1. Introduction

Quantitative precipitation estimation(QPE)isone of the most important applications for weatherradar. A comprehensive review of the reliability ofweather radar QPE products was conducted by Wilson andBrandes(1979), and it discussed in detailthe sources of uncertainty associated with radar-basedrainfall estimates. These include calibration, attenuation, anomalous propagation, bright b and , beamblockage, ground clutter, spurious returns, and r and omerrors. Moreover, the variability in the relationshipbetween reflectivity(Z) and rainfall rate(R)(Z–R relations)was also discussed. Since that time, therehas been great progress in radar hardware and QPEalgorithms. Polarimetric radar can measure multiplepolarization parameters, including differential reflectivityZ_DR, specific differential phase K_DR, and cross-correlation coefficient ρHV between two orthogonalradar returns. Z_DR reflects the median drop diameter.K_DR is immune to radar miscalibration, attenuationin precipitation, and beam blockage, while ρDRcan significantly improve the radar data quality, distinguishingrain echoes from the radar signals causedby other scatters such as snow, ground clutter, insects, birds, chaff, etc.(Zrnic and Ryzhkov, 1996; Krajewski et al., 2010). In recent years, radar meteorologistshave paid more attention to the polarimetric radar and its QPE products.

By combining a variety of polarization observationalparameters, several QPE algorithms such asR(Z_H), R(Z_H, Z_DR), R(Z_H, Z_DR, K_DP), and R(K_DP)have been investigated in the past decades(Bringi and Chandrasekar, 2001; Liu et al., 2002; Brandes et al., 2003; Ryzhkov et al., 2005; Hu et al., 2010). However, since polarimetric variables are the difference betweenhorizontal and vertical polarization, their values areone order of magnitude smaller than those measuredfrom only a single polarization direction. Moreover, polarimetric variables are easily influenced by noise, which needs to be removed before generating the polarimetricradar-derived rainfall estimation. For example, Jameson(1991) and Ryzhkov and Zrnić(1995)investigated the relationship between rainfall and polarimetricparameters. Their results indicated thatR(K_DP, Z_DR)was better relative to R(Z_H), R(Z_H, Z_DR), and R(K_DP)for moderate to heavy rain rates, however, K_DP and Z_DR needed to be smoothed forgreater accuracy of R(K_DP, Z_DR)relative to R(K_DP).Carey et al.(2000)developed a propagation correctionalgorithm utilizing the differential propagation phase(Φ_DP) and tested this algorithm using experimentaldata from the C-b and polarimetric radar. Their resultshowed that the algorithm greatly reduced theerror of R(K_DP, Z_DR). Cifelli et al.(2011)presentedan optimization algorithm to estimate rainfall on thebasis of Z_H, Z_DR, and K_DP, which performed betterin both tropical and extratropical regions. Moreover, the US National Severe Storms Laboratory(NSSL)conducted an operational demonstration for the polarimetricradar KOUN, and tested the reliability ofKOUN radar QPE for different seasons and rain typesusing a large dataset. The hourly rain estimate errorsusing polarimetric QPE were shown to decreasesignificantly compared to the conventional nonpolarimetricQPE(Ryzhkov et al., 2005). The US NationalWeather Service(NWS)upgraded the WeatherSurveillance Radar-1988 Doppler(WSR-88D)networkwith polarimetric capability in 2013.

The polarimetric parameter Z_DR is easily influencedby radar hardware such as the difference betweenthe two orthogonal polarization directions ofantenna gains, rotary joints, waveguides, and receivers(Melnikov et al., 2003). Thus, correction for Z_DR systembias is extremely complicated and a small errorof Z_DR can possibly cause a large bias of Z_DR-basedQPE. In addition, the reliability of Z_DR-based QPEis greatly affected by attenuation, especially for heavyrainfall. K_DP, which has been used more widely for polarimetricradar-derived QPE, is the slope of the Φ_DPprofile and is noisy and unstable in measurement, especiallyfor light rain. Thus, pretreatment of Φ_DP iscrucial for K_DP estimate quality(Liu et al., 2013).

