798JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGYVOLUMEI0Two-Dimensional Dcaliasing of Doppler -elocities ZHONGQI JING AND GERRY WIENERNational Center for Atmospheric Research, * Research Application Program, Boulder, Colorado(Manuscript received 22 September 1992, in final form 9 December 1992)ABSTRACT The Doppler velocity dealiasing problem has been discussed for many years. Because aliasing is easily identifiedby detecting abrupt changes in the data field, most existing algorithms use this technique to correct aliased data.Such algorithms are typically based on local expansion methods. Such methods make a dealiasing decision foreach gate based on the information of its dealiased neighbors and thus can be sensitive to scattered incorrectdata. This paper introduces a new approach that attempts to find all dealiased values for a given dataset bysolving a linear system involving the entire dataset and thus avoiding local expansion. Because the solution isglobal, the new technique is conceptually simple and displays good performance on a number of test cases, Thenew technique described here was implemented to support real-time dealiasing in an operational setting.1. Introduction The velocity measurement of Doppler weather radarsuffers from an aliasing problem that is analyzed, forexample, by Doviak and Zrnig (1984). If uniformlyspaced pulses are used and both I and Q samples areprocessed for resolving the sign of the Doppler shift, aradar can correctly measure the radial velocity providing that the velocity is known to be in the range of_+ V., where XF ['~ = -- (1) 4Here V, is the Nyquist velocity, and X and F are, respectively, the wave length and the pulse repetition frequency (PRF) of the radar. In general, the measuredvelocity is V,,, = V+ 2nVn, (2)where V is the true velocity, n is an unknown integer,and Vrn ~atisfies --Vn ~ Vm .~ Vn. Many Doppler weather radars operate with Vn ranging from 10 to 40 m s-~. The velocity measurementsmust be processed to recover V from V,, (a processcalled dealiasing) before they can be used for furtheranalysis. This paper describes a dealiasing techniquethat was developed at the Research Applications Program at the National Center for Atmospheric Research * The National Center for Atmospheric Research is sponsored bythe National Science Foundation. Corresponding author address: Dr. Zhongqi Jing, Research Ap.plieation Program, NCAR, P.O. Box 3000, Boulder, CO 80307-3000.(NCAR). The new technique has been implementedfor both data analysis and real-time processing. Preliminary tests and operational use of this technique onthe Mile-High Radar (MHR) and NCAR's CP-2 research radar data have shown encouraging results. Velocity aliasing can usually be identified by detecting abrupt velocity changes between neighboringmeasurements since true velocity fields must be continuous. Aliasing causes a discontinuity of about 2in the fields. Throughout this paper it is assumed thatthe velocity field is "smooth" such that the differencebetween neighboring measurements is smaller thanThe assumption of smoothness may not be valid inthe presence of strong wind shears in gust fronts, mesocyclones, microbusts, and storm-top divergence signatures. In such cases, the magnitude of the wind shearmay be comparable to the Nyquist velocity. We willdiscuss this issue later in the summary. A radar makesa set of measurements by pointing its antenna in aspecific direction (azimuth and elevation). We will callthe set of data gathered in a particular direction a rayof data. We will also assume that the radar directionchanges smoothly while generating rays of data. Thus,subsequent rays cover a continuous surface in threedimensional space. Within each ray, data occur at intervals called gates, whose spacing is determined byoperating characteristics of the radar. With these assumptions, we may consider the following basic dealiasing algorithm. For each alias-free or processed gate, check each ofits neighboring gates: If the neighbor has not been processed and the velocity difference between the neighborand the original gate is close to or bigger than 2 V~, weassume that this neighbor is aliased and add a suitablemultiple of 2 V, to it such that the difference is minimized. (Note that the difference will not be larger thanc 1993 American Meteorological SocietyDECEMBER 1993 JING AND WIENER 799V,,.) The neighbor is then labeled as proccssed whetherits value has been changed or not. This procedure isrepeated until no further correction can be performed. This algorithm will perform the dealiasing correctlyif there exists at least one gate that is known to be freeof aliasing, if all gates are connected, if the data arenoiseless, and if the data matches our assumption ofsmoothness. By connected, we mean that, for any twogates there exists a path of consecutive gates in rangeand azimuth connecting the two gates without crossinga gate with missing data. Note that the algorithm justdescribed can bc applied to either one-dimensional ortwo-dimensional data. It is clear that a two-dimensionalalgorithm has the potential of being superior to a onedimensional algorithm with respect to missing data,since it can