The possibility of rainfall estimation using R ( Z , Z DR , K DP , A H ) : 1 A case study of heavy rainfall on 25 August 2014 in Korea 2 3

11 To improve the accuracy of polarimetric rainfall relations for heavy rainfall, an extreme 12 rainfall case was analysed and some methods were examined. The observed differential 13 reflectivity (ZDR) quality check was theoretically investigated using the relation between the 14 standard deviation of differential reflectivity and cross correlation, and the light rain method 15 for ZDR bias was also applied to the rainfall estimation. The best performance for this heavy 16 rainfall case was obtained when the moving average of ZDR over a window size of 9 gates was 17 applied to the rainfall estimation using horizontal reflectivity (ZH) and ZDR and to the 18 calculation of ZH bias. The differential reflectivity calculated by disdrometer data may be an 19 alternative to the vertical pointing scan for calculating ZDR bias. The accuracy of the 20 combined rainfall relation, R(Z,ZDR,KDP,AH) was relatively insensitive to ZDR and ZH biases in 21 both observations and simulations. 22 23


Introduction
Weather radar is a very useful remote sensing instrument for estimating rainfall amount due to its high spatial and temporal resolution compared with other instruments.Calculations of radar rainfall are based on the relationship between reflectivity (Z) and rain rate (R) known as the Z-R relation (hereafter R(Z)).Experimentally measured drop size distributions (DSDs) have been extensively used to obtain both radar reflectivity and rain rate (Compos andHydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2015-515, 2016 Manuscript under review for journal Hydrol.Earth Syst.Sci.Published: 18 January 2016 c Author(s) 2016.CC-BY 3.0 License.control for differential phase shift (You et al., 2014).You et al. (2014) found that the accuracy of rainfall estimation using horizontal reflectivity (Z H ) and differential reflectivity (Z DR ) obtained by DSDs in the Busan area in Korea was better than that obtained with relations calculated by DSDs measured in Oklahoma in the US.A quality control algorithm and unfolding of differential phase shift (Ф DP ) for calculating specific differential phase (K DP ) were applied to the rainfall estimation (You et al., 2014).Recently, You et al. (2015a) proposed a relation combining many polarimetric variables of the form R(Z,Z DR ,K DP ,A H ) as a candidate for an optimum rainfall relation for S-band polarimetric data in Korea; this would allow a single relation to be used for different hydrometeor regimes in the absence of a stable hydrometeor classification algorithm.However, there are still issues to be resolved in improving Z DR data quality and the robustness of R(Z,Z DR ,K DP ,A H ) for the heavy rainfall case where error propagation from each polarimetric variable can occur.This paper discusses how to improve the accuracy of rainfall estimation using moving averaged differential reflectivity and examines the robustness of the R(Z,Z DR ,K DP ,A H ) relation for a heavy rainfall case in Korea.Sect. 2 describes the rain gage, DSD and radar dataset, together with the calculation of polarimetric variables from DSDs and the validation methods.
Sect. 3 provides Z H and Z DR bias correction, an examination of Z DR data quality, and the statistical results of rainfall estimation using observed and moving-average Z DR .Sect. 4 contains a discussion of a possible method for improving R(Z,Z DR ) accuracy and the robustness of the R(Z,Z DR ,K DP ,A H ) relation.Finally, we provide some conclusions in Sect. 5.

