Electromagnetic Coupling with a 

Collinear Array on a 

Two-Layer Anisotropic Earth

GEOLOGICAL SURVEY PROFESSIOJ«f AL PAyER 1JB77
Electromagnetic Coupling with a 

Collinear Array on a 

Two-Layer Anisotropic Earth

By JEFFREY C. WYNN

CONTRIBUTIONS TO GEOPHYSICS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1077

A study of the effects of electromagnetic 

coupling in dipole-dipole and pole-dipole 

induced polarization field measurements, with 

evaluation of a coupling-removal technique

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON 1979
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

H. William Menard, Director

Library of Congress Cataloging in Publication Data

Wynn, Jeffrey C.

Electromagnetic coupling with a collinear array on a two-layer anisotropic earth.

(Contributions to geophysics) (Geological Survey Professional Paper 1077)

Bibliography: p. 26

1. Induced polarization. 2. Anisotropy. I. Title. II. Series. III. Series:

United States Geological Survey Professional Paper 1077. 

QC820.W96 538'.7 78-606005

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402 

Stock Number 024-001-03169-8
CONTENTS

Page

Pa £6 Theoretical development Continued

Abstract ............................................................................................. 1 Method of calculation ................................................................... 6

Introduction ....................................................................................... 1 Model results ..................................................................................... 7

Acknowledgments .............................................................................. 1 Isotropic earth ............................................................................ 7

Theoretical development .................................................................... 1 Anisotropic earth ........................................................................ 9

Derivation of the general solution to EM coupling .................... 1 The real world: rock response and coupling removal ........................10

Horizontal component solution ........................................... 3 Conclusions and computer results .....................................................13

Vertical component solution ............................................... 3 Selected references .............................................................................26

ILLUSTRATIONS Page

FIGURE 1. Diagram of a two-layer anisotropic-earth model ..................................................................................................................... 2

2. Plots of behavior of the JJP(r)6SdS and Q functions .................................................................................................. 6

3. Diagram of collinear electromagnetic arrays ............................................................................................................................ 6

4. Diagram of an anisotropic earth with collinear dipole-dipole and pole-dipole arrays ............................................................. 6

5. Plot of homogeneous-earth electromagnetic coupling .............................................................................................................. 7

6. Plots of pole-dipole and dipole-dipole coupling ..................................................................................................................... 7

7. Diagram of complex resistivity relationships ........................................................................................................................... 8

8-17. Plots of 

8. Coupling as a function of N-spacing ......................................................................................................................... 8

9. Coupling as a function of D/A ratio ........................................................................................................................ 8

10. Coupling from Willcox Playa, Ariz. .......................................................................................................................... 9

11. Comparison of isotropic and anisotropic coupling .................................................................................................... 10

12. Anisotropic coupling for changing N ........................................................................................................................ 10

13. Anisotropic coupling for changing D/A ................................................................................................................... 10

14. Barren-rock spectrum in Cartesian coordinates ......................................................................................................... 11

15. Barren-rock spectrum in polar coordinates ............................................................................................................... 11

16. Altered-rock spectrum in Cartesian coordinates ........................................................................................................ 12

17. Altered-rock spectrum in polar coordinates .............................................................................................................. 12

TABLES

____________ Page

TABLE 1. IP parameters for a two-layer isotropic earth as a function of N-spacing ................................................................................ 8

2. IP parameters for a two-layer isotropic earth as a function of depth to the interface D/A ..................................................... 9

3. Comparison of coupling removal results with laboratory results for three isotropic-earth coupling cases ............................... 13

4. Comparison of coupling removal results with laboratory results for three anisotropic-earth coupling cases ........................... 13

5. Electromagnetic coupling computer-program listings ............................................................................................................... 13

6. Examples of electromagnetic coupling for isotropic and anisotropic two-layer earth models .................................................. 17

III

CONTRIBUTIONS TO GEOPHYSICS

ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON 

A TWO-LAYER ANISOTROPIC EARTH

By JEFFREY C. WYNN

ABSTRACT

The development of theoretical equations for calculating electromagnetic 

coupling commonly encountered in Induced Polarization (IP) field surveys 

is presented. For an isotropic earth with a resistive basement, the magnitude 

and phase angle of the electric field decrease as the induction parameter is 

increased. In the case of a conductive basement, the phase angle reaches a 

maximum negative shift, then increases with increasing induction 

parameter until it eventually becomes a phase lead. In this case the 

m&gnitude will also begin to increase at the higher frequencies.

For an anisotropic earth model, the magnitude may increase with 

increasing induction parameter in the low-frequency region, giving rise to a 

negative percent frequency effect (PFE). At higher frequencies, a "notch" 

may develop, but this latter feature will have little effect on field IP 

measurements.

A simple coupling-removal method is discussed and tested with model 

and laboratory data. In about half the cases this method worked reasonably 

well, its effectiveness diminishing as resistivities at the surface were reduced. 

For a conductive basement, the method fails consistently to remove the 

coupling contribution, due to the reversal of the phase shift. Because an 

anisotropic-earth model has phase shifts similar to isotropic models, the 

effectiveness of the removal method is not significantly reduced by the 

anisotropy.

Complete program listings in Fortran IV are included along with a 

representative suite of results plotted in the generalized Cartesian complexplane format.

INTRODUCTION

Induced polarization (IP) surveys are carried out with the 

purpose of measuring the polarization parameters of the 

earth (Sumner, 1976). Unfortunately, in such surveys 

electromagnetic (EM) coupling produces the same general 

effects on the measurements as do the polarization 

parameters of the earth. The EM coupling consists of 

wire-to-wire inductive coupling and coupling through induction within the earth. Unless the EM coupling contribution is 

accurately removed, the IP measurements can be incorrectly 

interpreted as being caused by the polarization of the earth.

The effects of EM coupling can be quite variable. Lateral 

conductive inhomogeneities, whether geologic or cultural in 

origin, have the greatest effects on the IP measurement. By 

contrast, in many sedimentary environments lateral inhomogeneities are not usually present, and the EM coupling 

effects are more subtle. Because strong coupling effects are 

more obvious, this report investigates instead the more

subtle aspects of EM coupling arising from a layered, 

anisotropic earth. These subtle effects of EM coupling must 

be accounted for in order to correctly interpret high-precision IP survey data.

This paper describes the theoretical development of 

electromagnetic-coupling calculations for an anisotropic 

two-layer earth, for two commonly used collinear arrays, the 

dipole-dipole and the pole-dipole array. A computer 

program was written to perform the calculations and is 

included along with tabulated spectra for many different 

earth models. The data are presented in a generalized form of 

normalized real and imaginary components and are plotted 

in the Cartesian complex plane. A method is described for 

the extension of these results to percent frequency effect 

(PFE) and phase angle (</>) representations. Time-domain-IP 

chargeabilities can be obtained from the phase angle results, 

and the method for this is also shown.

ACKNOWLEDGMENTS

The laboratory and field electrical measurements were 

made while the author was employed by Zonge Engineering, 

Tucson, Ariz., using equipment designed by Dr. Kenneth L. 

Zonge. The theoretical calculations were made using the 

University of Arizona CDC 6400 and the U.S. Geological 

Survey DEC-10 computer. 1

THEORETICAL DEVELOPMENT

DERIVATION OF THE GENERAL SOLUTION TO 

EM COUPLING

The original calculations for EM coupling on a two-layer 

isotropic earth were made by Sunde (1967). Derivations fora 

multi-layer isotropic earth are also available in Anderson 

(1975). The general solution for EM coupling over a 

two-layer anisotropic earth can be obtained from boundary 

conditions and Maxwell's equations for layered media as 

follows.

'Use of trade names in this report is for informative or descriptive purposes only and does not 

constitute endorsement by the U.S. Geological Survey.
ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

Let us assume a two-layer geometry as in figure 1.

PO eo

z=0

Plv

z=-d

P2v

FIGURE 1. Two-layer anisotropic-earth model. Pj v is the resistivity of the 

jth layer in the vertical direction;pjjjis the resistivity of the jth layer (0, 1, 

or 2 in this case) in the horizontal direction; x and y directions assumed to 

have the same resistivity; «j is magnetic permeability, andjuj is dielectric 

permittivity of the jth layer. Z is the vertical direction.

P /I \

Let Y^ - iuyt0 ( 757 +ituej J , where i= VTfor the jth layer.

In this equation, * j is the permittivity of the medium,//Q is 

the permeability of free space, o> is the angular frequency, 

P j is the resistivity, the inverse of the resistivity is the 

conductivity 0j , and 7 j is a propagation constant for 

electromagnetic waves in the medium.

Assuming the free space values *j=t o' Mj = ^o ' anc* 

Mo o= l-16xlO~ 17 in MKS units, then the quasistatic approximation (no displacement currents) can be made for the 

frequency range of interest and

(1)

of the Hertz

potential II (Sunde, 1968, p. 102), we have:

an, (2)

rl /air, ai

 "v ~~ I .. »

y \UIJLQ \ az a

F a / anx airz\ 

y "^y \ a x a z )'

F Ex-^x-n-x+ajA - y 2 TT ^-^-f-a

HZ =O,

/

aJt

a a

and

aTT

where we have setlly =0 for the x-z excitation case considered 

here and H and E refer to the magnetic and electric field 

components, respectively.

Eight equations may be developed from the condition that 

requires the tangential components to be continuous across 

the boundaries between the different media. Let the horizontal components be labeled h; then at the earth-air interface 

(z=0) and at a layer interface (z=-d) we obtain:

Earth-air interface 

2=0

Layer interface ' '

, (8b)

ail|h =r 2 flt2hi (9b)

(lOa) = (lOb)

^oh = ^lh (lla) 2 2 

rih 1Tlh = r 2t

(Equations 9a, b were simplified by equations 8a, b; 

equations lla, b were simplified using equations lOa, b.)

Let us assume a general solution for the Hertz potential 

(Sunde, 1968) of

where r is the total distance from the point electrode, <£ is an 

azimuthal angle, cos <?>=x/r, X is an integration variable, 

and Jn is an nth order Bessel function of the first kind (real
THEORETICAL DEVELOPMENT

argument). The function IL is a propagation constant, and

U U - jhfor the horizontal component, and

.

]v

for the vertical component. For 

horizontal components, 0 =0.

HORIZONTAL COMPONENT SOLUTION

at z=0,

where -dS Z^O, and

7r2h=/o°0 f3 eU2hZ J0 (rX)dX

wherez<-d.Note that fj and gj are functions used to satisfy 

the boundary conditions for the jth layer. Then, using the 

four boundary conditions 9a, 9b, lla, lib, we obtain the 

following equations at z=o and z=-d:

ySx(fo-go)=yfh u lh (f,-gl ) t (12)

(13)

and

At this stage we will simplify the subscript notation by: 

Ih -*1, lz-» 3, 2z -> 4, and 2h -^ 2. 

Equations 12 to 15 can be reduced by algebra to:

f

(16)

and

9

zx( Ur u2)e-2ui d

°rf (17)

where

If we make a temporary assumption of a homogeneous 

whole space, 7 0 =7,=7 2 =73 = 74 , by setting g,=0 and f0 =f, 

and solving the resulting equations, we obtain

r _IdS J "

For wires on the surface of the earth, this gives:

2X(u,+u2)

f = IdS 47T r2 A >

(19)

and

iu0o> 2X(ui-u2 ) -2 U . = IdS—— —————— e I 

4ir r2A

(20)

where dS is the infinitesimal segment of the source dipole 

carrying a current I.

VERTICAL COMPONENT SOLUTION

For the vertical components,

°° Po e"Xz J(r\)dX

at z=0,

where -d < z i- 0,

J,(rX)dX

where z < -d, and the p's and q's are functions used to satisfy 

the boundary conditions exactly like the f s and g's.

Using boundary conditions 8a, 8b, lOa, lOb, we can obtain

(21)

WU4d , (22)

and

(24)
4 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH 

We can then obtain the p's and q's by using algebra:

where

P,=

(25)

A ,, A ,, -u3 

A0A,e °+A0A,

and

.11 A u -2u,d A i A i 

Ao A,e 3 + A0A,

,(26)

where:

and

(27)

Now utilizing equations 4 and 5 we can obtain:

and

which reduces to

and

(28)

= X^/oCD (f,+g,)j-0 (rX)dA and

(29)

(30)

(31)

These results are the anisotropic analog of Riordan and 

Sunde's (1933) derivation. They can be generalized using a 

transformation similar to that suggested by Wait (1966): Let 

G= 5 A , B=r/5; and D=d/$, where

8 :

In addition let

XKH=— , ANI = -~, and AN2=-. 

