a. Instrument description

The TMI measures dual-polarized brightness temperatures at 10.65, 19.35, 37.00, and 85.50 GHz, and vertical polarization–only brightness temperatures at 21.3 GHz. Detailed descriptions of other TMI characteristics can be found in Kummerow et al. (1998) and Bauer and Bennartz (1998). For convenience, we henceforth denote vertically polarized brightness temperatures at TMI channels as T_10V, T_19V, T_21V, T_37V, and T_85V, and replace V with H for horizontal polarization.

As discussed by Petty (1994a), individual channel brightness temperatures have significant shortcomings as the primary observables in a physically based rain-rate retrieval algorithm. Each channel, regardless of whether it is horizontally or vertically polarized, will respond to a variety of environmental variables, as well as to a mixture of both emission and scattering from the rain cloud itself. Petty therefore proposed the use of linear transformations of dual-polarization brightness temperatures T_V and T_H at any given frequency. These effectively decouple the emission (attenuation) and scattering contributions into two separate variables, P and S, as well as factor out background variability resulting from variations in atmospheric water vapor, surface roughness, and so on.

b. Definition

We limit our attention here to the attenuation index (or normalized polarization) P, which is defined as

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where T_V and T_H are the vertically and horizontally polarized brightness temperatures; T_V,O and T_H,O are the estimated or modeled brightness temperatures in the absence of rain or cloud. Ideally, the value of P therefore falls in the range [0, 1], where 1 corresponds to a cloud-free pixel, and values approaching 0 represent a very opaque atmospheric condition generally associated with heavy precipitation.

The use of the attenuation index P in the retrieval algorithm has three advantages. First, unlike brightness temperature, it decreases monotonically with increasing rainfall intensity. Second, the attenuation index is not sensitive to the background variability because P is mainly determined by rain cloud optical thickness. Third, for the special case of a horizontally homogeneous rain layer, P and transmittance t obey an approximate power-law relationship, P ≅ t^α, with α ≈ 1.7. Therefore, in this limiting case, the P index yields a direct indication of the rain cloud transmittance. In this paper, our microwave observables are the attenuation indices at 10.65, 19.35, and 37.0 GHz. Our observation vector P is therefore given by (P₁₀, P₁₉, P₃₇).

c. Implementation for TMI

To convert satellite-observed brightness temperatures T_V and T_H at a given frequency to the attenuation index P, we require estimates of the cloud-free background brightness temperatures T_V,O and T_H,O. To a good approximation, these are functions of total column water vapor V and surface wind speed U. Both of these variables may be directly estimated from the passive microwave observations themselves, provided that care is taken to exclude FOVs contaminated by rain or land. The retrieved fields of V and U are then spatially interpolated into areas of precipitation and used as the basis for estimating T_V,O and T_H,O. It is not essential that these estimates be very precise, only that they account for most of the variation in the background polarization difference T_V,O – T_H,O. Uncertainties in the estimate of this quantity are treated as part of the inherent observational error in P.

We derived our own empirical algorithms for V and U from matchups between TMI radiances and surface observations. For column water vapor, we matched radiosonde observations to TMI overpasses in January and July 1999. The matchup procedure was similar to Alishouse et al. (1990) and Petty (1994b). Further details are given in Chiu (2003). The resulting statistical water vapor algorithm (kg m⁻²) is given by

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Matchups between the TMI measurements and surface buoy wind data from the National Oceanic and Atmospheric Administration (NOAA) Marine Environmental Buoy database (Chiu 2003) yielded the following algorithm for wind speed (m s⁻¹):

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Empirical expressions for the background brightness temperatures were derived from 489 TMI orbits in July 1999:

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where V and U were estimated by (3) and (4), and sea surface temperature T_S (°C) was obtained from a climatological database [obtained from the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center (GSFC)].

a. Simulated P for realistic rain fields

A key part of our algorithm is the specification of f (P|R), which depends on both the physical and statistical properties of rainfall, especially the variable effects of beam filling, which we take to be the single most important factor determining the normalized polarization (this is not true for the single-channel brightness temperatures). Because actual matchup data from microwave and rainfall measurements or detailed model simulations do not exist in a sufficient quantity, we rely on simulated data derived from high-resolution radar composites.

