## Introduction

The precipitation radar (PR) of the Tropical Rainfall Measuring Mission (TRMM) is an unprecedented tool for observing precipitation from space, in addition to the visible/infrared (VIS/IR) radiometer and the TRMM microwave imager (TMI) on board the platform. At the operating frequency (13.8 GHz), and cross-range resolution of the radar (about 4.2 km at nadir), it is necessary to correct for the two-way path-integrated attenuation (PIA), and nonuniform beam filling (NUBF) effects (Amayenc et al. 1993), in order to reduce bias in rain estimation. These challenges are the main ones that the PR standard rain-profiling algorithm, labeled 2A-25 in the TRMM nomenclature, has to face (Iguchi et al. 2000). Rain estimates also depend on the selected relations between the integrated rainfall parameters, and the way they are adjusted in the algorithm. The standard TRMM algorithms, including the 2A-25, were revised several times, and hopefully improved. The version-4 2A-25 was changed to the presently operational version 5 in midautumn of 1999. Accordingly, TRMM data acquired since satellite launching (at the end of November 1997) were reprocessed at the National Aeronautics and Space Administration (NASA)/TRMM Science Data and Information System (TSDIS) between November 1999 and April 2000. Tests performed with the version-4 2A-25 are reported to have a general tendency toward underestimating the rain rate *R* relative to other ground-based or space-based estimates such as monthly zonal averages derived from TMI or a TMI–PR combination (Iguchi et al. 2000; Kummerow et al. 2000). Modifications made in version 5, with respect to version 4, aim at alleviating these kinds of discrepancies. New upgraded versions of the algorithm are planned to be developed in future. Meanwhile, there is a need for TRMM experimenters to analyze possible deficiencies of the algorithm, to appreciate improvements brought by any new version, and to help to suggest new development.

In this context, the purpose of the present paper is twofold. First, possible improvements of the version-4 2A-25, obtainable from different adjustments of the prescribed initial rain relations, are explored using two alternatives to the standard rain rate. The second objective is to analyze improvements in *R*-estimates brought by the standard version-5 algorithm with respect to the standard version-4 2A-25 algorithm. With version 4, the first alternative rain rate exploits the concept of normalized rain drop size distribution (DSD) to scale the rain relations via a relevant parameter; the second one exploits the relation between *R* and the attenuation coefficient *k*, instead of reflectivity *Z* as in the standard. This allows us to point out effects of various error sources and limits on accuracy in rain retrieval expected from a single-frequency radar such as the TRMM PR. A preliminary study of such computable *R*-estimates was presented in Tani and Amayenc (1998), exploiting data of the airborne TRMM PR simulator, airborne rain-mapping radar (ARMAR; Durden et al. 1994) in TOGA COARE (Webster and Lukas 1992). The computational parameters can be easily obtained from the output file of the standard version-4 2A-25 without changing the physical concepts used in the algorithm. The alternative approaches could be used in the framework of the version-5 2A-25, but they would imply a full reprocessing after implementing additional specific coding, which is not attempted here.

In the framework of version 4, the reliability of the alternative rain estimates as compared with the standard one is tested from a TRMM PR dataset. The same PR dataset is also used to compare standard version-5 and standard version-4 results.

The paper is organized as follows. The basic concepts of the 2A-25 algorithm are outlined in section 2. The adjustment of rain relations, the alternative *R*-estimates, and a sensitivity study to various error sources are presented in section 3. Then, detailed results from PR data gathered in Hurricane Bonnie during Convection and Moisture Experiment 3 (CAMEX-3; 1998), along with the mean features of the results for a PR dataset over ocean and land, are discussed in section 4. A comparison of PR-derived *Z* and *R* fields with those gathered from airborne X-band dual-beam radar data in Hurricanes Bonnie (26 August 1998) and Brett (21 August 1999), for good cases of TRMM overpasses over the ocean, is presented in section 5. Conclusions and some prospects for future work are given in the last section.

## Basic concepts and rain estimate in the standard 2A-25

A detailed description of the 2A-25 algorithm is given in Iguchi et al. (2000). The basic concepts are described below with some emphasis on aspects and/or ingredients that are addressed in the present study. This outline concerns version 4 of the algorithm. Then, the main differences between versions 4 and 5 are briefly mentioned.

### Outline of the version 4 2A-25 algorithm

*α*in the chosen initial relation

*k*

*αZ*

^{β}

*δα*[defined in (12)]. In practice, this factor, which is further used to compute the two-way PIA factor for range

*r,*has the hybrid form

_{f}

*w*

*w*

_{0}

*w*is a normalized weight (0 ≤

*w*≤ 1), and ε

_{0}is the correction factor based on the surface reference (SR) technique (Meneghini et al. 1983; Meneghini and Nakamura 1990). The weight

*w*is a function of the total PIA factor to the surface (at range

*r*

_{s})

*A*

_{HB}(

*r*

_{S}), estimated from the solution of Hitschfeld and Bordan (1954, hereinafter HB), and the SR-based total PIA factor

*A*

_{S}(

*r*

_{S}), given respectively by

*A*

_{HB}

*r*

_{S}

*γS*

*r*

_{S}

^{1/β}

*γ*= 0.2 ln(10)

*β*= 0.46

*β,*and

*Z*

_{m}is the “apparent” or measured reflectivity factor. The SR-based correction factor ε

_{0}is computed from (5), according to

_{0}

*A*

^{β}

_{S}

*r*

_{S}

*γ*

^{−1}

*S*

*r*

_{S}

^{−1}

*A*

_{S}(

*r*

_{S}) is provided by the 2A-21 algorithm from surface echo measurements (Meneghini et al. 2000), and

*Z*

_{m}(

*r*) in

*S*(0,

*r*

_{s}) is provided by the 1C-21 algorithm (Iguchi et al. 2000). It is assumed that

*Z*

_{m}is not biased by an error in the radar calibration and

*A*

_{S}(

*r*

_{S}) is error free.

*ζ*=

*S*(0,

*r*

_{S}) = 1 −

*A*

^{β}

_{HB}

_{f}towards ε

_{0}in (2) increases with the magnitude of the HB-based PIA and the reliability of the SR-based PIA. The resulting PIA factor

*A*

_{f}(

*r*) is the hybrid of

*A*

_{HB}(

*r*) and

*A*

_{S}(

*r*) according to

The attenuation-corrected reflectivity factor at any range *r,* *Z*(*r*), is derived as the ratio of *Z*_{m}(*r*) to the PIA factor *A*_{f}(*r*). Therefore, the *Z*-profile is the hybrid of the HB-based solution that does not perform *α*-correction (ε = 1), and the SR-based solution with *α*-adjustment (ε = ε_{0}). The hybrid scheme, which provides *Z* retrievals close to the HB-based/SR-based solutions for low/large PIAs with *w* ≈ 0/*w* ≈ 1, respectively, avoids known potential divergence of the HB-based solution for high PIA or large error in the SR-based solution for low or unreliable PIA. Most of the time, however, the *α*-adjustment is hybrid since ε_{f} lies between 1 (HB case) and ε_{0} (SR case).

Also, for each beam the algorithm performs a range-free correction of NUBF effects based on a statistical scheme that involves PR data for the eight beams surrounding the angle bin in question to determine the index of nonuniformity (Kozu and Iguchi 1999).

*R*

_{std-V4}, is derived from the “true”

*Z*via a prescribed initial relation,

*R*

*aZ*

^{b}

*a*is corrected for NUBF effects.