P2 ^3 P4

These parameters XKH, AN1 and AN2 are used in the 

computer program in table 5. Then:

2 _2i_ 2 = 2I_ 

1 2 4 2

and

Y=SU4 =(G2 +2i-AN2)I/2 ,

and

ANI I 

AN2-1 '

; = 2j^rw _ Y AMi

I ON 3 L

where A is the receiver dipole length, and e is the generalized 

induction parameter.
THEORETICAL DEVELOPMENT

The results of the transformation are:

i-V e

.-2UD

P,=

2iG

-2WD

, and

o A" A" ~ 2WD A' A 

A0 A|6 + A0 A|

A' Mg+U)(U+V)+(g-U)(U-V)e~2UD .

Lastly:

(32)

and

Q(r)=f oo 

o

W

This transformation permits a broader application of the 

results of the numerical integration. Therefore, for instance, 

behavior of P(r) and Q(r) depends only on a resistivity ratio 

instead of on two separate values of resistivities. Results are 

always tabulated in terms of a generalized induction 

parameter,

so that with a given spectrum in the complex plane, results 

can be calculated for different values of the receiver dipole 

spacing, A, or surface resistivity, PI , or angular frequency,

CO.

At this point we should comment on the behavior of the 

two functions P and Q. For a homogeneous earth, Q is 

constant and real, is frequency-independent, and contributes 

the resistive component to the mutual coupling. Over a 

two-layer earth, the Q function becomes dependent upon 

frequency and varies with the distances between grounding 

points, the interface-depth-to-dipole-length, D/A, and the 

resistivity contrast, Pj/P2» between the two media. When 

evaluated over the two dipoles, Q yields only four terms 

dependent on distances between grounding points and is a 

simple scalar function.

The P function, on the other hand, is dependent upon the 

orientation of the dipoles. The mutual impedance includes a 

cosine term when integrated over the two dipoles:

P=/B /E P(r) cos<£dSds. 'A 'D (34)

Here, $ is the angle between the two elements dS and ds, 

referring to separate line segments. For these reasons, the P 

function can be called the inductive term, and the Q function 

can be called the grounding term. The behavior of these

A

functions is illustrated in figure 2. In this figure Q is a sum of 

Q terms each of which is calculated for a separate 

electrode-electrode distance. This will be examined in more 

detail later.

For two infinitely long wires, Q behaves asymptotically as 

1/r, and goes to zero as r becomes infinite. It goes to zero 

whether the wires are parallel or perpendicular. For two 

perpendicular wires over a one-dimensional earth, the P 

term is zero. Therefore for two infinite, perpendicular wires, 

irrespective of the layering beneath them, the total electromagnetic coupling becomes zero. For a three-electrode 

array, the Q function reduces from four to two terms, and for 

a two-electrode array with two infinite electrodes, Q is 

further reduced to a single term. Diagrams of these arrays 

are shown in figure 3. Because P must be evaluated by 

integrating over both dipoles by incremental lengths, the 

following results will not include coupling for three- or 

two-electrode arrays. The necessary computer time for the P 

term increases as the product of the incremental lengths over 

which the function must be integrated, and the author felt 

that the computer time necessary for accurate evaluation 

was not justified in light of the infrequent use of these 

geometries.
ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

1-

-0.5

-0.5 n

\V K

>/\ z ^\ o

//PdSds X<

\

^1 = 5.0 .5 = 1.0

P2 A

Sum

i / i

0 0.5 ( 

REAL \ \

-0.5 1.0

DIPOLE-DIPOLE ARRAY

FIGURE 2 Behavior of// P(f)dSdS' the inductive component of

A

EM coupling; Q, the conductive component of EM coupling; and their 

sum as a function of resistivity contrast for an isotropic earth. The 

functions are plotted in the Cartesian complex plane, f shows the 

direction of increasing frequency, and all values are normalized by the 

real dc (direct-current) component.

ARRAYS 

Dipole-dipole

NA

Pole-dipole

NA r JA

3-electrode 

1 NA {

2-electrode 

NA T

FIGURE 3. Collinear arrays commonly used in the field, presented in 

order of frequency of use. N and J are multipliers of dipole length, A.

A note should also be added concerning anisotropy. 

Anisotropic P and Q kernel functions (mathematical sense) 

are included in the program listing in table 5 of this report

and are called ANISOP(x) and ANISOQ(x). In general 

these functions require twice the computer time for 

anisotropic conditions as that required for isotropic P and Q 

kernel functions. The isotropic function subroutines used for 

the majority of the calculations are called PDP(x) and 

PDQ(x) and are also included in the computer program 

listings in table 5.

METHOD OF CALCULATION

In general, the integrals of equations 32 and 33 cannot be 

evaluated analytically, so numerical integration was 

employed using the following procedure. The general form 

of the integrals can be represented as: 

CO

where Jn ( Ar ) is the real component and F(X, r) is in 

general the complex component. The integral was evaluated 

as a sum of integrals between zeros of the Bessel function Jn. 

The first term, from zero (0.0) to the first zero of the Bessel 

function, was calculated using an adaptive Simpson's rule 

(Anderson, 1975), which divides the interval into smaller and 

smaller pieces until the iterated calculations repeat to within 

a user-specified precision. The next four terms were integrated using a sixteen-point Gaussian quadrature method, 

the ensuing series of terms were then integrated using an 

eight-point Gaussian quadrature method, and an Euler 

transformation was used to force convergence of the series. 

Precision was generally obtained to four decimal places by 

the sixteenth zero of the Bessel function.

In order to calculate the coupling for collinear 

dipole-dipole and pole-dipole arrays, the configuration 

shown in figure 4 was used.

P1\

P2h P2v

FIGURE 4. Model cross section of an anisotropic earth showing collinear 

dipole-dipole and pole-dipole geometries. N and J are multipliers of 

dipole length A; a, b, m, and n are electrodes.

In this figure and in subsequent equations, J is a multiplier 

indicating the length of the transmitter dipole with respect to 

the length of the receiver dipole.

A collinear dipole-dipole array is a specific subset of the 

collinear pole-dipole array, with J= 1. To obtain the mutual 

impedance,
MODEL RESULTS

one must evaluate the following: 

M JM ^2

S m=ll=l M 2 

and

(35)

(36)

27rA M JM 27rA A

Z Z P(r) + ———Q(r), 07) 

0 P\ m=l 1 = 1 ^>,

where Q(r)=Q(am)-Q(an)-Q(bm) + Q(bn).

In these equations, Zgs is the mutual impedance between the 

two dipoles S and s, and Z 0 is the dc (direct-current) 

coupling (resistive only) normalization factor. M is the 

number of segments that the dipoles are broken up into for 

purposes of integration.

It has been observed that four-place precision can be 

obtained for the pole-dipole configuration for any length of 

J greater than 7 to 10. In effect then, the pole-dipole array 

can be calculated with about 10 times the CPU (Central 

Processing Unit) time required for a dipole-dipole calculation.

An estimate of the convergence of Z§S /Z 0 against the 

number, M, of intervals that the dipoles should be broken up 

into can be obtained by comparing the homogeneous earth

(with Q constant) dipole-dipole results with the results using 

Milieu's (1967) equations for a summation parameter K=15 

(fig. 5).

-,-0.3

9=0.7

0.9 Millett (1967), Kmax=15 

+ This report M=4 

This report M=2 

This report M=1

1.0 0.0

FIGURE 5. Homogeneous-earth collinear array dipole-dipole electromagnetic coupling plotted in the Cartesian complex plane, showing 

convergence of the calculations with increasing number of intervals (M) 

that the dipoles are divided into.

MODEL RESULTS 

ISOTROPIC EARTH

A comparison of EM coupling for pole-dipole and 

dipole-dipole arrays for a homogeneous earth is shown in

figure 6. All the results shown have been normalized by the 

real or dc component. In this form, the pole-dipole curve has 

a somewhat smaller imaginary amplitude than the 

dipole-dipole curve; this is due to the diminished Q term, as 

the electrodes are moved farther away. However, the 

effective coupling for the pole-dipole array (especially for 

PFE's) is greater than for an equivalent (in everything except 

the J parameter) dipole-dipole configuration. This is because 

of the increased contribution from the P or inductive term.

6=0.7,

0.9

0.5

p,

Collinear dipole-dipole 

Collinear pole-dipole

0.4 0.6 0.7 

REAL

0.8 0.9 1.0

-0.3

-0.2 >

z 

o

-o.i i

0.0

FIGURE 6. A comparison of homogeneous-earth collinear dipole-dipole 

coupling with pole-dipole coupling plotted in the Cartesian complex 

plane.

As mentioned in the introduction, it is a relatively simple 

matter to obtain PFE's, phase angles, and chargeabilities 

from the more general, normalized results in the complex 

plane. This can be demonstrated graphically in figure 7. In 

this figure, M t and M2 are magnitudes at 0.1 and 1.0 Hz 

(hertz)respectively, and $, and<t> 2 are phase angles at the 

same corresponding frequencies.

Newmont standard chargeabilities can be derived empirically from the phase angle ($,) at 0. 1 Hz by multiplying 

the phase angle in milliradians by a constant factor of 1.2 

(Zonge and Wynn, 1975). Loss tangents may be calculated as

Figure 2 has already demonstrated the behavior of 

dipole-dipole coupling for various resistivity contrasts. In 

general, the phase lag (a negative phase angle) increases 

rapidly and monotonically for P, < P 2 in the frequency 

range of interest. For a resistivity contrast P( > P Z , the 

phase lag initially increases as frequency is increased, but it 

soon reaches a maximum, decreases, and then crosses over 

the real axis to a phase lead at the higher frequencies.

Figure 8 shows coupling curves as a function of N, where 

N is a multiple of dipole spacing. Another example for 

varying N (in terms of standard IP parameters) is shown in 

table 1 . The results here are similar to those obtained in the 

field, namely that both magnitudes and phase angles change 

more rapidly with increasing N-spacing. This gives rise to an 

apparent layering in field pseudosections, giving increasing 

coupling contribution with increasing depth or N-spacing.
ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

-0.3 i-

-0.2

-0.1

*=Tan

PFE x 100%, from 0.1 to 1.0 Hz

1.0 Hz

. 0.1 Hz

FIGURE 7. Graphic relationship between the polar coordinate parameters of frequency domain IP and the complex resistivity spectrum 

represented by the dots in the Cartesian complex plane. PFE is the percent-frequency effect.

TABLE 1. IP parameters for a two-layer isotropic earth as a function of

N-spacing

[PFE's are for the 0.1-1.0 Hz decade, phase angles are in milliradians, chargeability is in 

millivolt-seconds per volt, and loss-tangent is dimensionless. A-spacing is 300 m; depth to interface is 

60 m; first-layer resistivity is 50 ohm-meters, and second-layer resistivity is 10 ohm-meters]

1.0

FIGURE 8. Collinear dipole-dipole EM coupling as a function of 

N-spacing, for a conductive basement and a resistive basement, plotted 

in the Cartesian complex plane.

Figure 9 demonstrates the behavior of coupling for the 

dipole-dipole configuration as a function of D/A (depth to 

the layer interface). In the limiting case of an infinitely deep

PFE

0.1-1.0 

decade

Phase (MRAD)

1.0 Hz 0.1 Hz

Chargeability

Losstangent

1

3

5

10

0.8

7.3

17.1

43.2

-31 -3.9

-137 -21

-244 -46

-345 -130

4.7

25

55

156

256

47.6

21.7

7.6

N=3

-i -0.2

0.5

-I 0.2

FIGURE 9. Collinear dipole-dipole EM coupling as a function of D/A 

(depth to the interface) in the Cartesian complex plane.

or an infinitely shallow interface, the spectra approach the 

results of figure 6 (the homogeneous earth). An example for 

a different resistivity contrast, also in terms of standard 

IP parameters, is shown in table 2.
MODEL RESULTS

TABLE 2. IP parameters for a two-layer isotropic earth as a function of layer is given as, 

depth to the interface D/A

[Parameters are the same as in table 1. PFE's are for the 0.1-1.0 Hz decade, phase angles are in

milliradians, chargeability is in millivolt-seconds per volt, and loss-tangent is dimensionless.

A-spacing is 300 m; first-layer resistivity is 50 ohm-meters, and second-layer resistivity is 5

ohm-meters]

D/A

PFE

0.1-1.0 

decade

Phase (MRAD)

1.0 Hz 0.1 Hz

Chargeability

Losstangent

0.05

.10

.20

1.00

15.2

14.9

14.3

3.71

-240

-227

-202

-41.9

-42

-41

-38 

-9.

