Sensitivity of multiparameter radar rainfall algorithms

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. D2, PAGES , JANUARY 27, 2000 Sensitivity of multiparameter radar rainfall algorithms Eugenio Gorgucci and Gianfranco Scarchilli Istituto di Fisica dell' Atmosfera, Consiglio Nazionale delle Ricerche, Rome V. Chandrasekar Department of Electrical Engineering, Colorado State University, Fort Collins Abstract. The most commonly used polarimetric radar measurements in rainfall estimation are reflectivity at one polarization (Zr ), differential reflectivity (ZDR), and specific differential propagation phase (KDe). The reflectivity measurement requires knowledge of absolute power and is prone to calibration errors. Differential reflectivity is a relative power measurement and is not affected by absolute calibration errors. However, all the algorithms that use differential reflectivity also include reflectivity and therefore are prone to absolute calibration errors. Algorithms to estimate rainfall from specific differential propagation phase are immune to absolute calibration error and attenuation effects [Zrnic and Ryzhkov, 1996]. However, specific differential propagation phase is very noisy at low rain rates. In addition, specific differential propagation phase is estimated as the slope of differential propagation phase measurements over a path. Consequently, there is a trade-off between accuracy and resolution of KDe. Thus there are advantages and disadvantages of each multiparameter radar measurementhat translate into the error structure of algorithms involving multiparameter radar measurements. This paper presents a quantitative evaluation of the performance of five different polarimetric radar rainfall algorithms. The performance of the five algorithms is evaluated in the presence of physical variability in rainfall, radar measurement errors, and systematic calibration error. 1. Introduction The use of polarization diversity radar measurements to estimate rainfall is a topic of active research. The most commonly used polarimetric measurements in rainfall estimation are differential reflectivity, specific differential propagation phase, and reflectivity. Each of the above mentioned three parameters have their advantages and disadvantages. First, reflectivity factor (say, at horizontal polarization) Z r is the most commonly used measurement. However there are numerous reflectivity-based algorithms, and these algorithms always change with geographicalocation, storm type, season, etc. In addition, the reflectivity-based algorithms require knowledge of absolute power and are prone to calibration errors. Differential reflectivity (ZoR) is a relative power measurement, and it can be measured accurately without being affected by absolute calibration errors. However, all algorithms that use ZoR include reflectivity also and are prone to absolute calibration errors. Algorithms to estimate rainfall from specific differential propagation phase (Koe) have several advantages, such as they are immune to absolute calibration error and attenuation and are not significantly biased in the presence of hail contamination in the radar resolution volume. However, Koe measurement is noisy at low rain rates. Koe is typically estimated as the slope of differential propagation phase, and it needs to be estimated over a path. Consequently, there is a trade-off between accuracy and resolution of Koe. Thus there are advantages and disadvantages of each polarimetric measurement that translate into the error structure of algorithms Copyright 2000 by the American Geophysical Union. Paper number 1999JD /00/1999JD involving polarization diversity measurements. This paper presents a quantitative evaluation of the performance of five different polarimetric radar rainfall algorithms. The performance of the algorithms is evaluated in the presence of variability in rainfall, radar measurement errors, and systematic bias. Our paper is organized with the following structure. Section 2 describes the various multiparameter rainfall algorithms evaluated in this paper. Section 3 compares the accuracy of the various rainfall algorithms in the presence of variability in rainfall rate and the absence of radar system measurement errors and bias errors. In section 4 we compare the same algorithms with the measurement errors. Section 5 compares the performance of the radar rainfall algorithms when system biases are present. Section 6 summarizes the important results of the paper. 