Modeling Gaussian Random Fields by Anchored Inversion and Monte Carlo\n Sampling

It is common and convenient to treat distributed physical parameters as\nGaussian random fields and model them in an "inverse procedure" using\nmeasurements of various properties of the fields. This article presents a\ngeneral method for this problem based on a flexible parameterization device\ncalled "anchors", which captures local or global features of the fields. A\nclassification of all relevant data into two categories closely cooperates with\nthe anchor concept to enable systematic use of datasets of different sources\nand disciplinary natures. In particular, nonlinearity in the "forward models"\nis handled automatically. Treatment of measurement and model errors is\nsystematic and integral in the method; however the method is also suitable in\nthe usual setting where one does not have reliable information about these\nerrors. Compared to a state-space approach, the anchor parameterization renders\nthe task in a parameter space of radically reduced dimension; consequently,\neasier and more rigorous statistical inference, interpretation, and sampling\nare possible. A procedure for deriving the posterior distribution of model\nparameters is presented. Based on Monte Carlo sampling and normal mixture\napproximation to high-dimensional densities, the procedure has generality and\nefficiency features that provide a basis for practical implementations of this\ncomputationally demanding inverse procedure. We emphasize distinguishing\nfeatures of the method compared to state-space approaches and\noptimization-based ideas. Connections with existing methods in stochastic\nhydrogeology are discussed. The work is illustrated by a one-dimensional\nsynthetic problem.\n Key words: anchored inversion, Gaussian process, ill-posedness, model error,\nstate space, pilot point method, stochastic hydrogeology.\n