A common problem in meteorological data analysis is the estimation of a field of values--for example, over a grid--for some parameter. Statistical interpolation techniques have long been used to solve this type of problem. The most common formulation of this problem in meteorological data analysis, and its solution, is generally referred to as Optimal Interpolation (OI). The standard practice in OI is to start with an initial estimate field and a set of observations. An estimate of the error in the initial estimate at each observation is computed as the difference between the two. These point errors are interpolated, using models for the statistical behavior of the errors, to form an estimate of the error field. The initial estimate is then corrected using this estimate of the error field.
In winds analysis, multiple Doppler analysis techniques provide the most accurate estimates of wind fields. Given favorable viewing geometry (radars measuring in substantially different directions), only a change of basis is required to transform a pair of Doppler measurements into a horizontal wind estimate in east and north components. Given more than two Doppler measurements, the least squares solution is generally used.
For winds analysis, the standard formulation of OI requires both components of the wind vector; however, Doppler radars provide only the component along the radar beam. A further shortcoming to OI is that the derivative structure of the output field is intimately tied to the derivative structure of the initial estimate field, even in the case of Doppler analyses where observations are plentiful. Standard multiple Doppler techniques can be used only in regions with radar returns and favorable viewing geometry. Neither condition can be guaranteed in operational settings.
The Gauss-Markov theorem provides a framework for merging the ideas behind both OI and traditional multiple Doppler analyses that does not suffer from their noted shortcomings. This technique forms the basis for the FAA Integrated Terminal Weather Systems gridded winds analysis and for the wind profiling algorithms developed for the NASA Aircraft Vortex Spacing System. While this development was motivated by a desire to merge the best of OI and traditional multiple Doppler techniques for winds analysis, the method is directly applicable to any estimation problem where input estimates are linearly related to the unknown parameters.
* This work was sponsored by the Federal Aviation Administration and the National Aeronautics and Space Administration. The views expressed are those of the authors and do not reflect the official policy or position of the U.S. Government.
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are those of the authors and are not necessarily endorsed by the United
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