2. Radar Rainfall Algorithms tammop aiia a are of C½11 importance in determining the electromagnetic scattering properties of rain-filled media. These effects, in turn, are embodied in the radar parameters of interest, such as refiectivi factors (Z,, ) at H and C polarization states, differential refiectivi (Zoo), which is the ratio of refiectivities at the o polarization states [Seliga and B ngi, 1976], and specific differential phase (Koe), which is due to the propagation phase difference be een the o polarizations [Seliga and BNngi, 1978]. Both cloud models and measurements of raindrop size distribution (RSD) at the surface and aloft show that a gamma distribution model adequately describes many of the natural variations in the RSD [Ulb ch, 1983]: N(D) = NoD e - ø (m-3 mm - ) (1)

2 2216 GORGUCCI ET AL.: MULTIPARAMETER RADAR RAINFALL ALGORITHMS where N(D) is the number of the raindrops per unit volume troduced in the literature for estimation of rainfall using radar per unit size interval (D to D + AD), and No, A, and/z are measurements from a polarization diversity radar operating parameters of the gamma distribution. A physically meaningful linear polarization basis. parameter Do, known as the median volume diameter, can be In this paper we concentrate on the algorithms that have defined as been used extensively in the literature. The algorithms can be broadly classified into four categories, namely, (1) reflectivity- D3N(D) dd: D3N(D) dd (2) based algorithms; (2) algorithms that use reflectivity (ZH) and differential reflectivity (Zz, ): R(ZH, Zi R); (3) algorithms that use differential propagation phase (Kz,): R(Kz,); and Under this definition it can be shown that parameter A in (1) (4) algorithms that use differential propagation phase and difis related to/z and D O by ferential reflectivity: R (Kz v, Zz R). Gotgucci et al. [1995] showed that at S band the algorithm A = ( tz)/do (3) R(ZH, ZDR ) x 10-2Z 9110-ø.4ø3zoR (9) The equilibrium shape of a raindrop falling at its terminal fall speed is determined by the balance between the forces due to performs very well over a wide range of measurements. Sachisurface tension, hydrostatic pressure, and aerodynamic pres- danand and Zmic [1987] introduced the Kz v based rainfall sure from airflow around the drop. The shapes of raindrops estimate as have been studied theoretically by Green [1975] and Beard and R (K p) = 40.5 Chuang [1987], experimentally in wind tunnels by Pruppacher.- v qk 0'866 (10) and Pitter [1971], and in natural rainfall using aircraft probes by Scarchilli et al. [1993] introduced an estimate of rainfall rate Chandrasekar et al. [1988]. All of the above studies, as well as linearly related to Kz v; this algorithm for the S band is given polarimetric radar measurements at multiple polarizations as [Gotgucci and Scarchilli, 1997] [Holt et al., 1997], show that the shape of a raindrop can be approximated by an oblate spheroid with the axis ratio (b/a) of the drop approximated by the relationship R2(K p) = 39.8K p (11) b/a = De (4) Subsequently, Ryzhkov and Zmic [1995] introduced a rainfall algorithm that used both Kz p and Zz, given by where D e is the equivolumetric spherical diameter of a raindrop in millimeters, and a and b are the major and minor axes R i(kd?, ZDR) = 52. (I/( 0'967-0'447,-,xxDp Z-,DR (12) of the drop, respectively. Rainfall rate (R) and the radar parameters of the rain medium, namely, ZH, V, ZDR, and Kz p, can be expressed in terms of the RSD as follows: Gorgucci and Scarchilli [1997] derived a rainfall algorithm based on K v and Zv, making the form of the function robust in the presence of measurement errors given by R = 0.6z- x 10-3 f D3N(D)v(D) dd (mm h - ) (5) where v(d) is the fall speed of raindrop, which can be approximated as v(d) = CvDø'67; I ZH, V = rr i/c[2 rrh, v(d)n(d) dd (mm 6 m -3) (6) R2(gop, ZDR ) : / ø'9541 e 0-0'123Zø/ (13) In addition to the above algorithms, two commonly used Z-R relations are the Marshall and Palmer relation and the WSR- 88D algorithm given by R (ZH) = (3.65 x 10-2)Z 625 g2(z,) = (].70 x ]0-2)1 TM where O'H, V represents the radar cross sections at horizontal and vertical polarizations, respectively,, represents the wave- 3. Comparison of the