HU Zhiqunand LIU Liping

State Key Laboratory of Severe Weather,Chinese Academy of Meteorological Sciences,Beijing 100081

1.Introduction

Dual polarimetric weather radars will be widely used in the coming years.One of the reasons is that these radars are able to obtain the specif i c differential phase(KDP).KDPis def i ned as the slope of the range prof i les of the differential propagation phase shift(ΦDP)between the horizontal(H)and vertical(V)polarization states(?;Jameson,1985;Bringi and Chandrasekar,2001),and it is immune to rain attenuation,partial blockage and radar calibration Besides,KDPhas a nearly linear relationship with the rain intensity,even in the presence of hail(Aydin et al.,1995;Zrni′c and Ryzhkov,1996;Chandrasekar et al.,2008).For example,the coeff icientβin the empirical formula

for rainfall RDPestimation by KDPis given as 0.9056,andα as 44.806,23.918 and 15.060 in S-,C-and X-band radars,respectively(Zhang et al.,2001).These properties make KDPact as one of the important parameters of dual polarimetric weather radar.The value of ΦDPis the difference between the horizontal-and vertical-direction phases,and its magnitude is one order of magnitude lower than the phase obtained from only one direction.As a result,ΦDPis more easily affected by system noise,random f l uctuation,and ground clutters,and hereinafter all non-real ΦDPare collectively referred to as noises that need to be identif i ed and de-noised.In particular,in a small signal-to-noise ratio(SNR),ΦDPis always noisy and unstable(Hu et al.,2012).Besides,the actual radial prof i le of the total differential phase(ΨDP)contains both ΦDPand the differential backscatter phase shift(δ).Therefore,how to reduce the ΦDPnoise is crucial for polarimetric radar applications,which need not only effectively suppress all kinds of noises andδ,but also maintain the cloud and precipitation information as much as possible.

Ingeneral,therearetwoconsiderationsinde-noising,i.e.,smoothness and similarity,and they are performed differently in the time and frequency domains.For instance,using a median or mean f i lter in the time domain,the signal is more likely to ref l ect the similarity rather than the smoothness.On the other hand,using a Fourier transform in the frequency domain,it is easy to make the signal inf i nitely smooth by f i ltering out the high frequency noise,but some informative signals are inevitably f i ltered out for their small energy.For dual polarimetric radar,it is usually the indications of the loca-tion of clutter and large ref l ectivity gradient that causes ΦDPto oscillate rapidly in several gates,which are important in the clutter identif i cation,and echo track algorithm,so they should not be f i ltered out initially before the relevant tasks have been f i nished.

The traditional techniques of noise reduction include range f i ltering,linear f i tting,and a combination of the two.An iterative f i ltering technique separatingδfrom the propagation phase has been presented(Hubbert et al.,1993;Hubbert and Bringi,1995).Wang and Chandrasekar(2009)presented an algorithm to unwrap phase wrapping and keep the spatial gradients of rainfall for high-resolution KDP.He et al.(2009)introduced the Kalman f i lter method to separate ΦDPandδcomponents,and f i lter out the random noises.Hu et al.(2012)combined the ΦDPstandard deviation with the horizontal and vertical cross-correlation coeff i cient(ρHV)to verify the valid ΦDPvalue,and then smoothed ΦDPusing data collected in f i eld experiments.

Owing to the advantage of multiscale analysis,wavelet analysis has recently become popular as a de-noising method.The history of wavelets can be traced back to Harr’s work in 1909.However,from the modern viewpoint,wavelet analysis was not practicable until Caldero introduced a prototype in 1960,and the technique was not improved for 20 years until the work of Grossmann and Morlet(1984).Wavelets were widely used after 1986 because of the foundation developed by Meyer(1993),Mallat(1989,1992),Daubechies(1988),and other scientists.Subsequently,wavelet analysis has rapidly developed and become an emerging subject that is arguably the most signif i cant achievement in signal processing since Fourier analysis(Yang,2007).

