Inversion of 3D Electromagnetic Data in Frequency and Time Domain

8/10/2019 Inversion of 3D Electromagnetic Data in Frequency and Time Domain

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GEOPHYSICS, VOL. 69, NO. 5 (SEPTEMBER-OCTOBER 2004); P. 12161228, 9 FIGS., 2 TABLES.10.1190/1.1801938

Inversion of 3D electromagnetic data in frequency and time domainusing an inexact all-at-once approach

Eldad Haber1, Uri M. Ascher2, and Douglas W. Oldenburg3

ABSTRACT

Wepresentageneralformulationforinvertingfrequency-or time-domain electromagnetic data using an all-at-onceapproach. In this methodology, the forward modeling equa-tions are incorporated as constraints and, thus, we need tosolvea constrained optimization problem wherethe param-etersare the electromagnetic fields, the conductivity model,and a set of Lagrange multipliers. This leads to a muchlarger problem than the traditional unconstrained formu-lation where only the conductivities are sought. Neverthe-less, experience shows that the constrained problem canbe solved faster than the unconstrained one. The primaryreasons are that the forward problem does not have to besolved exactly until the very end of the optimization pro-cess, and that permitting the fields to be away from their

constrained values in the initial stages introduces flexibil-ity so that a stationary point of the objective function is

found more quickly. In this paper, we outline the all-at-once approach and apply it to electromagnetic problems inboth frequency and time domains. This is facilitated by aunified representation for forward modeling for these twotypes of data. The optimization problem is solved by find-ing a stationary point of the Lagrangian. Numerically, thisleads to a nonlinear system that is solved iteratively using aGauss-Newton strategy. At each iteration, a large, indef-inite matrix is inverted, and we discuss how this can beaccomplished. As a test, we invert frequency-domain syn-thetic data from a grounded electrode system that emu-lates a field CSAMT survey. For the time domain, we in-vert borehole data obtained from a current loop on thesurface.

INTRODUCTION

In this paper, we develop an inversion methodology for3D electromagnetic data in both frequency and time domains.This problem is of major interest in geophysics, medical imag-ing, and nondestructive testing [see, for example, Smith andVozoff (1984), Devaney (1989), Parker (1994), Borcea et al.(1996), Cheney et al. (1999), Vogel (1999), Haber and Ascher(2001a),and references therein].The forward model consists ofMaxwells equations in which the permeability is constant butelectricalconductivity can be highly discontinuous. Theparam-eter regimes considered give rise to highly stiff problems in thetime domain or, alternatively, low frequencies in the frequencydomain. The goal of theinversion is to recover theconductivitygiven measurements of the electric and/or magnetic fields.

There are many practical challenges to solving the inverseproblem. First, a fast, accurate, and reliable algorithm for 3D

Manuscript received by the Editor June 14, 2002; revised manuscript received March 31, 2004.1Formerly University of British Columbia, Department of Earth and Ocean Sciences, Geophysical Inversion Facility, Vancouver, British

Columbia V6T 1Z4, Canada; presently Emory University, Department of Mathematics and Computer Science, Atlanta, Georgia 30322.E-mail: [email protected].

2University of British Columbia, Department of Computer Science, Vancouver, British Columbia V6T 1Z4, Canada. E-mail: [email protected] of British Columbia, Department of Earth and Ocean Sciences, Geophysical Inversion Facility, Vancouver, British Columbia V6T

1Z4, Canada. E-mail: [email protected] 2004 Society of Exploration Geophysicists. All rights reserved.

forward modeling is required in frequency and in time. Sec-ond, thesensitivities forsuch problems aretoo numerous to beformed or stored in a reasonable amount of time and space. Fi-nally, finding the minimum of the objective function obtainedby matching thedata andincorporating a prioriinformationonthe conductivity field can be difficult due to the nonlinearityand sensitivity of the problem.

We use an inexact, all-at-once methodology (Haber andAscher, 2001b; Ascher and Haber, 2003; Biros and Ghattas,2004), solving the forward problem and the inverse problemsimultaneously in one iterative process. This approach allowsdevelopment of highly efficient algorithms. However, becauseit couples the solution of the forward problem with the solu-tion of the inverse problem, the forward problem cannot betreated as a black box. Care must be taken to properly fit theformulation and discretization of the forward problem withinthe inverse methodology.

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3D EM Inversion 1217

To be more specific, assume that the forward problem (incontinuous space) is written in the form

A(m)u q= 0 (1)where A(m) is a version of Maxwells equations (including

boundary conditions) either in time or in frequency,m = log()is the log conductivity,u stands for the fields, andq representssources and boundary values. We assume for simplicity of ex-position that A is invertible for any relevantm , i.e., there is aunique solution to the forward problem.

In the inverse problem, we measure some function of thefields and want to recover the modelm . Let us write the mea-sured data as

bobs = Qu + (2)

where Q is a measurement operator which projects the fields(or their derivatives or integrals) onto the measurement loca-tions in 3D space and possibly time, and is the measurementnoise.

The data are finite and contaminated with noise, and thereis no unique, true model. To obtain a unique model whichdepends stably on the data, we use a priori information andformulate the inverse problem (in continuous space) as a con-strained optimization problem of the form

minm,u

1

2Qu bobs2 + R(m)

subject to A(m)u q= 0. (3)

Here, > 0 is the regularization parameter, and R() is a reg-ularization operator reflecting our a priori information. Typ-ically, we know that m is a piecewise smooth function overthe spatial domain in three dimensions, so we assume thatR() involves some norm ofm over , e.g., weighted L2 orL

1 or a Huber combination (Huber, 1964; Farquharson andOldenberg, 1998). (Across jump discontinuities the directedderivative of m yields a Dirac -function which is integrablebut not square-integrable.) The data fitting term in expression3 involves theL 2norm over.