Algorithms in the Presence length, and k = (er - 1)/(er + 2) where e is the dielectric of RSD Variability constant of water; In this section we compare the five polarization diversity algorithms, in conjunction with the Z-R relations in the absence of measurement errors. We simulate the variability in ern(d) N(D) dd the drop size distribution to study the performance of the Z = (7) algorithms. The polarimetric algorithms (9)-(13) were obtained by minimizing the sensitivity to RSD variability. The f rrv(d ) N (D) dd comparison of the algorithms is done as follows. Rainfall values, ranging from 0 to 300 mm h-1, are simulated varying the parameters of the gamma RSD over a wide range as suggested Ko, = m Ifil(D) -- f (D)]N(D) dd (deg km - ) 180A f by Ulbrich [1983]. For each RSD the corresponding ZH, Zz, (8) and Kz e are evaluated using (6)-(8). Subsequently, the simulated ZH, Zz, and Kz e are used in algorithms (9)-(14) for where ffi refers to real part of a complex number and fh and fv comparison against the rainfall rate corresponding to the RSD. are the forward scatter amplitudes at H and V polarization, The fractional standard error (FSE) is used as the figure of respectively. The radar measurements used in the dual- merit to compare the performance of the algorithms. The polarization estimates of rainfall rate are Z H (mm 6 m-3), fractional standard error is the root mean square error nor- (db), and Kz p (deg km-1). Several algorithms have been in- malized with the mean rainfall value, defined as (14a) (14b)

3 GORGUCCI ET AL.' MULTIPARAMETER RADAR RAINFALL ALGORITHMS 2217 = 13 )- i=1 }1/2 1 M M Ri i=1 where M is the number of estimates, Ri( ) is the ith rainfall estimate using algorithms (9)-(14), and R i is the corresponding rainfail rate. Figure 1 shows the FSE of R i(zh), R2(ZH), R(Z q, ZDR), Ri(KDv), R2(K v), Ri(K v, Z R), and R2(KDp, ZDR ). We can see from the results of Figure 1 that Z-R algorithms have the worst performance and R(Kov, ZD, ) has the best overall performance. R(Zn, Zo ) and R(Kov, Zo ) algorithms have nearly the same error structure, whereas the performance of R(Kov) is between that of R(Zn) and those of R(Zn, ZD ) and R(Kov, ZoR). Other than R(ZH), all other algorithms try to account for variability in drop size distribution to estimate rainfall. The error in Figure 1 shows the residual sensitivity to RSD variability. Theoretically, this residual sensitivity can be made arbitrarily small by measuring more parameters. However, in practice each new measurement comes with its own errors and associated problems. The rest of the sections provide a quantitative evaluation of all random and systematic errors of each measurement and the impact on each rainfall algorithm considered here. 4. Comparison of Rainfall Algorithms in the Presence of Measurement Errors The three measurements Z, Zr,, and Kr v have completely different error structures. Z is based on absolute power measurement and has a typical accuracy of 1 db. Zr, is a relative power measurement and is the differential power estimate between Z and Zv-. It can be estimated to an accuracy of db. Kr v is the slope of the range profile of differential propagation phase r v, which can be estimated to an accuracy of 2 ø or 3 ø. The subsequent estimate of Kr v depends on the type of procedure used, such as a simple finite difference scheme or a least squares fit. A simple finite difference results in Kr v whose standard deviation can be expressed as 1 rr(km,)= x/ L (16) where rr(kr v) is the standard deviation in the estimate of Kr v, rr( r v) is the standard deviation in the estimate of r v, and L is the path length over which the Kr v value is estimated. The accuracy of K v given by a simple finite difference is not sufficient. A better way is to estimate the slope of r v using a least squares fit to the r v profile. Using a least squares estimate of the r v profile, the standard deviation of Kr v can be expressed as = x5 NAr X/' (N- 1)(N + 1) where Ar is the range resolution of the r v estimate and N is the number of range samples within the path. For large N we can see that rr(kr v) decreases N 3/2. For a typical 150 m range spacing, and 2.5 ø accuracy of r v, Kr v can be estimated, over a path of 2.1 km, with a standard error of 0.55 ø km -1. Thus the three measurements, Z, Zr,, and Kr v have completely different error structure. In