11 use ~42. response of a three-dimensional body · department of geology and geophysics,...

r- hys J Int (1989) 98 233-242 Grop

l

I A finite-difference algorithm for the transient electromagnetic 11 USe and ~42

work data response of a three-dimensional body If Artificial

Jopie 1 Adhidjaja and Gerald W Hohmann lull seism

1 PT Geoservices (Ltd) Il Sukawangi 5 Bandung 40153 Indonesia 2 Department of Geology and Geophysics University of Utah Salt Lake City UT 84112 USA

m-Wesley N 1985

Qutomatic Accepted 1989 February L Received 1989 January 26 in original form 1988 January 28 Report

Itof seismic

genee ap- SUMMARY

I We attempted to develop a direct time-domain finite-difference solution for the electromagshy~dge-based netic response of a 3-D model The algorithm is an extension of our 2-D modelling technique i~ Signal which uses the Du Fort-Frankel finite-difference scheme However the vector nature of the

-1208eds field makes the 3-D problem much more complicated than its 2-D counterpart and alalbos J t

I supercomputer is required for computations Unlike the 2-D case where we solve for the Artificial i

f electric field the solution is formulated in terms of secondary magnetic field to avoid dealing ~ i- with a discontinuous normal component of electric field However that difficulty is replaced

ly R D f with a problem involving the gradient of conductivity which is discontinuous at interfaces integrating

Suro U We experimented with both smoothing the conductivity variation so that the gradient is well defined and integrating the gradient terms which results in a tangential current density

e Systems contrast Our limited experiments indicate that the magnetic field step response is computationally more stable than the impulse response because it is a smoother function

mic signal with smaller dynamic range However the step response takes longer to compute because its ~ull seism I source which involves the primary fields in the earth requires very accurate numerical

nputational t integration Due to computer time and memory limitations on the supercomputer available to Consultant t

t

us we were not able to develop an accurate numerical solution We were able to carry out I only a few tests on small models for which results were not in good agreement with values l model of hitcctures computed using a 3-D integral equation solution

Ierstanding Key words Du Fort-Frankel finite difference step response 3-D modelling transient l 483-501 electromagnetic method (TEM) rnic Press

lore A 1 (SlAP case

INTRODUCTION calculating the response of one or a few small bodies

In many cases a numerical solution for a 3-D model is required for interpreting transient electromagnetic (TEM) data in terms of realistic Earth structures Although a 2-D model can be useful its application is limited to special cases and its response can be very different from its 3-D counterpart (Adhidjaja amp Hohmann 1988) However because there are conductivity boundaries in all directions and the unknown field is a vector quantity the 3-D problem is much more difficult

Methods that have been used for formulating numerical SOlutions are the differential equation (finite element and finite difference) integral equation and hybrid techniques Differential equations are easier to implement and are more suitable for modelling complex geology but they require more computer storage In contrast integral-equation formulations involve more difficult mathematics but the unknown fields only need to be found in anomalous regions Thus integral-equation solutions are less expensive for

(Hohmann 1983) Hybrid techniques arc attractive because they take the best features from both the differential- and the integral-equation methods but they have their own difficulties (Lee Pridmore amp Morrison 1981 Petrick 1984 Best et al 1985) A complete review of those methods and assessments of some of the solutions are given by Hohmann (1983)

Most EM solutions in the geophysical literature have been formulated in the frequency domain Integral-equation solutions were developed for example by Hohmann (1975) Weidelt (1975) Raiche (1974) and Das amp Verma (1982) Differential equations were solved by Lines amp Jones (1973) with a finite-difference method and by Reddy Rankin amp Phillips (1977) with a finite-element method applied to geomagnetic-field and magnetotelluric problems respecshytively which typically involve measurements at low frequencies Zhdanov et al (1982) presented an algorithm for computing the responses of 2-D and 3-D models with a finite-difference method but showed no 3-D results Pridmore et ai (1981) used a finite-clement method for

bull Formerly Department of Geology and Geophysics University of Utah

direct current (DC) and controlled-source applied to a 3-D model in a mining problem

EM methods

l 233

I 234 J I Adhidjaja

Two approaches have been used to obtain time-domain solutions The indirect method uses Fourier transformation of frequency-domain results (Lamontagne 1975~ Tripp 1982 Newman Hohmann amp Anderson 1986) One advantage of the indirect method is that it can be based on an existing and well-tested frequency-domain computer program The direct method requires solving an initial value problem directly in the time domain which can be done via a time-stepping method

The direct method may provide more insight into the transient (TEM) process within the whole spectrum from very early to very late times However the early-time solution must be calculated accurately to insure correct late-time results For a differential-equation method the computation can be very time consuming San Filipo amp Hohmann (1985) developed an efficient direct time-domain solution based on an integral equation formulation but their solution was for a simple prism in an otherwise homogeneous half-space

Yee (1966) presented a finite-difference scheme for solving the 3-D transient electromagnetic problem it was based on time-stepping Maxwells equations and solving for both electric and magnetic fields His scheme has been applied successfully to compute scattering by a perfect conductor Success seems to be due to an accurate representation of the boundary conditions of fields at interfaces which is achieved by carefully placing grid points The scheme utilizes a staggered grid the electric and magnetic fields at a node are defined at alternate time steps Later this scheme was adapted and used successfully for solving various 3-D electromagnetic scattering problems in free-space (Holland 1977 Kunz amp Lee 1978ab) The scheme has also been tested for a geophysical problem by Bergeal (1982) He solved for a 3-D body in a whole-space but his results computed for only 10 I-tS were not satisfactory The main difficulty involved propagating the electric field via the displacement current which we normally neglect in geophysical problems Including the displacement current requires a very small time step which is impractical in computing a solution to the later times of interest in geophysics

Our goal was to develop a direct time-domain solution based on a differential equation for simulating a complex 3-D model with realistic conductivity contrasts We formulated the solution in terms of the secondary field based on experience with a successful algorithm for the 2-D problem (Oristaglio amp Hohmann 1984 Adhidjaja Hohshymann amp Oristaglio 1985) In this paper we present the theoretical formulation the finite-difference scheme the limited numerical results we were able to compute and discussions of the problems encountered

THEORETICAL FORMULATION

Following Hohmann (1983) we begin with Maxwells equations

v x e(r t) = --to ab(r t) _ amP(r t) (1)at ~o at Yx her t) = ae(r t) (2)

with magnetic permeability fJ = P-o everywhere and with

Tdisplacement current neglected Here mP is the d - Ipole

moment per volume of an Impressed source In au cas bull - ~e We

sum vertical magnetic dipoles to create a horizO1tl I source a cop

As in the 2-D case we formulate the problem in term S of

primary and secondary fields through the decompositions

e=eP+es

and

h = hP + h

where superscript s denotes secondary and p denotes primary

Primary fields are defined as the fields in the earth if the earth were homogeneous They satisfy the equations

JbP Jnt v x e = - fJo at - Il0-----at (3)

and

YXhP=aeP (4)

where 0 is the conductivity of the half-space The secondary fields which are caused by in

homogeneities in the half-space satisfy

ahS V x e = -fJ0-at (5)

and

v x bS = oe + aacP (6)

where o = (0 - 0) is the anomalous conductivity Solving for b S and setting V h = 0 assuming no magnetic

permeability contrasts we obtain an inhomogeneous vector diffusion equation for the secondary magnetic field

ah ahP

V-h - a - = IJ a shyII at at10 10 a

+ av() x (7 x h) - av(~a) x eP (7) O 0

where the superscript s has been dropped for clarity The field is a vector quantity whose components in general are coupled which makes the 3-D equation much more complicated than the 2-D equation Inside a homogeneous medium the vector components become decoupled because the terms containing the gradient of conductivity vanish

FINITE-DIFFERENCE APPROXIMATION

For time stepping we use the Du Fort-Frankel finite-difference method The method was derived by Du Fort and Frankel (1953) for solving a homogeneous diffusion equation in one dimension It possesses the desirable properties of being unconditionally stable and explicit lt uses a staggered grid which when combined with a leap-frog method in time forms an efficient scheme for time-stepping computations

The Du Fort-Fran kel technique was used successfullyfO~ computing responses of 2-D conductors in geophySiCa environments by Oristaglio amp Hohmann (1984) and laterby Adhidjaja et al (1985) Oristaglio amp Hohmann made an

~

the dipole iur case we ontal loop

in terms of ositions

p denotes

arth if the ons

(3)

(4)

by in

(5)

(6)

y

) magnetic ous vector d

e (7)

arit y The meral are rch more iogencous j because an ish

ON

f -Frankel ed by Du f s diffusion

desirable xplicit It I leap-frog e-stepping

ssfully for eophysical rd later by mad e an t

IIi

alysis of a stability condition for th e method applied to ~ h d h f 2D problems w en compute wit UOI orm or non-uniform grid spacings Here we exte~ d their 2-D finite-difference formulation to a 3-D formulation

