A Modular Optimal Estimation Method for Combined Radar–Radiometer Precipitation Profiling

a. Optimal estimation theory

Optimal estimation (OE) is an inversion method that has been devised for retrieving a set of parameters x, which represent the true state of the system being observed, from a set of measurements y that are related through a forward model, y = f (x). This methodology is described in detail by Rodgers (2000); a brief summary is presented here. Optimal estimation seeks to find the set of parameters that minimizes the cost function

View Expanded

where 𝗦_y is the measurement covariance matrix. This cost function differs slightly from that of Grecu et al. (2004) in the inclusion of x_a, the a priori parameter set, and its covariance matrix 𝗦_a. Without x_a, the solution only minimizes the variance in the observations and can be underconstrained. The a priori parameter set represents the expected value of the retrieved parameters absent any information from the measurement and prevents extreme, unphysical values from being retrieved. Thus, the two terms in the cost function represent the weighted differences between the measurements and retrieved parameters from their forward-modeled and expected values, respectively. The relative weighting of these two terms that can substantially influence the location of the minimum cost, thus we describe the covariance matrices in more detail in section 2c. The diagonal elements in these matrices contain the variances of the elements of x_a and y. The off-diagonal elements represent the covariances between different measurements or the retrieval parameters. Nonzero values reduce the cost function if structures in the measurement or retrieval fields resemble those implied by the covariance matrices.

The solution that minimizes the cost function can be calculated iteratively if the forward model is moderately nonlinear, as is the case for radiative transfer. In each iteration, the forward model is linearized by calculating the Jacobian 𝗞, where K_ij = ∂y_i/∂x_j. Then, Newton’s method can be used to arrive at a solution once a convergence condition is satisfied. The iterative step is defined as

View Expanded

The convergence criterion is usually defined in terms of the closeness of subsequent iterations. One such criterion is that the standardized change in from one iteration to the next is much less than the number of retrieved parameters N:

View Expanded

b. Radar profiling algorithm

To apply OE theory to a radar–radiometer combined retrieval, one must first choose the parameters that compose x. Our retrieval begins with a radar-only solution, and then adjusts that solution with the additional information provided by TMI. The radar-only solution is provided by a rainfall profiling algorithm, similar conceptually to the TRMM operational radar rainfall algorithm (2A25). To convert the profile of radar reflectivity into a rain rate near the surface at the PR frequency, this algorithm needs to correct for attenuation. Although the full details of the stand-alone 2A25 algorithm are covered in great detail by Iguchi et al. (2000, 2009), it is necessary to understand the assumptions required by such an algorithm so they are reviewed briefly here.

An attenuation-correcting radar algorithm was described by Hitschfield and Bordan (1954) using internally consistent relationships between reflectivity Z, hydrometeor content W, and attenuation k to retrieve hydrometeor profiles. This method is numerically unstable; that is, small changes in the Z–k relationship can lead to large differences in the surface rain rate as errors in attenuation estimates amplify toward the surface. However, independent measurements of PIA provided by the surface reference technique (SRT; Meneghini et al. 2000) can provide a constraint on the Z–k relationship or the DSD model used to derive it.

Before describing the vertical hydrometeor model, it is worth noting that a potential source of error is the detection threshold of PR (17 dBZ). The 2A25 algorithm can afford to ignore this problem, since rain of this reflectivity has negligible attenuation, but may nevertheless contribute to a substantial fraction of the total LWP and is important for modeling microwave T_bs. Our algorithm fills the gaps in profiles by adding reflectivity at the lowest value measured by PR in each profile below the 17 dBZ threshold. This has a modest effect on simulated T_bs, with an increase of up to 5 K in the 37-GHz channel.

The vertical model of the precipitating cloud requires a description of hydrometeor phase, size distribution, ice density and morphology (Bennartz and Petty 2001), and melting layer structure (Olson et al. 2001) at each range gate in order to properly simulate radar reflectivity and upwelling microwave radiances. Because of the computational cost of retrieving multiple variables in each profile, our radar profiling algorithm is designed to capture the natural variability of these properties in as few parameters as possible.