addition, the measure Rainfall Rate (rnrn h 4) Figure 1. Fractional standard error, due to raindrop size distribution variability, for different radar rainfall estimates as a function of the rainfall rate. ment errors of Z H, Z D, and Kov are nearly independent. Therefore the three measurements, when used in rainfall algorithms, result in a very different error structure of the rainfall algorithms. In the following, we use simulations to quantify the error structure of rainfall algorithms discussed in section 2. The simulation is done as follows. Various rainfall values are sim- ulated varying the parameters of the gamma RSD over a wide range of values, as suggested by Ulbrich [1983]. For each RSD the corresponding Z, ZoR, and Kov are evaluated using (6)-(8). The random measurement errors are simulated using the procedure described by Chandrasekar et al. [1986]. The principal parameters of our simulation are as follows: (1) wavelength X = 10 cm, (2) sampling time PRT = 1 ms, (3) number of samples pairs M - 64, (4) Doppler velocity spectrum width tr, - 2 m s -1, (5) cross-correlation between H and V signals Pnv = 9, (6) range sample spacing of the path over which IkDp lb btllllktt L[, AF -- ' 1..) co n-l, dill[ ^-',/,) /'-t\ lxdp r..", c tilllat,u...,;...-..,.-.a as,g. -. us, slope of the line obtained from a least squares fit to the ci), profile. The following analysis compares the performance of the polarimetric radar rainfall algorithms under ideal conditions of uniform rainfall medium in the radar resolution cell. This assumption is particularly not valid when we integrate over long paths. Nevertheless, this procedure provides a basis for relative comparison of the algorithms under ideal conditions. Figure 2 shows the FSE of the rainfall estimate obtained over 15 range gates of 150 m spacing (2.25 km) versus the rainfall rate. The data in Figure 2 were obtained simulating uniform rainfall over the 2.25 km path, in the presence of measurement errors. The various curves shown are the FSE corresponding to the algorithms, namely, (1) R 1 (Zn), (2) n,.(z,,), (3)n(z,,, z,,,,), (6)

4 2218 GORGUCCI ET AL.: MULTIPARAMETER RADAR RAINFALL ALGORITHMS 1.00 l ' ' 'i'i I I I I... I....'75.25 ' 'd, / Rz(Zu) various algorithms computed for each region. In this paper we define light, moderate, and heavy rainfall as follows: (1) light rainfall, 0 < R _< 10 mm h- ; (2) moderate rainfall, 10 < R _< 90 mm h- ; and (3) heavy rainfall, R > 90 mm h -. The above definitions correspond to instantaneous rainfall rates along the measurement paths. Figure 6 shows the comparison of the FSE of various algorithms in light, moderate, and heavy rainfall, in the form of a bar chart. The result shown in Figure 6 correspond to range averaging of 20 bins (3 km). We can see from Figure 6 that in light rain if KDV is estimated over 3 km paths, then KDv-based algorithms cannot be used to estimate R. However, KDv-based algorithms may be improved in light rain if the path over which KDV is estimated is increased. The choice is between either R(ZH) or R(ZH, ZDR ). In moderate rainfall all algorithms have comparable performance, with R (ZH, Zo ) performing the best among them. In heavy rainfall the algorithms R(ZH, Zoo), R(Kov), and R(Kov, Zoo) all perform very well. Thus in the absence of any bias errors in ZH and Zo, R(ZH, ZDR ) performs uniformly well at all rain rates. O Rainfall Rate (mm h' 9 Figure 2. Fractional standard error of different radar rainfall estimates as a function of the rainfall rate. The estimates are computed over a path length of 2.25 km. R (KDv, ZDR), and (7) R2(KD,, ZDR ). We can see from the results that all KDv-based algorithms have large error at small rainfall rates and R(ZH, Z R) is better than any other algorithm at all rainfall rates. Figure 3 shows similar results, excepthe estimates of rainfall are averaged over 50 range gates, corresponding to a path of 7.5 km. A comparison of Figures 2 and 3 show significant improvement in all algorithms using KDe. In addition, averaging over 50 range gates, R (KDv, ZDR) is the best algorithm for rainfall rates higher than 100 mm h -. Once again, we note here that the above observations are made for a uniform rainfall path. It is obvious from the results of Figures 2 and 3 that the FSE of rainfall is dependent on rainfall rate as well as the path length over which the estimates