Spaced-Antenna Interferometry to Measure Crossbeam Wind

Spaced-Antenna Interferometry to Measure Crossbeam Wind, Shear, and Turbulence:Theory and Formulation

GUIFU ZHANG

School of Meteorology, University of Oklahoma, Norman, Oklahoma

RICHARD J. DOVIAK

National Severe Storms Laboratory, Norman, Oklahoma

(Manuscript received 11 April 2006, in final form 9 August 2006)

ABSTRACT

The theory of measuring crossbeam wind, shear, and turbulence within the radar’s resolution volume V6

is described. Spaced-antenna weather radar interferometry is formulated for such measurements usingphased-array weather radar. The formulation for a spaced-antenna interferometer (SAI) includes shear ofthe mean wind, allows turbulence to be anisotropic, and allows receiving beams to have elliptical crosssections. Auto- and cross-correlation functions are derived based on wave scattering by randomly distrib-uted particles. Antenna separation, mean wind, shear, and turbulence all contribute to signal decorrelation.Crossbeam wind cannot be separated from shear, and thus crossbeam wind measurements are biased byshear. It is shown that SAI measures an apparent crossbeam wind (i.e., the angular shear of the radial windcomponent). Whereas the apparent crossbeam wind and turbulence within V6 cannot be separated usingmonostatic Doppler techniques, angular shear and turbulence can be separated using the SAI.

1. Introduction

Wind, shear, and turbulence are important in quan-tifying and forecasting weather. The wind field is mea-sured either by Doppler or interferometric techniques(Doviak and Zrnic 2006; Doviak et al. 1996). Weatherradars, such as the Weather Surveillance Radar-1988Doppler (WSR-88D), measure the Doppler velocity(i.e., the radial component of the scatterers’ velocity)and its associated distribution (i.e., the spectrumwidth). But a spaced-antenna interferometer (SAI)such as the National Center for Atmospheric Re-search’s (NCAR’s) Multiple Antenna Profiler Radar(MAPR; Cohn et al. 2001) can, if wind is uniform, alsomeasure the crossbeam wind, as well as the along-beamwind component within the radar’s resolution volumeV6 (Doviak and Zrnic 2006, their section 4.4.4). Thespaced-antenna method was first applied to the mea-surement of crossbeam winds in the upper atmosphere

(Mitra 1949; Briggs et al. 1950). Alternatives to directmeasurements of crossbeam wind, numerical modelingtechniques have been applied to retrieve crossbeamwinds (e.g., Gao et al. 2001; Xu et al. 2001; Shapiro et al.2003) from single-Doppler radar measurements. So-phisticated data assimilation techniques, such as thefour-dimensional variational (e.g., Sun and Crook1997) and ensemble Kalman filter (e.g., Snyder andZhang 2003; Tong and Xue 2005) methods, are neces-sary for optimal retrievals. All of these retrieval tech-niques require multiple volume scans of radial velocitydata and often involve a number of assumptions; it isadvantageous to have direct measurements of the cross-beam winds, and the SAI is one such instrument thatcan, under certain limitations discussed in this paper,provide such measurements.

However, applications of the SAI have been limitedmainly to long-wavelength (i.e., �6 m) radars for windmeasurements in the mesosphere–stratosphere–troposphere (MST), and shorter-wavelength (i.e., 30cm) boundary layer profiling radars. The theory forthese radars has been developed assuming anisotropicBragg scatterers in uniform mean flow with isotropicturbulence (Briggs 1984; Doviak et al. 1996). Recently,

Corresponding author address: Dr. Guifu Zhang, School of Me-teorology, University of Oklahoma, 120 David L. Boren Blvd.,Suite 5900, Norman, OK 73072-7307.E-mail: [email protected]

MAY 2007 Z H A N G A N D D O V I A K 791

DOI: 10.1175/JTECH2004.1

© 2007 American Meteorological Society

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Unauthenticated | Downloaded 02/25/22 03:53 AM UTC

the SAI technique has received attention by theweather radar community. The MAPR was mountedon a pedestal to form a mechanically scanned beam(Brown et al. 2005). An X-band dual-polarization SAI(DPSA; also with a mechanically scanned beam) hasbeen built and data have been collected with it forcrossbeam wind measurement (Hardwick et al. 2005).Neither the scanning MAPR nor the DPSA has pro-duced satisfactory crossbeam wind measurements re-sulting from various limitations, some of which are ad-dressed herein. Turbulence and shear limit the accuratemeasurement of crossbeam wind. Turbulence is usuallycalculated from spectrum width data, which can be cor-rupted by wind shear (Fang and Doviak 2001; Fang etal. 2004). The effect of shear on crossbeam wind mea-surement has not been fully understood and is ad-dressed in this paper.

Another fundamental limitation is the requirementof a long dwell time at each beam direction for accuratecrossbeam wind measurements using an SAI. This iscontrary to the idea of fast-scanning weather radar forlarge volume coverage. A phased-array weather radarcalled the National Weather Radar Test Bed (NWRT),located at the National Severe Storms Laboratory(NSSL) in Norman, Oklahoma, offers an opportunityto explore SAI techniques for crossbeam wind mea-surements using electronically scanned beams. TheNWRT was developed by a government–university–industry team consisting of the National Oceanic andAtmospheric Administration’s NSSL, the Tri-Agencies’ (Departments of Commerce, Defense, andTransportation) Radar Operations Center (ROC), theU.S. Navy’s Office of Naval Research, the LockheedMartin Corporation, the University of Oklahoma’sElectrical and Computer Engineering Department andSchool of Meteorology, the Oklahoma State Regentsfor Higher Education, the Federal Aviation Adminis-tration’s William J. Hughes Technical Center, and Ba-sic Commerce and Industries, Inc., to facilitate the col-laborative study of the phased-array radar (PAR) as amultiuse radar for weather and aviation. The NWRTcharacteristics are described by Forsyth et al. (2005).The NWRT has a pulse-to-pulse beam-steering capa-bility that provides better utilization of radar resources(Orescanin et al. 2005) so that shorter dwell times fornormal weather surveillance can be interlaced withlonger ones for weather radar interferometry. Thus, aNWRT-type SAI can simultaneously survey weatherand measure crossbeam wind and shear, as well as tur-bulence along selected directions. A complete theoryfor such an application has not been developed, al-though a series of theoretical studies were performedfor SAI applications to wind profiling (e.g., Liu et al.

