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  • View in gallery

    R/A vs ZDR at C band for four gamma drop size distributions as a function of the shape parameter μ. The dotted line represents the inverse model deduced from the gamma DSD with μ = 2

  • View in gallery

    For a convective rain case, profiles along the radar beam of (a) apparent measured reflectivity Za (thin line) and corrected equivalent reflectivity Ze (heavy line); (b) measured differential reflectivity ZDRa (thin line) and corrected one ZDR (heavy line); (c) differential phase shift ΦDP, measured (thin line) and model adjusted (heavy line); the numbers are log10(N*0) and (*) symbols are boundaries of the ZPHI segmentation process; (d) retrieved rainfall rates from the classical estimate R(Za) (thin line) and from ZPHI (heavy line). Vertical lines in (a) and (c) result from jumps caused by the acquisition process and are treated as spurious points in ZPHI

  • View in gallery

    As in Fig. 2 but for a stratiform case

  • View in gallery

    Conical C-POL scan views (at 1.6° elevation, 19 Jan 1998 around 2240 UTC) of (a) the rainfall rate derived from ZPHI; (b) log10(N*0) retrieved by ZPHI (in gray when forced to the Marshall–Palmer value); (c) the rainfall rate R(KDP) derived from May et al.'s (1999) procedure

  • View in gallery

    Histograms of log10(N*0) derived from the Kowandi South BMRC disdrometer for five days of Jan 1998 (full line), and from 2D-P probes during 21 flights of TOGA COARE (dotted line)

  • View in gallery

    Histograms of log10(N*0) for rainfall rate greater than 10 mm h−1, derived (i) from ZPHI (dark and light gray filled), (ii) from the Kowandi South BMRC disdrometer for five days of Jan 1998 (full line), and (iii) from 2D-P probes during TOGA COARE (dotted line). The light gray filled histogram is from ZPHI after a calibration adjustment in Za of −1 dBZ, while the dark gray filled one is for ZPHI with the original Zas

  • View in gallery

    Scatterplots of RA,ZDR against RZPHI for r < 60 km and ZDR > 1 dB, after applying calibration corrections in Za of (a) −1 dB, (b) 0 dB, and (c) +1 dB

  • View in gallery

    C-scale rain gauge network spread around the C-POL radar, inside a 120-km-radius area. Position of the Kowandi South Disdrometer is also shown

  • View in gallery

    Weighting function used in the time-averaging process to compare radar- and gauge-derived rain rates

  • View in gallery

    Comparison of rain-rate time series on 2 Jan 1998. The rain rate, averaged over 19 rain gauge sites, is derived from rain gauge data (continuous line), and from radar data using the ZPHI algorithm (filled circle), or the classical estimate R(Za) (filled diamond). The daily integrated rainfall rates are specified for each time series. The algorithm index (heavy line) and the ZPHI quality index (thin line) are also shown in the bottom panel for each radar-derived point of the time series

  • View in gallery

    As in Fig. 10 but for 20 Jan 1998

  • View in gallery

    As in Fig. 10 but for 28 Jan 1998

  • View in gallery

    Point-by-point comparison of gauge rainfall rate (G(i) in the text) during seven days of Jan 1998, with collocated radar-derived estimates: (a) current ZPHI estimate (R(i)ZPHI in the text), (b) RA relation with fixed N*0 (R(i)ATT. in the text), (c) classical estimate RZa (R(i)CLASS. in the text)

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Validation of the Rain Profiling Algorithm “ZPHI” from the C-Band Polarimetric Weather Radar in Darwin

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  • 1 Centre d'Étude des Environnements Terrestre et Planétaires, Velizy, France
  • | 2 Bureau of Meteorology Research Centre, Melbourne, Australia
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Abstract

An extensive application of a rain profiling algorithm (ZPHI) employing a C-band polarimetric radar (the C-POL radar of the Australian Bureau of Meteorology Research Centre in Darwin) is presented. ZPHI belongs to the class of rain profiling algorithms that have been developed for spaceborne or airborne radars operating at attenuating frequencies. By nature, these algorithms are nonlocal: the full profile of the measured radar reflectivity is inverted to derive a retrieved profile of the rainfall rate. The retrieval accuracy lays in the imposition of an “external constraint” in the inversion procedure. In this case, that is supplied by the differential phase shift ΦDP. The primary products of ZPHI are the profile along the beam of the specific attenuation A, and the “normalized” intercept parameter N*0. The rainfall rate is further estimated through an RA relation adjusted for N*0. ZPHI solves automatically two problems met when operating at C band: the along-path attenuation and the variability of the raindrop size distribution. Moreover, its robustness with respect to radar statistical error allows ZPHI to operate with short dwell times, important for operational applications.

To provide high quality rain-rate retrieval, ZPHI requires careful radar calibration. Two techniques of calibration checking are investigated; both provide a calibration estimate to within 0.1 and 0.2 dB. One is based upon the climatological stability of the N*0 histogram. The second, which is purely radar based, uses a consistency test between the current rain-rate estimate by ZPHI and an estimate combining the specific attenuation A and the differential reflectivity ZDR.

Comparisons of rain rate, including an extensive dataset in the month of January 1998, show a remarkable agreement between rain gauge data and the ZPHI estimate, whereas the “classical” estimate (standard ZR relation applied without consideration of the attenuation) appears severely biased with respect to the rain gauges. In these comparisons, evidence for the crucial role of an N*0 determination to improve the rain-rate estimate is provided.

Corresponding author address: Dr. E. Le Bouar, Centre d'Étude des Environnements Terrestre et Planétaires, 10-12, Av. de l'Europe, 78140 Vélizy, France. Email: erwan.lebouar@cetp.ipsl.fr

Abstract

An extensive application of a rain profiling algorithm (ZPHI) employing a C-band polarimetric radar (the C-POL radar of the Australian Bureau of Meteorology Research Centre in Darwin) is presented. ZPHI belongs to the class of rain profiling algorithms that have been developed for spaceborne or airborne radars operating at attenuating frequencies. By nature, these algorithms are nonlocal: the full profile of the measured radar reflectivity is inverted to derive a retrieved profile of the rainfall rate. The retrieval accuracy lays in the imposition of an “external constraint” in the inversion procedure. In this case, that is supplied by the differential phase shift ΦDP. The primary products of ZPHI are the profile along the beam of the specific attenuation A, and the “normalized” intercept parameter N*0. The rainfall rate is further estimated through an RA relation adjusted for N*0. ZPHI solves automatically two problems met when operating at C band: the along-path attenuation and the variability of the raindrop size distribution. Moreover, its robustness with respect to radar statistical error allows ZPHI to operate with short dwell times, important for operational applications.

To provide high quality rain-rate retrieval, ZPHI requires careful radar calibration. Two techniques of calibration checking are investigated; both provide a calibration estimate to within 0.1 and 0.2 dB. One is based upon the climatological stability of the N*0 histogram. The second, which is purely radar based, uses a consistency test between the current rain-rate estimate by ZPHI and an estimate combining the specific attenuation A and the differential reflectivity ZDR.

Comparisons of rain rate, including an extensive dataset in the month of January 1998, show a remarkable agreement between rain gauge data and the ZPHI estimate, whereas the “classical” estimate (standard ZR relation applied without consideration of the attenuation) appears severely biased with respect to the rain gauges. In these comparisons, evidence for the crucial role of an N*0 determination to improve the rain-rate estimate is provided.