First we consider th e equation in the host where a = a so that 0 = 0 and V(l a) = O Hence equation (7) reduces to a homogeneous diffusion equation and in the Cartesian coordinate system each compon ent of the magnetic field satisfies a scalar equation

ah VZh - -loa- = o (8) at

To derive the finite-difference approxim ation we ext end the method used by Ori sta glio amp Hohmann ( 1984) Integrating equation (8) over a volume and applying the divergence theorem to the first te rm we get

(Vh bullda =fl oa ahdu (9) J s at v

For illustration we derive a finite-difference expression for one compon ent of the vector field at node i i k and at time t = n +1 Fig 1 sho ws one of the typical finite-difference molecules including the distribution of its physical properties for the gen eral case In the case we are considering all of the conductivities are th e same but we retain the general notation so th at the result s can be applied at boundaries later

First evaluate equation (9) over a sma ll volume ABCDEFGH as shown in Fig 1 to obtain

If gty dz - JJ dy dz + If h dx dz BFGC A EllD DCGH Y

- If h dx dz + JJ dx dy - JJ dx dy ABFE Y EFGH A B CD

IfJ -loa ~gtx dy dz A HC O E FGII

Then using central differences to approximat e de rivatives

r- x h k-I t

I gt --

(7 I 1 ii k + Imiddot

h- - I ~ ~ _- - middot - _ _ i

j 1 I rk

)( I J (gt I C 1 I 7

)1

y Z

B (7

1 2 A Z k

h hI i -i k X L i + Ik

1 2 6 Z k 1

0 i - I k ~2 6y t bull r jYj+ I i k + J+II ll

1 2 ly j l Hlt

1 2 Cx 1 2 L X G 01+1 i + I k e lI 1 + 1 I 1

vot u rne o f in tegrat io n

h j i k+ 1

Figure 1 A typical finite-difference computational molecule showing odd (0) and even (x ) nodes conductivity distr ibution and volume used in computing the field at the centre node

Transient electromagnetic response 235

and using supe rscr ipt n to denote the nth time we obtain

~ ( h(j i + 1 k ) - h(j i k)

4 1x i t-l

X (1Yj1zk + 1Yj+ t1 zk + 1Yj1Zk+1 + 1Yj +t1zk+ t )

h(j i k) - h(j i - I k)

Llxi

X (1Yj 1zk + 1Yj +l1Zk + 1Yj1Zk+1+ 1Y]+l Llzk+l)

h(j + 1 i k ) - h(j i k ) +

1Yj +l

X (Llxi1Zk + Llxi +1Llzk + Llxi1Zk+ 1+ Llxi +1Llzk+ 1)

h(j i k) - h(j -1 i k)

1Yj

X (Llx iLlzk + LlxI +11Zk + Llxi1Zk+ 1+ Llxi+ILlzk+ l)

h(j i k + 1) - h(j i k ) +

1Zk+ 1

x (Llxi1Yj + Llx+ 1 LlYj + LlxLlY]+1 + Llxi+ 1LlYj+ l)

h(j i k) - hn(j i k - 1)

1zk

x (LlxiLlYj + Llx+lLlYj + LlxjLlY+1 + Llx +1LlYj +1))

= flo ( h+ I (j i k ) - h -I(j i k ) (a Llx Ll Llz 8 2Llt ] t k bull Y] k

+ aji+l kLlx + 1 LlYj1Zk + aj +I kLlx iLlYj+1Llzk

+ aj +l i+I kLlx i +ILlYj +ILlzk + ajik + ILlxiLlYjLlzk+l

+ aj i+l k+ILlx+ILlYjLl zk +1+ aj + li k+1Llx LlYj +ILlzk+1

+ aj+ U +I k+lLlx+ ILlYj +l1Zk+d)

Applying the Du Fort-Frankel method by substituting

h+ l ( i k ) + h- I( i k)h(j i k ) j J

2

we obtain after some algebra

h+I(j i k) = (c1h(j i-I k) + c2 h (j i + 1 k)

+ c3h(j - 1 i k) + c4 h (j + 1 i k )

+ csh(j i k - 1) + c6 h(j i k + 1)

+ crh-I(j i k raquo 3 (10)

with

c1 = 2 Llx(Llx + Llx +I ) Cz = 2Llx i +I (Llxi + Llx+ I)

C3 = 21Yj (LlYj + LlYj +l) C4 = 2LlYj+l (1Yj + Llx + I)

Cs = 21Zk( 1Zk + 1Zk+I ) C6 = 2 1zk+1(Llzk + Llzk+ I)

a = ( flOajik2Llt - 1LlxiLlx+ 1 - 1 LlyjLlYj - 1 - 1LlzkLlzk+ I)

f3 = (flOaj k 21t + 1 LlxLlx i+ 1+ 1 LlyjLlYj +l + 1 1zk1Zk+ I)

o =8

aivl 8

Jlk LJ-- Vr = 2 V 1= 1 1=1v-

V = LlxLlyjLlzk V2 = Llx + I1yjLl zk

v 3 = LlxiLlYj+l1Zkgt V4 = Llxi +1LlYj +1Llzk

Vs = Llx1Yj1Zk+1 v 6 == Llxi+ 1LlYj1Zk+ 1

v7 = Llx 1Yj+I 1Zk+1 Vs = Llx+ 1LlYj + I Llzk+ I

236 J 1 Adhidjaja r Next we consider the eq uation inside an inhomogen eity

where a is con stant but a a and G a O Thus equation (7) reduces to the inhomogeneous diffusion equation where the terms involving gradien ts of co nductivities van ish In thi s case all components of the vector are dc coupled also Each component satisfies an equa tion simila r to equation (8) but with a simple source which is associated with a primar y field

ah ah P

V2h

- 1oa a = f-loObull at middot (11)

A finite -d ifference expression for a component of the vector field at nod e i i k needs only a simple modi fication from the previous one namely

all P

hn+ 1( bull k ) = middot h n- I(middot i k ) - fJ a - (12) I I 0 a at

where hn-rl(j i k) is computed similarly to equation (10) with a(j i k ) = co = [a(j i k ) - a] 1p and ah PI at is the corresponding component of primar y field co mputed analytically at node j i k and time t =n

Finally we con sider the equation in a general region specifically at interfaces All three vector components ar e coupled and satisfy equation (7) If we solved equation (7) directly we would face a mathematical problem because we would like to compute result s for models with large conductivity contrasts (1 10 to 1 1000) For a finite shydifference grid with a block-wise conductivity vari ati on th e gradient of the conductivity becomes infinite at interfaces

Brewitt-Taylor amp Weaver (1976) suggested smoothing such conductivity varia tions by averaging to avoid de aling with a discontinu ou s function This approach is very appealing if it works because it would be easy to compute a model with an arbitra ry conductivity distribution

Following Brewitt-Taylor amp Weaver (1976) we solved equ atio n (7) using a st ra ightforward appro ach The conductivities are averaged everywhere and assumed to vary smoothly and terms invol ving derivatives of conductivities are approxima ted with central differences With thi s procedure a finite -difference expression for the x component of the magnetic field at node i i k is

hn + 1( bull k) h n + t ( bull bull k ) ah ~ x I I = x I I - 1oa shy

t

+ [C7 [hxCj i k + 1) - hAj i k - 1)]

- C8[hAj i + 1 k ) - hz(j i-I k)J

x [RU i k + 1) - R(j i k - 1)]

- C9 [h j i + 1 k) - hy(j i-I k )]

- ClO[hxC j + 1 i k ) - hA j -1 i k)J

x [R (j + 1 i k ) - R(j -1 i k)]

+ Cll[a(j i k + l )R (j i k + 1)

- o(j i k -l )RU i k -l )] e~]Ip (13)

where h +I(j i k ) is compute d similarly to equ at ion (10) with a a = (Oi k - a)1p H ere ah~1 at is the x component of the primary magnetic field and e~ is the y component of the prim ary electric field computed ana lytically at nod e j i k

and time 1= n T he ot he r va riables an given by

C7 = aik[( 1zk + 1zk-t-l)(Ll zk + Llk ~ ) J

C8 = C7J bulliJ [(Llx + Llxh- I)(Llzk + Llzk 1)1

C = abullJ [(Llx i -t- Llxi bull ) ( Lli + Lly+ i )1

C IO = 0Jid[(Lly + Lly d(Llgt + Lly bull ill ell = aiid( Llzk + Llzk 1- )

aaU+ I i k ) = aJ t l i 1- - a

R(j i + 1 k) = I I 0Jbulli -rIk

Finite-difference expression s for the o the r components are derived in a simi lar manner

In our expe rime nt th is schem e dive rged for a model With the transmitting loop not centred over the body which Was 20 times as conductive as the surro unding ea rth We believe the probl em was caused by the inaccur acy in approximating the gradien t of the conduc tivity In contrast to frequency dom ain computation time-domain com putat ion demands higher accur acy because a small erro r in the solution at ear ly times is prop agated and co mpo unded at later times

Alternatively we can avoid averaging the conductivity if we conside r two medi a of differe nt cond uctivit ies with a bounding surface and perfo rm the V(l l o) operation resultin g in a 2-D delta function (6) normal to the inte rface Since the delt a fu nctio n is non- vani shin g only at the interface the corresponding volume integra tions collapse into su rface integrat ions

L(V2h- 1oa ~~) du

rshy alt J (1=J 1na - dv + a - shyv at a 2

1(0 2 01

- a 2 - 2 )n x eP da ) deg 2 0 1

whe re superscrip t and subscript

I ) - n X (V xh) du aI

(14)

I and 2 indicate the condu ctivities of media I and 2 respective ly and n is the no rmal unit vector point ing from medium I to medium 2

However this eq ua tion still poses a probl em because the conductivity a and the tan gential component of current density (V x h)l are discont inuous at the interface To circum vent this pro blem we perform the surface integ rations afte r taking the limiting values of those terms by approaching the interface from left (me dium 1) and right (med ium 2) Th en afte r pe rfor ming the cross-products

2 3h) J alt 1 2V h middot- fJ oa ---- dv = -loa - dv + (V X h) da1v (

0 1 v al

-L(V xh) da -1uc da + La~ e da (15)

whe re superscript s indicate the medi um in which the terms are eval uat ed and subsc ripts t indicate tangenti al comshypone nts on the inter face In effect We have applied the cond ition of continuity of h and e at the interface Terms on the right -hand side of equa tion (15) can be thought of as sources for the diffusion eq uation they co nsist of a volume current (the vo lume integra l) and anomalous surface curre nts (surface integrals) The source s involving primary

I I

I iee invc

Ai The

I

L( I

t I

=i I

whe mec bye Stol

Fmiddot equ curr h n ~

x

whe but

C7 =

jgt

1middot2 j -

=

area e~ is at n for man

Tl gent cone

hj

Figul sect ii averc

_ _- - - - - - - - - - - - - - - - - __ - - - _

-1

fields can be computed analytically while the sources volving secondary fields must be computed numerically Il1As an example consider an interface normal to the z axis Then the equation for the x component hx is

L(fi h - loa a~x) du

ah f J (16) = Lloa -at l

dv + s (j - j~ ) da - cu - a~)e~ da

where j~ and j~ are the Y co mponents of current densities in medium 2 and 1 respectively which can be computed either

tents are bv directly performing the curl operator on h or by applying Slokes the orem

xlel with For illustration we derive a finite-difference expression for

hich was equation (16) Use of Stokes theorem to compute the

= believe current density gives a finite-difference expression for hx

ximating h~ 1( k) - JIx n + l ( ) I k ) - C7[( 2 )y1) - Pj) I bull - )y - OaCy (17) =quency

where h+I(j i k) is computed similarly to equation (12)

lution at demand

-but with a j k as the volume averaged conductivity and times ictivity if C7=2(Llx + Llxi +I)(1Yj + 1Yj+I)IPVn

s witl a j =[-hxj i k - l)(Llx + Ax i bull 1)+hzj i - 1 k)1z k peranon nterface ~r +h()1-1k)Llx+h()I+l k)Llx+1

at the 1gt -hzji+1k)1zJarea1

collapse middot2 [h ( k 1)( - - ) h ( 1 k ) ly= x ) I + lMi + lM +1 - z ) 1 + 1zk + 1

-h(j i + 1 k)Llx +1-hj i -1 k)Llx

+ hzj i-I k )1zk + I ] area2

-tmiddot areal = (Llx + Llx +l)1Zk area = (Ax + Llx+l)1Zk+ l andla eis the Y component of primary field computed ana lytica lly at node j i k and time t = n Finite-difference expressions

(14) for the other components can be derived in a simil ar - manner

icate the The disadvantage of thi s analysis is that the solution is not I n is tbe general ie it applies to media involving only two different Hum 2 conductivities and generalization of it would lead to a very

cause the f current face To h IX j1k-cgrations

~

errns by z and right lucts 0shy0- 2

hjikfa hji-1k hJi+lk

jy into the paperda (15)

0shy0shy II4 3 the terms tial com -plied the Te rms on hJ i k+ I ight of as line integral path a volume

FlgUre 2 Finite-difference grid showing four conductivities intershys surface secting at a corner and the line-integration path to compute an g primal) averagecurrent density at the corner node


complicated numerical scheme As an approximation however it may be generalized rather simply by repl acing the anomalous current density U- j ) with an averaged current density

For example at a comer where four different conductivities intersect the average current density j can be computed by integra ting h along the line integral path shown in Fig 2 Here in fact we have done an averaging but we average the current densities inste ad of the conductivities

Time step and spatial step

The analysis given by Orista glio amp Hohmann (1984) modified for 3-D sug gests that the Du Fort-Frankel method retains its diffusive nature if the following condition is satisfied

l at)11211 $ _ 0_ Lis (18) ( 6

where 1t is the time step t is time range of interest and L1s is the spatial step The 1t is also the practical time step for the method with a uniform grid and spatial step Lis Discussions on the condition and time step are given in the Appendix

In our modelling where a typical grid is non-uniform with variable G we take both Lis and a as the smallest values in the model and increase 1t gradually during computation

1As examples for Lis = 10 m 0 = 001 S m- and t = 01 us 1t = 014 us for 1 = 1 ms 1t = 14 us

We use a discretization design similar to tha t of 2-0 modelling the grid is discretized finely and uniformly within the area of interest and gradually is made coarse and non-uniform toward the boundaries

BOUNDARY CONDITIONS

Inside the E arth we use a homogeneous Dirichlet boundary condition setting both the normal and tangential comshyponents of a vector field to zero This type of boundary condition has been tested for a 2-0 scalar field with good success (Oristaglio amp Hohmann 1984) It appears that in a conductive host th e amplitude of a diffusing field is attenuated (damped) strongly enough so that with a reasonably distant grid boundary a reflected field will not interfere in the area of interest

At the air-Earth interface we use an upward continuation technique to set the boundary condition assuming that the fields obey Laplaces equation in the air We perform upward continuation on the known field (at a given time step) on the surfa ce of the Earth on e grid step in the air and use it as a boundary value for the field in the future (field at the next time step)

The upward co ntinua tion equatio n is

h( ) - z f J h ( ~ 1- 0) x y z = 2 _~ ~ [(x _ ~)2 + (y _ lJI )2+ Z2J312 d~ dip

(19)

where h (~ 1- 0) is th e fie ld at the Earths surface and hex Y z) is the field at on e grid inter val z above the surface

In 2-0 modelling the corresponding equation is a I -D

l

-----238 J I Adhidjaja

integral that can be computed efficiently by interpolating with a cubic spline function to get evenly spaced samples and then Fourier transforming with a I-D fast Fourier transform (FFf)

In 3-D modelling we compute equation (19) directly with a trapezoidal rule For a short continuation distance (small z) the upward continuation operator has a narrow peak so that it can be approximated with a small operator The upward continuation is done point by point which better suits our non-uniform grid With this procedure however we need to interpolate the field to get the correct time level and in addition to get evenly spaced fields near the point For the interpolation first we used a bicubic spline function Because the bicubic spline function is very time-consuming to compute we replaced it with a linear function based on the three surrounding points However the interpolation is still time-consuming due to the pointwise upward continuation

Our test of the upward continuation indicates that for a

distance of about 10 m a 21 x 21 operator gives about 10 per cent error when applied to a smooth function A shorter operator such as 11 x 11 gives about 20 per cent error Because the longer operator is very time-consuming to compute we used a 5 x 5 operator and the bicubic spline function for interpolation for the earlier models and used an 11 x 11 operator and the linear function for the model in this paper

NUMERICAL RESULTS FOR STEP RESPONSE

To compute the step response we use fields due to a step current in the transmitter for the eP and abP at terms in equation (7) As a test we computed results for the coaxial model shown in Fig 3 It is a 100 x 100 x 100 m body with resistivity 1Q m -1 embedded 80 m deep in a 100 Q m- 1

half-space The transmitter is a 100 x 100 m loop carrying 1 A of current and centred over the body This model was designed to test the case where the response is dominated by vortex current effects but as it turns out it becomes an important model to test the galvanic current effect also For this model the terms in equation (7) which contribute to the galvanic-current response would be small but must be calculated accurately otherwise they may change the character of the response such as causing a sign change or a wrong decay rate

We discretized the model with the smallest cells being 10 x 10 x 10 m and used a uniform grid within the region of interest and a non-uniform grid in the region far from the body The time step ranged from 01 to 20 IlS As in the 2-D case the time step was increased gradually during the computation In the 2-D case the time step typically ranges from 01 to 40 JS

Comparison with integral-equation solution

For checking our results we used an integral-equation solution (Newman personal communication 1987) with which results are computed in the frequency domain and transformed to the time domain The frequency-domain solutions were computed in the frequency range 01shy

0-0-0 Finite - Difference - Solution

1ltgt-- 6 f ~c10- 11 I lt

I I 0gt I I raquo loteqrai EqJ lt7tirn

t j 0( Solu t ian II 1

10-12 l- II1 ~

M ~ ~ I hi I I

(T)

10013

I I

0 ~

10-14 f- ~ p

~~ ~

100 8011

10 -15 f- ~ h 100n-r

2 shy1Llll~~

~lQo ~(~

I 1 I 01 I

t(rn)

Figure 3 Comparison of decay curves of horizontal magnetic field at a point P 100m from the centre of the loop for a coaxial model with contrast 1 100

50 000 Hz and were sampled at five pointsdecade The body was discretized with cell size 125 x 125 x 125 m or 128 cellsquadrant

Figure 3 compares decay curves of the horizontal component bx computed with the finite-difference and the integral-equation methods The responses were computed for a point P 100 m from the centre of the loop and lying outside the loop The finite-difference solution is slightly higher than the integral-equation solution at early times but decays slightly slower at later times It also shows a sign change at about 20 us where the solution of the integral-equation method is not available However the overall agreement is reasonable

Figure 4 compares decay curves of the vertical component bz at the centre of the loop For this component also the finite-difference values are slightly larger than that of the integral equation at early times However at times later than 015 there is a large discrepancy in the results A sudden change in decay rate suggests that the finite-difference solution breaks down at this point

The magnetic field impulse response (emf) of a spheroidal body in a conducting whole-space decays as a power law t- 3

5

at late times (Kaufman 1981) Accordingly the step response of the body should decay as t- 25 at late times- Our results do not seem to have the correct decay rate but the responses plotted may not be late enough for determining the decay rate

s ~ z 7)

Fi th

fu ro

M

~

ili ~

~

~

~

ti

~

m rt

tl ~

I a [ a]

cl a b a

amp tl c

100

- - -

10 10 Finir -D ilf~ f c Solu t ion

101

I r_g ro t -Equa t ion shy

So lu ti o

l II

1011

10 middot1l

Of

If 10 shy

i f 01 10

netic field dal model

ade The 25 m or

iorizontal ~ and the omputed and lying s slightly imes but vs a sign

of the ever the

vert ical For this

~ slightly Iy times

a large ecay rate down at

pheroidal bullwer law

the step meso Our but the lcrm ining

[ Figure 4 Comparison of decay curves of vertical magnetic field at the centre of the loop for the model in Fig 3

Because of our limite d reso urces we did not co mpute further resu lts Th e computa tion of the solut ion to 15 ms look 5 hr of cpu time on a Cray X-MP48 supe rcomp uter

Magnetic field and current density plots

We examine the qual itative behavio ur of the finite shydifference solution by plo tti ng the mag ne tic field and current density distributions in t he Earth

Figure 5 sho ws sna psho ts of secondary magnetic fie ld patterns in a ver tica l cross-sectio n throu gh the cen tre of the body The patterns show a magnetic dipole field which starts developing on top of the co nductor and grows stronger with time The dipole has a positive pola rity as expec ted (z-axis posi tive downward) the same direction as the primar y field in the trans mitter loop show n in Fig 3 When the current in the transmitter is shu t off th e current in the Ea rth flows in the same direction as the curre nt in the t ransmitt er did so as to preserve the orig ina l magnetic field At 15 ms the Curvature of the magnet ic fields nearly disappears showi ng almost a uni form field inside the body which is characteris tic of the lat e- time response However 15 ms is an early time for thi s model It seems that the so lution breaks down at this time as evidenced also from the current-density plot s discussed later

Figure 6 sho ws snapshots of curre nt density distribution s aSsociated with th e dip ole in a horizontal section th rou gh the centre of the bod y Ex cept at very early tim e whe n the CUrrent density on this pla ne flows in the opposite di rec tion


I~~~~~~middoto ~-middot middot -~I I I

p -

IPLLU I

~co -J Hu i bull - O482 C4middot shy --~l - Ir ~

Imiddot - 1 PITI~middot middot middot middot 1 bull 1 t

1 i1 i lDmiddotf

- 1 I bull _ ~ bull t ) I 5C~L

Oll - 0 94 1 0 3

r- --~---~~ - ~- ~~

I I

r) ~ i 1Sl I ~

~ l - - j L i

i

-o---

r I

Imiddotbullbullbullbullbullmiddotbullbullbullbull bullbulli t I ~ bull bull

~ j j I ~ i I I

i i ~ i ~ i ~~ ~ I iL middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot- _ - - 500middot1 - - shy 0 - 00303

Figure 5 Secondary magnetic field patterns in a vertical cross-section through the centre of the model in Fig 3 at four times The field values are in A m-

to both the primary field an d the current on top of the conducto r the current density distributions are as expected flowing in a cloc kwise direction Th e early-time current de nsity distribution on the plane may be rea l because its amp litude is much smaller than that of the pr imary field and

r I

T~ ~ ~

~ ~ r ijJ~~~- Cj

- - - - ~ - o ~J

Q ~ _ O V middot v r

I

= - - -~- IF9-middotmiddotLdmiddot

~_~ ~~ ( o 2 4 ) 0

~ In i ~ gt------ - i I

Ij 1 middot middotmiddot 1 middot---------8

bull ~ 103S

bull _ O 1 6 ~ middot0 4

r---- - --- - --- shy

_

- ------ - - - - -

i ~~ - ~ middot1

0- bull _ _ bull bull r

-I j bull bull bull bullbull

-- - _ _ ~c J lt ~ - ~ 20 4 Q

Figure 6 Current density (secondary) in a horizontal section through the centre of the model in Fig 3 at four times The current density values arc in A m- 2

----240 J I Adhidjaja

~-shy

I

I - shylt _-1__----lt - ~ shy --- 1 _

~ 1 bull

~ I -

-- - - I j

-- - Tshy ---- - ------shy

I I omlt Il

M~0J4505 _oo_~ 0243middotC3

~-----------

- -~ - ~ - -------- ---

-- -- - ~ -

- - -

bull I ~ - j

I - ~- - - - ----- -----_------shy

I

150 ( L__~_~~ rlu trII~-oCJa-o~-- ~~J

111)( Om ~ 0392 -04

Figure 7 Current density (total) in a horizontal section through the centre of the model in Fig 3 at four times The current density values are in A m- 2

bull

also the distribution at 0035 IDS indicates a transition period when the current slowly switches direction Opposite flows of current on the two planes suggest that more complicated dipoles are induced in the conductor at early times

Figure 7 shows snapshots of current density distributions associated with the total electric field in a horizontal section through the centre of the body The current flows in the same direction as the electric field in the earth when the current in the transmitter is shut off However at 15 ms the ratio of tangential current density outside and inside the body is far from the ideal 1 100

Nevertheless the general pattern of both magnetic field and current density are reasonable

DISCUSSION

Initially we computed the magnetic field impulse response by substituting electric and magnetic fields due to an impulse current for the e and 8hPIat terms in equation (7) We used finite-difference schemes both with conductivity averaging and with current-density averaging and comshyputed results for two models a coaxial model which is designed to test the solution for a vortex-dominated case and a model where the body is located with its strike parallel to the long side of the loop designed to test the galvanic-dominated solution However the impulse reshysponses computed with both finite-difference schemes were not accurate They showed amplitudes that were too large compared with those of the integral-equation solution (San Filipo amp Hohmann 1985) and showed an unexpected sign change indicating that the solution oscillates in time

The problem may be caused by the rapidly varying primary fields in the earth which are the source in the diffusion equation Unlike that for the 2-D case the diffusion equation for the 3-D model has three source terms that interact For a single source such as in 2-D modelling the finite-difference method seems to propagate the field

correctly but it has a problem when more and interactin sources are involved g

Alford Kelly amp Boore (1974) presented an analysis for

accuracy of a finite-difference solution for the acoustic wave equation with regard to the frequency content of a source function They discussed the necessary number of samples per wavelength in order to avoid dispersion from undersarnpling which is known as grid dispersion

Another possible reason for the inaccuracy problem which is more serious is the mixture of numerical and analytical computation for the source terms Accuracy of the central differences in approximating the curl operation of the line integral for Stokes theorem is limited by the grid interval while the sources with analytical expressions can be computed very accurately If this is the case then we face a fundamental problem with our numerical scheme to achieve accuracy we may need to refine the discretization beyond what is practical Therefore we have to seek an altern~tive way to solve the problem

One alternative is to solve for the total field whose equation is simpler and without the analytical sources but as we learned from 2-D modelling the solution requires a larger grid and significantly more computation time The alternative that we pursued was to compute the step response ie the magnetic field due to a step current Computation of the step response involves smoother functions also the analytical source terms have fewer sign reversals which may alleviate the problem of inaccuracy due to coarse discretization

Our experiments show that in general the shapes of profiles or decay curves of the step response are simpler than those of the impulse response More importantly the solution shows no oscillation of sign as in the case of the impulse response confirming that the step response is better behaved However there is still a large discrepancy in amplitude for the vertical field component of the finite-difference solution and the integral-equation solution in the last experiment Possibly the computer codes may contain errors or may suffer from numerical inaccuracy because the source functions are still too sharp Smoothness of the source functions seems to be an important factor in our 3-D solution For the next experiment we should concentrate on analyzing the source function and try a smoother source waveform such as a step with ramp turn-off

In addition we face an efficiency problem At present the 3-D code is very time-consuming to run even for a small model However symmetry may be applied to some special-case models which could be useful for testing purposes

The computation for the boundary condition at the air-Earth interface is one of the most time-consuming parts of the solution because it involves interpolation and integration that must be done at each time step Use of a uniform grid although it requires very large memory would reduce the computation times because it would simplify the interpolations or it would permit a direct use of a highly optimized 2-D FIT routine to perform the convolution in the transform domain In addition it often makes the computer codes amenable for vectorization to take full advantage of the supercomputer features Utilization of a non-uniform grid makes the computer codes more complicated which often prohibits vectorization

I

r Transient electromagnetic response 241

teracting Another time-consuming computation that must be done at each time step is the computation of the analytical sources

dysis for of the diffusion equation especially for a source other than tic Wave an impulse current The step response takes longer to a SOurce compute than the impulse response because of extra samples numerical integrations required for computing the analytic

n from sources This problem might be reduced by tabulating the sourcefields and using a simple interpolation to compute the

Jroblem desired field Tabulation of 3-D space-time functions ical and however would require a large amount of memory such as cy of the that of a Cray-2 ation of The results presented here were computed on the Cray the grid X-MP48 of the San Diego Supercomputer Center (SDSC)

IS can be Unfortunately the maximum memory available is only three te face a million words running our program with a large amount of achieve memory is either very costly on a high priority or extremely beyond slow on a low priority For all models computed the

ternative maximum size of the grid used was 50 x 50 x 50 which required memory of just under one million words

whose With the present version of our 3-D code to compute the ces but solution for the small model shown in Fig 3 to 10 ms would quires a take 10 h (estimated) of cpu time on the Cray X-MP48 ne The supercomputer he step Faster computation might be achieved by changing the current grid several times as is commonly done in frequencyshymoother domain computations - fine discretization for high freshywer sign quencies and coarse for low frequencies However in the accuracy time domain this procedure would require an accurate 3-D

interpolation when changing the grid size rapes of We finally conclude that the development of the 3-D oler than finite-difference TEM program is not practical given our itly the available resources However the result of the stepshye of the response test although not completely satisfactory is is better somewhat encouraging In the future when adequate iancy in computational resources are more readily available the of the scheme may be worth pursuing solution des may ACKNOWLEDGMENTS accuracy

The authors thank Greg Newman for providing numerical oothness checks Financial support for this work was provided by the factor in following companies Amoco Production Co ARCO Oil should and Gas Co Chevron Resources Co CRA Explorationid try a PtyLtd Standard Oil Production Co and Unocal Corp th ramp

REFERENCESsent the a small Adhidjaja J I Hohmann G W amp Oristaglio M L 1985

o some Two-dimensional transient electromagnetic responses Geophysics 50 2849-2861 testing

Adhidjaja J I amp Hohmann G W 1988 Step responses for two-dimensional transient electromagnetic models

at the Geoexploration 25 13-35 ing parts Alford R M Kelly K R amp Boore D M 1984 Accuracy of

finite-difference modeling of the acoustic wave equationion and Geophysics 39834-842Jse of a

Bergeal J M 1982 The test of an explicit scheme in modeling y would transient electromagnetic fields in mining geophysics MS plify the thesis Yale University a highly Best M E Duncan P Jacobs F J amp Scheen W L 1985

Numerical modeling of the electromagnetic response oflution in three-dimensional conductors in a layered earth Geophysics

ikes the 50 665-676 ake full Birtwistle G M 1968 The explicit solution of the equation of ion of a heat conduction Comput 1 11317-323 s more Brewitt-Taylor C R amp Weaver J T 1976 On the finite

difference solution of two-dimensional induction problems

Geophys 1 R astr Soc 47375-396 Cushman-Roisin N 1984 Analytical linear stability criteria for

the leap-frog Du Fort-Frankel method J Comput Phys 53 227-239

Das U C amp Verma S K 1981 Numerical considerations on computing the EM responses of three-dimensional inshyhomogeneities in a layered earth Geophys J R astr Soc 66 733-740

Du Fort E C amp Frankel S P 1988 Stability conditions in the numerical treatment of parabolic differential equation Math Tables Other Aids Comput (former title of Math Computi 7 135-152

Hohmann G W 1985 Three-dimensional induced polarization and electromagnetic modeling Geophysics 40 309-324

Hohmann G W 1983 Three-dimensional EM modeling Geophys Sum 627-53

Holland R 1977 THREDE a free-field EMP coupling and scattering code IEEE Nucl Sci NS-24 2416-2461

Kaufman A A 1981 The influence of currents induced in the host rock on electromagnetic response of a spheroid directly beneath a loop Geophysics 46 1121-1136

Kunz K S amp Lee K M 1978a A three-dimensional finite difference solution of the external response of an aircraft to a complex transient EM environment Part I-The method and its implementation IEEE Trans Elec Comp EMC-20 328-333

Kunz K S amp Lee K M 1978b A three-dimensional finite difference solution of the external response of an aircraft to a complex transient EM environment Part Il-Comparison of prediction and measurements IEEE Trans Elec Comp EMCmiddot20 333-34l

Lapidus L amp Pinder G F 1982 Numerical Solution of Partial Differential Equations in Science and Engineering Wiley New York

Lee K H Pridmore D F amp Morrison H F 1981 A hybrid three-dimensional electromagnetic modeling scheme Geophysics 46 796-805

Lamontagne Y 1975 Application of wideband time domain EM measurements in exploration PhD thesis University of Toronto

Lines L R amp Jones F W 1973 The peturbation of alternating geomagnetic fields by three-dimensional island structures Geophys J R astr Soc 32 133-154

Newman G A Hohmann G W amp Anderson W L 1986 Transient electromagnetic response of a three-dimensional body in a layered earth Geophysics 51 1608-1627

Oristaglio M L amp Hohmann G W 1984 Diffusion of electromagnetic fields into a two-dimensional earth a finite-difference approach Geophysics 49870-894

Petrick W R 1984 A fully hybrid technique for calculating scattering from three-dimensional bodies in the earth PhD dissertation University of Utah

Pridmore D F 1978 Three-dimensional modeling of electric and electromagnetic data using the finite element method PhD dissertation University of Utah

Pridmore D F Hohmann G W Ward S H amp Sill W R 1981 An investigation of finite-element modeling for electrical and electromagnetic data in three dimensions Geophysics 46 1009-1024

Raiche A P 1974 An integral equation approach to 3-D modelling Geophys J R astr Soc 36363-376

Raiche A P 1983 Comparison of apparent resistivity functions for transient electromagnetic methods Geophysics 48 787-789

Reddy1 K Rankin D amp Philips R J 1977 Three-dimensional modelling in magnetotelluric and magnetic variational soundshying Geophys 1 R astr Soc 51 313-325

San Filipo W A amp Hohmann G W 1985 Integral equation solution for the transient electromagnetic response at a three-dimensional body in a conductive half-space Geophysics 50 798-809

Tripp A C 1982 Multidimensional electromagnetic modeling PhD thesis University of Utah

Ward S H amp Hohmann G W 1988 Electromagnetic theory for geophysical application in Electromagnetic Methods in Applied Geophysics Investigations in Geophysics 3 1 ed Nabighian M N Soc ExpI Geophys

pc lt 242 J I Adhidjaja

j

~

(

i

-

~ ~

Weidelt P 1975 Electromagnet ic induction in thr ee-dimensional structures J Geophys 41 85- 109

Yee K S 1966 Numerical solution of initial boundary value problems involving Maxwells equ at ion s in isotropic media IEEE Trans Antenna Propagat APmiddot14 302-309

Zhdanov M S Go lubev N G Spichack Y Y amp Varentsov Iv M 1982 The construction of effective methods for elcctromagnet ic modelling Geophys J R astr Soc 68 589-607

APPENDIX

Stability condition and time step for the Du Fort-Frankel finite-dilference method

The Du Fort- Frankel finite-di ffere nce method was derived for the so lution of a homogeneou s diffusion (parabolic) equation in I-D by Du Fo rt amp Franke l (1953) T hey also showed that the method is un conditionally stable Stability proofs of the method for higher dimensions are give n by Birtwi stle (1968) Analysis of stability criteria for the leapfrog Du Fort-Frankel method applied to an advectiveshydiffusive equation is given by Cushman-Roisin (1984) All of the an alyses wer e do ne for a homogeneous diffus ion equation discret ized with a uniform grid

Oristaglio amp Hohmann (1984) gave analysis of stability in 2-D as applied to the geophysical problem including generalizat ion of it for a non-uniform grid Th e Du Fort-Frankel method is unconditionally stable but it is still limit ed by the con sisten cy condition as the mesh spa cing is refined (La pidus amp Pinder 1982) A s shown by Du Fo rt amp Frankel (1953) the method approximat es a hyperbol ic equation instead o f a diffusion equation if Lit an d 1t approach zero with con sta nt ra tio

Fo llowing O ristaglio amp Hohmann (1984) we will sho w tha t the condition requ ired in the Du Fort- Frankel me thod for our pu rpose can be sat isfied This analysis is an approximation based on scalar wave and diffusion equatio ns in a who le-space which arc approximately ap plicable to ou r equation in the lar ger part of a typi ca l grid in th e Cartesian coo rdin ate

The complete scal ar wave equation is

oh o2h V-h - Iloa- - Eollo - 2 = O (A I)

al at

The Green function in a whole-sp ace associate d with (AI) is given in Ward amp Hohmann (1988) as

a l1 -u(rlc) ( r) a(r c)eshy

g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)

where a = a 2EO (j is a delt a functio n c2 = 1(Eolo) f l (z ) is the modified Bessel fun ction of the first kind and u(t - rIc) is the Heaviside ste p fun ction

For geophysical purposes the equa tion (AI) becom es a diffusion eq uation under a quas istatic approximation

2 ohV h -loO- = O (A3)

at

Th e Green function in a whole-space associated with (A3) is

given in Ward amp Hohmann (1988) as

(llu O ) I 2 - Or g(r I) = 8(m)3i2 e u( t) (A4)

where

e= ( Il~~a) tl2

T he qu asistatic ap proximation is valid onl y for late times when the expression (A2) is practica lly the same as the expression (A4) for

22Eo (2 r ) 12 _ laquo r - - bull2a c

or for our purpose

-2 pound0 laquo I (AS)a

Th e Du fo rt-Fra nkel finite-differe nce approximation for equa tion (A3) is

y 2h = h (i + I j k t) + h (i - I j k t) + h (i j + I k t )

+h(ij - l k 1) +Ir (i j k + 1 r) -+-Ir(ij k -I I)

+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp

looJh Jt = Iloa [h(i j k 1 + 1) - h (i j k t -1 )]1211

in a un iform grid wher e amp is the spati al ste p and 11 is the time step

If we use a Taylor se ries expa nsion and su bstitute for the derivatives

h (i plusmn I j k r) = h plusmn amp oh as + ( 1s 2 ~1r a2s

+ higher order terms

h (i j k t plusmn 1) =h plusmn poundIt ah al + ( 3 t) 22 ~h a2t


we get

2 Jh ( 1t)2 elaquoV h - u a- - 3 - - = o (A6) 0 al 11 ot-

Equation (A6) is hype rbo lic and its stability is governed by the Co ura nt-Friedrich s-Lewy (CFL) con dit ion

amp ~~c (A7)11v 3

where c is the propagat ion spee d The Du Fo rt- Fra nkel finite differen ce wou ld satisfy

equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo

Substituting (A8) into (A5) we get the co ndi tion which insur es both the sta bility as well as the diffusive nature of the Du Fort-Frankel fini te differe nce

6(11amp)2 t raquo ----~ (A9)

loa or

1t $ ( Il~or) 112 amp (AID)

which is the maximum practi cal time step for the Du Fort- Fr ankel finite -diffe re nce meth od

shy--

I 234 J I Adhidjaja

Two approaches have been used to obtain time-domain solutions The indirect method uses Fourier transformation of frequency-domain results (Lamontagne 1975~ Tripp 1982 Newman Hohmann amp Anderson 1986) One advantage of the indirect method is that it can be based on an existing and well-tested frequency-domain computer program The direct method requires solving an initial value problem directly in the time domain which can be done via a time-stepping method

The direct method may provide more insight into the transient (TEM) process within the whole spectrum from very early to very late times However the early-time solution must be calculated accurately to insure correct late-time results For a differential-equation method the computation can be very time consuming San Filipo amp Hohmann (1985) developed an efficient direct time-domain solution based on an integral equation formulation but their solution was for a simple prism in an otherwise homogeneous half-space

Yee (1966) presented a finite-difference scheme for solving the 3-D transient electromagnetic problem it was based on time-stepping Maxwells equations and solving for both electric and magnetic fields His scheme has been applied successfully to compute scattering by a perfect conductor Success seems to be due to an accurate representation of the boundary conditions of fields at interfaces which is achieved by carefully placing grid points The scheme utilizes a staggered grid the electric and magnetic fields at a node are defined at alternate time steps Later this scheme was adapted and used successfully for solving various 3-D electromagnetic scattering problems in free-space (Holland 1977 Kunz amp Lee 1978ab) The scheme has also been tested for a geophysical problem by Bergeal (1982) He solved for a 3-D body in a whole-space but his results computed for only 10 I-tS were not satisfactory The main difficulty involved propagating the electric field via the displacement current which we normally neglect in geophysical problems Including the displacement current requires a very small time step which is impractical in computing a solution to the later times of interest in geophysics

Our goal was to develop a direct time-domain solution based on a differential equation for simulating a complex 3-D model with realistic conductivity contrasts We formulated the solution in terms of the secondary field based on experience with a successful algorithm for the 2-D problem (Oristaglio amp Hohmann 1984 Adhidjaja Hohshymann amp Oristaglio 1985) In this paper we present the theoretical formulation the finite-difference scheme the limited numerical results we were able to compute and discussions of the problems encountered

THEORETICAL FORMULATION

Following Hohmann (1983) we begin with Maxwells equations

v x e(r t) = --to ab(r t) _ amP(r t) (1)at ~o at Yx her t) = ae(r t) (2)

with magnetic permeability fJ = P-o everywhere and with

Tdisplacement current neglected Here mP is the d - Ipole

moment per volume of an Impressed source In au cas bull - ~e We

sum vertical magnetic dipoles to create a horizO1tl I source a cop

As in the 2-D case we formulate the problem in term S of

primary and secondary fields through the decompositions

e=eP+es

and

h = hP + h

where superscript s denotes secondary and p denotes primary

Primary fields are defined as the fields in the earth if the earth were homogeneous They satisfy the equations

JbP Jnt v x e = - fJo at - Il0-----at (3)

and

YXhP=aeP (4)

where 0 is the conductivity of the half-space The secondary fields which are caused by in

homogeneities in the half-space satisfy

ahS V x e = -fJ0-at (5)

and

v x bS = oe + aacP (6)

where o = (0 - 0) is the anomalous conductivity Solving for b S and setting V h = 0 assuming no magnetic

permeability contrasts we obtain an inhomogeneous vector diffusion equation for the secondary magnetic field

ah ahP

V-h - a - = IJ a shyII at at10 10 a

+ av() x (7 x h) - av(~a) x eP (7) O 0

where the superscript s has been dropped for clarity The field is a vector quantity whose components in general are coupled which makes the 3-D equation much more complicated than the 2-D equation Inside a homogeneous medium the vector components become decoupled because the terms containing the gradient of conductivity vanish

FINITE-DIFFERENCE APPROXIMATION

For time stepping we use the Du Fort-Frankel finite-difference method The method was derived by Du Fort and Frankel (1953) for solving a homogeneous diffusion equation in one dimension It possesses the desirable properties of being unconditionally stable and explicit lt uses a staggered grid which when combined with a leap-frog method in time forms an efficient scheme for time-stepping computations

The Du Fort-Fran kel technique was used successfullyfO~ computing responses of 2-D conductors in geophySiCa environments by Oristaglio amp Hohmann (1984) and laterby Adhidjaja et al (1985) Oristaglio amp Hohmann made an

~



p denotes

arth if the ons

(3)

(4)

by in

(5)

(6)

y


e (7)


