Practical Aspects of Radar Rainfall Estimation Using Specific

JUNE 2000

945

GORGUCCI ET AL.

Practical Aspects of Radar Rainfall Estimation Using Specific Differential Propagation Phase EUGENIO GORGUCCI

AND

GIANFRANCO SCARCHILLI

Istituto di Fisica dell’Atmosfera (CNR), Rome, Italy

V. CHANDRASEKAR Colorado State University, Fort Collins, Colorado (Manuscript received 8 July 1998, in final form 21 June 1999) ABSTRACT Rainfall estimation using specific differential phase (KDP ) involves estimation of KDP over a propagation path. The choice of pathlength plays an important role in the performance of rainfall algorithms using KDP . The pathintegrated nature of KDP-based rainfall estimates may involve inhomogeneous paths, thereby having potential implications on the pathlengths used in algorithms for rainfall estimation and measurement errors. The effect of inhomogeneous rainfall paths (over which KDP is estimated) on the choice of rainfall rate algorithms is studied as a function of rain pathlength. Rainfall estimation is biased when algorithms relating rainfall rate and KDP are used over nonuniform paths. This bias is evaluated and compared for different rainfall algorithms. Radar and rain gauge data collected during the Convection and Precipitation/Electrification Experiment (CaPE) are used to evaluate the performance of KDP-based rainfall algorithms, for different pathlengths.

1. Introduction Estimation of rainfall rate using specific differential propagation phase (KDP ) is a topic of active current interest. Recent results of rainfall comparison between radar and ground instrumentation such as rain gauges provide evidence that KDP-based algorithms estimate rainfall fairly well (Chandrasekar et al. 1990; Aydin et al. 1995; Ryzhkov and Zrnic´ 1996; Gorgucci and Scarchilli 1997). Chandrasekar et al. (1990) suggest that a rainfall estimate based on KDP [R(KDP )] performs better than other rainfall estimators if the rainfall rate (R) is greater than 70 mm h21 . The KDP estimate is noisy at low rainfall rates and therefore imposes a lower limit on the applicability of KDP-based algorithms. However, at lower rainfall rates long-range profiles of differential phase (FDP ) can be used to improve the accuracy of KDP estimates in order to obtain accurate estimates of rainfall rate (Ryzhkov and Zrnic´ 1996; Scarchilli et al. 1993). One main difference between KDP and the other radar parameters that are used for rainfall estimation such as reflectivity and differential reflectivity (ZDR ) is that KDP is estimated as the slope of the range profile of differ-

ential propagation phase. This implies that rainfall estimates based on Z H and ZDR are point measurements (or measurements for a resolution volume), whereas a rainfall estimate based on KDP is an average value along the path over which KDP is estimated. Here, FDP can be measured to an accuracy of a few degrees (Sachidananda and Zrnic´ 1987; Hubbert et al. 1993; Liu et al. 1994). The KDP is often estimated as the slope of the FDP profile, based on a least squares fit to a line. For this estimate the variance of KDP can be written as (Gorgucci et al. 1999a) var(K DP ) 5

3 var(FDP ) , (N 2 1)N(N 1 1)Dr 2

where Dr is the range resolution, N is the number of range bins, and var (FDP ) is the variance of differential phase measurement FDP . It can be seen from (1) that the variance of KDP decreases with increasing path length (NDr 5 L). For a large number of range gates (N) the above equation can be reduced to (Ryzhkov and Zrnic´ 1996)

s (K DP ) 5 Corresponding author address: Dr. Eugenio Gorgucci, Istituto di Fisica dell’Atmosfera (CNR), Area di Ricerca Roma-Tor Vergata, Via del Fosso del Cavaliere, 100-00133 Roma, Italy. E-mail: [email protected]

q 2000 American Meteorological Society

(1)

!

3 s (FDP ) , N L

(2)

where s(KDP ) is the standard deviation in the estimate of KDP and s(FDP ) is the standard deviation of FDP . As an example for a uniform rainfall rate of 15 mm h21 , a path of 10.8 km is needed to estimate KDP to an accuracy

946

JOURNAL OF APPLIED METEOROLOGY

VOLUME 39

the fact that KDP is estimated over a path requires a different approach to analyzing and evaluating KDPbased rainfall estimates. This paper analyzes and presents issues related to the fact that KDP-based estimates of rainfall are path-averaged estimates. Our paper is organized as follows. Section 2 reviews algorithms for rainfall estimation using KDP . In section 3 pointwise bias in R(KDP ) algorithm is discussed. Section 4 discusses rainfall rate estimation using KDP over a path. Section 5 discusses potential bias in KDP due to increased noise at small path lengths. In section 6 the concepts developed in this paper are evaluated using KDP data in rainfall collected over Florida. Section 7 summarizes the key results of this paper. 2. Rainfall rate estimation using KDP FIG. 1. Maximum observed KDP shown as a function of pathlength over which KDP is estimated, for CP-2 radar data collected in rainfall over central Florida during CaPE field campaign.