As in many other rain profiling algorithms [e.g., TRMM 2A25 (Iguchi et al. 2000); Grecu and Anagnostou 2002], a gamma distribution is assumed to describe the rain DSD: N(D) = N₀D^μe^−ΛD, with an intercept parameter (N₀), shape parameter (μ), and slope parameter (Λ). In this model, the median volume diameter D₀ can be expressed as D₀ = (3.67 + μ)/Λ (Ulbrich 1983). This relationship formulates a power law relating D₀ and Z:

View Expanded

The constants can be grouped together to form a more simple power law D₀ = aZ^b, where a represents the Z-normalized D₀ and is dependent on N₀ and μ, whereas b only depends on μ. It should be noted that these relationships are only strictly valid if Z represents the sixth moment, that is, Rayleigh reflectivity, which is not true for large raindrops (>1-mm diameter) at the PR frequency. However, the error introduced by this approximation does not significantly affect the shape of the relationship, so no systematic error is introduced so long as N₀ and D₀ are derived from PR reflectivity values using Mie theory to calculate Z. To simplify the retrieval, the coefficients a and b are fixed by rain type (Table 1) and, along with the assumption μ = 3, are chosen to approximate the 2A25 default Z–R relationships.

To adjust the Z–D₀ relationship (e.g., in order to match the SRT PIA), we define a multiplicative factor ϵ_DSD, so that D₀ = ϵ_DSDaZ^b. This type of DSD adjustment is mathematically similar to the δN*₀ adjustment employed by Grecu and Anagnostou (2002) in their profiling algorithm and the α adjustment employed by 2A25 (Iguchi et al. 2000).

The ice phase is treated similarly. An exponential size distribution (i.e., μ = 0) is assumed for both snow and graupel above the melting layer, which is consistent with available aircraft measurements (e.g., Houze et al. 1979; Stith et al. 2002). The following size–density relationships for snow (ρ_s) and graupel (ρ_g), based on Heymsfield et al. (2004) are assumed:

View Expanded

View Expanded

where D is in millimeters and density is in kilograms per meter cubed and is not allowed to exceed the density of pure ice. The snow–graupel partitioning is assumed to be a function of temperature and rain type (Fig. 1) and derived from the database of CRM simulations used in radiometer-only retrievals (Kummerow et al. 2001). Improving the microphysics schemes in these simulations is an active area of research (e.g., Li et al. 2008) and part of planned prelaunch Global Precipitation Measurement (GPM) ground validation.

As with rain, a Z–D₀ power law (D₀ = aZ^b, where D₀ represents the mass-weighted average diameter of snow and graupel) is used to derive the ice PSD from the corrected reflectivity. From this PSD, the attenuation and scattering properties at all TMI frequencies can be calculated. In this study, the equivalent-mass sphere approach has been used to calculate the scattering properties of snow and graupel. This assumption can lead to significant errors in the scattering and extinction parameters for a given ice water content (Petty and Huang 2010), on the order of the difference between an all-snow and all-graupel column of the same reflectivity. Despite these numerous sources of error, there is minimal impact on the retrieved rainfall because attenuation at radar and lower TMI frequencies is insignificant (<0.01 dB km⁻¹ for 1 g m⁻³ at 13.8 GHz). The parameters of the Z–D₀ relationships for rain, snow and graupel are given in Table 1 and are consistent with the Z–ice water content (IWC) relationships reported by Black (1990). Like the rain DSD, these can be adjusted by a multiplicative factor, which we define as ϵ_ICE. This has the effect of increasing or decreasing the ice water path, for example, to match the scattering signature at 85 GHz.

Our melting layer model is based on the finding of Zawadzki et al. (2005) that the strength of the reflectivity peak is strongly related to the density of the melting ice particles. Our treatment begins by finding the peak corrected (2A25) reflectivity (Z_peak) within 0.5 km of the horizontally interpolated brightband height, provided by the standard TRMM 2A23 product. The brightband strength is given by the difference between Z_peak and the lowest 2A25 reflectivity within 1 km below Z_peak and is used to determine the initial snow density at the top of the melting layer. For a given density, there is a relationship between melt fraction and Z_peak − Z (Fig. 2), which is used to determine the dielectric constant and fall velocity ratio at levels up to 0.5 km above and 1 km below Z_peak. With this additional information, the PSD is derived using the same μ and Z–D₀ relationship as the rain, ensuring continuity between the melting and rain layers.