Gage, disdrometer and radar dataset
The rainfall data from rain gages operated by the KMA were used to evaluate the accuracy of radar rainfall.Rain gages located within the radar coverage area at distances from 5 to 95 km of the radar are included in the analysis.Fig. 1 shows the location of all instruments used in this study.The circle is the radar coverage, the solid rectangle is the centre of the Bislsan radar, the plus signs show the rain gages within the radar coverage and the open rectangle is the location of a PARSIVEL (PARticle Size VELocity) and POSS (Precipitation Occurrence Hydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2015-515, 2016 Manuscript under review for journal Hydrol.Earth Syst.Sci.Published: 18 January 2016 c Author(s) 2016.CC-BY 3.0 License.
Sensor System; detailed specifications are provided by Sheppard, 1990) disdrometer installed ~82 km away from the radar.
Relations for converting radar variables into rain rate are required because the radar does not observe rainfall directly.To calculate these relations, disdrometer data that can measure the DSDs are needed.One-min DSDs obtained by the POSS from 2001 to 2004 were used.To improve the accuracy of Z DR , DSDs observed by PARSIVEL on 25 August 2014 were used because POSS data were not available at that time.The PARSIVEL disdrometer is a laseroptic system that measures 32 channels from 0.062 to 24.5 mm (detailed specifications are given by Loffler-Mang and Joss, 2000).
Unreliable data, defined as belonging to the following categories, were removed: 1-min rain rate less than 0.1 mm h -1 ; total number concentrations of all channels less than 10; drop numbers counted only in the lower 10 channels (0.84 mm for POSS and 1.187 mm for PARSIVEL); and drop numbers counted only in the lower 5 channels (0.54 mm for POSS and 0.562 mm for PARSIVEL) (You et al., 2015b).
Radar data were collected by the Bislsan polarimetric radar installed and operated by the MoLIT in Korea since 2009.The transmitted peak power is 750 kW, beam width is 0.95°, and frequency is 2.791 GHz.The polarimetric variables are estimated with a gate size of 0.125 km.
The scan strategy is composed of 6 elevation angles with 2.5-min update interval.
Polarimetric variables for 0.5° elevation angle were extracted from the volume data every 10 mins for this study.

Calculation of polarimetric variables from DSDs
Polarimetric variables were calculated using T-matrix scattering techniques derived by Waterman (1971) and later developed further by Mishchenko et al. (1996).The following raindrop shape assumptions are used for the calculation of variables from the DSDs: Eq. ( 1) is for the equilibrium axis ratio derived from the numerical model of Beard and Chung (1987), which is in good agreement with the results from wind tunnel measurements.The actual shapes of raindrops in turbulent flow are expected to be different from the equilibrium shapes due to drop oscillations.Oscillating drops appear to be more spherical on average than drops with equilibrium shapes as shown by Andsager et al. (1999) in laboratory studies.They demonstrated that the shape of raindrops with diameter between 1.1 and 4.4 mm is better explained by Eq. ( 2).You et al. (2015a) found that combining Eq. ( 1) for drops less than 1.1 mm and larger than 4.4 mm with Eq. ( 2) for the drop diameter between 1.1 and 4.4 mm as proposed by Bringi et al. (2003) gave the best rainfall estimation compared with other drop axis ratio assumptions in Korea, and we use this combined formulation in this study.Other parameters in the T-matrix calculations include the temperature, which is assumed to be 20°C in this study.The distribution of canting angles of raindrops is Gaussian with a mean of 0° and a standard deviation of 7°, as determined recently by Huang et al. (2008).

Validation
The localized rainfall on 25 August 2014 was caused by a low pressure system that passed through southern Korea.Fig. 2 shows the time series of hourly rainfall and accumulated rainfall from the three gages, ID 255 (North Changwon site), ID 926 (Jinbook site), and ID 939 (Geumjeong-gu site) that recorded the highest rainfall within the radar coverage area.The daily accumulated rainfall values were 243.5 mm, 269.0 mm, and 244.5 mm for these gages.
The time period analysed was from 0900 LT to 1600 LT because the rainfall was concentrated in this period and radar data were available from 0900 LT.
The normalized error (NE), fractional root mean square error (RMSE), and correlation coefficients (CC) of the rainfall relations and 121 gages were used to investigate the performance of each rainfall relation: Hydrol.Earth Syst.Sci. Discuss., doi:10.5194/hess-2015-515, 2016 Manuscript under review for journal Hydrol.Earth Syst.Sci.Published: 18 January 2016 c Author(s) 2016.CC-BY 3.0 License.
where N is the number of radar rainfall (R R ) and gage rainfall (R G ) pairs, and R R and G R are the average hourly rain rates from the radar and gage, respectively.These statistical variables are calculated using hourly rainfall amounts derived from the radar and gage at the location of the gage.The radar rainfall at the rain gage was obtained by averaging rainfall over a small area (1 km × 1°) centered on each rain gage.The rainfall relations for calculating radar rainfall were obtained from the simulated polarimetric variables generated from DSDs and are summarized in Table 1.