of 10%, using 150-m range samples and with a s(FDP ) of 28 at S band. If s(FDP ) is larger than 28 the path length needed for the same accuracy in KDP increases proportionately. In summary, a rainfall estimate using KDP is naturally averaged over a path typically 2- to 10km long. The path-integrated nature of KDP-based rainfall estimates necessitates treating R(KDP ) differently from estimates of rainfall such as those obtained from reflectivity R(Z H ) or reflectivity and differential reflectivity R(Z H , ZDR ). In general if one averages rainfall fields, the maximum values encountered decrease. This is demonstrated in the following, where peak KDP values are studied in a rainstorm. Figure 1 shows the peak value of KDP observed as a function of pathlength over which KDP is estimated, as the slope of FDP measurements. The data shown in Fig. 1 were collected by the CP-2 radar over central Florida during CaPE. It can be seen from the results of Fig. 1 that the maximum observed KDP decreases as the pathlength increases. In addition the path over which KDP is estimated ceases to be homogenous if long paths are involved. Subsequently there is a compromise between accuracy and resolution of rainfall field. One more important aspect of path averaging is on the KDP algorithm itself. Even though KDP is proportional to rainfall, the relation is still not perfectly linear. Therefore parameterization done to estimate R of the form R(KDP ) 5 aK bDP

(3)

varies depending on whether the parameterization covers a wide range of rainfall rates or a smaller range of rainfall rates. However, this change is not as large as the variability in Z–R algorithms. This poses the question that perhaps a linear parameterization relating R and KDP may be better to use, because such an algorithm will be immune to inhomogeneities along the path. Thus

Seliga and Bringi (1978) introduced the concept of using differential propagation phase for rainfall estimation and suggested using a combination of KDP and ZDR for this application. Subsequently, Sachidananda and Zrnic´ (1987) introduced a power law equation of the form R(KDP ) 5 37.1K 0.866 DP .

(4)

It should be noted here that in the paper by Sachidananda and Zrnic´ (1987) they refer to twice the KDP (two way) as specific differential phase. Therefore, the equation is modified here with a constant of 2 0.866 , to allow comparison with the rest of the literature. Subsequently, Chandrasekar et al. (1990) derived an equation based on an error analysis as R(KDP ) 5 40.5K 0.85 DP ,

(5)

which is fairly similar to (4). Scarchilli et al. (1993) obtained a linear equation for rainfall rate in terms of KDP extending the range of rainfall rate to 300 mm h21 . This algorithm for S band is given as (Gorgucci and Scarchilli 1997) R(KDP ) 5 39.8KDP .

(6)

The linear equation has the advantage that it can be used directly to estimate average rainfall rate over an inhomogeneous path. On the contrary a nonlinear equation of the form given by (3) cannot be used directly to relate average KDP to average rainfall rate over an inhomogeneous path. However the nonlinear parameterization is better and has smaller mean square error in the estimation of rainfall. Thus due to the nature of estimation of KDP over a path, accuracy and resolution are two conflicting requirements and we need to find a compromise between them. In addition, since KDP is always estimated over a path, then the average and the spread of rainfall rate encountered are lower than those of pointwise rainfall rate. Figure 2 shows the maximum rainfall rate Rmax and the average rainfall rate R obtained by simulation over a 7.5-km long path (with a range resolution of 150 m), where the reflectivity variation

JUNE 2000

GORGUCCI ET AL.

947

FIG. 2. Maximum (Rmax ) (solid line) and average (R) (dashed line) rainfall rate in the path over which KDP is estimated shown as a function of the corresponding variation of reflectivity. The minimum reflectivity is 25 dBZ.

FIG. 3. Pointwise normalized bias in the estimate of R for the three rainfall algorithms R L (dotted line), R NL (dashed line), and R 50 (solid line) shown as a function of reflectivity factor (dBZ ).

ranges between 0 and 30 dB. For example, for a variation of 20 dB, the reflectivity varies linearly from 25 dBZ to 45 dBZ along the path. Variations of reflectivity and specific differential phase are simulated as follows: in each range gate we have chosen the parameters N 0 , D 0 , and m of the gamma dropsize distribution in order to obtain the fixed reflectivity variation. Here, N 0 , D 0 , and m are restricted to the limits suggested by Ulbrich (1983) as: a) 21 , m , 4, b) 10 3.210.216 m , N 0 , 10 4.510.55 m m23 mm212m , and c) 0.5 , D 0 , 2.5 mm. In correspondence with the values of N 0 , D 0 , and m the rainfall rate and all the other radar observables can be computed. Most of the parameterizations for R(KDP ) are obtained as one equation that works for a wide range of R (say 0–300 mm h21 ). If most of the estimates of KDP involve estimates over long paths, the range of variability in R is smaller and another parameterization for lower values of R (say 0–50 mm h21 ) can be a better fit. The best R(KDP ) algorithm obtained minimizing the mean square error and valid for rain rate between 0 and 50 mm h21 is given by