Although this model does not account for coalescence and breakup, these processes lead to errors of only 1 dB (Fabry and Zawadzki 1995), only slightly more than the noise in the radar itself. This approach has the advantage of mass consistency from top to bottom of the layer, which is an improvement over static Z–R and Z–k relationships in the melting layer, which are not valid at all densities. Furthermore, there is no need for explicitly separate convective and stratiform melting layer models since a weak Z_peak implies high particle density, that is, graupel, whereas a stronger value is associated with lower-density snow.

Cloud water has a small but nonnegligible contribution to attenuation at the PR frequency, and a stronger contribution to T_bs, especially at higher frequencies and in lighter rain. Like the snow–graupel partitioning, default cloud water profiles are derived from the CRM database (Fig. 1) for each rain type. Since the cloud liquid water path (cLWP) makes an important contribution to microwave T_bs, particularly in light rain, we define a third adjustable parameter, ϵ_CLW, which is used to multiply the default cLWP equally at all levels in the combined retrieval algorithm.

In summary, the profiling algorithm converts the observed reflectivity profile into a profile of hydrometeor PSDs with three adjustable parameters: ϵ_DSD, ϵ_ICE, and ϵ_CLW (Table 2). From the top down, an attenuation correction is made at each range gate, the PSD is derived from the corrected reflectivity and ϵ values, and the process is repeated to the lowest clutter-free bin, below which the same DSD is extrapolated to the surface. This process is similar to 2A25, although there are differences in details of the ice, mixed phase, and cloud water models.

Perhaps the most important difference is in incorporation of the SRT PIA. Since the standard deviation of the SRT can be greater than the PIA estimate itself in light and even moderate rain, the profiling algorithm needs to account for this to prevent unphysical profiles from being retrieved. The 2A25 algorithm uses a combination of the default and SRT solution that gives higher weight to the SRT PIA in heavy rain, where the SRT is most reliable. With the OE methodology, the default and SRT solution can be weighted with knowledge of the variance of the SRT, which is known, and the variance of ϵ_DSD, which has not been defined. When our profiling algorithm is used without radiometer input, we desire that the global rainfall estimates be unbiased relative to 2A25 both to serve as a standard reference to compare the radiometer-adjusted estimates and to indirectly utilize the data against which 2A25 itself has been validated. In this simple case, x consists of ϵ_DSD and y is the SRT PIA, with the profiling algorithm serving as the forward model to calculate the PIA from the measured Z profile (ϵ_ICE and ϵ_CLW are not considered in the radar-only retrieval because of their minimal impact upon the PIA). The only free parameter in this retrieval is the uncertainty in ϵ_DSD. An analysis of profiler-derived (Williams 2008) and polarimetric radar–derived (Matrosov et al. 2002) DSDs suggests a Z-normalized D₀ standard deviation of approximately 35%. One month of global retrievals over ocean provides a best match to 2A25 when 25% variance is assumed. Different variance levels in Fig. 3 illustrate the trade-off between matching the observations (SRT PIA) versus the constraint of the default DSD. When the constraint on the DSD is strong, it is difficult to adjust rain rates away from the default value, which tends to produce less rain than the standard 2A25 solution.

c. Combined retrieval framework

The flow of the overall retrieval is illustrated in Fig. 4. The parameters ϵ_DSD, ϵ_ICE, and ϵ_CLW required by the radar profiling algorithm form the retrieval vector x. In addition, there is another set of parameters that do not affect the radar solution but are necessary to model the T_bs. These parameters include SST and surface wind speed, which are necessary for emissivity calculations. Also, since most TMI FOVs cover some nonraining PR pixels, the cLWP, total precipitable water (TPW), and height of the freezing level need to be known for these pixels in order accurately model the microwave T_bs. These parameters are obtained either through ancillary sources or independent retrievals.

SSTs are derived from a multiday optimal interpolation (Reynolds and Smith 1994) of TMI-based SST retrievals (Gentemann et al. 2004). The SST data are gridded at 0.25° resolution, so a bilinear interpolation is used at each pixel. The freezing level at all PR pixels within 300 km of a reliable brightband height is interpolated using an inverse distance scheme with smoothing from the top of the bright band as given by the 2A23 product (rain characteristics) where a bright band is detected. For pixels more than 1000 km from a reliable height, the climatological freezing height level used by 2A25 is given instead, and a linear combination is used in between 300 and 1000 km.