1990; Woodman 1991; Doviak et al. 1996). Thesestudies have been based on the assumption of wavescattering from a statistically homogenous medium ofrefraction index fluctuations; no shear effects havebeen accounted, and particle scattering has not beenconsidered.

We formulate the theory of weather radar interfer-ometry for measurements of crossbeam wind, shear,and turbulence for an SAI scanning precipitation.Auto- and cross-correlation functions are derived basedon wave scattering by randomly distributed particles.The antenna separation, mean wind, shear, and aniso-tropic turbulence are all taken into account in the for-mulation.

This paper is organized as follows. Possible SAI con-figurations for the NWRT and their use are discussed insection 2. Spaced-antenna interferometry is formulatedbased on wave scattering from randomly distributedscatterers in section 3; in section 4 the composite ofmean crossbeam wind and crossbeam shear of the meanlongitudinal wind, the longitudinal wind, and turbu-lence are estimated from the auto- and cross-correlation functions. The separation of turbulencefrom wind and shear measurements is discussed in thissection.

2. Possible SAI configurations for the NWRT

The NWRT is the first phased-array weather radaroperating in the 10-cm-wavelength band (the same asWSR-88Ds). The NWRT has been adapted from amonopulse antenna used for target detection and track-ing. The antenna from an AN/SPY-1A radar (Brookner1988) and a transmitter from a WSR-88D weather ra-dar are used for the NWRT (Fig. 1a). The NWRTtransmits with all of the array elements uniformly ex-cited and receives signals with tapered weighting. Theantenna has three ports (i.e., a sum, azimuth difference,and elevation difference). Although the antenna’s dif-ference channels were disabled when it was transferredto the National Severe Storms Laboratory in Norman,these channels are presently being activated.

One of the research/development objectives for theNWRT is to make instantaneous and direct measure-ment of wind components along and across the beam ateach V6 along the beam. The NWRT is capable of pro-viding crossbeam wind and shear measurements withinthe beam while surveying the weather, a capability thatdoes not exist with mechanically steered beams. TheNWRT phased-array antenna allows SAI wind mea-surements without any change to the antenna hard-ware. The sum and azimuth difference signals can beused to form an azimuth SAI with two virtual receivers,

792 J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y VOLUME 24


one for each of the left and right halves of the antenna,as shown in Fig. 1b. Similarly, an elevation SAI can beconstructed (Fig. 1c) from the sum and the elevationdifference signals. A PAR in which one has access tomany more elements would allow the receiving anten-nas to be overlapped so that better performance forwind measurements can be achieved in low signal-to-noise environments (Zhang et al. 2004).

The antenna patterns for the monopulse sum anddifference channels are shown in Zhang et al. (2005).Here, the beam cross sections for the full and half-receiving apertures are sketched in Fig. 2. Figure 2ashows the full-aperture beam as a circle, and the beamof the azimuth half aperture is shown as an ellipse,corresponding to the azimuth SAI (Fig. 1b). The re-duced azimuth resolution is due to the reduced antennasize in the azimuthal direction. Rotating the antennapatterns in Fig. 2a by 90° leads to Fig. 2b, which showsthe beam cross sections for the zenith SAI (Fig. 1c).

FIG. 1. (a) The NWRT phased-array antenna (photo, courtesyof A. Zahrai, NSSL), and the configuration of receiving aperturesR1, R2 for SA weather radar interferometry for (b) azimuth SAIand (c) elevation SAI.

FIG. 2. Sketch of transmitting and receiving antenna beam-widths for the NWRT. The inner circles represent the transmittingbeam (also the beam associated with the receiving sum channel),and the outer ellipse is the beam associated with one of the SAIreceiving apertures: (a) azimuth SAI and (b) zenith SAI. Thetheoretical transmitting and receiving beamwidths are �1T �2.36��T � 1.53° and �1R � 2.36��R � 3.06°.



The elliptically shaped cross sections of the receivingantennas’ fields of view are taken into account in thefollowing formulation.

3. Formulation for spaced-antenna interferometry

Consider an SAI with the phase center of the trans-mitting aperture at T and that of the two receiving an-tennas at R1 and R2 (Fig. 3). A local Cartesian coordi-nate system in which (ax�, ay�, az�) are unit vectors has itsorigin at the center of V6, the radar resolution volumecircumscribed by the 6-dB contour of beam and range-weighting functions. The x�, z� vertical plane is definedas the one in which the beam lies. The x� axis is alongthe transmitting beam axis, and a location inside V6 is r� x�ax� � y�ay� � z�az�. Although the physical aperture

of the NWRT is on a plane tilted 10° from the vertical,the apertures of constant phase and the plane ofphase centers for the transmitting T and receiving R1

and R2 antennas are parallel to the (az�, ay�) plane,which is tilted relative to the physical aperture. In gen-eral, the two receiving antennas are separated by��(�y�12, �z�12).