ON




IIi












r- x h k-I t

I gt --


h- - I ~ ~ _- - middot - _ _ i

j 1 I rk

)( I J (gt I C 1 I 7

)1

y Z

B (7

1 2 A Z k


1 2 6 Z k 1


1 2 ly j l Hlt

1 2 Cx 1 2 L X G 01+1 i + I k e lI 1 + 1 I 1


h j i k+ 1




~ ( h(j i + 1 k ) - h(j i k)

4 1x i t-l


h(j i k) - h(j i - I k)

Llxi


h(j + 1 i k ) - h(j i k ) +

1Yj +l


h(j i k) - h(j -1 i k)

1Yj


h(j i k + 1) - h(j i k ) +

1Zk+ 1


h(j i k) - hn(j i k - 1)

1zk








h+ l ( i k ) + h- I( i k)h(j i k ) j J

2



+ c3h(j - 1 i k) + c4 h (j + 1 i k )

+ csh(j i k - 1) + c6 h(j i k + 1)


with






o =8

aivl 8

Jlk LJ-- Vr = 2 V 1= 1 1=1v-







ah ah P

V2h



all P







t

+ [C7 [hxCj i k + 1) - hAj i k - 1)]

- C8[hAj i + 1 k ) - hz(j i-I k)J

x [RU i k + 1) - R(j i k - 1)]

- C9 [h j i + 1 k) - hy(j i-I k )]


x [R (j + 1 i k ) - R(j -1 i k)]

+ Cll[a(j i k + l )R (j i k + 1)

- o(j i k -l )RU i k -l )] e~]Ip (13)












L(V2h- 1oa ~~) du


1(0 2 01

- a 2 - 2 )n x eP da ) deg 2 0 1



(14)




0 1 v al



I I

I iee invc

Ai The

I

L( I

t I

=i I

whe mec bye Stol


x

whe but

C7 =

jgt

1middot2 j -

=


Tl gent cone

hj

Figul sect ii averc

_ _- - - - - - - - - - - - - - - - - __ - - - _

-1




dv + s (j - j~ ) da - cu - a~)e~ da








lution at demand











~











l at)11211 $ _ 0_ Lis (18) ( 6





BOUNDARY CONDITIONS





(19)



l













1ltgt-- 6 f ~c10- 11 I lt



10-12 l- II1 ~

M ~ ~ I hi I I

(T)

10013

I I

0 ~

10-14 f- ~ p

~~ ~

100 8011

10 -15 f- ~ h 100n-r

2 shy1Llll~~

~lQo ~(~

I 1 I 01 I

t(rn)






5


s ~ z 7)

Fi th

fu ro

M

~

ili ~

~

~

~

ti

~

m rt

tl ~

I a [ a]

cl a b a

amp tl c

100

- - -


101


So lu ti o

l II

1011

10 middot1l

Of

If 10 shy

i f 01 10


ade The 25 m or


of the ever the

vert ical For this

~ slightly Iy times












p -

IPLLU I



1 i1 i lDmiddotf


Oll - 0 94 1 0 3

r- --~---~~ - ~- ~~

I I

r) ~ i 1Sl I ~

~ l - - j L i

i

-o---

r I


~ j j I ~ i I I




r I

T~ ~ ~

~ ~ r ijJ~~~- Cj

- - - - ~ - o ~J


I


~_~ ~~ ( o 2 4 ) 0

~ In i ~ gt------ - i I


bull ~ 103S


r---- - --- - --- shy

_

- ------ - - - - -

i ~~ - ~ middot1



-- - _ _ ~c J lt ~ - ~ 20 4 Q



~-shy

I

I - shylt _-1__----lt - ~ shy --- 1 _

~ 1 bull

~ I -

-- - - I j

-- - Tshy ---- - ------shy

I I omlt Il


~-----------

- -~ - ~ - -------- ---

-- -- - ~ -

- - -

bull I ~ - j

I - ~- - - - ----- -----_------shy

I

150 ( L__~_~~ rlu trII~-oCJa-o~-- ~~J

111)( Om ~ 0392 -04


bull




DISCUSSION











I














































j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--



p denotes

arth if the ons

(3)

(4)

by in

(5)

(6)

y


e (7)


ON




IIi












r- x h k-I t

I gt --


h- - I ~ ~ _- - middot - _ _ i

j 1 I rk

)( I J (gt I C 1 I 7

)1

y Z

B (7

1 2 A Z k


1 2 6 Z k 1


1 2 ly j l Hlt

1 2 Cx 1 2 L X G 01+1 i + I k e lI 1 + 1 I 1


h j i k+ 1




~ ( h(j i + 1 k ) - h(j i k)

4 1x i t-l


h(j i k) - h(j i - I k)

Llxi


h(j + 1 i k ) - h(j i k ) +

1Yj +l


h(j i k) - h(j -1 i k)

1Yj


h(j i k + 1) - h(j i k ) +

1Zk+ 1


h(j i k) - hn(j i k - 1)

1zk








h+ l ( i k ) + h- I( i k)h(j i k ) j J

2



+ c3h(j - 1 i k) + c4 h (j + 1 i k )

+ csh(j i k - 1) + c6 h(j i k + 1)


with






o =8

aivl 8

Jlk LJ-- Vr = 2 V 1= 1 1=1v-







ah ah P

V2h



all P







t

+ [C7 [hxCj i k + 1) - hAj i k - 1)]

- C8[hAj i + 1 k ) - hz(j i-I k)J

x [RU i k + 1) - R(j i k - 1)]

- C9 [h j i + 1 k) - hy(j i-I k )]


x [R (j + 1 i k ) - R(j -1 i k)]

+ Cll[a(j i k + l )R (j i k + 1)

- o(j i k -l )RU i k -l )] e~]Ip (13)












L(V2h- 1oa ~~) du


1(0 2 01

- a 2 - 2 )n x eP da ) deg 2 0 1



(14)




0 1 v al



I I

I iee invc

Ai The

I

L( I

t I

=i I

whe mec bye Stol


x

whe but

C7 =

jgt

1middot2 j -

=


Tl gent cone

hj

Figul sect ii averc

_ _- - - - - - - - - - - - - - - - - __ - - - _

-1




dv + s (j - j~ ) da - cu - a~)e~ da








lution at demand











~











l at)11211 $ _ 0_ Lis (18) ( 6





BOUNDARY CONDITIONS





(19)



l













1ltgt-- 6 f ~c10- 11 I lt



10-12 l- II1 ~

M ~ ~ I hi I I

(T)

10013

I I

0 ~

10-14 f- ~ p

~~ ~

100 8011

10 -15 f- ~ h 100n-r

2 shy1Llll~~

~lQo ~(~

I 1 I 01 I

t(rn)






5


s ~ z 7)

Fi th

fu ro

M

~

ili ~

~

~

~

ti

~

m rt

tl ~

I a [ a]

cl a b a

amp tl c

100

- - -


101


So lu ti o

l II

1011

10 middot1l

Of

If 10 shy

i f 01 10


ade The 25 m or


of the ever the

vert ical For this

~ slightly Iy times












p -

IPLLU I



1 i1 i lDmiddotf


Oll - 0 94 1 0 3

r- --~---~~ - ~- ~~

I I

r) ~ i 1Sl I ~

~ l - - j L i

i

-o---

r I


~ j j I ~ i I I




r I

T~ ~ ~

~ ~ r ijJ~~~- Cj

- - - - ~ - o ~J


I


~_~ ~~ ( o 2 4 ) 0

~ In i ~ gt------ - i I


bull ~ 103S


r---- - --- - --- shy

_

- ------ - - - - -

i ~~ - ~ middot1



-- - _ _ ~c J lt ~ - ~ 20 4 Q



~-shy

I

I - shylt _-1__----lt - ~ shy --- 1 _

~ 1 bull

~ I -

-- - - I j

-- - Tshy ---- - ------shy

I I omlt Il


~-----------

- -~ - ~ - -------- ---

-- -- - ~ -

- - -

bull I ~ - j

I - ~- - - - ----- -----_------shy

I

150 ( L__~_~~ rlu trII~-oCJa-o~-- ~~J

111)( Om ~ 0392 -04


bull




DISCUSSION











I














































j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--



ah ah P

V2h



all P







t

+ [C7 [hxCj i k + 1) - hAj i k - 1)]

- C8[hAj i + 1 k ) - hz(j i-I k)J

x [RU i k + 1) - R(j i k - 1)]

- C9 [h j i + 1 k) - hy(j i-I k )]


x [R (j + 1 i k ) - R(j -1 i k)]

+ Cll[a(j i k + l )R (j i k + 1)

- o(j i k -l )RU i k -l )] e~]Ip (13)












L(V2h- 1oa ~~) du


1(0 2 01

- a 2 - 2 )n x eP da ) deg 2 0 1



(14)




