Error analysis and modeling of C-band dual polarization radar quantitative precipitation estimation
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摘要: 双线偏振雷达定量降水估计精度受多种因素影响，为了更好地应用双偏振雷达估计降水并进一步提高降雨估测精度，需对雷达降水估计进行误差分析和建模。基于2015—2016年南京信息工程大学C波段双偏振雷达、雨滴谱仪观测资料以及南京地区雨量计数据，统计分析雷达估测降水的误差分布，分离雨量计代表性误差，并对随机误差和系统误差量化建模。首先对双偏振雷达数据进行预处理，并利用雨滴谱仪数据拟合测雨方程，通过对R（ZH）、R（ZH，ZDR）、R（KDP）、R（KDP，ZDR） 4个测雨公式反演结果与雨量计对比，分析每个测雨公式在不同降水强度时的估测性能。随后，利用雨量计数据估计空间相关函数，计算并分离由于雷达和雨量计空间不匹配造成的雨量计代表性误差，并分析测量误差和参数误差在雷达降水估计误差中的比重。基于雷达降水估计误差属性及分布规律，建立随机误差和系统误差量化模型。最后，基于4个测雨公式性能和误差分析，提出一种优化组合的双偏振雷达降水估计算法。结果表明，R（ZH）和R（ZH，ZDR）在弱降水时性能较好，当雨强阈值大于2.5 mm/h时，KDP的优势得以突显。由于空间不匹配造成的雨量计代表性误差不容忽视，在雷达分辨单元较大时进行定量降水估计需剔除点面误差。将雷达降水估计误差按照系统误差和随机误差进行建模，发现雷达近地面降水的系统误差和雨强成正比，呈线性函数形式，而双指数模型更好地表示随机误差分布。基于不同雷达测雨公式性能提出的优化组合降水算法在准确度和稳定性等方面优于单个测雨公式。Abstract: The quantitative precipitation estimation accuracy of dual-polarization radar is affected by many factors. In order to better use dual-polarization radar to estimate precipitation and further improve the rainfall estimate accuracy, error analysis and modeling of radar precipitation estimation are needed. Based on observations of the C-band dual-polarization radar of Nanjing University of Information Science and Technology from 2015 to 2016, raindrop spectrometer observation data and rain gauge data in Nanjing, the error distribution of radar estimation precipitation is statistically analyzed, the gauge representativeness error variance is separated, and the error quantitative model based on random error and systematic error is established. Firstly, the data of dual polarization radar is preprocessed, and the radar rainfall formula is fitted by the data of raindrop spectrograph in the observation base of Nanjing University of Information Science and Technology. By comparing the four radar rainfall formula R (ZH), R (ZH, ZDR), R (KDP), R (KDP, ZDR) with the rain gauge, the estimation performance of each formula under different precipitation thresholds is analyzed. Then, the spatial correlation function of rain gauge data is estimated, the gauge representative error caused by spatial mismatch between radar and rain gauge is calculated, and the proportion of measurement error and parameter error in radar precipitation estimation error is analyzed. Based on the attribute and distribution rule of radar precipitation estimation errors, they are divided into random errors and systematic errors, and a quantitative model is established. Finally, based on the performance and error analysis of four radar rainfall formula, an optimized combination of dual polarization radar precipitation estimation algorithm is proposed. The results show that R (ZH) and R (ZH, ZDR) have better performance for light precipitation estimation. When the rainfall threshold is greater than 2.5 mm/h, the advantage of KDP becomes obvious. The gauge representative error caused by spatial mismatch cannot be ignored. Therefore, the point to area error should be eliminated when the radar resolution unit is large. The radar error is modeled according to the systematic error and random error, and it is found that the systematic error of radar near surface precipitation is proportional to the rainfall intensity in the form of linear function, and the double exponential model better represents the random error distribution. Through the performance analysis of radar rain measurement formula and dual polarization signal analysis, it is found that the optimized combination is better than the single rain algorithm in accuracy and stability.