Improvement of Z DR data quality
Z DR is an important variable for hydrometeor classification and rainfall estimation.To check the quality of the Z DR measurements, the radial profile of Z DR was investigated as shown in where SD(Z DR ) is standard deviation of Z DR , N is the number of samples and where  is the cross correlation, v  is Doppler width, n is sample number, and s T is dwell time.
For a better comparison we display the correlations in L space, as proposed by Keat et al. ( where, ρ hv is cross correlation.The vertical pointing data were not available for the case considered here and the scan strategy with six elevation angles does not detect the melting layer.Therefore, light rain measurements close to the ground were used to calibrate the Z DR and Z H biases using the selfconsistency method in this study.Very light rain was defined by the thresholds 20 dBZ ≤ Z H ≤ 28 dBZ as proposed by Marks et al. (2011).The Z H bias was determined following Ryzhkov et al. (2005b).
The Z H biases calculated with the self-consistency method using observed Z DR and mZ DR are -1.95dB and -1.48 dB, respectively.The Z DR biases calculated by the very light rain method using observed Z DR (0.26 dB) and mZ DR (0.3 dB), respectively.

Validation
To investigate the performance of R(Z,Z DR ) and R(Z,Z DR ,K DP ,A H ), which is related to the Z H and Z DR bias, NE, RMSE, and CC were calculated using hourly rainfall from each relation Fig. 6 shows the scatter plot of 1 hour rainfall obtained using R(Z,Z DR ) and gage data.In Fig. 6 (a) the Z H bias was obtained from the observed Z DR bias and the Z DR biases calculated from observed Z DR (blue full circles) and mZ DR (red full circles).The RMSE, NE, and CC of the relation using mZ DR were as much as 8 mm h -1 , 0.1, and 0.18 better than those obtained using observed Z DR , respectively.In Fig. 6(b) the Z H bias is calculated from mZ DR ; the improved performance using mZ DR is clear.The accuracy of the rainfall estimate using Z H bias obtained by mZ DR is statistically more robust than that for the estimate based on observed Z DR .The RMSE, NE, and CC for the comparison of R(Z,Z DR ) rainfall obtained using different Z H and Z DR biases are summarized in Table 2.
Fig. 7 shows the scatter plots when R(Z,Z DR ,K DP ,A H ) is used for rainfall estimation.Fig. 7(a) shows the radar rainfall calculated using the Z H bias obtained from the observed Z DR bias and the Z DR biases obtained from observed Z DR (blue full circles) and mZ DR (red full circles).The RMSE, NE, and CC from each relation were not very different; differences of RMSE, NE, and CC in the two cases were 0.2 mm h -1 , 0.01, and 0, respectively.The statistics for the comparison of radar rainfall obtained using different Z H and Z DR biases are summarized in Table 3.These results show that R(Z,Z DR ,K DP ,A H ) is less sensitive to Z H and Z DR error than R(Z,Z DR ).This will be discussed further in Sect.4.2 using simulated data.

Impact of disdrometer data on radar rainfall
In the cases described in Sect.3.3, the accuracy of the R(Z,Z DR ) relation was improved when the moving-average Z DR (i.e., mZ DR ) was used to estimate rainfall.To improve the accuracy of rainfall estimation using R(Z,Z DR ), we examined the impact of Z DR bias (as obtained from disdrometer data) on the accuracy.The DSD data were quality controlled and polarimetric variables were calculated by T-matrix simulation with the same configuration as in Sect. 2.
Before applying the DSDs to rainfall estimation, 10-min rainfall amounts obtained by DSDs and gages were compared.According to these results, when moving average Z DR (i.e., mZ DR ) is used with the Z DR bias measured by PARSIVEL, the accuracy of rainfall estimation was improved and was more stable than that of other configurations using R(Z,Z DR ).
Fig. 11 shows the scatter plots for R(Z,Z DR ,K DP ,A H ) and gages.The statistical scores were not very different from Z H and Z DR biases.The differences of RMSE, NE, and CC between each relation were 0.4 mm h -1 , 0.01, and 0, respectively.These results were summarized in Table 3.

