Evaluation of Attenuation Correction Methodology for Dual-Polarization Radars: Application to X-Band Systems

Transcription

1 AUGUST 2005 G O R G U C C I A N D C H A N D R A S E K A R 1195 Evaluation of Attenuation Correction Methodology for Dual-Polarization Radars: Application to X-Band Systems EUGENIO GORGUCCI Istituto di Scienze dell Atmosfera e del Clima (CNR), Rome, Italy V. CHANDRASEKAR Colorado State University, Fort Collins, Colorado (Manuscript received 6 July 2004, in final form 19 January 2005) ABSTRACT Monitoring of precipitation using high-frequency radar systems, such as the X band, is becoming increasingly popular because of their lower cost compared to their S-band counterpart. However, at higher frequencies, such as the X band, the precipitation-induced attenuation is significant, and introduces ambiguities in the interpretation of the radar observations. Differential phase measurements have been shown to be very useful for correcting the measured reflectivity for precipitation-induced attenuation. This paper presents a quantitative evaluation of two attenuation correction methodologies with specific emphasis on the X band. A simple differential phase based algorithm as well as the range-profiling algorithm are studied. The impact of backscatter differential phase on the performance of attenuation correction is evaluated. It is shown that both of the algorithms for attenuation correction work fairly well, yielding attenuation-accurate corrected reflectivities with a negligible bias. 1. Introduction Corresponding author address: Dr. Eugenio Gorgucci, Istituto di Scienze dell Atmosfera e del Clima (CNR), Area di Ricerca Roma-Tor Vergata Via del Fosso del Cavaliere, Rome, Italy. gorgucci@radar.ifa.rm.cnr.it The extinction cross section of precipitation particles determines the power loss suffered by the electromagnetic waves resulting from absorption and scattering. For frequencies, such as the X band, absorption dominates the attenuation process. At X-band frequency, rainfall causes significant attenuation. As a consequence, radar measurements of reflectivity (Z) must be corrected for rain attenuation so that they can be used quantitatively. The early attempts to correct attenuation were applied iteratively, starting from a range close to the radar, correcting the attenuation one range sample at the time (Hitschfeld and Bordan 1954). The iterative techniques are known to be unstable because of potential errors in the measurements of Z and inevitable errors in specific attenuation reflectivity relations. By the early 1980s, researches recognized that the attenuation correction would be made stable by imposing constraints on the cumulative path attenuation (Marzoug and Amayenc 1994). The attenuation correction based on constraints was implemented in the Tropical Rainfall Measuring Mission (TRMM) program (Iguchi and Meneghini 1994). A simple attenuation correction procedure, studying the relation between differential phase ( dp ) and cumulative attenuation (A h ), was explored by Bringi et al. (1990). While the attenuation correction based on differential phase is useful in correcting the cumulative attenuation, it is not easy to produce a profile of specific attenuation. To estimate specific attenuation ( h ) one needs to compute the specific differential phase as the slope of the measured profile of differential phase, which has its own practical problems (Gorgucci et al. 2000). Recently, Testud et al. (2000) successfully adopted the range-profiling algorithm, used in spaceborne radars, with the cumulative attenuation estimate obtained from differential phase (Bringi et al. 1990) to develop an algorithm for obtaining the profile of specific attenuation. This algorithm was further evaluated 2005 American Meteorological Society

2 1196 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 22 for estimation of the rainfall rate at the C band by Le Bouar et al. (2001), and at the X band by Matrosov et al. (2002) and Iwanami et al. (2003). Bringi et al. (2001) analyzed this technique by evaluating the optimum conversion from differential phase measurements to cumulative attenuation that is used in the constraint. The objective of this paper is to evaluate the error structure of the attenuation correction methodologies using the differential phase at X-band frequencies. It is well known, based on the extensive studies done for the TRMM program, that the estimate of the specific attenuation profile is only as accurate as the constraint or the surface reference measurement (Iguchi and Meneghini 1994). Similarly, there is a one-to-one correspondence between the accuracy of the differential phase based cumulative attenuation estimate and the performance of the range-profiling algorithm. This paper quantitatively evaluates the attenuation correction methodologies based on differential phase at the X band. The paper is organized as follows. Section 2 briefly describes the range-profiling algorithm for the X-band frequencies, while section 3 establishes the accuracy of the cumulative attenuation estimate from differential phase measurements. Section 3 evaluates the attenuation correction process under different microphysical scenarios. The methodologies studied here are evaluated for realistic range profiles of attenuation and differential phase in section 4. Section 5 summarizes the important results of this paper. 