Some current uses of empirical orthogonal functions (EOF) are briefly summarized, together with some relations with spectral and principal component analyses. Considered as a mean square estimation technique of unknown values within a random process or field, the optimization of error variance leads to a Fredholm integral equation. Its kernel is the autocorrelation function, which in many practical cases is only known as discrete values of interstation correlation coefficients computed from a sample of independent realizations. The numerical solution in one or two dimensions of this integral equation is approximated in a new and more general framework that requires, in practice, the a priori choice of a set of generating functions. Developments are provided for piecewise constant, facetlike linear, and thin plate type spline functions. The first part of the paper ends with a review of the mapping, archiving and stochastic simulating possibilities of the EOF method. A second part includes a case study concerning precipitation fields, previously worked out by optimal interpolation methods.