In the two relations, *k*–*Z* [(1)] and *R*–*Z* [(8)], the initial coefficients *α,* *a,* and *b,* are functions of height due to changes in temperature, phase, and pressure (for *R*–*Z*). They also depend on rain type (as categorized by the 2A-23 algorithm) that is determined by the horizontal and vertical storm structure model (Iguchi et al. 2000). In (1), *β* is assumed range independent. In version 4, the initial coefficients for rain are modeled from ground-based distrometer data at Darwin (Australia), assuming Γ-shaped DSD with a shape parameter *μ* = 1. For ice, or mixed phase, the initial coefficients are modeled assuming prescribed ice density, or mixing ratio. Therefore, the height profile of all initial coefficients (except *β*) may change from beam to beam, according to the rain type and storm model.

Poorly adapted initial rain relations may induce significant errors in rain estimates, especially for low PIA (i.e., in stratiform light rain), since the HB-based solution does not perform an adjustment. For large PIA (i.e., in convective rain), the SR-based solution adjusts the *k*–*Z* relation. However, the initial *R*–*Z* relation is not modified. Hence, it was found useful to look for self-consistent scaling of the involved rain relations and alternatives to the standard version-4 rain-rate estimate.

### Main changes from version 4 to version 5

Significant changes in version 5, with respect to version 4, are as follows:

- modified inputs from other algorithms including an improved computation of
*σ*^{0}in clear air for the SR (from the 2A-21 algorithm), a better rain-type classification (from the 2A-23 algorithm), and a correction of the radar calibration, increasing*Z*_{m}by 0.52 dB (from the 1C-21 algorithm); - a better identification of the range of useful signal and noise elimination;
- a slightly modified vertical structure of the storm model;
- an improved calculation of the correction factor ε
_{f}based on a statistically objective, instead of arbitrarily prescribed, HB/SR weighting that takes into account the estimated uncertainties in the HB-based and SR-based total PIAs; - modifications of the initial rain relations,
*k*–*Z*and*R*–*Z,*relying on a worldwide-averaged empirical, instead of the Darwin-based, DSD model; and - the use of an adjusted instead of constant (initial)
*R*–*Z*relation with coefficients modified in accordance with*α*-adjustment in*k*–*Z*and the DSD model (Kozu et al. 1999).

Items 1–3 feature mostly “technical” improvements. Items 4–6 are the most important ones. Concerning item 6, the method used to adjust the rain relations and then to obtain the standard version-5 rain-rate estimate, further referred to as *R*_{std-V5}, differs from the *N*^{*}_{0}*N*^{*}_{0}

## Alternative rain estimates to the version-4 2A-25 standard

*R*-estimates arising out of previous works (Marzoug and Amayenc 1994; Iguchi and Meneghini 1994; Tani and Amayenc 1998; Durden and Haddad 1998) are considered. The first one uses a scaling of rain relations exploiting the concept of normalized DSD; the second one uses a constant

*R*–

*k*relation

*R*

*ek*

^{d}

*R*–

*Z*relation as in (8).

### Scaling and adjustment of rain relations

*X,*

*Y*) to be expressed as

*X*

*mN*

^{*(1−n)}

_{0}

*Y*

^{n}

*m*does not depend, and

*n*weakly depends, on the shape parameter

*μ*(Ulbrich 1983) of the DSD, provided that

*X*and

*Y*are close to DSD moments.

*N*

^{*}

_{0}

*μ*-free scaling parameter that identifies with the “classical”

*N*

_{0}intercept of the exponentially shaped DSD having the same water content, and mean volume diameter,

*D*

_{m}, as those of the Γ-shaped DSD. Using

*D*

_{m}instead of the median volume diameter

*D*

_{0}as in Dou et al. (1999a,b) is more convenient for practical application (Testud et al. 2001). In (10),

*m*and

*n*can be computed from fits to a scattering–attenuation model using experimental data. Thus, any rain relation can be scaled by

*N*

^{*}

_{0}

*n*approaches 1, the

*X*–

*Y*relation (10) becomes less sensitive to

*N*

^{*}

_{0}

The aforementioned property was exploited to compute the reference *N*^{*}_{0}*N*^{*}_{0}*μ* = 1 (in accordance with the underlying DSD model in 2A-25), and temperatures ranging from 0° to 20°C. The reference *N*^{*}_{0}*N*^{*}_{0}*Z* range of *Z.* Figure 1 displays such a comparison for the *k*–*Z* relation at 20°C. The reference *N*^{*}_{0}*k*–*Z* or *R*–*Z*) and temperature. Inferred initial *N*^{*}_{0}*N*^{*}_{0}^{6} m^{−4}) of Marshall and Palmer (1948). The TOGA COARE results refer to mean values derived from airborne DSD data gathered with 2D precipitation (2-DP) Particle Measuring Systems, Inc., (PMS) probes during 15 flights. The numbers given in Dou et al. (1999a,b) are slightly revised owing to the use of *D*_{m} instead of *D*_{0}. Results for total rain are only indicative since they are computed as the mean of convective and stratiform rain cases while ignoring any weighting by the relative volume (or area) of each rain type. The inferred initial *N*^{*}_{0}

*k*as

*X*and

*Z*as

*Y,*the

*δα*-correction of

*α*

_{init}(via the factor ε) in the

*k*–

*Z*relation may be interpreted as a

*δ*

*N*

^{*}

_{0}

*N*

^{*}

_{0init}

*k*

*α*

_{init}

*δαZ*

^{β}

*δα*

*δN*

^{*(1−β)}

_{0}

*δN*

^{*}

_{0}

^{1/(1−β)}

*N*

^{*}

_{0}

*N*

^{*}

_{0adj}

*N*

^{*}

_{0init}

*δN*

^{*}

_{0}

*δ*

*N*

^{*}

_{0}

*a*

*eα*

^{d}

*d*

*b*

*β*

*N*

^{*}

_{0}

The hybrid character of the *α*-adjustment in 2A-25, via ε_{f}, leads in practice to a hybrid *N*^{*}_{0}

It is useful to compare *N*^{*}_{0}*N*_{0}-adjustment of Kozu et al. (1999) used in the version-5 2A-25 (see section 2b). In the latter method, the assumed DSD model is built from a set of rain-type-dependent *Z*–*R* relations measured over the ocean at various places over the world. The *Z*–*R* relation is converted into the relation between *N*_{0} and Λ of a Γ-shaped DSD with a shape parameter *μ* = 3. The corresponding *k*–*Z* and *Z*–*R* relations at 13.8 GHz are calculated for rain, snow at different temperatures, and different mixing ratios. Initial coefficients *a* and *b* in *R* = *aZ*^{b} are adjusted when *α* in *k* = *αZ*^{β} is adjusted in such a way that the adjusted pair (*a,* *b*) is consistent with the pair (*α,* *β*) in *k* = *αZ*^{β} when they are both converted into the *N*_{0}–Λ of the DSD model. The value chosen for *μ* has weak impact on the derived *k*–*Z* and *Z*–*R* relations. In practice, the correction factors for *a* and *b* are defined as quadratic functions of log(ε_{f}). Therefore, both *N*^{*}_{0}*N*_{0}-adjustment allow self-consistent scaling of rain relations, but by different means. The two coefficients of the *Z*–*R* relations are adjusted in Kozu's approach, instead of only one in *N*^{*}_{0}

### Alternative rain estimates to the version-4 2A-25 standard

Let us now come to the basis of rain estimate computation taking into account potential error terms. As before, error in a given parameter *x* is defined as a unitless multiplying factor *δx,* thereby providing a “corrected” value, (*x* *δx*). The reasoning follows Tani and Amayenc (1998). It starts from the expression for the specific attenuation coefficient *k*(*r*), then comes to the expression(s) for the reflectivity factor *Z*(*r*), and finally the rain rate *R*(*r*). Related computations can also be found in Marzoug and Amayenc (1994), Iguchi and Meneghini (1994), Marécal et al. (1997), and Durden and Haddad (1998).