Next, the problem 3 is discretized using some finite-volumeor finite-element method over a finite grid representing thedomain in space and time, yielding the finite-dimensional op-timization problem

minm,u

1

2Qu bobs2 + R(m)

subject to A(m)u q= 0, (4)

where u, m,and q aregridfunctions ordered as vectors andcor-responding to their continuous counterparts above, andQ , A

and R/mare all large, sparse matrices. ThematrixA dependsonm and is nonsingular.

The common approach to solving this problem (Tikhonovand Arsenin, 1977; Madden and Mackie, 1989; Parker, 1994;Newman and Alumbaugh, 1995, 1997a, 1997b; Vogel, 1999),is to first eliminate the fieldu using the equality constraints,obtaining an unconstrained optimization problem of the form

minm

1

2Q A(m)1q bobs2 + R(m).

This approach is rooted in the vast amount of literature andmethods which have been developed for unconstrained opti-mization and the positive definiteness of corresponding ap-proximations to the Hessian matrix (Dennis and Schnabel,1996; Kelley, 1999; Nocedal and Wright, 1999). However, eachevaluation of the objective function requires a solution of the

forward problem, and evaluating the gradients requires thesolution of the adjoint problem. Evaluating the sensitivity ma-trix requires many more solutions of the forward and adjointproblems. Theresultingproceduremay therefore become verycomputing intensive.

For this reason a recent work using the unconstrained ap-proach employs a nonlinear conjugate gradient (CG) method(Rodi and Mackie, 2001). The cost of the inversion is then pro-portionalto thenumberof iterations takenin thenonlinearCGalgorithm times twice the cost of solving one forward problem.For large-scale problems, this can still be prohibitively expen-sive. Even if we replace the relatively slow nonlinear CG bya Newton or Gauss-Newton variant coupled with CG for thelinearized problem (Nocedal and Wright, 1999; Vogel, 1999)the cost of carrying out each such iteration remains very high

(Haber and Ascher, 2001b).Here, instead, we consider the constrained optimization

problem (4) directly. This allows balancing the accuracy of theiterationsfor solvingthe forward andthe inverse problems. Thework is a follow on to that presented in Haber et al. (2000b).It is important to note that this approach does not change thefinal results of the inversion but rather changes the way to getthere. The results of such inversion enjoys the same strengthsand suffer the same weaknesses as the usual unconstrained ap-proach, however, as shown in Haber and Ascher (2001b) it canbe carried out faster. For large 3D problems where computingsolutions may take days, this approach can be superior.

Let us form the Lagrangian

L(u, m, ) = 12Qu bobs2 + R(m) + T(A(m)u q),

where is the vector, or grid function of the same form andsize as u, of Lagrange multipliers. (We are abusing notationslightly byusingthe same symbols foru, m,and in thediscreteand the continuous cases. The meaning should be clear fromthe context.) The first-order necessary condition of optimalityfor problem 4 is that the gradient ofL vanish. This yields thenonlinear system of algebraic equations

L= A(m)u q= 0, (5a)

Lu= A(m) + Q(Qu bobs ) = 0, (5b)

Lm= R

m+ G(m, u) = 0, (5c)

where ()is t he adjoint operator (namely, the conjugate trans-pose of the argument matrix) and

G(m, u) = [A(m)u]m

.

The system 5 is solved by a Newton-type method. Within theouternonlineariteration,iterativemethodsare appliedto solvethe linearized problem approximately.

The system 5 is clearly a discretization of a system of bound-ary value partial differential equations (PDEs). The first PDEcorresponding to equation 5a is simply the forward problem


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1218 Haber et al.

(e.g., Maxwells equations in our present case). The secondPDE corresponding to equation 5b can be viewed as the ad-joint problem; that is, we can view it as an equation for theLagrange multiplier function

A(m) = Q(Qu bobs )where the right side involves the noise. This view is tightlyconnected to the adjoint method (Chavent, 1989).

Equation 5c can be viewed as discretizing a diffusion equa-tion with natural boundary conditions for the model of theform (Weickert, 1998),

(am) = G(m, u).Here, a could be a 3 3 matrix which in general depends onm. However, in this paper we restrict attention to a constant,diagonal-weight matrix (Ascher and Haber, 2003)

a= diag{1, 2, 3}.If we use the forward model to eliminate u, and then the

adjoint equation to eliminate , then the above diffusion equa-

tion is expressed solely in terms of the model m. This againexpresses the eliminate first, or unconstrained, approach.However, viewing equations 5 as discretizing the entire sys-tem of PDEs suggests an approach of simultaneously solvingfor all solution components. This is what we pursue here. It isimportant to note though, that for realistically small values ofthe regularization parameter , the PDEs are tightly coupled(Haber andAscher,2001b;Ascherand Haber, 2003). This leadsto complications in designing effective methods for solving thediscretized equations 5.

We emphasize that the formulation and discretization of theforward andthe inverse problemsshouldgenerategood, con-sistent forward, adjoint, and model equations in equations 5;otherwise, we may expect to have difficulties (Haber and As-cher, 2001b).

In the present article, we apply the methodology describedabove [which we developed earlier mainly in a simpler contextcorresponding to dc-resistivity problems (Haber and Ascher,2001b)] to Maxwells equations, both in time and in frequencydomains. In the nextsection,we discuss theformulation andthediscretization of theforward modeling in space andtime. Next,we reformulate the discrete inverse problem as a constrainedoptimization problem. After that, we discuss the solution ofthe systems which evolve from our formulation. Finally, wediscuss the optimization procedure. In all of our discussion, weconcentrate on what is new or different from our previous ex-positions, and only briefly describe that which is similar. Afterthe theory, we give some geophysical examples.

