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Arabian Journal of Geosciences ISSN 1866-7511 Arab J GeosciDOI 10.1007/s12517-013-0869-6

Investigating 2-D MT inversion codes usingreal field data

Reza Ghaedrahmati, Ali Moradzadeh,Nader Fathianpour & Seong Kon Lee

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ORIGINAL PAPER

Investigating 2-D MT inversion codes using real field data

Reza Ghaedrahmati & Ali Moradzadeh &

Nader Fathianpour & Seong Kon Lee

Received: 28 July 2012 /Accepted: 25 January 2013# Saudi Society for Geosciences 2013

Abstract There are currently a significant number of two-dimensional (2-D) and three-dimensional (3-D) inversioncodes available for magnetotelluric (MT) data. Throughvarious 2-D inversion algorithms suggested so far, the clas-sical Occam's inversion, the data space Occam's inversion,the nonlinear conjugate gradient (NLCG) method, and theGauss–Newton (GN) method are fundamental driving meth-ods to find optimum earth models, and OCCAM, DASOCC,NLCG, and MT2DInvMatlab are possible candidates onecan find in the public domain that implement these algo-rithms for 2-D MT inversions, respectively. In this study, weinvestigate the pros and cons (strength and weakness) ofthese codes to help one use them efficiently in practicalworks and, as an introductory guide, further develop (so-phisticate or extend) them, especially for the 3-D case. Toachieve this goal, we applied each one of the four afore-mentioned codes on a profile of real MT field dataset. Then,further investigations have been done by performing severalinversion tests to see how each code can find the appropriatemodel to reconstruct the subsurface resistivity structure.Numerical experiments show that the two parameters, reg-ularization and target misfit, in addition to the main criteriaof inversion (such as the forward and the sensitivities

calculation method, and the type of inversion algorithm),are very important to produce the expected model in inver-sion. The regularization parameter that acts to trade offbetween model norm and data misfit can affect the inversionprocess in terms of both the computational efficiency andthe accuracy of the obtained model. Also, lack of insuffi-cient precision to choose the target misfit can lead theinversion to produce and reach an incorrect model.

Keywords 2-D inversion code . Magnetotelluric .

Resistivity

Introduction

The magnetotelluric (MT) technique is a natural sourceelectromagnetic method by measuring time-varying surfacemagnetic and electric fields that contain information aboutvertical and lateral variations of electrical resistivity struc-ture of the Earth (Cagniard 1953; Vozoff 1991). For bothqualitative and quantitative interpretations, the significanceof inversion analysis of MT data has been increased in thepast few decades, and thus, many two-dimensional (2-D)and three-dimensional (3-D) MT inversion algorithms havebeen published (for 2-D case: deGroot-Hedlin and Constable1990; Smith and Booker 1991; Ogawa and Uchida 1996;Siripunvaraporn and Egbert 2000, 2007; Rodi and Mackie2001; Ogawa 2002; Lee et al. 2009; for 3-D case: Mackieand Madden 1993; Newman and Alumbaugh 2000; Zhdanovet al. 2000; Sasaki 2001, 2004; Siripunvaraporn et al. 2005;Han et al. 2008; Lin et al. 2008; Avdeev and Avdeeva 2009;Zhdanov et al. 2011). Though many 2-D MT inversion algo-rithms have been published, the main algorithm of each codefalls roughly into the category of the classical Occam's inver-sion, the data space Occam's inversion, the nonlinear conju-gate gradient (NLCG) method, or the Gauss–Newton (GN)method. The codes based on these algorithms which can be

R. Ghaedrahmati (*)Shahrood University of Technology, Shahrood, Irane-mail: r.ghaedrahmati@yahoo.com

A. MoradzadehFaculty of Mining, Petroleum and Geophysics,Shahrood University of Technology, Shahrood, Iran

N. FathianpourMining Engineering Department,Isfahan University of Technology, Isfahan, Iran

S. K. LeeKorea Institute of Geoscience and Mineral Resources (KIGAM),Daejeon, Republic of Korea

Arab J GeosciDOI 10.1007/s12517-013-0869-6

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available to the academic community in the public domainare OCCAM (Constable et al. 1987; deGroot-Hedlin andConstable 1990), DASOCC (Siripunvaraporn and Egbert2000), NLCG (Rodi and Mackie 2001), and MT2DInvMatlab(Lee et al. 2009).