*k*(

*r*) estimate is

*k*

*r*

*αZ*

_{m}

*r*

^{β}

*εA*

*r,*

^{−β}

*δαδC*

^{−β}

*α*

*Z*

^{β}

_{m}

*C*is the radar calibration constant, and

*A*(

*r,*ε) is the PIA factor at range

*r,*given by

*A*

*r,*

*γ*

*S*

*r*

^{1/β}

*k*,

*Z*, and

*R*estimates depend on the way by which

*A*(

*r,*ε), and the various error terms, are evaluated. The HB-based solution assumes ε = 1 and

*δC*=

*δα*=

*δa*=

*δe*= 1 (no adjustment) in all estimates and therefore may be corrupted by “uncorrected” errors. The SR-based solution allows, in essence, identifying the “measured” correction factor ε

_{0}(6) with the product of error terms ε (19). Using ε = ε

_{0}in (18) and (20) yields

*k*-estimate

*k*

*r*

*αZ*

^{β}

_{m}

*ε*

_{0}

*A*

*r,*

_{0}

^{−β}

*δC*or

*δα*) being specified, since (19) involves the product of errors only.

*Z,*the correction type must be specified. In

*C*-adjustment (ignoring

*δα*; i.e., ε

_{0}=

*δC*

^{−β},

*δα*= 1), (21) yields

*Z*

_{C}

*Z*

_{m}

*ε*

^{1/β}

_{0}

*A*

*r,*

_{0}

^{−1}

*α*-adjustment (ignoring

*δC*; i.e., ε

_{0}=

*δα,*

*δC*= 1), it yields

*Z*

_{α}

*Z*

_{m}

*A*

*r,*

_{0}

^{−1}

*Z*

_{C}

*ε*

^{−1/β}

_{0}

*Z*-estimate for the true SR case. Of course,

*Z*

_{C}and

*Z*

_{α}may be corrupted by

*δα*and

*δC*respectively via the term

*δα*

^{−1/β}in (24) and

*δC*

^{−1}in (25). The

*α*- and

*C*-adjustments are basically different despite the similarities of the two adjustment process. This is because adjusting

*α*(or

*N*

^{*}

_{0}

*C*in such a way is not, when considering the reported stability in the PR calibration (Iguchi et al. 2000).

*R*

_{C}associated to

*Z*

_{C}is derived from (22) with ε = ε

_{0}=

*δC*

^{−β}and

*δa*=

*δe*= 1, as

*R*

_{C}

*aZ*

^{b}

_{m}

*A*

*r,*

_{0}

^{−b}

*ε*

^{d}

_{0}

*Z*

_{α}, using (25) with

*δC*= 1. Assuming a constant

*R*–

*Z*relation (

*δa*= 1) yields

*R*

_{ZR}

*Z*

_{α}

*aZ*

^{b}

_{m}

*A*

*r,*

_{0}

^{−b}

*R*

_{std-V4}for the true SR case.

*k*–

*R*relation (

*δe*= 1) yields

*R*

_{kR}

*k*

*aZ*

^{b}

_{m}

*A*

*r*

^{−b}

*ε*

^{d}

_{0}

*R*

_{C}, though

*Z*

_{C}and

*Z*

_{α}are different. This is because using constant

*k*–

*R*relation is equivalent to adjusting

*R*–

*Z*(by

*δa*=

^{d}

_{0}

*α*-adjustment in

*k*–

*Z*(by

*δα*= ε

_{0}), to keep a self-consistent set of rain relations via (17). Hence,

*R*

_{kR}is formally identical to

*R*

_{C}computed with “unmodified” relations but with

*Z*

_{m}corrected for a calibration error,

*δC*=

^{−1/β}

_{0}

*N*

^{*}

_{0}

*δa,*or (16b) for

*δe,*yields

*R*

_{N0}

*aZ*

^{b}

_{m}

*A*

*r*

^{−b}

*ε*

^{(1−b)/(1−β)}

_{0}

*R*

_{kR}, and

*R*

_{N0}, provide alternative

*R*-estimates to the version-4 standard

*R*

_{std-V4}. The three rain estimates obtainable from

*α*-adjustment are related by

*β*= 0.761,

*b*≈ 0.65;

*d*=

*b*/

*β*≈ 0.854), the rain-rate magnitudes are ranked according to

For example, ε_{0} = 1.5 and 0.5 leads to *R*_{N0} ≈ 1.8 and 0.36 *R*_{std-V4}, and *R*_{kR} ≈ 1.4 and 0.55 *R*_{std-V4}, both largely different from the standard *R*_{std-V4}.

In the hybrid case, identifying ε_{f} with error terms [see (2)], which was possible for ε_{0} in the true SR case, is not correct. The mathematical formulation of hybrid estimates involves sophisticated expressions that are not shown here. The important points are (i) the alternative rain estimates are still given by (27), (28), and (29); (ii) their ratios still satisfy (30a,b); and (iii) their relative magnitudes still obey (31a,b), provided that ε_{f} is substituted for ε_{0} in all cases. The three *R*-estimates remain different if ε_{f} ≠ 1.

Using (13), (14), (30a), and (30b), with ε_{f} instead of ε_{0}, allows us to compute *N*^{*}_{0}*R*-estimates directly from the output parameter file of the standard version-4 2A-25 without the need for reprocessing. The basic structure of the algorithm, that is, the attenuation and NUBF corrections, and the corrected *Z* profile, are not modified.

The standard rain estimate of the version-5 2A-25 *R*_{std-V5} based on the *N*_{0}-adjustment of Kozu et al. (1999) cannot be expressed analytically in a simple manner (see section 3a).

### Analytical study of the sensitivity to errors

The *N*^{*}_{0}*α* coefficient (according to temperature, phase state, and rain type), but range-independent *β* exponent in the *k*–*Z* relation (11). The height structure of initial *α,* further interpreted in terms of initial *N*^{*}_{0}*δ**N*^{*}_{0}*N*^{*}_{0}*N*^{*}_{0}*S* as given by (4). This may have significant impact on the accuracy achievable in the computations of the PIA at any range *r,* and the correction factor ε, using (3) to (7). Therefore, the accuracy achievable in computing *δ**N*^{*}_{0}

*δ*

*N*

^{*}

_{0}

_{0}that is first interpretable as

*δα*-correction in

*k*–

*Z*relation, and then as

*δ*

*N*

^{*}

_{0}

_{f}instead of ε

_{0}, is equivalent to a hybrid

*N*

^{*}

_{0}

*δN*

^{*(1−β)}

_{0}

*w*

*w*

*N*

^{*(1−β)}

_{0}

*w*is a normalized weight as in (2) for ε

_{f}, and Δ

*N*

^{*}

_{0}

*δ*

*N*

^{*}

_{0}

*N*

^{*}

_{0}

*N*

^{*}

_{0}

*α*-adjustment ignores errors in the radar calibration. Referring again to the SR case, a calibration error

*δC*≠ 1 implies from (19) that ε

_{0}≡

*δαδC*

^{−β}=

*δ*

*N*

^{*(1−β)}

_{0}

*δC*

^{−β}. Thus, performing

*δ*

*N*

^{*}

_{0}

*δC*is equivalent to

*N*

^{*}

_{0}

*δN*

^{*}

_{0}

*N*

^{*}

_{0}

*δC*

^{−β/(1−β)}

*N*

^{*}

_{0}

*N*

^{*}

_{0}

*δC*= 1). With

*β*= 0.761 (version-4 2A-25), a calibration offset

*δC*of 1 dB (the typical uncertainty for the TRMM PR) may induce an error of 3.2 dB in

*N*

^{*}

_{0}

_{0}. This may result, for instance, from changes in surface roughness related to raindrop impacts, or surface wind over the ocean (Meneghini and Atlas 1986). The hybrid scheme usually mitigates this effect except for heavy rain cases (i.e., large PIA) that are dominated by the SR-based solution. The NUBF effect may also change the hybrid correction factor ε

_{f}, which has some impact on

*N*

^{*}

_{0}

*N*

^{*}

_{0}

*R*-estimates can be conveniently evaluated by comparing (27), (28), or (29) with (22) that provides the true (though not directly computable) rain rate,

*R*

_{t}. For the SR case, this yieldswhich shows how differences between

*R*-estimates and

*R*

_{t}, depend on the errors

*δC*and

*δ*

*N*

^{*}

_{0}

*δA*

_{t}in the SR-based PIA is computed in appendix A. Ignoring such an error

*δA*

_{t}, and using

*β,*

*b,*and

*d*values of version 4, yields

For the SR case, via ε_{0}, the three *R*-estimates have different sensitivities to uncorrected errors. The *R*_{kR} estimate, like the *k*-estimate, is immune to radar calibration error *δC*; whereas the *R*_{std-V4} and *R*_{N0} estimates, like the *Z*-estimate, are not. In essence, *R*_{N0} is immune to *δ**N*^{*}_{0}*R*_{kR} is much less sensitive to *δ**N*^{*}_{0}*R*_{std-V4}, which reflects a well-known feature of rain relations readily obtainable from (10).

