Diffusion of electromagnetic fields into a two-dimensional

GEOPHYSICS VOL 49 NO7 JULY 1984) P 87(HS94 16 fIGS 3 TABLES

Diffusion of electromagnetic fields into a two-dimensional earth A finite-difference approach

M L Oristaglio and G W Hohmann]

ABSTRACT

We describe a numerical method for time-stepping those for a half-space Maxwells equations in the two-dimensional (2-D) TEshy We use the 2-D model to simulate transient electroshymode which in a conductive earth reduces to the diffushy magnetic (TE) surveys over a thin vertical conductor sion equation The method is based on the classical embedded in a half-space and in a half-space with overshyDuFort-Frankel finite-difference scheme which is both burden At early times (microseconds) the patterns of explicit and stable for any size of the time step With diffusion in the earth are controlled mainly by geoshythis method small time steps can be used at early times metric features of the models and show a great deal of to track the rapid variations of the field and large steps complexity But at late times the current concentrates can he used at late times when the field becomes at the center of the thin conductor and with a large smooth and its rates of diffusion and decay slow down contrast (1000 1) between conductor and half-space

The boundary condition at the earth-air interface is produces the characteristic crossover and peaked anomshyhandled explicitly by calculating the field in the air from alies in the surface profiles of the vertical and horizontal its values at the earths surface with an upward continushy emfs With a smaller contrast (100 1) however the ation based on Laplaces equation Boundary conshy crossover in the vertical emf is obscured by the halfshyditions in the earth are imposed by using a large graded space response although the horizontal emf still shows grid and setting the values at the sides and bottom to a small peak directly above the target

INTRODUCTION

Most electromagnetic (EM) exploration for minerals is now done with transient ( time-domain tt) EM systems Although time-domain methods were first adapted to mineral explorashytion in the 19505 (Wait 1956 Kovalenko 1961 Kamenetzky 1976) their popularity rose mainly during the late 19605 and early 1970s when it became clear that future exploration would require broadband EM measurements-a realization which came both from advances in EM modeling (Ward et al 1973 Ward 1979) and from experience in areas like Australia (Spies 1976 Preston 1975 Lamontagne et al 1978) where ore bodies often lie under a thick conductive overburden The introduction to this Special Issue contains a short history of these developments

Here we describe a simple numerical method for simulating transient EM surveys in which the long sides of a rectangular loop are laid along the local geologic strike and the magnetic field or emf is measured along lines in the perpendicular direcshy

tion Examples of systems that use this configuration are the Newmont EMP system (Dickson and Boyd 1980 Boyd 1980) the UTEM system (Lamontagne 1975) and the Geonics EM37 system (McNeill 1982) Because their field layout resembles that of the frequency-domain Turam method (Parasnis 1979) these systems are commonly called time-domain Turam sysshytems though their transient measurements are of course quite different from classical Turam measurements

The models we study are strictly two-dimensional (2-D) but they illustrate many features of transient surveys such as the shapes and locations of typical anomalies and their approxishymate decay rates Our numerical method lS based on an explicit finite-difference scheme originally proposed by DuFort and Frankel (1953) for the one-dimensional (1-0) diffusion equation and later extended to two dimensions by Birtwistle (1968) We do however derive two new results which are needed for geophysical modeling with the Dufort-Frankel method The first is a straightforward generalization of this method to inhoshymogeneous models and irregular grids and the second is a

Presented at the 52nd Annual International SEG Meeting October 19 1982 in Dallas Manuscript received by the Editor Octoher 21 19R3 revised manuscript receivedJanuary 15 1984 middotSchlumberger-Doll Research PO Box 307 RidgefieldCT 06877 Department of Geology and Geophysics University of Utah Salt Lake City UT R4112(r 1984Society of Exploration Geophysicists All rights reserved

870

bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull

871 OIf1uslon of EM Fields Into a 2-D Earth

V 2EAIR = ay~X

zo=O 1

I

I I I I I I I I t

x J

1

z2

I

Xl x 2

FIG Two-dimensional model geometry including the finite-difference grid x) labels grid points in the x-direction z labels grid points in the z-direction The black dots represent one line of grid points in the air layer which are needed for the boundary condition at the earth-air interface

numerical boundary condition at the earth-air interface that avoids having to include the air layer in the grid

These results are derived after a short analysis of the role played by the diffusion equation in transient EM modeling Four model examples follow the description of the numerical method They are (I) a half-space (2) a half-space with a thin vertical conductor-large (1000 I) contrast (3) a half-space with both overburden and a thin conductor and (4) a halfshyspace with a thin vertical conductor-s-small (100 1) contrast

For each of these models we have computed the negative step response for the emf or the response obtained by switching off a steady current and observing the decaying magnetic field with a point coil receiver of unit area Most time-domain systems measure approximately this response and the response to a general waveform can of course always be obtained from the step response

Direct time-domain modeling is not new but our approach differs from the previous work by Kuo and Cho (1980) who also considered the 2-D Turam geometry and Goldman and Stoyer (1983) who dealt with axisymmetric models (see also Watts 1972 who computed magnetotelluric responses in the time domain) Some of the differences are in technique For example Kuo and Cho (1980) used a finite-element method for the damped wave equation we use a finite-difference method for the diffusion equation Goldman and Stoyer (1983) used an implicit finite-difference method for the diffusion equation we use an explicit method The most important distinction howshyever is that here we are mainly interested in the qualitative features of the models particularly in the patterns of diffusion

in the earth We have thus included a large number of contour plots showing how the earths electric field evolves in time To us this is the most appealing feature of numerical time-domain modeling since it gives a new perspective on EM induction in the earth

DIFFUSION EQUATION

Maxwells equations for the 2-D TE-mode where

E(x z t) = E y and

H(x z t) = flxx + Hzz

reduce to the following scalar equation for the electric field in the strike direction (letting E = E)

cxxE + czzE - jlcrctE - jlpoundcttE = jlcJ (1)

The model geometry is shown in Figure I The conductivity o = otx z) can vary blockwise over the (x z) plane while the permeability u and permittivity E are constant at their freeshyspace values jl 41t X 10- 7 Him and c = 8854 x 10- 12 Fyrn

J s is the density of the source current in the y-direction Equation (I) is a damped wave equation but in the earth

where a is normally much larger than e the wavelike features of the electric field vanish very quickly leaving only the diffusive behavior Dropping the term containing I in equation (1) ie neglecting displacement currents gives this diffusive limit with E satisfying the diffusion equation

872

120

OrlstagUo and Hohmann

~ a

100 I

80

] ~ 80 ~ ~

~ 40 U J1 w

20

00) 20

-00 90 80 70 80 -60

Log Time Isgt

I0

I L

~ 12000120

b

10000100 ( f

800080

II

~~

amp000110

2~

~

I J

U g 4000

l6 E 40

~ Ww

20 2000 1 II __JJ oo~~~~-~00 ~-2000

-40 -30 20 -00 -90 middot80 70 110 -~ 0 00 -30 20 100 90 80 70 -00 -~ 0 40 -30 20 20

Log Time (s) Log TIme (5)

FIG 2 Whole-space Green functions G (squares) and G (solid line) for the damped wave and diffusion equations at distances of (a) 3 m (b) 30 m and (c) 300 m from the source in which a steady current was shut off at l = O The conductivity of the whole space is 0001 Sm while the permeability and permittivity have their free space values The arrow indicates the singular first-arrival in Gw

Oxx E + oz E - Ilao E = J-lcJ (2)

For most geophysical models the diffusive limit begins at very early times Consider for example the whole-space Green funcshytion for equation (1) which has the units of electric field when defined as the solution of

AU Gw + oz Gw - 1-l00c l G - l-lecll Gw = - jJ18(x)8(z)b(c)

where the source is a delta function with amplitude -Ill Physically G corresponds to the electric field that is induced by shutting off a steady current I through a line source in a homogeneous medium that is by taking the source current to be

J(X z t) = lS(x)8(z)[1 - H(t)]

where H(t) is the Heaviside step function G is given by (Wolf 1979)

l a 2- 112pound) R 2 Jl e cosh -- ((2 - C )112J

(j(x Z t) - -~pound shy21t (t2 _~R-2-C~2-)1-2--- H(t - Rc)

(3)

where

R = (x 2 + z2)12

is the distance from the source and

c = (J-lE)-1I2

is the speed of light in free space This Green function is shown in Figure 2 at distances of 3 30 and 300 m from the source in a whole space of 001 Sm conductivity which is a typical value for rocks Also shown is the Greens function Gd for the pure diffusion equation

()xx Gd + cbullbull Gd - Jlao Gd = - JlI5(x)amp(z)D(t) (4)

which is given by (Wolf 1979)

jJl - R J d x z l = - e )-I bull r

41tt G ( ) (5)

Only at 3 m and at very early times do G and Gd differ appreciably in this example In general it is easy to show that they agree asymptotically when

2poundR 2C 2)l i2(t2 _ ~ - (6)

a

a condition which is always satisfied in transient EM surveys One difference between these Green functions is significant

however since it affects the numerical properties of the damped wa ve and diffusion equations Gw propagates the fastest signal with the speed of light and the leading edge of the pulse is always singular [at ( = Ric see equation (3)] even though this first arrival is squeezed into a very narrow time window when the ratio OE is large Because of this explicit finite-difference (or finite-element) methods for the damped wave equation are only stable ie they wont amplify rounding errors if the time step 1t and the distance between grid points ~ satisfy the Courant-Friedrichs-Lewy (CFL) stability condition for hypershybolic equations (Richtrnyer and Morton 1967 Mitchell and Griffiths 1980) For equation (I) the eFL condition is given by

1 1 1tlt-shy (7)- 12 c

v

which restricts the time step to be less than the speed-or-light propagation time between grid points

With the Green function Cd however the maximum in the signal at any distance R arrives at the time

JlcrR 2

t=-_middot 4

moreover the signal that arrives at times much before this is exponentially small and can simply be neglected In fact it is easy to show that the natural time interval for diffusion on a 2-D numerical grid with spacing ~ is given by

873 Diffusion of EM Fields Into a 2-D Earth

_I_ cu1+1

I----shy E ~ 1-11

(T 11 (Tij+1

611

A ___

I I I I

-----0 1 j

I

I

61 i + 1

Ej bull I shy 1 I I I I I I

BL - -shy

E IIj I I I I I

----- JC

E11+1

IJj+li IT+ 1 j+ 1

Ei+ 1bulli

FIG 3 Typical grid point Ei j in the finite-difference grid surrounded by its neighbors and blocks of constant conducshytivity ABeD is the rectangle formed by joining the midpoints of the blocks surrounding EI

M = ~cr~Z (8)4

which is called the grid diffusion time (Potter 1973) Nominally equation (8) sets the maximum time step for an explicit solution of the diffusion equation but even this estimate is overly reshystrictive as suggested by the following argument

Consider instead of Gd the negative of its radial derivative

crR -OR o = ~ Il 4 1

47tl ~ e-loR2

which is the electric field induced by a 2-D source of the form

J(X Z t) = - ~JdR8(R)[1 - H(tn

where IdR is the strength of the source and 8(R) stands for the radial derivative of the 2-D delta function 8x)amp(z) The reason for considering such a source is that the spatial maximum of its induced electric field -OR G moves outward from the origin as time progresses (the spatial maximum of Gd in contrast is always at the origin) At time l the maximum of - CR G is at the radial position

s = J2l ~cr

and moves through the conductor with a velocity

J HH 1

0 Rma x = 2I-

which decreases with time as the electric field in the conductor becomes very smooth This slowing down of the diffusion rate with time suggests that in a numerical simulation of diffusion it should be possible to increase the time step as time increases according to an equation of the form

