Geoscience Reference
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constraint and the square of the first or second gradient of the three-dimensional
wind field components, the latter representing the smoothness of the solution. The
cost functions are minimized by setting the first derivatives of the total cost func-
tion to zero and then solving for u,
v
, and w. When the wind components are
estimated from observational wind data only (e.g., radar data), with no use of a
numerical model, the analysis technique is broadly referred to as 3DVAR, or
three-dimensional variational analysis.
There are many sources of error in dual/multiple Doppler analysis, especially
in the vertical component of the wind (a component of which is only partly
resolved except at close range and high altitude), and in any other horizontal wind
component for which the radar beam measures only a small component. In addi-
tion to instrument error and problems with the non-uniformity of scatterers and
air motions within radar volumes, measurements made at low altitude, where
boundary-layer variability is likely to be great, are dicult owing to ground
clutter; thus, use of the continuity equation near the surface may suffer from a
lack of good observations. Perhaps just as significant is that it is very unlikely that
radar volumes are sampled at exactly the same time, and it is unlikely that radar
volumes are of exactly the same volume in space or same size. To account for the
non-simultaneity of radar observations horizontal advection schemes have been
used, but if the wind field is rapidly evolving there are errors introduced due
to time evolution. The use of rapid-scan (by ''rapid-scan'' we mean that the volu-
metric update time is small compared with the advective time scale) radars can
reduce errors due to advection and evolution. Dual/multiple Doppler analyses also
require data to be interpolated to a grid, thus introducing some filtering and
degradation of the intrinsic spatial resolution of the radar. As noted earlier, when
air motions are sharply curved scatterers may be centrifuged outward radially so
that air motion is not identical to scatterer motion. Finally, there may be substan-
tial errors introduced when the estimate of terminal fall speed is poor, for
example, when there is large hail having very high terminal fall speeds (
35m s
1
or greater), but the terminal fall speed for large raindrops is assumed (
10m s
1
).
This terminal fall speed error may be particularly serious when the target storm is
at close range and the elevation angle is high, so that vertical motions are well
resolved by the radar. Such a situation is likely when a mobile Doppler radar is
probing a tornadic supercell at close range, especially near a tornado. Dual/
multiple Doppler analyses must therefore be viewed with caution when making
quantitative measurements, even though there are a number of techniques
employed to mitigate some of the aforementioned problems; the reader is referred
to the current literature to find state-of-the-art solutions. Since commonly used
quantities derived from the Doppler wind field—such as the three-dimensional
vorticity vector, circulation, and trajectories—are very sensitive to the three-
dimensional wind field, these derived quantities must be viewed with caution.
To get around the need for two or more Doppler radars, spaced antenna
techniques are being tested so that the component of the wind parallel to the
radar beam is used in the conventional way, but the component of the wind
normal to the beam is estimated from a pair of closely spaced antennas. Such a
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