Wavelet analysis can localize a signal in both the time and frequency domain,and enables one to perform multiscale analysis to extract information effectively by means of signal zoom and translation.The signal information is not lost during the processes,only a new equivalent representation.Because the generating wavelet function is very f l exible a suitable generating wavelet function can be selected to enlarge and extract the interesting information from part of the signal,and lessen or keep the others for further analysis.Utilizing the multiresolution characteristic,the wavelet coeff i cients in different scales can be verif i ed as to whether they represent useful information or meaningless noise by a certain threshold.Wavelet-based noise reduction,especially the threshold strategy,is currently a very active f i eld,and has become the most popular de-noising method for its simplicity and effectiveness.Mallat and Hwang(1992)proposed a signal and image multiscale edge representation technique according to the signal singularity,and introduced a f i ltering method of maximum module reconstruction based on the mathematical description of the Lipschitz exponent on the multiscale signal,image and noise characteristics.Utilizing the signal correlation among the scales,Xu et al.(1994)proposed a spatially selective noise f i ltration(SSNF)algorithm.Donoho(1995)presented the soft and hard threshold functions during the signal reconstruction.In recent years,many new threshold strategies and functions have been proposed,which have greatly enriched the technique of wavelet de-noising.

However,the application of wavelet analysis in weather radar is rare,except for the occasional use in radar image recognition and processing.Jordan et al.(1997)developed an algorithm using wavelets to fi lter the ground clutters and noises in wind pro fi le radar.A wavelet-based approach that can improve the capability in mesocyclone discrimination was reported by Desrochers and Yee(1999).Liu et al.(2007)developed a wavelet-based algorithm to detect tornadoes from Doppler weather radar radial velocity.Despite these applications,wavelet analysis is still rarely used in polarimetric radar de-noising.

In the second section of this paper,the principles of wavelet noise reduction are introduced in detail,wherein a ΦDPpenalty threshold strategy is addressed according to its characteristic.In the third section,the de-noising process is described by means of a simulated radar beam that passes through two rain cells of different size,and the ΦDPis interposed in the fl uctuation in several gates to verify the ability of identifying clutter,added some degrees in a short and a long gates(distance)to contrast the de-noising effect to different scalesδnoises.Furthermore,a white noise of SNR 15dB is mixed into ΦDPto demonstrate the suppression to thermal noise,respectively In the fourth section,the de-noising effects with wavelet analysis are contrasted with mean,median,fi nite impulse response(FIR,Hubbert and Bringi,1995),and Kalman fi lters via two actual observational cases.Finally,a summary and discussion of wavelet de-noising are presented in the last section.

2.Wavelet analysis

The steps of wavelet de-noising are:an appropriate wavelet function is selected to deconstruct a signal into multiresolution signals;the detail coeff i cients(from the highpass f i lter)that generally represent noises are suppressed by a threshold strategy;and then the signal is reconstructed with a threshold function.

2.1.Wavelet functions

Similar to a Fourier transform,if a set of functions that is formed by a function that can be zoomed and translated to constitute dense orthogonal bases,then a signal with f i nite energy can be deconstructed into the bases.So,the signal is separatedintosignalswithdifferentresolutions,andtheinteresting parts in the signal can be observed in each resolution.This type of function,which requires a compact support set,i.e.,quickly decays to zero in a limited region,is known as the generating wavelet.The set of orthogonal bases formed by the generating wavelet is def i ned as the wavelet function.

Def i nition:ifψ(t)∈L2(R)and?ψ(0)=0,the functions{ψa,b(t)}

is called a continuous wavelet,whereψ(t)is the basic wavelet or generating wavelet in function space L2(R),t is the variable in real number f i eld R,a is the scale or striction coeff i cient b is the translation factor,and satisf i es the admissible condition

If a signal f(t)is deconstructed into the function set,the continuous wavelet transform(Wψf)is def i ned as

where“＜ ＞”indicates the inner product;(Wψf)(a,b)are the coeff i cients of wavelet transform corresponding to scale and location(a and b);ˉψrepresents the conjugations ofψ;This transform is known as a continuous wavelet transform(CWT)when a and b are continuously changing,and as a discrete wavelet transform(DWT)when a and b are discrete points.