THE FORWARD PROBLEM

In this section, we present our forward problem andMaxwells equations, and discuss solution procedures suitablefor the parameter regimes of interest. Most, but not all of thepresent development follows our previous work (Haber et al.,2000a; Haber and Ascher, 2001a). The following subsection, inparticular, is completely new.

The time-dependent Maxwell equations can be written as

E + Ht

=0, (6a)

H E Et

=sr(t), (6b)

over a domain [0, tf], whereE andH are the electric andmagnetic fields, is the permeability,is the conductivity,is the permittivity, andsris a source. The equations are given

with some initial and boundary conditions which we discussnext. In the frequency domain, reusing the same symbols forE,H, andsr(the context should make this unambiguous), thesame equations in the spatial domainread

E H = 0 (7a) H ( )E = sr(), (7b)

where is some discrete frequency. The boundary conditionsused for our experiments over the entire boundary of the spa-tial domain, , are

n H = 0, (8)although other boundary conditions could be used.

To put the equations in frequency and time on equal footingwe discuss first the discretization in time. We then treat the

equations for both equations 6 and 7 in a similar way for thespatial discretization.

Discretization in time

In order to select a method for the time discretization ofequations 6, we first note that given typical earth parameters,over very short time scales(0107 s) Maxwells equations rep-resent wave phenomena, whereas over longer times the equa-tionstend to haveheavydissipation.Considering alsothe rangeof conductivities(ground-air),there aremany timescales in thesystem. Thus, the equations are very stiff (Ascher and Petzold,1998). If explicit methods are used, one must take extremelysmall steps in order to retain stability for such a problem. Wetherefore turn to implicit methods.

Common methods for very stiff equations are based onbackward differentiation formulas (BDF) or on collocation atRadau points (Hairer and Wanner, 1991; Ascher and Petzold,1998; Heling, 1998; Bastian, 1999; Turek,1999). These methodshavethe propertyof stiffdecay, whereas conservative, centeredmethods such as midpoint or trapezoidal do not. Thus, the lat-ter methods exhibit oscillatory behavior in time unless the ini-tial, transient layer of the solution is resolved, whereas BDFor Radau methods strongly attenuate high frequencies of theerrorand so, even ifthe initial transientlayer isskipped(by tak-ing a time step which is larger than the transient layers width),an accurate solution may be obtained away from this layer.

In our case, resolving the initial, transient time layer wherethe fields change rapidly is not necessary for the inverse prob-lem because the measurements are typically taken at latertimes. Therefore, it is natural to use a BDF- or Radau-typemethod for the solution of the forward problem, as this will re-sultin moreefficientcomputingprocedures. Thenext issueis tochoose a specific member of these families of time integrationmethods.

Our choice is the simplest, lowest order member of boththese families of stiff integrators, namely, the backward Eulermethod. We justify this by noting that the backward Eulermethod is only first-order accurate. Most geophysical systems,however, produce sources which are merely continuous in


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time globally. This implies that the electric and magnetic fieldsare only once differentiable, and no advantage is obtained byusing a higher order discretization in time regions of lowersmoothness.

Even more interesting is the effect of the discretization ofthe forward problem (which generatesA) on the adjoint equa-

tion 5b. It is easy to show that using BDF for the forward prob-lem yields a forward differentiation formula for the adjointproblem, but with a terminal end condition rather than an ini-tial one. Thus, the adjoint equation is essentially integratedbackwards in time. If the adjoint equation had a smooth right-hand side,then these methods would generate a faithful,stablediscretization for the Lagrange multipliers. However, unfortu-nately, this is not the case. Note that the right-hand side forthe adjoint equation 5b is the noise which is further sampled atdiscrete points in time. (The operatorQ in equation 2 involvesa combination of-functions in time.) As such, the right-handside of the adjoint equation is not smooth, and the Lagrangemultipliers are therefore generally discontinuous, althoughbounded, at the observation times. Whereas the value of re-covering accurate Lagrange multipliers can be (and has been)

debated, there is hardly any incentive here to use a more com-plicated (and more expensive) method than backward Euler.

The above discussion may suggest using backward Euler atthe data points, and then switching to a more accurate BDF orRadau method between the data time locations. In our case,we consider measurements at most discretization times and,therefore, we simply use backward Euler for the discretizationof the problem. This leads to the following system of equationssemi-discretizing equations 6 and 8 over a time step [tn1, tn].Lettingn = (tn tn1)1, the equations become

En + nHn = nHn1 sH in , (9a) Hn (+ n)En = snr nEn1 sE in , (9b)

n Hn = 0 on . (9c)

The superscripts in equations 9 denote the time level, withsolution quantities atnbeing unknown while those atn 1 areknown.

This system requires initial conditions for bothE and H. Ifwe have a source which is zero before the initial simulationtime, then we set E 0 = H0 = 0. However, if the source is as-sumed static before time zero, then E0 =0, and we needto calculate [0, H0] by solving the electro- and magnetostaticproblems. This would yield a consistent initialization (Hairerand Wanner, 1991; Ascher and Petzold, 1998). Here, we haveused the method proposed in Haber (2000) for the solution ofthe static problems.