make use of both radial and azimuthalintbrmation. In practice, radar measurements are unreliable whenthe signal-to-noise ratio (SNR) is low, and are oftencontaminated by ground clutter, second-trip signals,and other ~hctors. Thus, in reality, the velocity measurements include incomplete and noisy data, whichmeans that some of the gates are not available andothers may have incorrect values. The basic algorithm described above suffers whenthe radar data are incomplete or noisy. The missingdata may cause the data to be separated into isolatedregions requiring addilional information for each region to be dealiased. Noisy data may cause incorrectdealiasing that may propagate in various ways depending on how the area is scanned in the algorithm. Several techniques have been proposed to partiallyovercome the difficulties discussed above. Local aweraging was introduced by Bargan and Brown (1980)to deal with the difficulty of missing and incorrect data.A two-dimensional algorithm was first introduced byMerritt (1984) and further developed by Boren, Cruz,and Zrnid ( 1986 ) and Bergen and Albers (1988). Thealgorithm, taking advantage of two dimensions, showedimprovcd performance over one-dimensional algo~rithms. However, because it uses a local expansiontechnique, it is sensitive to incorrect data, which mayresult in incorrect region separation. Moreover, thenumber of regions found may be quite large and introduce difficulties in the optimization step. Eilts andSmith (1990) introduced a technique called local environment dealiasing (LED), that incorporates localinformation on the current ray as well as previous raysto improve the dealiasing decision. LED substantiallyimproved a basic line expansion algorithm. Withproperly tuned algorithm parameters, thcir techniquewas applied to Doppler radar data used in the TerminalDoppler Weather Radar (TDWR) projects with success, and is the algorithm presently used in theNEXRAD (Ncxt Generation Weather Radar) system. In the following, a two-dimensional dealiasing algorithm is first described. The new technique attemptsto dealias a connected two-dimensional region by minimizing all detected discontinuities due to aliasing. Thedetection uses a conventional threshold method anddiscontinuity minimization is performed by solving aleast-squares problem. The technique is relatively insensitive to missing data because it is two-dimensional.We also believe it to be less sensitive to incorrect databecause it avoids local expansion and utilizes a globalsolution to the dealiasing problem, where effects ofscattered incorrect data will be locally limited. This two-dimensional technique, which dealiases aconnected region internally, is then combined with aglobal dealiasing algorithm using environmental windinformation, to form a complete dealiasing method.The new method is efficient and can be run in realtime. (It processes 100 512 gate rays per second on aSUN SPARC II workstation). It does not require anyspatial filtering or smoothing to be done on the radialvelocity fields, and thus single-gate features are preserved in dealiasing. And, finally, the method requiresno complicated feature-dependent rules or parametertuning, an advantage when applied to different climaticregions or radar systems. We will first describe the basic algorithm that dealiases a connected two-dimensional region. Then wewill present some of the details required for a real-timedealiasing implementation. A discussion on global dealiasing then follows. Examples are given to illustratethe use of the new technique. Finally, a few concludingremarks are made in the summary.2. The two-dimensional dealiasing algorithm (TDD) In this section, we introduce a technique that dealiases a two-dimensional connected region based onthe assumption of a smooth velocity field as describedin the previous section. The problem of dealiasing acomplete field containing multiple regions (global dealiasing) will be discussed later in section 4. Consider the two-dimensional radial velocity datasetas depicted in Fig. 1 (similar to B-scan formats). Thedataset contains m rays, each of which has n gates. Weassume that, as mentioned in the Introduction, the radar antenna direction changes smoothly as it measuresthese rays such that each of the gates in Fig. 1 is closeto its neighbors. Here, the neighbors of a particular,gate are the four gates that adjoin the given gate on theleft, right, above, and below. We further assume thatthe radar's gate spacing is a fixed constant. The missingdata may be due to ground clutter, low SNR, or otherdefects in the system. In Fig. 1, all the gates containingdata form a connected region with only single-levelaliasing, that is, Inl ~< 1 in (2). For the moment we assume that there are no incorrect data, and that shears are small enough such thatthe velocity differences between neighboring gates aresufficient to identify the existence of any aliasing. Laterin algorithm A we will relax the noiseless assumption.We apply a velocity difference check to each pair of800 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 10m2IRays gate nXXX3~ I~ -~-- .~-~- Range2 Gate spacing aliased gate non-aliased gate missing gate F16. 