Typically,a and bare taken as power series:

When a0=2 and b0=1,the scale and translation are dyadic discrete and the dyadic wavelet is obtained:

Hereafter,the DWT is represented as this dyadic wavelet transform,and the DWT coeff i cients are

cj,kis the kth largest value of the coeff i cients in the jth deconstruction level;

The reconstruction formula is

Different wavelet functions have different characteristics:(1)Compact support,which denotes the attenuation of the generating wavelet;the narrower the width of the support,the faster the attenuation,and the better the localization.(2)Orthogonality,which indicates the continuously differentiable number and the smoothness of the wavelet function;the better the orthogonality,the faster the convergence.(3)Symmetry,which relates to whether the wavelet f i ltering is in a linear phase that closely connects with the signal distortion after reconstruction.(4)Vanishing moments,which can be physically regarded as a convergence rate when the wavelet function approaches a signal,i.e.,when a signal is transformed by the wavelet,the wavelet is required to have compact support or acute attenuation in both the time and frequency domains;the higher the order of vanishing moments,the better the smoothness,and the stronger the ability to mirror the high-frequency details.(5)Regularity,which is related to the smoothness of a signal;the greater the regularity index,the smoother the signal,and the localization characteristics can be estimated according to the regularity index at each point in a signal.

However,such a generating wavelet function,which not only has compact support and symmetry but also has good orthogonality and vanishing moments,does not exist.The shorter the support width,the poorer the smoothness,so the compact support and the smoothness are two contradictory aspects.In addition,except for the Haar wavelet,which can meet the orthogonal and symmetric conditions at the same time but has poor localization performance,such a wavelet with both orthogonality and symmetry does not exist either,so orthogonality and symmetry are also contradictory.

2.2.Threshold strategies

(1)Penalty threshold strategy(Yang,2007):

Given that t?is the positive integer that makes function f(t)minimum:

where variable t∈[1,m];σis the noise intensity of the signal;αis an experience value,which is a real number greater than one;and m is the total number of coeff i cients;then the threshold value T corresponding to the value t?is:

To avoid the impact of a boundary effect on the wavelet coeff i cient calculation,the estimation of the standard deviation of noise level is calculated by the absolute value of the detail coeff i cients.When the signal is regular enough,the details of the signal are concentrated into the minority of the coeff i cients,so the penalty threshold is a suitable threshold strategy.

(2)ΦDPpenalty threshold strategy:

The largest deviation value of ΦDPin each deconstruction level can be estimated according to the scope of the polarmetric parameter in raindrop echoes,so the wavelet detail coeff i cients larger than the deviation can be considered as non-weather echoes,such as clutters,δ,or other noises,which need to be f i ltered out.For C-band radar,KDPis generally not larger than 6°km?1,taking into account f l uctuation,and the detail coeff i cients in the f i rst level will not exceed 8×0.3=2.4(°),where 21×0.15=0.3(km)is the resolution in the f i rst level.Because the size of the time-frequency window is constant to the wavelet function,the detail coeff icients over 2.4°can be considered as noise in each level,and the ΦDPpenalty threshold strategy is def i ned as:

(3)Default threshold strategy:

This is given by the formula

where n is the length of the discrete signal.

2.3.Threshold functions

After the detail coef fi cients are suppressed by a certain threshold strategy,a threshold function is needed to reconstruct the signal.Hard and soft threshold functions(Donoho,1995)are commonly used:

Hard threshold function:

Soft threshold function:

where?cj,kare the coeff i cients after processing and sgn()is the sign function.

