a moving finite element method for magneto tell uric modeling

8/2/2019 A Moving Finite Element Method for Magneto Tell Uric Modeling

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432 Physics ofthe Earth andPlanetary Interiors, 53 (1989) 432443Elsevier SciencePublishers B.V., Amsterdam Printed in The Netherlands

A moving finite element method for magnetotelluric modeling

Bryan J. Travis

Earth and Space Sciences Division, LosA lamosNational Laboratory, LosAlamos, NM87545 (U.S.A.)

Alan D. Chave *

Institute ofGeophysics and Planetary Physics, University ofCalifornia at San Diego, La Jolla, CA 92093 (USA.)

(Received January 29, 1987; revision acceptedAugust 20, 1987)

Travis, B.J. and Chave, A.D., 1989. A moving finite element method for magnetotelluric modeling. Phys. Earth Planet.

Inter., 53: 432443.

Finite element simulation ofthe electromagnetic fields in complex geological media is commonly used in interpretmg

field data. In this paper, recent, major improvements to finite element methodology are outlined. The implementationofa moving finite element technique, in which the mesh nodes are allowed to move adaptively to achieve an accuratesolution, is described. Efficient matrix solutions based on incomplete factorization and matrix ordering are also

discussed; these offer order ofmagnitude reductions in memory requirements and increases in execution speed. Finally,the advantages ofadapting finite element codes to modem supercomputers are emphasized. These topics are illustrated

by formulating a moving finite element forward model for the two-dimensional magnetotelluric problem. The result isvalidated by comparison with standard control models. While preliminary in nature, the result does indicate that

substantial improvements to geophysical modeling can be achieved through the use ofmodem approaches.

1. Introduction years for the numerical simulation of EM prob-lems. Examples include the use of FE to model

The experimental state-of-the-art in electro- magnetotelluric (MT) fields in 2-D structures

magnetic geophysics is currently in a condition of (Coggon, 1971; Lee and Morrison, 1985; Rodi,rapid evolution. This is owing in large part to the 1976; Wannamaker et a!., 1987), and standarddigital revolution, which has made reliable, porta- computer codes for this purpose are becomingble instrumentation widely available and increased available (e.g., Wannamaker et al., 1985). How-the utility of advanced data-processing methods, ever, FE modeling with either more complex (i.e.,even in the field. However, the capability to model controlled) sources or 3-D structures has notand interpret data in terms of electrical or geo- proven as satisfactory to date (e.g., Pridmore etlogic structure has lagged behind. This poses the al., 1981), and other approaches, especially in-single largest obstacle to the further development tegral equation methods, are in more general use.and wider application of electromagnetic princi- The advantages of the FE method include flexibil-pies in geophysics. ity and a capability to handle complex structures

The finite element (FE) method, among several in a straightforward manner. The principal disad-

others, has received increasing attention in recent vantage is the need for relatively extensive com-puting resources, especially storage. This has made

* Present address: AT&T Bell Laboratories, 600 Mountain FE somewhat difficult to implement on smallAve., Murray Hill, NJ 07974, U.S.A. computers.


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Major improvements to 2-D and 3-D electro- where the symbols have their usual meaning. Themagnetic modeling codes can be made by incorpo- role of electric charge in (1)(3) is often confused.

rating recent advances in FE methodology and By removing the displacement current to give (3),numerical algorithms which take full advantage of phenomena with time-scales shorter than that of

the structure inherent in FE matrices. First, the EM diffusion are filtered out, and charge appears

moving finite element (MFE) method incorporates to travel instantaneously. Electric charge is stilladaptively moving nodes into the FE equations, present, usually in association with conductivitysubstantially increasing the accuracy that is oh- gradients or discontinuities, and its fields are quitetamable with a given size mesh. Second, while important. This is especially true in 2-D and 3-Dmost MT modeling codes use Gaussian elimina- structures, and charge accounts for many of thetion or LU decomposition to solve the resulting important differences between simple 1-D andmatrix equations, considerably more efficient higher-dimensional realizations.methods based on incomplete factorization (~ehie For a 2-D structure in which a and p~areand Forsyth, 1984) are now available. This ap- independent of the ~ co-ordinate, it is well knownproach results in an order of magnitude reduction that the electromagnetic fields separate into twoin memory requirements, as well as a large in- independent modes if the sources are also free ofcrease in execution speed. Finally, while these ~ dependence. The first of these is called the

tools will yield markedly better performance on E-polarization or transverse electric (TE) mode,conventional, scalar computers, the vector archi- and is completely described by the field compo-

tecture of modern Class VI or VII supercomputers nents E~,B~,and B~.The second case is calledcan yield additional speed improvements of up to the B-polarization or transverse magnetic (TM)a factor of several hundred with properly designed mode, and involves only the B~,E~,and E~fields.algorithms. Following Rodi (1976), and assuming e depen-