Reformulation

The semi-discrete system9 andthe systemin frequency7 ap-

parently have the same form. Indeed, they can both be writtenas

E + H = sH in , H (+ )E = sE in ,

n H = 0 on ,where = in the frequency domain, and = (tn tn1)1for the time domain. Let us denote = + . As discussed in

Haberet al.(2000a)andHaberandAscher (2001a), this form isnot favorablefor iterative solvers, especially when|| is small(for example, in the air, with a large time step, or with a lowfrequency). We therefore reformulated the problem prior todiscretizing it further such that it is more amenable to applyingstandard iterative solvers.

A Helmholtz decomposition with Coulomb gauge is applied,decoupling the curl operator into its active and null subspaces:

E = A + , A = 0 in ,A n = 0 on .

After adding a stabilization term and differentiating (Haberand Ascher, 2001a), this leads to the diagonally dominant sys-tem

1 A 1 A + (A + ) = s, (10a) ( (A + )) = s, (10b)

in, subject to

n A = 0, n A = 0, (10c)n = 0, (10d)

on the boundary . This system is discretized next.

Discretization in space and solution of the discrete system

Following Haber and Ascher (2001a) and Haber et al.(2000a), we use a finite-volume approach for the discretizationof equations10 on an orthogonal, staggeredgrid.We choosetodiscretize A on cell faces andat cell centers. This is closely re-latedto mixed-type finite-element methods (Brezzi and Fortin,1991; Bossavit,1998;Haber,2000). Note that themodified con-ductivity is averaged harmonically at cell faces, whereas thepermeability is averaged arithmetically at edges (Haber andAscher, 2001a).

We write the fully discretized system asL + M Mhh M h Mh

A

=

s

h s

, (11)

whereh , h , andh are matrices arising from the dis-cretization of the corresponding continuous operators, Marises from the operator (), and L is the discretization ofthe operator (1 ) (1 ).

This linear system can be solved using standard iterativemethods (Saad, 1996) and effective preconditioners can be de-signed for it (Haber et al., 2000a; Aruliah and Ascher, 2002).Briefly, for small enough , the system is dominated by its di-agonal blocks and, therefore, a good preconditioner can be

obtained by using an approximation of the matrixL 0

0 h Mh

. (12)

It is possible to use one multigrid cycle (Aruliah and Ascher,2002) or an Incomplete LU factorization (ILU) (Saad, 1996;Haber et al., 2000a; Haber and Ascher, 2001a) of matrix 12 toobtain an effective preconditioner.


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1220 Haber et al.

For larger (i.e., higher frequency or smaller time step), theabove preconditioner may not suffice and a block precondi-tioner (or its approximation) of

L + M Mh0

h

M

h

(13)

is better. Here we have used t he ILU for the approximation ofthe main blocks of matrix 13. It must be understood, though,that our entire discretization is not suitable for high-frequencyparameter regimes or where wave phenomena dominate (inparticular, recall we are skipping the entire initial, transienttime layer in one step), and we do not propose to compensatefor such inadequacy by manipulating preconditioners.

Formulating the forward problem

In many applications of our techniques we are concernedwith multiple sources and with multiple frequencies or timesteps. For solving the inverse problem it is useful to formulatethe forward problem in a uniform way.

Formulation for the frequency domain.Assume we havea multiple source/frequency experiment. As explained in theintroduction, we consider an optimization problem which in-volves the forward problem as a system of equality constraints.We therefore formulate thekth experiment in real arithmetic,defining

Ak=

L 0 kM kMhh M h Mh 0 0kM kMh L 0

0 0 h M h Mh

,

qk=

0

h

sr

ksr0

.

The entire frequency system is then

A(m)u=

A1(m)

A2(m)

. . .

. . .

As (m)

u1

u2...

...

us

=

q1

q2

...

...

qs

= q. (14)

Formulation for the time domain.We treat the time do-mainsimilarly to themultifrequency domain experiment. How-ever, the system is no longer quite block diagonal, and the

blocks get somewhat larger. The source term for the time do-main problem is

ns=nTh Hn1 + 2n En1 nsnr= nTh Hn1 + 2n (An1 + n1) nsnr

and, from Maxwells equations, we have

1n M1h An+ Hn Hn1= 0.

Thus, we can write the problem in a block bidiagonal struc-ture forA, , Has

A1(m)

B2 A2(m)

B3 A3(m)

. . . . . .

Bs As(m)

u1

u2...

...

us

=

q1

q2...

...

qs

, (15)

where

un=

An

n

Hn

,

An(m)=

L + nM nMh 0h M h Mh 0

1n M1 h 0 I

,

qn=

nsnrh snr

0

,

Bn=

2n 2nh nTh h n h nh 0

0 0 I

.

In the case of multiple sources, we obtain a block diagonalsystemwhere each block hasthe same structureas equation 15.Note that only the diagonal blocks in equation 15 depend onthe conductivity. Also, once we have an efficient solver for oneblock Ak(m) as described in the previous subsections, solvingthe forward problems 15 or 14 is straightforward (with the costof solution increasing by a factor ofs compared to the cost ofsolving for one block).

To test ourforward solvers we made comparisons with otherexisting codes. For results, see Appendix A.


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THE DISCRETE INVERSE PROBLEM

Having defined the discrete forward problem 5a, we nextform the discrete constrained optimization problem 4. Forthis, we discretize the regularization operator R on the samegrid. Denote the result of discretizing the weighted gradient

(1mx , 2my , 3mz )T

byW m(i.e.,Wis a weighted differ-ence matrix). Then,

R(m) = mTWTW m. (16)

The matrixWTWis a discretization of the weighted Laplacianwith natural boundary conditions. The parameters j , whichare hidden by the notation in equation 16, are chosen usingour a-priori information and are incorporated into the matrix.With this discretization we next form the nonlinear system 5.

