Introduction
The research reported in this paper has been undertaken in the framework of the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE), an international experiment intended to document the interaction between ocean and atmosphere in the western Pacific warm pool (Webster and Lukas 1992). An intensive observing period occurred from 1 November 1992 through 28 February 1993, during which time the sampled domain extended from 10°N to 10°S latitude, and 140°E to 180°E longitude. It covered the warmest part of the warm pool region where interaction between ocean and atmosphere is the most intense. In the atmospheric component of TOGA COARE, one of the major goals was to observe the deep convection developing over the warm pool, with a particular insight on the evaluation of the associated rainfall since it determines the net latent energy transfer from the ocean to the atmosphere.
The investigation of the deep convection dynamics in TOGA COARE was conducted from an ensemble of Doppler weather radars. Two of them at C band (5-cm wavelength) were operated from research vessels: the Massachusetts Institute of Technology radar on board the Vickers (U.S. vessel), and the National Oceanic and Atmospheric Administration (NOAA) TOGA radar on board the Xiang Gyang Ghong Chinese vessel. Two others operating at X band (3-cm wavelength) were installed at Manus Island by the University of Hokkaido (Japan). Four were operated from research aircraft. Let us first cite the Ku-band Airborne Rain-Mapping Radar from the Jet Propulsion Laboratory. It was installed on the National Aeronautics and Space Administration DC8, scanning across track at about ±20° nadir. The three others were X-band systems, performing three-dimensional sampling from helical scanning. The Electra Doppler Radar–Analyse Stéréoscopique par Radar à Impulsion Aéroporté radar (ELDORA–ASTRAIA), developed in common by the National Center for Atmospheric Research (NCAR) and the Centre d’Étude des Environnements Terrestre et Planétaires (CETP), was installed on the NCAR–ELECTRA aircraft, while the two NOAA radars were on board the P3-42 and P3-43 aircraft. These last three systems used similar scanning strategies: dual beam for ELDORA–ASTRAIA (one antenna looking fore and the other aft) and pseudodual beam for the NOAA radar on the P3-43 (looking alternately fore and aft using a dual-beam antenna developed by CETP). The NOAA P3-42 radar operated the original antenna described in Jorgensen et al. (1996) fore–aft scanning technique methodology.
From ELDORA–ASTRAIA, or from one of the two NOAA P3 radars, a three-dimensional wind field synthesis is possible using the data of a single leg of aircraft trajectory through a dual-Doppler analysis such as Chong and Testud (1996), Roux (1998), or Jorgensen et al. (1996). By coordinating the operations of the three aircraft, large domains may be sampled. Actually, the three-aircraft experiment in TOGA COARE provided invaluable datasets to study the dynamics of the large convective systems typical in this region of the globe. However, a problem that is met in the radar data analysis at X band is related to the attenuation of radar waves through heavy precipitation. Consequently, the radar reflectivity (Z) measured by the radar may be severely biased (negatively), as would be the rainfall rate (R) subsequently derived from a Z–R relationship. If no correction is applied to the raw data, this situation may lead to a major misinterpretation of the weather system structure and dynamics.
The object of this series of two papers is the formulation and the application of algorithms that correct X-band reflectivities for along-path attenuation. Its final purpose is to better estimate physical quantities such as the rainfall rate and the precipitation water content, which are essential to understand the water and energetic budget of the TOGA COARE tropical weather systems.
Part I describes the development and exploitation of algorithms to correct the radar reflectivity for attenuation and, consequently, to retrieve the specific attenuation. Beyond the algorithm validation, its main focus is to study the K–Z relationship and what it implies for the physics of precipitation and for the retrieval of the rainfall rate in the rain cells.
Part II will present a more extensive application of the algorithms to the rainfall rate retrieval.
The stereoradar analysis, first proposed by Testud and Amayenc (1989) and later modified and validated for real data by Kabèche and Testud (1995, hereafter KT95), is the first algorithm considered in this paper. In its principle, the stereoradar analysis does not formulate any relationship between the radar reflectivity Z and the specific attenuation K, nor does it make any assumption about the type or size distribution of the hydrometeors. It is a mathematical technique allowing one to derive the “true” reflectivity Z and the specific attenuation K, as independent parameters, from the “apparent” (attenuated) reflectivities successively sampled along the two viewing angles of dual-beam radars. Note that the sampled rain cells are supposed to be stationary during the corresponding laps of time. In this paper, the stereoradar approach is revisited from two viewpoints. First, we adopt a new formulation in Cartesian coordinates, providing much more flexibility in the data analysis than that by KT95 in coplanar coordinates (this point is essential when dealing with data from several aircraft such as in TOGA COARE). Second, we present a variant of the stereoradar analysis, named quad beam, which exploits the four viewing angles provided by two aircraft equipped with a dual-beam radar and operating in coordination to sample the same convective event. This new approach is numerically more stable than the stereoradar technique, especially when calculating the K–Z relationship because of the overdetermination of the problem to be solved.
The behavior of these new algorithms is studied via realistic analytical representations of rain cells and analysis of real reflectivity datasets, which were simultaneously sampled from the two P3s at 1633–1648 UTC 9 February 1993. Comparing the corrected reflectivity fields, calculated from these two independent datasets, validates the new form of the stereoradar approach. Then, the quad-beam algorithm is validated through its comparison with the stereoradar technique.
