An MT response function derived from vertically incident, plane electromagnetic waves diffusing into a 1-D conductivity σ(z) must satisfy certain mathematical constraints. In practice, these theoretical constraints may be violated because measurement and processing of field records yield responses contaminated with errors. The constraints may also be violated because of the discrepancy between the idealized mathematical representation and the true physical situation. For example, in most areas the real Earth conductivity is 3-D, and the source fields are a complex superposition of time-varying waves propagating in different directions. Therefore, when interpreting MT responses in terms of a 1-D σ(z) model, it is important to answer first the question of existence; that is, does there exist a σ(z) profile which reproduces the observations? The question of existence is equivalent to determining whether or not the observations satisfy the theoretical 1-D constraints dictated by the mathematical model. We discuss first the mathematical form of these theoretical constraints.
The dispersion relations and inequalities are good theoretical tests for existence, but they are difficult to apply in practice because field data are discretely sampled within a finite bandwidth. The integrals and derivatives required to implement these existence tests cannot be calculated precisely.
Weidelt (1986) overcame this difficulty by deriving a set of existence tests which do not involve derivatives. The conditions require the positivity of 2N determinants derived from the discrete data measured at N frequencies. Importantly, these constraints are necessary and sufficient conditions for the existence of a 1-D conductivity model. Parker (1983) showed that the dispersion relations and inequalities are necessary but not sufficient conditions. Parker (1980) also examined the conditions on the response required for the existence of a solution by using a linear programming formulation.
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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.”