3.The de-noising steps by wavelet analysis

3.1.Construct a simulation signal

To illustrate the performance of wavelet function denoising,two neighboring precipitation cells are simulated with the gamma drop size distribution(DSD)(Ulbrich,1983,Chandrasekar et al.,1990,Scarchilli et al.,1993)

where N is the raindrops number per unit volume per unit size interval;D is the equivalent volume diameter of raindrops(mm);N0is the concentration parameter that is assumed to have the value of 8×103mm?1m?3;μis the distribution parameter and is assumed to be zero;and D0is the median volume diameter,and is assumed to be

where Dmaxis the maximum equivalent diameter of raindrops and is assumed to be 0.2 cm;rmaxrepresents the diameters of cells,with the f i rst one assumed to be 30 km and the next 15 km;and r is the distance between a raindrop to each cell center.

Assume that the radar wavelength is 5.6 cm,and the gate width is 150 m,so the beam passes through the simulation rain area with 300 gates.The particle scattering is calculated using the method of extended boundary conditions with the consideration of the relationship between the drop size and the ellipticity(Liu et al.,1989),and the range pro fi les of the horizontal re fl ectivity factor ZHand ΦDPare shown in Fig.1(the radar is located at 0 km).

3.2.Mixed signal noise

Because radar does not measure ΦDPdirectly,the total differential phase ΨDPis estimated from the co-polar covariance that consists of both forward propagation and backscattering phase shifts,δ:

Generally,ground clutters can cause severe change in ΦDPoveraveryshortdistance,andlargeoblateraindropsand melting hailstones can causeδ,which will increase the phase shifts from several degrees to about 30°for C-band radar.To simulate real situations,some phase shifts are intentionally increased into the ΦDPsimulation signal in Fig.1b,from 46 to 55 gates simulating clutters,from 96 to 105 gates simulating a long-distanceδ (hereafter referred to asδl),and from 249 to 252 gates simulating a short-distanceδ(hereafter referred to asδs).Simulating the short-and long-distanceδ is for contrasting the de-noising effect to different scales of noises by wavelet multiscale analysis.The added phase shift values(units:°)are listed in Table 1.For convenience,the no-noise ΦDPsignal in Fig.1b is hereafter represented by s;s polluted by clutters andδis represented by sp;and spfurther mixed by 15 dB SNR white noise is represented by sn.In actual radar observations,when the SNR is too small,the weak signals have no value because two receivers have diff iculty ensuring their consistency(Hu et al.,2012).

Fig.1.Images of the simulation results:(a)ZHand(b)ΦDP range prof i le.

Table 1.The added phase shift values(°)by clutters and δ at some gates in s to simulate an actual noisy signal.

Fig.2.Frequency distribution of sn.

Figure2showstheresultsoftheFourieranalysisofsn.As shown in Fig.2,when the frequency is greater than 15,i.e.,the distance is less than 3.0 km(300/15×150 m),the magnitude of the Fourier coeff i cients is less than 30 dB,so these high-frequency variations can be considered as noises that need to be suppressed,and then the signal is reconstructed by inverse Fourier transform.However,this method is too drastic because a large amount of useful information is inhibited and many energy components in the original signal are lost.

3.3.Deconstruct and analyze the wavelet coeff i cients

Without loss of generality,snis deconstructed into f i ve levels with the db5 wavelet,and the approximation and detail coeff i cients in each level are shown in Fig.3,wherein Figs.3a and g are spand sn,respectively;Figs.3b–f are the approximation coeff i cients represented by a1to a5,respectively;and Figs.3h–l are the detail coeff i cients represented by d1to d5from levels 1–5,respectively.

First,let us analyze the behavior of the approximation coeff i cients that represent the low-frequency change.The location of clutters and twoδare seen in the f i rst-and secondlevel approximation coeff i cients,a1and a2.The twoδare obvious,and clutters can be vaguely identif i ed in a3.The two δstill have weak responses,but clutter almost disappears in a4.The spatial scale in the f i fth level is 25=32 gates;that is,4.8 km,so a5is smooth enough to approximate s.

