The ST/MST (stratosphere troposphere/mesosphere stratosphere troposphere) clear air Doppler radar, or wind profiler, is an important tool in observational meteorology because of its capability to remote observe dynamic parameters of the atmosphere. There are difficulties in transforming the observed radial velocities into meteorological wind components. How this problem has been treated in the past is reviewed, and some of the analysis is recast to a form more suited to the high diagnostic abilities of a number of fixed beam configurations with reference to a linear wind field. The results, in conjunction with other works which treats problems such as the effects of finite sample volumes in the presence of nonhomogeneous atmospheric reflectivity, have implications important to the design of both individual MST/ST radars and MST/ST radar networks. The key parameters to uncoupling terms in the scaling equations are w sub x and w sub y. Whenever the stratiform condition, which states that these two parameters are negligible, is satisfied, a five beam ST radar may determine unbiased values of u, v, and w for sample volumes directly above the radar. The divergence and partial deformation of the flow may also be determined. Three beam systems can determine w and w sub z, but are unable to obtain u and v wind components uncontaminated by vertical sheer terms, even when the stratiform condition is satisfied

Clark, W. L.

Green, J. L.

Warnock, J. M.

English

NASA Technical Reports Server

N8 7 - 1 0502
7.1.3 DETERMINATION OF U, V, AND W FROM SINGLE STATION DOPPLER
RADAR RADIAL VELOCITIES
W. L. Clark, J. L. Green, and J. M. Warnock
Aer oncmy Laboratory
National Oceanic and Atmospheric Administration
Boulder, CO 80303
INTRODUCTION
The ST/MST clear-air Doppler radar, or wind profiler, is becoming an
important tool in observational meteorology because of its capability to
remotely observe dynamic parameters of the atmosphere. However, there are
difficulties, which have long been recognized in work with precipitation
sensitive Doppler radars, in transforming the observed radial velocities into
meteorological wind components. As WALDTNJFEL and CORBIN (1979) put it "...
the problem lies in the fact that one would like to know as much as possible
about the wind vector field, whereas a single-Doppler radar yields only one
wind component, the radial. Every attempt to gain some knowledge of the vector
field, therefore, must compound the data field with additional hypotheses or
simplifications. " In this paper, we review how this problem has been treated
in the past, and recast some of the analysis to a form more suited to the high
elevation angle, fixed beem ST radar profiling techniques. We then examine the
diagnostic abilities of a number of fixed beam configurations with reference to
a linear wind field. The results, in conjunction with other work which treats
problems such as the effects of finite sample volumes in the presence of
nonhomogeneous atmospheric reflectivity (e.g., KOSCIELNY et al., 1984), have
implications important to the design of both individual MST/ST radars and
MST/ST radar networks.
B ACKG ROUND
The use of pulse Doppler radar to measure horizontal winds seems to have
been first suggested in the literature by PROBERT-JONES (1960) in connection
with a 3-cm precipitation sensitive system. LHERMITTE and ATLAS (1961)
suggested the VAD (velocity azimuth display) method of retrieving the mean
horizontal velocity from radial velocity data taken in horizontal clrcles
centered along the vertical of the radar site. CATON (1963) and BROWNING and
WEXLER (1968) confined the analysis to a horizontal plane to deduce mean
convergence, stretching and shearing deformation as well as the mean horizontal
velocities from VAD observations. All of these authors treated only the
stratiform situation, where _w/_x = _w/_y = 0.0.
EASTERBROOK (1975) examined the processing of data in a conical sector,
wherein values are estimated at a point not centered on the radar. In this
case, although five parameters of the wind field can be extracted, the mean
horizontal velocity is contaminated by vorticity and cannot be independently
determined.
The papers by WOODMAN and GUILLEN (1974) describing results using the
Jicamarca incoherent-scatter radar, and GREEN et al. (1975) describing the