2. Attenuation correction using differential phase and rain-profiling algorithm a. Attenuation correction using differential phase Raindrop size distribution forms the building block for the theoretical description of the various parameters of the rain medium. The raindrop size distribution describes the number density of raindrops with size that can be written in terms of the probability density of the raindrops as N D n c f D D m 3 mm 1, 1 where N(D) is the number of raindrops per unit volume per unit size interval (from D to D D), n c is the concentration, and f D (D) is the probability density function (Chandrasekar and Bringi 1987). Several parametric forms of the drop size distribution (DSD) have been used in the literature, including exponential, lognormal, or gamma. Ulbrich (1983) argued that a gamma form of the DSD with three parameters, namely, the median drop diameter D 0, the scaling constant N 0, and the shape parameter, can adequately describe the observations from in situ sampling devices. Chandrasekar and Bringi (1987) proposed normalizing the DSD with n c (or N T ). Testud et al. (2000) fully developed a normalizing concept that was suggested by Srivastava (1971) and Willis (1984) to scale the variations resulting from the widely varying water contents, so that the inherent DSD shape can be observed. Testud et al. (2000) also showed that the reflectivity versus rainfall rate relation after normalization with N w takes a power-law form providing a physical basis for the Z R relations. The various parameters of the rain medium can be defined in terms of the DSD as follows. The reflectivity at horizontal polarization (Z h ) is defined as Z h 4 5 K w 2 hh D N D dd. 2 Similarly, the specific attenuation at horizontal polarization is given by h ext D N D dd. 3 The attenuation correction procedure, using a differential propagation phase, relates the specific attenuation and specific differential propagation phase. The specific differential propagation phase (K dp ) can be defined in terms of the DSD as K dp 180 Re f h D f v D N D dd deg km 1, where f h, f v are the forward-scattering amplitudes at horizontal and vertical polarization, respectively. Both attenuation correction procedures, namely, the simple dual-polarization attenuation correction procedure based on dp as well as the rain-profiling algorithm with the dp constraint, use the approximation that h a h K dp, where the coefficient a h depends on the behavior of h as well as K dp resulting from various factors, such as temperature, DSD, and the mean shape of raindrops. The variation of a h resulting from temperature comes from changing dielectric properties of water and is well documented in the literature (Jameson 1992). The impact of mean drop shape on K dp is also well understood (Bringi and Chandrasekar 2001). The effect of temperature as well as drop shape on h is essentially to scale the value correspondingly. Though there is a slight variation of a h with DSD, it is common to choose a bulk constant to represent most of the DSDs encountered. 4 5

3 AUGUST 2005 G O R G U C C I A N D C H A N D R A S E K A R 1197 Figure 1 shows the scatterplot of h normalized with N w estimated from each DSD that is shown as a function of D 0. The scatterplot consists of data with 10 3 N w 10 5 m 3 mm 1, D 0 in the range of 0.5 D mm, and in the range of 1 5. The data in Fig. 1 are also constrained by reflectivity in the range of dbz to include only regions where attenuation correction is typically done. In addition, it can be shown that ( h /N w ) varies nearly as D 5 0, though deviations from this approximation can be observed for large D 0. An average estimator of a h is obtained from a scatterplot of h and K dp. Figure 2 shows a scatterplot of h versus K dp for widely varying DSD, as shown in Fig. 1 for a temperature of 20 C. The normalized standard error (NSE; defined as the standard deviation normalized with respect to the mean) of h, estimated from K dp, is 24%. However, the actual error that is encountered will depend on the distribution of D 0 along the precipitation path. It should also be noted that errors of opposite signs would cancel each other out while converting to cumulative attenuation. Here, dp is twice the integral of K dp (r) over range; therefore, cumulative attenuation can be written as r 2 A r 2 A r 1 a h K dp r dr 2 r 1 a h dp r 2 dp r 1, where r 1 and r 2 are two ranges along a radar beam. If r 1 is set to zero, or the range to first radar echo, the estimate of A(r) can be obtained from dp (r). 