1. A two-dimensional region containing aliased, nonaliased,and missing gates. The aliased gates, nonaliased gates, and missinggates are depicted, respectively, with shaded, white, and crossed blocks.neighboring gates and illustrate the difference in Fig.2, where we depict the borders between aliased andnonaliased gates with thicker lines, and those betweenavailable and missing gates with dashed lines. Thethicker lines contour the aliased area with discontinuityat missing gates. We denote the velocity value of the jth gate on theith ray by V0, and define the. following smoothnessfunction, S = ~ d2(Vo., Vkl), (3)which serves as a measure of the discontinuity due toaliasing. In ( 3 ) the function d is defined for each pairof neighboring gates with velocity values Vo and Vkt by r2v., if IVo.- vk, I >~ tVn d(V0., V~t) =[ 0, otherwise, (4)wherek=i+ 1 and!=j, ork=iand!=j+ 1 (seeFig. 1 ). Here t is a threshold used for identifying differences due to aliasing from those due to normal gateto-gate velocity variations. A typical value for t is 1 asdiscussed later. The summation $ is performed overall possible pairs of neighboring gates containing validvelocities and thus velocities for missing gates are simply ignored. Note that S = 4kV 2~, where k is the number of pairsof neighboring gates that satisfy the aliasing criterion.In this way, S is a measure of the discontinuity due toaliasing. It is obvious that, if there is no aliasing, S= 0 where it reaches its minimum. Otherwise, $ willbe greater than zero as is the case in Fig. 1. Assume that the dealiased value for the gate i, j isVo + Yo. Here Yi~ must be a multiple of 2 Vn. To determine Yo', we introduce a correction term xo, whichmay not be a multiple of 2 Vn, and then we determinethe Yi~ based on the xii. We now set the smoothnessfunction $ to be the summation of the squares of thefollowing differences over all pairs of neighboring gatescontaining valid data:d(V0, Vat) [ 2V.+x0-xat, if Vo- V~> tV~ = 12V~-xi;+xat, if V,~- Val~<-tVn (5) [ 0, otherwise.Note that S is a function of x0 and that d(Vo Val)~- d( Vat, V0). The dealiasing problem can be solved byselecting suitable correcting values xi2 for all gates tominimize $. This optimization problem is equivalentto solving the following linear system of equations: OS -0. (6) OxoNote that this is a two-dimensional Laplace equationdescribing the problem of finding a homogeneous fieldwith known boundary values (or a two-dimensionalintegration problem). It can be shown that, since all data form a connectedregion, (6) has solutions, and any two of its solutionsdiffer by an additive constant. In the case of our example in Fig. 1, a trivial solution is I 2 V~, if gate i, j is aliased x~i = (7) [ 0, otherwise,where we assume that for the aliased gates, Vm = V-- 2 Vn. This solution gives correct dealiasing.Rays~ ..................................... ~ ....,,! ........ ; ll- .............; Rangeborder between border betweenaliased and available andnon-aliased gates missing gatesFlG. 2. The borders between aliased, nonaliased, and missing gates in Fig. 1.DECEMBER 1993 JING AND WIENER 801Histogram3718Ftcl. 3. The histogram of solution of Eq. (6), which features lines separated by a distance of 2 V,. Shown in Fig. 3 is a histogram plot of x0. We cansee that in a noiseless data case the histogram of thesolution of (6) features only two lines separated by adistance of 2 V,,. This feature is essential to the TDDalgorithm. In the general case where noisy data are present,precise identification of complete borders of aliasedareas (such as in Fig. 2) may become impossible.tlowever, if correct data are predominant and incorrectdata are randomly scattered, we can still identify mostparts of the border of an aliascd area even though sucha border may no longer be a connected simple curve. Analysis of simple cases and test results on actualradial velocity data has shown that the effect of theincomplete border of an aliased area causes the twolines in the histogram of solution (6) to split into twoclusters as depicted in Fig. 4. The distance between thetwo clusters tends to be slightly smaller than 2 V,. Because the dealias correction Y0 for each gate must be amultiple of 2 V~, we must discretize the correction termxij. An examination of the features of the histogramin Fig. 4 suggests the following scheme. We first identifythc two clusters in the histogram and then find a pointVa that is between thc two clusters and effectively separates the two clusters (see Fig. 4). Then the correctingvalue Yij for each gate may be determined by I :2V,, if xii --> VaY,:i = [0, otherwise.