50

49

46

11.

24

24

26

103

For a rough check on these theoretical results, a field 

measurement was made at Willcox Playa, Cochise County, 

Ariz. The field data are compared in figure 10 with a 

theoretical plot whose input parameters were derived from a 

conventional dipole resistivity sounding. The results are 

within the accuracy of the dipole-sounding inversion and 

show that the theoretical approach is in fact based on 

realistic assumptions.

ANISOTROPIC EARTH

The effects of anisotropy on the EM coupling spectra can 

be measured, where the coefficient of anisotropy for the jth

Jv

_ I horizontal resistivity 

~ I vertical resistivity jth layer.

1.20

1.50

2.00

5.00

2.55

1.76

1.28

.97

-35.4

-33.4

-35.4

-45.4

-7.3

-5.9

-5.4

-5.6

8.8

7.1

6.5

6.7

137

169

185

179

Nine examples of theoretical EM coupling for an anisotropic 

earth may be found in table 6. An initial examination (figure 

11) shows that an anisotropic model for moderate values of 

AJ will behave as one would intuitively expect. In the case 

shown here, an anisotropic model chosen somewhere 

between a homogeneous and an isotropic layered earth 

model gives results that fit between the homogeneous and 

isotropic cases.

As the resistive contrasts in the anistropic layer increase, 

several features begin to appear that are not obtainable from 

isotropic earth models. One of these features is an increase in 

magnitude with increasing frequency in the dc to 0.1 Hz 

range. This is noticeable in the shorter N-spacings in the 

earth model of figure 12; it is especially pronounced in the 

behavior of the Q term. In field measurements this peculiarity would be noticed as negative PFE's. If the earth 

polarization response were weak enough, this effect could 

mask the response enough to hide a significant polarization 

anomaly. The maximum effect in the model of figure 12, 

however, is only 2 percent on the N=3 curve. This would be 

significant only if one attempted to compensate by subtracting out isotropic-earth coupling derived from resistivity 

pseudosections. It should be noted that the phase shifts in the 

anisotropic case are not greatly different from those of most

32 Hz

128

7.3 Hz

4=0.2 £-=0.1 A 02

Field data 

Theoretical coupling

-0.5

- -0.4

— -0.3

— -0.2

- -0.1

0.1 0.2 0.3 0.4 0.5 

REAL

0.6 0.7 0.8 0.9 1.0

FIGURE 10. Collinear pole-dipole EM coupling at Willcox Playa, Cochise County, Ariz. Field data are compared with a theoretical model 

whose parameters were derived from curve-matching dc sounding data. Data plotted in the Cartesian complex plane.
10 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

Isotropic

homogeneous

earth

-0.3

0.9

FIGURE 11. Comparison of anisotropic two-layer-earth EM coupling curve with two isotropic-earth EM coupling curves in the Cartesian

complex plane.

f=1.0

0.5 0.6 0.7 0.8 0.9 1.0 <2

D/A =0.2 

A =305 meters 

110 Pi = 50ft-meters 

P2h = 1ft-meters 

P2v = 50ft • meters

0.1 ~

0.2

0.3

0.4

FIGURE 12. Anisotropic-earth EM coupling curves in the Cartesian 

complex plane, showing the effect of changing the N-spacing in the 

models. Note that anisotropy causes increasing magnitudes (and, 

therefore, negative PFE's) at the lower frequencies.

isotropic cases. Anisotropy, therefore, would not be readily 

identifiable in a purely phase measuring system, or in a 

time-domain system.

A final feature of interest may be observed in figure 13. In 

this case, curves for two values of D/A are plotted for a 

large, fixed anisotropy ratio. Compared with the curve for 

D/A=0.2, the magnitude and phase changes due to EM 

coupling diminish as expected in the curve for D/ A=1.0. A 

notchlike behavior appears, however, at the high-frequency 

end of the D/ A=0.2 curve. This high-frequency notch has 

been observed in field data and has been modeled in other 

work (Wynn and Zonge, 1975). In the frequency range

D/A=0.2

N =3

A =305 meters 

Plv=Pih = 50ft- meters 

P2h = 1ft • meters 

P2v = 50ft • meters

0.1 \0.2 0.3 0.4 0.5 0.6

FIGURE 13. Anisotropic-earth EM coupling curves in the Cartesian 

complex plane, showing the effect of changing the D/ A ratio.

normally used in IP (generally less than 1.0 Hz), this 

notchlike behavior would not be observed unless the earth 

resistivities were less than that of seawater, which is very 

unusual, but nevertheless has been encountered.

THE REAL WORLD: ROCK RESPONSE 

AND COUPLING REMOVAL

Several examples of complex-plane rock spectra and a 

discussion of the application of coupling removal from field 

data can be found in Zonge and Wynn (1975) and Wynn and 

Zonge (1975). In this section the contribution of the rock
THE REAL WORLD: ROCK RESPONSE AND COUPLING REMOVAL 11

-0.1 —

>cc

z 

o

Mil! II, !*•' — "^^"^~^~^7/

110 10 ^T^<

0.0 ———————————————————————

0.9

REAL

jj.>

1.0

COMPONENTS

FREQUENCY REAL IMAGINARY

0.01 1.0000 -0

.03 .9976

.05 .9959

.07 .9939

.09 .0028

.11 .9931

.10 .9927 

.30 .9863

.50 .9826

.70 .9798

.90 .9776

1.10 .9764

1.00 .9769

3.00 .9663

5.00 .9686

7.00 .9567

9.00 .9537

11.00 .9513

10.00 .9521

30.00 .9376 

50.00 .9308 

70.00 .9260

90.00 .9226

110.00 .9202

.0037

.0058

.0062

.0074

.0077

.0081

.0079 

.0107

.0124

.0131

.0142

.0152

.0135

.0165

.0175

.0182

.0186

.0189

.0187

.0193 

.0186 

.0175

.0169

.0161

Apparent resistivity = 1027.3 ohm-meters 

Phase at 0.1 Hz = 8.0 milliradians 

PFE for 0.1 to 1.0 Hz * 1.5

1.0 ——— ' —————

5

g 0.8 -

D

b

| 0.6 -

2

2 0.4 -

i 0.2 -

o

z

o.o — *—

0.01 0.10

—— ——— 1_ ____ L i , I,, 0.25 

M L~UJ W

- -0.20 IS

Jjp

oc

- -0.15 %

£

- -0.10 ri

- -0.05 $

* £ ————— — i — —— ' I ——————— I ————————— I ————————— I _ 010

1.00 10.00 100.00

FREQUENCY, IN Hz

FREQUENCY MAGNITUDE PHASE LOSS-TANGENT

0.01 1

.03

.05

.07 

.09

.11

.10

.30

.50

.70

.90 

1.10 

1.00

3.00

5.00

7.00

9.00

11.00

10.00 

30.00

50.00

70.00

90.00

110.00

.0000 -0.0037 270.7431

.9976 -.0059 170.8859

.9959 -.0062 160.0368

.9939 -.0074 134.3227 

.9926 -.0078 128.1578

.9931 -.0081 123.1535

.9927 -.0080 124.8938

.9864 -.0102 92.6125

.9827 -.0126 79.5342

.9799 -.0134 74.7817

.9777 -.0145 68.7416 

.9765 -.0155 64.4400 

.9770 -.0138 72.2468

.9665 -.0170 58.6813

.9608 -.0182 54.9466

.9569 -.0190 52.6324

.9539 -.0195 51.1506

.9515 -.0199 50.3428

.9523 -.0196 50.9916 

.9378 -.0205 48.6609

.9310 -.0199 50.1678

.9262 -.0189 52.8972

.9227 -.0183 54.6585

.9203 -.0175 57.2592

FIGURE 15. — Weak (barren) rock response spectrum of figure 14, in polar 

coordinates, plotted in normalized magnitudes and phase angles.

errors. In figures 14

1.0, and 10 Hz points

and 16 the triangles (A) mark the 0.1,

. Figures 16 and 17 show an example of

the electrical spectral response of an altered and mineralized 

rock; this is a type "A" response as named by Zonge, Wynn, 

and Young (1976).

FIGURE 14.—Weak (barren) rock response spectrum in the 

Cartesian complex plane, taken from laboratory 

measurements of a core sample of fresh igneous rock; 0.1, 

1.0, and 10 Hz are frequency points and PFE is percent-frequency effect.

response to electromagnetic coupling and a simple coupling 

removal technique will be discussed. Figures Hand 15 show 

an example of the electrical spectral response of a barren 

igneous rock. This spectrum is called a type "C" response in 

Zonge and Wynn (1975). The measurement was made in the 

laboratory in such a manner as to avoid coupling and other

These two laboratory data sets can be combined with three 

theoretical isotropic coupling data sets from table 6 to 

generate a group of synthetic field results. The data sets were 

combined by assuming that the second layer was polarizable; 

to be done rigorously, the coupling should be calculated for a 

different resistivity for each freqeuncy. The data sets of table 

6 that are used here are 2, 3, and 14. A coupling removal 

technique described by Hallof (1974) can be tested on the 

resulting data sets. This technique fits a straight line and a 

quadratic curve to the low-frequency phase angles, and 

extrapolates the results to give an estimated dc "coupling
12 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

-0.1 rFREQUENCY

COMPONENTS 

REAL IMAGINARY

0.01

.03

.05

.07

.09

.11

.10

.30

.50

.70

.90

1.10

1.00

3.00

5.00

7.00

9.00

11.00

10.00

30.00

50.00

70.00

90.00

110.00

0.9983

.9647

.9478

.9370

.9352

.9283

.9314

.9104

.9014

.8970

.9821

.8891

.8905

.8768

.8706

.8668

.8638

.8619

.8625

.8510

.8458

.8423

.8399

.8376

-0.0590

-.0480

-.0505

-.0439

-.0408

-.0399

-.0389

-.0291

-.0267

-.0237

-.0252

-.0227

-.0231

-.0195

-.0185

-.0179

-.0177

-.0173

-.0171

-.0149

-.0133

-.0120

-.0111

-.0095

0.01

Apparent resistivity =92.1 ohm-meters 

Phase at 0.1 Hz = 41.6 milliradians 

PFE for 0.1 to 1.0 Hz = 4.4

FIGURE 16. — Mineralized- and altered-rock response spectrum, plotted in 

the Cartesian complex plane; 0.1, 1.0, and 10 Hz are frequency points, 

and PFE is percent-frequency effect.

free" phase angle. This extrapolation is accomplished by the 

following formulas:

Extrapolated (linear) phase=[ — (J> \ - — <£ 3 ]

2 ' 2 '

for .1 and .3 Hz, and

1.0 •=- __ L

5

g 0.8-

D

g 0.6-

^

O s 0 ' A —

i 0.2 ^ — cc 

O

z 0.0 ——

0.01

-J —— — i

^^^

I

0.10 1

M ' l[l

i i

00 10.00

- -0.25

w

- -0.20 1

- -0.15 Z

>h

- -0.10 <31

- -0.05 gj 

—————— 0.00

100.00

FREQUENCY, IN »z

FREQUENCY

0.01

.03

.05

.07

.09

.11

.10

.30

.50

.70

.90

1.10

1.00

3.00

5.00

7.00

9.00

11.00

10.00

30.00

50.00

70.00

90.00

110.00

MAGNITUDE

1.0000

.9659

.9492

.9380

.9361

.9281

.9322

.9109

.9018

.8963

.8924

.8894

.8908

.8770

.8708

.8670

.8639

.8621

.8627

.8511

.8459

.8424

.8400

.8377

PHASE

-0.0591

-.0497

-.0532

-.0468

-.0436

-.0430

-.0417

-.0319

-.0296

-.0264

-.0283

-.0255

-.0260

-.0222

-.0213

-.0207

-.0205

-.0201

-.0198

-.0175

-.0157

-.0142

-.0132

-.0113

LOSS-TANGENT

16.9097

20.1062

18.7823

21.3454

22.9272

23.2516

23.9524

31.3272

33.7436

37.8633

35.3569

39.1500

38.5017

45.0012

46.9744

48.3297

48.8417

49.6880

50.5681

57.1147

63.7213

70.2471

75.4765

88.2485

FIGURE 17. — Mineralized- and altered-rock spectrum of figure 16, in pola

Extrapolated (quadratic) phase =[^-

8 

for .1, .3, and .5 Hz.