During the data analysis, a special effort has been devoted to intercalibration of the radars and connection to the “absolute calibration” derived from in situ microphysical probes’ sampling. Indeed, although the derivation of the specific attenuation K is not sensitive to an absolute calibration error of the radar, the K–Z relationships are. Therefore, the absolute calibration is essential when discussing the implication of our findings for the physics of precipitation.
The stereoradar algorithm in Cartesian coordinates
The sampling strategy for dual-beam radar is recalled in Fig. 1a. Both antennas in the tail of the aircraft are mounted back to back, one looking 18.5° forward and the other −18.5° aft for ASTRAIA (perpendicular to the aircraft’s heading), and, respectively, ±22° and ±19.5° for the P3-42 and P3-43 radar. The whole system rotates around a horizontal axis, which is collinear to the aircraft fuselage. Hence, the association of antenna rotation and aircraft motion provides two helical scans of the reflectivity from two different viewing angles, as shown in Fig. 1b. The following notations are used. Subscript 1 (2) refers to the fore (aft) beam of the radar.
ri radial distance to the sampled point (km), i = 1, 2;
αi azimuth angle defined in Fig. 2 (°), i = 1, 2;
θi elevation angle relative to horizontal (°), i = 1, 2;
Zi measured log-reflectivity scanned from viewing angle αi (dBZ), i = 1, 2;
Z true, nonattenuated log reflectivity (dBZ);
K specific attenuation (dB km−1); and
x, y, z standard Cartesian coordinates.
Equation (3) is the basic equation of the stereoradar algorithm in Cartesian coordinates. It has the same general structure as when written in coplanar coordinates: geometrical terms on the left-hand side of the equation and a measurement term on the right side. The variables A, B, and C depend only on the viewing angles from which each point has been sampled, while Mt depends only on the derivatives of the two apparent log-reflectivities Z1 and Z2.
A new concept of correction for attenuation: The quad-beam algorithm
Mathematical formulation
During TOGA COARE, there has been simultaneous sampling of precipitation cells from several aircraft equipped with dual-beam radar and flying on either side of a convective line (see Fig. 2). This opportunity allows direct calculation of the gradient of Z and then Z itself at each point of the Cartesian grid by way of a new algorithm called quad beam.
Actually, the first two terms in (11) force Z to be consistent (in the least squares sense) with the estimate of its gradient derived from (10). The second-order filter L(Z) ensures the stability of the integration and links the boundary conditions area to the rest of the domain, while the fourth term fixes the integration constant. From the Euler equation associated with (11), note that the 3-dB cutoff wavelength is related to μz through λZ = 2π(μZ)1/2.
Once Z is retrieved, four independent estimates of the specific attenuation field are available from (2) (one for each radar beam). Hence, the minimization of (7) following the procedure given in section 2, with i = 1 to 4 instead of i = 1 to 2, ensures the retrieval of K in a more stable way when compared to the stereoradar since more information is available.
Correction for polarization of the transmitted wave
The elevation angle θ can be expressed as sinθ = −cos(Tilt) cos(Spin). With Tilt = 20°, Fig. 3 shows that P is related to θ through a factor close to one-third for θ lower than 25°. Since most of the points that should be corrected for attenuation are sampled under elevations that do not exceed 20°, the variation of the polarization angle within the domain of interest stays lower than 7°, with 4° as mean value. This has negligible impact on the backscattering and attenuation cross sections and, consequently, on the calculation of ex and ey.
More serious is the correction when the radars on the two aircraft operate in a different polarization, as it is the case when the P3-42 radar (vertically polarized) is coupled with the P3-43 radar or with ASTRAIA–ELDORA (both of which are horizontally polarized). Since the above consideration allows one to assume the polarization is independent of the elevation angle, the formulation for merging data from the two radars, respectively, horizontally (subscript H) and vertically (subscript V) polarized, requires established relationships between the associated ZH, ZV reflectivities and the KH, KV attenuation fields. This formulation is done in the appendix for deformed drops having the shape of oblate spheroids. Their axial ratio (ratio of minor to major axis, noted a/b) is assumed to be related to the equivolumetric radius
Preprocessing of the radar data
It is crucial to correct the raw data for the aircraft navigational errors with a procedure such as Testud et al. (1995) and to systematically use the Global Positioning System positions (Matejka and Lewis 1997). The global advection of the rain cells is taken into account, translating the sampling frame into the advective frame moving with the rain cores. And finally, note that reflectivities can hardly be corrected beyond the 40-km range since both stereoradar and quad-beam approaches require a satisfactory cross-beam resolution (<1 km) and time lapses lower than a few minutes (<5) between the samplings from the various beams (because of the internal evolution of the rain cells).
The derivative of the apparent reflectivity with respect to r is calculated at each range gate, following an 11-point differentiation scheme during which a second-order polynomial least squares fit is applied to the series of plus or minus five adjacent gates. Then the derivative is defined as that of the adjusted polynomial expression at the range in question.
Once the preprocessing is done, a Cressman filter (influence radius 2 km horizontally, 0.8 km vertically) is applied to interpolate the apparent reflectivities (in dBZ), their derivatives with respect to r, and the viewing angles α and θ to a Cartesian grid where the x axis is defined as the aircraft trajectory.