(7), respectively. The subscripts NL, L, and 50 indicate nonlinear algorithm, linear algorithm, and the best algorithm when R , 50 mm h21 , respectively. The algorithms can be analyzed as a function of reflectivity allowing the range of variability of R between 0 and 300 mm h21 . The results of the analysis are plotted as a function of the reflectivity factor (Z H ) to show the variability with respect to a commonly measured as well as independent parameter. All the rainfall rates, for a given Z H , are compared against the values given by the three R(KDP ) algorithms and the estimates of bias are plotted as a function of Z H in Fig. 3. Some interesting observations can be made from Fig. 3. First, the linear algorithm R L has large biases for low reflectivity (or low rainfall rate) and small biases at high Z H (or R). The algorithm R 50 , that was obtained to yield best performance over the range 0–50 mm h21 , has negligible bias at low reflectivity regions (low rainfall rate) and large bias in high reflectivity regions, as expected. The algorithm R NL , commonly used in the literature, underestimates rainfall rate and has uniformly low bias over the full range of Z H or (R) though at a specific Z H another algorithm may have lower bias. The results of Fig. 3 indicate that if we are likely to have predominantly low values of R due to large-range averaging in the estimate of KDP , it may be better to use algorithm given by (5) and (7).

R 50 5 39.96K 0.81 DP .

(7)

Another issue specific to KDP-based rainfall estimation is the bias at low KDP values. In the following sections we evaluate the biases in KDP-based rainfall estimates due to the various causes. 3. Pointwise bias in the R(KDP ) algorithms due to the parameterization as a function of reflectivity In this section we study the bias in the R(KDP ) algorithms namely R NL , R L , and R 50 given by (5), (6), and

4. Rainfall rate estimation using KDP over a path The KDP is typically estimated as the slope of differential phase profile over a path. In the presence of a nonuniform path one average KDP is estimated and is

948

JOURNAL OF APPLIED METEOROLOGY

VOLUME 39

converted to average rainfall rate over the path. This procedure introduces biases in R(KDP ) algorithms. The normalized bias B T in KDP-based estimates of rainfall over a path, can be written as BT 5

s a(K DP ) b 2 ^R& , ^R&

(8)

where angle brackets ^ & indicate average or expectation, K sDP is the KDP estimation obtained as the slope of differential phase profile over a path, and a and b are the coefficients of the parameterization (3). The bias B T can be due to different reasons, namely, (a) bias B G due to the nonuniform path over which KDP is estimated, (b) bias B NL due to the nonlinearity in the estimation process, and (c) bias B P due to parameterization. The total bias B T can be written as B T 5 B G 1 B NL 1 B P ,

(9)

where the bias B G due to inhomogeneous rainfall path is given by s a(K DP ) b 2 a^K DP & b BG 5 , ^R&

(10)

the bias B NL due to nonlinearity is expressed as BNL 5

b a^K DP & b 2 a^K DP & , ^R&

(11)

and the bias B P due to parameterization is BP 5

b a^K DP & 2 ^R& . ^R&

(12)

a. Bias in the R(KDP ) algorithms due to nonuniform rainfall rate paths In this section we simulate reflectivity gradients along the path (7.5-km long with a resolution range of 150 m), over which KDP is estimated, to study the magnitude of bias introduced specifically by the use of the algorithms. One physical way to describe the variations is in terms of variability in the raindrop size distribution parameters along the propagation path. The RSD parameters can vary along the path to produce variations in reflectivity, differential reflectivity, and KDP . In our study the parameters (N 0 , D 0 , m) of the gamma drop size distribution are varied in agreement with the limits suggested by Ulbrich (1983) as described in section 2. We study nonuniform rainfall path in the form of a linear variation of reflectivity and of a step variation in reflectivity (dBZ) along the path. Linear variation of reflectivity in decibel scale can be used to approximate regions where there is a steady increase or decrease of reflectivity. Step variation in reflectivity can be used to describe regions in convective cells where an intense rainshaft is located adjacent in range to weak echo regions. A step variation of reflectivity is the limit of the linear variation when the gradient is infinity. Once the

FIG. 4. Normalized bias B G in R NL (dashed line), R 50 (solid line), and R L (dotted line) due to nonuniform rainfall path shown as a function of reflectivity variation (dB) along the path for a linear variation of reflectivity (dBZ ). The minimum reflectivity along the path is 25 dBZ.

