Outline of the version 4 2A-25 algorithm

The 2A-25 algorithm adjusts for every path the coefficient α in the chosen initial relation

kαZ^β(1)

by means of a range-independent correction factor ε = 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

_fww₀(2)

where w is a normalized weight (0 ≤ w ≤ 1), and ε₀ 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_HBr_SγSr_S^1/β(3)

where γ = 0.2 ln(10)β = 0.46β, and

Whether the total PIA is large is judged from an attenuation factor index ζ = S(0, r_S) = 1 − A^β_HB [see (3)]. The weighting of ε_f towards ε₀ 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 = ε₀). 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 ε₀ (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).

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:

1. modified inputs from other algorithms including an improved computation of σ⁰ 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);
2. a better identification of the range of useful signal and noise elimination;
3. a slightly modified vertical structure of the storm model;
4. 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;
5. 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
6. 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-scaling used in the present study with version 4 (see section 3a). Some prospects for exploiting N^*₀-scaling in the framework of version 5 are given in the last section.

Scaling and adjustment of rain relations

The aforementioned property was exploited to compute the reference N^*₀ implied by the initial relations used in the 2A-25 for stratiform or convective rain. For this purpose, the initial rain relations were compared with a set of normalized relations computed for different N^*₀ using Mie calculations, horizontal polarization at 13.8 GHz, μ = 1 (in accordance with the underlying DSD model in 2A-25), and temperatures ranging from 0° to 20°C. The reference N^*₀ was then determined, in each case, by identifying which N^*₀ produces the normalized relation best fitting (in the least squares sense) to the initial relation within the 20–50-dBZ range of Z. Figure 1 displays such a comparison for the k–Z relation at 20°C. The reference N^*₀ may change by about ±20% depending on the relation (k–Z or R–Z) and temperature. Inferred initial N^*₀ for the version 4 2A-25, given in Table 1 for the various rain types, agree fairly well with results from TOGA COARE, while N^*₀ for stratiform/convective rain is about one-half/2 times, respectively, the reference value (8 × 10⁶ 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 D₀. 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 results for version-5 2A-25 are also listed in Table 1. Though based on a different DSD model (see section 2b), they are fairly close to the version-4 results, except for stratiform rain.

According to (10), with 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 δN^*₀-correction to the initial N^*_0init. Using

kα_initδαZ^β(11)

with

δαδN^*(1−β)₀(12)

yields the correction

δN^*₀^1/(1−β)(13)

and the “adjusted” N^*₀

N^*_0adjN^*_0initδN^*₀(14)

Next, δN^*₀ is used to scale the coefficients of the other two relations,

according to

The hybrid character of the α-adjustment in 2A-25, via ε_f, leads in practice to a hybrid N^*₀-scaling that is discussed in section 3c.

It is useful to compare N^*₀-scaling (used here) with N₀-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 N₀ 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 N₀–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-scaling and N₀-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-scaling. Adjustment is performed via numerical functions in the former case, instead of the analytical approach in the latter one.

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).

The Z(r) estimate, using (19), is

Then, using either (15a) or (15b), and (11), (21), and (17), the true R_t(r) estimate becomes

Thus, 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

With the coefficients used in version 4 (i.e., β = 0.761, b ≈ 0.65; d = b/β ≈ 0.854), the rain-rate magnitudes are ranked according to

For example, ε₀ = 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 ε₀ 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 ε₀ in all cases. The three R-estimates remain different if ε_f ≠ 1.

Using (13), (14), (30a), and (30b), with ε_f instead of ε₀, allows us to compute N^*₀-scaling and alternative 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 N₀-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-scaling can be corrupted by several effects that may affect the various terms contributing to the computation of ε in (13). First of all, it must be recalled that the 2A-25 algorithm is based upon the hypothesis of a height-dependent 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, is determined by the prescribed storm structure model. The δN^*₀ correction, derived from the ε correction factor via (13), is a range-independent bulk adjustment of initial N^*₀. Any error in initial N^*₀, which may result from uncertainties in the storm structure model, the rain-type classification, the DSD characteristics, and so on, may corrupt the computation of 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 δN^*₀ from (13) may be affected as well. This fact is inherently attached to the use of attenuation-compensating algorithms based upon HB-like or SR-like methods.

The impact of various error sources on the three 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 yields

which shows how differences between R-estimates and R_t, depend on the errors δC and δN^*₀. The additional impact of a possible error 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 ε₀, 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 δN^*₀ error in the initial relations and R_kR is much less sensitive to δN^*₀ than 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 δN^*₀ and δC respectively. It can be shown, however, that (i) R_N0 and R_kR remain less sensitive to errors δC and δN^*₀ than the standard R_std-V4, and (ii) R_kR is less more sensitive to error δC and more sensitive to error δN^*₀ than R_N0.

Therefore, R_N0 is conceptually the most attractive estimate, but its reliability may be questioned because of potential errors in N^*₀-scaling. For rain estimates that do not use N^*₀-scaling, 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 N₀-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.

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

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), for the total rain. The results involve only selected PR beam paths where the SR-based total PIA was judged reliable. Such a criterion, associated in general to PIA > 3 dB, provides results that are significantly weighted towards the SR-based solution. The ε_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 δN^*₀ (not shown) from (13) also peaks for δN^*₀ > 1. The mean adjusted N^*₀ is higher than the initial value by a factor of about 4.

