2-D Niblett-Bostick magnetotelluric inversion

J.

Description

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, a self-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.

Informations

• Publié par: erevistas
• Publié le: 01 janvier 2010
• Nombre de lectures: 15
• Langue: English

Extrait

Geologica Acta, Vol.8, Nº 1, March 2010, 15-31
DOI: 10.1344/105.000001513
Available online at www.geologica-acta.com
2-D Niblett-Bostick magnetotelluric inversion
*J. RoDRíguez F.J. espa Rza and e. gómez-TReviño
CiCese/Ciencias de la Tierra
Km 107 carr. Tijuana-Ensenada, Ensenada, B.C., 22860, México
* Corresponding author E-mail: [e-mail]
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 profles.
The algorithm processes series and parallel magnetotelluric impedances and their analytical infuence functions
using a regularized Hopfeld artifcial 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 imped-
ances, a self-contained pair of invariants of the impedance tensor, avoids the need to decide on best angles of rota-
tion for identifying TE and TM modes. Field data from a given profle can thus be fed directly into the algorithm
without much processing. The solutions offered by the regularized Hopfeld 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. Hopfeld Neural Network. magnetotelluric.
INTRODUCTION imposed to construct realistic solutions that ft the data to a
given degree. Most commonly, the norm of the solution or
Electromagnetic inverse problems in geophysics are of its frst or second derivative are minimized together with
nonlinear. Even relatively simple cases, like the one- the misft to the data. This technique, frst developed for
dimensional (1-D) magnetotelluric (MT) problem, require 1-D problems, can readily be applied in higher dimensions.
special treatment to fully handle nonlinearities (e.g., Applications to the 2-D MT inverse problem include those
Bailey, 1970; Weidelt, 1972; Parker, 1983). Methods based of Rodi (1989), de Groot-Hedlin and Constable (1990),
on linearization can be applied iteratively to handle the Smith and Booker (1988) and Rodi and Mackie (2001).
nonlinearity of the problem (e.g., Oldenburg, 1979; Smith Minimizing roughness avoids the appearance of sharp
and Booker, 1988). In practice, as well as not being linear, features in the solution models that are not strictly required
electromagnetic inverse problems are ill-posed and severely by the data. That is, the resulting models are as smooth and
underconstrained. Sensible external constraints are usually even as the data permit.
15J. RODRÍGUEZ et al. 2-D Niblett-Bostick
It is also possible to obtain useful and somewhat 2-D: a) it captures the basic physics in a simple integral form,
more general information by slightly shifting the focus b) it relates the data directly to an arbitrary conductivity
of attention; instead of looking for a single model that fts distribution and, c) it relies on the relatively simple theory
the data, one can ask for general properties of all possible of a homogeneous half-space, but it handles heterogeneous
models that ft the data. The method of Backus and Gilbert media by adapting its conductivity according to the data,
(e.g., Backus and Gilbert, 1968, 1970) allows for the without the need for an initial or reference model. These
computation of spatial averages by means of averaging features are exploited by the 1-D extensions mentioned
functions constructed as linear combinations of the Fréchet above in relation to controlled source methods. Extensions
derivatives of the data. The averaging functions are made to higher dimensions for special types of electromagnetic
to resemble box-car functions for the averages to have the measurements also proft from them (e.g. Pérez-Flores and
usual intuitive meaning. The results are average models for Gómez-Treviño, 1997; Pérez-Flores et al., 2001; Brunner
given window sizes. The models are not intended to produce et al., 2003; Friedel, 2003). In this paper, we explore the
responses that ft the data. In fact, they seldom do better possibilities of the same approach for the two-dimensional
in this respect than models designed specifcally for ftting (2-D) MT problem.
purposes. It is perhaps for this reason that average models