2) Polarization calculations

Starting with the precipitation structures simulated from the radar measurements, we simulated satellite-observed polarization P via a simplified plane-parallel radiative transfer model. The mass extinction coefficient of suspended cloud water κ_e,l is listed in Table 1, which was computed from Liebe et al. (1991) assuming a temperature of 0°C. The relationship between the volume extinction coefficients of rain k_e,r and rain rates (R) was estimated from Mie theory, assuming spherical raindrops with a liquid water temperature of 10°C and Marshall–Palmer drop size distribution. A power-law form was found to approximate the relationship well (Petty 1994b),

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where coefficients a and b for each channel are shown in Table 1. The rain-layer optical depth τ_r was then modeled as

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representing the contribution of suspended rain to attenuation of the polarized ocean surface emission. Because there was no information about the values of the freezing height Z_f in the radar reflectivity product, a fixed value of 3 km was assumed for the purposes of this demonstration. The optical depth τ_l attributed to suspended cloud water was modeled as

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where variations in the suspended cloud water content L (kg m⁻²) were simulated via a lognormal random deviate applied to each grid cell.

Because depolarization of the ocean surface emission by rain clouds depends primarily on total path attenuation and not on the details of the vertical structure of temperature or hydrometeor properties, we may use a highly simplified 1D plane-parallel model to compute brightness temperatures at each grid point. The radiative transfer equation is written as

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where the simulated brightness temperature T_B,p at polarization p (vertical or horizontal) is determined by the transmittance t, the specular emissivity of the surface (ocean in the study) ε_p, the surface temperature T_S, and the air temperature T_A. The transmittance t is given by

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where θ is the incidence angle and τ is the total optical depth due to hydrometeors. The surface temperature was approximated from the extrapolation of the temperature profile, assuming a temperature at the freezing level of 0°C and a lapse rate of 6.5 K km⁻¹. Here T_A was estimated by the air temperature at the midpoint between the surface and Z_f . Because the emissivity of the ocean is polarized, both the vertically and horizontally polarized radiance was obtained.

We define t_a, t_l, and t_r as the transmittance attributed to the atmosphere, the suspended cloud water, and the rain, respectively, and t₁ as the total transmittance (the product t_a, t_l, and t_r). Based on (2) and (9), it can be shown that the attenuation index (normalized polarization) can be written as

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where T_A,O represents the air temperature when the suspended cloud water and rain are absent. We assume T_A ≈ T_A,O in the simulations. In addition, the order of the first term for both the numerator and denominator is much smaller than the second term. Therefore, the attenuation index can be approximated as

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Note that t_a cancels out in the calculation of P, and thus the absolute value of t_a has little effect on P. Therefore, cloud-free brightness temperatures can be easily approximated employing a transmittance of unity in (9), in which case T_B,O = εT_S.

Once the high-resolution field of P had been computed for each simulated rain-rate field, both the rain-rate field and the P field were spatially averaged. The rain-rate field was averaged using a moving window of 15 km × 15 km, representing the nominal retrieval resolution of the algorithm. The simulated P values for each TMI channel were spatially averaged using a Gaussian approximation to the effective field of view for that channel. Gaussian random noise with standard deviations of 0.01, 0.02, and 0.02 for the 10.65-, 19.35-, and 37.00-GHz channel, respectively, was then added to P in order to reasonably account for errors resulting from the highly simplified nature of the forward model, instrument noise, and similar variables. The above procedure yielded a large ensemble of matched FOV-averaged rain rates R and attenuation indices P₁₀, P₁₉, and P₃₇.

a. Conditional probability density function

The conditional likelihood f (P|R) is a multivariate probability distribution. We used two methods to characterize the conditional PDF. The first method is to use a covariance matrix to include linear relationships only between rain rate and microwave signature, which is similar to the parameterization of current available Bayesian algorithms. However, as shown in observations and simulations (Fig. 1), linear properties are not sufficient to approximate the multichannel dependency of microwave signature. Therefore, in the second method, we took a successive approach to describe the conditional PDF that included linear as well as nonlinear relationships. We will refer to these as “the linear model” and “the nonlinear model,” respectively.

