Conditional simulation of Gaussian stochastic processes

Generation of 2D and 3D normally distributed random fields conditioned on well data is often required in reservoir modeling. Such fields can be obtained by using three groups of methods: unconditional simulation with kriging interpolation (turning band or spectral methods), Sequential Gaussian Simulation (SGS) and Cholesky factorization of the covariance matrix. However, all these methods have limitations. First, it is known, that the second moment of the stochastic process conditionally simulated with the help of the kriging method is not identical to the target second moment (a priori known statistics). Second, SGS can't be calculated without limitation on the number of neighbors. As a result, SGS is only asymptotically exact. Third approach, which has the advantage of being general and exact, is to use a Cholesky factorization of the covariance matrix representing grid point’s correlation. However, for the large fields the Cholesky factorization can be computationally expensive. In this work we present an alternative approach, based on the usage of spectral representation of a conditional process. It is shown that covariance of two arbitrary spectral components could be factorized into functions of corresponding harmonics. In this case the Cholesky decomposition could be considerably simplified. The advantage of the presented approach is its accuracy and computational simplicity.