Table 3 shows the results of an experimental removal of 

coupling using Hallof s methods for three theoretical data 

sets from table 6 and for the three cases of (1) non dispersive 

rock, (2) type "C" rock, and (3) type "A" rock response 

added. For comparative purposes the actual phase responses 

as measured in the laboratory for 0.01 and 0.1 Hz are 

included. For coupling case 3, the results are quite good, 

with the dc "coupling free" phase angle falling somewhere 

between the 0.01 and 0.1 Hz result. Case 3 is theoretical 

coupling from a two-layer, isotropic, resistive basement 

environment. As the basement is made more conductive, as 

in case 2, the coupling removal technique begins to fail, but 

only for the case in which the inherent rock response 

included is weak or zero (type "C" or no rock response 

added). For a sharp resistive contrast, as in coupling case 2, 

the method fails entirely because the coupling-caused phase
CONCLUSIONS AND COMPUTER RESULTS 13

TABLE 3.—Comparison of coupling-removal results with actual laboratory 

results for three isotropic earth coupling cases

[All data are in milliradians (MRAD)]

Coupling case

2 3 14

Response of 

rock only

cfi.Ol c().l

Coupling removal with no rock response added

Coupling

Coupling

- —— -260.4

removal with

- —— -304.7

^r r Q

removal with

- —— -310.5

-0.4 

-.44

type "C 1

-7.0 

-6.7

type "A1

-47.0 

-49.9

-12.0 

-5.8

rock

1 Q £^

-12.0

rock

-58.6 

-55.2

0.0 0.0

response added

-3.7 -8.0

response added

-59.1 -41.7

shift begins to increase rapidly, eventually (by 0.5 Hz in case 

2) turning over to a phase-lead. This same failure occurs for 

the homogeneous earth case (though not nearly so seriously) 

as the earth resistivity becomes small (less than 20 

ohm-meters). In any configuration, the coupling becomes 

less amenable to removal as the N-spacing is increased.

Table 4 shows EM coupling removal tests for three 

anisotropic models (data sets 28, 29, and 33 of table 6).

TABLE 4.—Comparison of coupling-removal results with laboratory results 

for three anisotropic earth coupling cases

[All data are in milliradians (MRAD)]

Coupling case Response of 

rock only

28 29 33

Coupling removal with no rock response added

Linear————— 

Quadratic———

-1.4 +3.0 

+1.8 0.0 0.0

Coupling removal with type "C" rock response added

Quadratic ———

on "3 "7 A Q

i -i R n -7.1 -4.4 -6.5 3 ' 7 8 '°

Coupling removal with type "A" rock response added

Linear————— -48.0 -43.7 -46.9

Quadratic——— -50.3 -47.6 -49.7 "

In the previous section it was pointed out that an anisotropic 

earth affects principally the magnitude at low frequencies 

and not the phase angles. This observation is borne out in 

table 4, where coupling removal for three "typical"

anisotropic models gives results reasonably close to the 

actual rock response. The conclusion reached here, then, is 

that an anisotropic earth will not complicate the coupling 

removal task more than an isotropic earth.

CONCLUSIONS AND COMPUTER 

RESULTS

Electromagnetic coupling usually affects phase angle 

measurements more strongly than it does PFE-type 

measurements, in terms of the relative contribution of 

coupling as against rock response. This coupling contribution becomes significant when frequencies above 1.0 Hz are 

used. In one highly conductive environment, in fact, the 

coupling accounted for more than 75 percent of the phase 

angle measured at 0.1 Hz. Coupling can, of course, be 

minimized by using shorter dipole spacings.

The simple coupling removal technique described by 

Hallof (1974) can be effectively used in possibly half of the 

field conditions normally encountered. Theoretical modeling shows that introducing anisotropy into the environment 

can lead to negative PFE's for a collinear dipole-dipole array 

but does not appreciably alter the phase angles from those 

obtained over an isotropic earth. Since the method of Hallof 

utilizes phase angles, this coupling removal technique should 

work reasonably well in most anisotropic environments.

Further details of the theoretical calculations used in this 

study may be found in table 5, containing the computer 

listings, and in table 6, containing examples of electromagnetic coupling for both isotropic and 

anisotropic-earth models plotted in the Cartesian complex 

plane.

TABLE 5.—Computer program listings

[MAIN, Controlling main program; COMPAN, Integrating subroutine; READAN, 1 /O subroutine: 

ANISOP, Anisotropic P kernel function; AN1SOQ, Anisotropic Q kernel function; POP, 

Isotropic P kernel function; PDQ, Isotropic Q kernel function; BESJO, Bessel function of first 

kind, zeroth order; QG8, Eight-point Gaussian quadrature; QG16, Sixteen-point Gaussian 

quadrature]

MAIN PROGRAM

C PROGRAM TO CALCULATE EM COUPLING FOR A POLE-DIPOLE

C AND DIPOLE-DIPOLE ARRAY OVER A TWO-LAYERED EARTH.

C VAL(l) N-SPACING

C VAL(2) A SPACING IN FEET (CONVERTED TO METRIC)

C VAL(3) RHO-1, IN OHM-METRES

C VAL(4) RHO-2, IN OHM-METRES

C VAL(5) D, DEPTH TO THE INTER FACE IN FEET

C (CONVERTED ALSO TO METRIC INTERNALLY)

C VAL(6) =• W... IF W=0, IT WILL READ IN THE THETA VALUES,

C OTHERWISE IT GENERATES THE STANDARD FREQUENCY

C SPECTRA ONE MIGHT ANTICIPATE IN THE FIELD,

C (0.1 THRU 110 HZ IN THIS PROGRAM).

C USED LATER IN PROGRAM AS CARRIER FOR THE

C ANGULAR FREQUENCY W (OMEGA).

C VAL(7) - J, THE LENGTH MULTIPLIER FOR THE XMTR DIPOLE

C VAL(8) - M, # OF INTERVALS THAT THE DIPOLES ARE SUBC DIVIDED INTO. THIS IS FOR INTEGRATION IN

C EQUATIONS 36 AND 37 OF THE TEXT.

C VAL(9) - RHO-3, SECOND LAYER VERTICAL RESISTIVITY (OHM-M).

C _—._———^_«--^———.—.....-..-———._.----— — —.--—————......-_.-_.—.——

C FOR OPERATION, UTILIZE THE FOLLOWING:

C ASSIGN TTY - 4

C ASSIGN LPT - 5

COMPLEX RSLT,Z(16),ZN(16),PTEMP,QTEMP

RSLT IS RESULT FROM INTEGRATION, Z IS IMPEDANCE DERIVED 

FROM P & Q FUNCTIONS, ZN IS NORMALIZED Z, PTEMP & QTEMP 

ARE TEMPORARY STORAGE FOR P & Q CALCULATIONS.

COMMON /BLK1/ METHOD,VAL(9),NZERO,R,HEADER(20) 

DIMENSION W(16),TH(16),F(16) 

WRITE(4,10) 

WRITE(5, 10)
14 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

TABLE 5.—Computer program listings—Continued

6 10 FORMAT(//,' POLE- AND DIPOLE-DIPOLE ANISO EARTH EM COUPLING ',/

6 1 ' VERSION OF 1 JULY 1976 ',/, IX,46(1H*),//)

7 20 CONTINUE

8 CALL READAN

9 IVAL-4 

C

C READIN IS THE GENERALIZED READIN ROUTINE (HIGHLY SIMPLIFIED

C HERE) USED TO READ IN THE INPUT PARAMETERS AND STORE THEM

C IN THE COMMON ARRAYS. 

C

10 WRITE(4,30)

11 30 FORMATC INPUT METHOD OF INTEGRATION IN 12 FORMAT:',/,

11 1 ' 1 - GAUSSIAN QUADRATURE',/,' 0 - CONVOLUTION')

12 READ(4,40) METHOD

13 40 FORMAT(I2)

14 IF(METHOD.EQ.O) GO TO 60

15 WRITE(5,50)

16 50 FORMATC INTEGRATION BY GAUSSIAN QUADRATURE')

17 GO TO 80

18 60 WRITE(5,70)

19 70 FORMATC INTEGRATION BY HANKLE TRANSFORM CONVOLUTION')

20 80 CONTINUE

21 IF(METHOD.EQ.O) GO TO 100

22 WRITE(4,90)

23 90 FORMATC INPUT THE INTEGRATION PARAMETER IVAL IN 12 FORMAT:') 

C

C IVAL IS THE NUMBER OF INTERVALS INTEGRATED WITH QG16/QG8 

C BEFORE THE EULER-CONVERGENCE ROUTINE TAKES OVER. 

C

24 READ(4,140) IVAL

25 100 CONTINUE 

C

C PROTECT AGAINST ZERO-LENGTH TRANSMITTER: 

C

26 IF(VAL(7).GT.O) GO TO 110

27 VAL(7)-1.

28 110 CONTINUE 

C

C METRIC CONVERSION: 

C

29 VAL(2)-VAL(2)*0.30488

30 VAL(5)-VAL(5)*0.30488

31 IF(VAL(l).LE.O.) GO TO 340

32 PI=3.1415926

33 XNORM=1.

34 XMUO-PI*0.0000004

35 A-VAL(2)

36 IF(VAL(6).LE.O.) GO TO 130 

C

C GENERATE THE FREQUENCIES IF THETA, OPTION IGNORED... 

C

37 W(1)=(0.0001*VAL(3))/(A*A*XMUO)

38 W(2)-.l

39 W(3)-.3

40 W<4)-.5

41 W(5)-l.

42 W<6)-3.

43 W(7)-5.

44 W(8)-10.

45 W(9)-30.

46 W(10)-50.

47 W(ll)-70.

48 W(12)-90.

49 W(13)-110.

50 LIM-13

51 DO 120 J-l ,LIM

52 F(J)-W(J)

53 W(J)-6.2831852*F(J)

54 120 TH(J)-A*SQRT(W(J)*XMUO/VAL(3))

55 GO TO 230

56 130 CONTINUE

57 WRITE(4,150) 

C

C READ IN HOW MANY VALUES OF THETA ARE DESIRED 

C

58 TH(1)=0.01

59 READ(4,140) NOTHET

60 140 FORMAT(I2)

61 150 FORMAT(/,' TYPE IN THE NUMBER OF THETAS, 12 FORMAT:')

62 WRITE(4,160)

63 NOTHET-NOTHET+1

64 READ(4,170) (TH(I),1=2,NOTHET)

65 160 FORMAT(/,' INPUT VALUES IN F10.5 FORMAT:',/)

66 170 FORMAT(15(F10.5,/)) 

C

C FOLLOWING SECTION WILL TRUNCATE THETA ARRAY IF IT 

C ENCOUNTERS A ZERO IN IT, AND RESET LIM AUTOMATICALLY 

C

67 LIM-16

68 DO 180 1-2,16

69 IF(TH(I).GT.O.) GO TO 180

70 LIM=I-1

71 GO TO 190

72 180 CONTINUE

73 190 CONTINUE

74 IF(TH(2).GT.O.) GO TO 210

75 LIM-9

76 DO 200 K-l,8

77 KP-K+1

78 XKP-FLOAT(KP)

79 200 TH(KP)-(XKP-2.)*0.2+0.1

80 210 CONTINUE

C

C CONVERT THETAS TO ANGULAR FREQUENCY...

C 

81* DO 220 J-l ,LIM

82 220 W(J)-(TH(J)*TH(J)*VAL(3))/(A*A*XMUO)

83 230 CONTINUE

84 WRITE(4,240) (HEADER(I),1-1,20)

85 WRITE(5,240)(HEADER(I),1-1,20)

86 240 FORMAT(1X,62(1H*),/,lX,1H*,20A3,1H*,/,1X,62(1H*))

TABLE 5.—Computer program listings—Continued.

87 WRITE(4,250)

88 WRITE(5,250)

89 250 FORMATC/,' N A RHO-1 RHO-2 DEPTH',

89 1 W J M RHO-3 ',/,9(7H ******))

90 WRITE(4,260) (VAL(J),J-1,9)

91 »RITE(5,260)(VAL(J),J-l,9)

92 260 FORMAT(2F7.0,2F7.1,4F7.0,F7.1)

93 WRITE(4,270)

94 WRITE(5,270)

95 270 FORMAT(26H RESULTS ARE AS FOLLOWS...,

95 1 /,IX,5HTHETA,' FREQ REAL IMAG PREAL PIMAG'

95 2 20H QREAL QIMAG ,/,16H ***** ***** ,

95 3 2X,4H ,4X,4H** **,2X,2(4X,4H ),2X,2(4X,4H** **)) 

C

C CALCULATE THE IMPEDANCES, Z... 

C

96 DO 300 1=1,LIM

97 VAL(6)=W(I)

98 SKD=SQRT(2.*VAL(3)/{VAL(6)*P1*0.0000004)) 

C

C SKD IS THE SKIN-DEPTH USED IN GENERALIZING THE LINEAR 

C VARIABLES TO MAKE THEM DIMENSIONLESS. 