1) The linear model

Similar to a multivariate Gaussian distribution, the conditional PDF between P and R can be written as

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This closed-form function is essentially a normal distribution in the interior of the interval [0, a], but is forced to zero at the boundaries by the term of P(a − P) for each frequency. Based on observed ranges of actual TMI-derived values a is chosen to be 1.1; 𝗖 is the covariance matrix, and μ is

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where the subscripts 1, 2, and 3 describe the quantities at 10.65, 19.35, and 37.00 GHz, respectively. Subindex i is the corresponding channel. Based on the radar-radiative simulations, there parameters can be approximated by Table 2 and

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2) The nonlinear model

The more complete physical model is developed to account for the linear as well as nonlinear relations of P to rain rate. The distribution of P at a given rain rate R is approached hierarchically:

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where f (P₃₇|R) is the likelihood of P₃₇ at a given R, f (P₁₉|P₃₇, R) is the PDF of P₁₉ when P₃₇ and R are fixed, and the f (P₁₀|P₁₉, P₃₇, R) describes the PDF of P₁₀, while the other three variables are known. These conditional PDFs, for example, can be approximated as

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where μ₂, μ₃, σ₂, and σ₃ are all determined by fitting the same radar-radiative model simulations. Complete parameterizations can be found in Chiu (2003).

b. The prior distribution of rain rate

Surface rain-rate distributions have been commonly parameterized by lognormal functions (Houze and Cheng 1977; Kedem and Chiu 1987; Kedem et al. 1990, 1997; Sauvageot 1994; Nzeukou and Sauvageot 2002), although some observations showed departures from the lognormal distribution (Jameson and Kostinski 1999) and some suggested that rain rates followed gamma distributions (Ison et al. 1971; Swift and Schreuder 1981; Wilks and Eggleston 1992). Because Cho et al. (2004) found that both lognormal and gamma distributions were able to characterize the PDF of rain rates from TRMM data, we used lognormal functions as the prior distribution.

The lognormal density function for the prior distribution is denoted as logN(μ, σ) and defined as

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where μ and σ are the mean and standard deviation (mm h⁻¹) of the variable. These two parameters vary with different rain-rate observation datasets. Therefore, we will evaluate the effect of varying (μ, σ) on the simulated retrievals.

c. Estimators of the posterior distribution

When the conditional and prior likelihoods are specified, based on the Bayes theorem [(1)], the posterior distribution can be derived by

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Integrations with respect to rain rate in the four-dimensional space were performed numerically. Once the posterior distribution is known, the two most common estimators were taken: the mean and the maximum likelihood estimate of the posterior probability distribution, denoted as MEAN and MLE, respectively. The corresponding Bayesian estimates are stored in a three-dimensional lookup table for each P for single-pixel retrievals.

a. Experiment design

A number of experiments are designed for the sensitivity test by various combinations of the prior and conditional likelihoods (summarized in Table 3). The retrieval target was produced from a random number generator that followed all conditional PDFs of the nonlinear model [(17)−(19)]. Because one of our purposes for the sensitivity tests is to better understand the behavior of retrieval at higher rain rates, we generated this dataset along with the prior distribution logN(0, 2) to prevent a scarcity of high rain intensity.

The control run (R0) is the experiment in which the Bayesian retrieval method uses exactly the same conditional and prior PDFs with the retrieval target. An analysis of this control experiment can provide insight into the inherent uncertainty of the retrieval, because each PDF is perfect and no assumption is made in the algorithm. Furthermore, by comparing the control run with other experiments, the sensitivity of the algorithm to the prior distributions of rain rate can be evaluated by experiments R1 and R2, while R3 and R4 aim to understand the sensitivity to the conditional likelihoods. In experiment R3, we used a different parameterization for f (P₁₉|P₃₇, R) to investigate the sensitivity of the algorithm to various explicit functions. Experiment R4 applies the P−R relationships of the linear model (i.e., covariance matrix form) into the algorithm. This experiment is vital because it evaluates the adequacy of simple assumptions of the conditional PDFs in the retrieval algorithm, when in fact the dataset has a much larger degree of complexity.

b. Intrinsic uncertainty of the algorithm

As we have emphasized through this paper, a key property of the Bayesian algorithm is its ability to obtain complete posterior PDFs of retrieved rain rates. Based on the complete posterior distribution, a probability statement can be made about the realization of the retrieval. This advantage allows us to assess intrinsic uncertainties of the retrieval, when the ideal physical model and exact prior distribution are applied to the algorithm.