The rain-type-dependent means of ε_f and N^*₀/N^*_0init are summarized in Table 2 for the selected PR beam paths, and for the case involving all PR beam paths (note the large increase in the number of involved paths). For each rain type, the standard deviation of ε_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〉 exceeds the initial value, and the alternative rain rates are higher than the standard with 〈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〉 greater than N^*₀ implies that the initial N^*₀ is ill-adapted and/or N^*₀-scaling is contaminated by unknown error terms.

Figure 3 points out how the retrieved rain-type-dependent 〈N^*₀〉 from version 4 changes with respect to errors or effects that were discussed in section 3c. The standard hybrid algorithm results for all PR beam paths were considered as reference for comparisons. Then, the algorithm was reprocessed, after simulating separately some specific errors/effects. First, the stability of N^*₀-scaling results with respect to spatial sampling was checked by separately processing PR data included in the right-hand-side or the left-hand-side half-swath, and comparing them with the full-swath case. The 〈N^*₀〉 results (not shown) for the three configurations differ by less than 10%. The NUBF effects have low impact on N^*₀-scaling, especially for stratiform rain, since suppressing the 2A-25 NUBF correction in the simulation changes 〈N^*₀〉 by less than 15%. Adding a constant bias in the SR-based total PIA (±1 dB), radar calibration (±1 dB), or initial N^*₀ (±3 dB), may change the results by a factor of 2 with respect to the reference case. The sensitivity of N^*₀ to an offset δN^*₀ in N^*_0init is partly governed by the hybrid character of the adjustment (see section 3c). This prevents the offset in question to be compensated for in retrieved 〈N^*₀〉, as it would be for a true SR-based solution. The impact of errors amplifies when restricting the analysis to selected paths with large SR-weighting, except for δN^*₀-error that is better corrected via N^*₀-adjustment.

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〉 is closer to initial N^*_0init in all cases, and 〈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-scaling and 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-scaling and 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, the standard rain estimate 〈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 is larger than 1 for each rain type. It may reach a value of 3. For total rain, however, it does not exceed 1.8 owing to the fact that the reference N^*_0init is defined as the mean of the stratiform and convective rain cases. The ratio would increase if N^*_0init like 〈N^*₀〉 involved weighting by the relative numbers of PR beam paths in stratiform and convective rain. Referring to 〈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 to be adjusted to values higher than N^*_0init, and for the 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〉 (not used in the frame of version-5 data) is closer to the initial N^*_0init in all cases. Such changes likely result from the improved computation of ε_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.

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.

The typical sampling time of a hurricane by the airborne P3-radar is about 1 h owing to the aircraft flight speed of 120 m s⁻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/ρ₀)²], where 2ρ₀ = 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-scaled normalized relations R = aN^*(1−b)₀Z^b. The coefficients a and b were modeled to be representative of the P3 radar experiment conditions (frequency, polarization, and viewing geometry); N^*₀ was taken equal to the initial value used in the 2A-25 for stratiform or convective rain type (cf. Table 1), as categorized by the PR. Of course, this assumes that the PR and P3-radar data are perfectly collocated in space and time. The final 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 km²) 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 dBZ). The associated histogram of differences, ΔZ = (Z_TRMM − Z_P3), displayed in Fig. 9b, is sharply peaked. As seen in Table 5, the mean difference is small: 〈ΔZ〉 = −0.7 dB with a standard deviation σ_Z = 4 dB (for 1817 data points) for version-4 results, and 〈ΔZ〉 = −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 〈ΔZ〉 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 〈ΔZ〉_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 〈ΔZ〉_th ≈ +1 dB. The observed 〈ΔZ〉 (−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 for each rain type, which keeps the comparison fully coherent. This is not the case for R_N0, which implies N^*₀-scaling with point-to-point adjusted N^*₀ in version 4. Therefore, it is useful to evaluate the change of the reference R_P3 (to R^′_P3) when N^*₀-scaling is also used in the processing of P3 radar data, that is, in the calibration process and the estimation of R from an N^*₀-scaled 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 and decreasing 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-scaling is used in the two estimates: 〈R_N0〉/〈R^′_P3〉 = 1.27 instead of 〈R_N0〉/〈R_P3〉 = 1.11 (cf, Table 5). This points out the need for assessing first the reliability of PR-derived N^*₀-scaling results since any error in adjusted N^*₀ may bias the two estimates R_N0 and R^′_P3. The previous considerations also apply to the comparison of the standard version-5 R_std-V5 with R_P3, since the former relies on Kozu's N₀-adjustment (cf. section 2b) and the latter does not. However, there is no analytical way to test for the inclusion of such N₀-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 km²), 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 〈ΔZ〉 = 〈Z_TRMM〉 − 〈Z_P3〉 (and 〈ΔR〉 = 〈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 〈ΔZ〉 (2.1 dB for version 4, or 2.3 dB for version 5) in stratiform rain agrees fairly well with the theoretical computation 〈ΔZ〉_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 〈ΔZ〉 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, ΔR = (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.