VARIATIONAL ANALYSIS OF AIRBORNE DOPPLER OBSERVATIONS
John F. Gamache
Wen-Chau Lee [NCAR/ATD/RSF]
To improve dual- and multiple-Doppler analyses by incorporating the Doppler
projection equations, the three-dimensional continuity equation, and filtering
within a variational scheme.
Rationale : The solution of
a Doppler wind analysis involves two or more projection equations, and
one continuity equation. The solution of these three simultaneously is
difficult. In the past, the vertical wind was first assumed to be zero
and then the projection equations were solved for the horizontal wind components.
The divergence of the horizontal components was then integrated to determine
the vertical wind. The new vertical wind was then used in the projection
equations to determine a new horizontal wind field. This process was to
be iterated to a solution. The process was unstable for Doppler radials
more than approximately 45 degrees from horizontal. Thus these data must
be thrown out if the process is to be iterated to a convergent solution.
Throwing all three equations into a cost function, however, allows all
three equations to be solved simultaneously, and thus Doppler radials above
45 degrees may be kept in the analysis. This is because the continuity
equation is also solved in all three directions, and not just in the vertical.
Another advantage is that since the filtering is included in the cost function
at a much lower cost than continuity, a smoother wind field is found that
still satisfies the continuity equation closely.
Method : Several steps are
required in determining the wind field from dual- or multiple-Doppler observations
Doppler data are first interpolated
to a three-dimensional grid.
The projection portion of the
cost function is then determined from the difference between the projection
of the analyzed precipitation motion in the direction of the Doppler radial
and the measured Doppler radial velocity.
Next a grid point representation
of the anelastic three dimensional divergence is determined. This portion
of the cost function is determined from the value of the three-dimensional
divergence (which should be zero).
Similarly, a simple equation
that says that the value of six times the density weighted wind component
at a point is equal to the sum of the density weighted wind component at
the surrounding six points. This portion of the cost function is determined
from the difference between six times the central value and the sum of
the six surrounding values (which should be kept small). This is
the filtering portion of the cost function.
The cost function is represented
as a matrix equation with several bands. The matrix is symmetric positive
definite and the matrix equation is solved by the conjugate gradient method.
An option has been developed
to perform a wind analysis in a cylindrical grid, where the axis is assumed
to be the vertical central axis of the hurricane. The components
in this grid are thus radial, azimuthal, and vertical.
An option added to this analysis
in the last several years is a method called Multiple-Doppler Harmonic
Analysis (or MDHA) for short. In MDHA a fourier-component representation
of the azimuthal variation of the wind components is used, rather than
a grid-point representation. The radial and vertical variations are
still determined at discrete heights and radii.
In the past two months a version
of MDHA has been under development that could make a real-time
version available on the aircraft. This would permit a VTD-like
analysis to be performed from Doppler data collected while the antenna
scanned fore and aft of the fuselage. Data would not be required
to be collected in radial passes using scans normal to the flight track,
as in VTD.
A subroutine has been written
to detect those regions of the analysis that were not properly bounded,
so the option is there to remove them
Data from the 1995 VORTEX experiment,
on 7 May 1995, have been analyzed with the cartesian, boundary-checking
version of this method and a horizontal and vertical slice through the
analysis are shown in Figs. 1 and 2,
Synthetic wind fields were devised
and used to test the analysis method. New insights in how the analysis
actually handles difficult wind fields were found. The synthetic
fields also showed that a complicated wind field can be represented fairly
well by the technique, but not when errors are introduced. Thus the
major culprit in the analysis method is data collection and interpolation.
Errors of this sort can introduce high-frequency errors that are aliased
by the grid point solution.
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Last modified: 02/01/00