The random search, or Monte-Carlo, method has been used by many interpreters to construct conductivity models satisfying MT data. The method guesses a σ(z) profile and uses forward modeling to test its fit to the observed data. If the fit is acceptable, the proposed model is saved, otherwise the model is rejected and a new σ(z) is randomly selected. Monte-Carlo methods require no approximations because forward modeling is a simple, direct calculation. Moreover, by finding a large collection of acceptable yet diverse σ(z), the method explores to some extent the nonuniqueness of the problem.
The disadvantage of Monte-Carlo methods is that the time and expense required to propose and test thousands of σ( z) models can be prohibitive. A finite search can never completely explore the infinite-dimensional space of acceptable models. For L layers and M possible choices of σ for each layer, the total number of possible models is ML. To reduce the possibilities, the number of layers is usually limited to less than ten, and bounds are placed on the range of permissible layer conductivities and thicknesses. If these restrictions are well-founded then the method has a better chance of finding a model that is close to reality. Otherwise, arbitrarily limiting the search of model space to a smaller region decreases the chances of a correct interpretation. Thus, the method is heavily dependent on the model parameterization.
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In this short monograph, 1D inversion methods are examined collectively using a uniform notation. One-dimensional inversion methods are still important because there are geologic regions where lateral variation is small and 1D interpretation is directly applicable; 1D inverse solutions provide good starting models for 2D inversion; and understanding the 1D inverse problem provides a foundation for solving inverse problems in higher dimensions. The 10 chapters are “Introduction”, “Existence”, “Uniqueness,” “Asymptotic Methods,” “Linearized/Iterated Methods,” “Global Penalty Functional Methods,” “Monte Carlo Methods,” “Exact Methods,” “Appraisal Methods,” and “Conclusion.”