6 FIG. 2. Scatterplot of h vs K dp for widely varying DSDs, as shown in Fig. 1. b. Rain-profiling algorithm with dp constraint The rain-profiling algorithm with dp constraint was proposed by Testud et al. (2000) as an adaptation of the technique developed for TRMM precipitation radar using surface reference constraints. This procedure was termed ZPHI. A similar technique was also used by Tuttle and Rinehart (1983) for computing specific attenuation profiles using dual-wavelength radar measurements. The measured reflectivity in the presence of attenuation can be expressed as Z m r Z r e 4 0 r K im ds, where K im is the imaginary part of the propagation constant. If the specific attenuation h is written in units of decibels per kilometer, Eq. (7) becomes 7 Z m r Z r e r h s ds Z r A r. Invoking the concept of normalized DSD, a powerlaw relation can be introduced between Z and of the form 8 FIG. 1. Scatterplot of h /N w estimated as a function of D 0. The DSD parameters consist of data with 10 3 N w 10 5 m 3 mm 1, 0.5 D mm, and 1 5. The data are also constrained by reflectivity in the range of dbz. The dashed line represents a function of D 5 0. h a z Z b, where the coefficient b is fixed, and a z varies according to N w of the DSD (Testud et al. 2000; Bringi and Chandrasekar 2001). The solution to the two sets of Eqs. (5) and (9), with the constraint A(r) a h dp (r), is given by Z m r b ba h dp r 0,r m 1 h r I r 0 ; r m ba h dp r 0,r m 1 I r; r m, where 9 10

4 1198 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 22 r m I r 0 ; r m 0.46b Z m r r b dr, 11 0 where r 0 is the starting range of observations, r m is the final range where the dp observation is taken to place constraints on attenuation, and dp (r 0, r m ) is the change in differential phase from r 0 to r m. Equation (10) gives the solution for specific attenuation h at each range in terms of measured reflectivity, the measured dp, and the model coefficients a h and b. The corrected reflectivity profile can be obtained as r r h s ds r Z c r Z m r e 0 12 This algorithm described by (12) is sensitive to the following errors: (a) the error in the conversion from dp to A; (b) the error in the parameter b, which forms the exponent of reflectivity Z in Eq. (9); and (c) errors induced because of measurement fluctuations. The rain-profiling algorithm with dp constraint has been evaluated at the C band by Le Bouar et al. (2001) and Bringi et al. (2001) using data from the C-band dualpolarization Doppler radar (C-Pol), operated by the Bureau of Meteorology Research Centre of Australia. The coefficient b used by Le Bouar et al. (2001) is between 0.8 and 0.82 at the C band for rainfall at temperatures above 10 C, whereas Bringi et al. (2001) used a value of 0.8. The parameters involved in attenuation correction, such as a h and b, are relatively insensitive to DSD. A small variability still exists, however, and the performance of the algorithm depends on the prevailing DSD along the path. In practice, fixed mean values of a h and b are used. The residual variation of this mean estimate with respect to the optimum value for the prevailing DSD will result in some error in the attenuation correction algorithm. The impact of these three error sources is studied in the following sections. The value of b was estimated at the X band (using simulations similar to those in Bringi and Chandrasekar 2001), corresponding to the shape model of Andsager et al. (1999), as (for comparison, the same value at the C band is 0.8). 3. Evaluation of cumulative attenuation correction The estimate of cumulative attenuation to range r from the radar can be estimated in two ways. In the first method the dp up to that range is used to estimate the two-way path attenuation A(r) as A r a h dp r. 13 This simple procedure is, henceforth, referred as the differential phase (DP) method. The second way to estimate cumulative attenuation over a larger path, say r max, is to use the dp -based method to estimate the cumulative attenuation at r max, then construct the range cumulative attenuation profile for all ranges less than r max, using (10). This method is, henceforth, referred as the differential phase constraint (DPC) method (or ZPHI) based on the use of the dp constraint. It should be noted here that each procedure has unique features that impact their accuracy. The accuracy of the dp -based technique is fairly straightforward because it depends on the accuracy of (13), which comes from the accuracy of the parameterization in (5). However, as the pathlength increases, positive and negative errors in the parameterization of (5) could cancel each other out, and the cumulative attenuation estimate from dp is fairly accurate. Therefore, if the cumulative attenuation at a far-off range is used to constrain the attenuation profile, then the accuracy of the DPC algorithm primarily depends on the parameterization of h with Z. This parameterization is fairly accurate if N w is constant along the path. The following three hypothetical profiles are considered to evaluate and illustrate the comparison of the two methods. a. Case I: Uniform DSD path The following shows simulations of rain media of uniform DSD, namely, constant N w, D 0, and for 101 range bins, with the range spacing of 150 m. For each rainfall path, range profiles of Z h, K dp, h, and dp are computed. Subsequently, attenuated reflectivity profile Z m (r) and dp (r) are used to develop attenuationcorrected profiles of reflectivity. This process is repeated for a large number of paths, varying the DSD parameters in each path. The dp constraint is based on the observation at range bin 101. To understand the attenuation correction