THREE-DIMENSIONAL VARIATIONAL ANALYSIS OF
AIRBORNE DOPPLER OBSERVATIONS
John F. Gamache
To improve dual- and multiple-Doppler analyses by incorporating the
Doppler projection equations, the three-dimensional continuity equation,
and filtering within a variational scheme.
The solution of a Doppler wind analysis involves two or more projection
equations, and one continuity equation. The solution of these three
together is difficult, thus 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 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
Doppler data are first interpolated to a three-dimensional grid. The
cost function may then be determined for the difference between the
projection of the solution motion of the precipitation back on to the
Doppler radial and the measured Doppler radial velocity.
Next a grid point representation of the anelastic three dimensional
divergence is determined. The cost function for the difference between
that divergence and zero is determined.
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. The
difference between the weighted sum of those seven solution variables
and zero is also added to the cost function.
The representation of the cost function is a matrix equation with several
bands. The equation is symmetric positive definite and is solved by the
conjugate gradient method.
A bottom boundary condition of zero vertical wind is also imposed within
the cost function.
Last modified: 10/9/96