reflectivity profile along the path is defined, the other parameters of the rain medium can be obtained sampling the natural variation of raindrop size distribution under the constraint that the reflectivity value is given by the model. Thus the above procedure simulates propagation paths that result in linear or step variation of reflectivity. In addition, it should be noted that if the reflectivity variation was triangular, the bias due to nonuniformity can be positive or negative depending on the location of the reflectivity peak in the path (Gorgucci et al. 1999b). Therefore if a cell is fairly broad, so that the KDP estimates go through an increasing reflectivity, followed by a decrease, then the average bias obtained over the increasing region (of Z) followed by the decreasing region will be small. The average bias is small because the positive and negative biases cancel each other. Figure 4 shows the bias B G as a function of linear variation in dB along the path for the three algorithms R L , R NL , and R 50 . It can be seen that the bias is negative and it increases in magnitude by increasing the reflectivity variation in similar ways for the three algorithms, ranging from 0% for uniform reflectivity to about 225% for a 30-dB variation of reflectivity. In the following we model a step variation of reflectivity where the parameter of interest is the fraction ( f ) of the path over which the high reflectivity exists. In order to vary the fraction the reflectivity step is introduced at different location of the path from the beginning to the end, so that for example f 5 0.2 indicates that there is high reflectivity over 0.2 fraction of the path and low reflectivity over 0.8 fraction of the path. In the simulation f ranges between 0 and 1 with a resolution of 0.02. Figure 5 shows the bias B G

JUNE 2000

GORGUCCI ET AL.

FIG. 5. Normalized bias B G in R 50 (solid line) and R L (dotted line) due to nonuniform rainfall path shown in the presence of a step variation of reflectivity along the path as a function of the fraction of high reflectivity in the path. The various curves show the bias for step sizes of 10, 20, and 30 dB.

in R L and R 50 as a function of the fraction of path with high reflectivity at step sizes of 10, 20, and 30 dB. It can be seen that when the fraction f of high reflectivity in the path is small, the biases are negative and have a minimum increasing with the step size. When the half path has high reflectivity the bias is zero, as expected. For fraction of path greater than 0.5 the bias is positive and has a maximum for f 5 0.75. In general, R L shows lower B G for fraction of high reflectivity less than 0.5 and slightly higher for f . 0.5. The algorithm R NL presents similar results as R 50 .

949

FIG. 6. Normalized bias B NL in R NL (dashed line) and R 50 (solid line) due to nonlinearity of the algorithm shown as a function of reflectivity variation (dB) along the path for a linear variation of reflectivity (dBZ ). The minimum reflectivity along the path is 25 dBZ.

est. This bias can be as high as 40% for a 30-dB step but decreases sharply when increasing the fraction of the path over which high reflectivity exists. For example when half the path has high reflectivity and the other half has low reflectivity, the bias is on the order of 10% for 30-dB reflectivity step. It can be seen that the biases

b. Bias due to nonlinearity of R(KDP ) algorithms The bias due to nonlinear algorithms is different from zero only in the case of nonuniform path. Figure 6 shows the bias in the estimation of the rainfall rate over the path as a function of the variation of reflectivity along the path. The minimum reflectivity is chosen as 25 dBZ. The two curves shown in Fig. 6 are referred to the algorithms R 50 and R NL ; it can be noted here that the bias due to algorithm R L is zero. It can be seen from the results of Fig. 6 that the bias due to nonlinearity is about 10%–12% for a 30 dB variation of reflectivity along the path and is 5% for a 12 dB variation along the path. Figure 7 shows the bias B NL in R NL and R 50 in presence of a step variation of reflectivity as a function of the fraction of path with high reflectivity. The various curves show the step sizes of 10, 20, and 30 dB, respectively. It can be seen from Fig. 7 that, when the fraction of high reflectivity is small, the bias is the larg-

FIG. 7. Normalized bias B NL in R 50 (solid line) and R NL (dotted line) due to nonlinearity of the algorithm shown in the presence of a step variation of reflectivity along the path as a function of the fraction of high reflectivity in the path. The various curves show the bias for step sizes of 10, 20, and 30 dB.

