Regularized kernel PCA for the efficient parameterization of complex geological models

The use of geological parameterization procedures enables high-fidelity geomodels to be represented in terms of relatively few variables. Such parameterizations are particularly useful when the subspace representation is constructed to implicitly capture the key geological features that appear in prior geostatistical realizations. In this case, the parameterization can be used very effectively within a data assimilation framework. In this paper, we extend and apply geological parameterization techniques based on kernel principal component analysis (KPCA) for the representation of complex geomodels characterized by non-Gaussian spatial statistics. KPCA involves the application of PCA in a high-dimensional feature space and the subsequent reverse mapping of the feature-space model back to physical space. This reverse mapping, referred to as the pre-image problem, can be challenging because it (formally) involves a nonlinear minimization. In this work, a new explicit pre-image procedure, which avoids many of the problems with existing approaches, is introduced. To achieve (ensemble-level) flow responses in close agreement with those from reference geostatistical realizations, a bound-constrained, regularized version of KPCA, referred to as R-KPCA, is also introduced. R-KPCA can be viewed as a post-processing of realizations generated using KPCA. The R-KPCA representation is incorporated into an adjoint-gradient-based data assimilation procedure, and its use for history matching a complex deltaic fan system is demonstrated. Matlab code for the KPCA and R-KPCA procedures is provided online as Supplementary Material.