In recent decades, many Φ_DP de-noising methodshave been developed for increasing the accuracyof R(K_DP). Sachidan and a and Zrnić(1987)presentedan analysis of the accuracy of QPE from observed data and a simulation procedure, which indicated that errorscaused by sidelobe contamination significantly affectΦ_DP data. As such, large scale averaging is requiredto obtain reasonably accurate rain rate estimates.Chandrasekar et al.(1990)focused on theerror structure of K_DP, and simulated r and om errorsin ZH, Z_DR, and K_DP, which suggested that R(K_DP)is stable and insensitive to system calibration.As K_DP is crucial in polarimetric radar-derivedrain estimates, radar meteorologists have for manyyears investigated how to obtain accurate K_DP datafrom Φ_DP. An iterative filtering technique has beendeveloped to estimate K_DP, which can separate bothΦ_DP and the differential backscatter phase shift δ fromcomplicated echoes(Hubbert et al., 1993; Hubbert and Bringi, 1995). Ryzhkov and Zrnic(1996)presented analgorithm that uses K_DP exclusively, and suggestedthat R(K_DP)can be successfully applied to low rainrates, however, the Φ_DP fitting interval needs to beadjusted. May et al.(1999)described a K_DP estimationalgorithm, for which R(K_DP)produces higherquality data than R(ZH)for moderate to high rainrates. Gorgucci et al.(1999, 2000)similarly describedan algorithm to correct bias in the estimation of K_DPin nonuniform rainfall paths and evaluated K_DP-basedrainfall algorithms for different pathlengths. With thedevelopment of polarization radar, and the recent applicationof a new mathematical method, Φ_DP and K_DP process methods are constantly developed. Wang and Chandrasekar(2009)presented a robust algorithmto process Φ_DP, which is able to remain in sync withthe spatial gradients of rainfall and produce a highresolutionK_DP.Recently, an extended Kalman filter frameworkwas proposed, which determines rain-rate via a relationshipbetween R, K_DP, and Z_DR(Schneebeli and Berne, 2012; Grazioli et al., 2014). Vulpiani et al.(2012)demonstrated that R(K_DP)has a more seasonalrelationship than R(Z_H), except for the analyzed winterstorm. To estimate high-resolution K_DP, Hu et al.(2012)developed an algorithm that can smoothen Φ_DPas well as a synthetic Z_H/K_DP-based rainfall estimationmethod. They showed that the accuracy of R(Z_H, K_DP)is higher than the traditional R(ZH)methodwhen the rain rate is larger than 5 mm h⁻¹. Giangr and eet al.(2013)presented an application of linearprogramming for physical retrievals, in order toprocess measured Φ_DP by developing realistic physicalconstraints for the monotonicity and polarimetricradar self-consistency. Hu and Liu(2014)introducedwavelet analysis into the Φ_DP de-noising process and further addressed a Φ_DP penalty threshold strategybased on the attribution of weather echoes.

A variety of Φ_DP de-noising methods are usedextensively, yet there remains a lack of comprehensivecomparative analysis of these de-noising methods and their influences on K_DP-based QPE. In thepresent study, several new Φ_DP de-noising methodsare introduced in Section 2. In Section 3, a simulatednoisy Φ_DP profile is constructed and filtered with these and other traditional methods. The bias is comparedwhere K_DP is calculated from the simulated Φ_DP withvarious de-noising methods. In Section 4, an observedΦ_DP radial profile is selected as an example to determinethe differences between raw and de-noised data.In Section 5, K_DP-based rainfall estimates are verifiedagainst rain gauge measurements to evaluate theability of the Φ_DP de-noising methods. Finally, theconclusions and discussion are given in Section 6.2. Description of de-noising methods

With exception of the traditional running average and median filter methods, a host of new algorithmshave also been applied into the Φ_DP de-noising process.The following three methods, finite-impulse responsefilter(FIR), Kalman filter, and wavelet analysis, havebeen proved effective in removing Φ_DP noise.2.1 Finite-impulse response filter

An FIR filter system function H(z)may be givenas follows,

where coefficients b₀, b₁, · · ·, b_m−1 represent the systemunit impulse response h(0), h(1), · · ·, h(m−1); nis the time or the coordinates in space or distance, and indicates the ordinal number of gates in a radar radial;m is the filter window width, and here also refers tothe smoothing gate number in a radar radial. After asignal x(n)is processed with FIR, the output y(n)isshown aswhere ^“*” denotes the convolution. The FIR systemcan be calculated from the following difference equation,

Proakis and Manolakis(1988)used symmetric20th-order associated coefficients in the complex variableZ(Table 1). This method was used to smooththe Φ_DP radial profile for both the German AerospaceResearch Institute’s C-b and radar and the NCAR CP-2 radar(Hubbert et al., 1993; Hubbert and Bringi, 1995). In this study, the same coefficients in Table 1were substituted into bk in Eq.(3)to smooth the Φ_DPradial profile.