The surface wind, TPW, and cLWP are retrieved independently at each TMI FOV using the method of Elsaesser and Kummerow (2008) and interpolated to the nonraining PR pixels. This retrieval is considered valid where the χ² error statistic is less than 18, which is indicative of a homogeneous wind, TPW, and cloud water field with no precipitation. The temperature profile is calculated assuming a constant lapse rate that is calculated from the freezing height and SST (with a maximum of 7 K km⁻¹), and a scale height of 2.3 km is assumed for water vapor. In pixels with a radar echo, wind is interpolated as in the rain-free pixels, but the water vapor values are adjusted to 95% RH in range gates with an echo.

Since many of the TMI FOVs are only partially filled with raining PR pixels, it is necessary to have an accurate representation of the nonraining parameters outside the raining pixels. The chief cause of poor nonraining retrievals, aside from rain, is sub-FOV inhomogeneity of the cloud water field (Rapp et al. 2009) as well as contributions from drizzle, which has larger drops that create a different T_b signature than cloud water, but are too small to be detected by PR. Thus, we retrieve the cloud + drizzle LWP for all nonraining PR pixels that were within 30 km (approximately the size of a 19-GHz FOV) of any TMI nonraining retrieval that did not meet the χ² < 18 test or are adjacent to a PR pixel with rain. Cloud is assumed to transition to drizzle once a LWP of 150 g m⁻² is reached; this is based on a median value inferred from CloudSat and Moderate Resolution Imaging Spectroradiometer (MODIS) observations by Kubar et al. (2009). To maintain continuity among raining and nonraining pixels, the retrieved cloud LWP is interpolated into the raining pixels if it exceeds the value given by the default rainwater–cloud water relationships.

The next two routines retrieve the parameters used by the profiling algorithm (Table 2). Since modifying the ice PSD affects primarily the 85-GHz channels, where the emission signal is saturated even in light rain and therefore insensitive to cloud water–rain DSD modifications, this process is separated from the rain DSD and cloud water retrieval for computational efficiency. Rain and cloud water, on the other hand, must be retrieved simultaneously because of their similar, but not identical, contributions to upwelling microwave radiation at the TMI frequencies.

In each retrieval routine, the a priori covariance matrix 𝗦_a must be carefully constructed so that the algorithm can reproduce the variability observed in nature without giving unphysical profiles. To place an equal cost on any given increase–decrease of rain D₀, ice D₀, and cLWP, we assume the ϵ parameters are distributed lognormally. The variance of ϵ_DSD has already been established via matching to 2A25 and observations, but the covariance also needs to be defined, because the observational datasets (Matrosov et al. 2002; Williams 2008) used to determine the variance of ϵ_DSD also show spatial–temporal autocorrelation structure of the form

View Expanded

where 𝗦_a0 is the variance of ϵ_DSD, ΔZ is the reflectivity difference (dBZ), Z₀ is the reflectivity scale, ΔL is the distance between pixels i and j (km), and L₀ is the length scale. Based on the aforementioned datasets, a length scale of 10 km and reflectivity scale of 3 dBZ is used for ϵ_DSD and, lacking sufficient observations for independent determinations, also assumed for ϵ_ICE and ϵ_CLW.

There is less observational evidence for constraining ϵ_ICE than ϵ_DSD, but a slightly larger value (50%) is sufficient to match nearly all 85-GHz brightness temperatures with reasonable mass continuity within the column. It is likely that some snow–ice may fall below the PR detection threshold and still slightly depress 85-GHz brightness temperatures, which may lead to a low bias in retrievals of ϵ_ICE. This problem is effectively minimized by increasing 𝗦_y at 85 GHz.

The value chosen to represent the variance of ϵ_CLW is especially important, because rainwater and cloud water produce similar radiometric signatures, making it easy (with respect to minimizing χ²) to adjust one at the expense of the other. However, if the rainwater content is known, then the variance in ϵ_CLW can be set indirectly. This approach is used in section 3.