In this paper, the formulation of a 2-D MFE dence for all variables, the vector Maxwell equa-forward code for MT modeling is described. The tions (1)(3) reduce to the generic scalar setemphasis is placed both on the MFE formalism a i+~ j+yv = 0 (4and its implementation using modern numericaltechniques. The code is validated by comparison ~V qJ= 0 (5)

to standard control models proposed by Weaver et a ~ +~ = 6al. (1985, 1986). This result is considered as anintermediate one by the authors, and can be im- which may be combined to give the second-orderproved by using more sophisticated basis func- partial differential equation

tions, triangular elements, and similar enhance-ments. Ultimately, MFE will be used both in the a~(a~v/~)+a~(a~v/~)yV= 0 (7)development of a regularized inversion method for where the variables for the two modes are2-D MT data and as a step toward a fully 3-D TM TEmodeling code. V Br/fL E~

J E~ B~/1s

2. Governing equations I E~ Br/fts~ a

The physics of electromagnetic induction is de- y a

scribed by the Maxwell equations in the quasi-static or pre-Maxwell limit, in which the magnetic At horizontal contacts between media of differenteffect of displacement current is neglected conductivity or permeability, the quantities V,

v . B = 0 (1) ~V/i~, anda~v

are continuous, while at vertical

v x E +a B = 0 contacts, V. a~v,and are continuous. EM1 ~ ~ induction in a 2-D medium is governed by twov x (B/jz) aE= 0 (3) independent scalar equations for the principal


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fields E~and B~.The auxiliary fields E~,E2, B~, mesh FE, where the node spacing must be chosen

and B 2 are obtained from (5) and (6). Note also ahead of time to yield the required accuracythat the TE mode involves an electric field which throughout the solution region. More modern pre-is always parallel to changes in the conductivity, dictorcorrector algorithms are far more efficient,and hence involves no electric charge. This is varying the step size and order locally to achieve

manifest in the absence of terms involving the an accurate solution without using excessive com-conductivity gradient in the TE form of (7). By puter time. This is analogous to MFE, and cancontrast, boundary charge is important in de- handle more complicated equations in an auto-termining the behavior of the TM mode electric matic fashion with a substantial reduction in com-field. The eqns. (4)(7) may be simplified for most puter run-time. The MFE method was introducedreal Earth problems by taking fi as constant and by Miller (1981) and Miller and Miller (1981).equal to the free space value; this will be assumed Further developments and some illustrations of itsfor all of the results in this paper. operation are contained in Gelinas et al. (1981)

and Dukowicz (1984).

The representation used in the present MFE

3. MFE Solution formulation is a standard linear finite elementtype. The computational mesh is divided into rect-

The resolution with which (7) can be solved angular cells with horizontal or vertical sides. The

depends strongly on the numerical methodology field values are specified at the centers of thethat is applied. For a given simulation, there are zones. A linear element is defined for eachtwo basic ways to enhance performance. First, quadrant of a zone. Denoting the indices of themore accuracy can be achieved with a limited zone by i, j, where i is the z-index and jis thenumber of mesh nodes. Second, the computer y-index, yields the basic functionusage and memory requirements can be minimized v= V + a

2 (y y 1 2) b 2 (z z 2) (8)by taking advantage of the structure of the FE ~ / /equations; this is discussed in the next section. where the coefficients areSince most FE formulations use piecewise con- ~ J j_i)tinuous linear basis functions between fixed nodes, a