Figure 2 shows examples of the posterior distributions from experiment R0. Some posterior distributions have a single maximum over the entire rain-rate range, but some have two. A single maximum expresses the situation that a given observation vector P provides unambiguous information about the most likely rain rate when both physical relationships and prior knowledge are taken into account. A bimodal distribution implies that two distinct rain intensities are of comparable likelihood. The peak on the left indicates a scene of widespread stratiform precipitation associated with a smaller rain rate, while the second peak suggests the possibility of a strong convective cell within that 15 km × 15 km area. Based on the posterior PDFs of Fig. 2b, the MLE retrieval can increase from 5 to 60 mm h⁻¹ if P₁₀ decreases from 0.64 to 0.60 and P₁₉ and P₃₇ remain the same. However, in reality, this magnitude of the change in P value may not relate to a dramatic change of rain intensity of the scene, but rather arises from the variations of the instrument noise, atmospheric condition, and/or the calculations of brightness temperatures. Therefore, caution should be exercised in these cases when single-pixel retrievals are of interest. Such cases may be easily detected via their larger associated standard deviations.

Performance of the Bayesian algorithm at different rain-rate ranges for experiment R0 is demonstrated from the histogram of retrieved rain rates (as shown in Fig. 3). The title for each subplot is made of three components. The first component RR indicates the specific range of rain rates, where the retrieved rain intensity is drawn if its corresponding true rain rate in the training dataset is in this range. The second component is the sample size of the histogram. The third component, MEAN or MLE, describes the estimate used to interpret the posterior distribution. The pair of the numbers on the upper-right corner in each subplot is the mean and standard deviation of the histogram, while the percentage is the proportion of the retrieved rain rates that are in the same range as the actual rainfall intensities with respect to the whole histogram. Note that for the true range RR lower than 30 mm h⁻¹, the histogram is plotted in a logarithmic scale.

Results from experiment R0 using MEAN estimates suggest that the Bayesian algorithm is able to retrieve the light rain rate very well. For the moderate intensity (7–15 mm h⁻¹), the retrieval algorithm captures 46% of the data points, and the mean value is about right. The algorithm tends to underestimate the rain rates when the actual intensity in training data increases to a heavier range. Meanwhile, the associated standard deviation of the histogram starts to increase as well. Even so, the retrievals still encompass 30% data in the true range. For the extremely large rain rate (greater than 75 mm h⁻¹), there are 42% of data in the correct rain range, and the mode of the histogram falls within the true range. In this control experiment, the Bayesian algorithm shows the ability to retrieve rain intensity over all ranges, even for the case with extremely heavy precipitation.

The histogram of MLE retrievals for experiment R0 is shown in Fig. 4. Results from the light rain regime demonstrate a satisfactory performance. However, the underestimation for intermediate intensities (4–30 mm h⁻¹) is significant. Moreover, there are two distinct modes in the histogram when the true range RR goes up greater than 30 mm h⁻¹. The one associated with a lower rainfall rate dominates in the RR range of [30, 75] mm h⁻¹, and causes some retrievals at least 20 mm h⁻¹ smaller than the true rain intensity. The other mode becomes dominant under the extremely heavy precipitation, yielding 40% of the data points in the correct range.

In short, experiment R0 has demonstrated the retrieval ability of the Bayesian algorithm over various rain-rate ranges when the prior and conditional likelihoods are both idealized. The single-pixel retrieved rain rates might be associated with a bias because of the inherent uncertainty in the physical relationships and the interpretation using MEAN and MLE. The inherent uncertainty might be reduced via the inclusion of additional information concerning other atmospheric or microwave variables.

c. Sensitivity to the prior knowledge

Analyses of experiments R1 and R2 demonstrate how retrieved rain rates change to various specifications of the prior distribution when the physical model remains the same. In comparison with the control prior density function, the prior distribution in experiment R1, logN(0, 1), has relatively smaller probabilities at very light rain rates and beyond 5 mm h⁻¹. This property of the prior PDF leads the algorithm to retrieve reasonably for the true intensities between 0.2 and 7 mm h⁻¹ from both MEAN and MLE estimates. However, the smaller probability of the prior PDF at higher rain rates obviously limits the ability of the algorithm to retrieve heavy precipitation as shown in Fig. 5. Note that we also conducted other experiments changing μ and σ by 10% in prior PDFs, and found that slight variations of the prior distribution had no significant effects on the Bayesian algorithm.

A noninformative prior distribution is used in experiment R2, which assigns equal weight to all values over the parameter space. Based on Bayes’s theorem, it is clear that the posterior probability density is now proportional to the likelihood represented only by the data in this experiment, and thus the retrieval is dominated by the behavior of the physical model. Results demonstrate that a uniform prior PDF yields much higher retrieved rain rates by 7–15 mm h⁻¹ in light and moderate rain situations (not shown). For the true range with heavier rain rates, experiment R2 captures around 50% of the data points in the correct range. Because experiment R2 only includes the information of data with no prior knowledge, it is suggested that the P we used is sufficient to reflect the rainfall signal, even in heavy precipitating systems.