In order to calculate the matrixG in equation 5c, we needto differentiate the forward modeling matrix times a vectorwith respect tom. This may look complicated at first; however,note that the matrix Ain both frequency and time is made ofblocks and each block depends on m only through the matrixM. Therefore, if we know how to calculate

N(m, v) = [M(m)v]m

,

then we can differentiate any product involving M. Forexample,

m[h Mh w] = h N(m, h w).

To calculate this derivative, we recall that M operates onthe discrete A or h, which arecell-face variables. The matrixis diagonal, and each of its elements has the form

M(ii ) = 2

11 + 12

1,

where 1and 2are the values of at the two sides of the face

of the cell. From this form, it is clear that M1

is linear with

respect to 1 and, therefore, the matrix

Nr(v) =M1 v

[ 1]

is independent of and depends only on the vector fieldv at

each cell. Using this observation and the chain rule, we caneasily calculateN:

N(m, v)= [M(m)v]m

=

(M(m))11v

m

=

(M(m))11v

[ 1][ 1]

m

=[M1 ]2 Nr(v) diag(exp(m))

= M2 Nr(v) diag(exp(m)).We cannow proceed andsolvethe discrete nonlinearsystem

of equations5 bysomevariant ofNewtons method. Because ofthe chosen form ofR, we may use the Gauss-Newton method.

Thus, in a typical iteration for givenu, , andm, we differenti-ate equations 5 with respect to these variables and, droppingsecond-order information, obtain the following linear systemof equations for the correctionsu, , andm:

A 0 G

QT Q AT 0

0 GT WTW

u

m

=

L

Lu

Lm

(17)

(see, e.g., Dennis and Schnabel, 1996; Nocedal and Wright,1999; Haber et al., 2000b).

SOLUTION OF THE LINEAR SYSTEM

Thepermuted Karush-Kuhn-Tucker (KKT) system 17is very

large with possibly millions of unknowns. It is strongly coupled

Figure 1. Experimental setting for frequency-domain inversion. The cuboid indicates the earthvolume in which the inversion is carried out. Data are acquired within the dotted rectangle atthe surface.


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1222 Haber et al.

(because it discretizes a strongly coupled PDE system) andis indefinite. Therefore, special iterative linear algebra tech-niques are needed in order to solve it. As usual, the crux ofthe matter is designing an efficient preconditioner. One familyof preconditioners for the solution of this system is obtainedby approximating the block LU decomposition of its inverse

(Haber and Ascher, 2001b; Biros and Ghattas, 2004). For acareful development of this method, we point the reader toHaber and Ascher (2001b). Here is a synopsis.

It is easy to show that t he system 17 can be decomposed into

A 0 G

QT Q AT 0

0 GT WTW

1

=

A1 0 A1G H1red0 AT AT QTJ H1red0 0 H1red

I 0 0

QT Q A1 I 0JT Q A1 GTAT I

, (18)

where J= Q A1G is the sensitivity matrix and Hred = JTJ+WTWis the reduced Hessian.

One need not actually calculateA1,AT, and J, but rathergenerate an approximation in order to precondition the sys-

tem 17. If we have a matrix B such that, for any appropriatevectorv , Bv approximates A1v, and a matrix Mredsuch thatMredvapproximatesH

1redv, then we can calculate the action of

the preconditionerMfor the KKT system 3.17 as follows. Fora vectorv, obtainingx = Mv, we writevand x in their compo-nents formv

=[vT , v

Tu, v

Tm ]

T andx

=[x Tu,x

T,x

Tm ]

T, and obtain

the following preconditioning algorithm:

1) w1= B v.2) w2= B T(vuQ T Qw1).3) w3= vm GTw2.4) xm= Mredw3.5) xu= w1 B Gx m .6) x= B T(vuQ T Qx u ).

One option to generateB is to use the preconditioners dis-cussed earlier; that is, for the frequency domain we use thematrixB 1 = A, which has the structure

A1(m)

A2(m)

. . .

. . .

As(m)

, (19)

where Ak(m)1 represents the preconditioner of the forward

problems 12 or 13.

Figure 2. The accurate Ex ,Hydata for 512 Hz are shown in the top row, the error contaminateddata are shown in the middle row, and the bottom row displays the data predicted from theinverted model.


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For the time-domain formulation 15, we use the approxima-tion

A1(m)

B2 A2(m)

B3

A3(m)

. . . . . .

Bs As (m)

. (20)

As in Vogel (1999) and Haber and Ascher (2001a), we usethe sparse matrix WTWin order to approximate the reducedHessian. Suchan approximation workswell for large enough .

These preconditioners are stationary (that is, they do notchange with the iteration) and, therefore, standard iterativetechniques canbe used. In Haber andAscher (2001b), we usedthe symmetric QMR (Freund and Jarre, 1996).

A second option for the preconditioner, which is especiallysuitedfor parallel implementation in thetime domain, is to usean inexact solver with a very rough tolerance (here, we used102) in order to approximate A1 and AT. That is, we useanother Krylov method (BICGSTAB) with the preconditioner19 forboth frequencyand time.In this case,the preconditioneris decoupled for both frequency and time. The problem withthis type of preconditioner is that it is not stationary. That is,

Table 1. Optimization path for the frequency domain inver-sion. The number of iterations to solve the KKT system islisted in column 2. Column 3 indicates the relative accuracy towhich the data constraints are solved. Column 4 tabulates therelative gradient to assess progress in solving the optimizationproblem. At the stationary point, the gradient is zero.