Corresponding author address: Dr. E. Le Bouar, Centre d'Étude des Environnements Terrestre et Planétaires, 10-12, Av. de l'Europe, 78140 Vélizy, France. Email: erwan.lebouar@cetp.ipsl.fr

1. Introduction

With the conventional weather radars deployed in most operational networks, the rainfall rate R is normally estimated from the radar reflectivity Z, using an empirical ZR relation. These estimates are subject to large uncertainties, particularly because of the variability of the drop size distribution (DSD). Under the simple assumption that the DSD follows an exponential law of the form N(D) = N0 exp(−3.67D/D0) (where N is the concentration by drop size diameter D, N0 is the intercept parameter, and D0 is the “median volume” diameter), and with the hypothesis that the terminal fall velocity of raindrops Vt is described by a power law like VtD0.67 (Atlas and Ulbrich 1977), one may establish ZR relations of the form ZN−0.50 R1.5 or ZD2.330 R, parameterized by N0 and D0, respectively. Since N0 and D0 are random variables subject to large fluctuations (Testud et al. 2001), none of these relations is able to predict R accurately from the knowledge of Z.

To overcome this problem, the anisotropy of the medium produced by the oblateness of raindrops has been exploited by polarimetric radars to provide new observables such as the differential reflectivity ZDR and the specific differential phase shift KDP. Using the quasi-functional relation existing between ZDR and D0, the above ZR relation parameterized by D0 may be transformed into a “two-parameter” estimator R(Z, ZDR) much less dependent on the DSD variability (Chandrasekar et al. 1990; Gorgucci et al. 1995). Meanwhile the “quasi-linear” RKDP relationship found by Seliga et al. (1986), Sachidananda and Zrnić (1987), and Chandrasekar et al. (1990) enables an estimator R(KDP) less sensitive than R(Z) to DSD variability. The combination of KDP and ZDR, first suggested by Jameson (1991) and further investigated by Gorgucci and Scarchilli (1997), provides a two-parameter estimator R(KDP, ZDR) also immune from DSD variability. These algorithms were designed primarily for S-band polarimetric radars and their application at C or X band is limited by two effects usually negligible at S band: (i) Z and ZDR may be affected by a two-way path attenuation through the rain; and (ii) a backscattering effect, which may affect the measured differential phase shift ΦDP (Keenan et al. 1997) and as a consequence may impact the accuracy of the subsequent KDP determination. Another limitation of the above cited algorithms is the large number of independent samples required to obtain an estimate with an acceptable statistical uncertainty. This requires long radar dwell times, generally incompatible with typical operational applications.

The performance of any radar algorithm for rain-rate retrieval requires consideration of (i) the sensitivity of the estimate to the statistical variability of the radar signal, and (ii) the immunity of the estimator to the physical variability of the DSD. In the following, the uncertainties associated with each of these effects are denoted “statistical uncertainty” and “physical uncertainty,” respectively. Algorithm ZPHI developed by Testud et al. (2000) tends to combine the moderate statistical uncertainty of the classical R(Z) estimate and the reduced physical uncertainty provided by the above-mentioned polarimetric algorithms. ZPHI belongs to the class of rain profiling algorithms that have been developed for spaceborne or airborne radars operating at attenuating frequencies (Kozu et al. 1991; Iguchi and Meneghini 1994; Marzoug and Amayenc 1994; Testud and Oury 1997). By nature, these algorithms are nonlocal: the full profile of the measured radar reflectivity is inverted to derive a retrieved profile of the rainfall rate. The accuracy of the rain-rate retrieval is achieved by the application of an “external constraint” in the inversion procedure. In ZPHI, the external constraint is supplied by the differential phase shift ΔΦ measured between the two range bounds r1 and r2 along the radial used to retrieve a rain rate profile. Three characteristics of algorithm ZPHI are important:

  1. it may cope with a short radar dwell time;
  2. it automatically performs a correction for attenuation (important at C and X bands); and
  3. it determines, for each profile, a parameter N*0 (“normalized” intercept parameter of the DSD), which allows one to tune the rain relation.
These characteristics make ZPHI particularly suitable for operational applications, whatever the radar frequency (S, C, or X band). The potential capability of ZPHI has been investigated by Testud et al. (2000) using numerical simulations. The object of the present paper is to proceed to a validation of the algorithm with real data. For this purpose, ZPHI is applied to an extensive dataset from a polarimetric radar, and R and N*0 retrievals are systematically compared with those derived from rain gauge and disdrometer. The radar data are provided by the C-POL radar (C band, polarimetric) of the Bureau of Meteorology Research Centre (BMRC), located near Darwin, Australia, and operated as a ground validation site of the Tropical Rainfall Measurement Mission (TRMM), described by Kummerow et al. (2000). After a brief description of ZPHI in section 2, an illustration of its application to the C-POL radar data is given in section 3. Two radar calibration techniques based upon ZPHI are then presented: the first one uses the climatology of N*0 (section 4), the second, purely radar based, examines the consistency between the various polarimetric parameters, including ZDR (section 5). Finally, section 6 presents a validation of the rain-rate retrieval by ZPHI, using the rain gauge network associated with the Darwin TRMM validation site.

2. ZPHI algorithm principle

a. Inverse model

The inverse model in algorithm ZPHI consists of a set of three relationships between A and Ze, KDP and A, and R and A, respectively [where A is the specific attenuation for horizontal (H) polarization, in dB km−1; Ze is the equivalent radar reflectivity for H polarization, in mm6 m−3; KDP is the specific differential phase shift between H and vertical (V) polarization, in deg km−1; and R is the rainfall rate in mm h−1]. Each of these relationships is parameterized by N*0 referred to, in the following, as the normalized intercept parameter of the DSD. Here N*0 is defined as
i1520-0426-18-11-1819-e1
where LWC is the liquid water content, ρw is the density of liquid water, and Dm is the “mean volume diameter” (ratio of the fourth to the third moment of the raindrop size distribution). Whatever the shape of the DSD, N*0 represents the intercept parameter N0 of the exponential distribution having the same LWC and Dm as the actual one (Testud et al. 2001).
It was shown in Testud et al. (2000) that, after normalization by N*0, quasi-universal relationships are established between any couple of integral parameters of the DSD (as A/N*0 and Ze/N*0, for example). Moreover, the relationships between A/N*0 and Ze/N*0, KDP/N*0 and A/N*0, R/N*0 and A/N*0, respectively, as derived from T-matrix calculations, may be satisfactorily approximated by power laws, which leads to the following parametric representation of the “inverse model”:
i1520-0426-18-11-1819-e2
where a, b, α, β, p, and q coefficients are functions of the temperature (T) only. These coefficients are calculated in the conditions hereafter specified.

A composite oblateness law for raindrop is considered: Keenan et al.'s (1997) law for equivalent drop diameter D between 0 and 1.3 mm, Andsager et al.'s (1999) law for D between 1.3 and 4.2 mm, and Bringi et al.'s (1982) law for D > 4.2 mm. Andsager et al.'s law is probably the best documented for intermediate drop diameter range, but it is not valid for D > 4 mm and it prescribes prolate shape for D < 0.5 mm. Bringi et al.'s (1982) law, which predicts an oblateness of raindrop very close to the equilibrium shape, seems the best consensus for drop diameters greater than 4.2 mm, a range where no measurement is available. However, in natural conditions where the large drops may oscillate, the mean shape may possibly deviate from equilibrium, as shown by Gorgucci et al. (2000), and this is one of the uncertainties of the inverse model.

The terminal fall velocity law is Lhermitte's (1988), which is the best model presently available.