2.2 Kalman filter

For the Kalman filter, the observational data areregarded as an output from a state equation and theleast mean square error is used as an optimal estimationcriterion to estimate the state vector of the origi nal system. We assume that r and om signals a and V are defined as the process and measurement noises, which are mutually independence white noises withGauss distributions of p(a)~ N(0, Q) and p(V)~N(0, S), respectively. Here Q and S are the covariancematrixes. To remove the noise of Φ_DP, the process and measurement equations of the Kalman filterare presented as(He et al., 2009)

where the definitions of variables in Eqs.(4)–(9)arelisted in Table 2.2.3 Wavelet analysis

Wavelet analysis can localize a signal in both thetime and frequency domains. It performs multi-scaleanalysis using signal zoom and transform, in whichthe information of signal is kept well. Thus, it hasrapidly become a significant technology in signal processing.Hu and Liu(2014)employed this analysis toremove the noise of Φ_DP, and proposed a Φ_DP processstrategy based on the weather echo attributions. Thewavelet de-noising process generally includes the followingsteps:

(1)Deconstruction: a signal is deconstructed intoapproximate and detailed components of several levelsby a selected wavelet function.

(2)De-noising process: the coefficients of componentsdetailed in each level are suppressed by a selectedthreshold strategy.

(3)Reconstruction: the signal is reconstructed byapproximation and using the processed detailed coefficientswith a selected threshold function.

Based on Hu and Liu(2014), in this study, the db5wavelet function is used to deconstruct a signal intofive levels. The detailed coefficients are suppressed bythe Φ_DP penalty threshold strategy, and the signal isreconstructed by soft function.3. Evaluation with a simulated Φ_DP profile

To quantitatively compare the performance ofK_DP when Φ_DP is de-noised with alternative methods, a radar radial profile is simulated, which passesthrough two convective cloud cells that were assumedof a gamma drop size distribution(Ulbrich, 1983, Chandrasekar et al., 1990; Scarchilli et al., 1993)asfollows:

where N_w is the raindrop number for one unit volume and size interval, D is the equivalent volume diameterof raindrops(mm), N₀ is the concentration parameterwhich is assumed to be 8 × 10³ mm⁻¹ m⁻³, μ is thedistribution parameter(set to zero for this study), and D₀ is the median volume diameter, which is assumedto have the following distribution, where D_max represents the maximum equivalent diameterof raindrops(assumed to be 0.2 cm); r_max isthe diameters of two cells, for which the first is setto 30 km and the next 15 km; and r is the distanceof a raindrop from the center of each cell. We setthe radar wavelength to 5.6 cm and the gate width to150 m. Therefore, there are 300 gate measurementsas the beam propagates through the simulated rainarea. Particle scattering is calculated using the extendedboundary condition method which considersthe relationship between the drop size and the ellipticity(Liu et al., 1989).

Figure 1 shows the simulated radar radial profilesof Z_H, Z_DR, Φ_DP, and K_DP. The two symmetrical raincells are clearly evident in the profiles. As a result ofattenuation, the radial profile of Z_H is not symmetricto the center of the two convective cells. Z_H valuesbehind the centers are obviously smaller than thosein front. The profile of Z_DR reveals a similar behavior, for the same reason. However, Φ_DP is immuneto attenuation and the profile of Φ_DP increases monotonicallyfrom zero to 83.35◦. The K_DP curve derivedfrom Φ_DP shows symmetrical features for the two cells and changes from 0.07◦ to 0.45◦ km⁻¹, with similarfluctuations in Z_H and Z_DR.

In order to simulate a measured Φ_DP profile, white noise with a signal to noise ratio(SNR)of 25dB was added. Figure 2 shows the noisy Φ_DP profile and the de-noised profiles with mean filters(window widths of 7 and 13 points, respectively), median(7 and 13 points, respectively), FIR, Kalman, and waveletmethods. The curves de-noised by the complicate FIR, Kalman, and wavelet methods are obviously smootherthan those by the traditional mean and median methods.Namely, the FIR, Kalman, and wavelet methodshave better de-noised effects than do traditional methods.

