Dr. Alan Shapiro, Professor Emeritus, will be giving a seminar called “Spatially variable advection correction of Doppler radial velocity data” on Thursday, November 7th at 1:00pm in NWC 1350.
Spatially variable advection correction of Doppler radial velocity data
Alan Shapiro, Professor Emeritus
School of Meteorology, University of Oklahoma, Norman, OK
Techniques to mitigate errors arising from the non-simultaneity of data collection typically make use of advection correction procedures (space-to-time conversions) based on Taylor’s frozen turbulence hypothesis in which the analyzed field is idealized as a pattern of unchanging form that translates horizontally at constant speed. Mathematically, the frozen turbulence constraint (as applied to the reflectivity field R) is DR/Dt = 0, where D/Dt = ∂/∂t + U ∂/∂x + V∂/∂y, and U and V are pattern-translation components. These components can, in general, differ substantially from the actual air velocity components u and v. Advection correction procedures for the reflectivity field using this or related constraints are either supplied with independent estimates of U and V, or compute U and V as part of the overall analysis procedure.
The present study is concerned with spatially variable advection correction of the Doppler radial velocity field vr. This analysis problem is much more difficult than that for the reflectivity field because even if the vector velocity field satisfies the frozen-turbulence hypothesis, the radial component of the velocity field does not. However, Gal-Chen (1982) showed that if the velocity field satisfies the frozen turbulence hypothesis then the radial velocity field satisfies a second derivative constraint of the form D2(rvr)/Dt2 = 0, which expands out to a non-linear equation in U and V. Our radial velocity analysis problem is phrased as a variational problem in which errors in this second derivative constraint are minimized subject to smoothness constraints on U and V (U and V vary spatially, but they have to vary in a “smooth” manner). The Euler-Lagrange equations for this analysis are derived and a solution is proposed in which the trajectories of the pattern-motion, the U and V pattern-translation fields, and the radial velocity field are obtained simultaneously using a combined analytical and numerical procedure.
Results of the procedure are first presented from analytical data tests using radial velocities sampled from an array of counter-rotating vortices embedded in (and rotating with) a solid-body vortex. We then present results from real data tests using radial velocity data from the Atmospheric Imaging Radar (AIR), a rapid-scan X-band mobile Doppler radar maintained and operated by the Advanced Radar Research Center (ARRC).