These claims have to be somewhat revised for the current case since all *R*-estimates become sensitive to all error types as a result of hybrid adjustment via ε_{f}. In particular, *R*_{N0} and *R*_{kR} become sensitive to error *δ**N*^{*}_{0}*δC* respectively. It can be shown, however, that (i) *R*_{N0} and *R*_{kR} remain less sensitive to errors *δC* and *δ**N*^{*}_{0}*R*_{std-V4}, and (ii) *R*_{kR} is less more sensitive to error *δC* and more sensitive to error *δ**N*^{*}_{0}*R*_{N0}.

Therefore, *R*_{N0} is conceptually the most attractive estimate, but its reliability may be questioned because of potential errors in *N*^{*}_{0}*N*^{*}_{0}*R*_{kR} is expected to be a better candidate than the standard, *R*_{std-V4}, owing to a lower sensitivity to errors. Comparisons of the three estimates from PR data are discussed in sections 4 and 5. Let us note that, in version-5 2A-25, the *N*_{0}-adjustment (Kozu et al. 1999), and the standard rain estimate *R*_{std-V5} are also sensitive to the above-mentioned errors. However, an analytical study of corruption effects by error terms is not achievable.

## Results of *N*^{*}_{0} -adjustment and rain estimates from PR data

*N*

^{*}

_{0}

Computation of rain-type-dependent *N*^{*}_{0}*R*-estimates were performed for several PR observations. The rain classification involves convective and stratiform rain types, along with total rain that refers to the mixing of both rain types without sorting.

*N*^{*}_{0} -distribution and rain fields retrievals in Hurricane Bonnie

*N*

^{*}

_{0}

Detailed results obtained in Hurricane Bonnie over the Gulf of Mexico, as observed on 26 August 1998 (orbit 4283) during CAMEX-3 in 1998 are analyzed. Figure 2 displays version-4 histograms of ε_{f}, and log(*N*^{*}_{0}_{f}-distribution reaches a maximum for ε_{f} > 1 with a mean of 1.561 and a standard deviation of 0.195. The associated distribution of *δ**N*^{*}_{0}*δ**N*^{*}_{0}*N*^{*}_{0}

The rain-type-dependent means of ε_{f} and *N*^{*}_{0}*N*^{*}_{0init}_{f}, the areal mean of the standard rain rate (〈*R*_{std-V4}〉), and the ratio of each alternative rain rate (〈*R*_{N0}〉 or 〈*R*_{kR}〉) to the standard, at 2-km height, are also listed. At this altitude, the rain signal is not contaminated by surface clutter at off-nadir beam incidences. In all cases, 〈ε_{f}〉 exceeds unity, the adjusted 〈*N*^{*}_{0}*R*_{N0}〉 the largest. All these deviations increase when going from the full-hybrid solution (all PIA) to the SR-weighted case (reliable PIA only). Ideally, the algorithm should provide ε_{f}-distribution with 〈ε_{f}〉 ≈ 1, and a small standard deviation. The fact that 〈ε_{f}〉 is preponderantly larger than unity implies a systematic adjustment of the *k*–*Z* relation, in such a way that the path-attenuation correction is increased with respect to the HB-based estimate [see (20)]. An 〈*N*^{*}_{0}*N*^{*}_{0}*N*^{*}_{0}*N*^{*}_{0}

Figure 3 points out how the retrieved rain-type-dependent 〈*N*^{*}_{0}*N*^{*}_{0}*N*^{*}_{0}*N*^{*}_{0}*N*^{*}_{0}*N*^{*}_{0}*N*^{*}_{0}*δ**N*^{*}_{0}*N*^{*}_{0init}*N*^{*}_{0}*δ**N*^{*}_{0}*N*^{*}_{0}

Results from version 5 are shown in Table 3, like in Table 2 for version 4, except that the last two lines refer now to the areal mean of the standard version-5 rain rate (〈*R*_{std-V5}〉), and the ratio 〈*R*_{std-V5}〉/〈*R*_{std-V4}〉, at 2-km height. In each category, 〈ε_{f}〉 is much closer to unity and the standard deviation of ε_{f} is strongly reduced, especially for the “all paths” case. The adjusted mean 〈*N*^{*}_{0}*N*^{*}_{0init}*R*_{std-V5}〉 is always larger than 〈*R*_{std-V4}〉. Note that the number of paths in each category is slightly different for the two versions.

Figure 4a shows height profiles of the mean reflectivity factor, 〈*Z*〉, retrieved in Bonnie from version-4 standard 2A-25 above 1.5-km height, for each rain type. The 〈*Z*〉-profile for total rain (no sorting versus rain type) looks like the stratiform profile, owing to the prevailing number of paths in stratiform rain (cf. Table 2). Height profiles of the three mean rain rates are shown in Figs. 4b–d. Differences between the alternatives and the standard in the rain region (below 4.5-km height) are almost constant versus height whatever the rain type. Alternative rain rates are higher than the standard 〈*R*_{std-V4}〉 by about 50% and 30% for 〈*R*_{N0}〉 and 〈*R*_{kR}〉 respectively, for all rain types. Figure 5a (Figs. 5b–d) displays height profiles of 〈*Z*〉 (〈*R*〉) retrieved from version-5 standard 2A-25. The height profiles of 〈*Z*〉 and 〈*R*〉, respectively, have similar shapes to their version-4 counterparts (Fig. 4) for each rain type. However, 〈*Z*〉 is larger by about 0.5 dB, which mainly results from change in the radar calibration (see section 2b), and 〈*R*_{std-V5}〉 exceeds 〈*R*_{std-V4}〉 by about 15% for convective rain and 30% for stratiform or total rain.

The bulk characteristics pointed out in *N*^{*}_{0}*R*-estimate results (for both versions of the algorithm) can also be seen from the analysis of a PR dataset in section 4b.

### Statistical results for a set of events

The *N*^{*}_{0}*R*-estimate computations were performed for 13 TRMM PR observations in various meteorological conditions over ocean (nine cases) and land (four cases) listed in Table 4.

Figure 6 shows, for all events, the mean parameters obtained from version 4: the hybrid correction factor 〈ε_{f}〉, the ratio 〈*N*^{*}_{0}*N*^{*}_{0init}*R*_{std-V4}〉, and the ratio of each alternative estimate to the standard 〈*R*_{N0}〉/〈*R*_{std-V4}〉 and 〈*R*_{kR}〉/〈*R*_{std-V4}〉. For each event, all parameters are given for convective, stratiform, and total rain. For all events, 〈ε_{f}〉 is greater than 1. Typically, 〈ε_{f}〉 ≈ 1.2 to 1.4. The sample means are close to 1.3. Consequently, 〈*N*^{*}_{0}*N*^{*}_{0init}*N*^{*}_{0init}*N*^{*}_{0init}*N*^{*}_{0}*R*_{std-V4}〉 for each event/rain type, 〈*R*_{N0}〉/〈*R*_{std-V4}〉 ranges from 1.35 to 1.8 and 〈*R*_{kR}〉/〈*R*_{std-V4}〉 ranges from 1.18 to 1.4. For all rain types, the sample mean of the ratio is close to 1.5 for 〈*R*_{N0}〉 and 1.25 for 〈*R*_{kR}〉. On average, the systematic tendency for *N*^{*}_{0}*N*^{*}_{0init}*R*-estimates to satisfy the inequalities *R*_{N0} > *R*_{kR} > *R*_{std-V4}, is confirmed.