Validation of the algorithms with simulated data
The algorithms are first tested with simulated data, considering a theoretical rain cell model. The procedure generating these data consists of using real samplings performed by the two P3 radars and replacing the actual reflectivities with simulated ones. The selected TOGA COARE legs are the ones studied in section 6, that is, leg 1633–1644 UTC at 4.1-km altitude for P3-42 and leg 1633–1648 UTC at 0.3-km altitude for P3-43 on 9 February 1993. Hence, the stability of the algorithm is investigated in “natural” conditions on nonideal sampling, where the aircraft trajectory is not perfectly straight and real motions affect the attitude of the platform.
Simulated dataset from a theoretical precipitation model
The convection line is simulated by the superposition of two reflectivity cores of Gaussian shape (in mm6 m−3;i.e., parabolic shape in dBZ), about 9 km apart. Each core is characterized by the position (x0, y0), the intensity Zmax of its maximum reflectivity (in dBZ), and its“diameter” at −3 dBZ below Zmax. For vertical structure, we considered a constant reflectivity profile from the ground to the freezing level (4.2 km) and a decrease aloft at the constant rate of 5 dBZ km−1. A smooth transition zone around the freezing level is introduced using the Testud et al. (1996) scheme in order to avoid artificial jumps in the Z derivatives.
Retrieval of simulated data tainted with realistic radar speckle and receiver noise
The original, attenuated, and corrected fields are compared at 3.2-km altitude since this was the flight altitude of the Electra, which collected the microphysical data [from which the K–Z relationship given in (15) is derived]. All parameters used in this simulation are listed in Tables 1–4.
Figure 4a shows a horizontal cross section of the original (nonattenuated) reflectivity field for the abovementioned system of two rain cells (track of the Electra is not shown). At this altitude of 3.2 km, the two reflectivity cores [a (x = 63; y = 11) and b (x = 61; y = 20), respectively] peak at 48.6 dBZ. This value is slightly smaller than that of the analytical shape (peaking at 48.8 dBZ at this altitude) because of the smoothing effects of the Cressman interpolation (see section 4). Hereafter, Fig. 4a will be our reference for nonattenuated reflectivity. Figures 4c,d show the attenuated aft and fore simulated reflectivities sampled from the P3-42 radar. In these two fields, core a is barely visible (max = 43.7 dBZ), while core b has nearly disappeared. This situation underlines the necessity to correct Z for attenuation.
Both stereoradar and quad-beam algorithms are run with the same cutoff wavelength λZ = 3.5 km (see Table 4) in order to make the comparison meaningful. This value should not induce any significant extra smoothing of the Z fields, knowing that the cutoff associated with the Cressman filter is slightly more severe (λZ ≃ 4.4 km for an influence radius of 2 km for the Cressman filter).
Figure 4b shows that boundary conditions are available on the right- and on the left-hand side of the rain cells of interest when the PIA threshold is set to 0.8 dB (same threshold is used to process the real data in section 6c). In Fig. 5a, the stereoradar was able to correctly retrieve Z in structure and shape: a and b peak, respectively, at 47.7 and 47.3 dBZ. The slight underestimation of the peak value with respect to the reference (0.9–1.3 dBZ) is mainly due to the PIA threshold, which tends to induce too-low reflectivities within the E domain.
For quad-beam analysis, the boundary conditions region does not need to be as extensive as with the stereoradar since only a constant of integration is required. Now the PIA is set to 0.4 dB. Figure 5c shows that the quad-beam analysis delivers a corrected Z field remarkably well correlated in structure, shape, and intensity with the original one. Note that core b is less subject to the slight underestimation noticed for the stereoradar approach since the maximum reflectivity of cells a and b is about 47.8 dBZ. This emphasizes the improved robustness of the quad-beam technique with respect to the stereoradar.
The 3-dB cutoff wavelength of the filter in the retrieval of K was set to λk = 4.5 km, that is, the filter is slightly more severe than for Z. This is to mitigate the numerical instability associated with the differentiation of Z needed to calculate K. The specific attenuation cores are well retrieved by the stereoradar. Moreover, the corresponding K versus Z scatterplot well reproduces the original K–Z power law assumed in the rain cell model (Fig. 5b). The power-law fit to the retrieved data [K = 5.350 × 10−5 (Z0.894)] is within an 8% error margin from (15) (for 40 dBZ < Z < 50 dBZ), while its correlation coefficient is ρ = 0.941 for 175 points. The performance of the quad-beam analysis (Fig. 5) is superior: The power-law fit [K = 8.222 × 10−5 (Z0.852)] is closer to the original (within 5% error in the range 40 dBZ < Z < 50 dBZ) and the K–Z scatterplot is less scattered than for stereoradar, providing a higher correlation coefficient (ρ = 0.984) in the power-law fit.
Influence of cloud particles on the retrievals
It is important to recall that both algorithms retrieve the radar reflectivity Z and the specific attenuation K independently of any assumption about the size distribution of the hydrometeors or the type of the particles that induce the attenuation (raindrops, hailstones, melting snowflakes, cloud particles, water vapor, and other atmospheric gases). This means that in a situation where, for example, rain and cloud are coexisting, the measured (and then retrieved) specific attenuation field will be Krain + Kcloud. However, both algorithms postulate that the integrated attenuation is zero along the path between the radar and the farthest edge of the E domain. In this section, we aim to investigate the impact of this assumption on the accuracy of the retrieval.
First, the attenuation by atmospheric gases could be systematically taken into account, using the Doviak and Zrnic (1993) formulation. However, for paths of a few tens of kilometers that are to be considered presently at X band, the integrated attenuation should not exceed 0.3–0.5 dB, which is quite negligible in our application.