results from the prototype VHF ST wind profiling radar at Sunset, Colorado,
mark the arrival in the literature of VHF clear-air Doppler radars as important
wind profile measurement tools. These radars are generally fixed-multibeem
systems, utilizing large elevation angles consistent with wind height-profiling
through the troposphere and into the lower stratosphere, as opposed to the
conventional meteorological radars, which usually have a single rotating beam,
and utilize low zenith angles consistent with measuring winds at low heights.
https://ntrs.nasa.gov/search.jsp?R=19870001069 2020-03-23T14:36:47+00:00Z
386 _ "
Because precipitation sensitive Doppler radars work under conditions of
relatively high signal to noise, a practical radar may utilize a rotatable dish
antenna. Then it is natural to apply data-redundant least squares techniques
taking advantage of the easily produced circular symmetry, such as the VAD
technique (BROWNING and WEXLER, 1969). Although clear-air Doppler radars must
work with the much .poorer signal-to-noise ratio provided by echoes from
irregularities in the refractive index of the air itself, special high
performance clear-air radars are able to create sufficient power-aperture by
using very high power to allow use of a rotatable dish antenna. Under these
conditions, VAD techniques can be used in clear air, as demonstrated by
PETERSON and BALSLEY (1979). They used the Chatanika 23 cm Doppler radar to
compare the accuracy of VAD, VED (velocity elevation display), and direct
vertical probing in measuring w, the vertical wind component. They found
direct measurement with the vertical beam superior to the other two methods.
With typical ST radars, however, the power-aperture requirements couple
with economic considerations to dictate large, immobile antennas for economical
systems. It is not practical to steer these antennas by physically moving
them, and though electronic beam steering methods are available (GRE_ et al.,
1984, CLARK and GREEN, 1984, FUKAO et al., 1985), they have not yet been
generally applied. Thus, each beam position is expensive to implement, so that
economical ST systems are designed to obtain the minimum amount of data to
measure the wind components with as little bias as possible.
CLARK et al. (1985) show the significant bias reduction obtained in u, v
estimates over mountainous terrain obtained by adding a vertical beam to
measure w. It will be apparent from the next section that the addition of
additional beams can reduce the bias even further when stratiform conditions
prevail. However, as discovered in the previous work with precipitation
sensitive radars, it is not possible to eliminate bias completely with single-
Doppler radar techniques. The following discussion will try to clarify the
nature of this bias, and serve as an aid in design of economical ST radar
systems.
THE G_ERAL SCALING EQUATION FOR A LINEAR WIND FIELD
We adopt the usual Cartesian coordinate system x, y, z, representing
distances to the east, north and zenith, respectively, with origin at the
center of the radar antenna (Figure I). The primary assumption applied to the
wind field is linearity in the region about (x , y , z ), the point in
O O.
space at which the flow parameters are to be determlne_. The vector distance
to a measuring volume from the antenna is defined as
; = + y3+
where i, j, and k are the usual unit vectors. The wind vector at _ is
ul+ +
where it is assumed
u = u ° + (X-Xo)U x + (y-yo)Uy + (Z-Zo)Uz
v = v ° + (X-Xo)V x + (y-yo)vy + (Z-Zo)V z
w = Wo + (X-Xo)Wx + (Y-Yo)Wy + (Z-Zo)Wz
The x, y, z subscripts denote partial differentiation (e.g., u = _u/Bx);
u o, Vo, and w ° are the primary parameters we wish to measure, _eing the
values of u, v, and w at the point (Xo, Yo" Zo)"
(l)
387
Sampled
Volume
f/fJ
Antenna
jtt1_
J
e/
"-'W
Nor th
East
X
Figure i.
Often at this point in the analysis, a transformation to polar coordinates
convenient to the operation of a radar is made. Instead, we will stay in
Cartesian coordinates to facilitate the analysis. If we intend to make
measurements of wind velocity at points (Xo' Yo' Zo) for large ranges not
centered on the radar antenna, it would be necessary to consider the curvature
of the earth (DOVIAK and ZRNIC, 1984, p 261). Here we confine analysis to
profile measurement directly above the antenna, where z is identical to height.
