1. Introduction

Rainfall is the primary exchange process within the hydrological cycle. The latent heat released by tropical rainfall plays a crucial role in driving the low-latitude atmospheric circulation. Because of the huge extent of oceans, very little oceanic rainfall data are available from measurements by rain gauges, surface-based radar, and other conventional means. Satellite observation is considered the most reasonable method for oceanic rainfall measurement. Therefore, accurate estimates of rainfall based on satellite-derived data are necessary. They will improve the inputs to climate models and global circulation models and hence will improve our understanding of large-scale climate variability and our knowledge of the workings of the earth–atmosphere–hydrosphere system.

A series of satellites that carried visible–IR imagers or microwave radiometers have been used for detecting global rainfall. One of those is the Special Sensor Microwave/Imager (SSM/I) on the Defense Meteorological Satellite Program (DMSP) polar orbiting satellite, which was first launched in June 1987. It is a seven-channel, four-frequency (19.35, 22.235, 37.0, and 85.5 GHz) microwave radiometer that measures atmospheric–ocean surface brightness temperatures in both horizontal and vertical polarizations at all frequencies except 22.235 GHz, for which there is only a vertically polarized channel. The very clean design of the SSM/I has resulted in very stable, well-calibrated measurements. The operational nature of the DMSP satellite series has resulted in a reliable quasi-continuous data source. Although its spatial resolution at the lowest-frequency channel is rather poor, the SSM/I has been the most useful microwave instrument to date for rainfall estimation.

The proposed Tropical Rainfall Measuring Mission (TRMM) satellite (Simpson et al. 1988) is scheduled to be launched in 1997. The TRMM payload will include the TRMM Microwave Imager (TMI), which is essentially a copy of the SSM/I with a dual-polarized pair of 10.7-GHz channels added. One of the principal goals of TRMM is to determine the distribution and variability of rainfall and latent heat release on a monthly, 5° × 5° basis over the tropical oceans. It was this goal of TRMM that prompted us to develop the monthly rainfall estimation algorithm based on the SSM/I data.

Passive microwave rainfall-rate retrievals are usually based on one of two approaches (Wilheit 1986), although a few algorithms use both. The first method is emission-based, where absorption and emission by liquid cloud drops and raindrops cause rapid increases of brightness temperatures over a radiometrically cold background, such as the ocean. Most emission-based approaches rely on integration of the equation of radiative transfer for hypothetical models of raining atmosphere to predict the brightness temperature for a given rainfall rate. This method was first explored by Wilheit et al. (1977) and has been used by Weinman and Guetter (1977), Huang and Liou (1983), Wu and Weinman (1984), Olson (1987), Kummerow and Weinman (1988), Smith and Mugnai (1988), Mugnai and Smith (1988), and many others.

The second approach is scattering based, where scattering by ice hydrometeors in the upper parts of clouds causes brightness temperature decreases over land or ocean. Using airborne radiometers, Wilheit et al. (1982) first observed brightness temperatures as low as 140 K at 92 GHz and near 183 GHz over convective elements of Tropical Storm Cora. Scattering-based approaches are generally empirically based statistical relationships between satellite-measured brightness temperatures and surface rain rates, which are obtained from coincident ground-based radar measurements. Spencer et al. (1983) first used this approach to retrieve rain rates from the Nimbus-7 Scanning Multichannel Microwave Radiometer (SMMR) 37-GHz data. Rodgers and Siddalingaiah (1983), Spencer (1986), Spencer et al. (1989), and Petty and Katsaros (1990) have used different statistical relationships to retrieve rainfall over ocean and land.

Recently, Kummerow et al. (1989) and Kummerow et al. (1991) have developed a multichannel statistical–physical rain-rate retrieval approach. They established the statistical relationships between multichannel brightness temperatures and rain rates based upon data generated by a cloud radiative model for a wide range of conditions. The model simulates various atmospheric parameters, cloud structures, cloud microphysics, and surface characteristics and then computes their associated brightness temperatures in each of the available channels. Those simulated parameters are adjusted to obtain consistency between observed and calculated brightness temperatures. More recently, Kummerow and Giglio (1994a,b) improved this approach by explicitly accounting for diverse hydrometeor profiles. As a result, potential errors introduced into the theoretical calculations by the unknown vertical distribution of hydrometeors are reduced. Using similar techniques, Olson (1989), Smith et al. (1992), and Mugnai et al. (1993) have retrieved rain rates by employing different cloud models.

