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    TRMM @ JPL Rain Q&A TRMM combined algorithm GPM core reference algorithm
(last update: July 3, 2003)

The rain-profiling algorithm development and validation effort undertaken for TRMM has confirmed that one of the main difficulties surrounding the retrieval of rain rate profiles from spaceborne radar reflectivity measurements is the unknown drop size distribution (DSD). Indeed, if one starts with the assumption that the DSD is always an exponential (or, at worst, a gamma) distribution, whose dependence on the rain rate is known a priori, one can then derive a power-law relation k=a Zb which very adequately relates the 14-GHz radar reflectivity factor Z and the 14-GHz attenuation coefficient k. It follows that the one-way path-integrated attenuation PIA, integrated over a vertical rain column, must be related to the 14-GHz measured reflectivities Zm in that column by

PIA = ( 1 - 0.2 log(10) a b Zmb )b

Unfortunately, analysis of the TRMM data shows that this equation breaks down for moderate to light precipitation. The main reason seems to be the fact that current DSD models are not very good at all at moderate-to-low rain. The remainder of this page is a discussion of this issue, followed by some dual-frequency results from the CAMEX-4 campaign. You may jump directly to the CAMEX results by clicking here. spacer

The figure above is a plot of the -10 log10(PIA) values obtained from the TRMM radar measurements over the ocean during several orbits by comparing the rainy surface return with the average surface return from the nearby clear-air regions. This ``surface-reference'' PIA is shown on the horizontal axis, while the vertical axis represents the right-hand-side of our equation, calculated with the values that "a" and "b" are assumed to have a priori in convective cores. The right panel zooms in on the region of moderate attenuation.

If our PIA equation were verified exactly, one would expect much less scatter than is evident in the plots. Indeed, for one-way attenuations below 1.5 dB, there does not seem to be any correlation between the two PIA's, though there is a clear tendency for the Z-calculated values to be much smaller than the surface-referenced ones. This apparent failure of our equation is quite possibly caused in part by the change in the surface backscattering cross-section due to the variation of the wind from the clear air regions to the rainy area. But this effect is not sufficient to explain the large mismatch at moderate and low precipitation. To reduce the mismatch, the a-priori predicted values (the right-hand-side of the equation) must be increased. Barring a systematic "under-measurement" of Z, this would imply that the factor "a" must be increased. The figure below plots this coefficient as a function of the drop diameter assuming the precipitation is monodisperse.

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Clearly, increasing "a" is equivalent to decreasing the drop size. In other words, one reason for the mismatch between predicted and measured PIA's is that the a priori DSD's are not representative of the actual distribution of drop sizes in moderate-to-light rain, where the smaller drops are more numerous than in the a-priori DSD models (exponential or gamma), and where the larger drops are quite likely completely inexistent (contrary to the exponential or gamma model assumptions).

Moreover, the discrepancy between predicted and measured PIA's constitutes compelling evidence that the DSD parameters vary very significantly over a rain column. For if a single DSD applied throughout, only a slope correction would be required to bring both sides of the equation in agreement -- but no slope correction could reduce the large amount of scatter evident in the figures.

In the case of TRMM, this DSD problem has been dealt with in two ways. In the radar-only algorithm, the equation is used to adjust the coefficient "a" and thus reduce the ambiguity, at least in the case of heavier precipitation. In the Bayesian framework of the TRMM combined radar/radiometer algorithm, the equation is used to weight the candidate a-priori DSD's in favor of the better-matching ones, and the observed radiances are also used to further constrain the multiple possibilities for the DSD. The dual-frequency radar which the Global Precipitation Measurement (GPM) mission's core satellite will carry should prove a much more effective tool in sorting out at least part of this DSD-induced ambiguity. Indeed, with two radar reflectivity profiles, one would expect to be able to retrieve not just a single rain rate profile, but in addition at least one "first order" DSD profile, e.g. a profile of the (mass-weighted) mean drop diameter D*. Unfortunately, this expectation may turn out to be difficult to fulfill. That is because the reflectivity profiles at the two radar frequencies are far from independent. After all, lighter rain is made up mostly of small drops, and the backscattering cross-section of small drops is not significantly different at 14 and 35 GHz. One would therefore not expect large differences in the associated radar reflectivity factors. While the difference in the extinction cross-section appears more readily exploitable for small drops, its actual magnitude is unfortunately so small that the resulting attenuation is not significant for light precipitation. At the other extreme, while the attenuation will be appreciable (at both frequencies) for heavy rain, it is in fact likely to be so appreciable as to drive the back-scattered 35-GHz signal itself below the sensitivity threshold of that channel. Thus, the two frequencies are not very different at low rain rates, and they will in effect reduce to a single frequency at high rain rates, leaving a somewhat disappointing range over which the two frequencies can be realistically expected to resolve the DSD-induced ambiguity problem. That is why it is at least as important for GPM as it was for TRMM to develop an optimal approach to extract from all the GPM core satellite's measurements profiles of the best unbiased estimates of the means of the rain rate R and mass-weighted mean diameter D*. The rest of this discussion will try to quantify the effect of different plausible a-priori assumptions about the possible shapes of the DSD on the retrieved precipitation profiles using data from CAMEX-4.