1 =one way to improve the accuracy is to use a higher ~ +TJ~J~Yi_i

order (e.g., quadratic) approximation in the discre- 2 2~~,(J1,tization. This invariably leads to more com- a,j= /plicated FE equations and a larger bandwidth for ~ +~1,,j+i Y~+i

the FE matrices, and usually results in substan- 2~~1(J...11tially longer computer run-times. A better ap- b,~= ~ +proach involves an adaptive mesh, in which the t1nodes are allowed to move such that their loca- 2~~1(V1tions actually become a part of the solution. The b~1= ~ +

nodes will concentrate in regions of strong gradi- ~~ Z1 ~121~J Z~~1ents where they are needed for good resolution. with a,~= a~1,a~= a~,b~= b~,b~= b~,~y1An attractive feature of this method is that low- y~~ and ~z1 z~1.The cell quadrantorder basis functions may be used without a sig- indices n of 1 4 correspond to the upper left,nificant loss in accuracy. An analogy can be made upper right, lower left, and lower right corners;to two common methods for the numerical solu- the co-ordinates of the lower right cell corner arelion of ordinary differential equations. The stan- (y~,z); and the cell centers are located at (y 1 1/2~

dard RungeKutta method is not adaptive, and z11/2). In (8) and at horizontal cell interfaces, V

the order and step size must be chosen a priori to and B2V/~are continuous, while at vertical inter-yield an adequate solution in the most complex face centers, V and 8~V/i,are continuous, corn-part of solution space. This is similar to fixed mensurate with the discussion of the last section.


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Integration of (7) over a computational cell, solution of (7) (with boundary conditions) pro-with application of the divergence theorem to vides a minimum for the Lagrangian formtransform the volume integral to a surface type,leads to L=fdyfdzH(V, a~V,a 2 V )

jdZ[a~V/~I;+j~dy[a2V/~]~, = ~Re{fdyfdz[(ayV)2/n + (a2v)

2/~+ yV2]

_f~f 1 dzdyyV=0 (9) 2Z,.1 ~ . . + ~Jdy(_1)(atV2_2$hV)

The use of (8) in (9) yields the matrix equation 4

Av = f (10) + ~ Jdz( 1)(atV2 2$1V)} (12)

For this 2-D problem, A is a five-banded matrixwhose elements are This means that (7) is the Euler equation obtained

from the variation of L with respect to V that1 2 L~Y~ minimizes (12). For the FE formulation, the dis-

A~1= ~ + ~ ~ crete form of (12) becomes

2~z~ LFE=~Re{~ ~ ~ [(a~)2+(b~)2Im,1_ ~ y~1 ~i,J )~j i=1 j=1 J

2~z tV+ V

2~ J( 21 A + A tij ~ ~1 2 ~jf ij J Yj

~ y~+~m1 y 1

A5=a 2~y

1~ ~m-~-i,~~zi+i + Ti11 ~z1

1 / \2 / \2 2+ (a ) +(a ) (Ay)

A~31 (A~1+A~j+A~1+A~)y1, ~z1 ~y1 12 ~1 ~~

(11) + .~.[(b,~)2+ (b~j)2J(~zj)2

and

~z)2 ~ ~a,1 = 1 -y,~ii,~(

= X~y1 (13)8 The first variation of (13) with respect to the

The components of A must be modified ap- will give the FE set (10)(11). To get an MFE

propriately at the edges of the mesh to meet the formulation, additional equations for y 1 and z1boundary conditions. The vector f also incorpo- will be obtained in a similar manner.rates boundary values. There are several complications that can arise

The matrix eqn. (10) is similar to the usual when the FE mesh is allowed to move adaptively.form obtained in any fixed mesh finite difference The most prominent of these is the obvious re-(FD) or FE numerical approximation. An adap- quirement that two nodes not be allowed to oc-

tive mesh approach requires additional equations cupy the same location or pass each other. Tofor the movement of (y~,z.). From the calculus of eliminate these problems, note that arbitrary func-variations (Clegg, 1968), it can be shown that the tions in y and z can be added to the Lagrangian


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in (13) without affecting the variation with respect obtained by solving (15) alone, will not usuallyto V which gives (10). This results from the form satisfy (16) and (17), so the set is linearized and aof the Euler equations for the minimization of Newton iterative solution is found. This typically(12). The particular form of these functions can be requires three to five iterations. In practice, somechosen to control the movement of the nodes y~ simplifications can be made in the numerical solu-

and z,. Following Miller (1981), (13) is modified tion of(15)(17). Rather than solving all three setsto give of equations simultaneously, (15) can be solvedfor the current values of {J/

1, y~,z.}, then (16)N c 1 2 and (17) can be treated separately, neglecting lin-= LFE+ ~ (~ earization of the nonlinear terms Q 1 and Q 2, to

M 2 modify the node locations. This means that only+ ~ ~y1 -~~-) (14) tridiagonal equations are solved for y and z at

each iteration. This approximate form works wellfor the sample problems discussed later.