950

JOURNAL OF APPLIED METEOROLOGY

FIG. 8. Normalized bias B p in R NL (dashed line), R 50 (solid line), and R L (dotted line) due to parameterization shown as a function of reflectivity variation (dB) along the path for a linear variation of reflectivity (dBZ ). The minimum reflectivity along the path is 25 dBZ.

due to a step variation in reflectivity are slightly higher for R 50 than that for R NL . For example, when the half path has high reflectivity the bias for a 30-dB step in reflectivity is 12% for R 50 . It is to be noted that the linear algorithm R L has no bias due to a step variation of reflectivity along the path. c. Bias in the R(KDP ) algorithms due to the parameterization The path integrated nature of KDP estimate modifies the pointwise bias due to the parameterization described in section 3. Figure 8 shows the normalized bias B P in R L , R NL , and R 50 due to parameterization as a function of reflectivity variation along the path. For uniform reflectivity, which in our computation corresponds to 25 dBZ, the pointwise biases of Fig. 3 coincide with the biases B P shown in Fig. 8, as expected. By increasing the reflectivity variation, the bias B P can be obtained approximately as an average of the pointwise biases between the minimum and the maximum reflectivity considered along the path. The biases B P in R L and R 50 for the step model, due to the parameterization are shown in Fig. 9 as a function of fraction of path with high reflectivity at step sizes of 10, 20, and 30 dB. It can be observed that the biases are nearly independent of path fraction and can be obtained, at the same reflectivity variation, from the biases computed considering the model where the reflectivity varies linearly. Finally, the simulation results indicate that the value of the bias in R NL due to the parameterization is nearly the same as that seen in Fig. 3, because there is no significant variation with reflectivity.

VOLUME 39

FIG. 9. Normalized bias B p in R 50 (solid line) and R L (dotted line) due to parameterization shown in the presence of a step variation of reflectivity along the path as a function of the fraction of high reflectivity in the path. The various curves show the bias for step sizes of 10, 20, and 30 dB.

d. Comparison of algorithms in a nonuniform rainfall path In the previous section we have shown the bias in the R(KDP ) algorithms due to nonuniform rainfall path, nonlinearity, and parameterization. When KDP is estimated over a rain path that is nonuniform, the above errors are combined according to the distribution of the rainfall along the path. Figure 10 shows the total bias B T in the estimation of R, for the algorithms R L , R 50 , and R NL , for a nonuniform path where the reflectivity varies linearly along the path. It can be seen from a comparison of the three curves that R 50 has the least bias for all reflectivity variations up to 30 dB along the path. The R NL has a negative bias of about 10%–25% depending on reflectivity variation whereas R L has the largest variation in bias that is mainly due to parameterization. Figure 11 shows the total bias B T in R L and R 50 for a step type variation of reflectivity. The results indicate that for the entire range of high reflectivity fraction the algorithm R 50 has the least bias for all the reflectivity steps up to 20 dB. From simulation results the algorithm R NL presents similar performance as R 50 . It can be observed that for f , 0.25 the biases in the step model are negative and higher than the corresponding biases in the linear model, while for f . 0.5 the total biases are in magnitude less than the corresponding ones in the linear model. 5. Potential bias in R(KDP ) at low values of KDP The KDP estimate is relatively more noisy at low values of KDP . Based on (2) we can see that at low values

JUNE 2000

GORGUCCI ET AL.

951

FIG. 10. Total normalized bias B T in R NL (dashed line), R 50 (solid line), and R L (dotted line) shown as a function of variation of reflectivity (dB) along a path where the reflectivity (dBZ) is varied linearly. The minimum reflectivity is 25 dBZ.

FIG. 11. Total normalized bias B T in R 50 (solid line) and R L (dotted line) shown in the presence of a step variation of reflectivity along the path as a function of the fraction of high reflectivity in the path. The various curves show the bias for step sizes of 10, 20, and 30 dB.

of KDP , such as 0.38–0.58 km21 , s(KDP ) is comparable to KDP . Therefore, it is possible to have negative values for the estimate of KDP . Use of negative values results in negative rainfall rate (which is physically unacceptable). However, if we threshold KDP estimates to only positive values, then this yields a positively biased KDP and subsequently biased rainfall estimates. The bias in KDP mean and rainfall can be estimated assuming a Gaussian model for the error distribution of the KDP estimate. The mean value of KDP thresholding at zero is obtained as

6. Data analysis

P K DP 5 E [K DP U(K DP )],

Data analyzed in this paper were collected during the Convection and Precipitation/Electrification Experiment (CaPE) Program. CaPE was conducted in the central Florida region during the summer of 1991. The instrumentation for rainfall measurement experiment during CaPE primarily consisted of CP-2 multiparameter radar

(13)

where U( ) is the unit step function defined as U(x) 5

5

1 0

when x $ 0 when x , 0.

The relative bias in KDP using only positive estimates is ^K PDP&/^KDP &. This bias is shown in Fig. 12 where the bias in the mean of KDP due to truncation of estimates at zero is shown as function of fractional standard error (FSE) of KDP , where FSE is defined as the standard deviation normalized with respect to the mean. It can be seen from the result of Fig. 12 that when FSE of KDP is high, which is typical of small KDP , rainfall estimates can be significantly biased by using only positive values of KDP ; when FSE of KDP is small the ratio ^K PDP&/^KDP & approaches unity. This bias can be corrected using a Gaussian model for KDP estimate (which is reasonably accurate) based on the result of Fig. 12.

