Optimization of wind field retrieval procedures

doi:10.1016/j.amc.2005.01.033

Applied Mathematics and Computation

Volume 171, Issue 1, 1 December 2005, Pages 25-52

https://doi.org/10.1016/j.amc.2005.01.033 Get rights and content

Abstract

We study the formulation of the problem to retrieve wind fields from radar data. Our formulation allows us to consider the retrieved wind fields as a function of radar locations. We examine the properties of this function with the objective of determining “best” locations for observations. Problems are then posed to determine radar locations to minimize certain criteria involving retrieval errors over a class of test wind fields. A numerical study is presented illustrating the theory developed.

Introduction

The wind field retrieval problem seeks to estimate three dimensional vector fields representing wind fields in a three dimensional domain Ω from data consisting of measurements of radial velocities from n radar sites supplemented with constraints based on physical laws. The retrieval problem may be formulated as a minimization problem on a Hilbert space of admissible vector fields. The existence of a unique solution to this problem is then a consequence of classical Hilbert space theory. Typically, the retrieval problem depends on parameters that must be specified in its formulation. The solution of the retrieval problem thus depends on these parameters, and they may be considered as controls around which optimization problems may be designed. The parameter of interest in this initial study is n-tuple of locations of the radar observational sites. We view these problems as a member of a family of problems that are indexed by the radar site locations. Our objective here is to consider the determination of these parameters that minimize a general retrieval error defined over classes of possible test wind fields.

The synthesis of three dimensional vector wind fields from Doppler radar data is an important part of mesoscale research and operational meteorology, with particularly vital applications in hazard warning and nowcasting (e.g., tornado detection and prediction), and in numerical weather prediction. Techniques of single-Doppler velocity retrieval vary in complexity from the simple velocity azimuth display (VAD), in which the imposed model is a wind field that varies linearly with the spatial coordinates, to the full model adjoint techniques in which the radial wind obtained from time integration of the complete dynamical equation set of a numerical weather prediction model is fit to radial wind observations over a window of time. Dual-Doppler wind retrieval techniques may also be couched in an adjoint or other variational framework. Key developments in the history of single- and multiple-Doppler wind retrievals, and some of the remaining problems are summarized in Shapiro et al. [10].

In Section 2 a general formulation in a Hilbert space context that applies to our problem is presented. In Section 3 the simplified retrieval problem studied in this paper is posed. While the problem studied here is rather simple, in the sense that the radar models used are the simplest possible, it, nevertheless, captures the important features necessary for our considerations. In Section 4, the optimization with respect to locations is posed as a problem to minimize retrieval errors over a class of test wind fields. In Section 5 the results of a numerical study are discussed.

Section snippets

Some preliminary Hilbert space results

Let H and V be two Hilbert spaces over the real numbers such that V embeds continuously in H and let $R$ denote the set of real numbers. Distinguishing the respective inner products and norms with subscripts H and V there is a positive constant C such that for any u ∈ V $‖ u ‖_{H} ⩽ C ‖ u ‖_{V} .$

Let $a (\cdot, \cdot) : H \times H \mapsto R$ be a continuous symmetric positive semi-definite bilinear form on H. There is a positive constant K_a such that $a (u, v) ⩽ K_{a} ‖ u ‖_{H} ‖ v ‖_{H}$ and $a (u, u) ⩾ 0$ for all u and v in H. Under these conditions the mapping taking $V \times V \mapsto$

Retrieval of wind fields from radar data

The retrieval of wind fields from radar data is posed as a minimization problem that seeks wind fields matching data and models through a retrieval functional. Thus, to pose the problem of estimating wind field information from radar data requires the specification of a retrieval functional that includes terms involving data model (radar measurement model), the physics-based model, and the regularization for well-posedness. The model describing the relation between observed radar data and the

Optimal retrieval

In Section 3 the retrieval problem that attempts to find a three dimensional wind field to match radar data conditioned with physics-based side constraints is formulated. The approach uses a minimization problem based on the development in Section 2. The solution of the retrieval problem (2.5) depends on the collection of radar site locations. The collection of locations is expressed as a 3 × n matrix q of real numbers each column of which is the coordinates of a radar site. For the purpose of

Numerical experiments

To pose a numerical problem, let $Ω = (0, 150) \times (0, 150) \times (0, 3),$ where the numbers are understood to be in kilometers. The x, y and z dimensions are partitioned into subintervals of equal length 15, 15, and 3 subintervals, respectively. This coarse grid is used for computational convenience. Thus, using tensor products of piecewise linear functions [7] as basis elements there are 1024 elements in the basis of the approximating set. Test events are posed as 16 vortices spaced centered at x₀ points on a

Acknowledgments

This work is supported in part by the Engineering Research Centers Program of the National Science Foundation under NSF Award Number (EEC-0313747). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.

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Citation Excerpt :
Remaining terms in the model include the divergence free condition embodied in the continuity equation and a regularization term included so that the problem is well-posed. Discussion of these terms is given in [8,12,14]. For the present application the important feature is that the functional to minimized is quadratic and is associated with an inner product over a suitable Hilbert space.
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Optimization of wind field retrieval procedures

Abstract

Introduction

Section snippets

Some preliminary Hilbert space results

Retrieval of wind fields from radar data

Optimal retrieval

Numerical experiments

Acknowledgments

Elliptic Partial Differential Equations

Distributed Optimal Control of Systems Governed by Partial Differential Equations

Optimization by Vector Space Methods

Sobolev Spaces

Use of the vorticity equation in dual-Doppler analysis of the vertical velocity field

J. Atmos. Ocean. Technol.

Real Analysis