Continuing to analyze the detail coeff i cients that represent the high frequency change,to highlight the changes,note that the scale of the y-axis in each level is different.Because the clutters are high-frequency changes,the energy of clutters is much larger than that of other noise,so the location of clutters can be observed very clearly in the f i rst-level detail coeff i cient,d1,but the twoδ,which are drowned in noise,cannot be observed.The amplitude of the detail coeff i cients d2and d3are reduced in the second and third levels,wherein the clutter feature can still be identif i ed,and the position of δsbegins to emerge slightly,but theδldoes not appear as its lower frequency characteristic.The amplitude of d4and d5is further decreased,the position ofδlemerges obviously,and the clutter andδscan be recognized roughly in d4.The detail coeff i cient d5ref l ects the f l uctuations of the low-frequency signal in which the twoδcan vaguely be identif i ed,but the clutter disappears completely.

By the above demonstration,the different types and scale noises can be separated into each deconstruction level easily for their different frequency characteristics,and conveniently further processed.

3.4.The process of wavelet de-noising

As mentioned above,the basic noise model can be expressed as

where e(n)is the noise;σis the noise intensity that is generally determined by the standard deviation of coeff i cients in each deconstruction level;and n is the length of the discrete signal.The goal of wavelet de-noising is to suppress e(n)and restore s(n).The steps of wavelet de-noising are generally performed as below:

(1)Deconstruction:a signal is deconstructed into N levels with a selected wavelet function.

(2)De-noising:the detail coeff i cients in each level are suppressed with a selected threshold strategy.

(3)Reconstruction:the signal is reconstructed by means of the approximation and the processed detail coeff i cients with a selected threshold function.

According to the above discussion without loss of generality,hereafterthedb5 waveletfunctionisusedtodeconstruct a signal into f i ve levels,and two de-noising schemes are designed:(1)suppression by the penalty threshold strategy and reconstruction by the soft threshold function scheme(PSS);(2)ΦDPpenalty strategy and soft function scheme(PPSS).

4.Analysis of two cases

A mobile C-band dual polarimetric weather radar(POLC),which transmits and receives the horizontal andvertical polarization signals simultaneously,was f i rst developed in China in 2008.The radar operated at a frequency of 5.43 GHz and a 150-m gate width(Hu et al.,2012),and the main characteristics of the radar are summarized in Table 2.The following two representative actual cases are selected to examine the effects of noise reduction by the methods described above.Because the height of the antenna is only about 6.5 m,it is blocked by surroundings that cause some missing beams in the PPIs.

Table 2.Main characteristics of POLC radar.

Fig.3.The result of sndeconstructed into f i ve levels with the db5 wavelet:(a)sp;(b–f)approximation coeff i cients from a1to a5;(g)sn;(h–l)detail coeff i cients from d1to d5.

4.1.Squall line case

A squall line was detected by POLC radar in the afternoon of 17 July 2008 in Shouxian,Anhui Province.A total of 200 gates(30 km)from 40 to 239 were analyzed in the ΦDPradial data in the elevation of 1.5°and azimuth of 200°at 1828 LST.The radial prof i le of ZHalong with the gates is shown in Fig.4.

The ΨDPvalues(Fig.5a)from gates 86 to 89 are?119.5,?106.58,?102.22 and?121.33,respectively.It is shown that the values in gates 87 and 89 exhibited jumps of 13°and 19°due to backscatteringδand that the values from gates 105 to 108 exhibited jumps of over 20°.The section of the ΨDPsignal is deconstructed into f i ve levels with the db5 wavelet,and then the detail coeff i cients are suppressed by the ΦDPpenalty threshold.The detail coeff i cients before and after suppression are shown in Fig.5,where the images of approximation coeff i cients are not shown.

Fig.4.The range prof i le of ZHfrom gates 40–239(slope distance of 6–36 km)at elevation 1.5° and azimuth 200° at 1828 LST on 17 July 2008.

