Non-Linear Biased Estimation
In interpreting a scanty set of inexact data, conventional wisdom is to seek models that axe in agreement with a priori data derived from say, previous geophysical studies, borehole or geological data. These extraneous information help single out a plausible solution from amongst all possible ones admitted by the inexact practical data. In Chapter 5, we saw how the use of a priori data in linear inversion helps resolve the problem of non-uniqueness. Unfortunately, while the solution to the linear inverse problem incorporating a priori information is well known (see e.g., Jackson, 1979), there is no unequivocal technique as yet for resolving non-uniqueness in nonlinear inversion. A naive approach to using a priori information in nonlinear inversion involves holding constant the values of some of the sought parameters within an iterative inversion scheme. The formal approach is to incorporate the a priori data directly in the problem formulation. Much of the published formal treatment of a priori information in nonlinear inversion adopt a probabilistic approach (e.g., Gol’tsman,1971,1975; Tarantola b Valette,1982; Jackson k Matsu’ura,1985; Pous et al.,1987; Duijndam,1988) which, it may be argued, best characterises the huge variability in practical geophysical measurements (Meju, 1994d). We will adopt the same approach as in our treatment of liTiPar inversion since it involves: simple matrix algebra and minimal statistical commitment. Following Meju (1994d), the problem of inversion -with a priori information will be addressed within a unifying framework of biased estimation, with emphasis on simplicity and effective practical procedures. In this approach a distinction is made between starting models and a priori information for clarity. We shall aim to make the inversion scheme flexible enough to allow for the construction of a variety of least squares solutions using either reliable or diffuse a priori data thus making it a useful practical tool for exploiting particular geophysical situations. To achieve this objective, we will need to consider several forms of a priori constraints or solution simplicity measures and take the observational errors into account. Finally, we will show the relationships between the ensuing algorithms and various classical inversion algorithms. This strategy, it is hoped will enable us to analyse and understand the more rigorous algorithms for non-linear inversion (e.g., Tarantola & Valette,1982).
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“This publication is designed to provide a practical understanding of leastsquares methods of parameter estimation and uncertainty analysis. The practical problems covered range from simple processing of time- and space-series data to inversion of potential field, seismic, electrical, and electromagnetic data. The various formulations are reconciled with field data in the numerous examples provided in the book; well-documented computer programs are also given to show how easy it is to implement inversion algorithms.”