C

99 Z(I)-CMPLX(0.,0.)

100 IF(W(I).LE.O.) GO TO 300

101 MM-IFIX(VAL(8))

102 XMM-VAL(S)

103 MJ-IFIX(VAL(7)*VAL(8)) 

C

C MM IS THE NUMBER OF INTERVALS THAT THE RECEIVER DIPOLE IS 

C DIVIDED INTO, AND MJ IS THE NUMBER OF INTERVALS THAT THE 

C TRANSMITTER IS DIVIDED INTO FOR THE DOUBLE SUMMATION. 

C

104 DO 290 L-l.MJ

105 DO 280 M-1,MM

106 XL-FLOAT(L)

107 XM-FLOAT(M)

108 R-(1.+VAL(1)+(XL-XM)/XMM)*A 

C

C R IS THE DISTANCE BETWEEN SEGMENTS OF THE DIPOLES 

C BEING INTEGRATED OVER. 

C

109 CALL COMPAN(RSLT,1,IVAL) 

C

C FROM ZERO TO INFINITY;... 1 MEANS IT USES ANISOP, 2 MEANS 

C IT USES ANISOQ FOR THE KERNEL FUNCTION. RSLT IS RETURNED. 

C

110 Z(I)-Z(I)+RSLT

111 280 CONTINUE

112 290 CONTINUE 

C

C CONTRIBUTION FROM DOUBLE SUM OF P: 

C

113 Z(I)=Z(I)*2.*PI*A*A*A/(VAL(3)*XMM*XMM) 

C

C IT HAS JUST BEEN NORMALIZED BY Z(DC). 

C

114 PTEMP-Z(I)

115 CALL COMPAN(RSLT,2,IVAL) 

C

C CONTRIBUTION FROM Q: 

C

116 QTEMP-RSLT*2.*PI*A/VAL(3)

117 Z(I)=Z(I)+QTEMP 

C

C CALCULATE NORMALIZED IMPEDANCES ZN... 

C

118 IF(I.LE.l) XNORM=REAL(Z(I))

119 ZN(I)=Z(I)/XNORM

120 PTEMP-PTEMP/XNORM

121 QTEMP-QTEMP/XNORM

122 F(I)=W(I)/6.2831852 

C

C OUTPUT THE RESULT... 

C

123 WRITE(4,310) TH(I),F(I),ZN(I),PTEMP,QTEMP

124 WRITE(5,310) TH(I),F(I),ZN(I).PTEMP.QTEMP

125 300 CONTINUE

126 310 FORMAT(F7.3,F7.1,2F8.4,2X,2F8.4,2X,2F8.4)

127 WRITE(4,320) XNORM

128 WRITE(5,320) XNORM

129 320 FORMATC/,' NORMALIZING FACTOR =',E12.4)

130 WRITE(4,330)

131 WRITE(5,330)

132 330 FORMATOH1 ,//)

133 GO TO 20

134 340 STOP

135 END

COMPAN SUBROUTINE

SUBROUTINE COMPAN(RSLT,IFCTN.IVAL) 

GENERALIZED INTEGRATION SUBROUTINE

COMPLEX ANISOP,AN ISOQ,ANCONP,ANCONQ,ZHANKO

EXTERNAL ANISOP,ANISOQ.ANCONP.ANCOHQ

COMPLEX SSUM,GSUM,ESUM,SUMC,TEM,Y,SEVEN,SODD,FX,AMN,AMP,RSLT

COMMON /BLK1/ METHOD,VAL(9),NZERO,R,HEADER(20)

DIMENSION SUMC(7),Y(193)

DIMENSION V(4),ZERO(25)

DATA ZERO/2.40482555,5.52007811,8.65372791,11.7915344,

1 14.9309177,18.0710639,21.2116366,24.3524715,

2 27.4934791,30.6346064,33.7758202,36.9170983,

3 40.0584257,43.1997917,46.3411883,49.4826098,

4 52.6240518,55.7655107,58.9069839,62.0484691,

5 65.1899648,68.3314693,71.4729816,74.6145006,

6 77.7560256/ 

DATA ICALLD/1/ 

IF(ICALLD.EQ.1) TEM-CMPLX(0.,0.)
CONCLUSIONS AND COMPUTER RESULTS 15

	TABLE 5.—Computer program listings—Continued.

11 ICALLD-ICALLD+10

12 IF(IVAL.EQ.O) IVAL-3

13 NZERO*!

14 PI-3.1415926

15 MAX-8

16 SSUM-TEM

17 GSUM-TEM

18 ESUM-TEM

19 RSLT-TEM

20 NN-0

21 IF(METHOD.EQ.O) GO TO 320

C THE GAUSSIAN INTEGRATION METHOD:

C

C SIMPSON INTEGRATION OVER FIRST INTERVAL OF JO

C

22 DO 10 J=l,6

23 10 SUMC(J)-TEM

24 DO 20 L-l,193

25 20 Y(L)=TEM

26 J=0

27 30 J-J+1

28 NN-6*(2**J)

29 H-(2.4048256)/NN

30 IF(IFCTN-1)40,40,50

31 40 Y(l)-ANISOP(O.O)

32 GO TO 60

33 50 Y(1)=ANISOQ(0.0)

34 60 CONTINUE

35 NP-NN + 1

36 DO 90 1-2,NP

37 XI-FLOAT(I-l)

38 IF(IFCTN-l)70,70,8o

39 70 Y(I)-ANISOP(H*XI)

40 GO TO 90

41 80 Y(I)-ANISOQ(H*XI)

42 90 CONTINUE

43 SEVEN-TEM

44 DO 100 K-2.NN.2

45 100 SEVEN-SEVEN+4.*Y(K)

46 SODD-TEM

47 NM-NN-1

48 DO 110 K-3.NM.2

49 110 SODD-SODD+2.*Y(K)

C THE RESULT...

50 SUMC(J)-(H/3.)*(Y(1)+SODD+SEVEN+Y(NP))

C TEST SECTION...

51 IF(J.LE.1) GO TO 30

52 FRAC1-CABS(SUMC(J)-SUMC(J-l))

53 FRAC2-0.00001*CABS(SUMC( J))

54 IF(FRAC1.LE.FRAC2) GO TO 120

55 IF(J.LT.5) GO TO 30

56 120 SSUM-SUMC(J)

57 RSLT-RSLT+SSUM 

C

C GAUSSIAN 8-OR-16 POINT INTEGRATION OVER NEXT IVAL INTERVALS. 

C

58 DO 190 L-l.IVAL

59 NZERO-L

60 IF(L.LE.4)GO TO 150

61 IF(IFCTN-l) 130,130,140

62 130 CALL QG8(ZERO(L),ZERO(L+1).ANISOP,GSUM)

63 GO TO 180

64 140 CALL QG8(ZERO(L),ZERO(L+1),ANISOQ,GSUM)

65 GO TO 180

66 150 IF(IFCTN-l)16o,160,170

67 160 CALL QG16(ZERO(L),ZERO(L+1),ANISOP,GSUM)

68 GO TO 180

69 170 CALL QG16(ZERO(L),ZERO(L+1).ANISOQ,GSUM)

70 180 RSLT-RSLT+GSUM

71 190 CONTINUE

72 EPS-0.OOOOl*CABS(RSLT) 

C

C EULER TRANSFORMATION TO FORCE CONVERGENCE OF SERIES... 

C

73 NZERO-NZERO+1

74 DO 200 L-1,193

75 200 Y(L)-TEM

76 IF(MAX.LE.O) GO TO 310

77 1-1

78 M-l

79 XL-ZERO(NZERO)

80 XU-ZERO(NZERO+1)

C XL-LOWER BOUND, XU-UPPER BOUND FOR GAUSSIAN QUADRATURE...

81 IF(IFCTN-l) 220,220,210

82 210 CALL QG8(XL,XU,ANISOQ,FX)

83 GO TO 230

84 220 CALL QG8(XL,XU,ANISOP,FX)

85 230 CONTINUE

86 Y(1)-FX

87 ESUM-Y(1)*.5

88 240 J-0

89 250 I-I + l

90 IF(I.GT.MAX) GO TO 310

91 NZERO-NZERO+1

92 XL-ZERO(NZERO)

93 XU-ZERO(NZERO+1)

94 IF(IFCTN-1)270,270,260

95 260 CALL QG8(XL,XU,ANISOQ,FX)

96 GO TO 280

97 270 CALL QG8(XL,XU,ANISOP,FX)

98 280 AMN-FX

99 DO 290 K-l.M

100 AMP-(AMN+Y(K))*.5

101 Y(K)-AMN

102 290 AMN-AMP

103 IF(CABS(AMN).GE.CABS(Y(M))) GO TO 300

104 IF(M.GE.15) GO TO 300

105 M-M+1

106 Y(M)-AMN

107 AMN-.5*AMN

108 300 ESUM-ESUM+AMN

109

no

111

112

113

114

115

117

1 18

119

120

121

122

123

124

125

126

127

128

129

130

131

132

TABLE 5.—Computer program listings—Continued.

IF(CABS(AMN).GT.EPS*CABS(ESUM)) GO TO 240

J-J + 1

IF(J.LT.5) GO TO 250 

310 CONTINUE

RSLT-RSLT+ESUM

RETURN 

320 CONTINUE

CONVOLUTION/HANKLE TRANSFORM METHOD:

IF( IFCTN..EQ. 2) GO TO 330

P-FUNCTION CONVOLUTION

SKD=SQRT(2.*VAL(3)/(VAL(6)*PI*0.0000004)) 

R-R/SKD

RSLT-ZHANKO(ALOG(R),ANCONP,0.0001,L)/R 

RETURN 

330 CONTINUE

Q-FUNCTION CONVOLUTION

V(1)-(VAL(1)+VAL(7))*VAL(2) 

V(2)=VAL(1)*VAL(2) 

V(3)=(VAL(1)+VAL(7)+1.)*VAL(2) 

V(4)-(VAL(1)+1.)*VAL(2) 

DO 340 K-1,4 

R-V(K)/SKD

Y(K)-ZHANKO(ALOG(R),ANCONQ , 0. 0001 , L) /R 

340 CONTINUE

RSLT-Y(1)-Y(2)-Y(3)+Y(4)

RETURN

END

READAN SUBROUTINE

SUBROUTINE READAN

GENERALIZED I/O SUBROUTINE

COMMON /BLK1/ VAL(9).NZERO,R,HEADER(20)

WRITE(4,10) 

10 FORMAT(/,' GIVE ME A TITLE')

READ(4,20)(HEADER(J),J-l,20) 

20 FORMAT(20A3)

WRITE(4,30) 

30 FORMAT(/,' INPUT THE NINE VARIABLES ',/,

1 N A RHO-1 RHO-2 DEPTH TH:W-D

2 /,9(7H ******))

READ(4,40)(VAL(J),J-1,9) 

40 FORMAT(9F7.0)

RETURN 

50 STOP

END

ANISOP FUNCTION

COMPLEX FUNCTION ANISOP(X)

ANISOTROPIC TWO-LAYERED INDUCTIVE FUNCTION.

COMPLEX U,V,DELP,DELOP,DEL1P,DELOPP,DEL1PP,W

COMPLEX F1,G1,XI2,CC,XXKSK,WXKAN,XKON

COMMON /BLK1/ VAL(9),NZERO,R,HEADER(20)

PI-3.1415926

SKD=SQRT(2.*VAL(3)/(VAL(6)*PI*O.OOOOOD4))

XK-VAL(3)/VAL(4)

AN-VAL(4)/VAL(9)

D-VAL(5)*2./SKD

IF(X-0.0001)10,20,20

X-0.0001

CONTINUE 

RVAL-BESJO(X) 

BLM-R/SKD 

G-X/BLM 

G2-G*G

XI2-2.*CMPLX(0.,1.) 

U=CSQRT(G2+XI2) 

CC-CEXP(-U*D) 

SKD2-SKD*SKD 

SKD3-SKD2*SKD 

XKAN-XK*AN 

V-CSQRT(G2+XI2*XK) 

W-CSQRT(G2+XI2*XKAN) 

DELOPP-G*XI2/SKD3 

DELOP-DELOPP 

XXKSK-XI2*XK/SKD3 

WXKAN-W/XKAN 

DEL1P-XXKSK*(U+WXKAN) 

DEL1PP-XXKSK*(U-WXKAN) 

DELP-(G+U)*(U+V)+(G-U)*(U-V)*CC 

XKON-VAL(6)*0.0000001*G*SKD2/DELP 

F1-XKON*(U+V) 

G1-XKON*(U-V)*CC

ANISOP-(XI2/SKD3)*(F1+G1)*RVAL/BLM 

RETURN 

END

ANISOQ FUNCTION

COMPLEX FUNCTION ANISOQ(X)

ANISOTROPIC TWO-LAYERED CONDUCTIVE FUNCTION.