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图 2 2015年6月27日00时32分0.5°仰角
$ {Z}_{\rm H} $ （单位：dBz）数据PPI （a. C波段雷达原始$ {Z}_{\rm H} $ ， b. SA雷达原始$ {Z}_{\rm H} $ ，c. C波段雷达衰减订正后的$ {Z}_{\rm H} $ ，d. NUIST-CDP雷达和SA雷达$ {Z}_{\rm H} $ 的散点）Figure 2. PPI images of （a） original
$ {Z}_{\rm H} $ of NUIST-CDP，（b） original$ {Z}_{\rm H} $ of SA radar，（c）$ {Z}_{\rm H} $ of NUIST-CDP after attenuation correction at 0.5° elevation at 00：32 UTC 27 June 2015 and （d） scatter plot of$ {Z}_{\rm H} $ from NUIST-CDP and SA randar （unit：dBz）
图 3 NUIST-CDP雷达249°方位角上距离廓线对比 （a. ΦDP，b. KDP，c. ZH，d. ZDR；图c黄色线为对应S波段廓线数据）
Figure 3. Distance profile comparison of ΦDP （a）， KDP （b）， ZH （c） and ZDR （d） data before and after processing at 249° azimuth angle from NUIST-CDP （the yellow line in c is the corresponding S-band profile data）
图 4 测雨公式拟合结果与观测值散点 （a.
$ {R}_{1}=0.4722{Z}_{\rm H}^{0.6346} $ ，b.$ {R}_{2}=0.0034{Z}_{\rm H}^{0.9349}{Z}_{\rm{DR}}^{-0.75121} $ ，c.$ {R}_{3}=26.81{K}_{\rm{DP}}^{0.7224} $ ，d.$ {R}_{4}=32.2686{Z}_{\rm{DR}}^{-0.7487}{K}_{\rm{DP}}^{1.0181} $ ）Figure 4. Scatter plots of fitting results of radar rainfall formula and observed values from raindrop spectrograph （a.
$ {R}_{1}=0.4722{Z}_{\rm H}^{0.6346} $ ， b.$ {R}_{2}=0.0034{Z}_{\rm H}^{0.9349}{Z}_{\rm{DR}}^{-0.75121} $ ，c.$ {R}_{3}=26.81{K}_{\rm{DP}}^{0.7224} $ ， d.$ {R}_{4}=32.2686{Z}_{\rm{DR}}^{-0.7487}{K}_{\rm{DP}}^{1.0181} $ ）
图 5 NUIST-CDP雷达降水强度与地面雨量计对比 （a. R（ZH），b. R（ZH，ZDR），c. R（KDP）， d. R（KDP，ZDR））
Figure 5. Comparison of the NUIST-CDP radar calculated precipitation intensity and the ground rain gauge （a. R（ZH） method，b. R（ZH，ZDR） method，c. R（KDP） method，d. R（KDP，ZDR） method）
图 6 不同降雨阈值下的空间相关函数 （降雨阈值大于 （a） 0、（b） 0.2、（c） 1.0 mm/h）
Figure 6. Spatial correlation functions for different rainfall thresholds using 60-min accumulations （The rainfall threshold is greater than （a） 0，（b） 0.2，and （c） 1.0 mm/h ）
图 7 （a） 雨量计降水量与区域平均降水量散点，（b） var（Rg−Ra）随平均雨强的变化趋势
Figure 7. （a） Scatterplot between the true area-averaged rainfall
$ {R}_{\rm a} $ vs point gauge$ {R}_{\rm g} $ rainfall and （b） gauge representativeness error var（Rg−Ra） vs mean areal rainfall$ {R}_{\rm a} $ 
图 8 （a） 点面误差方差占雷达雨量计总误差方差的百分比，（b） 各测雨公式的 var（Rr−Rg）随雨强的变化趋势
Figure 8. （a） Ratio of point-to-area variance to the total error variance；（b） variance of （Rr−Rg） for the four algorithms used in the radar-based estimates，shown for various hourly thresholds of rainfall
图 9 四个测雨公式的雷达降水估计误差FSE估计值（实线） （a. R（ZH），b. R（ZH，ZDP），c. R（KDP），d. R（KDP，ZDP）；柱是式（15）的FSE′）
Figure 9. Estimates of the FSE （solid line） for the radar precipitation estimation error for the four algorithms shown for various hourly thresholds of rainfall （a. R（ZH），b. R（ZH，ZDP），c. R（KDP），d. R（KDP，ZDP）；bars represent FSE′ from Eq. （15））
图 10 四个测雨公式的系统误差模型 （a1—d1） 和相应的随机误差模型 （a2—d2）