Algorithms for estimating area–time-averaged rainfall from satellite measurements have been developed for climatological studies. Shin et al. (1990) retrieved seasonal 5° × 5° area-averaged rain rate over the tropical oceans by using single-channel microwave measurements from ESMR-5. Wilheit et al. (1991) developed an algorithm for the estimation of monthly rain totals for 5° × 5° cells over the oceans from SSM/I brightness temperatures. A linear combination of the 19.35- and 22.235-GHz channels was adopted to reduce the impact of the variability of water vapor. A mixed lognormal distribution was assumed as the probability distribution function of rainfall intensity. The coefficients of the mixed lognormal distribution were determined by matching the observed histogram of brightness temperature with the predicted histogram. Berg et al. (1992) computed monthly, seasonal, and annual oceanic rainfall from the SSM/I using a mixed lognormal distribution. They used the Hughes D-matrix algorithm to estimate the instantaneous rainfall and summed it over 1° × 1° box for each month. Then the ensemble of instantaneous rainfall values was fitted to a lognormal distribution by using a maximum likelihood method.

The previous work for estimating area–time-averaged rainfall has assumed that rain rates over whole dynamic range are available. However, this assumption is not true because an emission-based rain-rate retrieval has its limitations. First, the nonprecipitating clouds and water vapor will introduce the uncertainties in the estimates of low rain rates. Second, high rain rates cannot be retrieved due to the saturation of brightness temperatures. Even with a perfect measurement technique, the very high rain rates will also be poorly sampled with observations from a low orbiting satellite because of their relative infrequency. Thus, rain rate is reliably estimated only over a limited dynamic range so that only a portion of rain rates can be used in the estimation of monthly rainfall. In other words, when area–time-averaged rainfall is estimated, the instantaneous rain-rate samples are truncated. This study tries to get the best estimation of area–time-averaged rainfall from the truncated rain-rate samples.

The approach presented in this study still uses the emission-based method to retrieve instantaneous rain rates because the scattering-based algorithms work on a very tenuous physical connection. McGaughey et al. (1996) found that ice scattering in regions of convective precipitation over oceans during the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) was not as strong as those measured in convection over land. They also found that in tilted convective systems, there was ice aloft in stratiform regions with only light rain below and shallow heavily raining clouds with little ice above. Although scattering-based algorithms would extend the dynamic range of the measurements to higher rain rates, we do not have any confidence that the scattering can be related uniquely to rain rate.

The emission-based method gives the most physically direct estimate of rainfall over oceans because it represents observations of the liquid hydrometers. Its limitations can be overcome in the estimate of monthly rainfall by applying the maximum likelihood estimation algorithm to the truncated rain-rate samples; a brief description of data is given in section 2. The algorithm is described in detail in section 3. Section 4 applies this algorithm to the SSM/I data and provides the comparison between satellite estimated monthly rainfall with the Pacific atoll data. A summary and conclusions are given in section 5.

The algorithm discussed in this paper is a refinement of the Wilheit et al. (1991). It differs in the following ways.

1. The brightness temperatures are converted into rain rates (and freezing levels) on a pixel-by-pixel basis before accumulation into histograms.

2. The parameters of the lognormal distribution of rain rates are found by a maximum likelihood estimator that takes the limited dynamic range of the measurements into account as opposed to the ad hoc method of estimation used by Wilheit et al. (1991).

3. The 37-GHz observations are used, albeit in an approximate manner, to extend the dynamic range to somewhat lower rain rates.

Note that the algorithm is the same as Wilheit et al. (1991) in two important respects. The underlying radiative transfer model and the beamfilling correction are unchanged. Recent research (Tesmer 1995; Wang 1996) has suggested improvements in both these areas. Tests of these presumed improvements are under way and will be reported at a later date.

2. Data

The satellite data used in this study are brightness temperatures from SSM/I for the months of August through December 1987 and January through December 1989. The SSM/I data have been described in detail by Hollinger et al. (1987). Only data from 50°N to 50°S were extracted. The brightness temperatures with bad geolocation data have been removed. Bad brightness temperature data, such as data from a single scan line over ocean with unexpected high values, were also eliminated from the data file.