are averaged. To study this, we have repeated our analysis for different numbers of range gates of averaging between 5 to 50 in steps of 5, and the results are shown in Figure 4. Figure 4a shows the FSE contours for R (KDv), and Figure 4b shows the same for R2(KDv ). Similarly, Figures 5a and 5b show contours of R (KDp, ZDR ) and R2(KDp, ZDR ). Figures 4 and 5 show FSE for the KDp estimate as a function of rainfall rate and path length. A vertical section of these figures will yield FSE as a function of path length, and a horizontal section yields FSE as a function of R. From Figures 4 and 5 we can see that R(KDp, ZDR) performs better than R(KDp ) at all path lengths where these algorithms are usable. Often when rainfall algorithms are used, we are not sure of the exact value of rainfall, in order to choose which algorithm to use. Nevertheless, based on the radar observations, we are aware of the type of rain encountered such as light, moderate, Figure 3. or heavy. In the following, we classify the rainfall rate as light, moderate, or heavy and compare the average FSE of the 5. Sensitivity of Rainfall Algorithms to Bias in Zu and ZDR In the previou section the change in the performance of the rainfall algorithms due to random measurement errors was illustrated. The performance of the algorithms is also altered in the presence of bias in radar measurements. Among the three measurements Z, ZD, and KDp, Z is affected by radar absolute calibration errors, whereas ZD is affected by errors in characterizing differential gain of the radar system. The absolute gain is a difficult quantity to characterize accurately and can be known to an accuracy of 1 db. However, the gain 1.00 n,(z,).50 t _ ß,,. R (KDP) ' ' i" :" ' ' ' ' ' 2 ''' Rainfall Rate Fractional standard error of different radar rainfall estimates as a function of the rainfall rate. The estimates are computed over a path length of 7.5

5 _ ß ß GORGUCCI ET AL.' MULTIPARAMETER RADAR RAINFALL ALGORITHMS 2219 can change over time. The differential gain of a radar system, commonly termed Zz>R bias, can be determined more easily and can be known to an accuracy better than 0.1 db. Kz>r is a parameter that is based on measurements of phase and therefore is unaffected by amplitude calibration errors. This is major advantage of Kz>r measurement [Chandrasekar et al., 1990]. In this section we analyze the effect of measurement bias on rainfall algorithms, and compare it with the algorithm based on Kz>r only Sensitivity to Absolute Calibration Errors 3 Error in absolute calibration results in a bias on reflectivity. Therefore absolute calibration error affects all algorithms that use reflectivity. In the following we compare the rainfall algo- 2 rithms in the presence of measurement error and bias in Zu. Figures 7a-7c show the comparison of FSE in R(Kov), R(Zu), and R(Zu, Zr, R) for light, moderate, and heavy rainfall, respectively. The estimates of R are obtained over 20 range bins (3 km). Plots of FSE of R (Kr v) and R (Kr v, gr ) are not shown in Figure 7a because they are very high and lie outside the scale. We can see that it is better R(Z ) to use the best or R(ZH, Zr, ) in light rain. However, the bias in oo Rainfall Rate (ram h' 9 a O Rainfall Rate (mm h 4) Figure 5. Fractional standard error contours for (a) R (KDv, ZD ) and (b) R2(Kov, ZoR) as a function of rainfall rate and path length O Ram] all Rate (ram h 4) o8.. o6.' [] Light Ram -3 Moderate Rain _ Heavy Ram :: :: 'i..... ß ::... :.... ß... ' o4 o O Ram fall Rate (rnrn h- 9 Figure 4. Fractional standard error contours for (a) R (Kr r) and (b) R2(Kz>r) as a function of rainfall rate and path length. R,(Zu) RffZn) R(Zn, Zo R,(Kor) RffKor) R,(Ko Zo R (Kor, Zo Figure 6. Fractional standard error of different radar rainfall estimates for light, moderate, and heavy rainfall rates. The FSE values for R (Kz>e), R2(Kz>e), R (Kz> e, Zz> ), and R2(KDr,, ZDR ) are reported on the top of the light rainfall bar.