Different DSD models

We consider five well-documented DSD models. No discussion of DSD's can be complete without considering Marshall and Palmer's exponential form

N1(D) = N0 e-L D   m-3 mm-1
in which, if we assume a nominal terminal fall velocity of 9.56 ( 1 - e-0.53 D) m/s for drops of diameter D mm, the parameters N0 and L must be consistent with R, i.e. must satisfy
R = 0.11 ( L-4 - (L + 0.53)-4) N0   mm/hr
Thus, in addition to R, the exponential N1 has a single free parameter. As long as the rainrate equation is enforced, it makes no difference whether one chooses to identify this parameter as N0 or L. The 2nd, 3rd and 4th DSD models which we consider are special cases of the gamma DSD
N(D) = N0 D e-L D
This distribution effectively depends on two parameters in addition to R. There are several ways of "fixing" one of these parameters to end up with only two unknowns which can be solved for using the two measured radar reflectivity factors. One that has proved consistent with disdrometer and airborne 2D-probe (small-)sample statistics consists in re-expressing and L in terms of the mass-weighted mean drop diameter D* and the dimensionless relative mass-weighted r.m.s. diameter deviation s*, and enforcing on the pair (D*, s*) the rather restrictive joint behavior quantified by the sample statistics observed during the TOGA-COARE campaign and during the 1992-1993 Darwin field measurements. Roughly, these restrictions amount to requiring that D* R-0.155 have a mean of about 1.1 (with R in mm/hr and D* in mm) and a standard deviation of about 0.3, while s* D*-0.165 R-0.011, which has a mean of about 0.4 and a tiny standard deviation smaller than 0.05, is fixed at 0.4 so that D'' = D* R-0.155 is the independent DSD parameters in this case. We shall refer to the resulting "restricted" gamma DSD model as N2. It is the DSD model used in the TRMM combined radar/radiometer algorithm. The 3rd and 4th DSD models are similar "restrictions" of the gamma model, obtained by imposing a deterministic relation between N0 and . We chose the relation
N0 = 6734 e1.45 mm-1-   m-3
for the 3rd model N3, and the relation
N0 = 1500 e0.84 mm-1-   m-3
for the 4th model N4. Finally, we also consider a model which does not depend on any closed analytic form for the distribution function N. After all, there is an enormous wealth of sampled DSD's measured from various probes, and there is no reason not to use a large subgroup of such samples as an a-priori database in lieu of a model. Indeed, for our fifth DSD "model" N5, we chose the TOGA-COARE database of DSD samples collected by the NCAR 2-D PMS probes mounted on the NCAR Electra aircraft over the warm pool of the western equatorial Pacific between November 1992 and February 1993.

The next five panels show how the radar signatures of these five DSD's differ. Having calculated the Mie extinction and back-scattering efficiencies as a function of drop diameter, one can calculate for any (rain-rate,DSD) pair (R, N) in any one of our five models the corresponding radar reflectivity factors z14(R,N) and z35(R,N) (in mm6 m-3), and the corresponding attenuation coefficients k14(R,N) and k35(R,N) (in dB/km). The panels show the resulting "reflectivity manifolds" (to borrow a term dear to the passive radiometer community) for each of our DSD models:

In all but the last case, these "manifolds" were obtained by choosing a few representative values for the free DSD parameter and letting R vary from 0.2 to 200 mm/hr. In all cases, the value of the difference z14(R,N) - z35(R,N) is plotted versus z14(R,N). The first observation is that, for all five DSD models, when the 14-GHz reflectivities are small, the rain rate curves are almost horizontal, confirming our previous observation that for lighter precipitation there simply is no significant difference between the two frequencies.

