It is demonstrated that an ill-posed formulation can be improved by using special filter procedures based on fundamental properties of the noise to reconstructing signal ratio. These procedures do not require knowledge of any statistical properties of the noise. The approach can be applied to atmospheric and oceanic data sets as a useful first step prior to any analysis using traditional methods such as optimal interpolation or splines. Three filtering procedures are studied. It is shown that these methods can filter any type of stochastic and deterministic errors which can be represented in the form of a series of an appropriate basis functions.
All procedures use special transformations that reduce the noise to reconstructing signal ratio.
Numerical examples of the procedures are given. Section 3 Results of the test calculations These methods are illustrated in the reconstruction of the surface temperature field of the Black Sea. Calculations show that the filter procedures produce considerable improvement of the noise to reconstructing signal ratio for noisy data. In the examples used in this study the ratio ranged from 0 to 5 in the white noise limit. Filtering was also found to be useful for colored noise if the spatial correlation scale of the noise to minimum distance between observation stations ratio was less than 4.