COMPLEX U,V,XNUM,Q,DELP,DELOP,DEL1P,DELOPP,DEL1PP,W

COMPLEX DENOM,F1,G1,P1,Q1,XI2,CC,XXKSK,WXKAN,XKON

COMMON /BLK1/ VAL(9),NZERO,R,HEADER(20)

DIMENSION Y(4),Q(4)

PI-3.1415926

SKD-SQRT(2.*VAL(3)/(VAL(6)*PI*O.OOOOOD4))

Y(1)-<(VAL(1)+VAL(7))*VAL(2))/SKD

Y(2)-VAL(1)*VAL(2)/SKD

Y(3)-((VAL(1)+VAL(7)+1.)*VAL(2))/SKD

Y(4)-((VAL(1)+1.)*VAL(2))/SKD

XK-VAL(3)/VAL(4>
16 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

	TABLE 5.—Computer program listings—Continued.

13 AN=VAL(4)/VAL(9)

14 XI2-2.*CMPLX(0.,1.)

15 D=VAL(5)*2./SKD

16 IF(X-O.OOOl) 10,20,20

17 10 X=0.0001

18 20 CONTINUE

19 RVAL=BESJO(X)

20 DO 30 1-1,4

21 B=Y(I)

22 G-X/B

23 G2=G*G

24 U-CSQRT(G2+XI2)

25 COCEXP(-U*D)

26 XKAN=XK*AN

27 SKD2=SKD*SKD

28 SKD3-SKD2*SKD

29 V=CSQRT(G2+XI2*XK)

30 W-CSQRT(G2+XI2*XKAN)

31 DELOPP-G*XI2/SKD3

32 DELOP=DELOPP

33 XXKSK=XI2*XK/SKD3

34 WXKAN-W/XKAN

35 DEL1P=XXKSK*(U+WXKAN)

36 DEL1PP=XXKSK*(U-WXKAN)

37 XNUM=(G*XI2)/SKD3

38 DENOM-DEL1PP*DELOPP*CC+DELOP*DEL1P

39 DELP-(G+U)*(U+V)+(G-U)*(U-V)*CC

40 XKON=VAL(6)*0.0000001*G*SKD2/DELP

41 F1-XKON*(U+V)

42 G1-XKON*(U-V)*CC

43 P1-XNUM*(-(F1+G1)*DEL1P+(XK-1.)*(F1*CC+G1)*DELOPP)/DENOM

44 Q1=XNUM*((1.-XK)*(F1*CC+G1)*DELOP-(F1+G1)*DEL1PP*CC)/DENOM

45 Q(I)-(F1+G1-(U/G)*(P1-Q1))*RVAL/(B*SKD)

46 30 CONTINUE

47 ANISOQ-Q(1)-Q(2)-Q(3)+Q(4)

48 RETURN

49 END

	POP FUNCTION

1 COMPLEX FUNCTION PDP(X) 

C

C P KERNEL FUNCTION 

C

	COMPLEX DEL3,DEL,XI,U,V.CUD.D

	COMMON /BLK1/ ZERO (3 5) , VAL( 8 ) ,NZ ERO , R .'HEADER (2 0)

	PI-3.1415926

	SKD-SQRT(2.*VAL(3)/(VAL(6)*PI*0.0000004))

	XK-VAL(3)/VAL(4)

7 IFCX.LE.0.00005) X-0.00005

8 RVAL-BESJO(X)

9 BLM-RVSKD

10 G=X/BLM

11 G2-G*G

12 D=VAL(5)*2./SKD

13 XI=CMPLX(0.,2.)

14 U-CSQRT(G2+XI)

15 V-CSQRT(G2+XI*XK)

16 CUD=(U-V)*CEXP(-U*D)

17 DEL3=U+V+CUD

18 DEL=(G+U)*(U+V)+(G-U)*CUD

19 PDP-((DEL3/DEL)*RVAL*X)/(BLM*BLM)

20 RETURN

21 END

	PDQ FUNCTION

1 COMPLEX FUNCTION PDQ(X) 

C

C Q KERNEL FUNCTION 

C

2 COMPLEX XI,U,V.DEL,DELI,DEL2,DEL3,Q(4)

3 COMPLEX UXK,CUD,UMV,UXKV,UXKPV,D

4 COMMON /BLK1/ ZERO(35),VAL(8),NZERO,R,HEADER(20)

5 DIMENSION Y(4)

6 PI-3.1415926

7 SKD-SQRT(2.*VAL(3)/(VAL(6)*P1*0.0000004))

8 VSKD-VAL(2)/SKD

9 Y(1)-(VAL(1)+VAL(7))*VSKD

10 Y(2)=VAL(1)*VSKD

11 Y(3)=(VAL(1)+VAL(7)+1.)*VSKD

12 Y(4)-(VAL(1)+1,)*VSKD

13 XK-VAL(3)/VAL(4)

14 XI=CMPLX(0.,2.)

15 D-VAL(5)*2./SKD

16 IF(X.LE.0.00005) X-0.00005

17 RVAL-BESJO(X)

18 DO 10 1-1,4

19 B=Y(I)

20 G=X/B

21 G2-G*G

22 U-CSQRT(XI+G2)

23 V-CSQRT(XI*XK+G2)

24 UXK-U*XK

25 CUD-CEXP(-U*D)

26 UMV=(U-V)*CUD

27 UXKV=(UXK-V)*CUD

28 UXKPV-UXK+V

29 DEL1-UXKPV+UXKV

30 DEL2-(G+U)*UXKPV+(G-U)*UXKV

31 DEL3-U+V+UMV

32 DEL =(G+U)*(U+V)+(G-U)*UMV

33 Q(I)-(4.*(1.-XK)*G*U*U*CUD+DEL2*DEL3)*RVAL/(B*DEL*DEL1)

34 10 CONTINUE

35 PDQ-Q(1)-Q(2)-Q(3)+Q(4)

C PDQ=Q(AC)-Q(BC)-Q(AD)+Q(BD)

36 RETURN

37 END

TABLE 5.—Computer program listings—Continued

BESJO FUNCTION

10

FUNCTION BESJO(X) 

COMPUTES JO(X) FOR REAL X

IF(X.GT.3.) GO TO 10 

Y-X/3. 

Y2-Y*Y 

Y4-Y2*Y2 

Y6=Y2*Y4 

Y8-Y6*Y2 

Y10-Y8*Y2 

Y12=Y10*Y2

BESJO-1.-2.2499997*Y2+1.2656208*Y4-0.3163866*Y6 

1 +0.0444479*Y8-0.0039444*Y10+0.0002100*Y12 

GO TO 20 

Z-3./X 

Z2=Z*Z

Z3=Z2*Z

Z4-=Z2*Z2

Z5=Z3*Z2

Z6-Z3*Z3

FO=0.79788456-0.00000077*Z-0.00552740*Z2-0.00009512*Z3 

1 +0.00137237*Z4-0.00072805*Z5+0.00014476*Z6

THETA-X-0.78539816-0.04166397*Z-0.00003954*Z2+0.00262573*Z3 

1 -0.00054125*Z4-0.00029333*Z5+0.00013558*Z6

BESJO-FO*COS(THETA)/SQRT(X) 

20 RETURN

END

QG8 SUBROUTINE

SUBROUTINE QG8(XL,XU,FCT,Y)

8-POINT GAUSSIAN QUADRATURE, ADAPTED FROM IBM SSP.

COMPLEX FCT.Y

A-.5*(XU+XL)

B-XU-XL

C=.4801449*8

Y-0.05061427*(FCT(A+C)+FCT(A-C))

C=.3983332*8

Y-Y+.Ill 1905*(FCT(A+C)+FCT(A-C))

C-.2627662*8

Y-Y+.1568533*(FCT(A+C)+FCT(A-C))

C-0.09171732*8

Y-B*(Y+.1813419*(FCT(A+C)+FCT(A-C)))

RETURN

END

QG16 SUBROUTINE

SUBROUTINE QG 1 6(XL,XU,FCT,Y)

16-POINT GAUSSIAN QUADRATURE, ADAPTED FROM IBM SSP.

COMPLEX FCT.Y

A-.5*(XU+XL)

B-XU-XL

C-.494700467496*8

Y-.0135762297059

C-.472287511537*8

Y-Y+.031126761969

C-.43281560119*8

Y-Y+.0475792558412

C-.377702204178*B

*(FCT(A+C)+FCT(A-C))

*(FCT(A+C)+FCT(A-C))

*(FCT(A+C)+FCT(A-C))

*(FCT(A+C)+FCT(A-C))

Y=Y+.0623144856278 

C-.308938122201*8 

Y=Y+.0747979944083 

C-.229008388829*8 

Y-Y+.0845782596975 

C-.14080177539*8

Y=Y+.0913017075225 

C-.0475062549188*8 

Y=B*(Y+.0947253052275 

RETURN 

END

*(FCT(A+C)+FCT(A-C))

*(FCT(A+C)+FCT(A-C)) 

*(FCT(A+C)+FCT(A-C))

*(FCT(A+C)+FCT(A-C)))
CONCLUSIONS AND COMPUTER RESULTS 17

TABLE 6. — Examples with index of electromagnetic coupling for isotropic

and anisotropic two-layer earth models

[N, dipole separation in terms of dipole length A; D/A, depth to dipole-length ratio; fi/Pj^

resistivity ratio; J, transmitter-length multiplier; A2 , anisotropy ratio; 1MAG, imaginary axis;

REAL, real axis;P, , resistivity of layer 1;P2 , resistivity of layer 2;P 3 , resistivity of layer 3; W,

angular frequency; M, dipole multiplier; THETA, dimensionless coupling parameter; FREQ,

frequency= W/2 ir ; PREAL, real component of the inductive function; PIMAG, imaginary

component of the inductive function; QREAL, real component of the conductive function;

QIMAG, imaginary component of the conductive function. Tabulated data are facsimiles of

computer printout]

Example N D/A p n /p_, J A 1 2n 2

1 3 0.30 0.01 5 1

2 6 .50 50.0 1 1

3 6 .50 .02 1 1

4 6 .50 1.00 1 1

5 6 .50 20.0 1 1

6 1 .50 .04 1 1

7 1 .50 .02 1 1

8 2 .30 .11 5 1

9 3 .10 20.0 1 1

10 3 .20 .10 1 1

11 3 .05 10.0 1 1

12 3 .10 5.0 1 1

13 3 .10 10.0 1 1

14 3 .20 10.0 1 1

15 3 1.00 10.0 1 1

16 3 1.20 10.0 1 1

17 3 1.50 10.0 1 1

18 3 2.00 10.0 1 1

19 3 5.00 10.0 1 1

20 1 .20 5.0 1 1

21 5 .20 5.0 1 1

22 10 .20 5.0 1 1

23 1 .20 .1 11

24 1 .20 .01 1 1

25 5 .20 .01 1 1

26 3 .2 5.0 1 .20

27 3 .2 50. 1 -02

28 3 .2 1.0 1 50.0

29 1 .2 50. 1 .02

30 6 .2 50. 1 .02

31 3 1.0 50. 1 .02

32 3 .2 500. 1 -002

33 3 .03 .01 1 20.0

34 3 .03 .001 5 50.0

Example 1:

-0.5 r

X

' ' X ~^**^ ^^""'"-•"•^ / ^^^

< /

z /

2 x/ 1 /

/x

o.o A$c*_, —— , —— , —— , —— ,_, —— i —— ,

0.0 0.5

REAL

N A RHO-1 RHO-2 DEPTH W

3. 305. 1.0 100.0 91 1.

RESULTS ARE AS FOLLOWS. . .