The brightness temperatures from three channels (19-, 22-, and 37-GHz vertical channels) are applied in the retrieval of rain rate. For convenience, the brightness temperatures for those channels will be referred to T_B19V, T_B22V, and T_B37V, and rain rates retrieved from the corresponding channels will be R_19V, R_22V, and R_37V, respectively.

The validation data used here are the Pacific atoll rain gauge data provided by Morrissey et al. (1991). The atoll rain gauge data are little affected by topography and can represent open-ocean conditions. The monthly rain gauge totals were collected from 88 stations in the tropical Pacific and covered periods from January 1971 through December 1990. Only months with complete data were included. Months with missing data were excluded and accounted for about 30% of the total data. Unfortunately, the missing data made validation of monthly rainfall for 1989 difficult.

3. Algorithm

a. Retrieval of instantaneous rain rate

Instantaneous rain rates are estimated from brightness temperatures using T–R relationships derived from the Wilheit et al. (1977) radiative transfer model. Figure 1 shows the schematic diagram of the radiative transfer model. In this model, a Marshall–Palmer (M–P) drop size distribution is assumed from the surface to the freezing level. The lapse rate is assumed to be 6.5 K km⁻¹, and the relative humidity is assumed to be 80% at the surface, to increase linearly to 100% at the freezing level, and remain at 100% above the freezing level. Thus, the freezing level serves as a proxy variable for both the rain-layer thickness and for the atmospheric water vapor content. In addition to the M–P distribution of rain drops, a nonprecipitating cloud layer containing 0.5 g m⁻³ of cloud liquid water is assumed in the 0.5 km just below the freezing level.

An analytical approximation to the T–R relationships for different SSM/I channels has been developed by Wilheit et al. (1991) and can be expressed as follows:

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where

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where T(r) is measured brightness temperature, r is rain rate, F is the freezing level, and T₀ is the background temperature, which can be expressed as

T₀FF²

The parameters a, b, c, ta, tb, and tc vary with different channels and are given in Table 1. These numbers are fits to computational results from the Wilheit et al. (1977) model. Here, F can be derived from T_B19V, T_B22V, and the T–R relationships using an iterative technique (Wilheit et al. 1991).

Applying the T–R relationships, instantaneous rain rates can be retrieved from brightness temperatures for different channels. Figure 2 shows a freezing level–rain rate–brightness temperature (FLRRT) chart portraying a 2D histogram of brightness temperatures from the 19V and 22V channels over a 5° × 5° box in the intertropical convergence zone (ITCZ). Most pixels do not contain any rain, and the raining pixels are mainly located near the 4- and 5-km freezing levels.

Figure 3 shows the two T–R relationship curves for the 19V and 37V channels with a 4-km freezing level. As the rain rate increases, T_B37V increases faster at lower rain rates than does T_B19V. The T_B37V also saturates quickly at about 4.0 mm h⁻¹, whereas T_B19V reaches the maximum at about 12.0 mm h⁻¹. Thus, the 37-GHz channel is more sensitive to low rain rates, while 19 GHz has a larger dynamic range at the high-rain-rate end. To take advantage of both T_B19V and T_B37V, a composite best estimate of the rain rates is taken as the instantaneous rain rate as follows:

R_19VR_37V(3)

The use of maximum of two channel’s rain rates is based on the assumption that problems in the retrieval such as beamfilling, saturation, or scattering by ice particles will generally cause an underestimate of the rain rate. The larger of the two rain rates is presumably less affected by these problems and thereby a better estimate. The factor bf is the beamfilling error correction factor. Beamfilling error results from nonuniform rain rates within a field of view (FOV) and the nonlinear T–R relationship. It causes retrieved rain rates to be lower than true rain rates. Previous studies (Chiu et al. 1990; Short and North 1990) suggested that the beamfilling error could be corrected by multiplying the retrieved rain rate by a single factor. This study uses the factor of 1.8 for the 19.35-GHz channel, which was derived from simulation studies using the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) data (Wilheit et al. 1991). We have no intention here to improve individual pixel estimates with this beamfilling correction. The beamfilling correction can only correct for the average bias over an ensemble of rainfall estimates such as might be used for time–area averages.