FIG. 12. Normalized bias in KDP due to truncation at 08 km21 , shown as a function of normalized standard error of KDP , which is standard deviation divided by the mean.

952

JOURNAL OF APPLIED METEOROLOGY

FIG. 13. Histogram of observed KDP values estimated over a 4(dash-dot line), 8- (dotted line), 12- (dashed line), and 16-km path (solid line).

and a network of rain gauges located in the Kennedy Space Center. The gauges were located at ranges between 15 and 60 km from the radar, evenly distributed between these ranges. The rain gauge network consisted of 20 tipping bucket rain gauges with a recording resolution of 1 min. Toward the end of the CaPE project Colorado State University upgraded the CP-2 radar with an auxiliary signal processor that provided estimates of FDP (Chandrasekar et al. 1993). On 29 September 1991 a cluster of storms moved over the rain gauge network and the event lasted 4 h. The CP-2 radar collected data over these storms in a PPI, RHI, and SURVEILLANCE mode obtaining data over the rain gauge network. The data used in this paper were obtained from elevation angles over 2.08, because the radar transmission at lower elevation angle scans was blocked to prevent interference with space shuttle. The range spacing of the full set of multiparameter data collected with the new processor was 300 m. The KDP estimates were obtained from FDP measurements using a least square fit to the FDP range profile. a. Data analysis procedure The procedure to compute rainfall estimates from radar data is conceptually straightforward but numerous details are important. We need note here that we have not done any additional smoothing or applied any adjustment to the data. The steps involved in collecting radar data for each rain gauge location are as follows: (a) the location of each rain gauge is mapped on the radar PPI, and (b) the radar data are converted to rainfall rate using algorithms (5), (6), and (7). One of the ob-

VOLUME 39

FIG. 14. Mean reflectivity variation (dB) observed as a function of the pathlength over which KDP is estimated.

jectives of this paper is to investigate the bias in KDPbased rainfall estimates obtained over different path lengths. Therefore, we obtain the KDP estimates over seven different range profiles. The radar estimates are obtained over an area that has a dimension of 2 km along cross range (azimuthal direction) and variable length along the range. The different path lengths considered along the range are 4, 6, 8, 10, 12, 14, and 16 km that correspond to 13, 20, 27, 33, 40, 47, and 53 gates, respectively. Each of the different path lengths is centered over the gauge. The KDP is estimated over these various path lengths from range profiles of FDP as the slope of a straight line fit. The rainfall obtained from KDP-based algorithms is accumulated in time for the precipitation event and compared with gauge measurements. b. Experimental results Conventionally KDP values less than a threshold value such as 0.58 km21 are not used for estimating rainfall rates. Since the objective of this work is to evaluate the biases in the presence of different path lengths, we do not threshold the KDP values. Figure 13 shows the histograms of KDP estimates obtained over a 4-, 8-, 12-, and 16-km path length, respectively. It can be seen from Fig. 13 that the spread of KDP values decreases as the path length increases. Note the long tails of histograms for 4-km and 8-km paths. In addition, KDP estimates over smaller paths have negative KDP estimates due to errors in the estimation of KDP . The decrease of the spread of KDP indicates that the range of rainfall rates encountered in the path decreases as the path length increases. Figure 14 shows the mean reflectivity vari-

JUNE 2000

GORGUCCI ET AL.

FIG. 15a. Averaged (R/G) ratio for the three algorithms R L (dotted line), R NL (dashed line), and R 50 (solid line) shown as a function of the pathlength over which KDP is estimated for the CP-2 radar data.

ation, computed as the difference between the maximum and the minimum reflectivity over the path, as a function of path length. It can be seen from Fig. 14, that reflectivity variation as much as 14 dB can be observed in a 4-km path and a variation of 22 dB over a 16-km path. Data shown in Fig. 14 indicate that it is not uncommon to encounter a large variation of reflectivity and KDP in a path. c. Radar–rain gauge comparison Rainfall accumulations at each rain gauge location obtained from the three algorithms R L , R NL , and R 50 were evaluated using KDP estimated over seven path lengths of 4, 6, 8, 10, 12, 14, and 16 km, for comparison with rain gauge estimates. Two parameters are computed to describe the figure of merit of the radar–gauge comparison, namely: (a) the fractional standard error (FSE) defined as

FSE 5

[

1 M

O (R 2 G ) ] 1 OG M

1/2

M

2

i

i

i51

M

,

(14)

i

i51

where M represents the number of rain gauges, and R i and G i are the radar and rain gauge–based rainfall estimates; (b) radar-to–rain gauge–averaged ratio (R/G) of rainfall estimate defined as 1 M R R M i51 i 5 . (15) G 1 M Gi M i51

1 2

O O

953

FIG. 15b. Averaged (R/G) ratio corrected for the bias due to the truncation of KDP for the three algorithms R L (dotted line), R NL (dashed line), and R 50 (solid line) shown as a function of the pathlength over which KDP is estimated for the CP-2 radar data.