In summary, the prior rain-rate distribution plays a crucial role in the Bayesian algorithm retrieval, because the characteristics of the prior PDFs determine the retrieval ability and bias at different rain-rate ranges. However, the Bayesian retrieval is not sensitive to slight fluctuations of the parameters used in the prior PDFs. Therefore, when the prior rain-rate distribution is applied reasonably, the Bayesian algorithm is still robust.

d. Sensitivity to the conditional distribution

This section advances our understanding of the sensitivity of Bayesian retrievals to the characterization of conditional likelihoods that provide statistical and physical relationships between P and R. First, we attempted to understand whether different explicit functions characterizing conditional distributions would have a substantial impact on retrievals. In experiments R0 and R3, the conditional PDFs were both derived from the same radar-radiative simulations, and they presented similar relationships between P and R. As a result, experiment R3 yields consistent retrievals with experiment R0 for both MEAN and MLE estimates (not shown). It indicates that the specifications of the conditional distributions are not critical if the P−R relationships represented by those PDFs are not far from reality.

Second, we attempted to evaluate the adequacy of using simplified conditional PDFs to represent more complicated behaviors of data by using experiment R4. Figure 6 depicts the microwave multichannel relationships represented in experiment R0 (upper panel) and R4 (bottom panel). These relationships demonstrate similar patterns, including the orientation and spread of the contours. However apparently, the locations of the maximum likelihoods are shifted to lower P values in experiment R4.

Histograms of retrievals from the MEAN estimates of experiment R4 are shown in Fig. 7 (results of MLE are not shown because of similar responses). In general, for both MEAN and MLE estimates, the use of a linear covariance matrix to describe conditional likelihoods significantly overestimates rain rates, and the variations of the retrieval histograms are almost twice than those of the control experiment R0. More specifically, for very light rain rates, R4 only produces 0.1% of the data points in correct ranges, while the control run successfully retrieves almost half of data. The positive bias becomes more significant when true rain rates are of a moderate intensity. Under the condition that the true rain rates are only 2–15 mm h⁻¹, some retrievals from R4 are even greater than 50 mm h⁻¹. The large variation in retrieved intensity is also seen in the case of heavy precipitation.

One should not conclude that a simplified physical model is always unsuitable for the use of a Bayesian algorithm, because considerable differences exist in the multichannel relationships between the retrieval target dataset and the applied physical model for experiment R4. The retrieval bias may be reduced if the simple physical model is improved or tuned to be closer to the training dataset. However, we must note that the covariance matrix facilitating the linear model was estimated from the same radar-radiative simulations that were used in the explicit functional model. Therefore, it is clear that the simplified physical model cannot explain all of data behaviors, and this deficiency will lead to a significant bias in some ranges of rain rates.

a. PR–TMI matchup data

Since there is no long-term/wide-area dense rain gauge network or other reliable data to provide true rainfall intensity over the ocean, we used near-surface rain rates of the TRMM standard product 2A25 (retrieval from the PR) to develop and validate our TMI Bayesian retrieval algorithm. The 2A25 PR products provide retrieved rain rates with 4-km resolution. Because our retrievals represented rain intensity at a 15-km resolution, PR surface rain rates were averaged using a Gaussian weighting function and interpolated to the locations of TMI pixels. For convenience, the averaged rain-rate estimates are hereinafter referred to as PR rain rates. Note that PR reflectivity measurements can suffer from sidelobe contamination when the main beam of the PR is off nadir, and the resulting errors would propagate into rain-rate estimates. To minimize the likely impact of these errors, only matchups between TMI observations and near-nadir PR estimates are utilized.

