an estimate of thespread of random errors for a typical scene. If the te~nsin the sum for Es .... were statistically independent, wewould find the variance to be 0-2/N~. When the spatialdependence is taken into account we must include crossterms involving correlation; we find 0-2 0-2 - ~ ~ p(ij; i'j'), (7) M N~ (i,j)E.IC (i,,/,)E~where M is the equivalent number of the FOVs withstatistically independent BFEs. This quantity M is thedesired result of our calculations. Clearly, if p(ij; i'j')vanishes for i ~ i' andj :~ j', then M -- N~c. (It shouldbe noted that the data effects the results by determiningwhich FOVs are above the threshold.) Beford passing to the actual GATE clusters, considera case where there are only 9 rainy FOVs and these arein a 3 x 3 array. We used the above procedure to estimate M. Depending on where the correlation coefficient lies in the interval (0.35-0.50) we can show thatthe effective number of independent variates is roughly2.5-3.0 (instead of 9 in the case of no correlation between neighbors). This extreme situation shows the upper bound on the reduction of independent variates ina scene. Next we use actual GATE data in the computation. The results of the calculations for different p -- 0.1,0.2, .--, 0.9, are given in Table 5, which' shows thatthe mean number of the observed FOVs containing: rainrate above threshold varies from about 12 to 19, depending on the threshold value (the smaller the threshold, the greater the number of corresponding 'FOVsconsidered). This means that, when it is rainy somewhere in the GATE scene, only 12%-19% (on average) of FOVs had enough rain to be used in our calculations. Another important point is that the varianceof the number of rainy FOVs from scene to scene isWind DirectionThe adjacent FOVs withmaximum correlation~ The adjacent FOVs with~-~ minimum correlationFIG. 2. Dependence of the correlation values on the wind direction.OCTOBER 1996 NOTES AND CORRESPONDENCE 1105 T^BLE 5. Mean number of the FOVs (/~) with statistically independent BFEs and the mean number (~') of the observed BFEs [Eq. (2)] (averages over all rainy situations) for different spatial correlation values. Correlations p for lag d = I Mean number of Number of fields the observedThreshold (situations) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 FOVs, ~1 977 14.8 11.1 8.4 6.4 4.8 3.6 2.7 2.0 1.4 19.42 837 11.8 9.1 7.0 5.5 4.2 3.3 2.5 1.9 1.4 15.23 731 9.9 7.8 6.1 4.8 3.8 3.0 2.3 1.8 1.3 12.5very large; when it is raining in a scene, the rain tendsto be widespread, leading to an enlargement of the correlation effect we are studying here. The equivalent number of statistically independentFOVs in a scene, when the correlation coefficient valueis between 0.4 and 0.5, varies from about 4 to 6; it isabout a factor of 3 less than the total number of rainyFOVs. This result tells us that the degree of clusteringin a GATE scene is great.5. Final comments We have shown (theoretically and experimentally)that the BFEs of two adjacent FOVs are statisticallydependent. The most reliable, statistically significantestimates show that the correlation coefficient betweenBFEs of adjacent FOVs is about 0.35-0.5. The equivalent number of the FOVs with independent BFEs isabout three times less than the observed ones. In the ideal accumulation of data from FOVs over amonth, one might expect a great amount of cancelationof the random part of the measurement error. In thecase of the random part of the BFE, there is significantcorrelation between the error in an FOV and that in itsneighbors (when there is rain in each). Hence, the degree of cancelation of the errors is less. In the case ofGATE data it was found that the degree of clusteringof rainy FOVs is great, leading to an enhancement ofthe effect. In fact, for GATE-like rain data, the reduction in the number of independent random error termsis reduced by about a factor of 3. This means that inthe example of 12 000 readings with about 1200 ofthem rainy, we have only of the order of 400 independent error terms. This would lead one to think that thecontribution from the accumulation of random errors isabout a factor of ~-~ larger than previously estimated(few percent). As one moves away from the equatorand the associated heavy rain band, the number of rainyFOVs will be much less than the 1200 estimated here.It would be interesting to repeat the procedures invokedhere for subtropical conditions. Acknowledgments. We thank the anonymous reviewers for helpful comments and design of the figures thatled to an improved paper. We are grateful for the support of NASA through its TRMM Science Team. Themotivation for this work came from discussions withC. Kummerow and T. L. Bell. APPENDIX Theoretical Proof of the Dependence of the BFEs In this appendix we address the problem of whethera random variable like 6BF(i, j) should be expected toexhibit dependence between evaluations at (i, j) andnearby locations (i ', j' ) when the underlying