Another source of bias of the boundary condition is expected when the beam intercepts an undetected cloud along the path. To investigate this effect, the same simulation as in Fig. 5 has been performed with, in addition, a cloud acting as an undetected source of attenuation centered at x = 62 km, y = 5 km, that is, within the E domain. The cloud was characterized by a Gaussian axisymmetrical shape with maximum liquid water Wmax = 2 g m−3 and a “cell diameter” (or diameter of the W isocontour at Wmax/2) of 2 km. The induced specific attenuation was calculated from the relation K (dB km−1) = 0.06 W (g m−3) derived at 9.3 GHz from Meneghini and Kozu (1990).
Both algorithms were found to be almost unaffected by the presence of such a cloud. The results are summarized in Figs. 5b,d, where the long dashed lines represent the power-law fit of the K–Z plots “with cloud.” Only the correlation coefficients are slightly different: ρ = 0.923 for stereoradar and ρ = 0.931 for quad beam.
Application to real data on 9 February 1993
As previously stated, the two P3 legs from TOGA COARE currently selected to perform the stereoradar and quad-beam analysis are 1633–1644 UTC (P3-42) and 1633–1648 UTC (P3-43) on 9 February 1993. They correspond to a radar sampling favorable to the application of the stereoradar analysis, that is, with evidence of severe along-path attenuation and availability of low-reflectivity areas (boundary conditions) on either side of the rain cells in order to integrate the stereoradar equation. The two P3s flew southeast of the intensive flux array into a developing convection line oriented west-northwest–east-southeast. The echo top was up to 16–19 km and maximum reflectivity was up to 46 dBZ. The P3-42 and P3-43 simultaneously sampled the same rain cells by flying short parallel legs on both sides of the convective line. Recall that the flight altitude was 4.1 km for the P3-42 and 0.3 km for the P3-43.
Intercalibration of the P3s and absolute calibration with ELDORA–ASTRAIA
For the purpose of intercomparison and interpretation of the Z and K fields retrieved from different datasets and analysis, it is crucial to first intercalibrate the two airborne radars and to achieve a reasonable absolute calibration.
On 9 February 1993, between 1633 and 1636 UTC, the two P3s flew almost simultaneously two legs 25 km apart, sampling an area of low reflectivity (Z < 30 dBZ), presumably not subject to attenuation. This occurred just before sampling the main rain cell described in this paper. The area used for calibration is delimited by 10 km < x < 26 km and 5 km < y < 23 km in Figs. 7c and 7d (altitude between 1.4 and 9.4 km). For calibration, only the data points whose PIA [estimated from (15)] was lower than 1.2 dB have been selected. This threshold ensures that the attenuation is small enough not to bias the calibration, while the number of common points stays significant. After gridding the data of the two aircraft in a common Cartesian grid by way of the Cressman filter, ZP3-42 − ZP3-43 (in dB) is calculated at each point where data are available. The average value and the standard deviation of this difference are calculated at each altitude. They are both plotted in Fig. 6a. We can see that in spite of a standard deviation of about 1.7 dB at all levels, ZP3-42 − ZP3-43 remains quite stable at about −1.2 dB whatever the considered altitude is (variations < ±0.5 dB). It tends to demonstrate that the P3-43 sampled reflectivities are 1.2 dB lower than those of the P3-42.
The same operation, performed between the P3-43 and ELDORA–ASTRAIA between 1729 and 1737 UTC on 9 February 1993, shows that the P3-43 reflectivities are 1.2 dBZ lower than the ASTRAIA ones (see Fig. 6b). An absolute calibration of ASTRAIA was obtained by F. Marks from the onboard microphysical probes, using the procedure described in Marks et al. (1993). F. Marks (1996, personal communication) found that ASTRAIA was undercalibrated by 4 dB during TOGA COARE. This value is presently used to further calibrate the P3’s Doppler radar. The remaining error associated to the intercalibration procedure is probably less than 0.7 dB.
Description of the measured reflectivity fields
Figures 7a,b show a horizontal cross section at 3.2-km altitude of the observed reflectivities sampled by the P3-42 fore and aft beams at 1633–1644 UTC 9 February 1993 in the Cartesian grid. Behind the intense rain cell peaking at 47.5 dBZ and centered at x = 50 km, y = 7 km, reflectivities are nearly completely extinct. This occurs at ranges beyond 15 km, indicating a severe attenuation of the radar beams, while the measured reflectivities are still moderate. A secondary rain cell, which is less intense, is discernible near x = 70 km, y = 15 km. So far, the best available estimate of Z we have (using the P3-42s data) is the maximum of the aft and fore reflectivity fields in Fig. 7c. The need for a more accurate estimate becomes obvious when comparing Fig. 7c to the maximum of the reflectivities simultaneously sampled by the P3-43 from the other side of the convective line (Fig. 7d). There, the two rain cores described above are not discernible at all. Consequently, the scatterplot of the P3-42 reflectivities versus the P3-43 ones (see Fig. 7e) shows differences up to 29.6 dB.
Cross-validation of the retrieved reflectivities from both algorithms
Stereoradar analysis
To improve the stereoradar retrieval, the boundary conditions defined from the data of the two aircraft were merged. This was made possible by the new formulation in Cartesian coordinates. The whole E domain is shown in Fig. 7f, corresponding to PIA < 0.8 dB. It allows a full retrieval of Z and K from the P3-42 dataset, but only the Z retrieval from the P3-43 set (because of the gap on the right-hand side of the boundary conditions).