by the two methods, the cumulative attenuation profiles are plotted with range. The DSD parameters are uniform along the path. Therefore, the specific attenuation should be constant along the path, and the cumulative attenuation is a linear function of range. The dp -based specific attenuation estimate is also constant along the path, and the corresponding cumulative attenuation estimate is a linear function of range. The dp -based cumulative attenuation profile is based on the parameterization of a h using (5). The estimate of a h using (5) can be either larger or smaller than true a h, as shown in Figs. 3a,b. Figures 3a,b show the cumulative attenuation profiles for two examples when a h, estimated from (5), is larger and smaller than the true a h, respectively. The cumulative attenuation profile from the DPC algorithm is also shown in Figs. 3a,b. Two observations can be made about the profiles of cumulative attenuation from the DPC method, namely, (i) the profiles of cumulative attenuation are always nonlinear, whereas the true pro-

5 AUGUST 2005 G O R G U C C I A N D C H A N D R A S E K A R 1199 FIG. 4. The NSE and the normalized bias (NB) of the estimates of (a) A h and (b) h obtained from both the DP and DPC methods computed over paths with uniform DSD. FIG. 3. Cumulative attenuation profiles obtained from the differential phase (A DP ) using (5) and from the true value (A h ) for two examples that are (a) larger and (b) smaller than the true a h, respectively. The profiles are taken assuming a uniform DSD path. Cumulative attenuation profile from the DPC algorithm is also shown. file is linear; and (ii) the cumulative attenuation profile from DPC lies in between the true profile and that of the DP-based estimate for a constant DSD path (this result can be shown analytically). The error structure resulting from parameterization will be a direct result of this feature, as shown in the following. Figures 4a,b show the normalized standard error of the estimates h and A h obtained from both the DP and the DPC algorithms. The error of the cumulative attenuation at the last range is identical in both algorithms, as expected. However, the error in the cumulative attenuation estimate is less at shorter ranges for the DPC algorithm, whereas it is constant for the DP-based algorithm. The error structure of specific attenuation estimates is slightly more complicated, because it is a derivative of A(r). Figure 4b shows the NSE of h from both algorithms. The error in the DPC algorithm increases with range, whereas the DP-based technique is constant, about 20%. The error of h from the DPC algorithm is negligible at short ranges, whereas it is fairly high near the end of the path because of errors adding up at the last range bin. It should be noted here that the errors discussed herein are primarily because of the nature of the algorithms and do not include the effect of measurement error in dp or Z h. In addition to the errors, the biases in the two algorithms for the estimates of A(r) and h (r) are also shown. It can be seen that the biases are negligible for both algorithms b. Case II: Rain path with uniform N w The physical basis of the DPC algorithm is the concept of normalized DSD, and it should work the best when N w is constant along the path (Testud et al. 2000). This section evaluates both the estimates of h and A h for rainfall paths with constant N w, but the rest of the DSD parameters change randomly along the path. It

6 1200 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 22 The generic nature of the error structure is similar to the constant DSD profile, while the values are very different. The specific attenuation estimates can be used for potential rainfall rate estimates, whereas the cumulative attenuation estimates are used for correcting the reflectivity, which, in turn, can be used for a variety of applications. In terms of the impact on correcting reflectivity for attenuation, the absolute error (in attenuation correction) is more important for quantitative applications (than the percentage error); for example, a 10% error in cumulative attenuation will result in a 2-dB error in reflectivity if the cumulative attenuation is 20 db. However, the same 2-dB error will result when the cumulative attenuation is 10 db, with a 20% error in attenuation correction. FIG. 5. The NSE of the (a) cumulative attenuation estimate and (b) specific attenuation estimate, from the DP and DPC methods, as a function of range and computed over rain paths with uniform N w. should be noted that the DPC algorithm does not need any modification for application here. Figures 5a,b show the average NSE (averaged over a range of N w ) of both the cumulative and specific attenuation estimate from both the DP-based method and the DPC algorithm. Figure 5a shows the errors in the estimate of A(r) whereas Fig. 5b shows that of h (r). It can be seen from Fig. 5a that at the last range bin both algorithms have the same error as expected (because attenuation estimates are the same). However, at closer ranges the DPC algorithm has significantly less error than the DP algorithm. The error of the specific attenuation estimates of the two algorithms is