Nonlinear KKT Relativeit eration iterat ion Au q/q gradient

= 100 Final misfit = 0.061 4 310

2

2 101

2 4 2104 3 1023 3 2106 5 104

= 1 Final misfit = 0.031 8 1106 3 1032 6 8107 9 104

Table 2. Optimization path for the time-domain inversion.The number of iterations to solve the KKT system is listed incolumn 2. Column 3 indicates the relative accuracy to whichthe data constraints are solved. Column 4 tabulates the rela-tive gradient to assess progress in solving the optimizationproblem.

Nonlinear KKT Relativeit eration iterat ion Au q/q gradient

=1

101 Final misfit=

0.1

1 2 3103 1 1022 3 2104 4 1033 2 7106 1 1034 2 9107 3 104

= 1 102 Final misfit = 0.041 7 4106 2 1032 5 6107 7 104

= 1 103 Final misfit = 0.021 8 2106 3 1032 7 8107 9 104

the matricesBwhich approximateA1 change at eachiterationbecause they are an inexact solution of the forward problemusing a nonstationary iterative technique. As such, this kind ofpreconditioner cannot be used in a straightforward manner initerative Krylov space methods, and we turn to a special classof Krylov methods known as flexible methods, which allows

the change of the preconditioner at each iteration with thecost of extra storage. This family includes the flexible GMRES(FGMRES), FGCR (Saad, 1996), and FQMR. Here, we haveused a symmetric version of the FQMR and the FGCR forthe solution of the system. For implementation issues of thesealgorithms, the reader is referred to Saad (1996) and Freundand Jarre (1996).

OPTIMIZATION ISSUES

In this brief section, we return to the optimization problem.Within each nonlinear iteration, the KKT system 17 is solvedinaccurately and, thus, the sequential quadratic programming(SQP) algorithm is not strictly followed (Nocedal and Wright,1999). Therefore, care must be exercised so that at the end

of the process we achieve both optimality, (i.e., have a roughsolution to the optimization problem) and feasibility(i.e., solveMaxwells equations sufficiently well).

In order to achieve this goal, we usetwo safeguards.First,weuse the method of secondary correction. Thus, after each inex-act Newton step, we apply additional iterations to the solution

Figure 3. Three slices through the recovered 3D conductivitymodel obtained by inverting synthetic CSAMT data.

Figure 4. Borehole geometry for the time-domain problem.The dashed line at the surface is the loop source. All threecomponentsof the magneticfield areacquiredin the boreholes.The sphere has a radius of 15 m and its center is at 25-m depth.


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of the forward problem to reduce the residual. Beginning withthe updatedm and current value ofu , we solve A(m)u = q toreduce the misfit by a further order of magnitude. This has theeffectthat the computed solution convergestowards feasibility(solving Maxwells equations) faster than it converges towardsoptimality. We use this property in our convergence criteria

noting that, as in many inverse problems, we can take the op-timization goal less seriously than we take the constraints.Thus, we can terminate the optimization process at a relativelylarge tolerance (for example, 103) while fitting the constraintto a much smaller tolerance (say, 106).

After each such iteration, we test a decrease in a merit func-tion which is a combination of the optimality and feasibilitycriteria:

1= Qu bobs2 + R(m) + 1A(m)u q1.This merit function was suggested in Nocedal and Wright(1999), andwe have used it successfullyfor a simple1D inverseproblem (Haber et al., 2000b). The parameter 1is chosen asin Haber et al. (2000b). If the merit function decreases, thenthe step is accepted; however, if it does not, then we use a line

search on the updates of bothu and m .Unfortunately, there is no theory that guarantees decrease

in themerit functionfor an inexact solution of theKKT system.We therefore add a last safety mechanism. In the case wherethe linesearchfails, we turnto an unconstrainedGauss-Newtoniteration, which is outlined in Appendix B. This updates m. Thefields uand the Lagrangemultipliers are upgraded accordingto procedureN4 in (Haber et al., 2000b).

Another basic issue is the selection of the regularization pa-rameter. Here, we use the discrepancy principle; that is, weaim for a certain target misfit. To hit this target misfit, we usecontinuation in the regularization parameter. Thus, we startwith a guess which is obviously largerthan the true regularization parameterand solve the optimization problem. If

is large enough, then such a solution isachieved in 12 steps. We then decreasethe regularization parameter and solvethe problem again starting from the pre-viously obtained solution. To guaranteethat the first regularization parameter islarge enough, we use the estimate

0= 100Q BG v2/Wv2,

where v is a random vector and B isour approximation toA1. This selectionguarantees that, at the initial step, themodel objective function dominates theoptimization problem.

EXAMPLES

Inversion of frequency domain data

As a first example, we invert syntheticdata from a grounded source. The trans-mitterand receiver geometryis thesameas in an actual controlled source au-dio magnetotelluric (CSAMT) field sur-vey, but the conductivity model is sim-

plified compared to the true earth. Figure 1 shows the sur-vey geometry and the 3D model. The transmitter is a 1-kmgrounded wire that is a few kilometers west and north of thesurvey area. Within the survey area are 11 east-west lines witha line spacing of 100 m. On each line are 28 stations at in-tervals of 50 m. Five components (Ex ,Ey ,Hx ,Hy ,Hz ) repre-

sented as real and imaginary parts at three frequencies (16,64, and 512 Hz) result in 3080 data points. We have used thecode in Haber et al. (2000a) to generate the data. To simu-late realistic noise, we added Gaussian noise that was 2% inthe amplitude and 2 in phase. Two representative field com-ponents at frequency 512 Hz are shown in the top portionof Figure 2.