are not very popular among interpreters of feld data who
seem to prefer the assurance of a direct ft to the data. THE APPROXIMATION
Summarizing the above two paragraphs, we have on one In the magnetotelluric method, surface measurements
side an optimization process of ftting data with external of natural time-varying electric and magnetic felds are
constraints, and on the other side, an optimization process readily converted to four complex impedance values per
of ftting Fréchet derivatives to box-car functions. Although given angular frequency ω. In turn, these values are usually
the two processes are complementary, particularly for normalized to obtain apparent resistivities or, equivalently,
nonlinear problems, the latter is seldom applied, perhaps apparent conductivities, by referring the actual impedances
because it requires intensive computations, but most likely to those of a homogeneous half-space (Cagniard, 1953).
in view of the reason stated in the previous paragraph. In Here we use apparent conductivity s as derived from the a
this paper, we present an approach that combines features magnitude of a complex impedance Z which, for the mo-
from both methods. On one side, we keep the reassuring ment, represents any of the four elements of the impedance
feature of constructing models whose responses optimize tensor. The formula for apparent conductivity is simply
the ft to the data and, on the other, we maintain the concept given as
σ aof spatial averages. Spatial averages not only appeal to −2intuition, but when plotted against depth they resemble a σ ( x,ω) = ωµ |Z(x,ω) | , (1)(1) a 0σ aprofle of the property itself, fltered by the corresponding µ −20window. As shown in Gómez-Treviño (1996), a solution in where µ stands for the magnetic permeability of free-space σ (x,ω) = ωµ |Z(x,ω) | , (1)
0 a 0
σ (x,w) terms of averages represents a robust alternative to the well and x represents horizontal distance in an x-z coordinate a µ 0σ known 1-D Niblett-Bostick transformation. system whose z axis represents depth. s (x, w) represents a a2π
T = −2σ (x,w) the data at a given distance in a 2-D model with a fat a σ (x,ω) = ωµ |Z(x,ω) | , (1) a 0ωBetween linearization and nonlinear methods, there are topography and for a given angular ω. The data are usually 2πµ σ 0 a T = approximations simple enough to be handled analytically presented as individual sounding curves as a function of
ω 2 pσ (x,w) but that still keep some of the nonlinear features of the σ (x,w) period T = for different distances x, or in a pseudo-a a vσ aoriginal problem. Such is the case of the Niblett-Bostick section format contouring values of s as plotted over 2π a σ (,x z) T = approximate integral equation (e.g., Niblett and Sayn- (x,w) x-T coordinates. s (x, w) represents what is available; σ aω a Wittgenstein, 1960; Bostick, 1977; Jones, 1983) which what is required is s(x, z), the subsurface conductivity σ σ (,x z) a σ (x,T ) has inspired a number of generalizations. F a or instance: distribution.σ (x,w) a iteration of the corresponding exact equation (Esparza and σ (,x z)
σ (,x z) σ (x,T ) Gómez-Treviño, 1996) and, still within the approximation, A useful relationship between s (x, T) and s(x, z) for a a1 σ (x,T ) = F(x,xz', ',σσ,T ) (xz', ')dx'dz ', (2) a solution of the non-uniqueness problem by direct the problem at hand is (Gómez-Treviño, 1987a):σ (,x z) a ∫1−mσ (x,T ) computations of spatial averages (Gómez-Treviño, 1996). a 1d logσOther generalizations include the e σxtension of the basic (,x z) σ ( x,T ) = Fa(x,xz', ',σσ,T ) (xz', ')dx'dz ',(2)a m = ∫ , (3) 1−midea to 1-D inversion of controlled source electromagnetic d logT1
σ (x,T ) = F(x,xz', ',σσ,T ) (xz', ')dx'dz ', (2) d logσa ∫data (e.g., Boerner and Holladay, 1990; Smith et al., 1994; where: aF(xx, ',z ',σ ,T ) 1−m m = , (3)
Christensen, 1997). d logTd logσ aσ (x,T ) m = , (3) a (3)F(xx, ',z ',σ ,T ) d logT
The 1-D Niblett-Bostick approximation has the σ (,x z)
F(xx, ',z ',σ ,T ) σ (x,T ) afollowing basic features that are worthwhile exploiting in and F(x,x’,z’, s,T) represents the Fréchet derivative of σ (,x z)
σ (x,T ) σ (,x z) a
σ (x,T ) aσ (,x z) σ (,x z)
F(z ',σ ,)T = (1−mF) (z ',σ ,)T . (4) σ (,x z) haσ (x,T ) aFz( ',σ ,T )