The mean radial velocity v_ observed within a sample volume at a given
is related to the wind vector _ _here by the dot product
+ -_ (2)
rv = v,r = Ux + vy + wz
r
Inserting the linear flc_ relations of equation (1) into equation (2), we
get the general relation relating the paremeters of the linear wind field,
which we would like to measure, to the observed Vr'S:
rvr = (Uo + u(X-Xo) + uy(Y-Yo) + Uz(Z-Zo))X
+ (v ° + Vx(X_Xo ) + Vy(y_yo) + vz(Z_Zo))y (3)
+ (w o + Wx(X-X o) + Wy(y-y o) + Wz(Z-Zo))Z
388
Equation (3) is an arrangement of the general scaling equation
convenient for developing sample location strategies. Much of the rest of the
paper is concerned with describing seine of these strategies. First, however,
we will examine a second arrangement of the scaling equation in which general
characteristics of solution are more apparent:
rv
r = x [Uo-UXo-UyYo-UzZo]
+ y [Vo-VxXo-VyYo-VzZ o]
+ z [Wo-WxXo-WyYo-WzZ o]
+ x2[u x]
+ y2 [Vy]
+ z2[wz]
+ xy[Uy + vx]
+ xz[uz + Wx]
+ yz[vz + Wy]
(4)
This is a linear equation with nine parameters (the quantities in brackets),
which may be solved from simultaneous observation of v at nine spatially
independent locations. Of these nine parameters, onlyrthree correspond
directly to the twelve paremeters that specify the linear wind field. The
other six are couples of these parameters, so that, in general, only linear
combinations may be found. Specifically, solution of equation (4) for any
given position (x , y . z ) provides: ÷1)the parameters necessary to
describe the dive°gen°e o_ the flow V. v = [u ] + Iv ] + [w ]l 2) the
additional three par-meters [u + v ], [u +Xw ], a_d [v _ w ]
necessary to specify the defor_atio x of t_e fl_; and 3)Zpara_eters containing
Uo, Vo, and w ° coupled inextricably with spatial derivatives.
PROFILING: MEASUR_24ENT OF THE WIND FIELD ALONG THE VERTICAL
In volumes directly above the radar we have that x = Yo = 0.0, so
equation (3) becomes o
= + + (Z-Zo)Uz]rvr x[uo+ xux yUy
+ y[v o + xvx + yVy + (Z-Zo)V z]
+ + (Z-Zo)Wz]+ z [w° + xwx yWy
Once again, this equation may be rearranged to give
rv
r = x[u° - ZoUz]
+ y[vo - ZVz]
+ _[w° - ZoWz]
+ x2[ux]
+ y2 [Vy]
(5)
(6)
389
+ z2[Wz]
+ xy[Uy + Vx]
+ xz [uz + Wx]
+ yz [vz + Wy]
where the bracketed terms represent the nine parameters that may be evaluated.
The difference between this equation and equation (4) is the disappearance of
all but the vertical shear terms from the first three parameters, leaving Uo,
v , and w coupled only to u , v , and w , respectively. From the
o
last two %erms we see that z z z . .when the stratlform conditlon is present (i.e., w
and w may be neglected), u and v z may be evaluated, thus allowing x
unbiased solution for u ° an_ v O.
VERTICAL BEAM MEASUREMENT OF w and w
o z
Heasurements made with a vertically pointing beam are particularly simple
to analyze, since they only contain information on two of the linear flow
parameters : w and w . Inspection of the scaling equation with x = y = 0
O Z
shows that the observed v is identically w , the vertical velocity
O
component at height z . Furthermore, the vfhlues of v observed at two
• o . r
closely spaced helghts, z ° and Zl, yleld Wz, so that
W = V
O ro
and
w I = Vrl
V -
ro Vrl
W
z z - z_
O I
(7)
W
Z
Thus a vertlcal±y alrectea Deam allows unbiased determination of w
in the presence of linear flow. o
and
OBLIQUE BEAMS
Assuming that w and w have been determined using a vertically
z
pointing beam, we ma9 treat these values as knowns and move them to the left
side of the equation. It is then convenient to define the quantity V r as
V = rv - z[ - z ] - z2w
r r Wo oWz z
which contains only known quantities. Then we can write equation (6) as
V = X [U Or -- ZoUz]
+ y IVO --ZOVZ]
+ x 2 [Uxl
+ y2 [Vy]
+ xy[Uy + Vx]
(s)
390
+ = [uz + wx]
+ yz[vz + Wy]
This is the general overhead scaling equation when the vertical velocity and
vertical velocity shear are known or negligible. It has seven parameters,
requiring seven spatially independent observations for solution.
0RTHOGONAL VERTICAL PLANES
If measurement of the deformation of the flow is not required we can
require all beams to be either in the xz or yz vertical plane. Then either
y = 0 or x = 0 for all beams, so that the fifth term of equation (8) dis-
appears. Under these conditions, equation (8) divides into two independent
equations. For y = 0