Figure 8 compares the true reflectivity fields derived from the stereoradar analysis (applied, respectively, to the P3-42 data in Fig. 8a and to the P3-43 ones in Fig. 8b and from the quad-beam analysis (Fig. 8d). The corrections induced by both analyses often exceed 25 dBZ. Compare, for example, Figs. 7c and 8a around x = 50 km, y = 10 km and x = 75 km, y = 20 km.
The same stereoradar analysis applied to the P3-42 and the P3-43 data provides very similar true reflectivity fields (Figs. 8a, b). The corresponding scatterplot in Fig. 8c shows that the standard deviation between the two fields is 1.8 dBZ, instead of 9.3 dBZ, when comparing the maximum fore and aft apparent reflectivities from the two aircraft (Fig. 7e). Note that the P3-43 Z field (horizontally polarized) has been transformed following relation (A1-2) to a vertically polarized one in order to make the comparison more meaningful. The difference in the maximum reflectivity of the two fields (about 2.2 dBZ) is within the expected random error.
This cross-validation of the stereoradar retrievals shows that this algorithm, in its new Cartesian form, is an efficient way to correct the observed reflectivity for attenuation. However, it has the same limitation as the initial Kabèche and Testud’s formulation; that is, the boundary conditions are required at both ends of the radar echoes.
Quad-beam analysis
The quad-beam-corrected Z field in Fig. 8d shows reflectivities up to 49.9 dBZ, while the rain cell structure is identical to that of the stereoradar. In Fig. 8e, the scatterplot of ZQuad-beam versus ZStereo-P3-42 (for which the E domain is most favorable), gives a standard deviation of only 1.0 dBZ for 1535 points. The likeness of the two Z fields, including the main rain cell at x = 50 km, y = 10 km and the two secondary ones centered at x = 70 km, y = 20 km, directly confirms the very good behavior of the quad-beam algorithm for real data. This agreement between the two algorithms is all the more satisfactory because they differ in their basic approach to data analysis.
Note that the fact that boundary conditions are not absolutely necessary in the quad-beam analysis induces a good stability for Z at whatever altitude of the horizontal cross section we consider. However, this algorithm can be run only in the rather exceptional conditions where the two aircraft sample simultaneously the same rain cell. Another restriction is that the retrieval is not possible in areas where one of the four beams is extinct (Fig. 8d).
Comparison with the classical “max of the four” estimate of Z
To minimize the impact of along-path attenuation, a current approach when analyzing the TOGA COARE dataset is to consider the maximum measured reflectivity between all available beams (four beams in the present two-aircraft operation), without practicing any other correction scheme for attenuation. The “four-beam” estimate of Z is indeed better than the “two beam.” However, Fig. 8f shows that this estimate fails to reproduce the most intense part of the precipitation field if we refer to the corrected Z fields provided by both stereoradar and quad-beam techniques. As a matter of fact, all four measured reflectivities collapse in the middle of the rain cell at about 45 < x < 55 km, 5 < y < 15 km, which makes the four-beam estimate unable to reproduce the true structure of the rain field. A reflectivity hole is seen in Fig. 8f at the center of the heavy precipitation core retrieved by the stereoradar (Fig. 8a); there, the deviation between the two fields reaches about 20 dBZ, which means there was a tremendous underestimation of the rainfall rate when using the simple four-beam scheme.
Validation of the K–Z relationships from microphysical in situ measurements
The specific attenuation associated with the true reflectivities is calculated with the stereoradar technique for the P3-42 dataset (Fig. 9a) and then with the quad-beam algorithm (Fig. 9c). In both cases, the attenuation cores correspond to the high reflectivity areas with a good accuracy at x = 50 km, y = 10 km and x = 70 km, y = 20 km. The high values of K, respectively, peaking at 2.0 and 2.2 dB km−1 for stereoradar and quad beam, explain why the measured reflectivities are almost extinguished at only 15-km range from the radar antennas in Fig. 7c (x = 50 km, y =15 km).
The maximum relative difference between (16) and (17) for 39 < Z < 51 dBZ remains within a 7% error margin (<0.15 dB km−1), thus underlining the consistency between stereoradar and quad-beam results. However, both relationships overestimate K by about 21% when compared to the K–Z relationship derived from the in situ microphysical measurements (15). Nevertheless, this difference should be appreciated in referring to the K–Z relationship expected from the standard exponential drop size distribution N(D) = N0 exp(−ΛD) proposed by Marshall and Palmer (1948) as a function of the equivalent diameter of the rain droplets, D in meters, with N0 = 8 × 106 m−4, and Λ in inverse meters. Using the same scattering model as in the appendix at 5°C and 9.31 GHz, the corresponding K–Z relationship is found to be K = 1.55 × 10−4 Z0.751. It appears that this relation, also plotted in Figs. 9b,d, underestimates the specific attenuation as given by (15) by a factor larger than 220% for Z > 45 dBZ.
In fact, the discrepancy between in situ measurements and radar observations may be attributed to either a residual absolute calibration error of the radar (an underestimation of Z by 0.7 dBZ suffices to make it up with the two observational data), or a real effect associated to the fact the two types of data were neither collocated in space nor in time (the microphysical data were collected on board the Electra between 1700 and 2000 UTC).
