Exact Conditioning of Gaussian Fields on Wavelet Coefficients

Abstract

A new approach is presented to ensure exact conditioning of Gaussian fields on wavelet decomposition coefficients and hard data. The approach uses a matrix formulation of discrete wavelet decomposition enabling to compute all the required covariances and cross-covariances between hard data, points to simulate and the imposed wavelet coefficients at all scales. Two small examples show that the approach reproduces exactly the information provided, even when the wavelet information is unrelated to the true underlying image. For larger images, the approach of post-conditioning by cokriging is used to palliate the strong memory requirements of the matrix approach. The modified method remains exact and is shown to be computationally realistic for large problems. This marks a significant improvement over previously published methods where conditioning to wavelet coefficients was only approximate. After soft data calibration to wavelet coefficients at a given scale, the approach enables to simulate realizations fully consistent with the known wavelet coefficients and available hard data.