Inversion De Bostick

Geologica Acta, Vol.8, Nº 1, March 2010, 15-31 DOI: 10.1344/105.000001513 A v a i l a b l e o n l i n e a t w w w. g e o l o g i c a - a c t a . c o m

2-D Niblett-Bostick magnetotelluric inversion

J. RoDRíguez F.J. espaRza

*

and e. gómez-TReviño

CiCese/Ciencias de la Tierra Km 107 carr. Tijuana-Ensenada, Ensenada, B.C., 22860, México

* Corresponding author E-mail:fesparz@cicese.mx

ABSTRACT

A simple and robust imaging technique for two-dimensional magnetotelluric interpretations has been developed following the well known Niblett-Bostick transformation for one-dimensional profiles. The algorithm processes series and parallel magnetotelluric impedances and their analytical influence functions using a regularized Hopfield artificial neural network. The adaptive,weighted average approximation preserves part of the nonlinearity of the original problem, yet no initial model in the usual sense is required for the recovery of the model; rather, the built-in relationship between model and data automatically and concurrently considers many half spaces whose electrical conductivities vary according to the data. The use of series and parallel impedances, aself-contained pair of invariants of the impedance tensor, avoids the need to decide on best angles of rotation for identifying TE and TM modes. Field data from a given profile can thus be fed directly into the algorithm without much processing. The solutions offered by the regularized Hopfield neural network correspond to spatial averages computed through rectangular windows that can be chosen at will.Applications of the algorithm to simple synthetic models and to the standard COPROD2 data set illustrate the performance of the approximation.
KEYWORDS Niblett-Bostick inversion. Hopfield Neural Network. magnetotelluric.

INTRODUCTION Electromagnetic inverse problems in geophysics are nonlinear. Even relatively simple cases, like the onedimensional (1-D) magnetotelluric (MT) problem, requirespecial treatment to fully handle nonlinearities (e.g., Bailey, 1970; Weidelt, 1972; Parker, 1983). Methods based on linearization can be applied iteratively to handle the nonlinearity of the problem (e.g., Oldenburg, 1979; Smith and Booker, 1988). In practice, as well as not being linear, electromagnetic inverse problems are ill-posed and severely underconstrained. Sensible external constraints areusually

imposed to construct realistic solutions that fit the data to a given degree. Most commonly, the norm of the solution or of its first or second derivative are minimized together with the misfit to the data. This technique, first developed for 1-D problems, can readily be applied in higher dimensions. Applications to the 2-D MT inverse problem include those of Rodi (1989), de Groot-Hedlinand Constable (1990), Smith and Booker (1988) and Rodi and Mackie (2001). Minimizing roughness avoids the appearance of sharp features in the solution models that are not strictly required by the data. That is, the resulting models are as smooth and even as the data permit.

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2-D Niblett-Bostick

It is also possible to obtain useful and somewhat more generalinformation by slightly shifting the focus of attention; instead of looking for a single model that fits the data, one can ask for general properties of all possible models that fit the data. The method of Backus and Gilbert (e.g., Backus and Gilbert, 1968, 1970) allows for the computation of spatial averages by means of averaging functions constructed as linear combinations of the Fréchet derivativesof the data. The averaging functions are made to resemble box-car functions for the averages to have the usual intuitive meaning. The results are average models for given window sizes. The models are not intended to produce responses that fit the data. In fact, they seldom do better in this respect than models designed specifically for fitting purposes. It is perhaps for this reason that average...