(8)Clearly, the identification of clusters in the histogram,that is, finding Va, becomes more and more difficultas the percentage of incorrect (noisy) data increases,while for noiseless data it is trivial. At this point wehave not yet found a theoretically optimal way to analyze the histogram in order to determine Va. However,for typical data, preprocessed by SNR thresholding andsecond-trip and ground clutter removal, the percentageof incorrect data is small and the two clusters in thehistogram are clearly separable with a distance betweenthem close to 2 V,. We can find Va by a simple examination of the histogram. The above discussion can be directly extended tothe cases where multiply aliased areas exist, that is,there exist gates at which n > 1 or n < -1 in (2). Itturns out that the technique can be easily generalizedfor application to these more complex problems. Theonly difference is that the histogram will contain morethan two lines (or clusters for noisy data) separated by2 V, ( or approximately 2 V,), instead of two. Finally, the above dealiasing scheme can be summarized in the following algorithm.Algorithm A (basic TDD):Input: A two-dimensional dataset V0. (radial velocity,i = 1, - -., m,j = 1, - -., n) with missing data markedand the Nyquist velocity Vn.An algorithm constant: t--the threshold used for identifying the borders of aliased areas.Output: Dealiased (corrected) dataset V0. Step 1: Solve the system of linear equations: OS -0, (9)i = 1, - -., m,j = 1, - - o, n, excluding missing data,where S = ~ d2(V,;, Vkl). (10)The summation is performed on all pairs of neighboring data indexed by i, j and k, l, respectively, andd(Vo, vkt) [ 2V~ + xo- x~, if Vo- V~>~ tV~ = ~2V~-xo.+x~, if Vo- V~<-tV, (11) [ 0, otherwise. Step 2: Generate a histogram plot of x0 and find Va~,HistogramVa-- 2VnX FIG. 4. The histogram of solution of Eq. (6) when the data arenoisy and the exact identification of borders of aliased areas is impossible.802 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 10Va2 ..... that divide the histogram into clusters separated by a distance of 2 Vn. Assume that N clustersare found. Then for each xii we set I 2kV, if Vd,k ~ Xij ~ Vd, k+1 Yij = ~0, - if Xii~ Vd~ (12) [ 2(N- 1)V, if xil> Vd,~v-~. Step 3: Dealias the data:for all data.V,j is replaced by Vo. + YoThe best value oft is 1 if the noise (uncertainty) in thedata has a distribution that is not biased. If the dataare noiseless and the normal variation of radial velocityfrom gate to gate is less than Vn, the borders of aliaseddata are always correctly identified.3. Implementation issues of TDD The TDD algorithm, as described in the previoussection, requires the solution of the linear system (9).The number of unknowns in this system equals thenumber of gates in the two-dimensional region. Thesizes of our regions typically range from 30 x 100 gatesto 180 x 500 gates. Thus the number of unknowns in(9) may be quite large (3000-90 000). The time neededto solve (9) may significantly exceed the time neededfor the radar to scan the underlying rays ( 30-180). Toapply TDD to reaMime dealiasing, this problem mustbe addressed. In the following, several approaches arediscussed that improve the processing speed of TDDand make TDD useful for real-time dealiasing.a. Solving system (9) using a sparse matrix solver The linear system (9) is sparse since most of its coefficients are zero. Comprehensive theory and variouspractical solvers have been developed for sparse systems[see, e.g., Tewarson (1973)]. To solve (9) we use thepackage developed by Kundert and Vincentelli (1988),which is optimized for both CPU efficiency and memory utility. From 2 to 20 s are needed to solve a problemof 3600 (60 x 60) unknowns on a typical workstation(e.g., SUN SPARC II). This speed is useful for analysis, but not for realtime processing. Most of the computing time is usedin processing the sparse structure and inverting thematrix. To further reduce the processing time the following variation of the TDD algorithm, which doesnot require real-time matrix inversion, is introduced.b. A more efficient algorithm With a closer examination of(9 ), it can be rewrittenmore explicitly by ,O,x = b, (13)where x is the unknown vector, then matrix A dependson the locations of missing gates, while border information regarding aliased areas only appears in vectorb. If there is no missing gate, the matrix A will be afixed constant matrix independent of the data. Wepropose the following alternative algorithm, whichprocesses missing gates in a different way such that thematrix/~, remains constant regardless of missing gates.Algorithm B ( modified TDD): The same as algorithm A except 1 ) If gate i, j is missing, set Vil= 0 and mark gatei, j as missing. 2) Replace step 1 with the following: Step 1: Solve the system of linear equations OS - 0, (14) Oxijfori= 1, ...,re,j= 1, ...,n, where S = Y~ d2(V0, Vk~). (15)The summation is performed on all pairs of neighboring gates, indexed by ij and kl, respectively, includingmissing gates and d(Vi~, like) = Vv - l/~ + x~j - Xk~, (16)if at least one of the neighboring gates is missing. Ifboth gates i, j and k, l are available,d(Va, V~) = ~2V~ - xa + x~, [0,if Vo- Va; >~ tV.if Va- Vkt~< -tV,otherwise.