Using the least squares fitting method, K_DP isestimated with 13 consecutive Φ_DP in Fig. 2. Thecorresponding K_DP profiles are displayed in Fig. 3.Due to the noise and the various de-noised methods, the profiles of K_DP fluctuate to varying degrees. Themaximum bias of K_DP is 0.57◦ km⁻¹ for the procedurewithout any de-noising(Fig. 3a). After de-noising, themaximum bias of K_DP(Figs. 3b–h)is 0.44, 0.38, 0.46, 0.25, 0.20, 0.14, and 0.06◦ km⁻¹, respectively. Theprofiles of K_DP filtered by FIR, Kalman, and waveletmethods(Figs. 3g–h)are also found more similar tothe true K_DP(Fig. 1)than the profiles filtered by thetraditional methods(Figs. 3b–e). In particular, thewavelet method in Fig. 3h proves the most similar.

In order to quantitatively evaluate the Φ_DP denoisingeffect on the K_DP estimate, the mean absoluteerror ε and mean relative bias d are defined as

where K'_DP and K_DP represent the fitting values inFig. 2 and the simulated value in Fig. 1, respectively, n is the gate ordinal number, and G is 287, the totalnumber of fitted K_DP, i.e., the total Φ_DP gate numberof 300 minus a fitting width of 13.

The K_DP mean absolute error and mean relativebias results are listed in Table 3. Without de-noising, ε(0.19) and d(78.40%)are the largest among thoselisted in Table 3, indicating that all the methods selectedin this study de-noise K_DP and more closelyrepresent the simulated K_DP. The mean and medianfilters with more window points are an improvement, compared to those with fewer points. However, thisdoes not suggest that the more points the better. Thenumber of points of a window needs to be carefully selectedbased on the balance between smoothness and feature maintenance. The variables ε and d of FIR, Kalman, and wavelet methods are smaller than thosegenerated by the traditional methods, with the errorfrom the wavelet method the smallest.

4. Comparison with real Φ_DP radial profile

The following representative real Φ_DP radial profileat 0.5◦ elevation and 62.0◦ azimuth was detectedwith the mobile C-b and dual polarimetric weatherradar(POLC), which transmitted and received horizontal and vertical polarization signals simultaneously, at 0804 BT(Beijing Time)25 June 2013 in Dingyuan, Anhui Province. The main characteristics of the radar, which operated at a frequency of 5.43 GHz and a 150-m gate width, are summarized in Table 4.

Figure 4 shows the profiles of raw data and denoisedΦ_DP data from the previously mentioned methods.The corresponding K_DP are displayed in Fig. 5. Note that the scale of y-axis in Fig. 5a is 10 timeslarger than in Figs. 5b–h. Hence, the K_DP errorsare very large if Φ_DP data are not preprocessed beforeK_DP fitting. K_DP derived by the Kalman method(Fig. 5g)is smoother than by other methods, whichimplies that more detailed spatial features may be lostby the de-noise procedure. The FIR method(Fig. 5f)exhibits an evident boundary effect, which will influencethe K_DP values at the echo boundaries. In addition, the traditional mean and median methods areless smooth. The wavelet analysis, on the other h and , is sufficiently smooth, and at the same time preservesenough detail information of weather echo, which helps effectively locate the strong convection and precipitation.Figure 6 is the corresponding plan position indicators(PPIs)of Z_H and Φ_DP, and Fig. 7 is the K_DPPPIs corresponding to Fig. 4. K_DP noise in Fig. 7ais decreased by the Φ_DP de-noising(Figs. 7b–h). TheK_DP PPIs of traditional mean and median methods(Figs. 7b–e)barely differ from the visual views. TheFIR, Kalman, and wavelet methods(Figs. 7f–h)arerelatively smooth compared to the traditional methods, yet the Kalman method(Fig. 7g)is too smoothfor K_DP to locate the strong echoes that are easilyseen in ZH(Fig. 6a) and other K_DP PPIs. For example, in Fig. 4, the average reflectivity is 42.15 dBZ inthe strong echo area from gates 183 to 352, and theaverage value 1.17◦ km⁻¹ of K_DP using the Kalmanmethod is the smallest(Table 5).

5. QPE results of three real cases

In order to further verify the influence on K_DPbasedrainfall estimation, three large-scale rainfall processesduring 0200–0800 BT 25 June, 0900–1700 BT 5 July, and 0900–1700 BT 7 July 2013 are analyzed, and their Φ_DP values are de-noised with various methods.The hourly rainfall is measured by 160 raingauges around the radar within 70 km. Z_H is preprocessedby attenuation correction(Hu et al., 2012), and the ZH-based QPE equation is taken as

and the K_DP-based QPE equation is(Liu et al., 2002):In order to analyze the accuracy of QPE, the normalizederror(NE)is defined aswhere R_r and R_g represent radar estimated and gaugemeasured rainfall, respectively, and M represents thenumber of qualified data pairs when both the estimated and measured rainfall are larger than 0.0 mm.