There are two additional facts illustrated by the figure which are crucial to the retrieval problem. The first is that all the "curve crossings" correspond to retrieval ambiguities: they indicate that a pair of (14-GHz, 35-GHz) reflectivity factors can be explained by at least two rain rates (which can differ by a factor of two or more -- the two-dimensional manifolds could not be readily made to illustrate this ambiguitiy quantitatively), associated to different DSD parameter values. This implies that even in the absence of any observation noise, the dual-frequency retrieval problem can be ambiguous (and manifestly more so in the exponential case than in the other cases, though all the models have non-negligible ambiguities at low precipitation). Since these ambiguities are intrinsic to the dual-frequency observations, one would need to consider additional measurements to resolve them. The second point concerns the "blank" regions in the plots. These are most evident in the least ambiguous cases N2 and N3, though they are not entirely absent in the other models. Indeed, current technology cannot guarantee that the noise in the reflectivity measurements is less than about 0.3 dB r.m.s. at best. Thus, one's actual observations could quite easily fall outside the region covered by our manifolds, i.e. it is quite likely that with any DSD model one will face the situation where no rain rate can "explain" exactly a pair of (noisy) reflectivities. Therefore, when attempting a retrieval, one must have a rigorous mechanism to assess the plausibility of the various model pairs which are "close" to the measured pair. In summary, a dual-frequency radar cannot entirely avoid the ambiguities with which we have been all too familiar in the case of the TRMM radar, and the noise in the measurements (along with the unavoidable imperfection of any DSD model) will make it essential to allow for multiple inexact "matches" Both of these concerns make it highly desirable to use a Bayesian framework to make unbiased estimates of the precipitation underlying the measurements.

There is yet another problem which leads us to consider a sixth case. It is brought about by the need to account for the cumulative attenuation at both frequencies as one estimates the rain rate sequentially through the consecutive vertical range bins in the cloud. It is however easiest to describe this sixth case after we outline the Bayesian retrieval approach.

Dual-frequency Bayesian approach

In order to keep the problems associated with the specific retrieval procedure separate from the DSD ambiguities themselves, we applied the simplest Bayesian approach to the dual-frequency profiling problem. Let us start by fixing the notation. For a given vertical column of precipitation, call Z14(i) (respectively Z35(i)) the radar reflectivity factor measured from the ith vertical range bin at 14 (resp. 35) GHz, with i=1 for the first bin at the top of the rainy cloud and increasing downward. The equations that have to be solved for the (rain-rate,DSD) pair (R,N) at each range bin i are

Z14(i) = z14(R,N) - 2 A14(i-1) + noise14
Z35(i) = z35(R,N) - 2 A35(i-1) + noise35

where A14(i-1) (resp. A35(i-1) is the one-way 14 (resp. 35) GHz attenuation accumulated from the top of the cloud until the ith range bin, expressed in dB. To solve these equations for the unknowns R and N, one would thus need to track the accumulated attenuations A14 and A35. Assuming that the noise terms "noise14" and "noise35" are 0-mean Gaussian with variances v14 and v35, the simplest Bayesian approach consists in two steps repeated recursively for the consecutive range bins:

  1. starting at the top of the cloud (i=1), and setting A14(0) = A35(0) = 0, consider all realistic rain rates R and all DSD's N allowed by the a-priori model, and calculate for each pair (R,N) its mean-squared distance di from the two independent measurements:

    di(R,N) = ( Z14(i) + 2 A14(i-1) - z14(R,N) )2 / v14 + ( Z35(i) + 2 A35(i-1) - z35(R,N) )2 / v35

    These distances can then be used to define probability weights

    pi(R,N) = e-0.5 di(R,N),

    normalized so they integrate to 1, and the optimal unbiased estimate of the rain rate would then have to be the average R calculated using these probabilities.

  2. the corresponding accumulated attenuation up to and including the current range bin must then be updated, again by incrementing A14(i-1) and A35(i-1) by the probability-weighted incremental attenuation within the ith layer.
This is the Bayesian retrieval approach. It can be initialized at the bottom of the column if one has an estimate of the total path attenuation, in which case the recursion would proceed from the surface up.

Before applying this approach to the CAMEX measurements and comparing its retrievals with the five a-priori DSD cases, we shall now describe a sixth case which we had to consider for completeness. It comes about because rain is not the only source of attenuation of microwaves in the atmosphere. While absorption by oxygen and water vapor is relatively small and largely predictable, the attenuation due to cloud liquid water, especially at 35 GHz, is not negligible. Indeed, while clouds are not sufficiently reflective to be detectable, they will cast a "shadow", and this shadow may differ in "clear air" and within the rain. For example, a rather moderate two vertical kilometers of liquid cloud carrying 0.5 g/m3 of water will attenuate the 35 GHz signal by about 1 dB. This presents two problems. First, the surface cross-section in "clear air" (i.e. where the reflectivities from the atmosphere do not exceed the relatively high radar noise threshold), which is necessary to the proper estimation of the integrated attenuation within precipitation, would be under-estimated if no account is taken of the attenuation due to any undetected cloud. This would result in an underestimate of the PIA, and that is the main reason we chose not to use any a-priori information about the PIA in our retrieval approach. Second, within the precipitation, at each vertical resolution bin one must estimate (and "remove") the extra attenuation due to the cloud, and this cannot be done without biasing the estimate if one does not know how to apportion the attenuation between precipitating and non-precipitating liquid. We decided to test the effect of this "cloud-shadow" problem by considering a sixth case, where the DSD is the TOGA-COARE database, but where we systematically assume the existence of cloud liquid with liquid water content M (g/m3) equal to 20% of the precipitating liquid water in the given DSD sample and with an attenuation coefficient of 0.84 M dB/km. We shall refer to this DSD case as N6.

CAMEX-4 results -- part 1: Gabrielle

The following two panels show the rather low radar reflectivities measured at nadir over Tropical Storm Gabrielle on September 15, 2001. The system had just emerged off the Florida coast over the Gulf Stream (around 30oN 79oW), but had not re-intensified:

  • measured 14 GHZ reflectivities:

  • measured 35 GHZ reflectivities:

The following panels show the retrieved rain rates and mean drop diameters for the DSD models N1, N2, N5 and N6 in that order:

  • the retrieved rain rates:

  • the retrieved mean drop diameters:

The following panels show the one-way integrated attenuations corresponding to each of the models considered, along with the surface-reference PIA estimated from two models (a single average clear-air surface-cross-section reference value, and a polynomial-fitted model), at 14 GHZ (left) and 35 (right):

Finally, the following panels show the difference between the measured radar reflectivity factors and those reconstructed from the results of the Bayesian retrieval, in each of the four cases considered in this example:

  • at 14 GHZ:

  • at 35 GHZ:

CAMEX-4 results -- part 2: Humberto

The following two panels show the radar reflectivities measured at nadir over Hurricane Humberto on September 24, 2001. The cyclone was embedded in a strong southwesterly flow, and anticyclonic outflow from the convective region was quite obvious. The warm core in the eye was weak, about 2 to 3 K warmer than the surrounding environment. There was a large cirrus outflow extending several hundred nautical miles from the center near 37oN 63oW:

  • measured 14 GHZ reflecitivities:

  • measured 35 GHZ reflectivities:

The following panels show the retrieved rain rates and mean drop diameters for the DSD models N1, N2, N3, N4 and N5, in that order:

  • the retrieved rain rates:

  • the retrieved mean drop diameters:

The following panels show the various PIA's at 14 GHZ (left) and 35 (right):

Finally, the following panels show the errors in the case of N1, N2 and N5, in that order:

  • at 14 GHZ:

  • at 14 GHZ:

The main conclusion has to be that several quite different DSD models do indeed produce plausible dual-frequency precipitation estimates at least over the convective systems observed during CAMEX-4. The precipitation amounts do differ from model to model, but the general shape of the vertical variation of the retrieved drop size seems similar among the different models considered. Most important, it does appear that the a-priori decision about which drop size distributions should be considered plausible does have a determining effect on the eventual retrievals. In the future, such a-priori assumptions will therefore have to be justified by detailed DSD measurements at radar-sized resolutions.

Click here for a pdf version of the preprint summarizing these results.

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