shown in Fig. 5b. The average error in the specific attenuation estimate from DP is of the order of 27%; the error of h from DPC varies between 25% and 45%, with the error increasing with range. c. Case III: Rain path with randomly varying DSD Having a rain path with randomly varying DSD parameters is the most challenging for the DPC algorithm because the underlying methodology is based on a fixed N w. This large error can be mitigated by applying the DPC algorithm over short paths. Nevertheless, studying this scenario provides an idea of the upper limit of the error resulting from widely varying DSD parameters along the path. Similar to case I and case II, the NSE in cumulative and specific attenuation are plotted as a function of range in Figs. 6a,b. As expected, the error in the DPC algorithm is large compared to the DP-based algorithm for both cumulative attenuation and specific attenuation, as seen in Figs. 6a,b, respectively. Figure 6a shows that the cumulative attenuation estimates of both the DP and DPC algorithms converge at the farthest range, while at the shortest range the DPC algorithm is worse. The purpose of this exercise is to place an upper limit on the error structure for varying DSD parameters along the path rather than coming up with possible error values. 4. Analysis of realistic range profiles The detailed analysis of the attenuation correction at the X band presented so far suggests that the accuracy of the cumulative attenuation correction, as well as the specific attenuation estimate, are dependent on the distribution of the DSD along the path. In addition, the impact of the backscatter differential phase ( ) isessentially dependent on the variation of along the range, as well as in the last range bin. In a rangecumulative sense the errors of opposite signs can cancel each other out. One way to test the performance of the algorithms discussed here is to simulate X-band radar observations for realistic rain profiles. This can be accomplished by simulating observations at the X band

7 AUGUST 2005 G O R G U C C I A N D C H A N D R A S E K A R 1201 FIG. 6. The NSE of the (a) cumulative attenuation estimate and (b) specific attenuation estimate, from both the dp -based method and rain-profiling algorithm, as a function of range and computed over rain paths with randomly varying DSD. based on high-resolution dual-polarization radar observations at the S band. The appendix provides a brief description of the procedure to simulate X-band observations from S-band data (Chandrasekar et al. 2004). Using the procedure outlined in the appendix, about 120 range profiles of X-band observations are simulated. They are simulated from S-band observations where the attenuation is negligible. The data are simulated from observed radar data (at the S band); therefore, in addition to the realistic distributions of DSDs encountered along the path, many practical radar measurements that are related nuances are included. The S-band radar data used in the analysis were collected by the National Center for Atmospheric Research (NCAR) S-band dual-polarization Doppler radar (S-POL) during the Texas and Florida Underflight (TEFLUN) Experiment during August September The test data were obtained over range profiles where radar echo existed over at least 16 km (or 101 range bins). To choose cases that had sufficient attenuation, only profiles that had at least 10 of increase in dp at the S band were chosen. Figure 7 shows a typical range profile of reflectivity and attenuation at the X band. The three attenuation profiles correspond to the actual attenuation (simulated), as well as estimates from the DPC and DP algorithms. It can be seen that at the last range, both the DPC and DP profiles of A match, as expected. Using about 120 profiles, the following analysis is conducted to study the performance of the attenuation correction algorithm. The performance of the two specific attenuation estimates DP and DPC are now evaluated as a function of range. The bias and standard error of the two estimates are plotted as a function of range. Figure 8a shows the NSE versus range, whereas Fig. 8b shows the normalized bias versus range. This evaluation is done essentially to compare the error structure on realistic data profiles that fall between these models of Figs. 5b and 6b; namely, a constant N w profile needed for the best performance of the DPC algorithm and randomly varying N w along the path. The standard error varies between 25% and 80% for the DPC algorithm, whereas it is between 20% and 50% for the DP algorithm, which are in the ranges projected from theoretical simulations. The normalized bias for the two algorithms are shown in Fig. 8b, where it indicates that it is essentially very small, except for very short ranges (under 1 km). The specific attenuation estimates could be potentially used for rainfall estimation, whereas the cumulative attenuation estimates that are needed for attenuation correction are more robust. FIG. 7. Typical range profile of (left) reflectivity and (right) attenuation at the X band. The three attenuation profiles correspond to the actual attenuation (A h ), as well as estimates from the DPC and DP algorithms.