The various coefficients of relationships (2), (3), and (4) are tabulated between −4° and 25°C in Table 1. With this inverse model, the impact of the DSD variability on relationships (2)–(4) is described by the “equivalent” intercept parameter N*0 only. A verification of the pertinence of this model from a wide dataset of raindrop spectra can be found in Testud et al. (2001).

b. Algorithm ZPHI

Algorithm ZPHI operates as a ray-by-ray analysis. The primary radar observables considered in ZPHI are the apparent (attenuated) radar reflectivity Za, and differential phase shift ΦDP. Here Za is related to the equivalent radar reflectivity Ze through
i1520-0426-18-11-1819-e5
where r is the radial distance from the radar.
In ZPHI, a segmentation along the beam is considered defined by the n + 1 bounds {r0, r1, … , rn−1, rn} (where r0 = 0 and rn = rmax, maximum range of the radar). Along each segment [ri−1, ri], it is assumed that N*0 is constant, which implies, according to (2), that A and Ze are related through a power-law relationship. Thus, ZPHI is based upon a piecewise inversion of (5) based on Hitschfeld and Bordan's (1954) formulation. More specifically, within each segment [ri−1, ri], the solution of (5) is written with respect to A in the following manner (Testud et al. 2000):
i1520-0426-18-11-1819-e6
where
i1520-0426-18-11-1819-e7
This expresses the profile of specific attenuation A(r) along segment [ri−1, ri], as a function of the profile of the observed Za, and of the value of A at the far bound ri.
Meanwhile, the normalized intercept parameter N*(i)0 relevant to segment [ri−1, ri] may be derived as (see Testud et al. 2000)
i1520-0426-18-11-1819-e8
With the above formulation, all the inversion problem is conditioned by the determination of the A(ri)s, since subsequently A(r), Ze(r), and the N*(i)0s may be derived using (6), (5), and (8). The determination of the A(ri)s makes use of the measured differential phase shifts ΔΦ(i)s defined as
(i)DPriDPri−1
Assuming that the inversion has been processed to range ri−1, the integration of (3) between ri−1 and ri provides the following integral constraint:
i1520-0426-18-11-1819-e11
Equation (11) defines an implicit equation for the unknown A(ri) since N*(i)0 and A(s) are function of A(ri) through (8) and (6), respectively. Note that when β = 1, the dependence on N*0 vanishes in (11), and the integral on the right-hand side becomes merely the path attenuation between ri−1 and ri. In this particular case, the solution A(ri) expresses simply as
i1520-0426-18-11-1819-e12
In the general case where β ≠ 1, an iterative numerical procedure is used to solve (11) for A(ri). However, since β remains very close to 1 whatever the temperature between −4° and +25°C (see Table 1), the A(ri) given by (12) always constitutes a quite good “first guess” that allows one, when introduced as initial condition in the iterative procedure, to determine A(ri) in two or three iterations with a relative accuracy better than 10−5.

It is worth noting that when β = 1, A(ri) given by (12) is not subject to the radar calibration error C [if Za is multiplied by C, A(ri) is unchanged]; nor is A(r) given by (6). In the same condition N*0 given by (8) is subject to the radar calibration (if Za is multiplied by C, N*0 is multiplied by Cb/(1−b)). When β ≠ 1, the estimate of A(ri) resulting from the numerical solution of (11) depends on the radar calibration error, because of the presence of N*0 in this equation. However, this dependence is quite weak since (1 − β), the exponent of N*0 in (11), is at most 0.05 for temperatures ranging from −4° to 25°C.

The final step of algorithm ZPHI is estimating the rain rate R using (4). Thus the profile along the beam R(r) is derived through an RA relationship scaled by N*0.

c. Sensitivity of the rain-rate retrieval to measurement noise

For the purpose of an error analysis, it is sufficient to use the first-guess solution given by (12). From the A(r) derived from (12), Testud et al. (2000) established that
i1520-0426-18-11-1819-e13
Equation (13) shows that the relative error in A(r) or in R(r) is composed of an error random in space and time or “speckle” in Za (which is not correlated between any two range gates of the segment), and an error due to the uncertainty in ΔΦ (which is the same for all range gates of the segment). For Ni = 10 independent samples and with bq ≅ 0.8, the relative error due to the speckle in Za is ±25% for δA/A and ±20% for δR/R. The relative error due to ΔΦ decreases as 1/ΔΦ. It is ɛ2/ΔΦ for δA/A and (1 − qb)(1 − b)−1 ɛ2/ΔΦ for δR/R, where ɛ is the standard error in ΦDP. A typical value of ɛ for Ni = 10 is 2°. It is reasonable to require that the error in δR/R due to ΔΦ be less than or equal to 25%. This leads to ΔΦ ≥ 20°. In order to relax this constraint, ΔΦ is evaluated by averaging ΦDP over 11 adjacent gates about each bound ri [note that this averaging does not degrade the range resolution of A(r) or R(r), since ΔΦ only acts as a global constraint at each segment]. After this 11 range gate averaging, the constraint becomes
When (16) is satisfied, the error in N*0 is δ[log10(N*0)] ≤ 0.30. Thus the expected statistical error in N*0 is ±3 dB for ΔΦ = 6°, or ±0.6 dB (±15%) for ΔΦ = 30°. Meanwhile, the expected total statistical error in R is ±32% when ΔΦ = 6°, and ±21% when ΔΦ = 30°.

When (16) is not fulfilled, it is considered that the full version of the algorithm collapses. Nevertheless, to get a retrieval, N*0 is forced to the Marshall–Palmer (1948) value of 0.8 107 m−4. In this case, A(ri) is derived from (8).

d. Partition scheme

A basic assumption in ZPHI is that N*0 is constant along each segment [ri−1, ri] of the partition. In reality, N*0 is a highly variable parameter, subject to jumps (Waldvogel 1974) reflecting changes in the microphysical processes such as at the transition between “stratiform” and “convective” rain. From an extensive analysis of airborne microphysical data observed during the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE), described by Webster and Lukas (1992), Testud et al. (2001) confirmed the distinct behavior of N*0 according to rain type, stratiform or convective. They showed that in the climatic zone of the west equatorial Pacific, N*0 is, on average, 3 × 106 m−4 in stratiform rain, and 2 × 107 m−4 in convective rain. Large fluctuations in N*0 were observed for both rain categories. However, this variability is essentially from event to event. Within one particular event after separating convective and stratiform rain, the variability of N*0 is much more moderate.

Given the above, the partition scheme separately treats stratiform and convective rain. In convective rain, when several rain cells are encountered along the beam, they are treated separately. That way, it is expected that the N*0 variation along each segment will be minimized.

The partition scheme used herein is based upon the following approach.

  1. At each ray, a first run is performed over the full profile of the available {Za(r), ΦDP(r)}, which allows retrieval of a first-guess R1(r).
  2. A classification of the precipitation type along the beam is then undertaken using Testud et al.'s (2001) simple criterion. Namely, if R1(r) < Rth at the range r0 and at adjacent ranges from r0 − Δr to r0 + Δr, rain at range r0 is classified as stratiform; otherwise it is convective. Here Rth is an arbitrary threshold and Δr represents the “influence radius” of a convective cell. In this paper Rth is taken to be 5 mm h−1 and Δr to 3 km.
  3. Any convective segment defined by 2) is eventually resegmented if it includes more than one convective cell. The bounds of this new segmentation are chosen where R1(r) is minimum, but they do require that a minimum signal to noise ratio (>0 dB) be achieved in order to get a sufficient accuracy in the ΦDP estimate. This new segmentation is retained only if it leads to ΔΦ(i)s larger than 6°.
  4. Within each of the segments thus defined, a second run of the algorithm is then undertaken according to the procedure outlined in section 2b. The inverse model [defined by (2), (3), and (4)] is tuned according to the temperature of the midpoint of the segment (estimated from its altitude).