Assume there is a spatial distribution of pointscatterers, and that the nth scatterer is located at rn(t)at time t. Distances from the scatterer to the trans-mitting and receiving antennas are |r0 rn|, |r01 rn|,and |r02 rn| respectively. Following previous work(Doviak and Zrnic 2006, their section 4.2; Zhang etal. 2005), we have received signals at R1 and R2 ex-pressed by

Vr01, t1� � �n�1

N

A1nW1n exp{ jk |r0 rnt1�| � |r01 rnt1�|�} 1�

Vr02, t2� � �n�1

N

A2nW2n exp{ jk |r0 rnt2�| � |r02 rnt2�|�}, 2�

FIG. 3. The coordinate systems and parameters used for SA weather radar interferometry. The phase centers ofthe SAI receiving and transmitting apertures are R1, R2, and T, respecitvely. The beam axis, at a zenith �0, lies inthe x, z vertical plane (e.g., the east–west vertical plane). The y�, z� plane is tilted and parallel to the plane thatcontains the receiving and transmitting constant-phase apertures. The origin of the tilted Cartesian coordinatesystem is at the center of V6.



where A1n is the prefilter echo amplitude of the nthscatterer located at rn(t1) at time t1, r01 is the vectordistance from the center of V6 to the receiver at R1, andW1n is a range-dependent weight that is a function ofthe transmitted pulse width and the receiver filter’sbandwidth (Doviak and Zrnic 2006, their section 4.4).Similar definitions apply to (2). Here A1n is propor-tional to the product of the square root of the powerdensity patterns for the transmitting and receiving an-tennas. Because the range extent of V6 is usually smallcompared to r0, the small changes in the weighting func-

tion W(r) resulting from the 1/r20 factor can be ig-

nored.Thus, the cross correlation of signals from the two

receivers can be written as

C12t2 t1� � �V*r01, t1�Vr02, t2�� C12��, 3�

under the assumption that signal statistics are station-ary, where the angle brackets indicate time or ensembleaveraging, and t2 t1 � � � mPRT � mTs is the sampletime spacing, and PRT � Ts is the pulse repetition time.Therefore, substituting (1) and (2) into (3), we obtain

C12�� n�1

N

�n��1

N

A2nW2nA*1n�W*1n� exp{ jk |r0 rn�t2�| � |r02 rn�t2�| |r0 rnt1�| |r01 rnt1�|�}��

n�1

N

A2nW2nA*1nW*1n exp{ jk |r0 rnt � ��| � |r02 rnt � ��| |r0 rnt�| |r01 rnt�|�}��nr�A2 rt � ��W2 rt � ��A*1 rt��W*1 rt��

exp{ jk |r0 rt � ��| � |r02 rt � ��| |r0 rt�| |r01 rt�|�} dr� . 4�

Because scatterers are randomly located and move in-dependently of each other, the expectations of the off-diagonal terms of the matrix are zero (Doviak andZrnic 2006, their section 5.2.2); thus, the double sumhas been reduced in Eq. (4) to a single sum along thematrix diagonal. Furthermore, the weighted sum has

been replaced by the weighted integral of the numberdensity n(r) of scatterers at r. Henceforth n(r) is as-sumed to be uniform across V6, although it can be afunction of V6 location.

To perform the integral in (4), the angular and rangeweighting functions are needed; these can be expressed as

Ai rt�� A0gT1�2gR

1�2 exp�z�2t�

4r02��T

2 z�t� z�i�

2

4r02��R

2 y�2t�

4r02��T

2 y�t� y�i�

2

4r02��R

2 �, 5a�

Wi rt�� exp�x�2t�

4�R2 �, 5b�

where i � 1, 2 and (y�i , z�i ) are the phase centers of theSAI receiving apertures, �2

R is the second central mo-ment of the range-weighting function assumed equalfor both receivers, ��T and ��R are the square roots ofthe second central moment of the one-way power-weighting functions along the zenith for the transmit-ting and receiving antennas, respectively, and ��T and��R are, correspondingly, those beamwidths along theazimuth (Fig. 2). Referenced to the commonly usedone-way 3-dB beamwidth, �1T � 2.36��T, where ��T ��/D. Parameters �, D, and � are the wavelength, thediameter of the transmitting antenna, and a beamwidthfactor (of the order of 1⁄2) that depends on the weight-

ing of the aperture elements. We have assumed that thereceiving apertures could be located anywhere in the z�,y� plane, although, for the NWRT, the phase centers ofthe receiving apertures are constrained to lie eitheralong z� (i.e., �y�12 � 0 for �z� crossbeam wind measure-ments) or along y� (i.e., �z�12 � 0 for �y� crossbeam windmeasurements). Furthermore, although beamwidths ofthe NWRT are functions of the angular displacement ofthe beams relative to the boresight (i.e., the directionperpendicular to the plane of the physical aperture), weshall assume the beam is directed along or near theboresight.

A Taylor series expansion is applied to the phaseterms up to the second order in the location r � (x�, y�,z�) of the elemental scattering volume dr (Doviak andZrnic 2006, their section 11.5) to obtain



|r0 rt�| � r0 � x�t� �z�2t� � y�2t�

2r0;

|r01 rt�| � r0 � x�t� � z�1 z�t��2 � y�1 y�t��2

2r0,

6�

with similar expansions for the other terms in (4) validfor narrow beams. The location of the elemental scat-tering volume at time t � � is

rt � �� rt� � v�, 7�

where v � �x�ax� � �y�ay� � �z�az� is the velocity of thescatterers within the elemental volume. We assumescatterers are perfect tracers of wind (i.e., inertia effectsand terminal velocity are ignored). Strictly speaking, wemeasure the reflectivity-weighted vertical componentof the scatterers’ velocity (Doviak and Zrnic 2006, theirsection 9.2.1). If elevation angles are small, terminalvelocities can be ignored; in any case, the azimuthalcomponent of crossbeam wind is not affected by termi-nal velocity.

Substituting (5)–(7) into (4) yields

C12�� |A0|2nr0�gTgR�� exp� x�2 � x� � �x��2

4�R2

z�2 � z� � �z��2

4r02��T

2 z� z�1�2 � z� � �z�� z�2�2

4r02��R

2

y�2 � y� � �y��2

4r02��T

2 y� y�1�2 � y� � �y�� y�2�2

4r02��R

2 �exp� jk�2�x��

z� � �z��2 � z� � �z�� z�2�2 z�2 z� z�1�2

2r0

�y� � �y��2 � y� � �y�� y�2�2 y�2 y� y�1�2

2r0�dx�dy�dz�� . 8�

The integrations are straightforward for a uniform windfield. If turbulence and shear are present, the velocity isnot only random, but its mean is also a function oflocation. In general, each of the three wind componentscan have shear in three directions. For narrow beams,motion �x� parallel to the beam axis causes most phaseshifts and signal fluctuation; thus, it is justified that onlythe shears of �x� within V6 are considered. We assume�x� can be expressed as the Taylor series to first orderin r,

�x�r� � �x�0� � �tx� � s · r� � �cx� � s · r�� cx� � sx�x� � sy�y� � sz�z�, 9�

where �x�(0) is the mean wind component at the centerof V6 and parallel to the beam axis, �tx� is the corre-sponding turbulent component, �cx� is the combinedmean and turbulent wind along x�, and s � ��x�(r) is the

shear assumed uniform across V6. Because we onlyneed to consider the shear of the wind parallel to thebeam axis, henceforth shear will always refer to thegradient of �x�, unless otherwise noted.