The 3D volume for inversion (3350 3000 2000 m) wasdiscretized into 64 50 30 cells. The transmitter lies consid-erably outside this domain. To handle this, we have assumedthat the fields at the edges of the model volume are equal tothe primary fields in a homogeneous earth. We used 1D code(Routh and Oldenburg, 2001) to compute these fields. The in-version began with a homogeneous half-space that was equalto the true background conductivity. After five iterations, the

final misfit was 3.2%. Three slices of the recovered model areshown in Figure 3. The resistive and two conductive targetsare reasonably well recovered. The predicted data at 512 Hz(displayed in Figure 2) show good agreement with the truedata.

The convergence results for the experiment arepresented inTable 1. The table shows the regularization parameter andthe misfit which we got from solving the optimization problemwith that . For every , we record the number of nonlineariterations, the number of iterations needed to solve the KKTmatrix using the FGCR, the PDE residualA(m)u q/q,and the relative gradient. For this example, the starting value

Figure 5. Observed (Obs) and predicted (Pred) magnetic field data for all receivers inthe borehole survey.


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of the regularization parameter was = 100. Three nonlin-ear iterations were required to achieve an adequate solutionto the optimization problem, and the final misfit was 6.0%.Within each nonlinear iteration, only three or four iterationswere required to solve the large KKT system. The regulariza-tion parameter was then reduced to

=1.0, and the process

continued.We have done various comparisons of the all-at-once ap-

proach to the unconstrained approach. Although it is hardto make an exact comparison due to the many options ofparameters within the optimization process (for example, towhat tolerance should the forward problem be solved forin the unconstrained approach? to what tolerance shouldwe calculate derivatives and sensitivity-vector products?),our code was roughly two times faster when using the all-at-once approach compared with the usual unconstrainedapproach.

Inversion of time-domain data

We consider the case of a square loop with dimensions of

50 50 m located just above the earths surface. The trans-mitter current is a step-off at time zero, and responses aremeasured in 18 logarithmically spaced times between 104 and101 s. The earth model is a conductive sphere (=0.1 S/m,radius 15 m) buried in a uniform half-space (0.01 S/m), and

Figure 6. Result of inversion of time-domain data. The toppanel is a horizontal slice of the recovered conductivity at adepth of 25 m corresponding to the center of the sphere. Thetrue conductivity is shown in the bottom panel.

three components of the magnetic field are acquired at 20depths in each of four boreholes that surround the conductor(Figure 4).

Weuseagridof643 cells in space. The grid is uniform aroundthe loop area and stretched logarithmically at the boundary.For the discretization in time, we used 32 time steps, equally

spaced on a log-grid from 107 to 101 s.The inverse problem is performed on a smaller grid:

40 40 32 grid in space with the same grid in time. The in-version begins with the uniform half-space equal to the truebackground conductivity. The convergence results for the ex-periment are presented in Table 2. Overall, the data werefit to an average of about 2%, and a plot of the observedand predicted data from a representative station is shown inFigure 5. A cross-section through the inverted model is shownin Figure 6.

CONCLUSIONS

We have shown how time- and frequency-domain electro-magnetic data can be inverted with a procedure that simul-

taneously recovers the electrical conductivity model and thecorresponding fields. The forward modeling equations are in-cluded as constraints, but these are not satisfied until the op-timization is complete (that is, a stationary solution of a La-grangian has been found). Thus, in this procedure, it is notnecessaryto solvethe forward problem exactly at intermediateiterationsand, effectively, the forward problem is solved in tan-domwith theinverse problem. This haspotential for a solutionto be reached more quickly than in traditional unconstrainedoptimization approaches that are formulated to minimize afunction of the conductivity only. The all-at-once method-ology generates a large matrix that needs to be inverted,but the numerical example shows that such computations aretractable.

ACKNOWLEDGMENTSThis work was supported by an NSERC IOR grant and

an industry consortium Inversion and Modelling of Ap-plied Geophysical Electromagnetic data (IMAGE) project.Participating companies are Newmont Gold Company, Fal-conbridge, Placer Dome, Anglo American, INCO Explo-ration & Technical Services, MIM, Cominco Exploration,AGIP, Muskox Minerals, Billiton, and Kennecott ExplorationCompany.

APPENDIX A

TESTING THE FORWARD SOLUTION

Frequency domain

In order to verify the numerical solutions of the forwardproblem, we compared our codes with results from existingcodes. We do not attempt to describe these comparisons indetailhere as they arenot thefocus of this paper. However, forthe frequencydomain,we havemade comparisons with the MTcode in Madden andMackie (1989), theintegralequationcode(SYSEM) (Xiong, 1992) using both loop and wire sources, 1Dsolutions for electric sources obtained from the codes (Routhand Oldenburg, 2001), and 1D solutions for magnetic sources


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obtainedfrom thecodes in Farquharsonand Oldenburg(1993).In all cases, the results were acceptably close, and differenceswere most likely due to gridding issues and interpolation offields to the same locations. As an explicit test, we present thefollowing.

We generate the electromagnetic responses due to a con-

ductive block in a uniform host at a set of 31 frequencies.These responses are converted to the time domain using a dig-ital filter (Christensen, 1990). The comparison is made withtime domain fields generated from the integral equation codeSYSEM (Xiong, 1992). Those fields were also generated inthe frequency domain and transformed to time. The fact thatour converted frequency results match those from SYSEM isvalidation of our frequency modeling code. The advantage ofpresenting results in the time domain is that the data plot issimple and the example can be compared with the direct com-putation in the time domain presented in the next section ofthe appendix.

