Asymptotic methods are approximate inversion schemes which consider various limiting or approximate forms of the governing MT equations. All methods exploit the fact that low frequencies penetrate more deeply into the Earth than high frequencies, and therefore, successively lower frequencies determine successively deeper conductivities. Unfortunately, some of the approaches do not utilize the phase information. Moreover, because many do not incorporate the data errors the constructed σ(z) may not adequately fit all of the observations.
Despite their heuristic nature and innate deficiencies, these methods are easily programmable and can therefore be used in the field to obtain almost immediate information about the conductivity structure. Also these approaches are still used routinely in more sophisticated multidimensional interpretations and sometimes do generate conductivities which fit the data. Lastly, since one of our goals is to illustrate the numerous ways in which the MT inverse problem can be attacked, it is especially interesting to examine these methods in detail because many of them yield essentially the same end formula even though the derivation is considerably different. For instance, asymptotic analysis for several layers over a half-space, approximation of the induction equations, and low frequency correspondence with de resistivity all approach the MT problem from a different viewpoint. The fact that a similar algorithm is produced in each case is a strong testimonial to the overwhelming smoothing of the electromagnetic fields as they diffuse into the Earth. Physical understanding of this diffusion provides a firm foundation upon which to build a simple algorithm to yield information about the subsurface conductivity.
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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.”