The observation covariance (𝗦_y) matrices contain the measurement and modeling errors. In addition to the previously described PIA uncertainty, 𝗦_y represents the expected error in the brightness temperatures. This is considered to come from three sources: error in nonretrieved parameters, instrument noise, and radiative transfer model approximations. The first term is calculated by perturbing the nonretrieved parameters (SST by ±0.7 K, column water vapor by ±4 mm, wind by ±1.5 m s⁻¹ following Elsaesser and Kummerow 2008) in each pixel and convolving the perturbations to TMI resolution. Typical values of these errors range from 2 to 4 K, depending on channel and rain coverage. Instrument noise is much smaller, approximately 0.5 K (Kummerow et al. 1998), and is also added to the error estimate. If the sum of these is less than 3 K, then the variance is increased to 3 K to account for radiative transfer model error (Kummerow 1993). Higher minimum values of 5 K at 37 GHz, and 20% of the T_b depression from 280 K at 85 GHz, are used to account for errors in ice particle modeling as well.

Some hard limits are also set to prevent the profiling algorithm from retrieving unphysical profiles. A lower limit of either 0.3 or the minimum value necessary to perform the Hitschfield–Bordan adjustment to the surface is set on ϵ_DSD in reliable PIA pixels, and a lower limit that produces a maximum of 4 dB of attenuation is set in unreliable PIA pixels. This represents the maximum PIA that might be regarded as unreliable by the 2A21 algorithm (Meneghini et al. 2000) over ocean. Meanwhile, the upper limit on ϵ_DSD is set to 3.0, to prevent unrealistically large raindrops from being retrieved. The limits of ϵ_ICE are set at 0.2 and 4.0, mainly prevent unphysical mass concentrations. The cloud water limits are set from 1% of the original value to a maximum LWP of 10 kg m⁻², based upon autoconversion thresholds in a high-aerosol environment (Liu and Daum 2004).

For computational simplicity and continuity with operational algorithms, a modified plane-parallel Eddington approximation along the TMI slant path is used to calculate the upwelling microwave radiances in this particular study, although the overall framework is constructed so that interchangeability with other radiative transfer schemes is possible. The extinction, bulk scattering, and asymmetry properties of rain, snow, graupel, and melting hydrometeor size distributions are computed using Mie theory and stored in lookup tables. Like the radiative transfer model, these tables can be easily interchanged with others, such as those generated from T-matrix or discrete dipole approximations of the hydrometeors that would more realistically simulate scattering properties at multiple frequencies and are thus particularly needed for GPM.

Using Newton’s method to evaluate the nonlinear solution to minimize the cost function [Eq. (2)] requires a number of computationally intensive steps, which have different scaling relationships with the number of raining pixels n. The calculation of the Jacobian matrix (𝗞) requires a run of the profiling algorithm with perturbed values of ϵ_DSD, ϵ_ICE, and ϵ_CLW for each PR pixel, followed by calculation of the brightness temperatures and convolution to TMI resolution. While these steps are complex, they scale somewhere between O(n) and O(n²).

The inversion of the 𝗦_x covariance matrix in Eq. (2), meanwhile, takes O(n³) time steps to compute, and for large scenes becomes the limiting step in the retrieval. As a result of these scaling relationships, an orbit must be broken up into segments to process in real time or better on current hardware. The definition of an orbit segment is somewhat arbitrary but needs to be larger than a single 10-GHz FOV, and preferably, contain several such FOVs in order to take advantage of that channel’s sensitivity to heavy rain. Conversely, a segment should not be so large that, when filled with rain, the simultaneous retrieval of two parameters (ϵ_DSD and ϵ_CLW) results in prohibitive computation time. For simplicity, a segment size of 49 × 49 PR pixels was chosen for comparison with ground validation sites in section 3. For full orbits, a scheme that identifies continuous areas of rain as the segments may be a workable approach.

d. Sensitivity and information content

A particularly interesting outcome of this retrieval is the partitioning of rain and cloud water. This partitioning has always been either an explicit or implicit assumption in the TRMM 2A12 and 2A25 products, thus this retrieval offers additional insight by retrieving it directly. However, it is first useful to evaluate the extent to which the two variables can be determined from the measurements.