8 1202 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 22 FIG. 8. (a) Normalized standard error and (b) normalized bias of specific attenuation, relative to X-band profiles based on S- band observations, as a function of range using both the DPC and DP algorithms. One of the disadvantages of the DP method with a realistic range profile is the estimation of K dp. Because K dp is estimated as the slope of the range profile of dp, it is likely to have large errors. Any error in dp that is used for DP-based attenuation correction will also affect the DPC algorithm; however, the error in dp is fairly small and is constant, independent of the pathlength. This section evaluates the two techniques in the presence of measurements errors. The main advantage of the DPC algorithm is that there is no need to estimate range derivatives to estimate the cumulative attenuation. The measurement fluctuations were added to the range samples of Z h, Z dr, and dp, utilizing the procedure described in Chandrasekar et al. (1986). The copolar correlation and the Doppler spectral width were fixed at 0.99 and 2.5 m s 1, respectively. Though these could have been changed according to the observed values for the purpose of the objective of this section, it was not necessary. Using the range samples of Z h, Z dr, and dp with measurement error, both the DPC and DP algorithms were applied. Along the path, K dp was computed as the slope of the linear regression fits of dp profiles over a 3-km window. The standard error of h estimated from K dp over a 3-km path is less than 0.4 db km 1. Figure 9a shows a scatterplot of the specific attenuation h estimated from the DPC algorithm versus true h. The scatter in Fig. 9a has a correlation coefficient of 0.96 and a negligible bias. Figure 9b shows the similar scatter for the DP method. It is obvious that K dp that is estimated over a 3-km path has smoothed over the large value of h within the path. Therefore, the h that is estimated from the DP algorithm does not have large K dp ; the result was also reported by Gojara and Chandrasekar (2002). The data shown in Fig. 9b have a correlation of 0.88 and are biased for high values of h. The cumulative attenuation estimate does not have the same problem as specific attenuation. Both estimates are based on dp. Figure 10a shows the normalized standard error in the estimates of cumulative attenuation for both algorithms, plotted as a function of range. It can be seen that for ranges less than 2 km, the attenuation correction is not very accurate on a percentage basis. The main reason for this is that the cumulative attenuation values are small for ranges less than 2 km. Attenuation correction based on the measurements of dp, using both DPC and DP techniques, are comparable; nevertheless, the DPC method is mostly better. The results of Fig. 10a show the behavior of cumulative attenuation as a function of range without any consideration for how much attenuation is there. Three important observations from this analysis are in order. First, the DPC and DP algorithms have a similar error structure for correcting the path attenuation, while the DPC algorithm is slightly better. The DPC algorithm uses the measured reflectivity, and all of the analysis in this paper assumed that there is no absolute calibration error in the radar, because the DPC algorithm is immune to absolute calibration errors ( dp is not affected by attenuation). The application of the DPC algorithm was limited to a path with at least 10 of dp. The results of Fig. 10a are obtained without including the effect of. Figure 10b shows results that are similar to that of Fig. 10a, except it includes the impact of. It is clear from the results of Fig. 10b that the bias and the standard error of attenuation correction slightly increase because of. It is also obvious from Figs. 10a,b that the impact of on the DP method is more than that

9 AUGUST 2005 G O R G U C C I A N D C H A N D R A S E K A R 1203 FIG. 10. (a) (left) Normalized standard error and (right) normalized bias of cumulative attenuation, relative to X-band profiles based on S-band observations, as a function of range. The estimation of attenuation has been obtained using the DPC and DP algorithms in the presence of measurement errors. (b) Same as (a), except the effect of is included. FIG. 9. Scatterplot of specific attenuation estimates obtained from 120 profiles of simulated X-band data based on S-band observations. The estimates have been obtained using the (a) DPC and (b) DP algorithms in the presence of measurement errors. of the DPC method. This is obvious because the DPC method uses only dp from the last range, whereas the DP method uses dp from all ranges. The actual behavior of the error structure can be easily explained from a study of the