A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis

Stochastic analyses of groundwater flow and transport are frequently based on partial differential equations which have random coefficients or forcing terms. Analytical methods for solving these equations rely on restrictive assumptions which may not hold in some practical applications. Numerically oriented alternatives are computationally demanding and generally not able to deal with large three-dimensional problems. In this paper we describe a hybrid solution approach which combines classical Fourier transform concepts with numerical solution techniques. Our approach is based on a nonstationary generalization of the spectral representation theorem commonly used in time series analysis. The generalized spectral representation is expressed in terms of an unknown transfer function which depends on space, time, and wave number. The transfer function is found by solving a linearized deterministic partial differential equation which has the same form as the original stochastic flow or transport equation. This approach can accomodate boundary conditions, spatially variable mean gradients, measurement conditioning, and other sources of nonstationarity which cannot be included in classical spectral methods. Here we introduce the nonstationary spectral method and show how it can be used to derive unconditional statistics of interest in groundwater flow and transport applications.