The probability distribution of near-nadir 15 km × 15 km PR rain rates from data of January, April, July, and October 1998 was found to be comparable to (20). Because of the minimum detectable threshold (∼17 dBZ) of PR reflectivity, we applied a cutoff of 0.04 mm h⁻¹ to yield parameters (μ, σ) of (−2.8, 2.0). In addition, we used 2 months of PR–TMI matchups (January and July 1998) to adjust conditional PDFs of our Bayesian algorithms. Detailed parameterizations are given in Chiu (2003).

b. Validation datasets

We used a number of cases to evaluate the overall performance for all algorithms. In addition to looking at the quasi-global performance, we also considered specific classes of precipitating systems. These included tropical cyclones, frontal rainbands in extratropical cyclones, and some scattered strong convection cells. The purpose behind the selection of these cases is to evaluate the degree to which algorithm performance depends on the nature of the precipitation cloud system under consideration, especially when the latter is, in some sense, atypical of global precipitation.

c. Benchmark algorithms

Two benchmark algorithms are utilized in this study to provide intercomparisons for the purpose of validations. One benchmark algorithm is the GPROF, which is the official algorithm for TMI data (Kummerow et al. 1996, 2001; Olson et al. 1996, 2006). This algorithm introduced a database to represent the presumed probability distribution of rain rate and cloud profiles. Cloud profiles with microwave signatures that are close to satellite observations are picked from the database as candidates. The selected candidates are then averaged to yield the best surface rain rate and precipitation structure for each pixel, based on the relative occurrence of each cloud profile in the database.

The other benchmark algorithm is a linear regression model developed from PR–TMI matchup data in January and July 1998. Regression variables include the P₁₀, P₁₉, P₃₇, and scatter index at 37 and 85 GHz (S₃₇ and S₈₅; Petty 1994a). The linear model is formulated as follows:

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where R is retrieved rain rate (mm h⁻¹), and the asterisk symbol represents transformed variables. In this model, all pixels were retrieved without rain screening. Therefore, some constraints were needed for calculated rain-rate fields in order to exclude unrealistic precipitation. It is assumed that the pixel is not rainy when its retrieved rain intensity is smaller than 0.5 mm h⁻¹.

The linear algorithm may be regarded as the least sophisticated of the algorithms, in that it does not account for either nonlinearity in the relationship between R and the microwave observables or the highly non-Gaussian distributions of these variables. Hence, differences in the performance of this algorithm from the other algorithms may be regarded as a measure of the relative importance of these characteristics.

These two benchmark algorithms provide single-pixel retrieved rain rates only. To compare our retrievals with them, we used MEAN and MLE to represent the “best” retrieved rain rates, denoted as Bayesian-MEAN and Bayesian-MLE, respectively.

d. Validation metrics

There are several common validation statistics employed to characterize differences between retrieved estimates and true values. Of these, the difference between the mean [or mean bias (bias)], the root-mean-squared difference (RMSD), and the linear correlation coefficient are the most commonly used. It bears emphasizing that no performance statistic for any single algorithm is meaningful when considered in isolation, because it invariably depends on the statistical and physical properties of the validation dataset. We can therefore assess the “goodness” of any outcome only by comparing the results of several competing algorithms applied to identical datasets.

The linear correlation coefficient represents the strength of the linear relationship between observations and retrievals. Unlike either bias or RMSD, it is not affected by linear systematic errors in either the validation data or the retrievals and it is therefore a better measure of intrinsic (or potential) algorithm performance at discriminating between high and low rain rates. However, a systematic nonlinear bias, though potentially correctible, will adversely affect the correlation coefficient and therefore give a misleading indication of intrinsic algorithm skill.

A drawback to any of the traditional metrics is that they do not distinguish between performances at high and low rain-rate values; moreover, they do not necessarily distinguish between systematic errors resulting from correctable nonlinearities in an algorithm’s response and retrieval errors of a more random (and therefore noncorrectable) nature. We therefore also employ an adaptation of the Heidke skill score (HSS) proposed by Conner and Petty (1998). They specified separate rain-rate thresholds R_υ and R_r for the validation data and for the retrievals, respectively, and allowed these to vary independently of one another. The HSS may then be computed and plotted as a continuous bivariate function of the two thresholds. One may then identify those combinations of the two thresholds that maximize the skill score. The maximum skill found for any given value of R_υ is independent of any calibration bias (linear or nonlinear) in the retrievals. This method therefore offers a way of characterizing the intrinsic performance of an algorithm at discriminating between high and low (or zero) rain rates, independent of any biases, while also allowing the presence of any such biases to be inferred via the relationship between R_υ and R_r, which maximizes the skill score.

e. Results

The performance of all algorithms as measured using conventional validation statistics (bias, root-mean-square error, and correlation coefficient) are summarized in Table 4. For the purposes of this study, the outcome is deemed satisfactory if the performance of our Bayesian algorithm is found to be comparable to that of other algorithms, given that ours has the added advantage of producing posterior PDFs of surface rain rates.