profile of. The bias and standard error of the attenuation correction of the DP technique increased the most at ranges of 3 km and beyond 10 km. An analysis of the range profiles of showed that at these ranges the mean values and the variation of were the highest. The accuracy of the specific attenuation estimate from the DP technique is determined directly from the pathlength over which the estimates are made. Longer paths result in suppressing the peak values (averaging over a window), whereas shorter paths enhance the measurement error. Further discussion on this topic can be found in Gorgucci et al. (2000). The accuracy of the specific attenuation estimate from DPC for realistic profiles is of the order of 49%. The use of the cumulative attenuation estimate is in the correction of reflectivity. Because the error in the cumulative attenuation estimate increases with range, the probability of keeping the error in the estimate of corrected reflectivity within 1 db is also less. This is illustrated in Fig. 11a, where the percentage of corrected reflectivity remaining within 1 db of error is plotted as a function of range for both the DPC and DP algorithms without considering the impact of. The

10 1204 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 22 on the occurrence of along the path. Raising the error allowance to 1.5 db ensured that 90% of the corrected reflectivity from the DP algorithm stayed within this bound. In general, it appears that the impact of on the DP algorithm is, on average, to deteriorate the attenuation correction accuracy by another 0.5 db. FIG. 11. (a) Occurrence frequency of corrected reflectivity remaining with 1 db of error respect to the corresponding unattenuated X-band data based on S-band observation as a function of range for both the DPC and DP algorithms in the presence of measurement errors. (b) Same as (a), except the effect of is included. Also shown is the frequency of occurrence for reflectivity remaining within 1.5 db. results of Fig. 11a show that, for both algorithms, as the length of the precipitation path increases, the percentage of the corrected data that stay within 1 db decreases. In addition, the DPC algorithm performs slightly better than the DP algorithm. Similar results are shown considering the impact of in Fig. 11b. Figure 11b shows the percentage of time that the difference between the corrected reflectivity is within 1 and 1.5 db of the true value for both algorithms. It can be seen that for the precipitation path evaluated herein, about 80% of the attenuation-corrected reflectivity stayed within 1 db of error for the DPC method. However 90% of the reflectivity stayed within 1 dbof error for intense rain paths 10 km long. The DP algorithm, however, had a variable performance depending 5. Summary and conclusions The monitoring of precipitation using X-band radars is becoming increasingly popular because of the relatively low cost of these systems. However, at X-band frequencies precipitation-induced attenuation is very significant, and attenuation correction is required for any quantitative application of radar data, such as feature detection algorithms or rainfall estimation. Attenuation correction using reflectivity measurements alone, done iteratively from ranges close to the radars, were very unstable. Coherent polarization diversity radars brought significant advancements for attenuation correction in rain through the use of differential propagation phase. This ability has accelerated the consideration of X-band radars for rainfall monitoring. Currently two methodologies are available for attenuation correction, namely, (a) simple differential phase based attenuation correction, and (b) attenuation correction using differential phase constraint. The error structures of these two algorithms were analyzed in detail in this paper. In addition to algorithm errors and measurement fluctuations, the impact of backscatter differential phase was also studied. The differential phase attenuation correction procedure (DP) is fairly simple to implement. The DPC method is not as simple; nevertheless, it is easy to implement in modern processors. Regarding the correction of cumulative attenuation, both algorithms provide similar performance, with the differential phase constraint technique performing slightly better. The DPC algorithm could keep the corrected reflectivity within 1 db 80% of the time for the intense rain cells studied, while the DP algorithm is slightly worse. In the absence of measurement errors, the DP algorithm can provide better estimates of the specific attenuation estimates compared to the DPC method. However, practical issues that are associated with slope estimation reduce the resolution of the specific attenuation estimates, and suppress peaks. The backscatter differential phase is not negligible at the X band, and its impact is directly seen on attenuation correction algorithms. While the attenuation correction using the DPC procedure is affected by the backscatter differential phase only at the last range, the DP procedure is affected throughout. The effective impact of on the DP algorithm is a slight deterioration of the average standard