Turbulence vt is assumed to be statistically homoge-neous, and on average the scatterers’ displacements re-sulting from vt are zero. Thus, although the instanta-neous turbulent velocity vt is a function of r, turbulencecan be treated as a constant independent of the inte-gration variables because, on average, turbulence doesnot displace the scatterers. Thus, the integration can beperformed before the ensemble average. To integrate(8) we combine the mean and turbulent wind compo-nents (e.g., �cx�); the effect of turbulence becomes ap-parent after the ensemble average is performed. Sub-stituting (9) into (8), and performing integrationsshown in appendix A, we obtain (A4), repeated here as

C12�� |A0|2nr0�gTgR2��3�2�Rr02�e��e� exp 2k2�R

2 sx�2 �2�

� �exp 2jk�cx�� 2k2�e�2 r0sz�� cz�� z�12�2�2 2k2�e�

2 r0sy�� cy�� y�12�2�2��, 10a�

where

�cx� � �x�0� � �tx�, �cy� � �y�0� � �ty�, �cz� � �z�0� � �tz�

10b�

are the combined mean and turbulent velocities,

y�12 � y�2 y�1, z�12 � z�2 z�1 10c�

are the receiving antenna separations in y� and z� di-rections, and



�e� � ��T2 ��R

2

��T2 � ��R

2 , �e� � ��T2 ��R

2

��T2 � ��R

2

10d�

define the effective two-way beamwidths in zenith andazimuth, respectively; for the NWRT, ��T � ��T.

Because the SAI can be viewed as a pair of bistaticradars symmetrically located about the transmitter withFresnel zone centers separated a distance �y�12/2(Doviak et al. 1996), the measured Doppler shift, rep-resented by the phase term in (10a), is proportional to

the mean velocity along the bisector of the angle sub-tended by r01, r02. But, because the angle is very small,this phase shift is practically the same as that measuredwith a monostatic Doppler (e.g., the Doppler shift mea-sured in the sum-receiving channel).

Assuming that turbulent velocity components are in-dependent [i.e., the probability density function satis-fies p(�x�, �y�, �z�) � p(�x�)p(�y�)p(�z�)], and have a Gauss-ian probability distribution with standard deviations�tx�, �ty� and �tz�, the ensemble average can be per-formed (appendix B). Substituting (B1) to (B3) into(10a), the cross-correlation function can be written as

C12�� |A0|2nr0�gTgR21�2�3�2�Rr02�e��e� � exp 2jk�x�0�� 2k2�R

2 sx�2 � �tx�

2 ��2�

� exp{2k2�e�2 r0sz�� z�0�� z�12�2�2 2k2�e�

2 �tz�2 �2}

� exp�2k2�e�2 r0sy�� y�0�� y�12�2�2 2k2�e�

2 �ty�2 �2�. 11�

Because the factors �e�, �e� in (11) are much smallerthan 1 (for the NWRT �e� � 102), the crossbeam com-ponents of turbulence typically contribute to signaldecorrelation much less than the along-beam compo-nent does. On the other hand, if the beam is vertically,or nearly vertically, pointed (as it always is for windprofilers), and if turbulence is anisotropic (e.g., hori-zontally isotropic with significantly smaller verticalcomponent), crossbeam turbulence could be relativelysignificant. But, if turbulence is isotropic, or nearly so,

the contributions of crossbeam turbulence can be ig-nored as we henceforth assume.

If the receivers are matched and �z�12 � �y�12 � 0,C12(�) → C11(�) � C22(�), the autocovariances. At � �0, the autocovariance equals the signal power S. Thus,

C110� � S � |A0|2nr0�gTgR2��3�2�Rr02�e��e�,

12�

and the cross-correlation coefficient is

c12�� C12��

S� exp2jk�x�0�� 2k2�R

2 sx�2 � �tx�

2 ��2 2k2�e�2 { r0sz� � �z�0�� z�12�2}2

2k2�e�2 � { r0sy� � �y�0�� y�12�2}2�. 13�

If there is no shear, it can be shown that (13) is exactlythe same as the cross correlation of signals for Braggscatter from refractive index perturbations [Doviak etal. 1996, their Eq. (58)], under the condition that theBragg scatterers’ correlation length transverse to thebeam is small compared to the antenna diameter (i.e.,Bragg scatter is isotropic). Decorrelation of the signalby the mean wind advecting the scatterers out of V6 istypically small and has been neglected in deriving Eq.(13).

4. Interpretation of the formulation

Equation (13) constitutes the theoretical formulationfor SA weather radar interferometry in the presence ofmean wind, turbulence, and shear. In using interferom-

etry, we are primarily interested in the magnitudes ofthe auto- and cross-correlation functions, and hence-forth we shall focus on the magnitude of (13). The sec-ond and third exponents account for signal decorrela-tion caused by longitudinal shear sx� and the longitudi-nal component �tx� of turbulence. The remainingexponents account for signal decorrelation caused bythe baseline shear, baseline wind, and receiving an-tenna separation. Baseline wind and shear are thecrossbeam wind and shear parallel to a pair of SAIreceivers. As seen from (13), baseline shear combineswith baseline wind, and measurements of the correla-tion functions cannot separate the two.