The factor br, which is only applied to R_37V, is the beamfilling factor of R_37V relative to that of R_19V. Since the T–R relationship for the 37-GHz channel has a larger nonlinearity than that for 19-GHz channel, the beamfilling error of R_37V will be larger than that of R_19V even though the 37-GHz channel has a higher spatial resolution. Thus, the retrieved R_37V will be lower than R_19V. In order to estimate the ensemble difference between R_19V and R_37V, a statistical comparison of R_19V and R_37V was performed. Figure 4 shows the cumulative rain-rate probabilities for the 19V and 37V channels. The square points correspond to every 10th percentile of rain intensity. We can see that R_19V is approximately twice the value of R_37V up to the 90th percentile rain pixel. Since the actual rainfall is the same for both sets of retrievals, there is a factor of 2 for R_37V values relative to R_19V. The factor of 2 is a typical value for rain areas in the Tropics and thus is applied to R_37V. It may not be true in the subtropics and part of the midlatitudes where rainfall usually is not as heavy as that in the Tropics. Exactly how to combine the observations at multiple frequencies and resolutions is still an open issue; this can only be considered an initial attempt.

In application, the freezing level is determined by a simultaneous solution of the T–R relationships at 19.35 and 22.235 GHz (vertical polarization) for a given pair of observations. This solution is implemented via a look-up table for computational efficiency. With the freezing level so determined, it is then straightforward to solve for a rain rate using the 19.35- and 37-GHz observations. Because the T–R relationships decrease at very high rain rates, this solution is double valued; the lower rain-rate value is taken as the solution.

b. Estimate of monthly rainfall

Estimation of monthly rainfall totals over the oceans is important for climatological studies. Simply averaging the instantaneous rain rates for one month may not accurately represent the true distribution of a monthly rainfall because of the limited dynamic range of the underlying measurement technique. Moreover, even with a perfect measurement technique, the very high rain rates, being infrequent, would be poorly sampled with observations from a low orbiting satellite. However, if we have an estimate of the form of the probability distribution of rain rates, we can estimate the contribution to the rain rates outside of our dynamic range by extrapolation.

Examining data from GATE, Kedem and Chiu (1987) found that rainfall closely fits a mixed lognormal distribution both spatially and temporally. Kedem et al. (1990) also showed that a mixed lognormal distribution fits the rainfall data better than a gamma distribution. The assumption of a mixed lognormal distribution of rainfall intensity is adopted for the current study.

1) A mixed lognormal distribution

Since most pixels in a 5° × 5° box contain no rain and a lognormal distribution holds only for nonzero rainfall, a mixed lognormal distribution (the combination of a delta function at zero and a lognormal distribution for nonzero rainfall) has to be considered. Kedem et al. (1990) have discussed the mixed distribution of rainfall. It can be expressed as

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where p is the probability of a rain event and

View Expanded

where f(r) is the probability of a given rain rate, r; r₀ is the logarithmic mean rain rate; and σ is the standard deviation of the logarithm of rain rate.

The mean rain rate and variance of a mixed lognormal distribution are as follows:

View Expanded

where R indicates a rain event.

2) Estimation of parameters p, r₀, and σ

The coefficients p, r₀, and σ are determined by fitting the observed rain-rate distribution to the lognormal form using a maximum likelihood estimation method. This method has been described in detail by Aitchison and Brown (1957) and Cohen (1991). For untruncated rain-rate samples consisting of observations r_i, i = 1, . . . , n_t, the likelihood function of a mixed lognormal distribution is

View Expanded

where n₀: number of nonraining pixels; n_r: number of raining pixels; and n_t: total number of pixels, n_t = n₀ + n_r. To obtain maximum likelihood estimates of p, r₀, and σ, we take the logarithms of Eq. (8), differentiate with respect to these parameters, and equate to zero. The results are as follows:

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As mentioned before, a severe limitation of emission-based methods is that rain rates are reliably retrieved only over a limited dynamic range. The fit is likewise limited to the reliable portion of the dynamic range. Sample data that can only be observed or measured in certain regions of sample space are called truncated sample data. Figure 5 shows a lognormal distribution of rainfall with the fixed truncation points at r_T1 and r_T2. Truncated samples have three cases: (a) left truncation, in which sample data with values less than r_T1 are truncated; (b) right truncation, in which sample data with values larger than r_T2 are truncated; and (c) double truncation, which includes cases (a) and (b).