Discussion of the K–Z results
If we trust these K–Z relationships, either derived from microphysical measurements or from radar observations, how are these relationships related to the drop size distribution characteristics? Figure 10 is a RAIN Parameter Diagram (RAINPAD) of the sort proposed by Ulbrich and Atlas (1978). This RAINPAD displays in a KV versus ZV diagram, theoretical isocontours of the N0 parameter of the drop size distribution, assumed to be exponential (continuous line, in m−4), and the corresponding rainfall rate (dotted lines, in mm h−1). These theoretical curves are derived using the same scattering calculation as for the microphysical data (see the appendix). In the same diagram are plotted the “microphysical” and “radar-derived” K(Z). The slight discrepancy between these results was already discussed in the previous subsection. However, note that they do not cover the same dynamics in rainfall rate: 4–50 mm h−1 for microphysical and 20–150 mm h−1 for radar derived. Referring to these experimental relationships, what appears anyway is that the N0 parameter deduced from the RAINPAD increases with the rainfall rate from N0 = 5.5 × 106 for R = 5 mm h−1 to N0 = 5.5 × 107 for R = 100 mm h−1. Hence, N0 increases with R as R0.75. Moreover, it is noticeable that the radar-derived relationships make a relatively small angle with the isolines of D0 (the median drop diameter of the distribution; D0 = 3.67/Λ) when compared to the K–Z relationship calculated for the Marshall–Palmer DSD. Therefore, for R varying from 5 to 100 mm h−1, D0 varies from 1.34 to 1.5 mm (Λ from 27.3 to 24.5 cm−1) in experimental K–Z relationships and from 1.22 to 2.3 mm (Λ from 30.0 to 15.9 cm−1) in Marshall–Palmer relationships.
This quasi proportionality of N0 with R, associated with a weak variation of Λ, suggests that the drop size distribution in this convective cell tends to the equilibrium described by Hu and Srivastava (1995). These authors showed that in heavy rain, the conjugate action of collisional breakup and coalescence forces the drop size distribution to an equilibrium spectrum that is independent of the rainfall rate (after normalization) in a characteristic time inversely proportional to R. An implication of the concept of equilibrium spectrum is that all integrated parameters of the DSD (as Z and K) should be proportional. Hence, the exponent of the K–Z relationship should be 1 instead of 0.75 for a Marshall–Palmer DSD. The intermediate value (0.87) found in the present study may be another expression of this tendency of the DSD towards equilibrium.
As shown by Willis et al. (1995) and Tokay and Short (1996), the shape of the DSDs observed during TOGA COARE either from airborne microphysical probes or from a disdrometer on an island is not far away from exponential for moderate rain rates (as 1–5 mm h−1) but exhibits an increasing curvature at higher rain rates. These authors have shown that the DSDs were better represented by a gamma distribution [N(D) = N0Dμ exp(−ΛD)] and found that the μ parameter for very heavy rain is 4–5 (according to Willis et al. 1995) or 8 (according to Tokay and Short 1996). It is not meaningful to compare Λ and N0 found when fitting a DSD by gamma and exponential distributions. Nevertheless, Fig. 3 of Willis et al. (1995) displays an exponential fit of the average DSD for rainfall rates ranging from 50 to 70 mm h−1 and corresponding to Λ = 26 cm−1, a value very close to that deduced from our RAINPAD. Moreover, the Z–R relationship computed by the same authors for mean high-altitude spectra, when plotted in our RAINPAD, coincides perfectly with our experimental data.
Since the stereoradar and quad beam operate in three dimensions, Fig. 11 investigates the variation of the radar-derived K–Z relationship with altitude. Figure 11a displays the power-law fits of the K–Z scatterplot for a series of altitude between 2.2 and 4.4 km (the freezing level was determined as 4.6 km from a P3-43 dropsonde at 1506 UTC 9 February 1993). The step between each horizontal cross section is equal to 0.2 km. Below 2 km, the K retrieval is not reliable because the PIA is so intense that it leads to a complete extinction of the signal in a large portion of the storm. Note that the dispersion of the various K–Z relationships in Fig. 11a is quite small (±10% standard deviation) and that the average departs considerably from the K–Z relationship expected for a Marshall–Palmer DSD. This small dispersion tends to show that the precipitation through all this altitude range is rain and to confirm that the conditions of “equilibrium spectrum” are reached in this intense rain cell. It appears clearly in Fig. 11b that above the freezing level, the K–Z relationships are much more dispersed. The b coefficient suddenly drops from 0.83 at 4.4 km to 0.15 at 4.8 km; the attenuation continues to decrease as the altitude increases to reach the noise level in the K retrieval at 5.6 km. It seems very probable that the precipitation is slightly attenuating ice above 5.6-km altitude. The rain regime seems to establish below 4.6-km altitude. In the intermediate altitude range of 5.6–4.6 km, the specific attenuation is yet quite significant, and the interpretation is not obvious. In this convection line that has been processed for three-dimensional wind field by Roux (1997) and Lewis et al. (1998), upward vertical velocities up to 8–10 m s−1 at 5-km altitude were found. In such conditions, we may expect above the freezing level a complex situation with ice particles coexisting with supercooled raindrops. Actually, a large dispersion of the K–Z scatterplot was found at the altitudes in question (4.8, 5.0, and 5.2 km).
Conclusions
This paper takes advantage of a TOGA COARE dataset in which radar observations of a tropical squall line were collected simultaneously from two NOAA airborne platforms. This exceptional dataset allowed us to cross-validate two algorithms that correct reflectivity for attenuation: the stereoradar analysis, reformulated in Cartesian coordinates in order to improve its flexibility, and the new quad-beam analysis, which synthesizes the information of the four available viewpoints delivered by the two aircraft.