Most of the analysis presented earlier corresponded to KDP estimation over a path. However, comparison with rain gauges is slightly more complicated because rain gauge measurements are point measurements (typically over the center of the path) integrated over time, whereas radar measurements based on KDP are instantaneous estimates. Nevertheless, comparison with rain gauges provides a gross valuation of rainfall estimates obtained using various algorithms. Figure 15a shows the averaged ratio (R/G) as a function of path length for the three KDP-based algorithms. It can be seen that for small path lengths KDP estimates are noisy and introduce a positive bias, when truncated at zero. This bias due to measurement error in KDP decreases when path length is increased (which reduces error in KDP ). However, beyond a path of 8 km other effects discussed in this paper such as nonuniform rainfall path, linearity in algorithm, and parameterization enter. The results of Fig. 15a show that these effects are more pronounced in R L compared to the other two algorithms. The bias in the rainfall estimate using small path lengths is well characterized and can be eliminated. Figure 15b shows the bias in the form of (R/G) for the three algorithms R 50 , R NL , and R L removing the bias due to truncation. It can be seen that excessive bias over small path is eliminated. The biases of R 50 and R NL are similar. However, R L is larger by 10%, especially over long paths. In addition to the bias, the FSE provides a measure of the performance of the three algorithms. The analysis shows that for all three algorithms the FSE has a minimum when the bias is close to zero. At small path lengths the error in measurement of KDP dominates and

954

JOURNAL OF APPLIED METEOROLOGY

FIG. 16. Fractional standard error corrected for the bias due to truncation of KDP for the three algorithms R L (dotted line), R NL (dashed line), and R 50 (solid line) shown as a function of the pathlength over which KDP is estimated for the CP-2 radar data.

there is no difference between the algorithms. At path lengths of 8 km or larger the difference between the algorithms emerges, with R 50 giving the best performance. We can eliminate the contribution of the bias due to truncation of KDP in FSE and the results are shown in Fig. 16. The linear algorithm R L has higher FSE than R NL and R 50 for most path lengths. 7. Summary and conclusions Specific differential phase shift is estimated over a path typically a few kilometers long. For a fixed percentage accuracy in the estimation of KDP the path length required to estimate KDP increases as KDP decreases. As a result KDP is typically estimated over long paths. The path-integrated nature of the KDP estimate presents two issues, namely, (a) as the path length increases, the rain medium ceases to be homogeneous, and (b) the distribution of measured rainfall has a smaller variance as the path averaging increases. Presence of nonuniform rainfall path raises the question whether it is better to use a linear equation to relate rainfall rate and KDP in order to avoid errors due to averaging a nonlinear function. The path averaging decreases the spread in the distribution, indicating that the parameterization of R(KDP ) algorithm can be done over a smaller range. We have evaluated the bias in the various algorithms as a function of reflectivity. The algorithms evaluated are (a) best linear algorithm relating R and KDP , namely R L given by (6), (b) the algorithm that has uniformly low bias R NL given by (5), and (c) an algorithm optimized for minimum mean square error, over 0–50 mm

VOLUME 39

h21 , R 50 given by (7). Pointwise evaluation of the three algorithms showed that R L and R NL were biased for reflectivity values less than 45 dBZ and over the same range of reflectivities R 50 had the least bias. Here, R NL had uniformly low bias, though other algorithms were better than R NL at specific regions. Linear and step variation of reflectivity were simulated over a path to analyze the impact of nonlinearity on R(KDP ) algorithms in the presence of nonuniform rainfall paths. Simulation results indicate that in the presence of reflectivity variation along the path the bias in nonlinear rainfall algorithms was about 10%–12% and 5% for reflectivity variations of 12 and 30 dB, respectively. The step variation along the path was evaluated in terms of fraction of high reflectivity region for various step sizes ranging from 10–30 dB. The step variation is meant to evaluate situations in which strong rain cells are in the vicinity of weak echo regions. The analysis of step model showed that the bias due to steptype variation (or sharp gradients) can be as high as 40% if only a small fraction of high-reflectivity region is in the path. If the high- and low-reflectivity regions are over each half of the path, the bias is on the order of 10%. When KDP-based rainfall rate is estimated over a nonuniform path both the effects of the nonlinearity of the algorithm and of the parameterization exist. Evaluation of the R(KDP ) algorithms showed that when the reflectivity was varied linearly starting at 25 dBZ up to a maximum of 55 dBZ the resultant bias for R 50 was the least whereas R NL yielded a negative bias of 10%–25%. However, R L had large negative biases varying between 30% and 50%. In addition to the above issues, another point that is important in KDP-based rainfall estimation is the truncation of KDP at zero to avoid negative rainfall values. This truncation has the effect of introducing positive bias in KDP estimates. This bias was theoretically evaluated and was shown to depend primarily on the fractional standard error (ratio of standard deviation to mean) of KDP . This bias essentially dominates the rainfall estimation using KDP obtained over small paths. Radar and rain gauge data from the CaPE program was used to evaluate the topics discussed in this paper. Histograms of KDP were compared when KDP was estimated over different path lengths. The results show that as path length increases the standard deviation (or spread) of the distribution of observed KDP becomes smaller. Analysis of reflectivity variations along the path showed that on the average there was a 12-dB variation of reflectivity on a 4-km path and this increases to 22 dB along a 16-km path, indicating the amount of variation reflectivity encountered along a typical KDP estimation path. Bias analysis of the R(KDP ) algorithms showed that R NL and R 50 performed very similarly in agreement with the simulation results of this paper. The R L had lower bias at small pathlength, and the difference in the biases between R L and R NL progressively increased as the pathlength increased. Analysis of the FSE in the