Therefore, we can say the samples of instantaneous rain rates are doubly truncated at r_Tl and r_T2 for heavy rainfall. With the frequency range from 19 to 37 GHz, the r_T2 is approximately 20.0 mm h⁻¹ (after applying the beamfilling correction). Previous studies (Wilheit 1994) have suggested that r_T1 is about 1.0 mm h⁻¹. For light rainfall, the rain-rate samples are complete at the right side and hence are only left truncated at 1.0 mm h⁻¹. The maximum likelihood estimation method must be modified in the cases of truncated rain-rate samples.

For left truncated samples, the analytical solutions of r₀ and σ can be derived from maximum likelihood estimation equations. They are given by

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where

View Expanded

where n is the number of truncated rain-rate samples, θ(α) is the auxiliary estimation function, and ϕ(ξ) and Φ(ξ) are probability distribution function and cumulative distribution function of the standard normal distribution, respectively.

The table for the auxiliary estimation function θ(α) has been given by Cohen (1959). In application, it is necessary only that we calculate r̄ and s² from rain-rate samples. Then σ² and ln(r₀) are calculated from Eqs. (12) and (13). With r₀ and σ, the parameter p can be estimated by

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Unlike the singly left truncated rain-rate samples, an analytical solution is difficult for doubly truncated rain-rate samples. Therefore, only numerical solutions can be obtained. Here, σ, r₀, and p can be expressed as follows:

View Expanded

where

View Expanded

Here, ξ₁ and ξ₂ can be solved numerically from maximum likelihood estimation equations.

4. Application to SSM/I data

Histograms of instantaneous rain rates at 1.0 mm h⁻¹ increments from 0.0 to 40.0 m h⁻¹ have been collected in 5° × 5° boxes over oceans for each month from August through November 1987 and from January through December 1989. For each 5° × 5° box, there are about 16 000 pixels. If more than 25% of the pixels in one box are located over land, this box will be considered a land area and rainfall will not be calculated. If there are too few raining pixels in one box (≤100) to be used with any confidence, the monthly rainfall is calculated by simply averaging the observations. For normal situations, the r₀, σ, and p can be estimated using the equations described in section 3. Table 2 gives some partial results. Figure 6 shows one sample of the histogram and fitted curve.

Monthly rainfall totals are computed by multiplying the averaged rain rates estimated from Eq. (3) by the number of hours in one month. The global patterns of monthly rainfall for August, September, October, and November 1987 are displayed in Fig. 7. The figures clearly show the tropical large-scale features such as heavy rainfall along ITCZ and SPCZ.

Comparison of monthly rainfall between satellite estimates and the atoll data has been performed. Although the density of atoll rain gauge data is not sufficient for accurately validating the satellite observations, they are the best ground truth data available so far. Atoll data were first averaged into 5° × 5° boxes. In order to reduce the spatial sampling error between the satellite estimates and the atoll data, only those boxes that included at least two rain gauge data were utilized. Figure 8 displays the 5° × 5° boxes that have at least two rain gauges.

Since minimal atoll data are available for 1989, the comparisons are based on seasons [January, February, March (JFM); April, May, June (AMJ); July, August, September (JAS); and October, November, December (OND)], while monthly comparison statistics were calculated for 1987. There are approximately 10 5° × 5° boxes of atoll data for each case. Table 3 gives means of atoll rainfall and estimated satellite rainfall, mean error, root-mean-square (rms) error, and correlation coefficient between the atoll rainfall data and the satellite-estimated rainfall data. Two scatterplots of the SSM/I inferred rainfall versus the atoll rainfall data are displayed in Fig. 9. Except in JFM 1989, all cases have high correlation and reasonable mean and rms errors. These validation results indicate that this algorithm performs quite well in tropical areas, especially in September 1987 and JAS 1989.