(17) 3) The following modification is added to step 2:those xo that correspond to missing gates are discardedand are not to be used for contributing to the histogram.The idea behind this algorithm is that instead of excluding the missing gates from the equation, which isdifficult to process, the missing gates are assigned somearbitrary value and allowed to be freely adjusted in theprocess of minimizing the smoothness function. Thevalues xa for missing gates are discarded after step 1. The introduction of the free missing gates to theminimization problem causes additional residues dueto noisy data in the minimized smoothness measureand variance in the clusters in the histogram. As a resuit, the quality of the TDD output deteriorates slightly.However, we did not find noticeable difference in ourtesting and applications when compared with algorithmA. The advantage of algorithm B is that it has a fixedconstant coefficient matrix A, which can be pre-LUdecomposed and stored. In an implementation, onlyforward elimination and backward substitution areDECEMBER t993 JING AND WIENER 803computed. Refer to a textbook on solving linear systems, for example, Watkins ( 1991 ), for a detailed discussion on the LU decomposition. Because the I. and IJ matrices are sparse, the preceding computations and the memory requirement forstoring L and tJ do not cause problems for large twodimensional regions. For example, for a region with60 rows and 120 columns (7200 unknowns) it takesabout 2 MB of space to store [. and 0 and about 0.2 sto run algorithm B on a SUN SPARC II workstation.c. Incorporating subarea deatiasing If the two-dimensional region to be processed is toolarge to be processed in real-time, we may apply TDDto subareas instead of single gates. For example, wecan first divide the region into subareas, each of whichcontains 2 x 3 (6) gates. A dealiasing scheme is appliedto each of the subareas. Then by applying TDD tothese subareas, we can process a larger region (in thiscase larger by a factor of 6). The size of the subareas is often very small. A histogram technique described by Ray and Ziegler (1977)may be applied to dealias the subareas. Their techniquefinds the biggest gap in the histogram of the data inthe subarea and minimizes the width of the histogramby adding 2 Vn to data on the left-hand side of the gap.Note that in our case this technique is used only toremove the discontinuity due to aliasing in the smallsubarea. The difficulty of finding a true dealiased valueis not a concern at this stage in contrast to 1Zay'sscheme, where a complete dealiasing relies on the technique. The subarea size must be kept small. Otherwise,as the subarea size increases, the histogram width ofthe true velocity in the subarea may not be small whencompared with the Nyquist velocity (especially whenthere is strong wind shear in the area) and the techniqueof minimizing the histogram width will fail. If all the gates in a subarea are missing, we treat thesubarea as missing. A problem is encountered, however,in applying TDD to subareas that contain both dataand missing gates. In such subareas data and missinggates will share the same correction term. To minimizethe number of incorrectly dealiased gates a majorityrule is used: If most of the gates in the subarea aremissing, the subarea is processed as missing in TDD,otherwise it is considered to be available in TDD. Applying TDD to subareas provides a way of tradingdealiasing quality for processing speed. There is anotherway of doing this. That is, if a connected region is toolarge to process all gates, we can divide it into smallerregions and process each of them as an independentconnected region. This approach does not suffer fromthe problems of the subarea TDD, and should be usedif there are strong wind shears. Its shortcoming is thatknowledge of some of the data connections is lost. Thusthe probability of failure in the global dealiasing discussed in the next section is increased. In our real-timeimplementation these approaches are user selectable. With a more powerful workstation, for example, onewith 32 MB of memory and 50 MIPS (million instructions per second) processing speed, it is possible to process connected regions of 512 gates x 60 rays in realtime, and thus eliminate the need for a compromisedapproach in applying the TDD algorithm.4. Global dealiasing The TDD algorithm as discussed in previous sectionsremoves the discontinuity caused by aliasing in a singleconnected data region by correcting each gate using amultiple of 2Vn. Each region is either correctly dealiased or offset by a constant multiple of 2 Vn for allgates. Additional information must be provided to beable to correctly dealias all isolated regions. Hennington (1981 ) first introduced the idea o-dealiasing the Doppler velocity based on a known environmental wind field. Similar approaches were thenstudied and adopted by Merritt (1984), Boren et al.(1986), Bergen and Albers (1988), and Eilts and Smith(1990). The scheme works by establishing an environmental wind and performing the global dealiasing bycomparing the velocity data with it. The idea is basedon the following assumptions: 1 ) The wind field is composed of two parts, the environmental wind and the local wind. The local winddescribes fluctuations caused by gust fronts and stormsand acts as a local perturbation on the environmentalwind. 