V
r
_--= [u° + ZoUz] + x[ux] + z[uz +w x] (9)
and for x = 0
V
__r_r= + y[Vy] + z[v z + Wy]y [Vo + ZoWy]
In each plane, we now have three unknown parameters, requiring three spatially
independent observations for solution. Thus, considering both planes and the
two vertical beam observations used to evaluate V , eight independent
observations are needed altogether when using ort_ogonal vertical planes. This
does not mean we need eight different beams, since observations may be
independent if they are separated in range along a beam. This technique will
be considered in the next section.
SINGLE BEAM ANALYSIS
It would be very efficient if equation (9) could be solved by utilizing
observations in three sample volumes along a single beam. However, such a
scheme cannot lead to complete solution since z and x are not independent,
being related by
z = x ctn@
where 8 is the zenith angle of the beam. For example, in the xz plane
substituting for z using this relation yields
Vr = x [Uo-ZoUz] (i0)
+ x2[ux + ctne (uz + Wx)]
The number of parameters is thus reduced to two. While u is still biased byo
z u , the stratiform approximation no longer allows solution for u_,
• -> -+
w_iZh is now coupled to both u and u . Thus, nelther u nor V.v may be
measured in unbiased form in t_e presenceX of u . o
Z
CONSTANT HEIGHT ANALYSIS
Starting again with equation (5), the vertical shear terms may be
eliminated if we require that all observations be for z = z (i.e., constant
height analysis), giving o
391
rvr =x[u o+ XUx +yUy]
[v° + yVy]+y + xv x
+ Z o[W ° + xw x + yWy]
Gathering like terms
rvr = Zo[Wo]
+ X[Uo + ZoWx]
+ x2[Ux]
+ xy [Uy + vx]
+ y[vo + ZoWy]
+ y2 [Vy]
This is a linear equation with six unknowns. The bias to u and v is now
the z w and z w terms, respectively, so that once again t_e stra_i-
O X . •
form approxlma_1_n allows solution for u and v .
o o
USE OF THE VERTICAL BEAM
(11)
(12)
It is easy to see that for a vertically pointing bess (i.e., x = y = o),
v = w (13)
r o
Thus, w is measured directly, with no bias.
O
USE OF ORTHOGONAL VERTICAL PLANES
Restricting bess positions such that either x = 0 or y = 0 splits equation
¢4 _% " • •
rv r - ZoW o = x[u O + ZoW x] + x2[Ux ] (14)
and
rv r - ZoW O = y[v ° + ZoWy] + y2[Vy]
Thus, two besss in the xz plane plus two beams in the yz plane plus a vertical
bess yield [Wo], [u° + ZoW x], [v° + ZoWy], [Ux], and [Vy].
DISCUSSION AND CONCLUSIONS
The ability of ST radars to measure vertical velocity is unmatched at the
present time by any other technique. It is straightforward and without bias,
even if the linear wind field assumption is poor. Only under fiscal restraints
of the most severe kind, or for very special purposes, should this beam be left
out of ST radar systems, since this parameter is not only of general interest,
but is necessary to remove bias from u and v measurements (CLARK et el., 1985).
The key parsseters to uncoupling terms in the scaling equations are w
and w.. Whenever the stratiform condition, which states that these two x
parameters are negligible, is satisfied, a five-bess ST radar may determine
unbiased (with reference to a linear wind field) values of u, v, and w for
sample volumes directly above the radar. Furthermore, the divergence and
partial deformation of the flow may be determined.
392
Three-beam systems can determine w and w , but are unable to obtain u
.z e
and v wind components uncontaminated by vertlcal sheer terms, even when th
stratiform condition is satisfied.
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properties of a wind field using Doppler radar, J. Meteorol, 7, 105-113.
Caton, P. G. (1963), The measurement of wind and convergence by Doppler radar,
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Clark, W. L., and J. L. Green (1984), Practicality of electronic beam steering
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Clark, W. L., J. L. Green, and J. M. Warnock (1985), Estimating meteorological
wind vector components from monostatic Doppler radar measurements: a case
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Doviak, R. J., and D. S. Zrnic (1984), Doppler Radar and Weather Observa-
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Determination of U, V, and W from single station Doppler radar radial velocities

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