5. Summary and conclusions

A physical–statistical approach has been developed that uses an emission-based method to retrieve instantaneous rain rates and computes monthly rainfall employing a maximum likelihood estimate method on a limited rain-rate dynamic range. The beamfilling error is corrected by multiplying a factor of 1.8, which is generated from simulation studies of GATE data.

Global patterns of monthly rainfall clearly display tropical large-scale features such as the ITCZ and SPCZ. Comparison with Pacific atoll data indicates this algorithm performs very well in tropical areas. The good agreement with the atoll data, however, is imperfect. There are many potential reasons for the discrepancy. The atolls may well not be representative of the oceanic areas within the 5° × 5° boxes. A diurnal cycle of rainfall, if it contains harmonics of a single daily cycle, can introduce a bias if the rain is only sampled at two times of the day separated by 12 h, as is the case for tropical observations from a sun-synchronous satellite. The magnitude and even the sign of the bias will depend on the details of the higher-order harmonics in the diurnal cycle and the timing of the sun-synchronous observations. The future TRMM satellite will have a non-sun-synchronous orbit specifically to permit sampling of the diurnal cycle.

It would be tempting to adjust some parameter of the retrieval (for instance, the beamfilling correction) to make the retrievals agree with the atoll data or some other socalled ground truth data. We do not believe this would be wise or reasonable. Oceanic rain is extraordinarily difficult to measure by any means. There is no reason to believe that the ground truth represents the actual rainfall any better than do the retrievals. It seems more reasonable to make the physics and logic of the retrieval as realistic as we can and only depend on comparisons with ground truth to detect egregious errors.

Does this algorithm represent an improvement over other rainfall-retrieval algorithms? It would be convenient if there were some benchmark for algorithm performance. Experience at the various rainfall algorithm intercomparisons has made it abundantly clear that there is no such benchmark for oceanic rainfall. At this point, the best that can be achieved is to compare with datasets, such as the Pacific atoll set used here, to check for reasonableness and then attempt to judge the self-consistency of the physics used in the retrieval, which is a very subjective process. Here, we are attempting to address the limited dynamic range of the measurements by using the assumption that the rainfall intensity is lognormally distributed to estimate the contributions from above and below the dynamic range of the measurements by extrapolation. In examining the full set of our retrievals reported here, we find that for about 10% of the cells the contribution of rain rates greater than 20 mm h⁻¹ exceeded 35% of the total and in another 10% the contribution from rain rates lower than 1 mm h⁻¹ represented 10% of the rainfall total. Clearly, the contribution from high rain rates is important and must be more accurately approximated than the face value of the measurements will permit. On the other hand, the importance of the rain rates below the minimum measurable rate is marginal, at least for the latitude range studied here.

As an experiment, the cells that were used for the comparison with the atoll dataset were reanalyzed, but the dynamic range issues were ignored. The retrieved rain rates (after beamfilling corrections) were taken at face value in the computation of the lognormal parameters, that is, an analog of the Berg and Chase (1992) method. The rainfall totals were reduced in all cases by factors ranging from 0.80 to 0.83. Because of saturation effects, the contribution of high rain rates is systematically underestimated; the method discussed here addresses this underestimate.

With the launch of TRMM (planned for 1997), we will have observations at the same frequencies but with much better spatial resolution (roughly a factor of 6 in area). The beamfilling corrections will be smaller with less uncertainty. The improved resolution will also modify the observed rainfall distributions. The probability of rainfall p will decrease, and the log-mean rain rate r₀ will increase. The log-variance σ² may also change. As a result, the contribution of rain rates below the 1 mm h⁻¹ minimum will become even less significant, while the contribution from high rain rates will become even more significant. TRMM will also have a 10.7-GHz channel that will increase the upper limit on the dynamic range by roughly a factor of 4, but at a degraded spatial resolution.

More work is needed in order to improve this algorithm, such as modification of simple plane-parallel cloud models, further studies on beamfilling error, the addition of data from infrared radiometers, and other operational SSM/I to improve the poor spatial and temporal samplings. The selection of truncated rain-rate points will also affect the monthly rainfall because this algorithm works well only if a small percentage of the samples have been truncated. It should also be possible to generalize the maximum likelihood estimator to operate on several histograms simultaneously so that the information from all the channels could be used wherever their dynamic ranges overlapped.