rain-ratefield r(i, j) does exhibit spatially lagged correlation.Roughly speaking, is the "graininess" of the field fromplace to place correlated? To proceed we introduce a convenient matrix notation. The BFE (4) is proportional to 49 l__ ~ (Rk - [R]o)2 = XTX, (A1) 49 k= Iwhere T means transposition, and the random vector 1 rRa ~ ~vxzv (A2) X = ~ Rk -- t J0Jk=l 'where N x N is the number of GATE tiles in a FOV.The right-hand side of the formula (A1) is a quadraticform conveniently considered in statistics for a randomvector X. Let X~ and X2 be two vectors of the type (A2) corresponding to two adjacent FOVs. Then the problemthat we are going to consider can be expressed as follows: Are two quadratic forms, X ~Tx~ and X ~Tx2, statistically dependent? For normally distributed random vectors X~ andthe answer can be easily obtained from the theory presented, for example, in Rao (1973). According to thistheory two quadratic forms Z TA~Z and Z TA2Z (whereZ is a normal random vector, A~ and A2 are nonrandommatrices, elements of which are numbers) are independent if and only if A~A2 = 0 (A3)(~ is a covariance matrix of vector Z ). To apply this theory to our case, let us introducenotations for Z, ~, A~, A2 as follows: xl ) (A4)Z--- X2 'Z= -~2 -2J' (A5)1106 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY Vot,uME 13andwhere C~ and C2 are the covariance matrices of thevectors X~ and X2; C~2 is their cross-covariance matrix;! is the unit matrix of the N x N size; 0 is the matrixof the N x N size, all element of which are zeros. Because in our case ('0 0)A,SA2 = 0JtxC,2 C: 1[,0 I(A8)the two quadratic forms Z TAiZ = X ~TXland(A9) Z TA2Z: X 2Tx2 (A10)are independent if two sets of observations, X~ and X2,are independent (C~2 -- 0). Therefore, the two quadratic forms of-the vectors X~and X2 of the type (A2) are independent only if thevectors are independent. This result is really obviousand, in spite of the fact that rain-rate observations(when above threshold) are not normally distributed,one can expect that the theory will hold in our case aswell. Consider next the dependence of X~ and X2. As itwas shown, for example, by Polyak and North (1995)and by Nakamoto et al. (1990) rain-rate observationsare highly spatially correlated or C~2 :~ 0. Spatial correlations can be traced up to 100 km and more. Moreover, for distances of about 35 km, the spatial correlations of temporally averaged rain rate can be approximated by the exponential correlation function p(d)= 0.9d where d is lag in kilometers. This means that the above quadratic forms (and,therefore, the BFEs of two adjacent FOV) are stctistitally dependent. The reasoning and outcome presented in this appendix are confirmed by the findings based upon the observations in the GATE data as shown in the text. REFERENCES.Chiu, L. S., G. R. North, D. A. Short, and A. McConnell, 1990: Rain estimation from satellites: Effect of finite field of view. d. Geo phys. Res., 95, 2177-2185.Devore, J. L., 1994: Probability and Statistics for Engineering and Sciences. Duxbury Press, 743 pp.Ha, E., and G. R. North, 1995: Model studies of the beam-filling errors for rain-rate retrieval with microwave radiometers. J. At rnos. Oceanic Technol., 12, 268-281.Nakamoto, S., J. B. Valdes, and G. R. North, 1990: Frequency wavenumber spectrum for GATE phase I rainfields. J. Appl. Meteor., 29~ 842-850.Polyak, I., and G. North, 1995: The second-moment climatology of the GATE rain rate data. Bull. Amer. Meteor. Soc., 76, 535 550.Rao, R. C., 1973: Linear Statistical Inferences and Its Applications. Wiley & Sons, 547 pp.Short, D. A., and G. R. North, 1990: The beam filling error inNimbus-5 ESMR observations of gate rainfall. J. Geophys. Res.,95, 2187-2193.Simpson, J., R. Adler, and G. R. North, 1988: On some aspects of a proposed tropical rainfall measuring mission (TRMM). Bull. Amer. Meter. Soc., 69, 278-295.

## Abstract

In this paper the authors consider the possibility of correlations between the random part of the so-called beam-filling error between neighboring fields of view in the microwave retrieval of rain rate over oceans. The study is based upon the GARP (Global Atmospheric Research Program) Atlantic Tropical Experiment (GATE) rain-rate dataset, and it is found that there is a correlation of between 0.35 and 0.50 between the errors in adjacent rainy fields of view. The net effect of this correlation is reducing the number of statistically independent terms accumulated in forming area and time averages of rain-rate estimates. In GATE-like rain areas, this reduction can be of the order of a factor of 3, making accumulated standard error percentages increase by a factor of the order of √3. For the Tropical Rainfall Measuring Mission using the microwave radiometer alone. this could increase the accumulated random part of the beam-filling error for month-long 5°×5° boxes from about 1.2% to 2%. The effect is larger for less rainy areas away from the equatorial zone.