It is important to note that the algorithm is quite robust with respect to the segmentation scheme. Its output is not sensitive to the values of the arbitrarily chosen parameters Rth and Δr.

e. Correcting ZDR for path attenuation: An alternate R estimate

The differential reflectivity ZDR is an interesting parameter that may be used to obtain an improved rain-rate estimate and, together with the other polarimetric parameters, to produce classification of hydrometeors (Vivekanandan et al. 1999). However, at C band, the along-path differential attenuation tends to induce a negative bias in ZDR (because the horizontally polarized wave is more attenuated than the vertically polarized one). This negative bias may be quite severe after crossing a convective rain cell. Thus before any further use of ZDR to improve a rain estimate or within a microphysical classification scheme, it is essential to correct it for differential attenuation. The “true” ZDR may be expressed as a function of the “apparent” ZDRa, observed by the radar, as
i1520-0426-18-11-1819-e17
where ADP is the specific differential attenuation between H and V in decibels per kilometer.
The inverse model (see Testud et al. 2000) also specifies the relation between ADP and A (two integral parameters of the DSD) as
ADPN*0cAN*0d
Coefficients c and d are tabulated as a function of the temperature in Table 1. It should be noted that d is about 1.3 (thus the ADPA relationship does depend on N*0). After application of ZPHI, ADP may be estimated at any range from the retrieved A and N*0, which helps one to calculate the integral in (17).
The same inverse model shows that the relation between two intensive variables is independent of N*0 (and quasi-independent of the shape of the DSD). This is illustrated by the relation between ZDR and δ (backscattering phase shift) shown in Testud et al. 2000. For the purpose of establishing an alternate estimate of the rain rate, Fig. 1 investigates the relation between R/A and ZDR for different values of the shape parameter μ of the DSD. The dispersion between the various curves is quite moderate and the relationship appears linear (in log scales) from ZDR = 0.5 to 5 dBZ. This leads to a model like
RAeZDRf
which constitutes the alternate rain-rate estimate used in this paper.

Coefficients e and f are tabulated as a function of the temperature in Table 1 (assuming μ = 2).

3. Illustration of ZPHI application to real data

ZPHI has been applied to one month of data (January 1998) collected by the C-POL polarimetric radar operating near Darwin in Australia. Specific characteristics of this C-band radar are summarized in Table 2 (for more details, see Keenan et al. 1998). Most results presented hereafter were obtained after correcting the measured reflectivity by −1 dBZ, as a result of the recalibration procedures presented in section 5 and 6.

a. Illustration of ZPHI application to individual rays

As discussed in section 1, ZPHI operates on a ray-by-ray basis. Figure 2 illustrates its application to a particular ray at 1.6° elevation, where convective rain is sampled. At this elevation, the radar beam reaches 4.2-km altitude (center of the beam) or 5.2-km altitude (upper edge of the beam) at 120-km range, while the freezing level in January at Darwin is near 5-km altitude. Thus, except for the possible presence of hail in some convective cells, it is expected that rain is being sampled to 120-km range. From the profile along the radar beam of Za (thin line in Fig. 2a) and from the ΔΦs associated with the segmentation of the ΦDP profile (star symbols in Fig. 2c), the profiles of A and N*0 at each segment are derived. Further, Ze, resulting from the attenuation correction (heavy line in Fig. 2a) illustrates how significant along-path attenuation can be: at 118-km range, Za is underestimated by 15 dBZ with respect to Ze.

In Fig. 2, the automated partitioning distinguishes three segments: (22 km, 44 km), (44 km, 82 km), and (82 km, 118 km). For the first, which is classified as being stratiform precipitation, ΔΦ is below the threshold of 6°, hence N*0 is fixed at the reference value of 0.8 × 107 m−4. The other two (second and third) segments are classified as convective, and N*0 is retrieved to be 1.58 × 107 m−4 and 107 m−4, respectively.

The rainfall-rate profile R(r) derived from the combination of A(r) and N*0 through (4) is shown in Fig. 2d, together with a “classical estimate” (derived as a result of the RZe relationship with fixed N*0, defined in Table 1, applied to the “apparent” reflectivity Za without consideration of attenuation). At far range (third segment) it is seen that the rain-rate estimate by ZPHI is much higher than the classical estimate, and this is mainly due to the attenuation correction. But the ZPHI estimate is also much larger than the classical one for the rain cells of the second segment, which have undergone moderate integrated attenuation. This result is due to the tuning of the RA relation by the retrieved N*0.

Figure 2b illustrates the ability of ZPHI to correct the observed ZDR for attenuation. In this particular instance, the differential correction reaches 2 dB at far range. Such a correction is necessary for practical exploitation of ZDR in rain-rate estimation or the classification of hydrometeors at C band. It is interesting to note that the corrected ZDR profile is compatible with the presence of rain all along the profile (ZDR ranging between +1 and +4 dB).

Fig. 3 shows that ZPHI operates properly in stratiform rain as well. Here the automated partitioning leads to two segments of 40-km length. In this case the integrated attenuation is much weaker (nevertheless reaching 3 dBZ at the far bound). But even with the correction for attenuation, the ZPHI retrieval leads to rainfall rates smaller than the classical estimate. This again is an effect of N*0 tuning by ZPHI. For this particular ray, the retrieved N*0 for both segments is 0.25 × 107 m−4, much smaller than the Marshall–Palmer value prescribed in the classical estimate.

It should be emphasized that in the two rays illustrated in Figs. 2 and 3 (as in all rays processed by ZPHI in this paper), the dwell time (needed to collect 56 samples on both H and V polarization at 1000 Hz) is 112 ms.

b. Quality control

The inverse model, through (3), specifies the profile KDP(r) associated with A(r) retrieved by ZPHI. Hence a “theoretical” profile of ΦDP [denoted ΦthDP(r)] may be derived along any integration path [ri, ri+1] as
i1520-0426-18-11-1819-e20
It is important to recall that by construction, according to (11), ΦthDP(ri+1) = ΦDP(ri+1) in (20). Thus ΦthDP(r) constitutes the optimal interpolation of ΦDP between ri and ri+1, consistent with the profile of Za(r).

In Figs. 2c and 3c, ΦthDP(r) (heavy line) is shown superimposed on ΦDP(r) (thin line). This comparison enables one to objectively assess the A(r) retrieval. Differences between theoretical and “measured” ΦDP have five main sources: (i) statistical uncertainty (including thermal noise) in ΦDP; (ii) bias in the ΦDP measurement, due to antenna sidelobes; (iii) inadequacy of the inverse model due to the presence of hydrometeors other than rain; (iv) backscattering effect (δ) in the phase shift, affecting ΦDP but not included in the model to derive ΦthDP; and (v) inappropriate choice for the bounds of the segmentation.

Thermal noise in ΦDP increases rapidly when the signal-to-noise ratio (SNR) gets below 0 dB, thus the quality control should be restricted to data points with SNR > 0 dB. Biases due to antenna sidelobes are expected in regions of sharp reflectivity gradients (as between 67- and 72-km range in Fig. 2). The contribution of δ to ΦDP can be evaluated a posteriori thanks to the quasi-functional relationship between δ and ZDR (see Testud et al. 2000): δ is negligible for ZDR < 1 dB; δ ≅ 1°, 4.5°, and 9.5° for ZDR = 2 dB, 3 dB, and 4 dB, respectively. An inappropriate choice of the segmentation bounds ris induces inadequate estimates of the ΔΦis, which impacts the full profile of ΦthDP (and, of course, that of A and R).