THETA FREQ REAL IMAG PREAL PIMAG

0.01 0.0 1.0000 0.0015 -0.0000 -0.0005

0.27 0.1 0.9961 -0.0556 -0.0110 -0.0513

0.47 0.3 0.9637 -0.1485 -0.0580 -0.1238

0.61 0.5 0.9201 M). 2229 -0.1159 -0.1661 

0.86 1.0 0.7952 -0.3505 -0.2534 -0.1874

1.48 3.0 0.3880 -0.4589 -0.4300 0.0639

1.92 5.0 0.1851 -0.3853 -0.3180 0.2236

2.71 10.0 0.0363 -0.2125 -0.0724 0.2096

4.69 30.0 0.0144 -0.0377 -0.0235 0.0227

6.06 50.0 0.0331 -0.0014 -0.0426 0.0009

7.17 70.0 0.0486 0.0080 -0.0521 -0.0037

8.13 90.0 0.0578 0.0082 -0.0572 -0.0043

8.98 110.0 0.0624 0.0061 -0.0599 -0.0037

NORMALIZING FACTOR - -0.5509E+00

Example 2:

REAL

-1.0 '

o -*£ 1

\

\

\

a: V

z \

0+5.0 \

< V

- \

y\

\*

\

+10.0 1- \+

N A RHO-1 RHO-2 DEPTH W J

6. 305. 50.0 1.0 152. 1.

RESULTS ARE AS FOLLOWS... 

THETA FREQ REAL MAG PREAL PIMAG QREAL

0.01 0.0 1.0000 -0.1475 -0.1054 -0.2048 1.1054

0.04 0.1 0.8620 -0.1801 -0.2490 -0.3183 1.1110 

0.07 0.3 0.6464 -0.0141 -0.5470 -0.4033 1.1934

0.09 0.5 0.6320 0.1701 -0.6747 -0.4051 1.3067

0.12 1.0 0.7663 0.4288 -0.7901 -0.4482 1.5564

0.21 3.0 1.0748 0.8215 -1.0068 -0.8132 2.0816

0.27 5.0 1.2434 1.1481 -1.1777 -1.1336 2.4211

0.38 10.0 1.5818 1.8602 -1.5095 -1.8278 3.0913

0.66 30.0 2.6502 4.1755 -2.5495 -4.1003 5.1997

0.86 50.0 3.6901 6.1254 -3.5600 -6.0224 7.2501

1.01 70.0 4.7955 7.8439 -4.6354 -7.7220 9.4309

1.15 90.0 5.9764 9.3666 -5.7860 -9.2330 11.7624

1.27 110.0 7.2238 10.7071 -7.0047 -10.5685 14.2285

\

. . V

1.0

J M

5. 2.

QREAL QIMAG

1.0000 0.0020

1.0071 -0.0043

1.0217 -0.0247

1.0360 -0.0568 

1.0487 -0.1630

0.8180 -0.5228

0.5031 -0.6089

0.1087 -0.4221

0.0379 -0.0604

0.0757 -0.0023

0.1006 0.0117

0.1150 0.0125

0.1223 0.0097

M

1. 2.

QMAG

0.0572

0.1382 

0. 3892

0.5752

0.8770

1.6347

2.2816

3.6881

8.2758

12.1478

15.5660

18.5995

21.2756

NORMALIZING FACTOR - -0. 1113E-03
18 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

Example 3: 

-0.5 r

Example 5 : 

REAL

0.0

1.0

N A RHO-1 RHO-2

6. 305. 50.0 2500.0 

RESULTS ARE AS FOLLOWS... 

THETA FREQ REAL IMAG

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

0. 9993

0. 9991

0.9987

0.9972

0.9875

0..9734

0.9275

0.7014

0. 4986

0. 3444

0.2338

0. 1567

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0010

0030

0082

0135

0264

0742

1172

2083

4134

4740

4678

4331

3887

DEPTH

152. 

PREAL

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0000

0001

0004

0010

0030

0172

0374

0995

3326

4396

4438

3924

3195

W

1. 

PIMAG

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0.

0.

0.

0011

0026

0077

0127

0246

0665

1007

1596

1688

0401

0935

1918

2497

J

1.

(PEAL

1.0000

0.9994

0.9995

0. 9996

1.0003

1.0047

1.0108

1.0270

1.0339

0. 9382

0. 7882

0.6262

0.4763

M

2. 

QIMAG

0.0001

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0004

0005

0008

0018

0077

0165

0487

2447

4339

5614

6249

6383

NORMALIZING FACTOR - -0.3939E-01

+5.0 L

N

6.

RESULTS

THETA

0.01

0.09

0.15

0.19

0.27

0.47

0.61

0.86

1.48

1.92

2.27

2.57

2.84

A RHO-1

305. 10.

ARE AS

FREQ

0.

0.

0.

0.

1.

3.

5.

10.

0

1

3

5

0

0

0

0

30.0

50.

70.

90.

110.

0

0

0

0

FOLLOWS.

REAL

1.0000

0. 6654

0.6051

0.7119

0.8967

1.2442

1.5210

2.1514

4.6624

7.0231

8.9055

10.2276

11.0622

,0

-0.

-0.

0.

0.

RHO-2

0.5

IMAG

0803

1137

2127

3803

0.5681

1.

1.

2.

4.

5.

4.

4.

3.

1049

5459

4421

5685

2227

9992

3413

5410

-0

-0

-0

-0

-0

-1

-1

-2

-4

-6

-8

DEPTH

152.

PREAL

.0333

.4039

.6652

.7379

.8450

.1864

.4582

.0754

.5465

.8955

.7854

-10.1200

-10.9635

W

1.

PIMAG

-0.

-0.

-0.

-0.

-0.

-1.

-1.

-2.

-4.

-5.

-5.

-4.

-3.

1001

3587

3686

3996

5558

0947

5277

4134

5420

2197

0134

3608

5582

J

1.

(PEAL

1.0333

1.0692

1.2703

1.4498

1.7417

2.4306

2.9793

4.2268

9. 2089

13.9185

17.6908

20.3476

22.0257

M

2.

QIMAG

0.0198

0. 2450

0.5814

0.7799

1.1239

2.1996

3.0736

4.8555

9.1105

10.4425

10.0125

8.7021

7. 0992

NORMALIZING FACTOR - -0.2978E-03

0.0

0.4

REAL

N

6.

A

305.

RESULTS ARE AS

THETA

0.01

0.09

0.15

0.19

0.27

0.47

0.61

0.86

1.48

1.92

2.27

2.57

2.84

FREQ

0.0

0.

0.

0.

1.

3.

5.

10.

30.

50.

70.

90.

110.

1

3

5

0

0

0

0

0

0

0

0

0

RHO-1

10.

FOLLOWS.

REAL

1.0000

0.9827

0.9289

0.8724

0. 7458

0.4797

0.4177

0.4567

0.5066

0. 5026

0. 5028

0.5030

0.5029

.0

..

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0.

-0.

-0.

0.

-0.

-0.

RHO-2

10.0

MAG

0069

0642

1458

1967

2553

1980

0921

0169

0006

0005

0000

0000

0001

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

DEPTH

152.

PREAL

.0006

.0179

.0717

.1282

.2547

.5209

.5829

.5438

.4940

.4980

.4977

.4976

.4976

-0

-0

-0

-0

-0

-0

-0

0

-0

-0

0

-0

-0

W

1.

PIMAG

.0069

.0642

.1458

.1967

.2553

.1980

.0921

.0169

.0006

.0005

.0000

.0001

.0001

J

1.

(JIEAL

1.0006

1.0006

1.0006

1.0006

1.0006

1.0006

1.0006

1.0006

1.0006

1.0006

1.0006

1.0006

1.0006

M

2.

QIMAG

-0.0000

-0.0000

0.0000

0.0000

-0.0000

0.0000

-0.0000

-0.0000

0.0000

0.0000

0.0000

0.0000

-0.0000

NORMALIZING FACTOR - -0.5949E-02

IMAGINARY W ( 

O )&J

3* > 

)le6:

u.u 1 ————— 

0.7

N A

1. 305.

RESULTS ARE AS

THEIA FREQ

0.01 0.0

0. 04 0. 1

0.07 0.3

0. 09 0. 5

0.12 1.0

0.21 3.0

0.27 5.0

0.38 10.0

0.66 30.0

0. 86 50. 0

1.01 70.0

1.15 90.0

1.27 110.0

X\

\

\

REAL

RHO-1 RHO-2

50

FOLLOWS

REAL

1.0000

1.0013

1.0069

1.0067

1.0050

1.0039

1.0024

0. 9972

0.9639

0.9208

0.8741

0.8267

0. 7800

0

-0.

0.

0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

1250

IMAG

0001

0030

0005

0029

0063

0166

0272

0521

1360

2006

2505

2889

3180

0

\

1.0

DEPTH

152.

PREAL

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0000

0000

0000

0001

0003

0018

0041

0123

0633

1245

1853

2416

2915

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

W

1.

PIMAG

.0003

.0007

.0020

.0033

.0065

.0188

.0305

.0574

.1380

.1862

.2114

.2203

.2174

J

1.

QREAL

1.0000

1.0013

1.0069

1.0068

1.0052

1.0056

1.0065

1.0095

1.0272

1.0453

1.0595

1.0683

1.0715

M

2.

QIMAG

0.0001

0.0037

0.0025

0.0003

0.0001

0.0021

0.0033

0. 0053

0.0020

-0.0144

-0.0391

-0.0686

-0.1006

NORMALIZING FACTOR - -0.5707E-K>0
CONCLUSIONS AND COMPUTER RESULTS

Example 9: 

-0.3

19

o.o

0.7

H

1.

RESULTS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

A

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

RHO-1

50.

FOLLOWS.

REAL

1.0000

0.9951

1.0125

1.0227

1.0204

1.0129

1.0124

1.0075

0.9752

0.9327

0.8862

0.8389

0.7922

,0

, ..

-0.

0.

0.

0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

RHO-2

2500.0

MAG

0051

0035

0155

0070

,0083

0165

0266

0514

1353

2004

2508

2898

3193

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

DEPTH

152.

PREAL

.0000

.0000

.0000

.0001

.0003

.0017

.0039

.0121

.0638

.1264

.1889

.2470

.2987

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

W

1.

PIHAG

.0003

.0007

.0020

.0033

.0065

.0190

.0309

.0584

.1414

.1914

.2188

.2275

.2249

J

1.

<£EAL

1.0000

0.0051

1.0125

1.0228

1.0206

1.0146

1.0164

1.0197

1.0390

1.0590

1.0751

1.0858

1.0909

M

2.

QIMAG

-0.0048

0.0041

0.0175

0.0103

-0.0018

0.0025

0.0043

0.0070

0.0061

-0.0089

-0.0329

-0.0622

-0.0944

NORMALIZING FACTOR - -0.5695E400

1.0

H A RHO-1

2. 61. 0.4 

RESULTS ARE AS FOLLOWS. . .

THETA

0.01

0.09

0.16

0.20

0.29

0.50

0.65

0.92

1.59

2.05

2.42

2.75

3.04

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

REAL

1.0000

0.9995

0.9977

0.9951

0.9868

0. 9400

0. 8848

0.7435

0.3447

0.1636

0.0859

0.0549

0.0452

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

RHO-2

3.5

MAG

0006

0078

0223

0360

0675

1683

2424

3574

4209

3337

2462

1786

1293

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

DEPTH

18.0

PREAL

0000

0005

0026

0055

0151

0675

1245

2463

3821

3012

2084

1416

1003

W

1.

PIMAG

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0.

0005

0066

0186

0296

0539

1205

1545

1634

0411

0.1599

0.

0.

0.

1863

1717

1436

J

5.

OREAL

1.0000

1.0000

1.0003

1.0006

1.0019

1.0075

1.0093

0. 9898

0.7268

0.4647

0. 2942

0.1965

0.1455

M

2.

QIMAG

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0001

0012

0037

0064

0135

0478

0879

1940

4620

4936

4324

3503

2729

0.0

0.5

JD.5 1.0

N

3

RESULTS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

A

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

RHO-1

50.0

FOLLOWS. .

REAL

1.0000

0. 9838

0.9163

0.8499

0.7143

0.5020

OV5002

0.5972

0. 71 16

0.7710

0.8213

0.8651

0. 9048

RHO-2

2.5

QtAG

-0.0356

-0.0720

-0.1530

-0.1951

-0.2223

-0.0768

0. 0498

0.1570

0.2400

0.3224

0.3928

0.4565

0.5156

DEPTH

30.

PREAL

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0077

0238

0912

1583

2990

5586

6143

6147

6782

7391

7879

8304

8689

W

1.

PIMAG

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0379

0775

1700

2241

2811

2288

1585

1263

2375

3149

3823

4435

5005

J

1.

OREAL

1.0077

1.0076

1.0075

1. 0082

1.0133

1.0606

1.1145

1.2119

1.3898

1.5101

1.6092

1.6956

1.7737

M

2.

QIMAG

0.0023

0.0055

0.0171

0.0291

0.0588

0.1520

0.2083

0.2832

0.4775

0.6373

0.7751

0.9000

1.0160

NORMALIZING FACTOR - -0.1661E-02

0.0

0.0 0.5 

REAL

1.0

RHO-1 RHO-2

3. 305. 1.0 1 

RESULTS ARE AS FOLLOWS... 