Geophysical inverse problem is generally nonlinear andill posed, and so, there are also many sophisticated methodsthat extract the essential features of conductivity models(Van Beusekom et al. 2010), but all algorithms have non-uniqueness issues of the inverted model. A lot of experi-ments with these 2-D main algorithms have been done onsynthetic or real data (see deGroot-Hedlin and Constable1990, 1993; Ogawa 1997; Uchida 1997; Siripunvarapornand Egbert 2000; Bai et al. 2001; Uchida et al. 2002; Tunceret al. 2006; Lee et al. 2009; Bertrand 2010; to list a few), andthe comparisons between some of these algorithms havebeen made in some experiments and could be found, e.g.,in the good work by Rodi and Mackie (2001), in which thecomparison between GN and NLCG algorithms has beenexploited, but the main aim of this comparison is for com-putational efficiency.

Recently, Siripunvaraporn (2012) has provided a goodreview of several approaches of 3-D MT inversion. Thiswork reviews the many different 3-D inversion algorithms.Necessary components for 3-D inversion algorithm such asforward modeling, sensitivity calculation, and type of inver-sion algorithm have been considered in this literature. Itcould be a very good reference for users and developers,and comparisons of the mentioned algorithms with respectto computational efficiency have been done in it. Our inves-tigation, however, will focus on the practical aspects ofnumerical performance of the four inversion algorithms for2-D MT inversion with an identical real dataset: the Occamalgorithm with the code OCCAM, the data space algorithmwith the code DASOCC, the NLCG algorithm with the codeNLCG, and the GN algorithm with the code MT2DInvMat-lab. We have shown how these codes can be used to achievethe appropriate model for a complex geology structure.

We begin our study with a review of formulations ofthese codes. In the next section, we will describe a test setof real data obtained along a profile at a geothermal field.Then, each one of the four mentioned codes is performed onthis data.

Theoretical background

OCCAM method

The original OCCAM method is for one-dimensional (1-D)inversion of MT data described by Constable et al. (1987),and it is developed for 2-D case by deGroot-Hedlin andConstable (1990). This algorithm seeks the minimum

possible structure model subject to an appropriate fit to thedata. Mathematically, this objective is achieved by solvingan optimization problem as

Y m;λð Þ ¼ m�m0ð ÞTWm m�m0ð Þþ λ�1A d� F½m�ð ÞTWd d� F½m�ð Þ � c2Z ð1Þ

Here, d is the vector of observed data with dimension N,m is the resistivity model with dimension M, m0 is the priormodel, F(m) is the theoretical model responses or forwardfunction, Wd is the data-weighting matrix which is diagonaland consisting of the reciprocal of the data standard devia-tions, Wm is a second-difference operator used to define themodel roughness or model norm, χ2 is the target misfit, 1 isa Lagrange multiplier or regularization parameter, and thesuperscript T represents matrix transpose. The functionalΨ is also described by two main terms: the model norm[(m−m0)

T Wm(m−m0)] and the data misfit {[(d−F(m)]T

Wd[d–F(m)]}.The functional Ψ in Eq. 1 is minimized. Because the

forward function F(m) is nonlinear, the functional Ψ isnonlinear. After linearizing the forward function by consid-ering first-order Taylor expansion and minimizing the func-tional Ψ with respect to m, an iterative system of linearequations for finding model is obtained:

mkþ1 ¼ ½1Wm þ JTKWdJk� ���1JTKWd d� FðmkÞ þ Jk mk �m0ð Þ½ �

þm0

ð2Þ

Here, the subscript k denotes the iteration number, andJk=(∂F/∂m)k is the N×M sensitivity matrix or the Jaco-bian matrix calculated at mk.

In each of the inversion iterations, Eq. 2 is solved with aseries of trial values of 1 to find a model that minimizes themisfit to the data. Ideally, 1 should be chosen so that the misfitreaches to target misfit, but for nonlinearity MT inversionproblem, and because real data mostly have noise, it is notpossible. Thus, the inversion process finds the model withminimum misfit as the start model for the next iteration.

Inversion process in this algorithm includes twophases, which are called phase 1 and phase 2 (Parker1994; Siripunvaraporn and Egbert 2000). When inversion isstarted, phase 1 is begun until the misfit reaches the target levelby varying 1 in each of the iterations. Phase 2 is then begun tominimize the model norm. This is accomplished by again vary-ing 1 in each of the iterations to find the model with the smallestnorm while keeping the misfit at the target level of misfit.

DASOCC method

In order to reduce computational costs in Occam approach,Siripunvaraporn and Egbert (2000) developed the data space

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