We consider three reflectivity profiles which are a PR composites of stratiform, deep convective, and shallow convective rain types near Kwajalein in 2008 (Fig. 5). The ϵ_DSD and ϵ_CLW parameters were perturbed to produce a wide range of rain and cloud LWPs for each reflectivity profile, along with the associated T_bs. Given an uncertainty in the observations (i.e., 𝗦_y), we can then determine their constraint on the physical parameters. For each profile, the difference between the perturbed and default T_bs and PIA was normalized by standard deviations of 3 K and 0.7 dB, respectively, and are plotted in Fig. 6.

The constraint provided by PIA is poor in these profiles, which is not unexpected as they are composites of all rainfall, which skews toward light rain (only the deep convective profile barely exceeds 30 dBZ). The constraint provided by the T_bs is much narrower than that provided by the PIA. Although this represents an ideal scenario, that is, uniform rain within same-size FOVs, there is clearly some additional information present in the microwave radiances. Also note that the slope of the region constrained by the T_bs is not parallel to contours of constant total LWP, but instead represents a stronger response to rainwater than cloud water, a consequence of the increased absorption efficiency of raindrops relative to a distribution of cloud droplets with the same liquid water content due to Mie effects.

A similar exercise is performed to examine the sensitivity of the retrieval to ice water path (IWP) and snow–graupel partitioning. Only the deep convective and stratiform profiles are examined here since no ice is present in the shallow convective profile; however, reflectivities were increased by 10 dBZ for each profile in order to create a significant ice scattering signature. The 85V T_bs are contoured in IWP–graupel fraction space in Fig. 7, with the lines enclosing the region defined by the default T_bs ± 5 K. The most important feature of these plots is that, while T_b does vary with graupel fraction, a greater range can be achieved by altering the total IWP via ϵ_ICE. In some cases it may be necessary to adjust the graupel fraction to match the coldest observed T_bs, although this is not done in the current implementation.

a. Kwajalein

The Kwajalein Ground Validation site, located on a small atoll in the central Pacific Ocean, is an ideal site to test the combined algorithm because of its oceanic location, where the radiometer has the potential to add the most information to the radar. Although GV rain products are available since 1998, only recently have calibration problems been addressed and dual-polarization capability added (Silberstein et al. 2008; Marks et al. 2009). With these additional capabilities, a robust dataset of polarimetric rain retrievals has been developed by D. Wolff (2009, personal communication) using the method of Bringi et al. (2004).

A free parameter in the combined retrieval is the variance of ϵ_CLW, which determines the cloud water–rainwater partitioning. The rain probability density functions (PDFs) and cumulative distribution functions (CDFs) (Fig. 8) show that a reasonable range of uncertainty in ϵ_CLW provides estimates that are in line with those of the GV radar, with an optimal uncertainty of 100%. Higher values reduce the cost of changing ϵ_CLW, thus, the rain DSD is modified to a lesser extent; similarly, lower values constrain cloud water more strongly, and the rain DSD is modified more readily. The GV-tuned value results in a 10% increase of rainfall over 2A25, which is in line with the PR–GV bias at Kwajalein reported by Wolff and Fisher (2008). These PDFs also show the general agreement of 2A25 and the PIA-only profiling algorithm, the low rain bias of the default DSD in this region, and general agreement of all methods in heavy rain, where the SRT measurement of PIA is most reliable.

b. Melbourne, Florida

An independent validation test was performed by comparing retrieval results to those from the standard 2A53 GV radar products over Melbourne for the years 2006–08. The mean rain rates from each retrieval are plotted in Fig. 9. It is apparent that there is much month-to-month and year-to-year variability as to which retrieval method best matches the collocated GV totals. The GV Z–R relationship is based on monthly gauge accumulations, whereas there are only 10–20 overpass events each month, with just a few of these dominating the total rainfall. Thus, the average rain DSD that determines the monthly Z–R relationship may be quite different than that in the overpasses, especially considering the gauges are all on land whereas the retrievals compared here are all over the ocean. Nevertheless, the combined algorithm at the very least does no worse than the PIA-based radar algorithms in making adjustments from the default DSD. This is especially clear during January–April, when all algorithms reduce rainfall considerably relative to the default, implying larger drops as might be expected in the predominately frontal, stratiform rain that falls this time of year. The total accumulated rainfall is only 1.6% higher than GV, while 2A25 underestimates the GV total by 5% and the default DSD overestimates it by 14%. Thus, the combined method is able to match the Melbourne rainfall while also being unbiased at Kwajalein, where 2A25 underestimates GV significantly.

c. Brightness temperature statistics

While agreement between the retrieved and observed brightness temperatures does not guarantee that correct DSD parameters are being retrieved, and cannot be considered validation in a strict sense, such internal consistency does indicate that the vertical model described in section 2 is adequate to explain the observed T_bs. Moreover, the degree to which simulated T_bs improve in the combined algorithm relative to a radar-only algorithm (such as 2A25) can also be considered a measure of the effectiveness of the retrieval.

