# THREE-DIMENSIONAL VARIATIONAL ANALYSIS OF AIRBORNE DOPPLER OBSERVATIONS

Principal Investigator: John F. Gamache
Collaborating Scientist:  Wen-Chau Lee [NCAR/ATD/RSF]

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 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.

Accomplishments:
• 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.

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• 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.

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• 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

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• 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, respectively.

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• 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.