~l2 = c-~-R = (2~(Jt)z~ (9) 1 ma~

Such an approach is very useful for geophysical modeling where the response is first governed by the rapid diffusion of electric field through the weak conductors but is later conshytrolled by the slow decay of energy in the strong conductors As we show below the DuFort-Frankel method (DuFort and Frankel 1953) involves a time step closely related to Al 2

although it does require an unusual difference approximation to the diffusion equation

FINITE-DIFFERElCE EQUATIONS

Spatia) terms

A finite-difference approximation for the spatial terms in the diffusion equation (omitting source terms for now)

~1Cjt E = tu E + (1 E

can easily be developed by the integration method (Varga 1963 Vemuri and Karplus 1981 Hermance 1982) On a recshytangular grid this method produces discrete equations that closely resemble those of the finite-element method

Consider Figure 3 which shows a typical grid point Er i

surrounded by its nearest neighbors Ei T1 j Ei-Ij Eibull j + l

E j i I Integrating equation (1) over the rectangle ABeD (formed by joining the midpoints of the four rectangles in the figure) gives

If dx dz ~crc E = II dx dz ux E + Cz E)

ABeD ABeD

= r dx Cz E - JI dx 0 E JBC AD

+ ( dz ex E - r dz ex E (10)Joe JAB

The integrals above can be approximated as follows

II dx dz ~crcE ~ (crjjAZi~Xj + c Ijflz j + lAx)

ABeD

+ (jij+J~Z~Xj+1 + cri+l)+IAzl+l~x)+daEij

i dx (Axmiddot +1x ) (Emiddot - E )x cE ~ )+1 lj ij Be 2 ~Zi~ I

r dx az E ~ (Axj + ~x) + d (E i j - E j - J j)

JAD 2 ~z

r dz axE ~ (1z1 + flz i + l) (E1)+ - Etj) Joe 2 ~Xj+ I

and

_(6Zi+~Zi+I(Eij-Eij_l) z ex E I d AB 2 Ax)

Finally substituting these expressions into equation (10) and rearranging we obtain

1 (2~Zi+ IE ~ -I)~cJi)cEj= l1Z6zi+1 6z+6z1

26zmiddot )+ Ei+lj-2E i bull J -j + 1_ I

874 Orlstaglio and Hohmann

1 (MX j + 1+ poundjj-ILlxJ 6 x j + 1 6xj + 6xj 1

+ 26xj ) ( 11)Ax + Ax EibullJ+ 1 - 2E i J ) 1

Here crt J is the area-weighted average of the conductivities surrounding Ebull j

~rmal = 11 rmn (o-ij)12 (16)

where min (0) is the lowest value of Cr i j in the model This maximum time step is just the classical grid diffusion time ir 1

that is associated with a conductivity min laquo) and a scale length ~ in two dimensions [equation (8)]

6 = (Jij1Zi tiXj + (JjILZi+I1Xj + (Jjj+lI1Z j LXj + 1 -I- (Ji+ 1)+ ILZi+ lLX)+1

I (hZ j + LZj + 1KtiX + hx j + tlJ

while the right-hand side is just the standard (S-point) differshyence approximation to the Laplacian on an irregular grid

Time-stepping

The last step in discretizing the diffusion equation is to approximate the time derivative in equation (11) This topic has a vast literature Richtmyer and Morton (1967) for example describe 14 different schemes for just the 1-0 diffusion equashytion In this section we discuss some features of the timeshystepping for geophysical applications A more general treatshyment including both two- and three-dimensional models is given by Lapidus and Pinder (1982) Similar issues for timeshystepping the clastic wave equation are discussed by Emerman et al (1982)

Consider first a model with equal spacings Lx = tJz = A so that equation (11) becomes

c E = pound7+1) + pound7- 1 + E~ +1 + pound~ J - 1 - 4pound7 ( 12) t I IlcrjjLl 2

where we have added the superscript n to indicate the electric field at time t = ntir

The simplest approximation to the time derivative is a forshyward difference between times [ = ntu and t = (n + 1)AI

pound+1 _ E a pound -- J J + O6t) (13) I J -- At

which as indicated is accurate to first order in Ll Substituting into equation (12) and solving for pound7) 1 gives the explicit Euler method for marching the diffusion equation in time (Richtmyer and Morton 1967)

where

pound7 = (1 - 4r i)Ei i + rij(poundi+ i]

+ Ei- I j + Ei j + 1 + poundi j - d (14)

r _ hc I- -shy

j1ai j A2

(15)

is a dimensionless quantity called the local mesh ratio For a homogeneous model it is easy to show that the Euler

method is stable if the mesh ratio is less than or equal to t (Appendix A) when the mesh ratio exceeds i numerical errors grow exponentially and eventually swamp the correct solution Inhomogeneous models are more difficult to analyze for stabilishyty but keeping r j less than or equal to t everywhere is usually a safe approach Thus the maximum time step for equation (14)

is set by

The problem with the Euler method is that most geophysical models contain both weak and strong conductors and thus have both fast and slow diffusion times In a typical geoelectric section for example min (ai j) corresponds to the conductivity of the host rock or half-space which is usually about 001 Sm For a grid with 10 m spacing equation (l4) then gives maxishymum time step of about 3 x 10- 8 s Ore bodies however respond on a much longer time scale (Spies 1980 Mishra et al 1978) For example the natural modes of a sphere in free space decay exponentially (Wait and Spies 1969 Nabighian 1970) and the slowest decaying mode has a time constant given by

lar1=7

where (J is the conductivity of the sphere and r is its radius Ins

addition a simple argument given in Appendix B indicates that the largest time constant for 2-D rectangular block with sides L and L= is probably close to

~al 12D = -shyrr2

where I is the harmonic sum of L and L 1 I 1 ~=L2+2 ~ x Lz

Both of the above expressions give time constants on the order of milliseconds for typical ore bodies So the Euler method would require an enormous number of steps to compute the response of an are body in a half-space

There are many schemes for the diffusion equation that are stable for any time step ( unconditionally stable ) and are therefore more efficient than the Euler method The most accushyrate are the implicit methods such as the Crunk-Nicolson alternating direction and backward-difference methods which require matrix inversions at each time step (Lapidus and Pinder 1982 Goldman and Stoyer 1983) Even with implicit methods however large steps will only give accurate results at late times after diffusion has smoothed the electric field and the slower response of the strong conductors dominates If the early response of the model is interesting as we believe it is for geophysical models then a time step determined by equation (16) must be used at early times to track the rapid diffusion of the electric field through the weak conductors

The DuFort-Frankel method described below is well-suited to an approach in which the time step is changed as the computation proceeds Although less accurate than implicit methods the DuFort-Frankel method is both unconditionally

875 Diffusion 0 EM Fields Into a 2-D Earth

stable and explicit and is therefore very easy to program Still our numerical experiments indicate that the DuFort-Frankel method requires many time steps for accurate results [f only the late time response is desired an implicit method will probshyably be more efficient (Goldman and Stoyer 1983)

DuFort-Frankel metbod

The DuFort-Frankel method is the simplest of several methshyods for the diffusion equation that are explicit and unshyconditionally stable (Lapidus and Pinder 1982 Birtwistle 1968) On a regular grid the DuFort-Frankel method is also equivalent to the transmission-line-matrix or TLM method (Johns 1977) which is based on a network analogy to Maxshywells equations All of these methods achieve unconditional stability by adding a hyperbolic term to the diffusion equation

The DuFort-Frankel method involves only a slight modifishycation of the Euler method Consider instead of eq uation (13)

the more accurate centered difference approximation

e 1 _ e C E ~ + O(Llr2 ) (17)I J I J

I I J 211t

Although the higher accuracy seems promising it is easy to show that the combination of equations (17) and (12) which is called the leapfrog method is always unstable DuFort and Frankel (1953) noticed however that equation (17) and the

approximation

E+ 1 + pound-1 pound7) ~ J 2 t j + O(~(Z) (18)I

give an unconditionally stable method if substituted into (12)

(Both of the above results are derived in the Appendix A) The classical Dufort-Frankel method for a regular grid is

thus given by

En+1 _ poundn-l I j I

2M

pound7+ I) + pound7- 1 j + pound7) 1 + pound7 j_ I - 2(pound7~ I + pound7 j 1)

110 11 2 r I

(19a)

or solving for pound71I we have

1 - 4rmiddot 2rE~I = I e + ___1

11 1 + 4r I) 1 + 4rmiddot I I

(pound7 1) + E7-1J + pound7)+ 1 + pound~j- dmiddot (l9b)

The reason for the unconditional stability of the DufortshyFrankel method is quite simple Equation (19a) is actually the classical difference formula for the damped wave equation

211t 2

~ ~1 ~ 20 ~ DIIE + j1OcI E = DuE + ozrE ( )

where the coefficient of the hyperbolic term all E depends upon the grid spacings (To show this use centered difference apshyproximations for both 0rl and aI the terms involving pound~ J

cancel) The stability of the DuFort-Frankel method is thus governed by the classical Courant-Friedrichs-Lewy (CFL) conshydition (Richtrnyer and Morton 1967) for equations of the genshy

eral form

v- 2( E + a-zcE = iE + cE (21)

where r is the wave velocity and a is the diffusion constant As mentioned before the CFL condition implies that an explicit method for this hyperbolic equation is stable only if the space step L1 the time step I1rand the wave velocity rsatisfy

1 11 rlt-shy- v2 ~t

Comparing equations (20) and (21) shows that the OuFortshyFrankel method approximates a wave equation in which the wave speed v is exactly IJ 2At In other words the CFL condition is always satisfied by the DuFort-Frankel method changing the time step or the space step simply translates into approximating a new equation with a different wave speed Equation (20) also indicates however that the DuFort-Frankel method must be used with care because the wave-like solutions of this equation will obviously dominate the diffusive behavior if the time step is too large

Our previous analysis of the damped wave equation suggests some guidelines for the Dufort-Frankel method The Greens function for equation (20) follows directly from equation (3) by setting

1 2M 2 euro=-=--

IlC Z 1111 2

Equation (6) then indicates that diffusive behavior will domishynate equation (20) when

2 I ~ 4M (22)

11011 2

and the DuFort-Frankel method should give increasingly accushyrate results for the diffusion equation as equation (22) is satisshyfied to a greater and greater extent Making expression (22) an equality and solving for I1t gives an estimate of the maximum practical time step for the DuFort-Frankel method namely

112 ~ I1tmax = ()lCJt) 2 (23)

The DuFort-Frankel method thus allows large time steps if accuracy is only required at late times or large values of t for accuracy at early times equation (23) just gives ordinary grid diffusion time by setting t = 11t In modeling EM surveys a reasonable choice for I is 1 ms which is about when most transient EM systems begin recording For CJ = 10- ~ Sjm and ~ = 10 m as before equation (23) gives a time step of 177 x 10- 6 s which is still small but is about 50 times larger than the maximum step for the Euler method The examples given below show that this estimate is realistic

The DuFort-Frankel method can easily be generalized to an irregular grid by substituting equations (17) and (18) into equashynon (I I) The result is similar to eq uation (l9b) if we define the following averaged grid spacings

~ ~ _ I1zi + I + I1zmiddot i - I

2

- Llx)+1 + I1xJAxmiddot=----~ 2

and the following local mesh ratios for diffusion in the x and z

876 Orl81agllo and Hohmann

directions

~t rj j - _ A

lJ(Jij~ZiU=i+ 1

6t rf J llOij11X j11X j + 1

The generalized DuFort-Frankel method is then given by

- 4f tj En- 1ET 1

I J + 4f II

2r J (AZi n ~Zt + 1 n )+ =--Ei~I)+--=-Ei-l)

