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 Z^b 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 Z_m in that column by

PIA = ( 1 - 0.2 log(10) a b Z_m^b )^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.

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.

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.

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

N(D) = N₀ D e^{-L D}

N₀ = 6734 e^1.45 mm^-1-   m^-3

N₀ = 1500 e^0.84 mm^-1-   m^-3

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:

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.

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 i^th 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 i^th 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 d_i from the two independent measurements:

   d_i(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

   p_i(R,N) = e^{-0.5 d_i(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 i^th layer.

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.

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 30^oN 79^oW), 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:

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 37^oN 63^oW:

• 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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