The comparison between apparent (directly observed), and true (corrected) reflectivity fields emphasizes the importance of the correction for attenuation, which is often larger than 20 dBZ. While the apparent reflectivity fields seen from the two aircraft are very different, their reconciliation after correction by the stereoradar analysis appears quite remarkable (1.8-dB standard deviation between the two fields). The picture obtained in the Z field by the quad-beam analysis is completely consistent with that of the stereoradar (indicating that the standard error of the two analyses is of the order of 1 dB). The better numerical stability of the quad-beam analysis leads to improved accuracy of the retrieved specific attenuation field.
The derived K–Z relationships from stereoradar and quad beam are very close to each other and were found quite compatible with that derived from the in situ microphysical data collected by the NCAR Electra within the same convective event (a careful intercalibration of the radar on the three airborne platforms was nevertheless necessary to assess this result). This radar-derived relationship is very far from what would be expected from a Marshall–Palmer rain DSD. In the Z range where both stereoradar and quad-beam algorithms operate (i.e., Z > 39 dBZ), this relationship predicts a specific attenuation two times larger than that for a Marshall–Palmer rain. When fitted by a power law such as K = aZb, it leads to an exponent b = 0.85, instead of 0.75 for Marshall–Palmer rain.
The implication of our findings concerning the raindrop size distribution was investigated using a rain parameter diagram (under the assumption of an exponential DSD). It was found that the “equivalent” N0 parameter of the DSD increases with R as R0.75 and that our radar-derived K–Z relationship is fully consistent with the Z–R relationship previously derived by Willis et al. (1995) for a larger set of TOGA COARE airborne microphysical data in the same altitude range. The invariance of the K–Z relationship (adjusted to a power law) along the vertical in the rain layer, added to the fact that its exponent b is larger than that for a Marshal–Palmer DSD, suggests that the observed heavy rain tends to the “equilibrium” described by Hu and Srivastava (1995). It is consistent with previous observations of tropical rain made by Zawadzki and Agostinho (1988).
Acknowledgments
We are indebted to A. Kabèche and J.-B. Henry for their help in developing parts of the original stereoradar and quad-beam algorithms. The absolute calibration of the P3s radar on 9 February 1993 was made possible by F. Marks from NOAA/AOML (Miami, Florida) and P. Hildebrand from NSCAR (Boulder, Colorado), who kindly performed the processing of the ELDORA–ASTRAIA data. We also express our appreciation to F. Marks, P. Willis, and R. Black from NOAA/AOML (Miami, Florida) for providing the experiment drop size spectra used in the microphysical calculation. We are particularly indebted to NOAA for flying the P3s in TOGA COARE and to INSU, which partially funded the corresponding flight hours.
This research was supported in part by the U.S. National Science Foundation under a cooperative agreement with the University Corporation for Atmospheric Research, by the French Institue National des Sciences de l’univers (INSU), and by the French Centre National d’Études des Télécommunications (CNET).
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APPENDIX
Results of the Microphysical Data Analysis
The in situ microphysical measurements were analyzed for two purposes. First, they were used to estimate the corrections for polarization in the reflectivity and specific attenuation retrievals, which are required in the quad-beam algorithm. Second, they were analyzed to calculate a relationship between KV and ZV for comparison with both stereoradar and quad-beam algorithm results. The PMS probe data collected during TOGA COARE aboard the two P3s suffered from serious deficiencies; the most critical one was a very small sample volume. Thus, we only used 2DP images of raindrops collected aboard the Electra (NCAR) aircraft. This instrument measures two-dimensional images of particles with diameters ranging from 0.2 to 6.4 mm (32 classes of 0.2 mm). At 9.31 GHz, the raindrops within this diameter range provide the dominant contribution to specific attenuation and reflectivity. Drop size distributions (DSDS) were derived from the 2DP measurements following Black and Hallet (1986). Each DSD is determined from a 6-s sample time corresponding to a trajectory length of about 0.7 km. We selected microphysical data between 1730 and 2000 UTC 9 February 1993 since the Electra crossed the convective line described in section 6 several times during this period. The Electra altitude was 3.2 km above sea level.
Reflectivity and specific attenuation values were calculated for each DSD as in Marécal et al. (1996) and Marécal et al. (1997). To take into account the raindrop flattening for increasing diameter, raindrops are assumed to be spheroids. The backscattering and extinction cross sections used to calculate reflectivity and attenuation were determined from the T matrix approach (Waterman 1965)
Effect of polarization on K and Z
These results are not very sensitive to the incidence angle with respect to the horizontal plane and to the temperature of the raindrops. When the variations of the angles are confined between 0° and 8° in the considered radar geometry and T varies between 0° and 10°C (temperature of raindrops at 2.2 km), the maximum variations on p and q are lower than 0.1% and 0.4%, respectively.
K–Z relationship
Since the stereoradar and quad-beam analysis provide a KV–ZV relationship for convective rain, we only selected the 349 DSDs that corresponded to ZV ≥ 32 dBZ in vertical polarization at 5°C drop temperature and 4° incidence angle with respect to the horizontal plane. This threshold eliminates nearly all the samples corresponding to stratiform precipitation, which exhibits a slightly different behavior when compared with convective precipitation (not shown). Since KV and ZV variables are both naturally scattered, an orthogonal least squares fit was considered as given in Marécal et al. (1996) and Marécal et al. (1997) to determine the KV–ZV relationship given in (15). The scatterplot and the fit are shown in Fig. A1c (correlation coefficient of 0.94). Although only 2 % of the selected DSDs correspond to Z ≥ 39 dBZ, the fit appears good for the whole chosen ZV range (ZV ≥ 32 dBZ). It was also verified that the effects of drop temperature uncertainty and of particles smaller than 0.2 mm are negligible (<1%).
