Linearized and Iterated Methods
Two fundamental deficiencies of the asymptotic methods of Chapter 4 are that they do not incorporate observational errors directly in the inversion, and they cannot guarantee that the constructed model reproduces the observations. To correct these problems, numerous authors have linearized the MT equations and developed algorithms which iterate to an acceptable σ(z). Unfortunately, as shown in the excellent paper by Parker (1980), the conductivity with the least-squares misfit is the geophysically unreasonable D+ delta function model. Although other models may exist which fit the data equally well, the fact that a D+ model always achieves the minimum χ2 misfit is of fundamental importance to iterative algorithms which attempt to find a model that reproduces the data. D+ solutions imply that the iterative formulation is inherently unstable if considered in terms of misfit alone, and our conjecture is that linearized algorithms will naturally tend to an oscillatory model which in some sense resembles a D+ solution. Such oscillatory models are generally undesirable, and consequently artificial stabilization or smoothing must be applied. This stabilization may hinder convergence.
For linearized approaches, additional problems are the choice of starting model and the sensitivity of the final σ(z) to the starting model. Anderssen (1975) discussed some of the hazards of linearization such as oscillating iterations and convergence to an incorrect result. He concluded that linearization is only appropriate if the starting model is sufficiently close to a possible solution. Nevertheless, despite these difficulties, the lure of linear equations has resulted in numerous linearized and iterated MT inversion schemes. Many of these schemes work exceedingly well.
The first linearized approach discussed in Section 5.1 is the Born approximation of Coen et al. (1983). These authors solved a linearized integral equation written in terms of the normalized electric field. Section 5.2 shows that the inversion method of Schmucker (1972) described by Larsen (1975) is quite similar to a Born inverse. This method is derived from the layer to layer recursion relation of a response approximately equal to r(w). Larsen (1981) extended this method and made it more precise. The parametric inversion discussed in Section 5.3 is used extensively to interpret MT data in terms of a layered σ(z). Finally, Section 5.4 presents the Frechet derivative approach of Oldenburg (1979). This linearization is a continuous version of the parametric method, and it constructs smooth σ(z) models.
Figures & Tables
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.”