Optimum Smoothing of Two-Dimensional Fields

The problem of smoothing out nonsystematic errors in a two-dimensional field of measurements has been studied from the standpoint of finding the type of weighted area average for which the RMS difference between the true field and the weighted average of the field of observations is the least. For fields whose space-autocorrelation functions are invariant with rotation and have a simple and rather typical form, the optimum weighting function is a linear combination of Bessel functions, whose rate of decrease away from the origin depends partially on the so-called “signal-to-noise” ratio, but primarily on the ratio of the scales of the true field and error field. A comparison of optimum averaging with the analyst’s subjective process of smoothing indicates that the former is significantly superior in its ability to distinguish between random small-scale fluctuations and minor synoptic features of only slightly greater scale. Finally, the minimum RMS error of linearly smoothed fields is expressed in terms of the statistical properties of the true fields and the observing system itself.