2) The environmental wind can be estimated insome way. 3) The magnitude of an averaged local windis less than the Nyquist velocity. These assumptions are true for most real situationsand therefore widely accepted. We adopt the sameglobal dealiasing idea and incorporate it into the following algorithm:Algorithm C (environmental wind adjustment):Input: A set of radial velocity data di located at xi,zi, i = 1, - -., n, and an environmental wind fieldE(x, y, z). Output: A modified radial velocity dataset. Step t: Evaluate the average difference between thedata and the environmental wind,D=-~ [di-E(xi,Yi,Zi)]. (18) 1Step 2: Add the correction value, c= 2V, X round(-~) ,(19)804 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME I0to all data di, where round( ) denotes the operationof rounding to the nearest integer.In this algorithm we simply compare the data with theenvironmental wind and minimize the difference byadding a suitable correction of multiple 2 V,. In practice, a rough estimate of the environmentalwind is mostly sufficient. For example, the wind canbe assumed to be a constant wind depending on thealtitude, and with the wind intensity and direction givenat a few altitudes, linear interpolation can be used toestimate the whole field. As time goes on, data sources useful for estimatingthe environmental wind are becoming more prevalent.Currently, sounding and forecast model data are available nationwide and the wind profiler data are availablein selected areas and will soon be available nationwidefrom the National Weather Service and other sources. The environmental wind can also be estimated viaVAD (velocity-azimuth display) analysis (Browningand Wexler 1968 ) using Doppler radar data. However,if we simply use dealiased data to estimate the environmental wind and then, in turn, use it to dealias thedata, serious incorrect, although self-consistent, dealiased results may be generated. Thus, any radar data used for VAD analysis mustbe free of aliasing. In the following situations, the environmental wind can be estimated from Doppler radardata: 1 ) The radar makes a high Nyquist velocity environmental wind scan (EWS) (using high PRF or long)~) once in a while, and thus generates data for environmental wind estimation. Because the environmental wind changes slowly, the EWS does not need to beperformed very often. It can also be done more quicklythan a normal volume scan since high spatial resolutionis unnecessary. 2) If the Nyquist velocity is known to be comparableto the magnitude of maximum radial velocity, it ispossible to remove aliased data by simply discardingall data that are close to Vn and - Vn. A VAD analysisusing a least-mean-squares fitting technique (Rabin andZrni~ 1980) can be applied to the remaining incomplete data for estimating the environmental wind. Thiscase may be difficult to establish in practice. The environmental wind, which must be continuous,provides a natural mean that enforces the continuitybetween isolated regions that are close in space. If mostof the data in a connected region are not aliased, a zeroenvironmental wind estimate, E(x, y, z) = 0, will oftenbe adequate for correct global dealiasing. Isolated regions can also be dealiased by examining other nearbyregions (Bergen and Albers 1988). Combining the modified TDD with the global dealiasing algorithm, we have developed a real-time dealiasing program. A flow diagram of the program isdepicted in Fig. 5.Update environment wind data IIRead N and form 2D data rays a set IIdentify all connected regionsFor each region I Apply Algorithm 13 ('I~D) to it II ^pply^lgo~~hmCtoit IIOutput the dealiased rays IFIG. 5. The flow diagram of a real-time implementationof the two-dimensional dealiasing method. The radial velocity data are assumed to be processedbefore input to the program. In the preprocessing, theunreliable data are detected using the SNR field andthen marked as missing. A ground-clutter map is usedto mark the missing data due to the ground-cluttercontamination. Before applying the two-dimensional dealiasing algorithm, a set of N rays must be buffered. The biggerN is, the more delay introduced in a real-time environment. We selected N = 60 or 120. The set of raysmust be adjacent in space. If there is a discontinuityin the radar scan, such as shifting from one elevationto another in a volume scan, the data buffering muststop and fewer than N rays must be passed forward forprocessing. Because algorithm B and algorithm C can only beapplied to a connected region, all isolated connectedregions in the two-dimensional data must be identifiedand processed separately. A fixed processing size of 60 beams X 120 gates isused in the implementation of algorithm B. If a singleconnected region is larger than 60 x 120, a suitablesubarea size is determined and algorithm B is appliedto the subareas after they are processed by the histogramwidth minimization algorithm. If a single connectedregion is small (less than 40 gates), it is processed bythe histogram width minimization algorithm insteadof TDD. The environmental wind estimate is obtained usinga data source different from the primary radar. A VADanalysis program is used to estimate the environmentalwind if no other data source is available.DECEMBER 1993 JING AND WIENER 805~DvC ~ ~ :-'~::~"C' ~": ~ -~"~ ::'~ ~. ~:' ~:~ :~::~ "".:::::::: ~ :':, - 'C-~ ~ ~ . s' ~-~ ' ~':: :~ ~:::~:::::: ~-~ ~ ,~ c~--:~ ~.. - :~ ~. :~: - ~:~'~::: /:~ - ~ ~-.:~:::~::::':'~..-~.< ~: ~ :::4:::: ...... : :: :7::: :.:..:::::: .