6 2220 GORGUCCI ET AL.' MULTIPARAMETER RADAR RAINFALL ALGORITHMS '7 0.6 o o o Rt fiectivity Bias (db) Figure 7a. Fractional standard error of different radar rainfall estimates, in the light rainfall regime, as a function of the bias error in Z/_/. The estimates are computed over a path length of 3 km. R (Kt,), R2(Kt p), R,(Kt,, Zt R), and R2(KDp, ZDR ) are not shown because they lie outside of the scale Sensitivity to Errors in Differential Gain Calibration Differential gain of a polarimetric radar system is easier to be calibrated in comparison with absolute gain. However, errors in differential gain can still occur due to maintenance problems or drift in component characteristics that have not been accounted for. In this section we study the impact of erroneous differential gain calibration on rainfall algorithms. The effect on differential gain errors will be seen only on algorithms that use Zt, namely, R(Zi-i, Zt R) and R(Kt,, Zt ). In this section, as before, we show rainfall estimates averaged over a 3 km path that corresponds to 20 range gates. Figures 8a-8c show the FSE of rainfall algorithms R(Zi-i, Zt ), R (Kt ), and R2(Kt,) for light, moderate, and heavy rain, respectively. The figures also show the FSE of R (Kt ) for reference, since FSE of R (Kt ) does not change with bias in Zt. In light rain all algorithms that use Kt p measurements (estimated over a 3 km path) have large error and are not shown in Figure 8a, because they are outside the scale of the figure. It can be observed that in light rain the performance of R (Z/_/, Zt ) does not deteriorate too much in the presence of Zt R bias errors. Figure 8b shows FSE of R(Z/_/), R(Zi-i, Zt ), R (Kt,, Zt ), R:(Kt,, Zt ), and R(Kt,) in the presence of Zt bias errors and in moderate rainfall. The following observations can be made from Figure 8b: (1) The performance of R(Zi-i, Zt R) is worse than R(Kt,) when the bias error in Z t is negative and larger than -5 db; however, in the presence of positive bias error in Zt, R(Zi-i, Zt ) is still better than R(Kt,); and (2) the algorithm R (Kt,, Zt ) becomes worse than R(Kt,) if the negative Zt R bias error is larger than - db, but the algorithm R(Zi-i) will remain as the data are averaged over time and space. As stated in the previous section, Kt e-based algorithms may be improved in light rain if the path over which Kt e is estimated is increased. On the basis of the results shown in the previous section, we know that in moderate rainfall and in the 0.7 absence of any bias R(Zi-i, Zt ) was better than R(Kt,) (with Kz>e estimated over 3 km). However, that can change in the presence of bias, and the results are shown in Figure 7b. From Figure 7b we can see that if the bias in Z/_/is larger than db or -2.5 db, then the algorithm R(Zi-i, Zt ) can be O.5 worse than R(Kt,). In addition, we can see that R (Zi_i) is always worse than R (Kt e). The value of R, when R (Kt e) is 0.4 better than a biased R(Zi_i, Zt ), depends on many factors such as rainfall rate and range averaging of Kt e. The data in Figure 7b correspond to 20 bins. If the range averaging is more 0.3 than 20 bins, then with a smaller bias in Z/_/( smaller than 1.5 db or -2.5 db), R(Zi-i, Zt ) will be worse than R(Kt,). Similarly, if the range averaging is less than 20 bins, the algorithm R(Zi-i, Zt R) will be better than R(Kt p), even for a 0.1 large bias in Z/_/. Figure 7c shows the FSE of R(Kt e), R (Z/_/, Zt ), and R (Z/_/) for heavy rainfall. We know from i the previous section that even in the absence of any bias errors R(ZH) is worse than R(Kt,). It can be seen from Figure 7c that even with a slight bias in Z/_/(=0.4 db) the performance Reflectivity Bias (db) of R(Zi-i, Zt ) is worse than R(Kt,) for estimates averaged Figure 7b. Fractional standard error of different radar rainover 20 bins. However, it takes a bias of about + 1 db or -1.5 fall estimates, in the moderate rainfall regime, as a function of db in Z/_/ for R(Zi-i, Zt ) to be worse than R(Kt e), if the bias error in Z,. The estimates are computed over a path averaging was done over 20 range bins. length of 3 km. 0.8

7 GORGUCCI ET AL.: MULTIPARAMETER RADAR RAINFALL ALGORITHMS 2221 R2(KDp, ZDR ) becomes worse thanr(kdp) even with a slight negative bias error in ZDR. BothR(KDp, ZDR) algorithms still perform better than R(KDp) in the presence of positive ZDR bias error. Figure 8c shows FSE of R(ZH), R(ZH, ZDR), R(KDp, ZDR), and R(KDp) in the presence of ZDR bias errors in heavy rainfall. In heavy rain, R(Z H, ZDR) becomes worse than R(KDp) even with small negative ZDR bias error, but it takes 0.15 db of positive ZOR bias error to make R(ZH, ZDR) worse than R(KDp). Similarly, R (KDp, ZDR) is worse than R(KDp) when ZDR negative bias error is larger than -5 db, whereas R2(KDp, ZDR ) is worse than R(KDp) when ZDR negative bias error is larger than db. Both R(KDp, ZD ) algorithms maintain FSE comparable to R(KDp) under positive ZDR bias errors. In addition, the R(ZH, ZDR) algorithm is more sensitive to ZDR bias errors compared with R(KDp, ZDR) in heavy rainfall. 