where , c1, and c2 are constants. The additional The FE solution of (4)(7) requires the specifi-entries in (14) are penalty or regularization ones cation of a on a finite size grid z1 ~ z ~ z2, y~ywhich ensure a stable solution by controlling the y~.The conductive Earth is assumed to lie inallowed node spacing. The terms involving are the region 0 z z2, while the half space z


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should be forced to satisfy as many of these typesof constraints and continuity conditions as possi- 0 ~ oble to ensure an accurate simulation. This gener- 0 a ~ally requires FE representations higher than lin- a ~th step of ~ear. In practice, a decision is made, explicitly or Gaussian

implicitly, to favor some of these constraints over 0 etiminationothers to avoid the added complexity of banded matrixhigher-order approximations. Neglect of any ofthese conditions can allow spurious numerical a \~\ ~modes to appear, although they may be of noharm or may be damped by other physical _______processes included in the model equations. For the 0 th \\\o ~\\2-D MT problem, the condition (1) and the vector a 0 steP at ~identity V V V= 0 lead to the requirement that8~V=8~V,a condition that holds for (sufficiently banded matrixdifferentiable) continuous variables but may not Fig. 1. Sketch illustrating the differences between ordinaryfor a given FE formulation. Gaussian elimination and an ILU method on a banded matrix.

The solid diagonal lines indicate the matrix bands, which

contain nonzero matrix elements, while the large zeroes indi-

cate regions where the matrix contains no entry. The top part4. Efficient matrix solutions ofthe figure shows that Gaussian elimination fills most oftheempty locations with an intermediate result during computa-

A given FE scheme can also be improved tion. The bottom part ofthe figure indicates that ILU addre-through minimization of the usage of computer sses many fewer such elements in the matrix.time and memory by taking advantage of the

inherent structure of the FE equations. Both FDand FE discretizations lead to matrix equations of entry value originally was zero. With ILU, fewerthe form Av = b, where A is sparse (i.e., only a locations are filled, or a particular entry is re-com-small fraction of its elements are nonzero), and puted fewer times, leading both to faster execution(usually) symmetric and banded. The computa- and to a smaller memory storage requirement.tional focus is then placed on optimizing a factori- Some new variants have been developed to im-zation of A to reduce the cost of solving the prove on ILU schemes. For example, variable ILUequations (Behie and Forsyth, 1984; Zyvoloski, (VILU) changes the number of Gaussian elimina-

1986). A considerable improvement over conven- tion steps per iteration in different parts of thetional Gaussian elimination or LU decomposition matrix depending on local conditions and thecan be achieved. For background on standard structure of the matrix problem. Behie and For-numerical methods for solving matrix problems, syth (1984) and Zyvoloski (1986) discussed ILUsee Golub and Van Loan (1983). and VILU for solving FE and FD matrices. De-

The class of matrix methods known as incom- tails of the ILU or VILU scheme depend criticallyplete factorization (ILU) has become quite popu- on the form of the equations being solved and on

lar for FD and FE calculations in many disci- the discretization used to parameterize the prob-plines. ILU is an iterative procedure in which only lem. A real advantage of this approach is itsone or (at most) a few Gaussian elimination steps relative insensitivity to the dimensionality of a

are taken per iteration. Figure 1 illustrates the simulation because of the partial fill-in. Thisdifference between the ordinary methods and ILU. should become very significant in 3-D applica-In Gaussian elimination on a banded matrix, most tions.

of the entries between the original bands will be Another enhancement in solving matrix equa-occupied at some time in the solution process by a tions arises from changing the order in which thecomputed, intermediate result even though the nodes are processed. Red/black partitioning re-


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19 20 21 22 23 24 7 1 1 15 19 22 2/. ai r

10 ii 1 2 ~2 1 6 20 23 yfl-a 2 ya

1 2 3 4 5 6 1 3 6 1 0 14 18 Fig. 3. The geometry of the control model of Weaver et al.(1985, 1986). The conductive structure consists ofthree regions

natural ordering red / block ordering ofdifferent conductivity and uniform thickness, d, separated

to ) (b) at the interfaces y = a and y =a. The conducting media areoverlain by nonconducting air and underlain by a perfect

Fig. 2. A comparison of a typical ordenng for finite elementconductor.

matnces with red/black ordenng: (a) shows a standard order-ing, in which the node addresses are set up in a straightforward

manner (b) shows the red/black ordering, in which the node red/black squared, which involves two orderingsaddresses have been redefined to facilitate reduction of the , .