Because weather radar beamwidths are small, it canbe shown that factors sz� � �z�(0)/r0 � s� and sy� ��y�(0)/r0 � s� are approximately angular shears of the



mean radial wind. Angular shears are defined as theradial velocity change per differential arc length (e.g., r0

sin�0d� ). Thus, s� � (1/r0 sin�0)(��r/�� ) is the azi-muthal angular shear of the radial wind component �r.Angular shears can also be determined from Dopplermeasurements at two or more angles in each direction;this measurement is called Doppler beam swinging[DBS; a variant of the velocity–azimuth display (VAD)technique commonly used to determine vector windsfrom weather observations with a single Doppler ra-dar]. Either the DBS or SAI method can be used toestimate the angular shear of the radial velocity. But,given the same resolution and a high signal-to-noiseratio, the SAI method provides a more accurate mea-sure than the DBS method (Doviak et al. 2004a). TheDBS method is employed by wind profilers to measurewind, typically under the assumption that wind is uni-form over the region scanned by the beams, and thatshear of the vertical wind can be ignored. Ignoring hori-zontal shear of the vertical wind should be a reasonableassumption for fair weather conditions, but under dis-turbed conditions this shear could cause significant er-rors in wind measurements. The effects of shear onwind measurements with profilers was evident in ex-periments in which six measuring systems were com-pared; on days of relatively unperturbed flow, esti-mated winds agreed very well, but the agreement wassporadic on days in which the convective boundarylayer was active (Flowers et al. 1994).

As stated above, cross-correlation measurements ofweather signals cannot distinguish baseline shear frombaseline wind. Thus, shear bias measurements of cross-beam wind are within V6. To provide a physical expla-nation as to why baseline shear acts like baseline wind,consider a pair of scatterers a, b at range r0 (Fig. 4). TheSAI can be viewed as a pair of bistatic radars, TR1 andTR2, symmetrically located about the transmitter T,having phase centers at TR1 and TR2 separated by adistance �y�12/2 (i.e., the separation of the Fresnel zonecenters; Doviak et al. 1996). Fresnel zones are thoseannuli at range r0 within which the phase pathlengthfrom T to the scatterer within the annulus and from thescatterer to the receiver remains within �/2 for all scat-terers within the annulus. Thus, if scatterer “a” advectsalong y� from position a to a� (Fig. 4a), it [or any otherscatterer being advected by �y�(0) such as b] will occupythe same relative position within the Fresnel zones ofTR1, because scatterer a� occupies in the Fresnel zonesof TR2. Thus, the time series of weather signals from adistribution of scatterers received in bistatic receiver R2

will be exactly the same as those received in bistaticreceiver R1, but delayed by �y�/2�y�(0). The key con-cept is that the phase path k(Ta� � a�R2) (i.e., from T to

a�to R2), after the scatterer has advected a distance�y�12/2, is the same as the initial phase path k(Ta � aR1).Alternatively, because �y�12/2 K r0, the bistatic radarsTR1, TR2 can be replaced with a pair of monostaticradars at the respective bistatic phase centers. For ex-ample, the DPSA (Hardwick et al. 2005) is a pair ofclosely spaced monostatic radars. In this case, for sig-nals in receiver R2 to be identical to those in R1 after alag of �y�/2�y�(0), the phase path kLa� needs to be theidentical to kLa.

Now let the scatterers only be moved by shear sy�

[i.e., �y�(0) � 0; Fig. 4b]. In order for the signal in R1 tobe the same as in R2, the paths La and La� must be thesame (or equivalently Lb � Lb�). If the scatterers,spaced p apart, are symmetrically located about theorigin, they would be equally displaced �� in oppositedirections, where �� sy�p/2. Using the geometry inFig. 4b,

La � �d y�12�4�2 � r0�2; La�

� �r0 ��2 � d � y�12�4�2. 14�

FIG. 4. A bistatic radar schematic to explain how baseline shearacts the same as baseline wind. Time delay caused by (a) cross-beam wind and (b) shear.



For narrow beams d K r0, and typically d k �y�12, d k

��. Expanding the radicals to first order in �� and�y�12, respectively, and setting La � La�, we obtain

�� dy�12

2r0. 15�

Substituting �� sy�p/2 and d/p � 1/2, the maximumcross correlation of the signals occurs if r0sy�� x/2,which is in agreement with that shown by (13).

Equation (13) has also been verified with numericalsimulations of wave scattering by a layer of randomlydistributed particles moving across the beam (Zhang etal. 2005). Cross-correlation functions are estimatedfrom the simulated time series data. Indeed, the base-line shear of along-beam wind causes the cross-correlation peak to move toward positive or negativetime lags, depending on whether shear is positive ornegative.

Figure 5 shows the cross-correlation coefficient as afunction of �. The parameters of the NWRT were usedfor the calculation. Both auto- and cross-correlation co-efficients at various ranges are shown in Fig. 5a. At nearranges, signal decorrelation is mainly caused by the an-tenna separation, crossbeam wind, and turbulence. Pre-vious studies that ignored shear showed that turbulenceplays a significant role in decorrelating signals becauseits effect on the correlation time, compared to thatcaused by baseline wind, is multiplied by the ratio of thetransmitting antenna diameter to wavelength [Doviaket al. 2004b, their Eq. (3)]. But, shear is also a signifi-cant factor in decorrelating the signals, especially atlong ranges where beamwidths are large. Because �R isindependent of range, and turbulence is assumed to beuniform, the change in correlation time with range isonly due to crossbeam shear.

Figure 5b shows the correlation coefficients forweather signals from scatterers at a fixed range of 30km, and for various shears along the baseline. Increasesof baseline shear not only decrease the correlationtime, but it also shifts the time lag to the peak of thecross-correlation function.

