Abstract

A meaningful solution to an inversion problem should be composed of the preferred inversion model and its uncertainty and resolution estimates. The model uncertainty estimate describes an equivalent model domain in which each model generates responses which fit the observed data to within a threshold value. The model resolution matrix measures to what extent the unknown true solution maps into the preferred solution. However, most current geophysical electromagnetic (also gravity, magnetic and seismic) inversion studies only offer the preferred inversion model and ignore model uncertainty and resolution estimates, which makes the reliability of the preferred inversion model questionable. This may be caused by the fact that the computation and analysis of an inversion model depend on multiple factors, such as the misfit or objective function, the accuracy of the forward solvers, data coverage and noise, values of trade-off parameters, the initial model, the reference model and the model constraints. Depending on the particular method selected, large computational costs ensue. In this review, we first try to cover linearised model analysis tools such as the sensitivity matrix, the model resolution matrix and the model covariance matrix also providing a partially nonlinear description of the equivalent model domain based on pseudo-hyperellipsoids. Linearised model analysis tools can offer quantitative measures. In particular, the model resolution and covariance matrices measure how far the preferred inversion model is from the true model and how uncertainty in the measurements maps into model uncertainty. We also cover nonlinear model analysis tools including changes to the preferred inversion model (nonlinear sensitivity tests), modifications of the data set (using bootstrap re-sampling and generalised cross-validation), modifications of data uncertainty, variations of model constraints (including changes to the trade-off parameter, reference model and matrix regularisation operator), the edgehog method, most-squares inversion and global searching algorithms. These nonlinear model analysis tools try to explore larger parts of the model domain than linearised model analysis and, hence, may assemble a more comprehensive equivalent model domain. Then, to overcome the bottleneck of computational cost in model analysis, we present several practical algorithms to accelerate the computation. Here, we emphasise linearised model analysis, as efficient computation of nonlinear model uncertainty and resolution estimates is mainly determined by fast forward and inversion solvers. In the last part of our review, we present applications of model analysis to models computed from individual and joint inversions of electromagnetic data; we also describe optimal survey design and inversion grid design as important applications of model analysis. The currently available model uncertainty and resolution analyses are mainly for 1D and 2D problems due to the limitations in computational cost. With significant enhancements of computing power, 3D model analyses are expected to be increasingly used and to help analyse and establish confidence in 3D inversion models.

Highlights

  • In geophysical electromagnetic induction methods (Nabighian 1991; Berdichevsky and Dmitriev 2008; Chave and Jones 2012), natural or artificial source currents generate the primary electromagnetic fields which propagate through the Earth

  • We first provided a concise exposition to the theory of linearised and nonlinear model analysis tools to estimate model uncertainty and resolution

  • The model resolution matrix offers a way to examine the contributions of the unknown true model to the preferred inversion model and can be considered a blurring filter through which we see the former in the latter

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Summary

Introduction

In geophysical electromagnetic induction methods (Nabighian 1991; Berdichevsky and Dmitriev 2008; Chave and Jones 2012), natural or artificial source currents generate the primary electromagnetic fields which propagate through the Earth. Linearised model analysis tools can offer quantitative measures for both resolution and uncertainty of the preferred inversion model. It is valuable to mention another quantitative measure of model uncertainty in terms of funnel functions (Oldenburg 1983; Menke 2012) In this approach, the upper and lower bounds of the model parameters (or the local averages of model parameters) for both linear and nonlinear optimisation problems are estimated. Considering the uncertainty estimates, we can establish confidence levels for the various structures observed in the preferred inversion model (Fig. 1) Those parts of a model space that have low uncertainties and resolution matrix entries with little spread around the main diagonal might be used for assisting the geological interpretation or the design of drilling campaigns. We conclude this paper with general recommendations both on inversion and model analysis and an outlook on future research directions

Methods of Model Analysis
Linearised Model Analysis
Effects of Data Quality on Model Uncertainty and Resolution
Effects of Model Discretisation on Model Uncertainty and Resolution
Effects of Model Regularisation on Model Uncertainty and Resolution
Partially Nonlinear Model Analysis
Nonlinear Model Analysis
Deterministic Methods
Stochastic Methods
Computationally Efficient Approaches
Parallel Direct Solvers
Partial or Truncated SVD Analysis
Krylov Subspace Methods
Iterative Update Formula
Model Reduction Techniques
Sensitivity Analyses
Uncertainty and Resolution Analyses of Models Inverted from Multiple
Examples of Nonlinear Model Analyses
Survey Design
Inversion Grid Design
Conclusions and Suggestions
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