Figure 6 shows the de-noising results of the ΨDPsection using the PPSS,Kalman,mean(15 points),median(15 points)and FIR(once)f i lters.All these f i lters can de-noise to a certain extent,but the Kalman f i lter has a larger signal distortion,and the median is rougher than the others.Because thecolor-keyresolutionofΦDPisabout30°,itishardtoshow the differences among these methods,but KDP,obtained with 13-gate f i tting,is selected to illustrate the differences(Figs.7a–f).Figure 7 shows part of the PPI images of the squall case at the same time and elevation angle as in Fig.4,in which the de-noising methods of Figs.7a–f correspond to Fig.6,Figs.7g and h are the raw ZHand ΦDP,respectively,and the interval of the range ring is 15 km.From Fig.7h,the ΦDPincreases rapidly in the south-southwest.All the denoised pictures(Figs.7b–f)are smoother than the raw image(Fig.7a).Focusing on the area between azimuth 240°and 270°and the range between 15 and 30 km,the PPSS(Fig.7b),mean(Fig.7d),and median(Fig.7e)methods can preserve more details.In other words,using these methods it is still easy to determine the center of the heavy rainfall after de-noising.The Kalman f i lter(Fig.7c)method is smoother than the others,but lost more details.The FIR approach has a more obvious boundary effect(Fig.7f),which can cause great errors at the edges of rain clouds.

4.2.Typhoon Koppu case

Fig.5.The section of the ΨDPsignal in Fig.6a is deconstructed into f i ve levels with the db5 wavelet,and then the detail coeff i cients in each level are suppressed by the ΦDPpenalty threshold.The left-hand panels(a–e)are the details without suppression,and the right-hand panels(f–j)are after suppression.

Fig.6.The(a)ΨDPsection corresponding to Fig.4 and de-noised by the(b)PPSS,(c)Kalman,(d)mean,(e)median,and(f)FIR f i lters,respectively.

Fig.7.Part of the KDPPPI images of the squall case at the same time and elevation angle as in Fig.4.The de-noising methods of Figs.7a–f corresponding to Figs.6,7g and h show the raw ZHand ΦDP,respectively,in which the interval of the range ring is 15 km.

The landing typhoon Koppu was observed by POLC radar from 14 to 15 September 2009 in Zhuhai,Guangdong Province,and it caused serious backscatter phase shiftδbecause of carrying lots of large raindrops on ΦDPmeasurements.The beam at elevation 1.5°and azimuth 237°at 0443 LST passed through hills ranging from 15 to 25 km,and through a heavy precipitation area from about 75 to 100 km behind the typhoon eye.Therefore,its ΦDPdata are a good example to illustrate the mitigation of multiscale mixed noises.Since the maximum continuous contaminated ΦDPdistance can be more than 5 km in such a violent typhoon,the ΦDPdata are deconstructed into six levels with the wavelet function db5;namely,the maximum distinguishable noise scale is 26/2×0.15 km=4.8 km.The mean and median fi lters are performed with 31 points(31×0.15 km=4.65 km).The pictures of wavelet deconstruction coef fi cients are not shown.The radial pro fi les of raw ZH,ΨDP,and ΨDPafter de-noising by the above methods are shown in Fig.8,and their corresponding PPIs are shown in Fig.9,in which the interval of the range ring is 30 km.

Fig.8.The range prof i les of raw(a)ZH,(b)ΨDP,and ΨDPafter de-noising by the(c)PPSS,(d)PSS,(e)Kalman,(f)mean,(g)median,and(h)FIR f i lters at elevation 1.5°and azimuth 237°at 0443 LST 15 September 2009.

Fig.9.The PPI images corresponding to Fig.8,in which the interval of the range ring is 30 km,and mainly focusing on the area enclosed with red ellipses in(b–h).