Although there appears to be no practical method toseparate baseline shear from baseline wind, measure-ment of the apparent baseline wind [e.g., �ay�(0) � r0sy�

� �y�(0)], obtained simultaneously everywhere alongthe beam, might provide useful constraints to sophisti-cated retrieval algorithms (e.g., Shapiro et al. 2003) thatcan generate the fields of vector winds over large do-mains. Henceforth, further discussion is focused onmeasurements of the apparent baseline wind and tur-bulence. We need to distinguish a definition of appar-ent wind used in early observations with SAIs (e.g.,

Briggs 1984). That apparent wind is baseline wind bi-ased by turbulence and/or cross-baseline wind (Hollo-way et al. 1997), whereas the apparent wind definedherein is baseline wind biased by baseline shear. Ap-parent baseline winds in this paper are directly relatedto the angular shears of the radial wind (i.e., s� � �ay�/r0,and s� � �az�/r0).

5. Estimation of the apparent baseline wind andturbulence

a. Estimation of wind

We can calculate the apparent baseline winds fromthe cross-correlation function using a cross-correlation

FIG. 5. Auto- and cross-correlation coefficients vs � at r0 �10,20, 40, and 80 km for NWRT parameters: � � 0.0938 m, �y�12 �1.46 m, �z�12 � 0.0 m, ��T � 0.65°, ��R � 1.30°. Meteorologicalparameters are �y�(0) � 20, �z�(0) � 5, �tx� � �ty� � �tz� � 0.5m s1, and sx� � 0. (a) Dependence on r0; sy� � 0, sz� � 0.002 s1

and (b) dependence on shear sy� at r0 � 30 km; sz� � 0.0.



ratio (CCR) method (Zhang et al. 2003), or the full-correlation analysis (FCA) method (Briggs 1984). Forexample, the apparent azimuth baseline wind compo-nent [i.e., �ay�(0)] can be calculated from the cross-correlation function for signals from SAI’s azimuth re-ceivers (Fig. 1b) for which �z�12 � 0 in (13). The loga-rithm of the cross-correlation magnitudes at equalpositive and negative lags leads to

L�� ln|c12

��||c12

��|� 4k2�e�

2 y�12�ay�0��. 16�

Thus, the apparent baseline wind �ay�(0) is given by

�ay�0� �L��

4k2�e�2 y�12�

. 17�

In a similar way, if two SAI-receiving antennas areseparated in the zenith direction, the apparent baselinewind component �az�(0) can be calculated. Hence, an-gular shears s� and s� within V6 are obtained.

Because baseline shear combines with baseline wind,the accuracies of measuring the apparent baseline wind(or angular shear) can be directly derived from theo-retical error analysis developed for baseline wind mea-surements in the absence of shear (e.g., Zhang et al.2004; Doviak et al. 2004b). Figure 6 shows the standarddeviation of the estimates of the apparent azimuthalbaseline wind as a function of range and turbulence.The error increase with range is due to the increased

effect of the vertical shear. Compared to measurementsof the along-beam wind component using Dopplermeasurements, baseline wind measurement requireslonger dwell times. For example, if the turbulence �tx�

� 0.5 m s1, about 5 s of data collection time is requiredto achieve an azimuthal apparent baseline wind �ay�(0),with a measurement accuracy of 2.0 m s1 using theNWRT at near ranges. On the other hand, Dopplervelocity (i.e., the radial wind component) can be mea-sured with accuracies better than 1 m s1 in a collectiontime two orders of magnitude shorter. The crossbeamwind estimated using DBS is degraded by a factor of1/��, where �� is the angular separation, leading topoor measurement accuracy.

b. Estimation of turbulence: The SAI method

Turbulence can estimated from spectrum width (or,equivalently, the correlation time of the autocorrela-tion function) using single-beamwidth (1BW) Dopplerweather radars such as WSR-88D. Such an estimation isneither accurate nor reliable because beam broadeningand shear (Doviak and Zrnic 2006, their section 5.3)bias the turbulence estimates. Melnikov et al. (2003)show that layers of unusually large spectrum widths(e.g., larger than 8 m s1), suggestive of turbulence haz-ardous to safe flight (Lee 1977), are seen in stratiformprecipitation. But, these large widths are principallydue to shear, and are not necessarily hazardous to safeflight. It is difficult to separate between the shear andturbulence contributions to spectrum width, especiallyif strong shear is confined to layers that are thin com-pared to V6. Thus, finding an accurate way to estimateturbulence could be advantageous. With the SAImethod, turbulence can be calculated from (13), as alsoshown by Holloway et al. (1997).

Examination of (13) also shows that longitudinalshear sx� combines with turbulence �tx� and thus turbu-lence measurements are biased by sx�. Because rangeresolution is fine, longitudinal shear is typically smallcompared to turbulence and often can be neglected. Ifnot, methods must be found to separate the two. Forexample, the longitudinal shear sx� can be determinedby using different range resolutions, and thus measure-ments of turbulence can be extracted from (13). Butbaseline shear cannot be determined by using differentbeamwidths. Comparisons between turbulence mea-surements using the SAI method and ones obtainedusing a beam sufficiently narrow that the transverseshear of the radial wind can be ignored showed goodagreement (Chau et al. 2000), confirming the robust-ness of the SAI method for the measurement of turbu-lence.

FIG. 6. Normalized standard deviation of apparent wind esti-mates �ay�(0) � r0sy� � �y�(0) vs range for various levels of turbu-lence and NWRT parameters: � � 0.0938 m, �y�12 � 1.46 m, �z�12

� 0.0 m, ��T � 0.65°, ��R � 1.30°. The mean apparent baselinewind is �ay�(0) � 20 m s1 . Other meteorological parameters are�x�(0) � 0, �z�(0) � 5 m s1, sx� � 0, sz� � 0.002 s1.



c. The dual-beamwidth method

Recently, a dual-beamwidth (2BW) radar methodwas applied to a VHF (i.e., 6-m wavelength) profilingradar to measure turbulence using the DBS method(VanZandt et al. 2002). The dual-beamwidth methodseparates the effect that vertical shear of the horizontalwind has on turbulence measurements when off-vertical beams are used. Off-vertical beams are re-quired at long wavelengths, because vertical incidencebackscatter from Bragg scatterers, with correlationlengths not significantly smaller than the antenna diam-eter, introduces an unknown multiplicative factor thatcombines with turbulence �t (Chau et al. 2000). The useof an off-vertical beam mitigates the backscatter fromBragg scatterers of unknown correlation lengths. Here,we extend the dual-beamwidth method to an anisotro-

pic beam pattern to estimate shear and turbulence witha weather radar interferometer.