Figure 7 shows similar results for version 5, except that the last two plots at bottom now display 〈*R*_{std-V5}〉 and the ratio 〈*R*_{std-V5}〉/〈*R*_{std-V4}〉. Comparison with results of Fig. 6 for version 4 shows that, for all events and rain types, 〈ε_{f}〉 (ranging from 1 to 1.07) is closer to 1; also, the standard deviation from the mean (not shown) is reduced by a large factor. All sample means are close to 1. Correlatively, 〈*N*^{*}_{0}*N*^{*}_{0init}_{f} and/or a better-adapted initial *k*–*Z* relation in version 5 (see section 2b). They clearly point out a better functioning of the version-5 algorithm. Meanwhile, observing 〈*R*_{std-V5}〉/〈*R*_{std-V4}〉 above unity in all cases confirms a systematic increase in rain-rate estimates from version 4 to version 5. This relative increase is smaller for convective rain (sample mean ≈ 1.2) than for stratiform or total rain (sample means ≈ 1.3), except for event 2.

In contrast to the findings of Iguchi et al. (2000) from results of the standard version-5 2A-25, differences between data over ocean or land (events 4–6, and 8) are not evident. However, all “land” cases refer to data taken in the vicinity of the Darwin site (Australia) during the wet season. This site is likely not very representative of typical continental conditions for the observed storms.

## Tests of PR-derived rain parameters using coincident airborne radar data

Rain products derived from the TRMM PR have to be compared with external data taken as reference although no rain measurement of any kind can be considered as “truth.” Small-scale rain observations over large areas from ground-based or airborne radar are quite useful provided that datasets are acquired in space–time coincidence with PR data. Here, we used data gathered by the Doppler dual-beam X-band radar on board National Oceanic and Atmospheric Administration (NOAA) P3 aircraft, in Hurricanes Bonnie and Brett. In each case, the P3 radar data were close in time to a TRMM overpass that occurred at 1137 UTC on 26 August 1998 (orbit 4283) for Bonnie and at 2240 UTC on 21 August 1999 (orbit 9967) for Brett, over the Gulf of Mexico. They are the best cases that we could select. Good coincidence between the P3 radar and PR observations allowed us to perform point-to-point comparisons of retrieved *Z*- and *R*-fields.

### Processing of the P3 radar data

A four-step procedure was used to correct the P3 radar data for path-attenuation and radar calibration error, take into account rain pattern advection, scale the retrieved *Z*-field at the PR beam resolution, and finally estimate the reference rain rate field, *R*_{P3}.

*Z*

_{m}-field of the P3 radar was corrected for path attenuation using the “hybrid” stereo–radar/dual-beam algorithm (Oury et al. 1999, 2000). The absolute error in the radar calibration Δ

*C*which may induce large bias in the

*R*-retrieval, was corrected according to Oury et al. (2000) as

*C*

*α*

*α*

_{0}

*β,*

*α*is the coefficient of the

*k*=

*αZ*

^{β}relation at X band provided by the analysis, and

*α*

_{0}is a reference value computed from a

*N*

^{*}

_{0}

*k*–

*Z*relation derived from a microphysical model. The model involves normalized DSD fitting to experimental conditions encountered in P3 radar measurements, that is, X band, vertical polarization, and nearly horizontal viewing of oblate raindrops. The

*k*–

*Z*relation was scaled by initial

*N*

^{*}

_{0}

*C.*Note that changing

*N*

^{*}

_{0}

*C*by 0.5 dB, and increasing/decreasing

*N*

^{*}

_{0}

*Z.*The above procedure provides

*Z*-fields from P3 radar that are corrected for path-attenuation effects, and calibration error.

^{−1}, whereas it is 1–2 min for the TRMM PR. Therefore, the nearly instantaneous rain pattern observed by the PR may evolve and move during the airborne radar sampling time. Such an effect was, one hopes, reduced by restricting the region used for comparisons to a domain in which the maximum time lag between the two samplings did not exceed 10 min. Also, the rain pattern was corrected for advection using a model for the horizontal tangential speed (

*V*

_{t}, m s

^{−1}) of individual rain cells versus the radial distance (

*d*, km) to the eye center (F. Marks, 2000, personal communication):

Considering the “degraded” PR cross-range resolution (≈4.2 km in a nearly horizontal plane at 350 km range) with respect to that of the P3 radar (≈1.6 km in horizontal direction, for typical maximum range of 50 km), a PR beamlike smoothing was applied to the P3 radar data in order to get significant comparisons of *Z* from both instruments. Accordingly, *Z*-fields retrieved from the P3 radar were interpolated horizontally on the PR grid, then averaged with a Gaussian beam-weighting gain function, *P*(*ρ*) = exp[−2 ln 2(*ρ*/*ρ*_{0})^{2}], where 2*ρ*_{0} = 4.2 km is the PR half-power cross-range resolution, and *ρ* is the radial distance of any involved P3 radar data point to the nearest data point of the PR grid. The difference in the PR range resolution (250 m) and the equivalent vertical resolution of the P3 radar were ignored.

For computing the P3-radar reference *R*-field, the vertical storm structure depending on the PR-derived rain-type and the *R*–*Z* relationships in ice were taken similar to those used in the 2A-25, for self-consistency. In rain, *R*-field was derived from *Z*-field using rain-type-dependent *N*^{*}_{0}*R* = *a**N*^{*(1−b)}_{0}*Z*^{b}. The coefficients *a* and *b* were modeled to be representative of the P3 radar experiment conditions (frequency, polarization, and viewing geometry); *N*^{*}_{0}*R* was corrected for change in air density with height according to Foote and du Toit (1969).

### Analysis of the TRMM PR/P3 radar comparison results in Hurricane Bonnie

Horizontal cross section, at a 2.8-km height, of raw reflectivities *Z*_{m} measured by the TRMM PR and P3 radar in Bonnie are shown in Fig. 8. The P3 radar *Z*_{m}-field (Fig. 8a) is underestimated when compared with the PR-derived one (Fig. 8b). The P3 radar calibration correction, derived from (36), leads to increased *Z*_{m} by 6.5 dB. The *Z*-fields corrected for calibration error and path-attenuation (version-4 2A-25 for the PR), are displayed in Figs. 8c,d. The two P3 radar fields in Figs. 8a,c are corrected for advection. The corrected *Z*-field is averaged at the PR beam resolution in Fig. 8c. Both corrected *Z*-fields agree much better. The “comparison domain” (≈112.5 × 100 km^{2}) is also drawn. In the area outside this domain, which constitutes a large part of the hurricane, the time lag between the two samplings exceeds 10 min, and so the area cannot be used for point-to-point comparisons. Besides, the small number of paths in convective rain in the comparison domain prevented us from collecting reliable results for this rain type. Thus, the following results refer to total (mostly stratiform) rain only.

