Kernel Principal Component Analysis for Efficient,Differentiable Parameterization of Multipoint Geostatistics

This paper describes a novel approach for creating an efficient, general, and differentiable parameterization of large-scale non-Gaussian, non-stationary random fields (represented by multipoint geostatistics) that is capable of reproducing complex geological structures such as channels. Such parameterizations are appropriate for use with gradient-based algorithms applied to, for example, history-matching or uncertainty propagation. It is known that the standard Karhunen–Loeve (K–L) expansion, also called linear principal component analysis or PCA, can be used as a differentiable parameterization of input random fields defining the geological model. The standard K–L model is, however, limited in two respects. It requires an eigen-decomposition of the covariance matrix of the random field, which is prohibitively expensive for large models. In addition, it preserves only the two-point statistics of a random field, which is insufficient for reproducing complex structures. In this work, kernel PCA is applied to address the limitations associated with the standard K–L expansion. Although widely used in machine learning applications, it does not appear to have found any application for geological model parameterization. With kernel PCA, an eigen-decomposition of a small matrix called the kernel matrix is performed instead of the full covariance matrix. The method is much more efficient than the standard K–L procedure. Through use of higher order polynomial kernels, which implicitly define a high-dimensionality feature space, kernel PCA further enables the preservation of high-order statistics of the random field, instead of just two-point statistics as in the K–L method. The kernel PCA eigen-decomposition proceeds using a set of realizations created by geostatistical simulation (honoring two-point or multipoint statistics) rather than the analytical covariance function. We demonstrate that kernel PCA is capable of generating differentiable parameterizations that reproduce the essential features of complex geological structures represented by multipoint geostatistics. The kernel PCA representation is then applied to history match a water flooding problem. This example demonstrates that kernel PCA can be used with gradient-based history matching to provide models that match production history while maintaining multipoint geostatistics consistent with the underlying training image.     

Stochastic simulations of ocean waves: An uncertainty quantification study

The primary objective of this study is to introduce a stochastic framework based on generalized polynomial chaos (gPC) for uncertainty quantification in numerical ocean wave simulations. The techniques we present can be easily extended to other numerical ocean simulation applications. We perform stochastic simulations using a relatively new numerical method to simulate the HISWA (Hindcasting Shallow Water Waves) laboratory experiment for directional near-shore wave propagation and induced currents in a shallow-water wave basin. We solve the phased-averaged equation with hybrid discretization based on discontinuous Galerkin projections, spectral elements, and Fourier expansions. We first validate the deterministic solver by comparing our simulation results against the HISWA experimental data as well as against the numerical model SWAN (Simulating Waves Nearshore). We then perform sensitivity analysis to assess the effects of the parametrized source terms, current field, and boundary conditions. We employ an efficient sparse-grid stochastic collocation method that can treat many uncertain parameters simultaneously. We find that the depth-induced wave-breaking coefficient is the most important parameter compared to other tunable parameters in the source terms. The current field is modeled as random process with large variation but it does not seem to have a significant effect. Uncertainty in the source terms does not influence significantly the region before the submerged breaker whereas uncertainty in the incoming boundary conditions does. Considering simultaneously the uncertainties from the source terms and boundary conditions, we obtain numerical error bars that contain almost all experimental data, hence identifying the proper range of parameters in the action balance equation.     

A stochastic approach to nonlinear unconfined flow subject to multiple random fields

In this study, the KLME approach, a moment-equation approach based on the Karhunen–Loeve decomposition developed by Zhang and Lu (Comput Phys 194(2):773–794, 2004), is applied to unconfined flow with multiple random inputs. The log-transformed hydraulic conductivity F, the recharge R, the Dirichlet boundary condition H, and the Neumann boundary condition Q are assumed to be Gaussian random fields with known means and covariance functions. The F, R, H and Q are first decomposed into finite series in terms of Gaussian standard random variables by the Karhunen–Loeve expansion. The hydraulic head h is then represented by a perturbation expansion, and each term in the perturbation expansion is written as the products of unknown coefficients and Gaussian standard random variables obtained from the Karhunen–Loeve expansions. A series of deterministic partial differential equations are derived from the stochastic partial differential equations. The resulting equations for uncorrelated and perfectly correlated cases are developed. The equations can be solved sequentially from low to high order by the finite element method. We examine the accuracy of the KLME approach for the groundwater flow subject to uncorrelated or perfectly correlated random inputs and study the capability of the KLME method for predicting the head variance in the presence of various spatially variable parameters. It is shown that the proposed numerical model gives accurate results at a much smaller computational cost than the Monte Carlo simulation.