JUNE 2000

GORGUCCI ET AL.

comparison between radar and gauge estimates of rainfall shows that there is a broad minimum in R 50 and R NL indicating the relative insensitivity to choice of path between 3 and 8 km. However, beyond 8-km pathlength, the effects of nonuniform rainfall path increased the FSE steadily. Among the algorithms R L had the largest error and R 50 was progressively becoming better than R NL as the pathlength increased. In summary, the data indicate that, in convective rainfall, pathlengths on the order of 3–8 km provide a good compromise in the choice between spatial resolution of rainfall estimate and lowering the measurement error for rainfall estimation using KDP . Acknowledgments. This research was supported partially by the National Group for Defense from Hydrological Hazard (CNR, Italy), by the Progetto Strategico Mesoscale Alpine Program (CNR, Italy), Italian Space Agency (ASI), and by the National Science Foundation. The authors would like to acknowledge Drs. D. S. Zrnic´ and A. V. Ryzhkov of NSSL for their comments that improved the clarity of presentation in the manuscript. The authors are grateful to P. Iacovelli for assistance rendered during the preparation of the manuscript. REFERENCES Aydin K., V. N. Bringi, and L. Liu, 1995: Rain-rate estimation in the presence of hail using S-band specific differential phase and other radar parameters. J. Appl. Meteor., 34, 404–410. Chandrasekar, V., V. N. Bringi, N. Balakrishnan, and D. S. Zrnic´ ,

955

1990: Error structure of multiparameter radar and surface measurements of rainfall. Part III: Specific differential phase. J. Atmos. Oceanic Technol., 7, 621–629. , G. R. Gray, and I. J. Caylor, 1993: Auxiliary signal processing system for a multiparameter radar. J. Atmos. Oceanic Technol., 10, 428–431. Gorgucci, E., and G. Scarchilli, 1997: Intercomparison of multiparameter radar algorithms for estimating rainfall rate. Preprints, 28th Conf. on Radar Meteorology, Austin, TX, Amer. Meteor. Soc., 55–56. , , and V. Chandrasekar, 1999a: A procedure to calibrate multiparameter weather radar using properties of the rain medium. IEEE Trans. Geosci. Remote Sens., 37, 269–276. , , and , 1999b: Specific differential phase estimation in the presence of nonuniform rainfall medium along the path. J. Atmos. Oceanic Technol., 16, 1690–1697. Hubbert, J., V. Chandrasekar, and V. N. Bringi, 1993: Processing and interpretation of coherent dual-polarized radar measurements. J. Atmos. Oceanic Technol., 10, 155–164. Liu, L., V. N. Bringi, V. Chandrasekar, E. A. Mueller, and A. Mudukutore, 1994: Analysis of the copolar correlation coefficient between horizontal and vertical polarizations. J. Atmos. Oceanic Technol., 11, 950–963. Ryzhkov, A. V., and D. S. Zrnic´, 1996: Assessment of rain measurement that uses specific differential phase. J. Appl. Meteor., 35, 2080–2090. Sachidananda, M., and D. S. Zrnic´, 1987: Rain rate estimates from differential polarization measurements. J. Atmos. Oceanic Technol., 4, 588–598. Scarchilli, G., E. Gorgucci, V. Chandrasekar, and T. A. Seliga, 1993: Rainfall estimation using polarimetric techniques at C-band frequencies. J. Appl. Meteor., 32, 1150–1159. Seliga, T. A., and V. N. Bringi, 1978: Differential reflectivity and differential phase shift: Application in radar meteorology. Radio Sci., 13, 271–275. Ulbrich, C. W., 1983: Natural variations in the analytical form of raindrop size distributions. J. Climate Appl. Meteor., 22, 1764– 1775.