Dual-beamwidth signals are indeed obtained fromthe NWRT SAI (i.e., the sum channel gives a narrow-beamwidth signal, and each half of the receiving aper-tures gives broad-beamwidth signals). Thus, by letting�y�12 � �z�12 � 0 and applying (13) to the sum channel,the narrow-beam autocorrelation coefficient magni-tude is

|cN��| � |css��| � exp 2k2�tx�2 �2 k2r0

2��T2 s�

2 � s�2 ��2�.

18�

To obtain a measurement of the angular shear s� weuse the azimuth SAI to obtain the azimuth broad-beamautocorrelation coefficient

cB��|� � |c11��| � exp 2k2�tx�2 �2 k2r0

2��T2 s��2 2k2r0

2�e�2 s��2�. 19�

To arrive at (19) we substituted �e� � ��T /�2 because,for the azimuth SAI, ��R � ��T. In deriving these twoequations we have assumed that longitudinal shear isnegligible but, if is not, it can be determined as pointedout in the previous section. Using (18) and (19) s2

� iscalculated to be

s�2 �

1

k2r02�22�e�

2 ��T2 �

ln� |cN��||cB��|�

�. 20�

Likewise, we can solve for the within-beam shear s� inthe zenith direction. These two shear values, substi-tuted into (18), lead to the solution

�tx�2 �

1

2k2�2 � ��T2

2�e�2 ��T

2 ln�|cB��|�|cN��|��

��T2

2�e�2 ��T

2 ln�|cB��|�|cN��| � ln|cN��| 21�

for turbulence in terms of the measured autocorrelationfunctions of signals received through the broad azimuthand zenith beams, and the narrow sum beam of theNWRT. These results reduce to those of VanZandt etal. (2002). When s� and s� are calculated, these valuesare substituted into (19) to obtain a measure of turbu-lence �tx�. In the typical case that azimuthal shear s� isnegligible compared to zenith (i.e., vertical) shear, thesecond term in (21) can be ignored. Thus, dual-beamwidth-derived shear and turbulence can be com-pared with those estimated using SA interferometry.

6. Summary and conclusions

We develop and discuss weather radar interferom-etry to measure, with a spaced-antenna interferometer(SAI), crossbeam wind, turbulence, and shear withinthe radar’s resolution volume V6. Although the SAI hasprincipally been applied to measurements of the cloud-free atmosphere, it has potential application to futureweather radars. We formulate the problem based on

scattering from randomly distributed particles, allowthe receiving beams to have elliptical cross sections (re-quired if full receiving apertures are used), and con-sider anisotropic turbulence and shear, whereas previ-ous formulations assumed Bragg scatter, circularbeams, uniform wind, and isotropic turbulence. Cross-and autocorrelation functions of the weather signals re-ceived by a pair of SAI receivers are derived. If turbu-lence is isotropic the theory shows that turbulencetransverse to the beam contributes negligibly to the cor-relation functions.

It is shown that mean baseline wind (i.e., the cross-beam wind parallel to the baseline connecting a pair ofSAI receivers) within V6 cannot be separated frombaseline shear of the mean longitudinal (i.e., along thebeam) wind. That is, baseline wind and shear combineto form the angular shear of the radial velocity. Never-theless, the SAI can separate turbulence from shearwithin V6, whereas this separation cannot be made us-ing Doppler techniques. SAI turbulence measure-



ments could improve quantification of atmospheric tur-bulent kinetic energy and measurements of turbulencewithin V6.

Meteorologists have been developing methodswhereby crossbeam winds can be retrieved from Dopp-ler weather radar data combined with numericalweather models (e.g., Gao et al. 2001; Shapiro et al.2003). These researchers demonstrate that the retrievalof crossbeam winds from dynamic models is improvedif there are observational constraints (e.g., actual mea-surements of the radial component of wind) in additionto those imposed by the model. For example, a velocityvolume processing (VVP; Waldteufel and Corbin 1979)method has been used to estimate the crossbeam windat selected regions and is added as a constraint to im-prove the accuracy of the retrieved winds (Shapiro et al.2003). It is suggested that SAI simultaneous measure-ments of the angular shear along the beam at chosenbeam directions could provide additional observationalconstraints on the numerically derived wind field. But,

angular shear measurement with SAI requires longdwell times, whereas DBS methods (e.g., the VVP) canmake this measurement, albeit with less resolution, inan order or two less dwell time.

Acknowledgments. The authors greatly appreciatehelpful discussions with Drs. D. S. Zrnic, Q. Xu, and D.Forsyth of the National Severe Storms Laboratory, andDrs. M. Xue, R. Palmer, and H. Bluestein of the Uni-versity of Oklahoma. Many thanks go to LockheedMartin and NSSL’s engineers for developing theNWRT. This work was partially supported by NOAA/NSSL under the Cooperative Agreement NA17RJ1227.