As can be seen from Figs.8a and b,the data of ZHand ΨDPfl uctuatedrasticallyaroundthe100thgatefortheground clutter.In the area between the 500th and 700th gates,approximately,the ΨDPdata have several peaks that inf l uence gates 5–30 byδ,and the measured ZHreaches about 40 with themaximumvalueof42dBZ.TheSNRsaresmallerthan20 dB behind the 700th gate,so the data quality becomes poor and will not be discussed here.Figures 8c–h are the radial prof i les of ΨDPde-noising by the PPSS,PSS,Kalman,mean,median,and FIR methods,respectively.It can be seen that the PSS method retains the largest amount of clutter information for further identif i cation.Both PSS and PPSS de-noise well,especially in the hybrid noise area behind the typhoon eye,and this means that more effective KDPdata can be obtained.As in Fig.6,the Kalman f i lter has larger distortion,and the others have a few remains of pollution.

From Fig.9,all the methods perform well,and at f i rst sight it seems there is no difference between these pictures because the color-scale interval is too large to the change value after de-nosing.However,when we focus on the area enclosed by red ellipses in images of ΦDP(Figs.9b–h),the features of color distribution and gradation in Figs.9c and d(PPSS and PSS)are more similar to Fig.9b(raw ΦDP)than the other f i gures,which suggests that wavelets have less distortion.In particular,the clutters are almost coincident between Figs.9b and d,which fully demonstrates the stronger ability of retaining details by the PSS method compared to the other ones.This is useful for further processing in some dual radar algorithms,such as quantitative precipitation estimation,toavoidthecontaminationofcluttersandthelocation of heavy rainfall.

5.Summary and discussion

As polarimetric parameters are easily affected by noise effective applications of polarization radar have been limited for many years and noise mitigation methods have been introduced since the invention of polarization radar.This paper introduces a wavelet multiscale analysis method,which has been widely used in many other fi elds.Using simulations and actual ΦDPdata as examples,the processes and results of wavelet de-noising are presented in detail and,further,a ΦDPpenalty threshold strategy is proposed according to the characteristics of the polarimetric parameter.

Owing to multiscale analysis,and depending on the purpose of de-noising,wavelet analysis can not only smooth a signal very well,but can also retain enough detail to accurately indicate the clutters and strong echoes,even if they have a small amount of energy in the frequency domain.

According to the characteristics of precipitation echoes,the deconstruction scales typically do not overrun more than fi ve levels,which have a satisfactory de-noising effect,and avoid causing extra errors when small-scale noises are deconstructed into a large number of levels.

The noise reduction can be obviously in fl uenced by the selection of threshold strategies and functions.As described in this paper,the ΦDPpenalty is a very effective strategy for ΦDPdata de-noising,which can retain more details after being combined with the hard-threshold function,but a smoother signal with the soft one.

In this paper we have attempted to introduce the wavelet analysis approach in polarimetric radar data quality control,and the results are encouraging.With the swift development of computing technology,a volume span data can be processed within one minute by wavelet analysis with a personal computer.Therefore,further work could be carried out in the future to select,or even construct,a particular wavelet generating function,and design a pertinent threshold strategy and function for better de-noising performance for each polarimetric parameter.

Acknowledgements.This work was funded by National Natural Science Foundation of China(Grant No.41375038),and China Meteorological Administration Special Public Welfare Research Fund(Grant No.GYHY201306040，GYHY201306075).The Coef fi cients of the FIR Method and Process of the

APPENDIX

Kalman Filter

The coef fi cients of the FIR are shown in Table A1,wherein the 20th-order fi lter is symmetric,and the coef ficients are shown for the complex variable Z(Proakis and Manolakis,1988).The magnitude of the response of the FIR fi lter is shown in Fig.A1.

Table A1.FIR f i lter coeff i cients.

Fig.A1.Magnitude of the response of the FIR f i lter used in the de-noising.

Table A2.Meaning of each term in Eq.(A1)–(A6).

For ΦDP,the process and measurement equations of the Kalman f i lter are:

where the meanings of each term in Eqs.(A1)–(A6)are listed in Table A2(He et al.,2009).

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