Figure 10. Systematic error models （a1—d1） and random error models （a2—d2） for the four rainfall algorithms
图 11 （a） R（ZH）、（b） R（ZH，ZDR）、（c） R（KDP）和 （d） R（C）在ZH-ZDR空间中的每小时定量估测降水的归一化误差 （黑线代表三种降雨估计值的阈值，用以计算综合的R（C））
Figure 11. Hourly normalized QPE errors of （a） R（ZH），（b） R（ZH，ZDR），（c） R（KDP） and （d） composite algorithm R（C） for the three cases in ZH-ZDR space （black lines represent the thresholds of three rainfall estimators to calculate composite R（C））
图 12 双线偏振雷达降水估测综合方法
$ R\left(C\right) $ 算法流程Figure 12. Block diagram illustrating the composite algorithm
$ R\left(C\right) $ for rain-rate estimation by dual linear polarization radar
图 13 2015年6月26—28日个例NUIST-CDP雷达定量降水估测公式对比散点 （a） R（ZH），（b） R（ZH，ZDR），（c） R（KDP）以及（d） 优化组合算法散点对比 （其中图a、b、c中黑色点是Chen 等 （2017） 拟合的雷达公式结果）
Figure 13. Scatter plots of （a） R（ZH） method，（b） R（ZH，ZDR） method， （c） R（KDP） method，and （d） composite algorithm R（C） for 26—28 June 2015 （black points in （a），（b） and （c） are the results of radar rainfall formula fitted by Chen，et al （2017））
表 1 测雨公式拟合结果
Table 1. Fitting results of rain measurement formula
测雨公式 $ {a}_{x} $ $ {b}_{x} $ $ {c}_{x} $ 相关系数 $ {R}_{1}={a}_{1}\times {Z}_{\rm H}^{{b}_{1}} $ 0.04722 0.6346 \ 0.97272 $ {R}_{2}={a}_{2}\times {Z}_{\rm H}^{{b}_{2}}\times {Z}_{\rm{DR}}^{{c}_{2}} $ 0.003436 0.93485 −0.75121 0.99278 $ {R}_{3}={a}_{3}\times {K}_{\rm{DP}}^{{b}_{3}} $ 26.81 0.7224 \ 0.98152 $ {R}_{4}={a}_{4}\times {Z}_{\rm{DR}}^{{b}_{4}}\times {K}_{\rm{DP}}^{{c}_{4}} $ 32.2686 −0.7487 1.0181 0.99667 
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表 2 不同阈值下各测雨公式统计性能分析和统计使用的样本数
Table 2. Key statistics of radar-gauge comparison for four thresholds along with the number of samples used in computing the statistics
阈值（mm/h） 算法 FSE（%） MAE（mm/h） RMSE（mm/h） ρ points 0.2 R（ZH） 102.42 2.177 4.572 0.796 47291 R（ZH，ZDR） 111.11 2.981 5.467 0.734 R（KDP） 96.51 2.670 4.252 0.765 R（KDP，ZDR） 100.47 2.475 4.255 0.740 R（ZH原） 113.20 2.360 5.071 0.721 1.0 R（ZH） 87.62 2.795 5.320 0.774 34669 R（ZH，ZDR） 93.60 3.914 6.378 0.712 R（KDP） 83.61 2.863 4.694 0.749 R（KDP，ZDR） 87.41 3.070 4.913 0.724 R（ZH原） 96.83 3.043 5.905 0.688 2.5 R（ZH） 69.95 4.196 6.837 0.745 20634 R（ZH，ZDR） 73.48 5.788 8.200 0.681 R（KDP） 65.33 3.175 5.262 0.748 R（KDP，ZDR） 69.97 4.125 6.029 0.719 R（ZH原） 77.43 4.626 7.606 0.643 7.6 R（ZH） 45.71 9.237 11.590 0.682 6701 R（ZH，ZDR） 47.13 11.518 13.574 0.622 R（KDP） 42.32 4.702 6.827 0.729 R（KDP，ZDR） 46.84 6.726 8.620 0.704 R（ZH原） 50.50 10.322 12.935 0.564
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表 3 R（C）统计性能分析
Table 3. Statistical performance analysis of R（C）
阈值（mm/h） FSE（%） MAE（mm/h） RMSE（mm/h） $ \rho $ 0.2 85.94 2.023 3.620 0.823 1.0 75.86 2.609 4.191 0.805 2.5 62.46 3.330 5.086 0.780 7.6 42.42 4.772 6.870 0.730
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