The segmentation is inappropriate when one (or several) of the ris is set where the radar reflectivity Ze is high (with a risk of significant backscattering phase shift), where Ze is too low (predominance of thermal noise in ΦDP), or where Ze shows important gradients (risk of bias due to antenna side lobes). Our automated partition scheme is built to avoid such situations. Moreover, the fact that δ is negligible at any bound may be checked a posteriori by inspecting the retrieved ZDR. If a bound is selected (as, for instance, the far bound) where δ is significant, it is always possible to solve the inversion problem in the iterative way described in Testud et al. (2000), that is, perform a first run of ZPHI ignoring δ, then calculate δ from ZDR, then correct ΔΦ for δ and use it in a next run of ZPHI, and so on. In practice, in this paper, the need for such a procedure never occurred.

A “quality index,” founded upon the calculation of the rms of (ΦthDP − ΦDP) between ri and ri+1 is established. If this rms value is less than 8°, the quality index is set to 1 for all points of the retrieved R profile. Otherwise it is set to zero.

An “algorithm index” also labels the retrieved R profile along each segment. It is set to 1 when R is derived from the full version of ZPHI (e.g., when ΔΦ > 6°), and to 0 otherwise (where the version of ZPHI with N*0 forced to the Marshall–Palmer value is practiced).

c. Comparison with a conventional RKDP approach

The primary parameter retrieved by ZPHI is the specific attenuation A. Since A and KDP are almost proportional, it could be argued that ZPHI is a new approach to retrieve KDP. This new approach avoids ΦDP differentiation, and by integrating the measured reflectivity profile in the process, it preserves the original range resolution of the radar and leads to a reduced statistical uncertainty. But the additional strength of ZPHI lays in the determination of N*0, which allows automatic tuning of the RKDP relation for the intrinsic DSD variability.

Figure 4 illustrates a full 360° conical scan (PPI) at 1.6° elevation, for which the product delivered by ZPHI is compared with that derived by the more conventional approach of May et al. (1999), consisting of deriving KDP from ΦDP and subsequently, estimating R from an RKDP relationship [referred to in the following as the R(KDP) estimate]. May et al.'s (1999) method for estimating KDP is based on a differentiation of ΦDP after application of a nine-point filter, followed by a six-point consensus estimate procedure aiming to eliminate spurious effects introduced by occurrence of significant δ.

In Fig. 4, a stratiform rain structure is evident in the northwest quadrant, whereas a cluster of intense convective cells are observed in the southwest quadrant. Quite reasonable consistency is observed between the R retrievals of ZPHI (Fig. 4a) and the R(KDP) estimate (Fig. 4c) in the convective cell cluster. In particular, heavy rain cells, which could be hidden by the attenuation effect, are evident away from the radar in both ZPHI and R(KDP) retrievals. In contrast, in the stratiform area of the northwest quadrant, the difference between the two retrievals appears quite clear: while the R(KDP) retrieval is very noisy, ZPHI produces a rain-rate field with a statistical error similar to that with the conventional R(Z) estimate. Such a large statistical uncertainty with the R(KDP) algorithm at low rain rate is not surprising. It represents the classical limitation of this type of algorithm.

The N*0 product produced by ZPHI is illustrated in Fig. 4b. The gray areas are where N*0 cannot be derived with sufficient accuracy (because ΔΦ < 6°) and ZPHI is run with N*0 set to a fixed value (0.8 × 107 m−4). In its full version including N*0 retrieval, ZPHI operates over about 42% of the rainy area. But in terms of “area-integrated rainfall,” it captures a higher proportion of the rainfall (63%) because it treats preferentially the most intense rain areas. Where N*0 is retrieved, Fig. 4b shows distinct N*0 jumps between convective and stratiform (low values in stratiform, high values in convective) in agreement with the observations of Testud et al.'s (2001) obtained from the TOGA COARE airborne microphysical data.

The ray pattern in the rain field in the northwest quadrant of Fig. 4a (stratiform rain) is due to the statistical error in the retrieved N*0. According to (14), the relative error in N*0 is proportional to the relative error in ΔΦ. Thus it may be quite significant in stratiform rain where ΔΦ is small. The modeling error in R associated with an error in N*0 is obviously correlated along each individual segment, which produces the ray pattern in question. The only way to improve the picture is to reduce the error in ΔΦ. This could be achieved by filtering ΦDP across beam (in addition to the along-beam filtering already practiced).

4. Calibration check from N*0 statistics

a. N*0 statistics at Darwin

The precipitation regime at Darwin in January is usually tropical oceanic. The similarity of this precipitation with that observed during TOGA COARE is attested to by the strong resemblance between the N*0 histogram observed by Dou et al. (1999) and that derived from the ground-based disdrometer from near Darwin (the Kowandi South BMRC station) during five rainy days of January 1998. Both histograms are bimodal, each mode corresponding to stratiform and convective rain, respectively (Fig. 5). The following describes how the stability of the N*0 histogram may be utilized to fine-tune the radar calibration.

b. A calibration procedure founded upon N*0

The sensitivity of the N*0 retrieval to the radar calibration error was already mentioned. A simple analysis of (8) shows that if C is the radar calibration error (in dBZ), then N*0 is biased by a factor 10−[b/(1−b)]C/10. Thus, if one considers a histogram of log10(N*0), derived from an extensive application of ZPHI on a large dataset, for a calibration error C, that histogram is expected to be shifted by −[b/(1 − b)]C/10 with respect to the actual one. For C = 1 dBZ and b = 0.8, the shift is 0.4. If one admits that the position of the peaks in the log10(N*0) histogram is stable to ±0.1 for a sufficiently large dataset, it is conceivable to adjust C to obtain Ze within ±0.25 dBZ accuracy.

However, the practical application of this method requires some caution. As specified in section 2b, the N*0 retrieval by ZPHI is only applied for ray segments whose ΔΦ is greater than 6°, which roughly represents a rain rate greater than 6.4 mm h−1 over 20 km or greater than 2.7 mm h−1 over 60 km. With such thresholds, it is expected that the statistics of the N*0 histogram provided by ZPHI are biased for stratiform rain. Thus for meaningful comparison between N*0 histograms derived from ZPHI and those observed from airborne microphysical probes or ground-based disdrometers, it is appropriate to select high rainfall rates only. A threshold R > 10 mm h−1 is presently considered, which practically selects only convective rain.

Figure 6 compares such histograms (with the threshold R > 10 mm h−1) from three different data sources:

  1. TOGA COARE airborne microphysical data (all flights);
  2. Kowandi South disdrometer measurements, for five rainy days of January 1998; and
  3. ZPHI outputs restricted to data falling within a 10-km radius centered at the disdrometer site (indicated in Fig. 8), for the same five days of January 1998.
The convective histograms from TOGA COARE (dashed line) and Kowandi South disdrometer (full line) are closely coincident, confirming the stability of the N*0 statistics in a given precipitation regime. The general shape of the histogram derived from ZPHI (light gray) is consistent with the other two. However, to make the peak of the ZPHI histogram coincident with that of the ground-based disdrometer, an adjustment of the radar calibration by −1 dBZ is necessary. This leads one to consider that there is a residual calibration error C in the Darwin radar data of +1 dBZ. Otherwise the histogram provided by ZPHI (backward histogram in dark gray) is shifted leftward by −0.45 (in log scale). This technique of tuning the radar calibration by adjusting the peak of the N*0 histogram derived by ZPHI to a reference (from ground truth or climatological) seems very accurate (probably to a few tenths of dB), while the limitation of any calibration procedure by hardware technique is about ±1 dB.

