Theory and applications of multi-dimensional stationary stochastic processes

The theory of stationary stochastic processes in several dimensions has been investigated to provide a general model which may be applied to various problems which involve unknown functions of several variables. In particular, when values of the function are known only at a finite set of points, treating the unknown function as a realisation of a stationary stochastic process leads to an interpolating function which reproduces the values exactly at the given points. With suitable choice of auto-correlation for the model, the interpolating function may also he shown to be continuous in all its derivatives everywhere. A few parameters only need to be found for the interpolator, and these may be estimated from the given data. One problem tackled using such an interpolator is that of automatic contouring of functions of two variables from arbitrarily scattered data points. A "two-stage" model was developed, which incorporates a long-range "trend" component as well as a shorter-range "residual" term. This leads to a contouring algorithm which gives good results with difficult data. The second area of application is that of optimisation, particularly of objective functions which are expensive to compute. Since the interpolator gives an estimate of the derivatives with little work, it is simple to optimise it using conventional techniques, and to re-evaluate the true function at the apparent optimum point. An iterative algorithm along these lines gives good results with test functions, especially with fuactions of more than two variables. A program has been developed whicj incorporates both the optimisation and contouring applications into a single peckage. Finally, the theory of excursions of a stationary process above a fixed level has been applied to the problem of modelling the occurrence of oilfields, with special reference to their spatial distribution and tendency to cluster. An intuitively reasonable model with few parameters has been developed and applied to North Sea data, with interesting results.