APPENDIX A

Integrations over Radar Resolution Volume

To complete the integrations in (8), we perform theintegrals over x� first, defining

Jx� � � exp�x�2 � x� � �x��2

4�R2 2jk�x�� dx�

� � exp�x�2 � x� � �cx�� sx�x�� sz�z�� sy�y��2

4�R2 2jk�cx�� sx�x�� sz�z�� sy�y�� dx�

� 4��R2

1 � 1 � sx��2 exp��cx�� sz�z�� sy�y��2

4�R2 2jk�cx�� sz�z�� sy�y��

� exp� 4jk�R2 sx�� 1 � sx��cx� � sz�z� � sy�y��2

4�R2 1 � 1 � sx��2�

→sx��K1 �2��R

� exp��cx�� sz�z�� sy�y��2

4�R2 2jk�cx�� sz�z�� sy�y�� exp� 4jk�R

2 sx�� cx� � sz�z� � sy�y��2

8�R2

� �2��R exp 2k2�R2 sx�

2 �2 2jk�cx�� sz�z�� sy�y��. A1�

Two assumptions, sx�� K 1 and [(�cx� � sz�z� � sy�y�)2�2/8�2

R] K 1, were made in arriving at (A1); these assump-tions should be valid for weather radar applications.For example, consider � � p�c, where �c is the correla-tion time of weather signals and p is a multiplying fac-tor. Typically, correlation functions only have signifi-cant values if � is 2 or 3 times �c. For a 10-cm-wavelength radar, �c 0.02 s (i.e., spectrum widths �� /4��c are typically larger than 0.5 m s1). Thus, if sx�

20 m s1 km1, a rather large value results, sx�� 103.Further argument can be made to show that if beam-widths are less than a few kilometers across, range reso-lution is about 100 m, and �cx� 100 m s1, the secondassumption is acceptable.

If receiving antennas are symmetrically located aboutthe transmitting antenna, the z� integration in (8) canbe written as

Jz� � � exp�z�2 � z� � �cz��2

4r02��T

2 z� z�1�

2 � z� � �cz�� z�2�2

4r02��R

2 2jksz�z�� jk

2r02z� � �cz��2�cz�� z�12��dz�,



where �z�12 � z�2 z�1. Carrying out the integration, we arrive at

�� 1

2r02��T

2 �1

2r02��R

2 �1�2

exp��cz�

2 �2

2 � 1

2r02��T

2 �1

2r02��R

2 �z�1

2 � z�22 2z�2�cz��

4r02��R

2 jk

2r0�cz��2�cz�� z�12��

� exp��jksz�� jk

2r02�cz�� z�12�

�cz��

2 � 1

2r02��T

2 �1

2r02��R

2 ��2 � 1

2r02��T

2 �1

2r02��R

2 �.

Under far-field conditions (i.e., r0 � 2D2/�) and similar assumptions used to arrive at (A1), it can be shown theabove expression reduces to

� �2��e�r0 exp 2k2�e�2 r0sz�� cz�� z�12�2�2 jksz��cz��, A2a�

where the effective beamwidth �e� is

�e� �1�1

��T2 �

1

��R2 � ��T

2 ��R2

��T2 � ��R

2 .

A2b�

Similarly, we have

Jy� � � exp�y�2 � y� � �cy��2

4r02��T

2 y� y�1�2 � y� � �cy�� y�2�2

4r02��R

2 2jksy�y�� jk

2r02y� � �cy��2�cy�� y�12��dy�

� �2��e�r0 exp 2k2�e�2 r0r0sy�� cy�� y�12�2�2 jksy��cy��, A3a�

with

�e� � ��T2 ��R

2

��T2 � ��R

2 . A3b�

Substituting (A1)–(A3) into (8), we obtain

C12�� |A0|2nr0�gTgR2��3�2�Rr02�e��e� exp 2k2�R

2 sx�2 �2�

� �exp 2jk�cx�� 2k2�e�2 r0sz�� cz�� z�12�2�2 2k2�e�

2 r0sy�� cy�� y�12�2�2��. A4�

APPENDIX B

Ensemble Average Velocity Fluctuations

The three components �tx�, �ty�, �z� of turbulence areassumed to be independent [i.e., p(�tx�, �ty�, �tz�) �p(�tx�), p(�ty�), p(�tz�)], and each has a Gaussian-shapedprobability distribution with, in general, different rmsvalues �tx�, �ty�, �tz�; that is, we allow turbulence to be

anisotropic. Substituting (11) into (10a), the ensembleaverage over the three components of turbulence re-duces to the product Ix�Iy�Iz� where Ix� is defined as

Ix� � �exp2jk�cx��.

Because only �tx� is a random variable, the above equa-tion can be expressed as



Ix� � exp 2jk�x�0��exp2jk�tx��

� exp 2jk�x�0�� exp2jk�tx��p�tx�� d�tx�

�1

�2��tx�

exp 2jk�x�0�� exp2jk�tx�� exp��tx�

2

2�tx�2 � d�tx�.

Performing the integration, we obtain

Ix� � exp 2jk�x�0�� exp2k2�tx�2 �2�. B1�

Now turning our attention to the crossbeam terms,

Iz� � �exp k2�e�2 r0sz�� cz�� z�12�2�2�� exp{2k2�e�

2 r0sz�� z�0�� tz�� z�12�2�2��

�� exp 2k2�e�2 r0sz�� z�0�� tz�� z�12�2�2�p�tz�� d�tz�

Iz� �1

�2��tz�

� exp{2k2�e�2 r0sz�� z�0�� tz�� z�12�2�2} exp�

�tz�2

2�tz�2 �d�tz� .

Performing the integration, we obtain

Iz� �1

1 � 4k2�e�2 �tz�

2 �2�1�2 exp� 8�tz�2

1 � 4k2�e�2 �tz�

2 �2 k4�e�4 �2 r0sz�� z�0�� z�12�2�2

� exp{2k2�e�2 r0sz�� z�0�� z�12�2�2}.

Because 2k2�2e��

2tz��

2 K 1 for typical weather radar pa-rameters, we convert the square root factor in the de-nominator to an exponential function, and by keepingterms to second order in �2, this equation can be ex-pressed more simply as

�exp{2k2�e�2 r0sz�� z�0�� z�12�2�2 2k2�e�

2 �tz�2 �2}.

B2�

In a like procedure, the integral Iy� is

Iy� � �exp{2k2�e�2 r0sy�� y�0�� ty�� y�12�2�2}�

�� exp{2k2�e�2 r0sy�� y�0�� ty�� y�12�2�2}p�t�� d�t�

� exp{2k2�e�2 r0sy�� y�0�� y�12�2�2 2k2�e�

2 �ty�2 �2}. B3�

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Spaced-Antenna Interferometry to Measure Crossbeam Wind