5. Internal calibration check with ZPHI

The possibility of internally calibrating a dual polarization radar was previously investigated by Gorgucci et al. (1992) and Goddard et al. (1994). Gorgucci et al.'s approach is founded upon the examination of the consistency between two polarimetric rain-rate estimates (one using ZH and ZDR, the other KDP), while Goddard et al.'s technique determines the radar calibration by checking directly the consistency between the three polarimetric variables ZH, ZDR, and ΦDP. These techniques are certainly efficient at S-band where the path-integrated attenuation is negligible, but they cannot be used at C band without adaptation because of the attenuation affecting both ZH and ZDR.

The technique proposed in this section is connected to the previous approaches cited, but it takes advantage of the ability of ZPHI to correct ZH and ZDR for path attenuation. It is based on the principle that the two R estimates defined by (4) and (19) have different sensitivities to the radar calibration. Equation (4) provides the standard ZPHI estimate and is referred to in the following as RZPHI. This RZPHI combines A (parameter independent of the radar calibration error C) and N*0 (dependent on C as 10b/(1−bC/10). The alternate estimate given by (19) combines A and ZDR (referred to in the following as RA-ZDR). If there were no attenuation, ZDR would be independent of the radar calibration. But because of the differential attenuation, ZDR depends on the calibration in a complex way due to the fact that ADP in (18) depends on N*0 as ≈[N*0]−0.3 (see Table 1). Hence the path-integrated differential attenuation (in dB km−1) that corrects ZDR varies with the calibration error C as ≈100.3b/(l−bC/10. This represents quite a nonlinear dependence. If a short range (r < 60 km) is employed, the integrated differential attenuation is normally moderate, thus the sensitivity of RA-ZDR to radar calibration would be small. Then the dominant feature in the comparison between RZPHI and RA-ZDR is the RZPHI sensitivity to C. Figure 7 displays three scatterplots of RZPHI against RA-ZDR obtained for r < 60 km, ZDR > 1 dB, and with three corrections to the radar calibration of −1 dB, 0 dB, +1 dB. In the three cases the scatterplot is well represented by a linear relationship (the linear correlation coefficient is higher than 0.985 in the three cases), but the best comparison is clearly obtained for the (−1 dB) correction, with a coefficient of proportionality between RA-ZDR and RZPHI of 1.06 (while it is 1.17 for 0 dB, and 1.34 for +1 dB) and a correlation coefficient of 0.993 (instead of 0.989 for 0 dB and 0.986 for +1 dB). This result brings independent confirmation that the residual calibration error of the Darwin radar data is +1 dB. Figure 7 also shows that the accuracy of this internal calibration technique is far better than ±1 dB (probably of the order of ±0.2 dB).

6. Validation of ZPHI from rain gauge data

This section describes and applies a procedure to compare rainfall rates retrieved by ZPHI and those observed by the tipping-bucket rain gauges within the so-called C-scale network at Darwin. This network consists of 25 rain gauges spread within a 150-km radius (see Fig. 8). The current radar operation of the C-POL consists of volume scanning, 360° in azimuth, and from 0.5° to 42.2° elevation; the complete cycle requires 12 min. For this comparison exercise, we have considered the radar data at the second tilt angle only (elevation of 1.6°). It could be argued that the lowest tilt angle is more appropriate for comparison with rain gauges. Nevertheless, at 0.5°, the N*0 retrieval may not be reliable, because of beam blockage effects (which do not affect the A retrieval but do affect the Ze retrieval). As already mentioned in section 3, in January at Darwin, with a 1.6° elevation, the radar beam stays below the 0°C isotherm as long as the range is restricted to less than 120 km. With this restriction, 19 of the 25 rain gauges of the C-scale network are usable.

For the comparison, it is necessary to take into account the fundamentally different sampling characteristics of the weather radar and of the rain gauge network.

  1. Space and time resolution: With a rain gauge, the collection area is very small, but the integration time is long (typically 1 min); with the radar, each measurement is quasi instantaneous (each sample is collected in 0.12 s), but the sampling volume is large (typically 0.1 km3).
  2. Sampling strategy: From the rain gauge network, the sampling is continuous in time, but discontinuous in space (typical spacing with the C-scale network is 20 km; see Fig. 8); from the radar the sampling is continuous in space but discontinuous in time (revisit time is 12 min).
  3. Representation: The radar measures precipitation at altitude and the rain gauge on the ground; during its fall, the precipitation may drift horizontally with the wind; it may also partially evaporate in dry ambient air or enhance by collecting water from low-level clouds. This general problem of representativeness in the comparison between rain gauge and radar data was treated in depth by Kitchen and Blackhall (1992) and Anagnostou et al. (1999).

a. Comparison protocol

The comparison protocol considered presently is derived from that used by Willis (1999, personal communication) with the Florida Nexrad weather radar and neighboring rain gauges. It includes both spatial and temporal filtering.

At each rain gauge site, the radar-derived rainfall rate is averaged horizontally within a circle of 2-km radius. The distance of 2 km approximates the scale of the horizontal drift associated with precipitation falling from 1 km altitude under a wind of 10 m s−1.

Temporal filtering is also employed to compensate for the difference between the radar revisit time (12 min), and the 1-min sampling rate of the rain gauge. Both measurements are smoothed over a window of 30 min according to the bell-shaped weighting function W(t) illustrated in Fig. 9. This weighting function is an empirical approach to optimize the time averaging, taking account of the temporal correlation of the rain rate.

More specifically, the “filtered” rain gauge estimate G(i) by the ith rain gauge is calculated each minute as
i1520-0426-18-11-1819-e21
where Gi is the 1-min estimate, Δtg = 1 min, Ng = 15.
The filtered radar estimate is calculated every 12 min, at a time t centered at the averaged time of each 1.6° PPI sweep as
i1520-0426-18-11-1819-e22
where 〈R(i)〉 is the spatial averaging of the radar estimate (within 2-km radius) at the ith rain gauge site, and Δtr = 12 min, Nr = 1.

Three radar estimates are derived by the filtering defined by (22):

  1. the ZPHI retrieval (corresponding filtered value is referred to as R(i)ZPHI);
  2. the result of the R(A) relationship of Table 1, but at fixed Marshall–Palmer N*0 (corresponding filtered value is referred to as R(i)ATT.); and
  3. the classical estimate (e.g., standard ZRe relationship applied to Za ignoring the along-path attenuation) (corresponding filtered value is referred to as R(i)CLASS.).
As mentioned previously, the primary parameter retrieved by ZPHI is the specific attenuation A. Since A and KDP are almost proportional, R(i)ATT. may also be considered as the result of an R(KDP) estimator with fixed N*0. The comparison of R(i)ATT. and R(i)ZPHI allows one to evaluate the improvement brought by adjusting N*0 in the RA relationship.