11 AUGUST 2005 G O R G U C C I A N D C H A N D R A S E K A R 1205 FIG. A1. (a) Range profile of reflectivity and differential reflectivity from S-band polarimetric radar collected over central Florida. (b) Simulated range profile of reflectivity at the X band (solid line) and the corresponding attenuated profile (dashed line). (c) Simulated range profile of differential reflectivity at the X band (solid line) and the corresponding attenuated profile (dashed line). (d) Differential phase profile at the S band (solid line) and the corresponding profile at the X band (dashed line). error of attenuation correction, depending on the variation of the backscatter differential phase along the rain path. In summary, the quantitative analysis shows that attenuation-corrected reflectivity can be estimated from X-band dual-polarization radars accurately most of the time. Acknowledgments. This research was supported partially by the National Group for Defense from Hydrological Hazard (CNR, Italy), by the Italian Space Agency (ASI), and by the National Science Foundation (ERC and ATM ). APPENDIX Methodology to Simulate Range Profiles of X-Band Radar from S-Band Rain Observations Simultaneous observations of rain by dualpolarization radars at the S and X bands were discussed by Chandrasekar et al. (2002). Their study established the relation between the intrinsic dual-polarization radar observations (in the absence of propagation effects) at the S and X bands. Simple scaling principles can be used to translate the observations between the S and X bands. For example, at the Rayleigh approximation the reflectivity factor will not change with frequency. However, at the X band this assumption is strictly not valid. Similarly, Z dr values differ slightly because of non- Rayleigh scattering. The differential propagation phase, however, scales directly with frequency (valid up to 13 GHz). The simple argument discussed above can be made more sophisticated with the underlying DSD structure. It was shown by Scarchilli et al. (1996) that the triplet of measurements Z h, Z dr, and K dp nearly lie on a threedimensional surface. Therefore, once Z h and Z dr are specified, the choice of a possible K dp falls in a narrow range. As a result, if we choose a value of K dp corresponding to Z h and Z dr, then the K dp value at the X band can be obtained by direct frequency scaling. This procedure to choose K dp avoids the problem of direct K dp estimation. The above technique is implemented in a detailed manner, as explained in the following.

12 1206 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 22 A large dataset of Z h, Z dr, and K dp values are generated, corresponding to a wide range of DSD parameters, 3 log 10 N w 5 m 3 mm 1, 0.5 D mm, A1 1 5, under the constraints of Z h 55 db and R 300 mm h 1. For a given set of Z h and Z dr, a search of this database provides possible choices of DSD that satisfy the observations. One of those DSDs is randomly chosen to compute the X-band observation. Because it is structured on the DSD, the specific attenuation can also be computed to obtain the observed reflectivity as r s ds Z m r Z r e A2 The same simulation structure is also used to construct the phase shift on backscatter, and the observed dp gets modified by. Simulation example Figure A1a shows the range profile of reflectivity and differential reflectivity from the S-band polarimetric radar observed over central Florida. The simulated X- band profiles of Z m h (r), Z h (r), as well as Z m dr(r), Z dr (r), are shown in Figs. A1b,c whereas the profiles of dp at the S and X bands are shown in Fig. A1d. A cursory glance of Fig. A1 shows that the simulation procedure produces reasonable range profiles of X-band radar observations. Once again, the purpose of this simulation is to simulate realistic range profiles of X-band dualpolarization observations in order to maintain the spatial correlation structure of the DSD from naturally occurring rainfall. It should be noted that our procedure is not to simulate the exact observation, but to simulate profiles that fall within the possible range of observations. REFERENCES Andsager, K., K. V. Beard, and N. F. Laird, 1999: Laboratory measurements of axis ratios for large raindrops. J. Atmos. Sci., 56, Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 648 pp.,, N. 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