A FAST AND EXACT METHOD FOR MULTIDIMENSIONAL GAUSSIAN STOCHASTIC SIMULATIONS

To generate multidimensional Gaussian random fields over a regular sam pling grid, hydrogeologists can call upon essentially two approaches. The first approach covers methods that are exact but computationally e xpensive, e.g., matrix factorization. The second covers methods that a re approximate but that have only modest computational requirements, e .g., the spectral and turning bands methods. In this paper, we present a new approach that is both exact and computationally very efficient. The approach is based on embedding the random field correlation matri x R in a matrix S that has a circulant/block circulant structure. We t hen construct products of the square root S1/2 with white noise random vectors. Appropriate subvectors of this product have correlation matr ix R, and so are realizations of the desired random field. The only co nditions that must be satisfied for the method to be valid are that (1 ) the mesh of the sampling grid be rectangular, (2) the correlation fu nction be invariant under translation, and (3) the embedding matrix S be nonnegative definite. These conditions are mild and turn out to be satisfied in most practical hydrogeological problems. Implementation o f the method requires only knowledge of the desired correlation functi on. Furthermore, if the sampling grid is a d-dimensional rectangular m esh containing n points in total and the correlation between points on opposite sides of the rectangle is vanishingly small, the computation al requirements are only those of a fast Fourier transform (FFT) of a vector of dimension 2(d)n per realization. Thus the cost of our approa ch is comparable with that of a spectral method also implemented using the FFT. In summary, the method is simple to understand, easy to impl ement, and is fast.