Simulation of R(Z,Z DR ,K DP ,A H ) with error propagation from each variable
With the relation using combined polarimetric variables, R(Z,Z DR ,K DP ,A H ), error propagation can affect the accuracy of radar rainfall estimation.To examine the contribution of errors from each variable, simulated polarimetric variables such as Z, Z DR , K DP , A H , were generated with dimensions of 960 sizes of bins and 360 radials.To investigate the extent of contamination of the rainfall amount by propagation of errors in each polarimetric variable for R(Z,Z DR ,K DP ,A H ), the errors of Z, Z DR , and K DP ingested to simulated data were 0 to 5 dBZ with interval 0.25 dBZ, 0 to 0.6 dB with interval 0.03 dB, and 0 to 0.2 degree km -1 with interval 0.01 degree km -1 , respectively.The rain rate was calculated by same R(Z,Z DR ,K DP ,A H ) as applied to real data in the previous Sect..The RMSE and NE were calculated for rainfall amount with and without error-ingested polarimetric variables.
The rainfall amount obtained using the raw simulated variables was used as a reference.The theoretical approach to investigate the observed Z DR quality used the relation between the standard deviation of Z DR and hv  using the scan strategy parameters of the Bislsan radar.
The result showed that more samples were required to achieve the theoretical accuracy in Z DR .
The best performance was obtained when a moving average Z DR with window size of 9 gates was applied to the rainfall estimation using R(Z,Z DR ) and to the calculation of Z H bias.The Z DR quality check should be performed before using Z DR for quantitative applications like rainfall estimation and hydrometeor classification for the Bislsan radar.We also expect that the light rain method for obtaining the Z DR bias may be used as an alternative to the vertical pointing scan method, because the rainfall estimation using this method performed well in our case.Using DSD data for the calculation of Z DR bias might give more accurate rainfall estimation with R(Z,Z DR ).
Finally, the accuracy of R(Z,Z DR ,K DP ,A H ) was not very sensitive to Z DR and Z H biases in both observations and simulations.Thus R(Z,Z DR ,K DP ,A H ) is expected to be less sensitive to Z DR and Z H errors and could be used to estimate rainfall for heavy rainfall cases in Korea until an accurate hydrometeor classification algorithm is developed.