It should be noted that with such a protocol, there is no mutual adjustment of the radar to the rain gauges. Both types of data are processed independently.

b. Comparison of time series

In this first series of comparison, we consider a fictitious watershed consisting of the assembly of the 19 circular catchments of 2-km radius centered at the rain gauge sites. The average rainfall rate in this watershed is denoted G19(t) [average of the Gi(t)s] when estimated from the rain gauge network, and R19(t) [average of the Ri(t)s] when estimated from the radar. Figures 10–12 illustrate the comparison of G19 and R19 time series during three days of January 1998 (2, 20, and 28 January). In each figure, two R19 retrievals are shown: ZPHI (filled circle) and the classical estimate R(Za) (filled diamond). In the bottom panel, the heavy-line time series labeled “algorithm index” is the 19-site average of the algorithm index (AI, defined in section 3b); it represents, as a function of time, the fraction of points where application of the full version of ZPHI (with retrieval of N*0) was met in the averaging process. Similarly, the thin-line time series labeled “quality index” is the 19-site average of the quality index QI (defined in section 3b). For instance, in Fig. 10 at 1000 UTC, an algorithm index of 0.25 means that 25% of the points used to obtain the average ZPHI estimate were processed with the full version of ZPHI. The corresponding quality index of, say, 0.9 means that, among the 25% points for which the full version of ZPHI applies, 90% have a quality index of 1.

A quick look through Figs. 10–12 shows that the ZPHI estimate follows remarkably well the estimate from the rain gauge network, while the classical estimate R(Za) is unable to reproduce the peaks in G19, especially when their intensity exceeds 2 mm h−1. This systematic underestimation of the classical estimate points out the importance of the ZPHI correction, and how precise its retrieval is, especially in case of heavy rain. The algorithm index indicates that the full version of ZPHI operates 40%–50% of the time on average, and more frequently in heavy rain rather than light rain (as expected). Most of the quality index is close to one. Local discrepancies between the R19 estimate by ZPHI and G19, as observed in Fig. 10 around 1240 or 1540 UTC, are probably due to the representation error (as, e.g., that due to evaporation, making the rainfall rate measured at ground weaker than at altitude). It is also remarkable that the daily integrated rainfall obtained from ZPHI for the three days in question (31.2, 35.5, and 32.1 mm, respectively) is quite consistent with that calculated from rain gauge measurements (29, 35.9, and 34.3 mm), while the classical estimate R(Za) leads to a severe underestimation (20.3, 22.7, and 18.5 mm).

c. Point-by-point comparison

Figure 13a is a scatterplot representing a point-by-point comparison between R(i)ZPHI and G(i), covering seven rainy days during January 1998 for the various available times and rain gauge sites. Figures 13b and 13c display a similar scatterplot but between R(i)ATT. and G(i), and between R(i)CLASS. and G(i), respectively. In these scatterplots, in order to isolate the improvement brought by algorithm ZPHI in its full version (i.e., with N*0 retrieval), only data points where both ZPHI algorithm and quality indexes are 1 are retained. This corresponds to 41% of the total number of data points or 55% of the total area-integrated rainfall. Also, the comparison is restricted to data points within 60-km range in order to reduce the representation error between rain gauge and radar estimate.

Figure 13c confirms that the classical radar rain-rate estimate is severely biased with respect to the rain gauge: the slope of the linear fit is 0.37 (which means that, on average, the classical estimate R(i)CLASS. is only 37% of the gauge estimate G(i)). The use of the R(A) estimate with fixed Marshall–Palmer N*0 sensibly reduces the bias. The slope increases to 0.58, which is a serious improvement with respect to the classical estimate. With the ZPHI estimate [where the R(A) relationship is tuned for the retrieved N*0] the slope jumps to 0.84. This demonstrates that not only the correction for attenuation, but also the N*0 adjustment, is critical to retrieve good rainfall rate estimates at C band.

Another important statistical feature is the regular improvement of the “linear correlation coefficient” from Fig. 13c to Fig. 13a: ρ = 0.871 with the classical estimate; ρ = 0.904 with the R(A) estimate with fixed N*0; ρ = 0.918 with ZPHI. This increase in ρ with R(i)ATT. and R(i)ZPHI should be interpreted as a reduction in the relative standard deviation (proportional to 1/ρ2 − 1) with respect to the classical estimate (R(i)CLASS.), of 20% and 30%, respectively.

When comparing rainfall rate estimates from rain gauge and radar, the bias and random errors are due to five effects:

  1. the statistical error of the radar estimate;
  2. the along-path attenuation (for R(i)CLASS. only);
  3. the N*0 variability (for R(i)CLASS. and R(i)ATT. estimates);
  4. the representation error between rain gauge and radar; and
  5. the rain gauge error.
The last two effects are the same for the three estimates. The statistical error of the radar estimate was theoretically analyzed in section 2c. It was shown that R(i)CLASS. is subject only to the speckle noise in Za, while R(i)ATT. and R(i)ZPHI are in addition subject to the statistical error in ΦDP. The along-path attenuation acts as a bias (because it systematically leads to rain-rate underestimation) and as a noise source (because of its random character). The N*0 variability acts to introduce “noise,” but it also introduces a bias because the intense rainfall rates have N*0 values systematically higher than Marshall–Palmer value. The improvement of the correlation coefficient ρ mentioned above demonstrates that in spite of the increase in the statistical error due to their ΦDP dependence, estimates R(i)ATT. and R(i)ZPHI bring finally a reduction of the total statistical error because they are corrected for along-path attenuation and for N*0 variability (for R(i)ZPHI).

According to Kitchen and Blackall (1992) and Anagnostou et al. (1999), the representativeness error of a rainfall estimate G by an individual rain gauge, with respect to the average rainfall A in a pixel of 3 × 3 km2 or 4 × 4 km2 (centered on the rain gauge), may be characterized by the variance of log(A/G). For hourly estimates, Kitchen and Blackall and Anagnostou et al. found that σ2[log(A/G)] ≅ 0.14. The same characterization of the error may be considered in this paper using 30-min estimates R(i) and G(i). It is found that, for rain gauges at range smaller than 40 km from the radar:
i1520-0426-18-11-1819-eq1
Thus ZPHI allows us to approach the variance of 0.14 associated with the representativeness error of the individual rain gauge, which should be considered as a “theoretical limit” that cannot be passed in such comparison exercise.

It is interesting to further comment on the difference between R(i)ATT. and R(i)ZPHI. As mentioned previously, R(i)ATT. is derived from the specific attenuation A and is representative of what would be obtained from an RKDP relationship since KDP and A are nearly proportional. One could propose to adjust the coefficient of the RA relation in order to obtain the same slope for the R(i)ATT.G(i) scatterplot as in the R(i)ZPHIG(i) approach. Such an adjustment would be equivalent to increasing N*0 by a factor of 5 (with respect to Marshall and Palmer), which is statistically valid for high rain rates (corresponding to convective rain), but not for low rainfall rates observed in stratiform rain. Moreover, the only way to proceed with such an adjustment would be by comparison with a rain gauge network, while ZPHI operates the N*0 adjustment ray by ray and segment by segment, without any reference to rain gauge.

7. Summary and conclusions

This paper presented an extensive application of algorithm ZPHI to the polarimetric data from the C-POL radar of BMRC, operating near Darwin, Australia. This should be considered as first results from real data, which need to be confirmed by a wider dataset. The primary data processed by ZPHI are the apparent (attenuated) reflectivity and the differential phase shift. The different steps in the rainfall-rate retrieval were discussed, showing the various capabilities of the algorithm, which include the following.