matrix, yielding a reduction in the size of the problem byup to a factor of three. Figure 2 compares a

typical ordering for a FD or FE discretizationorders the nodes into red types, which have no red with a red/black ordering. The increased corn-nearest neighbors, and black types, which con- puter time required to sort the matrix must bestitute the remainder. The red terms are placed at balanced against the savings in time from factori-

the top of the matrix, and their associated equa- zation; this will yield a substantial improvementtions can be removed by simple matrix transfor- for large problems.mations. Only the black node linear system has to The use of acceleration methods can also in-be solved by ILU. This results in a reduction in crease the speed of matrix computations. At eachthe size of the problem by up to a factor of two, iteration in the ILU method, an estimate can bedepending on the discretization used. Wolfe and made of the direction and relative magnitude thatZyvoloski (1987) discuss a new partitioning, the next correction to each dependent variable

TMMODE T=300S TEMODE T=300Sxx,nuruerical. real xxx numedual. realflu uumericxt.~mag xxx uumericul.imxg

0 400

100

300

2 00

2003 00

-~ ~

400 100 ~ll0cn=_~0.

0

50 40 3 0 20 10 0 10 20 30 40 50 50 40 30 20 10 0 10 20 30 40 50

(a) Y(KM) (b) Y(}O.4)

Fig. 4. Comparison ofthe analytic solutions ofWeaver et al. (1985, 1986) for the (a) TM and (b) TE mode response functions with

the MFE numerical results at a period of300 s at the Earths surface. The solid and dashed lines are the real and imaginary parts of

the analytic E/B response function parameterized by horizontal distance, y; the interfaces separating the conductive regions are at 10 and 10 km and are delineated by the heavy vertical lines. The discrete symbols show the numerical results, with Xs for the real

and circles for the imaginary parts.


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should have to minimize the current matrix equa- ated to each. This performs much like an assemblytion squared residuals. For symmetric matrices, line, so that independent parts of the hardwarethe procedure is a conjugate gradient scheme with operate simultaneously on different parts of the

a carefully chosen step-size (Kershaw, 1978). For data and pass the result on to the next station. Aasymmetric matrices, the matrix equation residual result can be delivered at each machine clock cycle

is still minimized, but an additional orthogonaliza- using pipelining. These principles allow dramatiction step is required which transforms the matrix improvements in machine performance to beto its principal axes before determining the direc- achieved, with peak execution rates in excess oftion and magnitude of the optimal increment for 100 million floating point operations per second.the dependent variable vector. A popular proce- Matrix calculations are particularly well-suited todure for asymmetric matrices is the ORTHOMIN vector machines, and the execution time can bemethod of Vinsome (1976). This also controls the nearly independent of the size of the problem.condition number of the matrix. Dongarra et al. (1984) discuss the principles of

Finally, vectorization on a modern super- vector pipeline computers and their use for linearcomputer can dramatically improve computa- algebra computations.

tional efficiency. A Cray-class supercomputer de-rives its enhanced performance both by the use ofsignificantly faster hardware and by implementing 5. Validation of the MFE codea number of new principles. The first of these is

the vector instruction, in which a single machine The current version of the magnetotelluric MFEinstruction allows data vectors (as opposed to code, named MTAM2D, has been validated bysingle elements) to be processed. The Cray allows comparison to the standard analytic control mod-vectors of 64 words to be treated in this way. els for the TM and TE modes developed by WeaverCray-class hardware also implements pipelining, et al. (1985, 1986). Figure 3 displays the geometryin which a single operation is split into smaller of their test case. A conductive region of constantpieces and separate parts of the machine are alloc- thickness, d, overlies a perfect conductor, and

TM MODE T = 30,000S TE MODE T =30,000S

0 ~oo_~-.x. ~ 12

5 8

4

10 __0____0___~___0___0___0___c_

0 ~ wx03C*010 III.15 ~~0 4

20 i 8 I

50 40 30 20 10 0 10 20 30 40 50 50 40 3 0 20 10 0 10 20 30 40 50

(a) Y(KM) (hi Y(KM)

Fig. 5. Comparison ofthe analytic solutions ofWeaver et al. (1985, 1986) for the (a) TM and (b) TE mode response functions withthe MFE numerical results at a period of30000 s at the Earths surface. The solid and dashed lines are the real and imaginary parts

ofthe analytic E/B response function parameterized by horizontal distance, y; the interfaces separating the conductive regions areat 10 and 10 km and are delineated by the heavy vertical lines. The discrete symbols show the numerical results, with Xs for the

real and circles for the imaginary parts.