(a) Schematic of the sampling strategy with the NOAA P3s and ELDORA–ASTRAIA. Units denote rotation per minute (rpm). (b) Schematic of the viewing angles in a horizontal plan at flight altitude. The along-track data density of intersecting beams is equal to 1.6 km for the P3-43 and P3-42 (aircraft ground speed ≈ 120 m s−1) and 0.8 km for ASTRAIA, which uses two radars to transmit the fore and aft beams simultaneously.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
Sampling strategy of the aircraft for stereoradar (one aircraft required) and quad-beam processing (two aircraft required). The retrievable area is the one where reflectivities are sampled from two or four beams. The boundary conditions are the regions where the along-path attenuation of at least one beam is negligible and, consequently, where Z = Zi is available.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
The solid line is the exact polarization angle of a horizontally polarized radar (tilting = 20°) vs the spin angle of the radar antenna (defined from nadir). The dashed line is its linear approximation, which is good for an elevation <25°.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
Horizontal cross section of the simulated rain cells interpolated in the Cartesian grid at 3.2-km height. (a) The original and nonattenuated reflectivities (considered as reference). (b) The boundary conditions, that is, the nonattenuated regions. (c) and (d) show the attenuated reflectivities sampled by the fore and aft beams, respectively. (e) The maximum of the aft and fore attenuated reflectivities.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
(a) The Z field calculated with the stereoradar approach. (b) The K vs Z scatterplot of the simulation without clouds. The corresponding orthogonal fit is the dotted line. The long dashed line corresponds to the K–Z fit of the simulation with clouds (for Z > 40 dBZ). (c) and (d) The same as (a) and (b) except for quad-beam fields.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
(a) Display of the mean value of ZP3-42 − ZP3-43 as a function of the altitude for each horizontal plane. This difference is calculated only in the areas where the two P3s radars were not attenuated at 1633–1638 UTC 9 Feb 1993. The horizontal plain line gives the global (3D) intercalibration error. The error bars are the standard deviation of the fits for each altitude. (b) Same as (a) except for the P3-43 vs ELDORA–ASTRAIA reflectivities at 1729–1737 UTC.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
(a) The horizontal cross section of the real reflectivities of the rain cells at 3.2-km height measured from the P3-42 fore beam at 1633–1648 UTC 9 Feb 1993. (b) Same as (a) except for the aft beam. (c) Maximum of the P3-42 fore and aft beams. (d) Same as (c) except for the P3-43 apparent reflectivities in the same Cartesian grid. (e) Scatterplot of the P3-43 vs the P3-42 maximum apparent reflectivities. The dotted line is the reference y = x, and sigma is the standard deviation of the scatterplot. (f) Union of the entire boundary conditions available in the area.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
(a) The P3-42 reflectivities corrected with the stereoradar approach. (b) Same as (a) except for the P3-43 dataset. (c) The scatterplot of the (a) vs the (b) field. (d) The reflectivities corrected with the quad beam. (e) The scatterplot of the (a) vs the (d) field. In (c) and (d), the dotted line represents the reference y = x, and sigma is the standard deviation of the scatterplots. (f) Classic estimate of Z based upon superposition of all four measured reflectivities.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
(a) The K field calculated with the stereoradar approach from the P3-42 real dataset. (b) The corresponding K vs Z scatterplot. Its orthogonal fit is the solid line. The long dashed line corresponds to the K–Z relationship given by (15), while the dotted one is the one derived from the Marshall–Palmer DSD. (c) and (d) The same as (a) and (b) except for the quad-beam fields.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
RAINPAD of KV vs ZV for an exponential drop size distribution with the associated isopleths of N0 (m−4, solid lines) and R (mm h−1, dashed lines). The dashed–dotted line is the N0 = 8 × 106 m−4 isopleth. The solid bold line corresponds to the relation derived from the microphysic data analysis. The dashed bold line and dotted bold line are, respectively, the stereoradar and quad-beam relationships.
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
(a) K–Z relationships derived from independent horizontal cross sections between 2.2- and 4.4-km height. Each K and Z field is calculated using the stereoradar algorithm. The dotted line is the K–Z relationship derived from a Marshall–Palmer exponential DSD at 5°C and 9.31 GHz. (b) Same as (a) between 4.4- and 5.6-km altitude (the altitude of the curves are labeled).
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
Fig. A1. (a) Scatterplot of KH vs KV. The solid line corresponds to the fit chosen where KH = KV for KV = 0 dB km−1. (b) Same as (a) but for ZH and ZV. (c) Scatterplot and corresponding orthogonal least squares fit (solid line) of KV vs ZV (for ZV ≥ 32 dBZ).
Citation: Journal of Applied Meteorology 38, 2; 10.1175/1520-0450(1999)038<0156:EOPFTD>2.0.CO;2
Flight characteristics of the two P3s.
Characteristics of the Cartesian grid.
Characteristics of the simulated rain cells.
Input parameters for the stereoradar and quad-beam algorithms.