6. Summary and Conclusions The multiparameter radar measurements ZH, ZD, and KDp are used in various algorithms to estimate rainfall rate. The error structures of the three measurements are independent and completely different. In addition, measurements of Z H and ZDR can be biased due to errors in absolute and differential calibration of the radar system; meanwhile, KDp measurement is immune to calibration errors. However, measurements of KDp are noisy at low rainfall rate. Thus the errors of different multiparameteradar measurements manifest themselves as errors in rainfall rate algorithms. We have evaluated the error structure of rainfall rate esti ß,..,,,, 0.8 L 0.'7- o D!.fferential Reflectivity Bias (db) Figure 8a. Fractional standard error of different radar rainfall estimates, in the light rainfall regime, as a function of the bias error in ZDa. The estimates are computed over a path length of 3 km. R (KDe ), R2(KDe ), R (KDp, ZDR ), and R2(KDe, ZDR) are not shown because they lie outside of the scale o.4 o.3 - R, (Ko.,,) / R2(Ko Zo Rqfiectivity Bias (db) R,(Kot,,Zod i i Figure 7c. Fractional standard error of different radar rainfall estimates, in the heavy rainfall regime, as a function of the bias error in ZH. The estimates are computed over a path length of 3 km. mates from five multiparameteradar algorithms, namely, R(ZH, ZDR), R (KDp), R2(KDp), R (KDp, Zoo), and R2(Kop, ZD ) at S band. In the absence of random measurement errors and systematic gain biases, R(KDp, Zoe) is uniformly the best algorithm at all rainfall rates. The next best is R(ZH, ZOR), followed by R(KDp). However, in the presence of measurement errors, the inference changes dramatically. First, the error structure of all algorithms involving KDp changesignificantly with rainfall rate and path length over which KDp is estimated. In the presence of only random measurement errors, R(KDp, ZDR ) is better than R(ZH, ZDR ) at 100 mm h- for a 7.5 km path average. For the same 7.5 km path, R(KDe, ZDR) becomesignificantly better than R(KDe ) above DU IlllIl 11. noweyre, for billrill [JdLLL, UCU as z. m, R(ZH, ZDR) is always better than R(KDp) or R(Kop, ZDn). Biases in absolute gain and differential gain affect R(ZH, ZDn) and R(KDp, ZDn) algorithms. All these effects are dependent on rainfall rate. We divide the rainfall regions into three classes, namely, (1) light, (2) moderate, and (3) heavy, where light rain is less than 10 mm h - and moderate rainfall is between 10 and 90 mm h -, and heavy rain is above 90 mm h -. In light rain, R(ZH) or R(ZH, ZDn) algorithms are better than algorithms using KDp (estimated over a 3 km path). However, any bias that is in R(ZH) will remain as the data are averaged over time and space. As noted before, KDp-based algorithms can be improved by longer estimation paths. Similarly, in moderate rainfall (for a 3 km path estimate of KDp), R(ZH, ZDR) can be worse than R(KDp) or R(KDp, ZDn)

8 2222 GORGUCCI ET AL.: MULTIPARAMETER RADAR RAINFALL ALGORITHMS = ,. 0.3 D!fferential Reflectivity Bias (db) Figure 8b. Fractional standard error of different radar rainfall estimates, in the moderate rainfall regime, as a function of the bias in ZDR. The estimates are computed over a path length of 3 km. 1 with a bias larger than 1.5 db or -2.5 db. In a well-maintained radar system, bias errors of the order of 1 db can be expected, but errors larger than 2 db are unusual (it is not well maintained anymore). However, biases due to attenuation may be introduced. In heavy rain for the same 3 km path estimates, R(Z, ZDR) becomes worse than R(KDe, ZDR) algorithms even with slight bias in Z H (0.5 db). In addition, R(ZH) becomes worse than both R (KDe) algorithms with 1 db bias in reflectivity. Thus on the basis of the above observations, KDebased algorithms R(KDe) or R(KDe, ZDR) are best for estimation of rainfall rate in heavy rainfall regions. Biases in differential gain of the radar system affect all algorithms using ZDe, namely, R(ZH, ZD ) and R(KDe, ZD ). In light rain, even in the presence of biases in differential gain, R (Z q, ZDe) and