MODEL ASSESSMENT FOR NON-LINEAR GEOPHYSICAL INVERSE PROBLEMS

Model assessment is arguably one of the most important aspects of geophysical inversion. The inherent non-uniqueness of the geophysical inverse problem, coupled with the reality of incomplete and inexact data, frustrate recovery of a true-earth model. Artifacts in inverse models arise from a range of issues, including regularization, model parameterization, linearization of the inverse problem, data density, error specification, and data sensitivity. These factors may also give rise to an absence of structure, which can be equally misleading. Model assessment can be loosely broken down into: 1) data fit statistics and formal measures of sensitivity, 2) the effects of regularization, 3) methods for exploring the permissible model space, and 4) investigation of spatial and geometric resolution.

Formal approaches to model assessment include normalized Χ² statistics and transformations or decomposition of the sensitivity matrix, which relates changes in the data to linear perturbations of the model parameters. Regularization concerns the effects of trade-off parameters, model norm, and starting and prior models on the resulting inverse model. For example, multiple starting models can be used to help assess the non-uniqueness of the inverse problem, and are also useful in determining the region of model space that is strongly influenced by the data (Oldenburg and Li, 1999). A number of approaches are taken to explore the permissible model space. These range from linear sensitivity analysis, valid in assessing the sensitivity to small model perturbations, to global search methods which employ a probabilistic approach to exhaustively sample the model space. In practice, a combination of forward modeling and constrained inversion are commonly used to investigate alternate models. This is generally a hypothesis-driven approach in which the geometry or physical properties of parts of the model are altered. A final aspect of model assessment is spatial resolution - what is the minimum-size structure that can be resolved, and with what accuracy can the geometry and physical properties be recovered? Some inverse problems, such as seismic refraction tomography, employ checkerboard tests (Zelt, 1998) to determine the minimum resolvable structure. Others, including electrical inverse problems, are plagued by the strong non-linearity between the modeled physical property and the measured data. Analysis of the model resolution matrix, a function of the sensitivity and data errors, is one approach to assessing resolution, but can be cumbersome for large inverse problems.

Oldenburg, D.W. and Li, Y., 1999, Estimating depth of investigation in DC resistivity and IP surveys: Geophysics, v. 64,p. 403-416.

Zelt C.A., 1998, Lateral velocity resolution from three-dimensional seismic refraction data: Geophys. J. Inter.. v. 135, p. 1101-112.