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consists of three distinct zones with different con- GRID FOR TE MODE, F.M., T300Sductivities ai(y < a), a2(a y a), and a3(y 100> a). For this comparison, the variables have

been chosen as follows

a=lOkm 50 -

d=SOkm 25

a1=0.1Sm1 0

a2=1.0Sm

1 j : :a

3 = 0.5 S m~ 30 110 90 70 50 30 1 0 10 30 50 70(a) Y(KM)

Solutions for two penods Tof 300 and 30000 s for

both modes were computed. 100 -Figures 4 and 5 compare the analytic and -

numerical results for the ratio E/B as a function 50

of y at the Earths surface z = 0 for the TM and 25TE modes at the two periods. The size of the

0

25

GRID FO R TM MODE, F.M., T=300S5 0 -

130 1 1 0 9 0 7 0 5 0 3 0 1 0 10 30 50 700 iiHI__________________________________________ (b) Y(KM)

1 0 _____________________________________________ Fig. 7. The (a) initial and (b) final meshes for the MFE___________________________________________ solution ofFig. 4b for the TE mode at 300 s period. The initial

_____________________________________________ mesh was uniform, with 32 zones in the vertical (12 in air and___________________________________________ 2 0 within the model Earth), and 49 zones in the horizontal (25,

30 ________________________________________________ 12, and 12 zones in the three conductive regions). The final

________________________________________________ mesh has been adaptively modified to concentrate the nodes

40 __________________________________________________ where the electromagnetic field is changing most rapidly, espe-

50 cially the boundaries between conductive regions and the 130 1 10 9 0 7 0 5 0 30 1 0 10 30 50 70 Earthatmosphere contact at z 0.

(a) Y(KM)

initial FE mesh (i.e., the number of nodes) was

~ adjusted empirically to the minimum required toachieve acceptable accuracy for each case. The

-20 -~-~ ~M~fi initial mesh for the TM mode 300 s numerical-30 - solution consisted of 30 uniformly sized zones in

the vertical direction and 60 zones in the horizon-

tal30, 15, and 15 uniformly spaced zones in thethree conductive regions, respectively. At 30000 s

-130 -110 -90 -70 -50 -30 -10 10 30 50 70 period, the initial mesh contained only 20 zones

(b( y~ vertically and 45 horizontally20, 10, and 15

Fig. 6. The (a) initial and (b) final meshes for the MFE zones in the three conductive regions. For the TEsolution ofFig. 4a for the TM mode at 300 5 period. The initial case at the shorter period, the vertical zoning wasmesh was uniform, with 30 zones in the vertical and 60 zones modified to include 12 sections in air (z 0), while the horizontalregions). The final mesh has been adaptively modified to . .

zomng included 49 umformly spaced regions25concentrate the nodeswhere the electromagnetic field is chang-ing most rapidly, especially the boundaries between conductive 12, and 12 zones in the three conductive regions.regions and the Earthatmosphere contact at ~ = 0. At 30 00 0 s, the TE mesh was reduced to 10 and


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20 vertical zones in air and Earth, and 45 uniform GRID FOR TE MODE, F.M., T=30,000Shorizontal zones20, 10 and 15 sections in the

100three regions. The side boundanes were located aty = 130 and y = 70 km for all of the models,sufficiently far from the conductive transitions to S o

not affect the solution, and the horizontal deriva- ~tives of the fields were taken to vanish at thosepoints (periodic boundary conditions). At the bot-tom of the model for the TM mode, B~B~was set 25to zero, while B~= 1 at the surface z = 0. As for -sothe TE mode, E~was set to zero at the bottom of -130 -1)0 90 -70 -50 -30 -10 10 30 50 70the model, and a~E~was set to iw~tat z = z

1. The ~ I ~regularization parameters in (14) were set as = 100i0

5, c1 = c 2 = 5 x 102 for the TM mode and - - - - - =

= 10~,c 1 = c 2 = 5 x i0~for the TE mode. The - - - - -50 - -

quantity being minimized in (14) takes on consid- ~. - - - -erably different values for the TE and TM mode ~ - -cases, varying over a range of several hundred. 0

Accordingly, the values of the penalty term con- 25stants , c 1 , and c 2 must be changed to keep the50- r 1 -T

1 3 0 1 10 9 0 7 0 5 0 3 0 10 10 30 50 70

GRID FOR TM M O D E, F.M., T=30,000S ) b) Y (1 (M )

Fig. 9. The (a) initial and (b) final meshes for the MFEo solution of Fig. Sb for the TE mode at 30000 s period. The

initial mesh was uniform, with 30 zones in the vertical (10 and

-10 20 in air and Earth) and 45 zones in the horizontal (20, 10 and

15 in the three regions).