R (ZH) are better than any KDe-based algorithm. In moderate rainfall, for a typical 3 km path averaging for KDe estirnates, R(ZH, ZD ) becomes worse than R(KDe) when bias in ZDe is larger than -5 db or 0.5 db. However, in heavy rain, R(ZH, ZDR ) becomes worse than R(Kop) when Zo bias errors are larger than +5 db. When there are simultaneous bias errors in Z u and ZoR, they affect R (Z v, Zo ) adversely compared to any other algorithm. In the presence of calibration bias, errors in Z u and ZoR biases of the same sign compensate each other and R(ZH, ZDR ) estimates are slightly affected. However, if the biases are of opposite sign, then the R(ZH, ZDR ) algorithm is significantly affected. Thus, assuming the worst case scenario, R(Zu, Zoie) will be comparable to R(Koe ) in moderate and heavy rainfall, if the bias errors in ZH can be kept to within I db and the bias error in Zo to within 5 db. Acknowledgments. This research was supported partially by the National Group for Defense from Hydrological Hazard (CNR, Italy), by Progetto Strategico Mesoscale Alpine Programme (CNR, Italy), the Italian Space Agency (ASI), and NASA (TRMM). The authors are grateful to P. Iacovelli for assistance rendered during the preparation of the manuscript. L 0.3 R,(Koe) R (Koe, Zod / O. 1 R(ZmZo o -0.3' -' - o'. 1 0 ' o'.1 0 i 2 0 i Differential Reflectivity Bias (db) Figure 8c. Fractional standard error of different radar rainfall estimates, in the hea rainfall regime, as a function of the bias in ZD. The estimates are computed over a path length of 3 km. References Beard, K. V., and C. Chuang, A new model for the equilibrium shape of raindrops, J. Atmos. Sci., 44, , Chandrasekar, V., V. N. Bringi, and P. J. Brockwell, Statistical properties of dual polarized radar signals, paper presented at 23rd Conference on Radar Meteorology, Snowmass, Colo., Am. Meteorol. Soc., Chandrasekar, V., W. A. Cooper, and V. N. Bringi, Axis ratios and oscillation of raindrops, J. Atmos. Sci., 45, , Chandrasekar, V., V. N. Bringi, N. Balakrishnan, and D. S. Zrnic, Error structure of multiparameter radar and surface measurements of rainfall, III, Specific differential phase, J. Atmos. Oceanic Technol., 7, , Gorgucci, E., and G. Scarchilli, Intercomparison of multiparameter radar algorithms for estimating rainfall rate, paper presented at 28th Conference on Radar Meteorology, Austin, Tex., Am. Meteorol. Soc., Gorgucci, E., G. Scarchilli, and V. Chandrasekar, Radar and surface measurements of rainfall during CAPE: 26 July 1991 case study, J. Appl. Meteorol., 34, , Green, A. W., An approximation for the shapes of large raindrops, J. Appl. Meteorol., 14, , Holt, A. R., R. J. Watson, M. Chandra, S. Nanni, and P. P. Alberoni, A comparison of C-band observations using different polarization schemes, paper presented at 28th Conference on Radar Meteorology, Austin, Tex., Am. Meteorol. Soc., Pruppacher, H. R., and R. L. Pitter, A semi-empirical determination of the shape of cloud and raindrops, J. Atmos. Sci., 28, 86-94, Ryzhkov, A. V., and D. S. Zrnic, Comparison of dual-polarization

9 GORGUCCI ET AL.: MULTIPARAMETER RADAR RAINFALL ALGORITHMS 2223 radar estimators of rain, J..4tmos. Oceanic Technol., 12, , Sachidananda, M., and D. S. Zrnic, Rain rate estimates from differential polarization measurements, J..4tmos. Oceanic Technol., 4, , Scarchilli, G., E. Gorgucci, V. Chandrasekar, and T. A. Seliga, Rainfall estimation using polarimetric techniques at C-band frequencies, J..4ppl. Meteorol., 32, , Seliga, T. A., and V. N. Bringi, Potential use of the radar reflectivity at orthogonal polarizations for measuring precipitation, J..4ppl. Meteorol., 15, 69-76, Seliga, T. A., and V. N. Bringi, Differential reflectivity and differential phase shift: Application in radar meteorology, Radio Sci., 13, , Ulbrich, C. W., Natural variations in the analytical form of raindrop size distributions, J. Clim..4ppl. Meteorol., 22, , Zrnic, D. S., and A. V. Ryzhkov, Advantages of rain measurements using specific differential phase, J..4tmos. Oceanic Technol., 17, , V. Chandrasekar, Colorado State University, Fort Collins, CO E. Gorgucci and G. Scarchilli, Istituto di Fisica dell' Atmosfera, Consiglio Nazionale delle Ricerche, Area di Ricerca Roma-Tor Vergata, Via del Fosso del Cavaliere, 100, Rome, Italy. ( (Received July 25, 1998; revised April 12, 1999; accepted April 29, 1999.)