  1. The segmentation of the analysis along each radar beam, compensating for the basic hypothesis of ZPHI, that is, N*0 constant along each segment.
  2. The retrieval of the A profile (A: specific attenuation) and of parameter N*0 characterizing the raindrop size distribution.
  3. The retrieval of profiles of the equivalent radar reflectivity Ze and differential reflectivity ZDR, both corrected for effects of along-path attenuation.
  4. The retrieval of the rainfall rate through an RA relationship tuned for local value of N*0.
The mathematical simplicity of the algorithm, its robustness with respect to measurement noise, and its ability to check internally the quality of the retrieval make it particularly suitable in a real-time operational application. The main shortcomings of ZPHI seem to be presently the following.
  1. ZPHI is based upon the hypothesis of constant N*0 along the path. If N*0 is not constant (which probably occurs along a path in stratiform rain), ZPHI adjusts an averaged N*0. Thus there may be a modeling error related to local deviation of N*0 with respect to the adjusted average.
  2. The inverse model of ZPHI depends on the oblateness law for raindrops. Some uncertainty remains about the averaged shape of large drops (with diameter greater than 4 mm), assumed close to equilibrium in this paper, but that may differ from equilibrium in natural conditions, due to drop oscillations.
  3. When hail is met along the beam, the inverse model (designed for rain) is not appropriate. The hail region has to be identified and treated selectively, according to a process that remains to be specified.
The question of the radar calibration should be addressed carefully to get high quality products with ZPHI. In this paper two calibration techniques were investigated: one is founded upon the climatological stability of the N*0 histogram; the second, purely internal to the radar, consists of a consistency test between the current rain rate estimated by ZPHI and an estimate obtained by combining A and ZDR. Both techniques are very simple, provide a calibration estimate within 0.1–0.2 dB, and diagnose a consistent calibration error of −1 dB for the C-POL data of January 1998. The C-scale rain gauge network neighboring the Darwin radar was used to proceed to a systematic validation of the rain rates produced by ZPHI. Spatial and temporal filtering were applied to both data types in order to compensate for the quite different sampling characteristics of the radar and the rain gauges. A fictitious catchment consisting of an assembly of nineteen 2-km-radius circles, centered on the rain gauge sites, was first considered. Comparing time series of the average rain rate on this fictitious catchment, remarkable agreement was found between the rain gauge and the ZPHI estimate, while the classical estimate (standard ZR relationship applied without consideration of the attenuation) appeared unable to represent properly the rainfall rate peaks. A point-by-point comparison was also performed including all available data from the month of January 1998. It confirmed that the classical estimate is severely biased with respect to the rain gauges (slope of linear correlation is 0.37). The estimate based upon A only (at fixed N*0) improves the comparison but the bias remains important (slope of linear correlation is 0.58). The ZPHI estimate (founded upon A and N*0) approaches best the rain gauge estimate with a slope of 0.84.

The A profile derived by ZPHI is also representative of a KDP profile, since KDP and A are nearly proportional. As compared to a conventional KDP algorithm, ZPHI presents two advantages:

  1. it allows derivation of the A profile with much less statistical error than that available with KDP profile derived from conventional ΦDP differentiation; and
  2. it improves the bias and standard deviation of the rain-rate estimate by N*0 tuning ray by ray, and segment by segment.

The Ze and ZDR retrievals are by-products of ZPHI that warrant particular interest. In combination with KDP (derived from A) and the correlation coefficient ρHV, they may be used in a classification of the hydrometeors as performed by Keenan (1999) with the Darwin radar data.

Acknowledgments

This study was conducted in the framework of a contract between Météo-France and the Centre National de la Recherche Scientifique (CNRS). We acknowledge TSDIS of the TRMM office for providing the Darwin radar data and rain gauge data.

REFERENCES

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Fig. 1.
Fig. 1.

R/A vs ZDR at C band for four gamma drop size distributions as a function of the shape parameter μ. The dotted line represents the inverse model deduced from the gamma DSD with μ = 2

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 2.
Fig. 2.

For a convective rain case, profiles along the radar beam of (a) apparent measured reflectivity Za (thin line) and corrected equivalent reflectivity Ze (heavy line); (b) measured differential reflectivity ZDRa (thin line) and corrected one ZDR (heavy line); (c) differential phase shift ΦDP, measured (thin line) and model adjusted (heavy line); the numbers are log10(N*0) and (*) symbols are boundaries of the ZPHI segmentation process; (d) retrieved rainfall rates from the classical estimate R(Za) (thin line) and from ZPHI (heavy line). Vertical lines in (a) and (c) result from jumps caused by the acquisition process and are treated as spurious points in ZPHI

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 3.
Fig. 3.

As in Fig. 2 but for a stratiform case

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 4.
Fig. 4.

Conical C-POL scan views (at 1.6° elevation, 19 Jan 1998 around 2240 UTC) of (a) the rainfall rate derived from ZPHI; (b) log10(N*0) retrieved by ZPHI (in gray when forced to the Marshall–Palmer value); (c) the rainfall rate R(KDP) derived from May et al.'s (1999) procedure

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 5.
Fig. 5.

Histograms of log10(N*0) derived from the Kowandi South BMRC disdrometer for five days of Jan 1998 (full line), and from 2D-P probes during 21 flights of TOGA COARE (dotted line)

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 6.
Fig. 6.

Histograms of log10(N*0) for rainfall rate greater than 10 mm h−1, derived (i) from ZPHI (dark and light gray filled), (ii) from the Kowandi South BMRC disdrometer for five days of Jan 1998 (full line), and (iii) from 2D-P probes during TOGA COARE (dotted line). The light gray filled histogram is from ZPHI after a calibration adjustment in Za of −1 dBZ, while the dark gray filled one is for ZPHI with the original Zas

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 7.
Fig. 7.

Scatterplots of RA,ZDR against RZPHI for r < 60 km and ZDR > 1 dB, after applying calibration corrections in Za of (a) −1 dB, (b) 0 dB, and (c) +1 dB

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 8.
Fig. 8.

C-scale rain gauge network spread around the C-POL radar, inside a 120-km-radius area. Position of the Kowandi South Disdrometer is also shown

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 9.
Fig. 9.

Weighting function used in the time-averaging process to compare radar- and gauge-derived rain rates

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 10.
Fig. 10.

Comparison of rain-rate time series on 2 Jan 1998. The rain rate, averaged over 19 rain gauge sites, is derived from rain gauge data (continuous line), and from radar data using the ZPHI algorithm (filled circle), or the classical estimate R(Za) (filled diamond). The daily integrated rainfall rates are specified for each time series. The algorithm index (heavy line) and the ZPHI quality index (thin line) are also shown in the bottom panel for each radar-derived point of the time series

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 11.
Fig. 11.

As in Fig. 10 but for 20 Jan 1998

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 12.
Fig. 12.

As in Fig. 10 but for 28 Jan 1998

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Fig. 13.
Fig. 13.

Point-by-point comparison of gauge rainfall rate (G(i) in the text) during seven days of Jan 1998, with collocated radar-derived estimates: (a) current ZPHI estimate (R(i)ZPHI in the text), (b) RA relation with fixed N*0 (R(i)ATT. in the text), (c) classical estimate RZa (R(i)CLASS. in the text)

Citation: Journal of Atmospheric and Oceanic Technology 18, 11; 10.1175/1520-0426(2001)018<1819:VOTRPA>2.0.CO;2

Table 1.

Rain relations used in this paper. The inverse model in algorithm ZPHI consists of relations 1, 2, and 3. The ZDR correction and alternate rain estimator use 4 and 5. The relation used in the so-called classical estimate is 6 (in which N*0 is fixed to 0.8 e7 m−4)

Table 1.
Table 2.

Specifications of the C-POL radar during the January 1998 TRMM validation experiment

Table 2.
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