Comparison of 3D PR and P3 radar corrected reflectivities, for all data points within the 2–4-km height range in the comparison domain, is shown in Fig. 9a, for version-4 PR results. The comparison involves only points where *Z* is above the PR detection threshold (18 dB*Z*). The associated histogram of differences, Δ*Z* = (*Z*_{TRMM} − *Z*_{P3}), displayed in Fig. 9b, is sharply peaked. As seen in Table 5, the mean difference is small: 〈Δ*Z*〉 = −0.7 dB with a standard deviation *σ*_{Z} = 4 dB (for 1817 data points) for version-4 results, and 〈Δ*Z*〉 = −0.2 dB with *σ*_{Z} = 4 dB (for 1765 data points) for version-5 results. The 0.5-dB shift is likely due to the change of 0.52 dB in the PR calibration in version 5 (cf. section 2b). Thus, 〈*Z*〉 retrieved from the PR, for both versions, is slightly lower than 〈*Z*〉 retrieved from P3 radar. This feature is almost constant with height in the rain zone (below 4 km), as shown by the mean horizontally averaged vertical *Z*-profiles in Fig. 10.

It is not expected that 〈Δ*Z*〉 be zero owing to differences in frequencies (*X*- and *K*_{u}-bands) and scanning geometries for both instruments. A simple data-based model of the expected difference 〈Δ*Z*〉_{th}, and the standard deviation *σ*_{Z,th} is described in appendix B. It is shown that the difference should be positive, that is, the PR value above the P3 radar one, with 〈Δ*Z*〉_{th} ≈ +1 dB. The observed 〈Δ*Z*〉 (−0.7 dB for version 4, and −0.2 dB for version 5) is slightly different from theoretical predictions; the best agreement is obtained for version 5. Anyway, the 1.7-dB and the 1.2-dB offsets for version 4 and version 5, respectively, are compatible with the uncertainty margin due to residual calibration errors of both radars (about 1 dB for each one). The observed large standard deviation (*σ*_{Z} = 4 dB) as compared with theory (*σ*_{Z,th} ≈ 0.2 dB) may come from combined effects of measurement noise, residual uncertainties in data collocations, and evolution/advection of the hurricane structure.

The mean (horizontally averaged) vertical *R*-profiles retrieved from the PR, and P3 radar (*R*_{P3}), are shown in Fig. 11. The mean differences, standard deviations, and ratios of all PR-derived estimates with respect to the P3 reference, 〈*R*_{P3}〉 = 3.4 mm h^{−1}, in the 2–4-km height range, are listed in Table 5. Histograms of point-to-point differences in the 2–4-km height range, for the alternative version-4 estimate *R*_{kR}, and the version-5 standard *R*_{std-V5}, are shown in Fig. 12. The version-4 alternative estimates, *R*_{kR} and *R*_{N0}, are larger than the standard *R*_{std-V4} by 19% and 39%, respectively, in accordance with previous findings (see section 4). Also, the standard *R*_{std-V5} is higher than *R*_{std-V4} by 15%. Clearly, *R*_{kR} and *R*_{std-V5} show the best agreement with the reference *R*_{P3}, within a 5% and 8% margin, respectively; *R*_{std-V4} underestimates *R*_{P3} by 20% and *R*_{N0} overestimates it by 11%. Therefore, the reported deficiency of *R*_{std-V4} seems rather well alleviated by *R*_{kR} or *R*_{std-V5}.

Among the involved rain rates, *R*_{std-V4}, *R*_{kR}, and *R*_{P3} are computed with a similar hypothesis concerning the DSD model. The computation relies on the use of constant initial *N*^{*}_{0}*R*_{N0}, which implies *N*^{*}_{0}*N*^{*}_{0}*R*_{P3} (to *R*^{′}_{P3}*N*^{*}_{0}*R* from an *N*^{*}_{0}*R*–*Z* relation at *X* band (see section 5a). The reference rain rate decreases slightly: 〈*R*^{′}_{P3}*R*_{P3}〉 ≈ 0.87 for total rain in the 2–4-km height range. The change is not as large as could be expected because effects of increasing *N*^{*}_{0}*Z* (as a result of the P3 radar calibration correction that becomes 5.5 dB instead of 6.5 dB) partly compensate for each other in the normalized *R*–*Z* relation at X band used to obtain *R.* The comparison between *R*_{N0} and the P3 radar reference estimate deteriorates when *N*^{*}_{0}*R*_{N0}〉/〈*R*^{′}_{P3}*R*_{N0}〉/〈*R*_{P3}〉 = 1.11 (cf, Table 5). This points out the need for assessing first the reliability of PR-derived *N*^{*}_{0}*N*^{*}_{0}*R*_{N0} and *R*^{′}_{P3}*R*_{std-V5} with *R*_{P3}, since the former relies on Kozu's *N*_{0}-adjustment (cf. section 2b) and the latter does not. However, there is no analytical way to test for the inclusion of such *N*_{0}-adjustment in the computation of *R*_{P3}.

### Analysis of the TRMM PR/P3 radar comparison results in Hurricane Brett

For Hurricane Brett, the P3 data were processed in the same manner as for Bonnie. The P3 radar calibration correction, derived from (36), leads to increased *Z*_{m} by 2.7 dB (instead of 6.5 dB in Bonnie, 1 yr earlier). Figure 13 displays horizontal cross sections of attenuation-corrected *Z*-fields retrieved from the P3 radar (Fig. 13a) and the PR (Fig. 13b) at 3.2-km height. Both fields are shown at the PR beam resolution, and the P3 radar field is corrected for advection. The comparison domain (90 × 80 km^{2}), centered on the hurricane eye, is also shown. Figure 13c shows the rain-type classification derived from the PR (from the 2A-23 algorithm). Brett has a smaller size than Bonnie (see Fig. 8). In the comparison area, however, large regions of convective rain exist, in contrast to Bonnie where stratiform rain prevailed. Hence, rain-type-dependent comparisons of *Z* or *R* could be achieved in Brett. Though the two reflectivity patterns have similar shapes, the PR-derived reflectivities are higher than the P3 radar–derived ones, especially in convective rain. The version-5 mean horizontally averaged vertical *Z*-profiles, depending on rain type, are shown in Fig. 14. The version-4 *Z*-profiles (not drawn) are weaker than the version-5 profiles by 0.5 dB at most. Results for stratiform rain show good agreement between the PR and P3-radar estimates. In contrast, for convective rain and total rain, the PR estimate deviates more and more from the P3 radar estimate as altitude decreases below about 4.5 km.

Bulk results of the rain-type-dependent mean difference 〈Δ*Z*〉 = 〈*Z*_{TRMM}〉 − 〈*Z*_{P3}〉 (and 〈Δ*R*〉 = 〈*R*_{TRMM}〉 − 〈*R*_{P3}〉), along with the associated standard deviation from the mean, in the 2–4-km altitude range, are listed in Table 6. The ratio of the mean *R* estimates 〈*R*_{TRMM}〉/〈*R*_{P3}〉 in the same altitude slab, for each rain type, is also indicated. Results are shown for version 4 and version 5, as in the Bonnie case. The small observed 〈Δ*Z*〉 (2.1 dB for version 4, or 2.3 dB for version 5) in stratiform rain agrees fairly well with the theoretical computation 〈Δ*Z*〉_{th} ≈ 1.1 dB (cf. appendix B), though the observed standard deviation is much larger than expected from theory (as in the Bonnie case). The residual offset (1 and 1.2 dB) lies within the margin of calibration errors of both radars. The observed 〈Δ*Z*〉 for convective rain (5 and 5.5 dB) and for total rain (3.4 and 3.7 dB) far exceeds the theoretical prediction; the residual offsets that exceed 2 dB are outside the margin of radar calibration errors. As for rain-rate estimates, for stratiform rain, the PR version-4 mean estimates range from −15% below the reference (〈*R*_{P3}〉 = 5.4 mm h^{−1}) for the version-4 standard to 24% and 59% above the reference for 〈*R*_{kR}〉 and 〈*R*_{N0}〉, respectively. The version-5 standard is only 4% above 〈*R*_{P3}〉. For convective rain (〈*R*_{P3}〉 = 12.8 mm h^{−1}) or for total rain (〈*R*_{P3}〉 = 8.7 mm h^{−1}), all PR estimates are largely above the reference by a factor of 1.5 to 2. The histogram of rain rate differences, Δ*R* = (*R*_{TRMM} − *R*_{P3}), for the PR version-5 standard and stratiform rain, which provides the best agreement, is shown in Fig. 15. Examining differences with the Bonnie case for stratiform rain, shows that the various PR estimates are ranked almost similarly with respect to the 〈*R*_{P3}〉 reference. However, the range of variations from the lowest to the highest estimate is larger in the Brett case. In particular, 〈*R*_{kR}〉 and 〈*R*_{N0}〉 deviate more from 〈*R*_{P3}〉 in Brett (by 59% and 24%, respectively) than in Bonnie (by 5% and 12%, respectively), as a result of the increased correction factor ε_{f} that may reveal quite ill-adapted initial rain relations in Brett. In both cases, 〈*R*_{std-V4}〉 is below 〈*R*_{P3}〉 (by 20% for Brett and 15% for Bonnie) whereas 〈*R*_{std-V5}〉 provides results that are close to 〈*R*_{P3}〉 (by 8% for Bonnie and 4% for Brett).

