Modeling Gaussian Random Fields by Anchored Inversion and Monte Carlo Sampling

TL;DR

This paper introduces a flexible anchored inversion method combined with Monte Carlo sampling to efficiently estimate Gaussian random fields from diverse datasets, handling nonlinearity and errors systematically.

Contribution

The paper presents a novel anchored parameterization approach that reduces the problem dimension and improves inference and sampling in Gaussian random field modeling.

Findings

Reduces problem complexity via anchor parameterization.

Provides a systematic Monte Carlo sampling procedure.

Demonstrates effectiveness on a synthetic one-dimensional example.

Abstract

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