~ -30corresponding terms in (14) small relative to the

-40 LFE term. Variations of a factor of 10 about the

-50 - specified values do not affect the result apprecia- 130 110 90 7 0 5 0 30 10 1 0 30 50 70 bly. Computer run times for all calculations were

(~) y~


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TM MODE T =300S ~ one, the analytic solutions involve only variations

1 ______ ox, xumrro,nt.,x,x~ in conductivity in the 9 direction, and the changesin conductivity between the regions are not large.In addition, this first version of the code, in the

05 interest of simplicity, required a specified numberof nodes in each region. When this restriction isrelaxed to simply require specification of the num-

0 ~ ~ ber of nodes for the entire mesh, greater mesh

deformation will be observed. Another factor isthe nature of the numerical mesh. Because the

05 mesh uses rectangular elementswith square cornersin the current version of the MFE code, all points

on a vertical or horizontal grid line are con-

strained to move simultaneously. The mesh is-50 -40 -30 -20 -10 0 10 20 30 40 50 actually responding to the average gradient of the

Y ( 1 ( 1 . 0 solution along a y o r z grid line. This restrictionFig. 10. The magnetic field component B~,for the TM mode at results in less node movement than is desirable.300 s period and a depth of15 km inside the Earth. Note the

The rectangular elements used here will be re-points at which the fields are changing most rapidly and .

compare with Fig. 4 placed with tnangular types in the next version ofthe code, improving its ability to handle non-horizontal interfaces. This will also allow the nodes

the zoning changes in an asynmietrical fashion. to move individually and improve the adaptabilityFor example, in the 300 s case in Fig. 4, the zones of the result. Nevertheless, the principle of MFE ishave shifted to the centre between y = 10 and well-illustrated by the examples in Figs. 4 and 5 .y =10 km. The reason for this becomes apparent The advantages of an adaptive mesh in mini-after inspection of Fig. 10, which shows the field mizing mesh design problems and maximizing thecomponent B~against the horizontal co-ordinate resolution for a given size mesh are obvious, al-y inside the model Earth at z = 15 km. That curve though careful design of the mesh using a fixedis fairly steep in the left part of the middle con- mesh model would probably give answers that areductive region, but flattens out on the right-hand just as good. Adaptive procedures should reducesection. Curvature is greatest around z=0. The the human time involved in solving forward prob-zoning will tend to reflect the structure of the lems, and will be of definite advantage for inver-

electromagnetic field component that is actually sion when the locations ofconductivity boundariessolved for, not the response function that is ob- may not be known a priori.tamed from the fields. This accounts for the ap-parent discrepancies in node locations betweenFigs. 6 9 and the shapes of the response function Referencescurves shown in Figs. 4 and 5. The adaptive Behie, GA. and Forsyth, PA., 1984. Incomplete factorizationmeshes are sometimes not as smooth as one would methods for fully implicit simulation of enhanced oil re-expect, as in Fig. 9. This is owing to choosing poor covery. SIAM J. Sci. Stat. Comp., 5: 543561.

Clegg,- J.C., 1968. Calculus of Variations. Oliver and Boyd,values for the constants in (14). If regulanzation Edinburgh.parameters are too small, the system will tend to Coggon, J.H., 1971. Electromagnetic and electrical modellingbecome ill-conditioned. Practice with the model is by the finite element method, Geophysics, 36: 132155.required to find the optimal range of regulari.za- Dongarra, J.J., Gustavson, F.G. and Karp, A., 1984. Imple-tion parameter values. menting linear algebra algorithms for dense matrices on a

vector pipeline machine. SIAM Rev., 26: 91112.In retrospect, the mesh changes obtamed for Dulcowicz, J.K., 1984. A simplified adaptive mesh technique

the test cases are significant but not dramatic, derived from the moving finite element method, J. Comp.There are several factors acting to cause this. For Phys., 56: 324342.


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