Scatterplots of observed and simulated T_bs are shown in Fig. 10. At the lower-frequency channels it appears that a slight cold bias is evident in the default DSD and PIA-adjusted products (both 2A25 and the PIA-only version of the combined retrieval), particularly at 10 GHz. These biases are improved, but still exist to a slight degree in the combined product. Meanwhile, the fairly large scatter in the T_bs simulated by the radar products at 37 and 85 GHz is reduced significantly in the combined retrieval.

Quantitative measures of these errors are presented in Table 3. The largest errors and biases occur when no adjustments from the default DSD are made. When adjustments are made in heavy rain to match the PIA, errors are reduced slightly, by about 1 K, on average in the T_bs and by about 0.6 dB in the PIA. A further reduction in the T_b errors is seen with the combined algorithm but biases and RMS errors are only reduced by about one-third to one-half of the default values, approaching the expected errors defined in the 𝗦_y covariance matrix. It is notable that the reduction in T_b errors has occurred without increasing the PIA errors from the radar-only algorithm. The residual RMS errors are a consequence of the OE methodology, which does not give a high value to matching observations better than the a priori error. However, the residual biases remain significant, particularly in the 19-GHz channels, and may indicate a deficiency in the forward model or calibration of these TMI channels.

d. Disdrometer comparisons

As a final reference point, the retrieved DSDs are compared to ground-based disdrometer measurements. Because the disdrometer represents a point measurement and the number of TRMM overpasses during rain events would yield an extremely small sample size, it is more meaningful to compare summary statistics than attempt to collocate TRMM overpasses with the GV radar and disdrometer. Disdrometer data from Kwajalein are available from 2003 to 2004, and Melbourne from 2006 to 2009. Although the former do not overlap the TRMM observations we processed, the lack of a significant interannual change in DSD at Kwajalein (Kozu et al. 2009) should not prevent a statistical comparison.

Following Rosenfeld and Ulbrich (2003), we plot the liquid water content as a function of drop diameter in Fig. 11 for reflectivities representing light (20 dBZ), moderate (30 dBZ), and heavy (40 dBZ) rain. Because of errors introduced by calculating D₀ from binned disdrometer data, the mass-weighted mean diameter D_m is used instead. For a gamma distribution, assumed in both ground radar and PR retrievals, D_m = (4 + μ)/(3.67 + μ)D₀. Deviations along the lines of constant reflectivity represent changes in the DSD, going from maritime to continental as D_m increases. Deviations perpendicular to these lines represent deviations in shape parameter or from the gamma distribution itself, in the case of disdrometer data.

There is a relatively tight clustering of the DSDs derived from the TRMM data at low reflectivities that diverges as Z increases. This represents the increase in information (stronger PIA response) at higher rain rates that leads to a change from the default DSD. Curiously, the disdrometer data also show this divergence, despite the lack of any a priori assumptions in that dataset. There is also a clear increase in D_m and the decrease in W from Kwajalein to Melbourne, particularly evident in the disdrometer data, and both radar and combined retrievals. There is an apparent offset of the disdrometer data toward larger drop sizes, which was also noted by Kozu et al. (2009) in the Kwajalein dataset, where it was suggested that noisy conditions may have contributed to an undercounting of small drops. Notwithstanding this offset, the change in D_m from Kwajalein to Melbourne is nearly identical between the combined algorithm and disdrometer at all reflectivities. This ability of the combined algorithm to distinguish DSDs is particularly evident in moderate rain where PIA-only retrievals such as 2A25 must rely on default DSD assumptions. Even the polarimetric radar algorithm Bringi et al. (2004) relies upon an assumed DSD in the lightest rain, although the small contribution of these rain rates to the total should not impact the cloud water constraint that was derived from the Kwajalein GV radar dataset.