A possible explanation to the fact that large discrepancies with respect to the reference are observed in convective rain, then in total rain, can be suggested. Strong surface winds over ocean in hurricanes can modify the surface roughness below rain, thus corrupting the SR-based total PIA estimate derived from surface echo measurements. In stratiform rain, the corrected *Z*-profile retrieved from the 2A-25 algorithm is weakly-to-not weighted toward the SR-based solution. Conversely, in convective rain, the highly SR-weighted solution may suffer from the mentioned error in the SR-based PIA estimate. For low off-nadir beam-pointing angles (less than 17° for the PR), an increase in surface roughness due to surface wind (Ulaby et al. 1982), possibly enhanced by the effect of drop impact, leads to an overestimate of the total SR-based PIA, thus inducing an artificial increase in *Z* toward the surface, as observed in PR *Z*-profile for convective rain (Fig. 13b). Observing good agreement between *Z* retrievals aloft (above 5-km height) where path-attenuation is low, along with discrepancies increasing downward with the penetration depth into rain, supports the idea of an overestimated attenuation correction. Such a behavior does not appear in the *Z*-profile for stratiform rain (Fig. 13a) but is partly transferred into the *Z*-profile for total rain (Fig. 13c) via the contribution of convective rain. According to (A5) with *b* ≈ 0.65 in appendix A, overestimating the SR-based PIA by 5 dB may increase *R* by a factor of 2 near the surface, for the true SR case with large PIA. Such characteristics were not depicted in Bonnie because stratiform rain was predominant in the comparison domain. This points out a potential deficiency inherent to the 2A-25 algorithm: a possibly large overestimation of *Z* (and *R*) toward the surface in convective rain above ocean in the presence of strong surface winds, as is usually encountered in hurricanes.

## Conclusions and prospects

Testing improvements brought by changes in the standard versions of TRMM algorithms, and/or suggesting modifications aimed at improving these algorithms, is a sound work for TRMM experimenters. Potential improvements in rain-rate estimates from the TRMM PR standard version-4 2A-25 profiling algorithm were explored using different ways to adjust the involved rain relations. Also, changes from the previous standard version 4 (*R*_{std-V4}) to the presently operating standard version 5 (*R*_{std-V5}) were analyzed. Two alternatives to the standard version-4 *R*-estimate were derived. They rely on using either constant *R*–*k* relation (*R*_{kR}), or *N*^{*}_{0}*R*_{N0}) exploiting the concept of normalized Γ-shaped DSD, instead of constant *R*–*Z* relation as in *R*_{std-V4}. The computational parameters can be easily derived from the standard 2A-25 output file without need for reprocessing the algorithm. Analysis of errors in *N*^{*}_{0}*R*-estimates was conducted. The alternative *R*-estimates appear less sensitive to unknown errors in radar calibration or initial relations than the standard. Conceptually, *R*_{N0} is the most attractive, but its reliability is questionable owing to inherent effects of errors in *N*^{*}_{0}*N*^{*}_{0}*R*_{kR} is expected to be more reliable than the standard. Examination of such alternatives is useful to assess effects of various error sources and to determine limits on accuracy expected from a single-frequency radar such as TRMM PR.

A detailed analysis of the above-mentioned approaches was performed from PR observations in hurricane Bonnie, and the mean features were pointed out from a set of PR data (13 orbits) over ocean and land. The version-4 2A-25 yields adjusted *N*^{*}_{0}*R*-estimates, higher than *R*_{std-V4} (by about 25% for *R*_{kR}, and 50% for *R*_{N0}) for total rain, may correct for some reported underestimation of the rain rate by the standard (Iguchi et al. 2000; Kummerow et al. 2000). The new standard version 5 points out an improved functioning with respect to the standard version 4: the correction factor is much less spread and its mean is closer to unity, the need for large *N*^{*}_{0}

For better evaluating TRMM PR products, 3D PR-derived *Z*- and *R*-fields were compared with reference fields derived from airborne X-band dual-beam radar, on board a NOAA P3-42 aircraft, in Hurricanes Bonnie and Brett, for good cases of TRMM overpasses over the ocean. Special attention was brought to respect proper conditions for the comparisons. This involved, in particular, a small time lag between both datasets, rain-type-dependent estimations from P3 radar, and averaging of the P3 radar data at the PR beam resolution. Results deteriorate significantly when such conditions are not fulfilled.

For Bonnie, dominated by stratiform rain in the comparison domain, the observed mean difference, 〈Δ*Z*〉 = 〈*Z*_{TRMM} − *Z*_{P3}〉, in the 2–4-km height range is weak: −0.7 and −0.2 dB for version 4 and version 5, respectively. As compared with 〈Δ*Z*〉 ≈ 1 dB, as expected from the different frequencies and scanning geometries of the two instruments, the residual offsets are compatible with the uncertainty margin due to residual errors in the calibration of both radars. Comparison of mean *R*-profiles for total (mainly stratiform) rain shows that *R*_{kR} and *R*_{std-V5} agree with the P3 radar reference *R*_{P3} within a margin of 5% and 8%, respectively, and the *R*_{std-V4} and *R*_{N0} are respectively smaller and higher than *R*_{P3} by 20% and 11%. Therefore, the alternative version-4 estimate, *R*_{kR}, or the version-5 standard estimate, *R*_{std-V5}, corrects rather well for the identified deficiency of the version-4 standard estimate.

For the Brett case, comparisons could be made for convective and stratiform rain separately. In stratiform rain, as in the Bonnie results, the observed mean difference 〈Δ*Z*〉 = 〈*Z*_{TRMM} − *Z*_{P3}〉, in the 2–4-km height range, remains weak—2.1 dB (2.3 dB) for version 4 (version 5)—and is close to that (≈1.1 dB) expected from theory. The residual offsets still lie within the margin of calibration errors of both radars. Besides, all rain estimates are ranked almost in the same manner as in Bonnie with respect to the reference *R*_{P3} but show somewhat larger deviations from it. The best agreement is found for *R*_{std-V5}; it only differs from *R*_{P3} by 4%. This is not the case for convective or total rain. The mean differences 〈Δ*Z*〉 = 5–5.5 dB and 3.4–3.7 dB for convective and total rain respectively, depending on the algorithm version, leave residual offsets that are largely outside the margin of errors in the radar calibration, and the various rain estimates exceed *R*_{P3} by a factor ranging from 1.5 to 2.4. A possible explanation of such discrepancies is a corruption of the SR-like solution of the algorithm in convective rain, by overestimated SR-based total PIA due to changes in surface roughness in the presence of strong surface winds. This points out an inherent limit of the 2A-25 algorithm in such conditions.