0 1 v al



I I

I iee invc

Ai The

I

L( I

t I

=i I

whe mec bye Stol


x

whe but

C7 =

jgt

1middot2 j -

=


Tl gent cone

hj

Figul sect ii averc

_ _- - - - - - - - - - - - - - - - - __ - - - _

-1




dv + s (j - j~ ) da - cu - a~)e~ da








lution at demand











~











l at)11211 $ _ 0_ Lis (18) ( 6





BOUNDARY CONDITIONS





(19)



l













1ltgt-- 6 f ~c10- 11 I lt



10-12 l- II1 ~

M ~ ~ I hi I I

(T)

10013

I I

0 ~

10-14 f- ~ p

~~ ~

100 8011

10 -15 f- ~ h 100n-r

2 shy1Llll~~

~lQo ~(~

I 1 I 01 I

t(rn)






5


s ~ z 7)

Fi th

fu ro

M

~

ili ~

~

~

~

ti

~

m rt

tl ~

I a [ a]

cl a b a

amp tl c

100

- - -


101


So lu ti o

l II

1011

10 middot1l

Of

If 10 shy

i f 01 10


ade The 25 m or


of the ever the

vert ical For this

~ slightly Iy times












p -

IPLLU I



1 i1 i lDmiddotf


Oll - 0 94 1 0 3

r- --~---~~ - ~- ~~

I I

r) ~ i 1Sl I ~

~ l - - j L i

i

-o---

r I


~ j j I ~ i I I




r I

T~ ~ ~

~ ~ r ijJ~~~- Cj

- - - - ~ - o ~J


I


~_~ ~~ ( o 2 4 ) 0

~ In i ~ gt------ - i I


bull ~ 103S


r---- - --- - --- shy

_

- ------ - - - - -

i ~~ - ~ middot1



-- - _ _ ~c J lt ~ - ~ 20 4 Q



~-shy

I

I - shylt _-1__----lt - ~ shy --- 1 _

~ 1 bull

~ I -

-- - - I j

-- - Tshy ---- - ------shy

I I omlt Il


~-----------

- -~ - ~ - -------- ---

-- -- - ~ -

- - -

bull I ~ - j

I - ~- - - - ----- -----_------shy

I

150 ( L__~_~~ rlu trII~-oCJa-o~-- ~~J

111)( Om ~ 0392 -04


bull




DISCUSSION











I














































j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--

-1




dv + s (j - j~ ) da - cu - a~)e~ da








lution at demand











~











l at)11211 $ _ 0_ Lis (18) ( 6





BOUNDARY CONDITIONS





(19)



l













1ltgt-- 6 f ~c10- 11 I lt



10-12 l- II1 ~

M ~ ~ I hi I I

(T)

10013

I I

0 ~

10-14 f- ~ p

~~ ~

100 8011

10 -15 f- ~ h 100n-r

2 shy1Llll~~

~lQo ~(~

I 1 I 01 I

t(rn)






5


s ~ z 7)

Fi th

fu ro

M

~

ili ~

~

~

~

ti

~

m rt

tl ~

I a [ a]

cl a b a

amp tl c

100

- - -


101


So lu ti o

l II

1011

10 middot1l

Of

If 10 shy

i f 01 10


ade The 25 m or


of the ever the

vert ical For this

~ slightly Iy times












p -

IPLLU I



1 i1 i lDmiddotf


Oll - 0 94 1 0 3

r- --~---~~ - ~- ~~

I I

r) ~ i 1Sl I ~

~ l - - j L i

i

-o---

r I


~ j j I ~ i I I




r I

T~ ~ ~

~ ~ r ijJ~~~- Cj

- - - - ~ - o ~J


I


~_~ ~~ ( o 2 4 ) 0

~ In i ~ gt------ - i I


bull ~ 103S


r---- - --- - --- shy

_

- ------ - - - - -

i ~~ - ~ middot1



-- - _ _ ~c J lt ~ - ~ 20 4 Q



~-shy

I

I - shylt _-1__----lt - ~ shy --- 1 _

~ 1 bull

~ I -

-- - - I j

-- - Tshy ---- - ------shy

I I omlt Il


~-----------

- -~ - ~ - -------- ---

-- -- - ~ -

- - -

bull I ~ - j

I - ~- - - - ----- -----_------shy

I

150 ( L__~_~~ rlu trII~-oCJa-o~-- ~~J

111)( Om ~ 0392 -04


bull




DISCUSSION











I














































j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--













1ltgt-- 6 f ~c10- 11 I lt



10-12 l- II1 ~

M ~ ~ I hi I I

(T)

10013

I I

0 ~

10-14 f- ~ p

~~ ~

100 8011

10 -15 f- ~ h 100n-r

2 shy1Llll~~

~lQo ~(~

I 1 I 01 I

t(rn)






5


s ~ z 7)

Fi th

fu ro

M

~

ili ~

~

~

~

ti

~

m rt

tl ~

I a [ a]

cl a b a

amp tl c

100

- - -


101


So lu ti o

l II

1011

10 middot1l

Of

If 10 shy

i f 01 10


ade The 25 m or


of the ever the

vert ical For this

~ slightly Iy times












p -

IPLLU I



1 i1 i lDmiddotf


Oll - 0 94 1 0 3

r- --~---~~ - ~- ~~

I I

r) ~ i 1Sl I ~

~ l - - j L i

i

-o---

r I


~ j j I ~ i I I




r I

T~ ~ ~

~ ~ r ijJ~~~- Cj

- - - - ~ - o ~J


I


~_~ ~~ ( o 2 4 ) 0

~ In i ~ gt------ - i I


bull ~ 103S


r---- - --- - --- shy

_

- ------ - - - - -

i ~~ - ~ middot1



-- - _ _ ~c J lt ~ - ~ 20 4 Q



~-shy

I

I - shylt _-1__----lt - ~ shy --- 1 _

~ 1 bull

~ I -

-- - - I j

-- - Tshy ---- - ------shy

I I omlt Il


~-----------

- -~ - ~ - -------- ---

-- -- - ~ -

- - -

bull I ~ - j

I - ~- - - - ----- -----_------shy

I

150 ( L__~_~~ rlu trII~-oCJa-o~-- ~~J

111)( Om ~ 0392 -04


bull




DISCUSSION











I














































j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--

100

- - -


101


So lu ti o

l II

1011

10 middot1l

Of

If 10 shy

i f 01 10


ade The 25 m or


of the ever the

vert ical For this

~ slightly Iy times












p -

IPLLU I



1 i1 i lDmiddotf


Oll - 0 94 1 0 3

r- --~---~~ - ~- ~~

I I

r) ~ i 1Sl I ~

~ l - - j L i

i

-o---

r I


~ j j I ~ i I I




r I

T~ ~ ~

~ ~ r ijJ~~~- Cj

- - - - ~ - o ~J


I


~_~ ~~ ( o 2 4 ) 0

~ In i ~ gt------ - i I


bull ~ 103S


r---- - --- - --- shy

_

- ------ - - - - -

i ~~ - ~ middot1



-- - _ _ ~c J lt ~ - ~ 20 4 Q



~-shy

I

I - shylt _-1__----lt - ~ shy --- 1 _

~ 1 bull

~ I -

-- - - I j

-- - Tshy ---- - ------shy

I I omlt Il


~-----------

- -~ - ~ - -------- ---

-- -- - ~ -

- - -

bull I ~ - j

I - ~- - - - ----- -----_------shy

I

150 ( L__~_~~ rlu trII~-oCJa-o~-- ~~J

111)( Om ~ 0392 -04


bull




DISCUSSION











I














































j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--


~-shy

I

I - shylt _-1__----lt - ~ shy --- 1 _

~ 1 bull

~ I -

-- - - I j

-- - Tshy ---- - ------shy

I I omlt Il


~-----------

- -~ - ~ - -------- ---

-- -- - ~ -

- - -

bull I ~ - j

I - ~- - - - ----- -----_------shy

I

150 ( L__~_~~ rlu trII~-oCJa-o~-- ~~J

111)( Om ~ 0392 -04


bull




DISCUSSION











I














































j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--

I














































j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--


j

~

(

i

-

~ ~




APPENDIX







al at



g(r t) =- e ) 1- - + ( 2 2 2) 1124Jlr c 1 - r c

X f l[a(t2_ ~) tl2]U(t_ ~) (A2)




at




where

e= ( Il~~a) tl2



or for our purpose





+ 3[h(i j k 1 + 1) + h (i j k 1 - 1)J amp








we get



amp ~~c (A7)11v 3


equation (A 7) fo r

Eo =3(11 3S)2 (A8) lo


6(11amp)2 t raquo ----~ (A9)

loa or



shy--

11 use ~42. response of a three-dimensional body · department of geology and geophysics,...

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