1 + 4r j j 11zi ~Zi

2rf (11X) n ~Xj+ 1 n )+ _ =- Eij+ 1 + -=- pound)-1 (24)1 + 4r i j Ax) Sx

where

- r1 J + rL r = 2

is the average of the mesh ratios in the x and z directions Equation (24) with the boundary conditions described below was used for all the computations in this paper Although we have not proved it our numerical experiments indicate that the generalized DuFort-Frankel method given by equation (24) is stable for any time step

Finally we note that the DuFort-Frankel difference equation can be applied to only half the grid points at any time level because the central point in the finite-difference stencil (Figure 3) is defined only by the time average equation (18) In two dimensions the method can be implemented as follows At times n111 where n is an odd number the difference equation (24) can be used to advance E7i I where (i + j) is odd to the time level (n + I) The new values can be used to advance E i

where (i + j) is even to the level (n + 2) and the cycle can then be repeated This requires that field be given initially on two time levels n = 0 and n = 1 but these values can easily be obtained by physical arguments or by using the Euler method to advance the values at n = 0 (which are always required for the diffusion equation) by one time step

BOUNDARY CONDITIONS

Since the electric field and its gradient are continuous at all boundaries in the 2-D TE-modc the only boundary conditions that need special treatment are the radiation conditions in the air and at the bottom and sides of the grid As we show in this section the radiation condition in the air is easily handled in the time domain (see also Goldman and Stoyer 1983) there is however no easy way to truncate the grid in the earth except to make it very large

Consider first the boundary condition in the air where under the quasi-static approximation the electric field satisfies Lashyplaces equation

GuE+czzE=O

The electric field in the air E(x z lt 0 t) can thus be computed from its value at the earth-air interface E(x Z = 0 t) by an upward continuation

E(x z lt 0 t) = _ ~ fX dx E(x Z = 0 t) (25) or 7( - o (x - X)2 + Z

Q

1 f dKc elxl z = 0 t) (26)Elx z lt 0 t) = -2 ilaquoraquo E(Kc Z

1t - ltL

where E(K x Z = 0 r) is the Fourier transform of the electric field at the earth-air interface

EIKc Z = 0 t) = Loc dx laquo iK x E(x z = 0 r) (27)

Q

When the Fourier integral in equation (26) is uniformly convershygent this equation implies that at z = 0

( ~ + IK x I) E = 0 (28) UI

which is an exact boundary condition on the norma] derivative of the field in the spatial-frequency domain (ti is the outward pointing normal derivative so G = -c z at z = 0) Noting that

IK x 1= (iK x)[ -i sign (K x )]

we obtain the following boundary condition in the space domain at z = degby an inverse Fourier transform of equation (28)

~ 1 fX cxE(xz=Ot) c Eiraquo z = 0 r) + - P dx = 0 (29)

1t a x-x-r

which relates the norma) derivative of E to the Hilbert transshyform of its tangential derivative (P stands for a principal value integral)

The boundary condition given by equation (28) or (29) can be approximated numerically in the following way At the first time step the electric field at z = 0 is given by the initial conditions and equation (25) or (26) can be used to compute the field a distance 6z above the interface Once these values are found the regular finite-difference equations (24) can then be used to advance the field at z = 0 to the next time level This cycle can obviously be repeated for any number of steps it corresponds to a standard approach for normal derivative boundary conditions (Lapidus and Pinder 1982) In our comshyputer program the upward continuation is done by fast F oushyrier transforms In addition since we always use a graded grid (Figure 1) we first interpolate the electric field to a uniform

spacing ~x with a cubic spline (deBoor 1978) The resulting upward continuation is very accurate but it is probably less efficient than a direct approximation of equation (25)

Goldman and Stoyer (1983) used a method simi liar to ours to handle the air layer in axisymmetric models but since their finite-difference solution was implicit it was necessary to solve an integral equation for the upward continuation Upward continuation is also closely related to the asymptotic boundary conditions used for frequency-domain modeling by Zhdanov et al (1982 and Weaver and Brewitt-Taylor (1978) In frequencyshydomain modeling a direct upward continuation is impossible since the field at the earth-air interface is not known until the problem has been completely solved These authors have thus developed local difference approximations of equation (28) in the x domain which can then be incorporated as ordinary boundary conditions on the difference equations

The method outlined above can be generalized to give a

-- --


boundary condition for the diffusion equation in the earth but the resulting equation is difficult to implement numerically Consider for example the boundary condition at the bottom of the grid (z = z) which we can assume is below all the inhomoshygeneities It is easy to show using Greens theorem that the field below the depth z can be computed from its value at z by the integral (Stakgold 1968 p 199)

~O(z - z ) I JCCE(x Z gt z t) = b de dx 41t 0 - 00

e -11ltx - XI) ( - b)2)4(1 - II)

) E(x z r)(t - t

In addition the boundary condition equivalent to equation (28) involves the Fourier transform of the electric field over both x and t

E(K z==z (j)=Joo dtfOC dxe-i(Kxx-OgttIE(x Z=Z t)x b b

- a - ao

and is given by

(n + Jillaro - K) E = 0

where the square root must be chosen to give decay as z goes to infinity Neither of these equations however has a simple approximation in the space-time domain since they both reshyquire values of the electric field at all times from degto t (see also

Israeli and Orszag 1981) To impose a boundary condition in the earth we have thus

simply used a large graded grid to move the boundaries far from the region around the source and have set the values at the bottom and sides equal to the analytical solution for a half-space which should be valid at suitably large distances

MODEL EXAMPLES

Half-space

As a first check of the numerical method we computed the response of a homogeneous half-space to the shut-off of a steady current in a double line source at the surface This example is especially useful since an analytical solution is available for both the electric field at the surface and in the half-space (Oristaglio 1982 Lewis and Lee 1981 Wait 1971) The analytical expressions are considerably simplified if we

define a normalized time variable T by

4t T=shy

~a

which has units of m For a single positive line source the

electric field E = E for t 0 is then given by

I (Z2 - x 2 2z2) e~R2T 2ze--2IT

E(xzt)=- ---+- -----shyR 2 R 2

1[0 T fiR 2

[_1 _2XF(XT- 1 2 )( +) JT l l 2 T R 2

2+ - x _ Z2 rro ~ erfc (ZT-

l 1

) (30)

Table 1 Node spcings for the finite-difference grid This grid was used for all the models described in the text

X -direction Z-direction

Nodes Spacing Nodes Spacing

1-10 240 I-54 10 10-14 120 54-59 15 14--19 60 59--64 30 19-25 30 64-69 60 25-33 15 74-79 240 33-167 10

167-171 15 171-177 30 177-182 60 182-186 120 186-195 240

where R = (x 2 + 2)12 is the distance from the source and the function F is Dawsons integral (lebedev 1972j

ld ( u) -- -Ill 2eve o

At the surface z = 0 it is easy to show that equation (30) reduces to the much simpler expression

1 1Etx Z = 0 t) = - 2 (l - e-~2T)

7tCf x

(X Z)III 1 agt (-1) (31)

= 41t ~ ~O (n + III T

From equations (30) and (31) and Faradays law in differshyen tial form

cIB = - cxE

IiB = Cz E

we also obtain the following expressions for the time derivashytives of vertical and horizontal magnetic fields at the surface

2

Cr Hz == - III 4 [_ xZT (- -- e 21tx ~(jX2

X1 + shyT

) ]- I

_ ~J 1 -x----2 2TCX t n=)

(-1) 111 (X 2) -

(n + 1) T

(32)

and

Table 2 Time steps and number of iterations at each step used for tbe blf--space model tbe half-space model (large contrast) and the halfshyspace and overburden model For the smaU contrast model each rime step was multiplied b) 3

Time steps

Iterations 1t

1000 400

10000

10472 x 10472 x 20944 x

10- 7

10- ti

10 - 6

878 Orl8tagllo and Hohmann

The agreement between the analytical and numerical solutions~l 2 1[T 1 2 (T)JoBlt=---- --F(xT- 1 2 ) 1+- 2 seems to be satisfactory the maximum error at any point is21tx ~ t X x v about 13 percent but the average error is considerably less

= ~~ ~ ox (-1)(2 + IX + I) (4X 2) 12 The grid for the numerical solution consisted of 79 nodes in

21tXJ1tt~o (2n+3) T the z-direction and 195 nodes in the x-direction with the grid spacings shown in Table I The sources were at nodes 48 ( - )

(33) and 98 (+ ) and thus were not placed symmetrically on the grid

For each of these equations the series expansion is much easier The time steps that were used are listed 10 Table 2 The initial

to evaluate for large T step is just the grid diffusion time for a 300 nom half-space at

In Figure 4 we have plotted the analytical and numerical the shortest grid spacing (10 m) Eventually this time step was

solutions for the vertical emf on the surface of a 300 n-m increased by a factor of 20 The computation required 8t hours

half-space By the vertical emf we mean - 0B or the emf of CPU time on a VAX 11780 computer

measured by a point coil receiver of unit area with its axis The only unusual feature of the numerical solution is that

vertical The source was a double line source with the positive sources were replaced by initial conditions on the electric field

limb located at x = 0 and the negative limb at x = - 500 m (for a justification of this replacement see Stakgold 1968 p

2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000

Distance from (+)-source (m) Distance from (+)-source Irn)

0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000

Distance from (+)-source (rn) Distance from (+)-source (m)

FIG 4 Comparison of analytical (solid line) and numerical solutions (squares) for the vertical emf ( - 0 B) induced by a double line source on a half-space Profiles are at (al 007 ms (b) 09 ms (c) 3 ms and (d) 21 ms after source current was switched off The positi ve line source is at the origin negative line source is at - 500 rn


203-204) To ensure a smooth start the conditions were set on Table 3 Late-time variation of the EM fields for a balf-space

the top rows of the grid a short time after the current was shut Single line source Double line source off but before the field had penetrated very far into the earth In

this case the starting time was 10 times the initial time step I~ ( _crll-~2) ltJ~2 (w 2 + 2xw)Electric heldThis procedure can also be used if the model is inhomogeneous 4rc I 8[2 32rc [2

because equation (30) with 0 set to the conductivity of the top layer will still be valid at early times

It is clear from equations (30H33) that early times for a half-space means

R2 1 or i laquo 1OR (34a)T~ 4

while late times implies

R2 2 or lP IlOR (34b)Tlt 4

This simple distinction based on the half-space conductivity

Elapsed Time = 1000 ms

o


(E)

_ltJ~2 x ltJ1l 2 wVertical emf ~ ifut

(-crBJ

-lcrL21132 I -lcr311151 (w 2 + 2xw)Horizontal emf 61t 3 1 [TI 201(32 [51

(-cB x )



FIG 5 Contours of electric field in a half-space after the current in a double line source at the surface was switched off Positive values indicate an electric field which points toward the reader negative values point into the page Contour values are in IlVrn The tic marks are 250 m apart The zero contour intersects the surface at the middle of the double line source which is 500 m wide

880

76

Oristagllo and Hohmann

10~

1 6ms e-e-e-e-e-e_e_e

e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms

middote e 0-0--0-0 9ms d 0- 0 - 9ms e-e-e-e-e_e eo --0-0

-e e _0middotV15ms e 0 bull0 15ms -e-e-e-e_e -bullbull P0-- (ro-o--O-OE -e-e-a_ _0 -0--0 - 0gt 103 20ms e-e_e e eOp 0-0_-0 0 20ms

3shy e e e 6 - rgtU E w middot~t g bull10shy(p gt

500m 300m bull bull I bull bull--i__bull

100m IT h =- 0033 Slim I 300mI10amp

IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0

Distance from source (m)

FIG 6 Profiles of the vertical emf (- at B ) for the half-spaceshyzconductor with a 1000 1 contrast Model geometry is shown at the bottom of the figure The positive line source is on the right Open circles indicate a negative value dark circles a positive value The convention is that B is positive if it points downshyward here positive emf thus corresponds to a magnetic field that points downward and is decreasing

and distance from the source is also useful for general models because the total response must tend asymptotically to that of the half-space at large enough times (Lee 1982) A convenient choice for R is 1000 m which is approximately the scale of an exploration survey With a 300 O-m half-space the division between early and late times then falls at about 1 ms which is the nominal value used for the discussion below

Since most transient EM surveys are recorded during the late-time regime of the half-space we have listed in Table 3 the leading order terms of half-space response for single and double line sources The vertical emf decays at the same rate (I - 2) for either sourcebut the horizontal emf switches from a slower decay than the vertical emf for a single line source (t - 32) to a faster decay for the double line source (r 52) It is interesting to note that these results do not match those for the voltages induced by a finite loop on the surface of a half-space where the vertical emf decays as t - 5r~ and horizontal emf decays as t 3 at late times (McNeill 1980) This discrepancy is discussed in more detail by Nabighian and Oristaglio (1984 this issue)

The electric field in the earth for a single line source is eventually given by

I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)

which for large enough 1 is approximately constant in any local region about the source 1t can also be shown from equations (30) and (35) that the maximum of current for a single line source is always directly below the source (ie x = 0) moreshyover at late times the maximum is at the depth

8 Jt hmax 9 ~(J

and moves with velocity

(I h 4 J 1 max 9 1t ~l(JC

Finally for the double line source the electric field in the earth at late times is

I 2 (1 1 8 z)E(x z t) - (w + 2xw) - 2 + ---r --sl (36) 10 2 T 5Jn T

Figure 5 shows snapshots of the electric field in the earth for a double line source as computed from the analytical solution These 2-D patterns closely resemble cross-sections through the smoke-ring that is generated by a finite loop on the surface of a half-space in three dimensions (Nabighian 1979) although there are differences in both the directions and rates at which the 2-D and 3-D smoke rings move through the earth (Oristashyglio 1982)

Half-space with conductor (large contrast)

Our second example of the finite-difference solution is shown in Figures 6 through 9 and consists of a 300 l-m half-space that contains a thin rectangular orebody 1000 times more conducshytive than its surroundings For this model the most interesting feature of the vertical emf profiles (Figure 6) is the crossover from positive to negative values which is the characteristic response of a thin vertical conductor (Lamontagne 1975 Boyd 1980 Kuo and Cho 1980) In transient surveys the crossover point is often used to indicate the approximate horizontal position of the body Here the crossover initially develops about 50 m downwind of the body (ie on the far side of the conductor away from the source) and then gradually moves back to the correct horizontal position

Figure 7 compares the finite-difference solution for this model with a solution obtained by a Laplace transform of the deeay spectrum (Weidelt 1982) In the transform method which is described by Tripp (1982) the electric field integral equation (Hohmann 1971) is first solved in the frequency domain for a broad band of real frequencies The response function is then analytically continued to the imaginary freshyquency axis where a Laplace transform converts to the time domain Overall the agreement between the finite-difference and transform methods is very encouraging

Figure 8 shows profiles of the horizontal emf (- Dt Bx) for this model which have the characteristic peaked anomaly caused by the currents in the thin vertical conductor According to the conventions used here the peak is really in the absolute value of the field since the horizontal emf is negative over the conducshy


-I 10

10 Fnte Oi flerenee Time 5 eppIng

------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(

r Ii (negrol Equation Founeraquo Tronform )(

OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500

DISTANCE FROM SOURCE (lftj

FIG 7 Comparison of the finite-difference and integralshyequation solutions for the large contrast half-space model

10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000

0 lt)bull 0 _-0 q0 lt)-o--o-)-middotoo 0- 00 -OO gms

UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r

600m 300mI bull I bull -_

100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600


FIG 8 Profiles of the horizontal emf ( - c Bx) for the large contrast model Open circles indicate a negative value ie Bx is negative (points to the left) and is decreasing in absolute value

r

tor ie B is negative-points to the left-and is decreasing in absolute value In contrast to the crossover point of the vertical emf the peak in the horizontal emf is always directly above the target in the millisecond time range and thus gives a better indication of the position of the target This feature of the horizontal profiles is caused in part by the rapid decay of horizontal emf for the pure half-space (Table 3) and indicates that in practice vector emf measurements are useful even for simple models In principle of course the horizontal and vertishycal emfs at the surface are not independent Since they are both derived from the electric field which satisfies Laplaces equation in the air one can be obtained from the other by a Hilbert transform [equation (29)]

Figure 9 shows the electric field in the earth for this model with snapshots running from very early (0006 rns) to late times (20 ms) Only the centra] uniform portion of the grid is disshyplayed The grid lines in the figure are 50 m apart although the numerical grid had a 10 m spacing in this region The color scale for the contours is shown on the left of each snapshot while the cross-hatching on the contours indicates a negative value a positive electric field points out toward the reader and a negative electric field points into the page In addition at the top of each snapshot we have plotted the surface profiles of vertical emf on a linear scale which reveals a little more strucshyture than the logarithmic scale used in Figure 6

The 2-D smoke ring along with its characteristic double crossover in the vertical emf dominates the early patterns The contours are distorted however by the inability of the electric field to penetrate the conductor at early times This is easily understood as the time-domain version of the skin effect-at very early times the rate of diffusion into the conductor is small lmpared to the rate of diffusion in the half-space (in the ratio

(Jcr) so the contours must flow around the body The snapshot at 06 ms shows part of the transition from

early to late times On the left the smoke ring has become very diffuse while on the right it has interacted with the conductor and generated a large response from the top of the body which is starting to dominate the electric field in the earth A crossshyover associated with the body response has appeared on the surface about 50 m downwind of the body Moreover on these linearly plotted profiles it is clear that the current in the target body also causes an inflection in the profile shape which stays nearly fixed over the target The late time snapshot at 37 ms shows a fully developed target response The inflection point on the profiles has merged with the crossover and the further evolution of the electric field involves its gradual equalization and decay within the conductor

Overburden and half-space with conductor

It is well-known both from field work (Lamontagne et aI 1978) and model studies (Lee 1975 Hurley 1977 Kaufman 1981 Spies 1980) that overburden blanks out the response of underlying targets at early times Qualitatively this effect can be understood from Figure 10 which shows the evolution of the electric field in a model with a 10 nom overburden that is 100 rn thick The half-space resistivity in this model is still 300 Q-m but the body resistivity was lowered to 03 nom to give a large contrast with the overburden

The most obvious effect of overburden is to slow down the development of electric field in the earth At 01 ms for examshy

_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I

Elapsed Time = 0036 ms F Iaps ed Time -- 01--- - - -- middotmiddot middot~- middot middot _ middot--~middot middot~middot-~-middot ---r -- -rt1--middot _ middot 1-Ir- bull shy

b bull ~ 1 1 r 0shybull~ Q

z

Ibullo

lI ~

FIG9 Cross-sections of the electric field in tbe earth for large contrast model Only tbe central uniform region of the numerical grid is shown The grid lines are 50 m apart and comprise 5 points on the numerical grid The positive source is marked by the cross The color intensity scale for tbe electric field values is coded to the color bar on the left of the figure the values on the scale are the logarithm of the electric field (absolute value) in Vm Cross-hatching on the contours indicates a negative value which points into the page At the top of eaeh snapshot is the profile of the vertical emf on a linear scale the values are in IlV1m2

bull

188

c f 51 III IE

-IIt Jbull ~ c i

()

FIG 10 Cross-sections of the electric field in the earth for the overburden model Sec the caption on Figure 9 for further details on the plotting conventions E

t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i

l88 IDJII3 o-e II Olul Pla~ ft3 10 UOftIIIO

888 Orlatagllo and Hohmann

100 OJ

-0 - o-)middot-0 0-00000

o- o- tgt0-0O ltgt010 o om

9ma10-2 middot-e-e-middot-middote

Sm ~e-e~ - bull -e-e-middot--~e 15m ~

E

a-gt bull ISmae-e-e-e- Sms ) _OmiddotO--)--Ou ~e OJ 34m

~ 34ms lt~~~8~gsm8 ij 0 bull J~ 0 emsamp

lt0 6bull ~ 10middot

6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~

lIb = 333 Slim --20m

1500 2100 2700 3300 3900 4600


FIG 11 Profiles of the vertical emffor the overburden model in the ms range (open circles are negative values dark circles are positive values)

100

00 00

10 0

-e-middot_middot~middot-__~mabull b

~ )-o-o--omiddotmiddotomiddoto 5m Omiddot-OmiddotO-)middotmiddotCJmiddoto 0-- 0 -0- 0 0 -0 0 ~ 102

(j-o -)-o-o -00U

_0 -000 1Sm~ otY 0--0--00-00-00 00 1ij J) 0middotmiddot(7 00-0-0-00 Qo 25m g o 0 00 -00~-O

0 0 0 00-0 O ryomiddoto- 34momiddotmiddot~cmiddot

~ 10 0

~

SOOmet-2 4

AI HI (j 0=-1 sim10 IIh 0033 sim

lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600

Distance from source (ml

FIG 12 Profiles of the horizontal emf for the overburden model (open circles are negative values dark circles are negative values)

ple most of the electric field is still near the surface where it diffuses very slowly through the conductive overburden In addition the different rates of diffusion through the overburden and half-space initially cause a strong refraction of the contours at their interface

The slow response of the overburden also complicates the surface profiles At 47 ms for example the crossover is posishytioned near the body but it is still due mainly to currents in the overburden The interaction between the overburden and body is very strong between 47 and 68 msbut by 89 ms the electric field in the body has begun to dominate Its crossover appears nearly 100 m downwind and gradually moves back but much more slowly than in the model without overburden Nevertheshyless the profiles in Figure 11 show that by 34 rns which is within the range of most transient systems the crossover is only 15 m away from the target

The early profiles of the horizontal emf (Figure 12) are also complicated by overburden response but there is a distinct anomaly on the later profiles which gives an accurate indicashytion of bodys position In fact the characteristic peak above the target first appears at about 7 ms although this profile is not shown in the figure

Half-space conductor (small contrast)

In both of the preceding examples the surface profiles were eventually simple enough that the target could be identified by a qualitative interpretation In general the appearance of a clear target response depends upon both the relative strengths of the response from orebody half-space and overburden and on the interaction between these responses Our final example which is shown in Figures 13 through 16 is a small contrast (100 1) version of the second model and illustrates some of these features For this run the half-space resistivity was 100 n-m while the body resistivity was 1 n-m

Although delayed in time the early evolution of the electric field (Figure 13) is very similar to the large-contrast example The snapshots at 032 and 095 ms show some additional details of the interaction between the smoke ring and the conductor which occurred very quickly in the large-contrast example More interesting however is that the body response never really dominates the surface profiles The crossover in the vertishycal emf between 1 and 10 ms seems to be caused by the currents flowing in the target but it moves to the right away from the target which indicates a strong contribution from the currents in the half-space Moreover by 167 ms the crossover has disappeared and as shown by the profiles in Figure 14 it does not reappear by 35 ms which was the latest time computed for this example

The profiles shown in Figure 14 are difficult to interpret directly but the effect of the body can easily be seen from Figure 15 which shows the vertical emf over the half-space to the left of the loop Here the crossover due to the smoke ring in the half-space is present at 1 ms but by 5 ms it has moved far to the left leaving relatively fiat profiles compared to those over the target

As in the previous examples the profiles of the horizontal emf (Figure 16) have a simple structure These profiles still peak directly above a low-contrast target but size of the peak is small and it could easily be obscured by noise

Elapsed ie ~ O ~08 ms Elapsed 4 middot_middot~--- f-----~-middot ~lI _ ~ ~ ~ ~ ~ ~ ~ CQrmiddot -

~--~- f ~ ~ bull ~

---- - ~kt () middot1-lt H i l-+=Y- ~ f ~ t ~ -v t 1 -i1 t

~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)

FIG 13 Cross- sections of the electric field in the earth for the small-contrast (100 I) model See the caption on Figure 9 for further details on the plotting conventions I

UU8WllOH pUB 0lla-IJO 068

891

105

Diffusion of EM Fields Into 8 2-D Earth

10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe

lms----shy ~ 0middot0middotmiddot00lt)-0

- 0 5ms F ------- 6 5ms

middot~lt)o~~------ o~ lt ~ms15ms P o oshy

------_-- O

- ~ 025ms bull --t - t- e- - --- 15ms 35ms bull bullbull-e__ j 0

middot -middot -middot -middot middotmiddot -middot -middot -middot -middot _ -t_~ ___~_ - - ---~-

p

bull 500m I 300m o

ltll---i-~100mIr h=Olsim

~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500

Oistsnce from source (rn)

fIG 14 Profiles of the vertical emf for the small-contrast (100 1) model (open circles are negative valuescdark circles are positive values)

10shy _bullbullbullbull e bullbull eshybull-eshy

lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-

~ 3shy 9mB_ __----e-ebull-e- u

~ LlJ e 15ms bull ---- - bull -- - - bull- t - - e -i6 10middotl g

25m5 bull t___t_t__t-t-t-t-t~ bull35m5 t ____e___e_ e_ e_

10middot

rr =01 SImh

3600 3000 2400 1800 1200 600

Distance from left line source (m)

FIG 15 Profiles of the vertical emf over the region to the left of the loop for the small-contrast (100 1) model shown in Figure 14 (open circles are negative values dark circles are positive values)

DISCUSSION

Most features of the models presented above have been observed both in field work and other model studies What is surprising at least to us is how strongly these features are controlled by induction effects that seem to occur independentshyly in host rock overburden and orebody as illustrated by the electric field in the earth Only in the last example was there significant interaction between the target and its surroundings that persisted through late times This lack of interaction is

caused in part by the 2-D approximation where the vector nature of the EM field does not come fully into play and important effects like current channeling are absent However it also seems to be a fundamental properly of time-domain induction which is connected to the large-time asyrnptotics of the diffusion equation (Kaufman 1981 Lee 1982) At present most practical transient EM interpretation is based on the late-time response because of its simplicity but also because the response is very difficult to measure accurately at times much before the millisecond range Nevertheless the early-time reshysponse deserves closer attention since it obviously depends strongly on the geometric features of the model

The finite-difference method given here is easy to program and gives accurate results but it is much too slow for routine modeling middotthat would allow a detailed analysis of the features mentioned above There are several ways by which the speed of the method could be increased The best way would be to solve the finite-difference equations for the secondary electric field in

the earth defined as the difference between the total field and the field of a backgroundmodei For example defining Eh as

10 0 middoto--0-middotJmiddot Q -Q 00omiddot o_omiddotmiddot)middotmiddotr 0-00- 000 1ms

0--o- )-J o- -00 Omiddot() O -0- _JmsJ

bully (Y-0 Y deg - 0- 0- - 0- - c- -0- -0 _0- -0 -0--0- _gm0 CrO

e 0-ltgt- _~_o-middot 0- -0-- 0middot0--0- -0 _0- -0 0- 0- _8 bull~ IJO

J o middot ma- -Cr--0 cmiddotu

~ w o_-o-gto IJmiddot 0 )-cr Omiddot ) bullbull o 0 -8bull () 15ms

O-Cr- - 0 -00

pound

] 103 sect ~ _-o-oltJ--middotO)-cr 0 -00-00_0 25ms

c-gt C 0-0-0

0 lt)_ - omiddot -o--0 () ) () Omiddot -0-0 35msoo-() bullo -0--00

50Om 300mI 11~O-lt I middot t 100m

(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500


FIG 16 Profiles of the horizontal emf for the small-contrast (100 1) model (open circles are negative values dark circles are positive values)

10-5

892 Orlst_gUo _nd Hohmann

the electric field for a half-space of conductivity cr we obtain the following differential equation for the secondary electric

field E == E - pound10

~cra E - an E$ - au E = -Il(cr - cr)o E (37)

Sources for E now exist in the inhomogeneities where c f 0 and are proportional to the time derivative of the half-space field These source terms must be included directly into the difference equations but otherwise the discretization of equashytion (37) is straightforward Upward continuation of E can be used for the boundary condition at the earth-air interface while E can simply be set to zero at distant boundaries of the grid in the earth Moreover since the secondary field varies more slowly than the total field in both space and time it can be modeled on a relatively coarse grid Our initial tests indicate that the computation time can be reduced by about a factor of five without any loss in accuracy

Another obvious improvement would be introduction of an aysmptotic boundary condition in the earth that allows the use of small grids In two dimensions the improvement would not be dramatic because the storage requirements for the timeshydomain solution are modest Only one value of the field is stored at each grid point so the 79 x 195 grid used here reshyquires only about 15 K real words of storage In three dimenshysions however storage becomes a problem because of both the extra dimension and the vector nature of the EM field A 79 x 195 x 195 grid for example requires about 10 million real words of storage which would tax even a very large computer Nevertheless the storage needed for a time-domain solution is still substantially less than that for a frequencyshydomain solution on the same grid (see eg Pridmore et al 1981 and Lee et al 1981)

The most general approach to numerical boundary conshyditions is that of Bayliss and Turkel (1980) and Kriegsrnann and Morawetz (1980) who have developed extended versions of the Sommerfeld radiation condition that allow the grid boundshyaries to be set very close to the inhomogeneities Both papers however deal with wave equations and further work is needed to adapt these boundary conditions to the diffusion equation

Time-domain modeling in three dimensions also involves a number of issues which have yet to be resolved even for frequency-domain modeling Among these are whether it is best to reduce Maxwells equations to a second-order equation for the electric field or for the magnetic field (Pridmore 1978) or to work directly with the coupled first-order equations The last approach has been used with great success in computing EM scattering from objects in free space (Tafiove 1980 Yee 1966) but at much higher frequencies than are involved in EM exshyploration

ACKNOWLEDGMENTS

This research was begun while one of the authors (MLO) was a postdoctoral research associate at the University of Utah most of the work was completed while this author was at Yale University We thank the National Science Foundation which provided much of the support for the research described here under NSF grant EARmiddot7912447

The project benefitted from discussions with many colshyleagues In particular we thank M Nabighian of Newmont Exploration Ltd (Tucson Arizona) and S H Ward of the

University of Utah both for their encouragement and for nushymerous helpful suggestions throughout the project

Finally we thank 1 Roberts of Schlumberger-Doll Research who wrote the software for displaying the contour plots of the electric field in the earth

REFERENCES

Bayliss A and Turkel E 1980 Radiation boundary conditions for wave-like equations Comm Pure App Maths v 23 p 707-725

Birtwistle G 1 1968 The explicit solution of the equation of heat conduction Comput J v 11 p 317

Boyd G W 1980 Newrnont EMP surveys over Elura The geophysshyics of the Elura orebody D M Emerson Ed Sydney Austral Soc Expl Geophys p 153-160

de Boor C 1978 A practical guide to splines New York SpringershyVerlag

Dickson G 0 and Boyd G W 1980 Newmont transient electroshymagnetic system Bull Austral Soc Expl Geophys v 11 p 47-51

DuFort E C and Frankel S P 1953 Stability conditions in the numerical treatment of parabolic differential equations Math Tables and Other Aids to Cornput (former title of Mathematics of Computation) v 7 p 135-152

Emerrnann S H Schmidt W and Stephen R A 1982 An implicit finite-difference formulation of the elastic wave equation Geophysshyics v 47 p 1521-1526

Goldman M M and Stoyer C H 1983 Finite-difference calculashytions of the transient field of an axially symmetric earth for vertical magnetic dipole excitation Geophysics v 48 p 953-963

Hermance1 F 1982 Refined finite-difference simulations using local integral forms Application to telluric fields in two dimensions Geophysics v 47 p 825-831

Hohmann G W 1971 Electromagnetic scattering by conductors in the earth near a line source of current Geophysics v 36 p 101-131

Hurley D G 1977 The effect of a conductive overburden on the transient electromagnetic response of a sphere Geoexpl v 15 p 77-85

Israeli M and Orszag S A 1981 Approximation of radiation boundary conditions J Camp Phys v 41 p 115-135

Johns P B 1977 A simple explicit and unconditionally stable numerishycal routine for the solution of the diffusion equation Int 1 Num Meth Eng v II p 1307-1328

Kamenetzky F M Ed 1976 Instructions in the use of transient methods in mining geophysics Nedra (in Russian)

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

Kovalenko V F 1961 An applied method for recording transitional processes in the south Urals Soviet Geol v 4 p 89-101

Kriegsrnann G A and Morawetz K 1980 Solving the Helmholtz equation for exterior problems with variable index of refraction I Siam J Sci Stat Cornput v 1 p 371-385

Kuo J T and Cho D-H 1980 Transient time-domain electroshymagnetics Geophysics v 45 p 271-291

Lamontagne Y 1975 Applications of wideband lime domain EM measurements in mineral exploration PhD dissertation University of Toronto

Lamontagne Y Lohda G Macnae J and West G F 1978 Toshywards a deep penetration E-M system Bull Austral Soc Expl Geophys v 9 p 12-17

Lapidus L and Pinder G F 1982 Numerical solution of partial differential equations in science and engineering New York 1 Wiley and Sons

Lebedev N N 1972 Special functions and their applications New York Dover Publ Inc

Lee T 1975 Transient electromagnetic response of a sphere in a layered medium Geophys Prosp v 23 p 492-512

--- 1982 Asymptotic expansions for transient electromagnetic fields Geophysics v 47 p 38-46

Lee K H Pridmore D F and Morrison H F 1981 A h~brid three-dimensional electromagnetic modeling scheme GeophYSICS v 46 p 79fr805

Lewis R and Lee T 1981 The effect of host rock on transient electromagnetic fields Bull Austral Soc Expl Geophys v 12 p

5-12 McNeill J D 1982 EM37 Ground transient electromagnetic system

Design features Technical Notes Geonics Ltd Ontario --- 1980 Applications of transient electromagnetic techniques

Technical Note TN-7 Geonics Ltd Ontario Mishra D C Murphy K S R and Narain H 1978 Interpretation

893 Diffusion of EM Field Into a 2-D Earth

of time-domain airborne electromagnetic (IN PUT) anomalies Geoexpl v 16p 203-222

Mitchell A R and Griffiths D r 1980 The finite difference method in partial differential equations New York John Wiley and Sons

Nabighian M N 1979 Quasi-static transient response of a conducshyting half-space An approximate representation Geophysics v 44 p 1700-1705

--- 1970 Quasi-static transient response of a conducting sphere in a dipolar field Geophysics v 35 p 303-309

Nabighian M N and Oristaglio M L 1984 On the approximation of finite loop sources by two-dimensional line sources Geophysics v 49 this issue p 1027-1029

Oristaglio M L 1982 Diffusion Of electromagnetic fields into the earth from a line source of current Geophysics v 47p 1585-1592

Parasnis D S 1979lt Principles of applied geophysics London Chapman and Hall

Potter D 1973 Computational physics New York John Wiley and Sons

Preston B 1975 Review-Difficulties for the electromagnetic method in Australia Geoexpl v 13 p 29-43

Pridmore D F 1978 Three-dimensional modelling of electric and electromagnetic data using the finite-element method PhD dissershytation Univ of Utah

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

Richtmyer R D and Morton K W 1967 Difference methods for initial value problems New York John Wiley and Sons 2nd ed

Spies B R 1976 The transient electromagnetic method in Australia Bur Min Resources 1 Austral Geol Geophys v I p 23-32

--- 1980 Interpretation and design of time-domain EM surveys in areas of conductive overburden il The geophysics of the Elura orebody D MEmerson Ed Sydney Austral Soc Expl Geophys p130-139

Stakgold I 1968 Boundary value problems of mathematical physics Volume II New York John Wiley and Sons

Taflove A 1980 Application of the finite-difference time-domain method middotto sinusoidal steady-state electromagnetic-penetration probshylerns Inst of Electrical and Electronic Engineers Trans Electromag Cornpat v EMCmiddot22 p 191-202

Tripp A c 1982 Multidimensional electromagnetic modeling PhD dissertation Univ of Utah

Varga R S 1963 Matrix iterative analysis Englewood Cliffs Prentice-Hall Inc

Vernuri V middotand Karplus W J 1981 Digital computer treatment of partial differential equations Englewood Cliffs Prentice-Hall Inc

Wait J a 1956 Method ofgeophysicalex~lorationUS Patent No 2735980 (To Newmont Mining Corporation February 211956) -- 1971 Transient excitation of the earth by a line source of

current Proc Inst of Electrical and Electronic Engineers Lett v 59 p1287-1288

Wait J R and Spies K P 1969 Quasi-static transient response of a conducting permeable sphere Geophysics v 34 p 789-792

Ward S H 1979 Ground electromagnetic methods and base metals Geophysics and geochemistry in the search for metallic ores P J Hood Ed Ottawa Geological Survey of Canada p 45-62

Ward S H Peeples W 1 and Ryu J 1973 Analysis of geoshyelectromagnetic data Meth Comput Phys v 13 B A Bolt Ed New York Academic Press p 163-238

Watts R 1972 finite element models of rnagnetolluric fields over axial structures PhD dissertation Univ of Toronto

Weaver J T and Brewitt-Taylor C R 1978 Improved boundary condition for the numerical solution of Espolarization problems in geomagnetic induction Geophys J Roy Astr Soc Y 54 p 309shy317

Weidelt P 1982 Response characteristics of coincident loop transient electromagnetic systems Geophysics v 47 p 1325-1330

Wolf K B 1979 Integral transforms in science and engineering New York Plenum Press

Yee K S 1966 Numerical solution of initial boundary value probshylems involving Maxwells equations in isotropic media Institute of Electrical and Electronic Engineers Trans Ant Prop v AP-14 p 302-309

Zhdanov M S Golubev N G Spichak V V and Varentsov Iv M 1982 The construction of effective methods for electromagnetic modelling Geophys J Roy Astr Soc v 68 p 589-607

APPENDIX A

The stability of most finite-difference equations can be anashy

lyzed by a Fourier method first used by yon Neumann This

method is not the most general since it assumes constant coefficients and periodic boundary conditions bu t it is simple

to apply and always yields a necessary condition for stability which in many cases is also a sufficient condition (Lapidus and Pinder 1982)

If we let stand for the sum over nearest neigh bors

r E j = pound+ 1 J + pound7- 1 j + pound7 j + 1 + pound j - 1

then the Euler leapfrog and DuFort-Frankel methods for a

homogeneous model with equal grid spacing in x and z can be written

Euler

pound7 1 = (1 shy 4r)pound~j + rtEL (A-I)

Leapfrog

pound7 J 1 = pound7) 1 + 2r(t poundL shy 4pound1 )) (A-2)

DuForr-Frankel

+ 1 1 - 4 - 1 2 ~ E A 3) Ej ) = -- Ej j + -- L- j igt ( shy 1 + 4r 1 + 4r

where r = ~r(cr~~2) is the mesh ratio

To analyze the stability of these equations we substitute for pound7 )a Fourier mode of the form

E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)

where An is the amplitude of the mode at time t = nM It is then

easy to show by standard trigonometric identities that

tE7 j =i fE7j bull

where

A = 2cos Kx~ + cos K ~

Thus substituting equation (A-4) into the Euler difference equation (A-I) we obtain

pound7 = (l - 4 + ))E7)middot (A-5)

The quantity in parentheses which is a function of r K and K is the amplification factor of the Euler method

gf == I - 4r + rt (A-6)

A necessary condition for stability of the Euler method is that

-1 ~ gf ~middot1 (A-7)

for all K x K z since any mode for which I IEI gt 1 will obvishyously grow without limit in time Since r gt 0 and ) ~ 4 the

upper bound in equation (A-7) is always satisfied The lower bound gives the stability condition

T min(4 ~ J or

1 r lt- (A-8)- 4

since the minimum value on the right-hand side occurs when K = K = TC~ and A = -4 Note that this stability condition

[equation (A-8)J is determined by the stability of the checkershy

894 OrlatagUo and Hohmann

board mode of plus and minus values

pound7 j = An cos in cos jt

= A n ( -Ir( -tV

Although this mode may not be present in the actual initial conditions on E it will always be generated by rounding errors in the computer so the stability condition (A-8) holds in genershyal

Both the leapfrog and Dufort-Frankel methods are threeshylevel schemes involving pound7 J 1 as well as pound7~ I and E7 j and can be treated by a slight modification of the previous analysis Consider first the leapfrog method We can again use equation (A-4) for pound7 i but we now set

pound7 ~ = 9 L pound7 J I

and

E- = 9L 1e j

Substituting these expressions into equation (A-2) we obtain a quadratic equation for the amplification factor (h of the leapshyfrog method

gi - 2r(A - 4)gL - 1 = 0

which has the solutions

4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)

For stability both of these solutions must have a magnitude less than 1 for all K x K z But this is impossible no matter how small the mesh ratio r is since

g + gZ = r(A - 4)

and

gZ laquo = -l

Therefore the leapfrog method is always unstable Finally the same method as above gives the following amplishy

fication factors for the DuFort-Frankel method

+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r

A straightforward check shows that both IgF I and IgVFI are bounded by 1 for all K x K z regardless of the value of r so the Dufort-Frankel difference equation is unconditionally stable

APPENDIX B

To estimate the decay constant of a 2-D rectangular block we can consider the simple model of a block of sides L~ and L and conductivity o which is surrounded by a perfect conducshytor We then have for the electric field inside the block

V 2E - cr~C E 0 (B-l)

and the boundary conditions

pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)

For these homogeneous boundary conditions an arbitrary inishytial condition can be represented by a double sum over the sinusoidal modes

mrrx nnz SIn -- SIn-

t t

where m and n are arbitrary positive integers Since these modes are orthogonal over the block we can

consider them independently Thus taking the initial conshyditions to be

mnx mrz E(x z 0) = Am SIn -- SIn -- (B-4

L L

and setting

Eix Z mnx nnz

t) = f(t)A m n sin - sin -i i

we obtain forf(t)

f(t) = Am n e - t r (B-S)

where

r _ CJ~[2 mn - n 2 middot n (B-6)

and

2 21 m n~=-+2 (B-7)L n l- L

The first-order mode m = n 1 decays at the slowest rate and gives the estimate for the largest time constant of the twoshydimensional rectangular block

tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i

Since homogeneous boundary conditions at the sides of the block are unrealistic for bodies in free space equation (B-8) should only be taken as a crude estimate of 2middotD deeay constant Nonetheless it is interesting to note that in the limit as the block becomes a finite slab (L -- (0) equation (B-8) gives for the largest time constant for a I-0 conductor of width L

cr~e tID=~

which is exactly the time constant of the slowest decaying mode of a sphere of radius L in free space (Nabighian 1970)

bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull

871 OIf1uslon of EM Fields Into a 2-D Earth

V 2EAIR = ay~X

zo=O 1

I

I I I I I I I I t

x J

1

z2

I

Xl x 2

FIG Two-dimensional model geometry including the finite-difference grid x) labels grid points in the x-direction z labels grid points in the z-direction The black dots represent one line of grid points in the air layer which are needed for the boundary condition at the earth-air interface

numerical boundary condition at the earth-air interface that avoids having to include the air layer in the grid

These results are derived after a short analysis of the role played by the diffusion equation in transient EM modeling Four model examples follow the description of the numerical method They are (I) a half-space (2) a half-space with a thin vertical conductor-large (1000 I) contrast (3) a half-space with both overburden and a thin conductor and (4) a halfshyspace with a thin vertical conductor-s-small (100 1) contrast

For each of these models we have computed the negative step response for the emf or the response obtained by switching off a steady current and observing the decaying magnetic field with a point coil receiver of unit area Most time-domain systems measure approximately this response and the response to a general waveform can of course always be obtained from the step response

Direct time-domain modeling is not new but our approach differs from the previous work by Kuo and Cho (1980) who also considered the 2-D Turam geometry and Goldman and Stoyer (1983) who dealt with axisymmetric models (see also Watts 1972 who computed magnetotelluric responses in the time domain) Some of the differences are in technique For example Kuo and Cho (1980) used a finite-element method for the damped wave equation we use a finite-difference method for the diffusion equation Goldman and Stoyer (1983) used an implicit finite-difference method for the diffusion equation we use an explicit method The most important distinction howshyever is that here we are mainly interested in the qualitative features of the models particularly in the patterns of diffusion

in the earth We have thus included a large number of contour plots showing how the earths electric field evolves in time To us this is the most appealing feature of numerical time-domain modeling since it gives a new perspective on EM induction in the earth

DIFFUSION EQUATION

Maxwells equations for the 2-D TE-mode where

E(x z t) = E y and

H(x z t) = flxx + Hzz

reduce to the following scalar equation for the electric field in the strike direction (letting E = E)

cxxE + czzE - jlcrctE - jlpoundcttE = jlcJ (1)

The model geometry is shown in Figure I The conductivity o = otx z) can vary blockwise over the (x z) plane while the permeability u and permittivity E are constant at their freeshyspace values jl 41t X 10- 7 Him and c = 8854 x 10- 12 Fyrn

J s is the density of the source current in the y-direction Equation (I) is a damped wave equation but in the earth

where a is normally much larger than e the wavelike features of the electric field vanish very quickly leaving only the diffusive behavior Dropping the term containing I in equation (1) ie neglecting displacement currents gives this diffusive limit with E satisfying the diffusion equation

872

120


~ a

100 I

80

] ~ 80 ~ ~

~ 40 U J1 w

20

00) 20

-00 90 80 70 80 -60

Log Time Isgt

I0

I L

~ 12000120

b

10000100 ( f

800080

II

~~

amp000110

2~

~

I J

U g 4000

l6 E 40

~ Ww

20 2000 1 II __JJ oo~~~~-~00 ~-2000

-40 -30 20 -00 -90 middot80 70 110 -~ 0 00 -30 20 100 90 80 70 -00 -~ 0 40 -30 20 20







J(X z t) = lS(x)8(z)[1 - H(t)]




(3)

where

R = (x 2 + z2)12


c = (J-lE)-1I2





41tt G ( ) (5)


2poundR 2C 2)l i2(t2 _ ~ - (6)

a



1 1 1tlt-shy (7)- 12 c

v



JlcrR 2

t=-_middot 4



_I_ cu1+1

I----shy E ~ 1-11

(T 11 (Tij+1

611

A ___

I I I I

-----0 1 j

I

I

61 i + 1


BL - -shy

E IIj I I I I I

----- JC

E11+1

IJj+li IT+ 1 j+ 1

Ei+ 1bulli


M = ~cr~Z (8)4




47tl ~ e-loR2


J(X Z t) = - ~JdR8(R)[1 - H(tn


s = J2l ~cr


J HH 1

0 Rma x = 2I-


~l2 = c-~-R = (2~(Jt)z~ (9) 1 ma~




Spatia) terms


~1Cjt E = tu E + (1 E






ABeD ABeD





ABeD




JAD 2 ~z


and









~rmal = 11 rmn (o-ij)12 (16)






Time-stepping








where



r _ hc I- -shy

j1ai j A2

(15)



is set by


lar1=7



~al 12D = -shyrr2











e 1 _ e C E ~ + O(Llr2 ) (17)I J I J

I I J 211t


approximation




thus given by


2M


110 11 2 r I

(19a)


1 - 4rmiddot 2rE~I = I e + ___1

11 1 + 4r I) 1 + 4rmiddot I I



211t 2




eral form

v- 2( E + a-zcE = iE + cE (21)


1 11 rlt-shy- v2 ~t



1 2M 2 euro=-=--

IlC Z 1111 2


2 I ~ 4M (22)

11011 2


112 ~ I1tmax = ()lCJt) 2 (23)




2




directions

~t rj j - _ A

lJ(Jij~ZiU=i+ 1



- 4f tj En- 1ET 1

I J + 4f II


1 + 4r j j 11zi ~Zi


where

- r1 J + rL r = 2




BOUNDARY CONDITIONS



GuE+czzE=O



Q


1t - ltL



Q


( ~ + IK x I) E = 0 (28) UI





1t a x-x-r






-- --




e -11ltx - XI) ( - b)2)4(1 - II)

) E(x z r)(t - t



- a - ao

and is given by





MODEL EXAMPLES

Half-space



4t T=shy

~a



I (Z2 - x 2 2z2) e~R2T 2ze--2IT

E(xzt)=- ---+- -----shyR 2 R 2

1[0 T fiR 2

[_1 _2XF(XT- 1 2 )( +) JT l l 2 T R 2


l 1

) (30)




1-10 240 I-54 10 10-14 120 54-59 15 14--19 60 59--64 30 19-25 30 64-69 60 25-33 15 74-79 240 33-167 10

167-171 15 171-177 30 177-182 60 182-186 120 186-195 240




1 1Etx Z = 0 t) = - 2 (l - e-~2T)

7tCf x

(X Z)III 1 agt (-1) (31)

= 41t ~ ~O (n + III T


cIB = - cxE

IiB = Cz E


2


X1 + shyT

) ]- I

_ ~J 1 -x----2 2TCX t n=)

(-1) 111 (X 2) -

(n + 1) T

(32)

and


Time steps

Iterations 1t

1000 400

10000

10472 x 10472 x 20944 x

10- 7

10- ti

10 - 6














2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000


0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000














o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


872

120


~ a

100 I

80

] ~ 80 ~ ~

~ 40 U J1 w

20

00) 20

-00 90 80 70 80 -60

Log Time Isgt

I0

I L

~ 12000120

b

10000100 ( f

800080

II

~~

amp000110

2~

~

I J

U g 4000

l6 E 40

~ Ww

20 2000 1 II __JJ oo~~~~-~00 ~-2000

-40 -30 20 -00 -90 middot80 70 110 -~ 0 00 -30 20 100 90 80 70 -00 -~ 0 40 -30 20 20







J(X z t) = lS(x)8(z)[1 - H(t)]




(3)

where

R = (x 2 + z2)12


c = (J-lE)-1I2





41tt G ( ) (5)


2poundR 2C 2)l i2(t2 _ ~ - (6)

a



1 1 1tlt-shy (7)- 12 c

v



JlcrR 2

t=-_middot 4



_I_ cu1+1

I----shy E ~ 1-11

(T 11 (Tij+1

611

A ___

I I I I

-----0 1 j

I

I

61 i + 1


BL - -shy

E IIj I I I I I

----- JC

E11+1

IJj+li IT+ 1 j+ 1

Ei+ 1bulli


M = ~cr~Z (8)4




47tl ~ e-loR2


J(X Z t) = - ~JdR8(R)[1 - H(tn


s = J2l ~cr


J HH 1

0 Rma x = 2I-


~l2 = c-~-R = (2~(Jt)z~ (9) 1 ma~




Spatia) terms


~1Cjt E = tu E + (1 E






ABeD ABeD





ABeD




JAD 2 ~z


and









~rmal = 11 rmn (o-ij)12 (16)






Time-stepping








where



r _ hc I- -shy

j1ai j A2

(15)



is set by


lar1=7



~al 12D = -shyrr2











e 1 _ e C E ~ + O(Llr2 ) (17)I J I J

I I J 211t


approximation




thus given by


2M


110 11 2 r I

(19a)


1 - 4rmiddot 2rE~I = I e + ___1

11 1 + 4r I) 1 + 4rmiddot I I



211t 2




eral form

v- 2( E + a-zcE = iE + cE (21)


1 11 rlt-shy- v2 ~t



1 2M 2 euro=-=--

IlC Z 1111 2


2 I ~ 4M (22)

11011 2


112 ~ I1tmax = ()lCJt) 2 (23)




2




directions

~t rj j - _ A

lJ(Jij~ZiU=i+ 1



- 4f tj En- 1ET 1

I J + 4f II


1 + 4r j j 11zi ~Zi


where

- r1 J + rL r = 2




BOUNDARY CONDITIONS



GuE+czzE=O



Q


1t - ltL



Q


( ~ + IK x I) E = 0 (28) UI





1t a x-x-r






-- --




e -11ltx - XI) ( - b)2)4(1 - II)

) E(x z r)(t - t



- a - ao

and is given by





MODEL EXAMPLES

Half-space



4t T=shy

~a



I (Z2 - x 2 2z2) e~R2T 2ze--2IT

E(xzt)=- ---+- -----shyR 2 R 2

1[0 T fiR 2

[_1 _2XF(XT- 1 2 )( +) JT l l 2 T R 2


l 1

) (30)




1-10 240 I-54 10 10-14 120 54-59 15 14--19 60 59--64 30 19-25 30 64-69 60 25-33 15 74-79 240 33-167 10

167-171 15 171-177 30 177-182 60 182-186 120 186-195 240




1 1Etx Z = 0 t) = - 2 (l - e-~2T)

7tCf x

(X Z)III 1 agt (-1) (31)

= 41t ~ ~O (n + III T


cIB = - cxE

IiB = Cz E


2


X1 + shyT

) ]- I

_ ~J 1 -x----2 2TCX t n=)

(-1) 111 (X 2) -

(n + 1) T

(32)

and


Time steps

Iterations 1t

1000 400

10000

10472 x 10472 x 20944 x

10- 7

10- ti

10 - 6














2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000


0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000














o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~



_I_ cu1+1

I----shy E ~ 1-11

(T 11 (Tij+1

611

A ___

I I I I

-----0 1 j

I

I

61 i + 1


BL - -shy

E IIj I I I I I

----- JC

E11+1

IJj+li IT+ 1 j+ 1

Ei+ 1bulli


M = ~cr~Z (8)4




47tl ~ e-loR2


J(X Z t) = - ~JdR8(R)[1 - H(tn


s = J2l ~cr


J HH 1

0 Rma x = 2I-


~l2 = c-~-R = (2~(Jt)z~ (9) 1 ma~




Spatia) terms


~1Cjt E = tu E + (1 E






ABeD ABeD





ABeD




JAD 2 ~z


and









~rmal = 11 rmn (o-ij)12 (16)






Time-stepping








where



r _ hc I- -shy

j1ai j A2

(15)



is set by


lar1=7



~al 12D = -shyrr2











e 1 _ e C E ~ + O(Llr2 ) (17)I J I J

I I J 211t


approximation




thus given by


2M


110 11 2 r I

(19a)


1 - 4rmiddot 2rE~I = I e + ___1

11 1 + 4r I) 1 + 4rmiddot I I



211t 2




eral form

v- 2( E + a-zcE = iE + cE (21)


1 11 rlt-shy- v2 ~t



1 2M 2 euro=-=--

IlC Z 1111 2


2 I ~ 4M (22)

11011 2


112 ~ I1tmax = ()lCJt) 2 (23)




2




directions

~t rj j - _ A

lJ(Jij~ZiU=i+ 1



- 4f tj En- 1ET 1

I J + 4f II


1 + 4r j j 11zi ~Zi


where

- r1 J + rL r = 2




BOUNDARY CONDITIONS



GuE+czzE=O



Q


1t - ltL



Q


( ~ + IK x I) E = 0 (28) UI





1t a x-x-r






-- --




e -11ltx - XI) ( - b)2)4(1 - II)

) E(x z r)(t - t



- a - ao

and is given by





MODEL EXAMPLES

Half-space



4t T=shy

~a



I (Z2 - x 2 2z2) e~R2T 2ze--2IT

E(xzt)=- ---+- -----shyR 2 R 2

1[0 T fiR 2

[_1 _2XF(XT- 1 2 )( +) JT l l 2 T R 2


l 1

) (30)




1-10 240 I-54 10 10-14 120 54-59 15 14--19 60 59--64 30 19-25 30 64-69 60 25-33 15 74-79 240 33-167 10

167-171 15 171-177 30 177-182 60 182-186 120 186-195 240




1 1Etx Z = 0 t) = - 2 (l - e-~2T)

7tCf x

(X Z)III 1 agt (-1) (31)

= 41t ~ ~O (n + III T


cIB = - cxE

IiB = Cz E


2


X1 + shyT

) ]- I

_ ~J 1 -x----2 2TCX t n=)

(-1) 111 (X 2) -

(n + 1) T

(32)

and


Time steps

Iterations 1t

1000 400

10000

10472 x 10472 x 20944 x

10- 7

10- ti

10 - 6














2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000


0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000














o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~






~rmal = 11 rmn (o-ij)12 (16)






Time-stepping








where



r _ hc I- -shy

j1ai j A2

(15)



is set by


lar1=7



~al 12D = -shyrr2











e 1 _ e C E ~ + O(Llr2 ) (17)I J I J

I I J 211t


approximation




thus given by


2M


110 11 2 r I

(19a)


1 - 4rmiddot 2rE~I = I e + ___1

11 1 + 4r I) 1 + 4rmiddot I I



211t 2




eral form

v- 2( E + a-zcE = iE + cE (21)


1 11 rlt-shy- v2 ~t



1 2M 2 euro=-=--

IlC Z 1111 2


2 I ~ 4M (22)

11011 2


112 ~ I1tmax = ()lCJt) 2 (23)




2




directions

~t rj j - _ A

lJ(Jij~ZiU=i+ 1



- 4f tj En- 1ET 1

I J + 4f II


1 + 4r j j 11zi ~Zi


where

- r1 J + rL r = 2




BOUNDARY CONDITIONS



GuE+czzE=O



Q


1t - ltL



Q


( ~ + IK x I) E = 0 (28) UI





1t a x-x-r






-- --




e -11ltx - XI) ( - b)2)4(1 - II)

) E(x z r)(t - t



- a - ao

and is given by





MODEL EXAMPLES

Half-space



4t T=shy

~a



I (Z2 - x 2 2z2) e~R2T 2ze--2IT

E(xzt)=- ---+- -----shyR 2 R 2

1[0 T fiR 2

[_1 _2XF(XT- 1 2 )( +) JT l l 2 T R 2


l 1

) (30)




1-10 240 I-54 10 10-14 120 54-59 15 14--19 60 59--64 30 19-25 30 64-69 60 25-33 15 74-79 240 33-167 10

167-171 15 171-177 30 177-182 60 182-186 120 186-195 240




1 1Etx Z = 0 t) = - 2 (l - e-~2T)

7tCf x

(X Z)III 1 agt (-1) (31)

= 41t ~ ~O (n + III T


cIB = - cxE

IiB = Cz E


2


X1 + shyT

) ]- I

_ ~J 1 -x----2 2TCX t n=)

(-1) 111 (X 2) -

(n + 1) T

(32)

and


Time steps

Iterations 1t

1000 400

10000

10472 x 10472 x 20944 x

10- 7

10- ti

10 - 6














2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000


0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000














o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~








e 1 _ e C E ~ + O(Llr2 ) (17)I J I J

I I J 211t


approximation




thus given by


2M


110 11 2 r I

(19a)


1 - 4rmiddot 2rE~I = I e + ___1

11 1 + 4r I) 1 + 4rmiddot I I



211t 2




eral form

v- 2( E + a-zcE = iE + cE (21)


1 11 rlt-shy- v2 ~t



1 2M 2 euro=-=--

IlC Z 1111 2


2 I ~ 4M (22)

11011 2


112 ~ I1tmax = ()lCJt) 2 (23)




2




directions

~t rj j - _ A

lJ(Jij~ZiU=i+ 1



- 4f tj En- 1ET 1

I J + 4f II


1 + 4r j j 11zi ~Zi


where

- r1 J + rL r = 2




BOUNDARY CONDITIONS



GuE+czzE=O



Q


1t - ltL



Q


( ~ + IK x I) E = 0 (28) UI





1t a x-x-r






-- --




e -11ltx - XI) ( - b)2)4(1 - II)

) E(x z r)(t - t



- a - ao

and is given by





MODEL EXAMPLES

Half-space



4t T=shy

~a



I (Z2 - x 2 2z2) e~R2T 2ze--2IT

E(xzt)=- ---+- -----shyR 2 R 2

1[0 T fiR 2

[_1 _2XF(XT- 1 2 )( +) JT l l 2 T R 2


l 1

) (30)




1-10 240 I-54 10 10-14 120 54-59 15 14--19 60 59--64 30 19-25 30 64-69 60 25-33 15 74-79 240 33-167 10

167-171 15 171-177 30 177-182 60 182-186 120 186-195 240




1 1Etx Z = 0 t) = - 2 (l - e-~2T)

7tCf x

(X Z)III 1 agt (-1) (31)

= 41t ~ ~O (n + III T


cIB = - cxE

IiB = Cz E


2


X1 + shyT

) ]- I

_ ~J 1 -x----2 2TCX t n=)

(-1) 111 (X 2) -

(n + 1) T

(32)

and


Time steps

Iterations 1t

1000 400

10000

10472 x 10472 x 20944 x

10- 7

10- ti

10 - 6














2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000


0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000














o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~



directions

~t rj j - _ A

lJ(Jij~ZiU=i+ 1



- 4f tj En- 1ET 1

I J + 4f II


1 + 4r j j 11zi ~Zi


where

- r1 J + rL r = 2




BOUNDARY CONDITIONS



GuE+czzE=O



Q


1t - ltL



Q


( ~ + IK x I) E = 0 (28) UI





1t a x-x-r






-- --




e -11ltx - XI) ( - b)2)4(1 - II)

) E(x z r)(t - t



- a - ao

and is given by





MODEL EXAMPLES

Half-space



4t T=shy

~a



I (Z2 - x 2 2z2) e~R2T 2ze--2IT

E(xzt)=- ---+- -----shyR 2 R 2

1[0 T fiR 2

[_1 _2XF(XT- 1 2 )( +) JT l l 2 T R 2


l 1

) (30)




1-10 240 I-54 10 10-14 120 54-59 15 14--19 60 59--64 30 19-25 30 64-69 60 25-33 15 74-79 240 33-167 10

167-171 15 171-177 30 177-182 60 182-186 120 186-195 240




1 1Etx Z = 0 t) = - 2 (l - e-~2T)

7tCf x

(X Z)III 1 agt (-1) (31)

= 41t ~ ~O (n + III T


cIB = - cxE

IiB = Cz E


2


X1 + shyT

) ]- I

_ ~J 1 -x----2 2TCX t n=)

(-1) 111 (X 2) -

(n + 1) T

(32)

and


Time steps

Iterations 1t

1000 400

10000

10472 x 10472 x 20944 x

10- 7

10- ti

10 - 6














2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000


0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000














o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


-- --




e -11ltx - XI) ( - b)2)4(1 - II)

) E(x z r)(t - t



- a - ao

and is given by





MODEL EXAMPLES

Half-space



4t T=shy

~a



I (Z2 - x 2 2z2) e~R2T 2ze--2IT

E(xzt)=- ---+- -----shyR 2 R 2

1[0 T fiR 2

[_1 _2XF(XT- 1 2 )( +) JT l l 2 T R 2


l 1

) (30)




1-10 240 I-54 10 10-14 120 54-59 15 14--19 60 59--64 30 19-25 30 64-69 60 25-33 15 74-79 240 33-167 10

167-171 15 171-177 30 177-182 60 182-186 120 186-195 240




1 1Etx Z = 0 t) = - 2 (l - e-~2T)

7tCf x

(X Z)III 1 agt (-1) (31)

= 41t ~ ~O (n + III T


cIB = - cxE

IiB = Cz E


2


X1 + shyT

) ]- I

_ ~J 1 -x----2 2TCX t n=)

(-1) 111 (X 2) -

(n + 1) T

(32)

and


Time steps

Iterations 1t

1000 400

10000

10472 x 10472 x 20944 x

10- 7

10- ti

10 - 6














2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000


0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000














o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~















2000 Ti-----------------~ 100

a

1000 -1

J

~ j

3shy Mu B B r

~ 00 LJJ

co g 1 ~ gt ~

-1000

-2000 41 ~

Imiddot~

b

50

~ gt3shyu ~ 00 UJ

iii c

cent gt

-50

J -100 I

-4000 -2000 0 2000 4000 middot4000 -2000 0 2000 4000


0010 ------------------- ~o 20 -r--------------------

c d

000amp 10

~ ~ 3shy

gt3shy

u ~ 0000

u

~ DC W UJ

iij U

iij U

G5 gt

Q gt

-0005 -10

-0 010 I -2 a I I -4000 -1000 0 20CO 4000 -4000 -2000 0 2000 4000














o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~













o


(E)


(-crBJ


(-cB x )




880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


880

76


10~


e 16m5

5ms _0 0- 0 -0- 0102 -e--e-e-e_ 0 5ms






IT b

= 3 3 SLm 20m

i

1500 2100 2700 330 0 3900 450 0






I [1 4 iE(x z t) - - + r T 32 ncr T 3 nv

1 x 2 -+- 3z2) 4 (2x2 + ZZ)zJ

- 2 T 2 - 5-) T 5 2 (35)


8 Jt hmax 9 ~(J











-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~



-I 10


------ shy102 5 )( ~ Jl )(

9 x-x ~ ~-------il )Ie x _

_ -x- _-x

$ x--)( )(gt shy

i

X I - _-shy 20~~ IX _---tr )(~ 10-) 1)( -~_--i---X)C

x--bullI r

)( I LL ( 11 I )(


OJ

x U

~ 104 w gt

MOOlL

500 II ~O Ii10-5

1003000- M

20 Con4uetor O W

106 I i i

1500 2100 2700 3300 3900 4500



10deg shy

10

0 0 -000000 -0 shy0

0 O _0- --0--0 -0--0-0-_0_0__0_ 0_0--0-0

0O 0O-0_~~o6ms ~ ~ u

10 O

00- 0 0 0- 0 0-0-0 00 m

o 0 -000


UJ 0 ltgtO

0 0-0-0 0 bull 00-0 _0 o ~m

iij 0- 0 0--0- 20m0 0 0 0C 0-0

g 10 3

(5r


100m

(T h -0033 Slm I10~ I 300m

rT = 33 Slm--b

20m

1500 2100 2700 3300 3900 4600



r









_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


_L 4 ~ p _ _ _~ ~ ~ _ __~_ __ M_ _ ~ -- __J~_ ~_~__- ~ 1- 1 J 1 tJ

o

I



z

Ibullo

lI ~


bull

188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


188

c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


c f 51 III IE

-IIt Jbull ~ c i

()


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


t middotmiddot ~

I J

i N10

-- I ~( ] IJ1

) shy

i-I

l bull t l

iII 1

~ t f-

I

bull i

1 ~ IP IN -

~

i i



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~



100 OJ

-0 - o-)middot-0 0-00000




E




6 bull4 500m 4 300m _~

101 II 0 0033 51m fa I 300m

h ~


1500 2100 2700 3300 3900 4600



100

00 00

10 0



(j-o -)-o-o -00U



~ 10 0

~

SOOmet-2 4


lIb =333ilm 20m

1amp00 2100 2700 3300 31100 4600












~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~



~--~- f ~ ~ bull ~


~ 1 ~~~~~ ~ __ ~ _~ J ~___ ~ ~~ ~ __~

I g

5l m E

ibull i bullo

g l g

(a)



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~



891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


891

105


10

102

E-~ 10l U

~ B ~ 10middot gt

1Oe


- 0 5ms F ------- 6 5ms


------_-- O



p

bull 500m I 300m o


~~ 300m 1

ITO = 10 Slm t 20m~

1500 2100 2700 330 0 3~OO 4500




lms

0 0

bull

102_ ~ _0___ _ bull _ - _ - - e - e - e-




10middot

rr =01 SImh

3600 3000 2400 1800 1200 600



DISCUSSION











O-Cr- - 0 -00

pound


c-gt C 0-0-0



(Th Dlsim I 300m

(Tb 1 0 Slm--shy 106 0

1500 2100 2700 3300 3900 4500



10-5









ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~










ACKNOWLEDGMENTS





REFERENCES





























































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~

































APPENDIX A









Euler


Leapfrog


DuForr-Frankel




E7 j = E(x = j~ 2 = j~ t = n~t)

= A cos (l(j~) cos (K i~) (A-4)



tE7 j =i fE7j bull

where





gf == I - 4r + rt (A-6)





T min(4 ~ J or

1 r lt- (A-8)- 4






= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~





= A n ( -Ir( -tV




and

E- = 9L 1e j


gi - 2r(A - 4)gL - 1 = 0


4) + jr2(Ag = r(A - - 4)2 + 1 (A-9a)

and

gZ = r() - 4) - jr-2(J - 4)2 + 1 (A-9b)


g + gZ = r(A - 4)

and

gZ laquo = -l



+ rA + jr2(J - 4)(1 + 4) + 1 (A-lOa)

gDF 1 + 4r

and

_ rA - jr2(A - 4)(1 + 4) + 1 (A-lOb)

gDF = 1 + 4r


APPENDIX B


V 2E - cr~C E 0 (B-l)


pound(0 Z r) = E(L x Z t) = 0 (B-2)

and

E(x 0 t) = E(x L t) = O (B-3)



t t




L L

and setting

Eix Z mnx nnz


we obtain forf(t)


where


and

2 21 m n~=-+2 (B-7)L n l- L


tw = cr~[2 2 (B-8)

Tt

where

I I 1 ~=2+2 (B-9)L i i


cr~e tID=~


Documents

Diffusion of electromagnetic fields into a two-dimensional