It is too early to claim that the above-mentioned preliminary results have a general character, owing to the small number of cases that were processed. Additional data have to be studied before reaching definite conclusions. However, getting proper cases of good coincidence (i.e., short time lag) between airborne radar and TRMM PR observations, as is required to perform significant comparisons, is quite difficult. Occurrence of good coincidences increases by considering also ground-based radar data. Comparisons involving C-band polarimetric radar data gathered on the TRMM ground validation site of Darwin (Australia) are under way. An additional interest is the capability to check directly the reliability of PR-derived *N*^{*}_{0}*R*–*Z* relation for that based on *N*^{*}_{0}

The present work was performed at Centre d'étude des Environnements Terrestre et Planétaires (CETP) in the framework of Euro TRMM program, which involves a consortium of scientists from CETP (France), European Centre for Medium-Range Weather Forecasts (United Kingdom), German Aerospace Research Establishment (Germany), Instituto di Fisica dell'Atmosfera (Italy), Max Planck Institute for Meteorology (Germany), Rutherford Appleton Laboratory (United Kingdom), University of Essex (United Kingdom), Université Catholique de Louvain (Belgium), and University of Munich (Germany). Euro TRMM is funded by European Commission and European Space Agency. We acknowledge NASA/TSDIS and the Distributed Active Archive Center for free access to the TRMM data. We thank Dr. T. Iguchi (Communications Research Laboratory, Tokyo, Japan) and Dr. F. Marks (NOAA/Hurricane Research Division, Miami) for providing us with airborne radar data from Hurricanes Bonnie and Brett.

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# APPENDIX A

## Impact of an Error in the SR-Based PIA on Rain-Rate Estimates

*A*

_{s}(

*r*

_{s}; hereinafter noted

*A*

_{t}) measured from the surface echo attenuation is error free, the true rain estimate

*R*

_{t}is given by (22) where ε = ε

_{0}in the PIA factor at range

*r,*

*A*(

*r,*ε), given by (20). The ratio of each rain estimate (

*R*

_{std-V4},

*R*

_{kR}, or

*R*

_{N0}) to

*R*

_{t}was obtained by comparing (22) to (27), (28), or (29), providing (34a–c). In the presence of an error

*δA*

_{t}, the SR-based PIA becomes

*A*

_{t}

*δA*

_{t}, and ε

_{0}from (6) becomes

^{′}

_{0}

^{′}

_{0}

*A*

^{β}

_{t}

*δA*

_{t}

*γ*

^{−1}

*S*

*r*

_{S}

^{−1}

*R*

_{t}in (22) becomes

*R*

^{′}

_{t}

*A*(

*r,*

^{′}

_{0}

*A*(

*r,*ε

_{0}). Therefore, using the proper expressions for

*R*

^{′}

_{t}

*R*

_{t}from (22) yields

*R*

_{t}

*R*

^{′}

_{t}

*A*

*r,*

^{′}

_{0}

^{b}

*A*

*r,*

_{0}

^{−b}

*A*(

*r,*ε

_{0}) and

*A*(

*r,*

^{′}

_{0}

*S*(0,

*r*) = [

*S*(0,

*r*

_{s}) −

*S*(

*r,*

*r*

_{s})], and expressing

*S*(0,

*r*

_{s}) versus

*A*

_{t}from (6), yields,

*R*

_{t}by (A3) in (34a–c) points out the impact of an error,

*δA*

_{t}, on the rain-rate estimates, in addition to errors

*δC*and

*δ*

*N*

^{*}

_{0}

*R*

^{′}

_{t}

It may be verified that *E* decreases when the PIA and/or the distance (*r*_{s} − *r*) to the surface increases. Near the surface (*r* ≈ *r*_{S}) and for large PIA (i.e., *A*_{t} ≪ 1), *E* ≈ *δ**A*^{b}_{t}

# APPENDIX B

## A Data-Based Model Simulating Expected Differences in Z at X and K_{u} Bands

The observed distribution of differences in *Z*-fields between the airborne P3 radar (X band) and the PR (K_{u} band) may result from effects of sampling geometries, effects of DSD and phase variability at the two different frequencies and/or polarizations, errors in the calibration of each radar, and statistical uncertainties in *Z* measurements.

The following model simulates effects of sampling geometry, difference in frequency and polarization, and variability in the DSD, while assuming no error in the radar calibrations. It starts from the reference *R*-field derived from the P3 radar over a selected 3D domain, associated to *Z*_{X}-field (at X band) at the TRMM PR beam resolution, from which it is computed via rain-type-dependent *R*–*Z*_{X} relationships. The *R*–*Z*_{X} relations rely on a normalized Γ-shaped DSD model fitting to the airborne P3 radar measurement conditions, tuned with proper rain-type-dependent *N*^{*}_{0}*R*-field is used, as well, to generate the expected *Z*_{Ku}-field (at K_{u} band) via relevant normalized *Z*_{Ku}–*R* relations fitting, this time, to the TRMM PR measurement conditions. Apart from differences in frequencies, the conditions to be fulfilled when simulating the airborne and the spaceborne radar measurements are the nearly horizontal versus vertical viewing, and the vertical versus horizontal polarization, for the P3 radar versus the PR radar. Thus, the expected distribution of (*Z*_{Ku} − *Z*_{X}) over the selected 3D domain, can be easily computed. That gives the “expected” mean, 〈Δ*Z*〉_{th} = 〈*Z*_{Ku} − *Z*_{X}〉 and the standard deviation *σ*_{Z,th} to be compared with the measured ones.

This data-based model applied to data points in the rain region (2–4-km height range) of the comparison domain provides: 〈Δ*Z*〉_{th} ≈ +1 dB with *σ*_{Z,th} ≈ 0.2 dB for Hurricane Bonnie, and 〈Δ*Z*〉_{th} ≈ +1 dB (convective rain) to 1.1 dB (stratiform rain) with *σ*_{Z,th} ≈ 0.2 dB for Hurricane Brett.

Initial *N*^{*}_{0}^{6} m^{−4}) inferred from the 2A-25 algorithm initial relations (versions 4 and 5) compared with other reference values, for convective (C), stratiform (S), and total rain (mean of C and S rain)

Mean values (〈〉) of *ϵ*_{f} (and the standard deviation from the mean), *N*^{*}_{0}*N*^{*}_{0}_{init}, *R*_{std-V4}; and the ratio of each alternative (*R*_{N0} or *R _{kR}*) to the standard rain rate for stratiform (S), convective (C), and total (T) rain, as derived from version-4 2A-25 in Hurricane Bonnie (26 Aug 1998, orbit 4283). Rain rates refer to a 2-km altitude. Here,

*N*

^{*}

_{0}

_{init}stands for initial

*N*

^{*}

_{0}

_{s}is the SR-based total PIA

Mean values (〈〉) of *ϵ*_{f} (and the standard deviation from the mean), *N*^{*}_{0}*N*^{*}_{0}_{init}, *R*_{std-V5}; *R*_{std-V5/}/*R*_{std-V4} for stratiform (S), convective (C), and total (T) rain, as derived from version-5 2A-25 in Hurricane Bonnie (26 Aug 1998, orbit 4283). Rain rates refer to a 2-km altitude. Here, *N*^{*}_{0}_{init} stands for initial *N*^{*}_{0}_{s} is the SR-based total PIA

PR observations, and their main characteristics, involved in the results of Fig. 6; MCC stands for mesoscale convective complex. All observations are made over the ocean except 4, 5, 6, and 8, which were made over land

*Z*- and *R*-estimates, from the TRMM PR and P3 radar in the comparison domain of Hurricane Bonnie (see Fig. 7). Results refer to data points within the 2–4-km height range and total rain

Same as Table 5 but for Hurricane Brett, and results given separately for convective (C), stratiform (S), and total (T) rain