Fig. 3 .
Fig. 3. Fig. 3(a) shows the spatial distribution of Z DR at 0.5° elevation at 1401 LT on 25 August 2014.Fig. 3(b) shows the radial profile of observed Z DR (red line) and the standard deviation of Z DR (black line) calculated using 9 gates along the line A-B shown in Fig. 3(a).The average standard deviation of Z DR along the line was 0.615 dB.Fig. 3(c) shows the radial profile of the cross correlation; the average cross correlation was 0.982.To find the accuracy of the observed Z DR value, we use the theoretical relation between the standard deviation of Z DR and the cross correlation following Bring and Chandrasekar (2003):
Fig. 5 shows the results for Z DR measurements at 1401 LT on 25 August 2014.Fig. 5(a) shows the spatial distribution of a moving average Z DR from 9 gates.Fig. 5(b) shows the radial profile of the Z DR (red line) and its standard deviation (black line) calculated for 9 gates along the line A-B shown in Fig. 5(a).The average standard deviation of Z DR along the ray was 0.169 dB.Fig. 5(c) shows the radial profile of the cross correlation; the average cross correlation was 0.985.Both the standard deviation of Z DR and the averaged hv  values are Hydrol.Earth Syst.Sci.Discuss., doi:10.5194/hess-2015-515,2016   Manuscript under review for journal Hydrol.Earth Syst.Sci.Published: 18 January 2016 c Author(s) 2016.CC-BY 3.0 License.and from the gages.For the comparison of rainfall amount, two different Z H and Z DR biases were applied to observed variables as mentioned in Sect.3.2.Each bias was calculated using observed Z DR and mZ DR .
Fig.8shows the scatter plot of 10-min rainfall amount measured by PARSIVEL and the gage located less than 100 m away from PARSIVEL.The daily accumulated rainfall amounts were 116.0 mm for the gage and 129.4 mm for PARSIVEL.The RMSE, NE, and CC were 0.52 mm, 0.26, and 0.99, respectively.For the comparison the Z DR of the radar was averaged over 3 km × 3° as shown in Fig.9.The calculated Z DR biases were 0.26 dB for observed Z DR and 0.30 dB for mZ DR .The Z H biases described in Sect.3.3 were used.Fig.10shows the scatter plots of 1-hour rainfall obtained by R(Z,Z DR ) and gages.The radar rainfall was calculated after Z DR bias correction using the bias result in the comparison between radar Z DR and PARSIVEL Z DR .The Z DR biases were -0.05 dB for observed Z DR and -0.07 dB for mZ DR .In Fig.10(a) the Z H bias was obtained from the observed Z DR bias and Z DR biases calculated from observed Z DR (blue full circle) and mZ DR (red full circle).The radar rainfall using mZ DR was better than that using observed Z DR by as much as 5.5 mm h -1 for RMSE and 0.36 for NE.In Fig.10 (b) the Z H bias was calculated from mZ DR ; the improved rainfall estimation using mZ DR is clear.This result shows the better scores compared with the statistics shown in Fig.6that were obtained using Z DR biases extracted from the radar Z DR only.When the observed Z DR , which fluctuates considerably along the ray, was applied to the rainfall estimation, the rainfall amount was much more variable with Z H bias values (blue full circle) than that with mZ DR (red circle) as shown in Fig.10(a) and (b).

Fig. 12
Fig.12shows the distribution function of the polarimetric variables generated assuming a Gaussian distribution in each case.Fig.12(a)  shows the occurrence frequency of Z H generated

Fig. 13
Fig.13shows the RMSE and NE distribution of different polarimetric rainfall relations with ingested error.The magenta, black, red, green, blue, and purple lines show RMSE and NE obtained by the rainfall relations R(Z), R(K DP ), R(Z,K DP ,A H ), R(Z,Z DR ), R(K DP ,Z DR ), and R(Z,Z DR ,K DP ,A H ), respectively.The threshold rainfall was from 0 to 300 mm h -1 for calculating statistical scores.Fig.13(a) shows the RMSE distribution of each rainfall relation with different ingested error step.The RMSE of R(Z,K DP ,A H ) is the largest of all the rainfall relations.The RMSE of R(Z,Z DR ,K DP ,A H ) is higher than that of R(Z), R(Z,Z DR ), and R(K DP ,Z DR ) but less than that of R(K DP ).It means that not all errors from Z, Z DR , and K DP propagate into the R(Z,Z DR ,K DP ,A H ).Fig.13(b)  shows the corresponding distributions for NE.The value of NE increases in the order R(Z,Z DR ,K DP ,A H ), R(Z,K DP ,A H ), R(K DP ), R(K DP ,Z DR ), R(Z,Z DR ), and R(Z).In Sect.3.3 and 4.1, the statistical scores of R(Z,Z DR ,K DP ,A H ) did not change significantly with respect to different Z H and Z DR biases.The results of the simulation and observations suggest that the accuracy of R(Z,Z DR ,K DP ,A H ) is relatively weakly affected by errors in each polarimetric variable.

Figure 8 .
Figure 8. Scatter plot of 10 min rainfall amount measured by PARSIVEL and gage for 24 hours.

Figure 9 .Figure 10 .Figure 11 .Figure 12 .
Figure 9. Schematic diagram for the comparison of radar and PARSIVEL Z DR .The numbers refer to azimuth angle.

Table 1 .
Polarimetric radar rainfall relations used in this study.

Table 2 .
Statistics of the comparison of hourly rainfall amount between R(Z,Z DR ) and gages.