# THREE-DIMENSIONAL VARIATIONAL ANALYSIS OF AIRBORNE DOPPLER OBSERVATIONS

Principal Investigator: John F. Gamache
Objective: 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 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 closely.
Method: 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.

Accomplishment: