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Interpolation and Filtering of Spatial Observations Using Successive Corrections and Gaussian Filters

M. A. PedderDepartment of Meteorology, University of Reading, Reading, United Kingdom

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Abstract

This paper describes a simple empirical analysis system based on Bratseth's method of successive corrections applied to detrended field data, which approximates an optimal interpolation of fields with a spatially variable mean sampled within a limited domain by scattered observations. As in other empirical interpolation schemes, the influence function that determines the weights applied to increment variables is chosen such that unresolvable scales tend to be strongly damped, even if the contribution from observation error is not represented in a formally equivalent correlation model for observed increment variables. Unlike most empirical successive correction schemes, the number of iterations is not necessarily considered as a prescribed analysis parameter. Instead, the number of iterations can be chosen on a judgemental, posterior basis such that the analysis approximates the observed field to within some acceptable limit.

The analysis generated by this form of successive corrections can be represented by a continuous function of location variables. Consequently, it is possible to express the result of posterior filtering of the analysis field as a weighted sum of “filtered” influence functions, each of which is defined by the convolution of an increment autocorrelation function with a continuous linear filter. If both the autocorrelation model and filter are based on a simple Gaussian function of spatial lag, then these convolution integrals can be solved analytically. This leads to a numerically inexpensive method of scale selection analysis that is in some ways less ambiguous than methods based on applying filters directly to the observed data.

The performance of the analysis system is demonstrated by applying it to simulated observations sampling two-dimensional fields.

Abstract

This paper describes a simple empirical analysis system based on Bratseth's method of successive corrections applied to detrended field data, which approximates an optimal interpolation of fields with a spatially variable mean sampled within a limited domain by scattered observations. As in other empirical interpolation schemes, the influence function that determines the weights applied to increment variables is chosen such that unresolvable scales tend to be strongly damped, even if the contribution from observation error is not represented in a formally equivalent correlation model for observed increment variables. Unlike most empirical successive correction schemes, the number of iterations is not necessarily considered as a prescribed analysis parameter. Instead, the number of iterations can be chosen on a judgemental, posterior basis such that the analysis approximates the observed field to within some acceptable limit.

The analysis generated by this form of successive corrections can be represented by a continuous function of location variables. Consequently, it is possible to express the result of posterior filtering of the analysis field as a weighted sum of “filtered” influence functions, each of which is defined by the convolution of an increment autocorrelation function with a continuous linear filter. If both the autocorrelation model and filter are based on a simple Gaussian function of spatial lag, then these convolution integrals can be solved analytically. This leads to a numerically inexpensive method of scale selection analysis that is in some ways less ambiguous than methods based on applying filters directly to the observed data.

The performance of the analysis system is demonstrated by applying it to simulated observations sampling two-dimensional fields.

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