The normalized errors(NEs)of R(Z_H) and R(K_DP)are listed in Table 6, in which <R_g> representsthe average rainfall measured with all gauges, and the bold numbers indicate the minimum NE ofR(K_DP)with different de-noising methods. As expectedR(ZH)is the best estimator when all data pairsare statistically counted, regardless of rainfall intensity(Hu et al., 2010, 2012). When only R_g values greaterthan 5, 10, 15 mm h⁻¹ are considered, the advantage ofK_DP-based estimation is evident: the heavier the precipitation, the higher the accuracy of K_DP-based QPE.The SNR of the echo signal increases with greater rainfall, and the quality of the K_DP data improves rapidly.Even without any pretreatment, the K_DP-based estimateshave improved 10.61%(49.09% minus 38.48%) and 19.85%(51.88% minus 32.03%), in comparison toZ_H-based methods, when the average rainfall is heavierthan 10 and 15 mm h⁻¹ in Table 6, respectively.In Table 6, the wavelet method is the best of these denoisingmethods. When the average rainfall is largerthan 5 mm h⁻¹, after de-noising, the NE of R(K_DP)decreases from 55.41%(raw K_DP-based estimates)toa minimum of 36.30%(Kalman method), i.e., the NEof R(K_DP)has improved from –7.43% to 11.68% ofR(Z_H). In addition, the filter window widths, i.e., 7or 13 points of Φ_DP, of traditional mean and medianfilters seem to have little influence on the results ofK_DP-based QPE in all three cases.6. Discussion and summary

Since the polarimetric variable is one order ofmagnitude smaller than those measured from a polarizationdirection, they are easily influenced by noise.This is particularly the case in low SNR conditions forwhich the useful echo information is always submergedin noise. As K_DP is one of the polarization parameters, how Φ_DP measurements are processed and obtainingbetter K_DP will directly influence the accuracyof K_DP-based QPE. In recent decades, many types ofΦ_DP preprocessing methods have been employed; however, the specific effects of these de-noising methods and K_DP-based QPE are yet to undergo comprehensivecomparative analysis.

In this paper, the Φ_DP filtering effect with variouscommon methods is assessed and compared byusing numerical simulation, and finally demonstratedby real Φ_DP radial. The K_DP-based rainfall estimatesare also evaluated by refitting K_DP to further analyzethe influence of the different Φ_DP de-noising methods.

Φ_DP de-noising clearly reduced the errors ofR(K_DP), with the improvement more obvious whenΦ_DP de-noising used the more complicated Kalman, FIR, and wavelet methods, and the rainfalls were heavierthan 5 mm h⁻¹. However, for heavier precipitation(R > 10 mm h⁻¹), with an increased SNR, thedifferences between these methods are not significant.Therefore, for strong precipitation, the filteringmethod is not the primary consideration. Since the efficiencyof the complicated Kalman, FIR, and waveletmethods is significantly lower than that of the traditionalmethods for different precipitation types and rain rates, an efficient Φ_DP de-noising program needsto be deliberately designed. For instance, the thresholdvalue of average Z_H in a radial can be set todetermine whether the radial data are de-noised withfast traditional methods or more complicated ones.

In addition, it is found that the filter windowwidth of traditional filters on Φ_DP had less impacton K_DP-based QPE in the three large-scale rainfallprocesses. In other words, if the rain rate is greaterthan 10 mm h⁻¹, these de-noising methods are basicallysimilar. Overall, the performance of the waveletmethod is better than other methods. Nonetheless, these results need to be appropriately verified withmore observational data, especially for convectivecloud rainfall processes.

It has been found that the value of K_DP is less representativethan those of Z_H and Z_DR, as K_DP is fittedby a number of gates over a gauge, and Z_H and Z_DRare detected only over a gate width. As such, the accuracyof Z_H- and Z_DR-based rainfall estimations shouldbe higher than the K_DP-based estimation. Aside fromthe algorithm and radar system errors, precipitation isa complex dynamic, thermodynamic, and microphysicalprocess. Due to the air flow, time and space areneeded before the echoes detected by radar are transformedinto rainfall measured with ground gauges.The results in this paper further suggest that the accuracyof K_DP-based rainfall estimation is higher thanthe Z_H-based estimation for heavy rainfall(R > 10mm h⁻¹), and that after Φ_DP de-noising, the accuracyof K_DP-based QPE is more accurate than ZH-basedQPE, even for light to moderate rain(R >5 mm h⁻¹).