Despite the nonlinearity of the MT equations, several exact methods exist which calculate a σ(z) consistent with the data. For these approaches there is no need for any approximations other than those required for computer implementation. The solution is not iterative so no starting model is required and there are no convergence problems; however, these algorithms are not entirely without problems. Some do not incorporate the data errors, others have numerical solutions which are unstable, and still others may not guarantee a positive σ(z) model.
All exact methods subdivide into at least two stages. The first major stage is to complete the measured data somehow to obtain realizable responses at all frequencies. The second stage maps these completed responses to a unique conductivity profile. The different approaches to the completion and mapping problems account for the different inversion methods. Some techniques are more sensitive to noise than others. We have divided the exact methods into rough categories based on their completion and mapping schemes; however, there are overlaps between categories in many cases.
Bailey (1970,1973) derived an exact inversion scheme for finding a radially symmetric conductivity σ(r). As a response, the method uses the frequency-domain ratio of the induced to the inducing magnetic field of any spherical harmonic mode. This response satisfies a Riccati equation as well as dispersion relations which guarantee causality. Integrating the Riccati equation over frequency and using the causality condition gives the conductivity in a shell adjacent to the level at which the responses are known. The Riccati equation can then be used to downward continue the responses through this layer. These new responses define the next deeper conductivity value, and the process is repeated. Importantly, Bailey showed that this inversion technique is unique for the class of all nonzero, bounded, infinitely differentiable σ( r). That is, there is only one conductivity profile which keeps the responses causal at all radii.
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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.”