~S: ~? ' ~:~"--~:~::"~,~.:3..:::~...., '. ? :.: ~... - : .~ .~ ~ ~ ~ ::~ . ~ -~ --~.~ ? ~: ~ ~ ..~ ' ~ ~ .. ~4 ~ ~ 4 ~ . l ~ ~ : ~ , ~150 Range (Km)~32 m/s 32 m/sFIG. 6. The original (upper) and the dealiased (lower) images of a sector scan with elevation 4.0-. The Nyquist velocity is 12.8 m s-~.0 ~ 15 ~ 0 ~ 15 Range (Km) ::: --32 m/s32 m/sFIG. 7. Area A in Fig. 6. The image on the left-hand side is the original, and that on theright-hand side is the dealiased. All gates in this aliased area are correctly dealiased.806 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME l0v 115 Range (Km) .... ~. _:~ -32 m/s 32 m/s FIG. 8. Area B in Fig. 6. The upper image is the original and the lower is the processed. Allaliased gates are corrected except for three isolated gates that are incorrectly processed using onlyglobal dealiasing. The dealiased data are output to other programs,which display or further process the data. Once thedealiased data are output, the real-time program loopsback to process the buffered data.5. Sample results The new real-time dealiasing tool was applied to bothMHR and NCAR's CP-2 research radar data inNCAR's 1992 summer field observation projectRAPS92 (real-time analysis and prediction of storms)from 1 June to 8 August. The processed data were sentto a real-time dual-Doppler analyzer, from which thethree-dimensional wind was generated and displayedin near-real time (the analyzer has a 30-s delay). TheCP-2 radar worked mostly in a dual polarization moderesulting in a quite small Nyquist velocity of 12.8m s-~. Thus the velocity data were often seriouslyaliased and compromised the quality of the dualDoppler results. Because of the particular location ofthe CP-2 radar, there was extensive ground-cluttercontamination. As a result, the dealiasing task was achallenging one. Shown in Fig. 6 is a sample of the results from ournew dealiasing tool. The CP-2 radar data were collectedat 1819:00 UTC 26 June 1992. The radar was operatingin a dual polarization sector scan mode. The radar datawere preprocessed with a threshold of SNR = -5 dBand a ground-clutter map was utilized to remove incorrect data. A rough estimate of the environmentalwind was made from radar data. The estimates usedwere: 10 m s-I, 330-, at 700-m altitude, and 10 m s-j,180-, at 3000-m altitude. The white areas in Fig. 6 are missing data. It can beseen in the figure that the dealiasing tool worked quitesatisfactorily. Most of the aliased gates were correctlydealiased except for a few isolated gates and small areas.The details of areas A, B, and C are shown in Figs. 7,8, and 9, respectively. Areas A and B (Figs. 7 and 8) are heavily aliasedwith an unclear border. Correct data and aliased gatesare interleaved. There are numerous missing gates,mostly due to ground clutter in those areas, that mayintroduce difficulties in a local expansion technique,especially for a scheme that starts from the radar andtravels away from it. The two-dimensional nondirectional TDD algorithm had no problem dealiasing thedata in these areas. There are three isolated gates inarea B that are not correctly dealiased since they donot have any neighboring continuity information. Asa result, their dealiased values depend solely on globaldealiasing, which uses a very rough estimate of the environmental wind. It is seen in Fig. 91 that the remote isolated areasare correctly dealiased except again for a few isolatedgates. Note that there are small-scale velocity featuresin the storms. These features are correctly dealiased Color images of Figs. 6-9 can be acquired by sending e-mail tojing@rap.ucar.edu.DECEMBER 1993 JING AND WIENER 807 I 60 ~ 80 Range (Km) 'T"it ,~ '.t' '"' ~ ~:.;~- '~' ~q' -32 m/s 32 m/s F~o. 9. Area C in Fig. 6. The upper image is the original and the lower is the processed. Notethat the remote isolated areas at ranges from 60 to 90 km are correctly dealiased. The complexfeatures in the areas are revealed. A few isolated gates are not processed correctly as in Fig. 8.and are clearly revealed. Some of the features are smalland comparable to the gate size. The thct that isolated gates or small regions tend tobe incorrectly dealiased is probably due to the followingreasons. FirsL these data are usually more noisy thanothers. Second, the environmental wind estimate is notaccurate. And, third, the difference between the environmental wind and the local wind averaged in a smallarea may not be small when compared with the Nyquistvelocity. The last two reasons justify the need for adealiasing technique, such as TDD, to use the datacontinuity information, in addition to an environmental wind comparison method for solving the dealiasing problem.6. Summary This work describes a new dealiasing technique forDoppler radar radial velocity data. It uses aliasing-induced discontinuity information and finds correctionvalues for all gates by solving a two-dimensional leastmean-squares problem, while other existing techniquesare based on local comparisons and regional expansions. The technique is conceptually simple (no complex rules and feature~dependent parameters are required) and is believed to work well on data contaminated by randomly scattered noise (such effects arelimited locally due to the nature of the least-meansquares solution). Environmental wind information isused for the global dealiasing of isolated regions. Thenew technique was applied to real-time operational radar data processing applications. It is believed to beuseful for the postanalysis of radar data as well. An assumption of a smooth wind field was made inour discussion. This assumption, however, may not bevalid in the presence of strong wind shears. If the difference in radial velocities of adjacent gates is largerthan V., falsely aliased border segments are detected.In the cases of mesocyclones and microbursts, thesefalse border segments tend to be short and isolated. Insuch cases, these false border segments, like thosecaused by random errors, will not result in incorrectdealiasing. However, if the segments are not isolatedor if they are long as in the case of a strong gust front,a long or organized false border may be created. Thiscan lead to incorrect dealiasing. In these cases, additional information must be used to identify these falseborders and remove them before solving the linear system in the TDD algorithms. A gust-front detection algorithm, for example, may be considered for this purpose. The technique proposed by Albers (1989) provides another approach to identify such false borders. Because the technique tries to minimize the internaldiscontinuity caused by aliasing in a connected twodimensional dataset, it may fail if the region is connected by incorrect data. For example, if a region hastwo parts that are only joined by a "thin (e.g., one gatewidth) bridge," then incorrect data on the "bridge"may cause one of the parts to be incorrectly dealiased.This situation is rarely encountered. A dilation-anderosion technique may be used to break these kinds of"weak" connections before TDD, as well as to connectclosely located isolated gates or small regions, whichmay not be correctly dealiased based on comparisonto the environmental wind. This is a subject of furtherstudy.808 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME I0 Acknowledgments. The authors want to express theirappreciation to Dr. Peter Neilley, Cathy Kessinger andDr. Andrew Crook for their invaluable discussions andsuggestions. This research was sponsored by NSFGrants DTFA01-91-Z-02039 under NSF-FAA Interagency Agreement. REFERENCESAlbers, S. C., 1989: Two-dimensional velocity dealiasing in highly sheared environments. Proc. 24th Conf on Radar Meteorology, Tallahassee, FL, Amer. Meteor. Soc., 411-414.Bargen, D. W., and R. C. Brown, 1980: Interactive radar velocity unfolding. Proc. 19th Conf on Radar Meteorology, Miami, Amer. Meteor. Soc., 278-283.Bergen, W. R., and S. C. Albers, 1988: Two- and three-dimensional de-aliasing of Doppler radar velocities. J. Atmos. Oceanic Tech nol., 5, 305-319.Boren, T. A., J. R. Cruz, and D. S. Zrni~, 1986: An artificial intel ligence approach to Doppler weather radar velocity dealiasing. Proc. 23rd Conf on Radar Meteorology, Snowmass, CO, Amer. Meteor. Soc., 107-110.Browning, K. A., and R. Wexler, 1968: The determination of kine matic properties of a wind field using Doppler radar. J. Appl. Meleor., 7, 105-113.Doviak, R. J., and D. S. ZrniO, 1984: Doppler Radar and Wealher Observations. Academic Press, 458 pp.Eilts, M. D., and S. D. Smith, 1990: Efficient dealiasing of Doppler velocities using local environment constraints. J. Atmos. Oceanic Technol.. 7, 118-128.Hennington, L., 1981: Reducing the effects of Doppler radar ambi guities. J. Appl. Meteor., 20, 1543-1546.Kundert, K. S., and A. S. Vincetelli, 1988: Sparse User's' Guide, A Sparse Linear Equation Solver, Version 1.3a. Dept. of Elect. Eng. and Comp. Science, University of California, Berkeley, 1 PP.Merritt, M. W., 1984: Automatic velocity dealiasing for real-time applications. Proc. 22rid Coq- on Radar Meleorology, Zurich, Amer. Meteor. Soc., 528-533.Rabin, R., and D. Zrni~, 1980: Subsynoptic-scale vertical wind re vealed by dual Doppler radar and VAD analysis. J. Atmos. Sci., 37, 644-654.Ray, P. S., and C. Ziegler, 1977: Dealiasing first moment Doppler estimates. J. Appl. Meteor., 16, 563-564.Tewarson, R. P., 1973: Sparse Matrices. Academic Press, 83 pp.Watkins, D. S., 1991: Fundamentals of Matrix Computations. John Wiley, 20 pp.

## Abstract

The Doppler velocity dealiasing problem has been discussed for many years. Because aliasing is easily identified by detecting abrupt changes in the data field, most existing algorithms use this technique to correct aliased data. Such algorithms are typically based on local expansion methods. Such methods make a dealiasing decision for each gate based on the information of its dealiased neighbors and thus can be sensitive to scattered incorrect data. This paper introduces a new approach that attempts to find all dealiased values for a given dataset by solving a linear system involving the entire dataset and thus avoiding local expansion. Because the solution is global, the new technique is conceptually simple and displays good performance on a number of test cases. The new technique described here was implemented to support real-time dealiasing in an operational setting.