The source is a 1 1 km loop that is 500 m from the edgeof the block. The conductivity of the block is 1 S/m, and the

half-space is 0.005 S/m. The geometry

is shown in Figure A-1. The verticalcomponent of the magnetic field, gen-erated from a step-off current, is plot-ted at a sequence of times ranging from20 s to 200 ms. Figure A-1 shows themodeland magnetic fieldvaluesalong aneast-west traverse over the conductiveblock. The results agree well with thoseobtained from the integral equationcode.

Time domain

We do not have access to other di-rect solvers for the time-domain prob-

lem; thus,for verificationof ourcode, wehave used codes that compute responsesin the frequency domain, which we thenconvert totime. Here,we usetheintegralequation codeSYSEM (Xiong, 1992) fora 3D example andresponses from a loopover a layered space obtained from thecodes in Farquharson and Oldenburg(1993) for our 1D example.

1D conductive and permeable

earth.As a first example, we computethe vertical component of the magneticfield due to a step-off current in a loopsource on the surface of a conductiveand magnetic permeablehalf-space. Theconductivity is 0.01 S/m, and the mag-netic susceptibility 1.0 SI. In Figure A-2,we show the vertical magnetic fieldresponses at the center of the loopfor the two cases of a magnetic andnonmagnetic earth. Our time domainresults are compared to those generated

from a frequency-domain 1D algorithm (Farquharson andOldenburg, 1993) in which theconversion to time hasbeencar-ried out by using a digital filter (Christensen, 1990). The agree-ment is good, and we regard this as a solid test for our time do-main code being able to handle both conductive and magneticunits.

3D conductive earth.In a second example, we gener-ate the time-domain responses due to a conductive blockburied in a homogeneous half-space. The geometry is thesame as that used to test the frequency-domain algorithm.The source is a 1 1 km loop that is 500 m from theedge of the block. The vertical component of the magneticfield, generated from a step-off current, is plotted at a se-quence of times ranging from 20 s to 200 ms. Figure A-3shows the magnetic-field values along an east-west traverseover the conductive block. The results agree well with thoseobtained from the integral equation code SYSEM (Xiong,1992).

Figure A-1. Validation of frequency-domain forward modeling. The conductivitymodel is shown in the upper figure. The dark square at the surface is the source loop.Data are acquired along the solid line shown above the buried prism. Frequency-domaindata at 31 frequencies aregenerated andthen convertedto time.The earliesttime channel 20 s is shown by circles, and the latest time channel at 200 ms isindicated by crosses.


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APPENDIX B

UNCONSTRAINED OPTIMIZATION FORPARAMETER ESTIMATION IN MAXWELLS

EQUATIONS IN THREE DIMENSIONS

In this appendix, we briefly discuss the solution of electro-magnetic inverse problems using an unconstrained, inexactGauss-Newton formulation.

As explained in the text, this approach is complementaryto our constrained approach, and it serves a few purposes.

Figure A-2. Validation of time-domain forward modeling: acomparison between our 3D code and responses from a 1Dcode. Solid lines represent the results from the 3D code, sym-bols refer to the 1D code. The upper curve corresponds to ahalf-space with unit susceptibility, the lowercurve correspondsto a half-space with zero susceptibility.

Figure A-3. A comparison between our 3Dtime-domain code and SYSEM over a 3Dmodel. Data are acquired along the solidline shown in Figure A-1. The earliest timechannel 20 s is shown by circles and thelatest time channel at 200 ms is indicated bycrosses.

First, it is a straightforward procedure to implement, and it al-lows us to examine methods for noise estimation, data weight-ing, model weighting, and other practical aspects of the in-version procedure without addressing the more involved nu-merical issues of the constrained approach. Having decidedupon these components of the inversion, the final large in-

version is carried out using the constrained methodology.The second reason for implementing the unconstrained ap-proach is that there is no proof of convergence for the con-strained approach, and it is possible that the constrained stepfails. In this case, we may resort to taking an unconstrainedstep in the constrained inversion algorithm. The last reasonfor investigating the unconstrained methodology is that thecode serves as a base for comparison with the constrainedapproach.

In theunconstrainedapproach theconstraints areeliminatedand problem 4 is transformed into an unconstrained optimiza-tion problem of the form

minm1

2Q A(m)1q b2 + 1

2W(m mre f2. (B-1)

Differentiatingproblem B-1 with respect tom, weobtainthenonlinear gradient system

g(m)= GTAT QT(Q A1q b)+ WTW(m mre f) = 0. (B-2)

Note that in order to evaluate the gradient, one must solve theforward and the adjoint problems.

Also, comparing equation B-2 with the usual unconstrainedformulation, we see that the sensitivitymatrix can be expressedas

J(m) = Q A1G, (B-3)


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This is a key observation in the solution of the inverse problemusing the unconstrained approach because, upon applying aniterative method for the linear system, we need not calculatethe large and dense sensitivity matrix but merely evaluate itsproduct with vectors in order to carry out a Gauss-Newtoniteration (Haber et al., 2000b; Haber and Ascher, 2001b). The

Gauss-Newton iteration can be written as

(JTJ+ WTW)m= g(m). (B-4)In order to solve system B-4, we use a preconditioned conju-gate gradient method, and thus only products of the form Jvand JTw are required. Using decomposition B-3, this can beachieved involving only sparse matrix operations.

As a preconditioner forour systemwe useWTW. This worksreasonably well as long as is large enough. In order to fur-thersave computational time, we use an inexact Gauss-Newtonformulation (Kelley, 1999) and solve equation B-4 to a roughtolerance (typically 101 to 102), which usually requires onlyfew conjugate gradient iterations.

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Inversion of 3D Electromagnetic Data in Frequency and Time Domain

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