THETA FREQ REAL IMAG QREAL QIMAG

0.01

0.27

0.47

0.61

0.86

1.48

1.92

2.71

4.69

6.06

7.17

8.13

8.98

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

0.9928

0.9657

0.9316

0. 8354

0.4797

0. 2491

0.0342

0.0687

0. 0806

0.0757

0.0735

0.0738

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0005

0480

1259

1893

3055

4658

4341

2386

0259

0237

0192

0126

0073

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0000

0080

0382

0751

1709

4007

4078

1828

0617

0747

0713

0706

0714

-0.0004

-0.0382

-0.0938

-0.1318

-0.1771

-0.0676

0. 1060

0.2397

0.0328

0.0249

0.0179

0.0112

0.0065

1.0000

1.0008

1.0040

1.0068

1.0063

0.8804

0.6570

0.2170

0.1304

0.1553

0.1470

0.1441

0.1452

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0001

0098

0322

0575

1284

3982

5401

4784

0587

0486

0371

0238

0138

NORMALIZING FACTOR - -0.6572E400

NORMALIZING FACTOR - -0.1987E400
20 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH

Example 11: 

-0.3

1.0

0.2 +0.4

N

3

RESULTS

THETA

0.01

0.04

0.07

0.09

0. 12

0.21

0.27

0.38

0.66

0.36

1.01

1.15

1.27

H

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

RHO-1

50.0

FOLLOWS. . .

REAL

1.0000

0.9933

0. 9625

0.9279

0.8425

0.6057

0.4999

0.4623

0.5776

0.5989

0.6123

0.6260

0. 6388

RHO-2

5.0

IMAG

-0.0198

-0.0420

-0.1013

-0.1434

-0.2060

-0.2261

-0.1538

-0.0014

0.0867

0.0946

0.1120

0.1287

0.1435

DEPTH

15.

PREAL

-0.0029

-0.0096

-0.0403

-0.0749

-0.1606

-0.4046

-0.5227

-0.5938

-0.5534

-0.5707

-0.5890

-0.6034

-0.6158

W

1.

PIMAG

-0.0203

-0.0433

-0.1053

-0.1501

-0.2199

-0.2673

-0.2169

-0.0987

-0.0624

-0.0938

-0.1119

-0.1266

-0.1403

J

1.

OREAL

1.0029

1.0029

1.0028

1.0028

1.0031

1.0103

1.0226

1.0561

1.1310

1.1696

1.2014

1.2294

1.2545

M

2.

QIMAG

0.0005

0.0013

0.0040

0.0068

0. 01 39

0.0412

0.0631

0.0972

0.1491

0. 1884

0.2239

0.2554

0.2838

N

3

RESULTS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

A RHO-1 RHO-2

305. 50. 0 5. 0

ARE AS FOLLOWS. ..

FREQ REAL IMAG

0.0 1.0000 -0.0192

0.1 0.9934 -0.0407

0.3 0.9630 -0.0976

0.5 0.9289 -0.1374

1.0 0.8455 -0.1954

3.0 0.6171 -0.2026

5.0 0.5186 -0.1222

10.0 0.^943 0.0436

30.0 0.6408 0.1666

50.0 0.6834 0.1998

70.0 0.7139 0.2389

20.0 0.7424 0.2752

110.0 0.7687 0.3082

DEPTH

30.

PREAL

-0.0029

-0.0094

-0.0397

-0.0737

-0.1578

-0.3997

-0.5215

-0.6099

-0.6123

-0.6487

-0.6832

-0.7121

-0.7378

W

1.

PIMAG

-0.0203

-0.0432

-0.1054

-0.1507

-0.2227

-0.2837

-0.2468

-0.1512

-0.1436

-0.1969

-0.2361

-0.2699

-0.3010

J

1.

QREAL

1.0029

1.0029

1.0027

1.0026

1.0033

1.0168

1.0401

1.1042

1.2531

1.3321

1.3971

1.4545

1.5065

M

2.

QIMAG

0.0011

0.0025

0.0078

0.0132

0.0273

0.0811

0.1246

0. 1948

0.3102

0.3968

0.4750

0.5451

0.6093

NORMALIZING FACTOR - -0.3327E-02

NORMALIZING FACTOR - -0.3337E-02

1.0

N

3

RESULTS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

A

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

RHO-1

50.0

FOLLOWS. . .

REAL

1.0000

0.9975

0.9847

0. 9693

0.9270

0. 7696

0.6558

0.5137

0.5340

0.6067

0. 6391

0.6570

0.6713

RHO-2

10.0

IMAG

-0.0101

-0.0222

-0.0575

-0. 0859

-0.1385

-0.2220

-0.2211

-0.1299

0.1031

0.1387

0.1483

0.1604

0.1752

DEPTR

30.

PREAL

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0011

0036

0163

0316

0738

2335

3536

5208

6052

5957

6059

6229

6401

W

1.

PIMAG

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0106

0233

0608

0916

1502

2585

2814

2394

1004

1073

1338

1562

1742

J

1.

OREAL

1.0011

1.0011

1.0010

1.0009

1.0008

1.0032

1.0094

1.0345

1.1391

1.2024

1.2451

1.2800

1.3114

M

2.

QIMAG

0.0005

0.0011

0.0034

0.0057

0.0117

0.0365

0.0603

0.1095

0.2035

0.2460

0.2821

0.3166

0.3494

Example 14: 

-0.2

0.0

0.5 •

N

3

A

305.

RESULTS ARE AS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

FRB

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

5

1

RHO-1

50.

FOLLOWS.

REAL

.0000

0.9936

0.9642

0

0

0

0

0

0

0

0

0

1

.9315

.8520

.6399

.5544

.5526

.7602

.8500

.9193

.9829

.0424

0

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0.

0.

0.

0.

0.

0.

RHO-2

5

IMAG

0180

0380

0900

1255

1743

1575

0617

1325

3397

4327

5226

6051

6807

0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

DEPTH

61

PREAL

.0028

.0091

.0382

.0707

.1512

.3870

.5140

.6326

.7189

.7979

.8690

.9321

.9901

-0

-0

-0

-0

-0

W

1

PIMAG

.0200

.0428

.1048

.1508

.2264

-0.3131

-0

-0

-0

-0

-0

-0

-0

.3030

.2545

.3183

.4243

.5128

.5914

.6636

J

1.

OREAL

1.0028

1.0027

1.0024

.0022

.0032

.0269

.0685

.1852

1.4791

1.6479

1.7883

1.9150

2.0325

M

2.

QIMAG

0.0020

0.0048

0.0148

0.0253

0.0521

0.1556

0.2413

0.3871

0.6580

0.8571

1.0354

1.1965

1.3443

NORMALIZING FACTOR - -0.6686E-02

NORMALIZING FACTOR - -0.3381E-02
CONCLUSIONS AND COMPUTER RESULTS 21

Example 15: 

-0.1

0.5

N

3

RESULTS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

A

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

RHO-1

50.

FOLLOWS.

REAL

1.0000

0. 9982

0. 9902

0.9816

0. 961 2

0.9074

0.8834

0.8752

0.9707

1.0884

1.2077

1.3214

1.4243

0

-0.

-0.

-0.

-0.

-0.

-0.

0.

0.

0.

0.

0.

0.

0.

RHO-2

5.0

IMAG

0046

0097

0222

0303

,0403

0284

0033

0803

2660

3691

4280

4529

4515

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-1

-1

-1

DEPTH

305.

FREAL

.0008

.0024

.0097

.0177

.0373

.1012

.1488

.2385

.5233

.7975

.0546

.2799

.4650

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

W

1.

PIMAG

.0063

.0135

.0343

.0512

.0843

.1668

.2221

.3273

.6092

.7635

.8240

.8153

.7589

J

1.

(JIEAL

1.0008

1.0006

1.0000

0.9993

0.9985

1.0086

1.0322

1.1137

1.4941

1.8859

2.2623

2.6013

2.8893

M

2.

QIMAG

0.0016

0

0

0

0

0

0

0

0

1

1

1

1

.0038

.0121

.0209

.0440

.1384

.2254

.4076

.8752

.1327

.2520

.2682

.2104

Example 17: 

-0.2 r

-0.1

0.0

0.1

0.8

N

3.

RESULTS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

A RHO-1

305 50. 0

RHO-2

5. 0

DEPTH

457.

W

1.

J

1.

M

2.

ARE AS FOLLOWS. . .

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

REAL

1.0000

0. 9991

0.9954

0. 991 3

0. 981 5

0.9522

0.9328

0.8999

0.8104

0.7531

0.7285

0.7269

0. 7373

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

MAG

,0028

,0059

,0146

,0211

,0328

,0554

,0665

,0833

,1038

0798

0423

0092

0.0136

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

FREAL

.0004

.0011

.0044

.0079

.0165

.0467

.0727

.1331

.3830

.6168

.7915

.8993

.9513

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

PIMAG

.0032

.0069

.0178

.0270

.0461

.1012

.1436

.2284

.4204

.4558

.4003

.3060

.2062

(JIEAL

1.0004

1 . 0003

0.9997

0. 9992

0.9981

0.9990

1.0055

1.0330

1.1934

1.3699

1.5201

1.6262

1.6885

QIMAG

0. 0004

0. 0009

0.0032

0.0059

0.0133

0.0458

0.0771

0. 1451

0.3166

0.3759

0. 3580

0.2968

0.2198

NORMALIZING FACTOR - -0. 2159E-01

NORMALIZING FACTOR - -0.1090E-01

Example 16:

-0.1

0.0 

> 0

oc

z

a

|

0.1

0.2

8

_

N A

3 305

RESULTS ARE AS

THETA FREQ

0. 01 0. 0

0. 04 0. 1

0.07 0.3

0. 09 0. 5

0.12 1.0

0. 2 1 3. 0

0.27 5.0

0.38 10.0

0.66 30.0

0. 86 50. 0

1.01 70.0

1.15 90.0

1.27 110.0

•*•/

r~

REAL

\

\

RHO-1 RHO-2

50. 0 5. 0

FOLLOW

REAL

1.0000

0.9987

0.9933

0.9873

0.9732

0.9343

0.9138

0.8940

0.8997

0.9296

0.9705

1.0164

1.0611

3...

IMAG

-0.0035

-0.0073

-0.0173

-0.0242

-0.0345

-0. 0403

-0.0313

-0.0041

0.0685

0.1175

OV1513

0.1692

0.1724

^S*

V

1.0

<\t—

DEPTH W

366. 1.

FREAL PIMAG

-0.0005 -0.0044

-0.0017 -0.0096

-0.0066 -0.0246

-0.0119 -0.0369

-0.0250 -0.0618

-0.0690 -0.1283

-0.1037 -0.1761

-0.1755 -0.2698

-0.4358 -0.5061

-0.6902 -0.6036

-0.9126 -0.6049

-1.0863 -0.5454

-1.2080 -0.4543

J

1.

(JIEAL

1.0005

1.0004

0.9998

0.9992

0.9983

1.0033

.0175

.0695

.3355

.6197

.8832

2.1026

2.2691

M

2.

QIMAG

0. 0009

0.0023

0.0073

0.0127

0.0273

0. 0880

0.1449

0.2657

0.5746

0.7211

0.7562

0.7145

0.6266

Example 18: 

-0.3 r

>-0.2

oc 

<

z

o 

<

1-0.1

0.0

0.5

REAL

3 305. 50.0 

RESULTS ARE AS FOLLOWS. .. 

THETA FREQ REAL PIMAG (JIEAL

NORMALIZING FACTOR - -0. 3855E-01

1.0

QIMAG

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

1.

1.

1.

,01

,04

07

,09

,12

,21

,27

,38

,66

,86

,01

,15

,27

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

0. 9994

0.9967

0.9937

0.9866

0.9618

0. 9404

0.8891

0.6984

0.5789

0.5261

0.5123

0.5172

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0025

0054

0138

0207

0349

0740

1030

1569

2272

1857

1226

0690

0309

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0003

0008

0030

0054

0112

0334

0552

1135

3741

5764

6846

7220

7187

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

,0024

,0053

,0138

,0212

,0372

,0871

,1277

,2093

,3528

,3191

,2239

,1281

,0536

1.0003

1.0002

0.9997

0.9991

0.9978

0.9952

0.9955

1.0026

1.0725

1.1553

1.2107

1.2344

1.2358

-0.0001

-0. 0001

0. 0000

0. 0005

0.0023

0.0131

0.0247

0.0524

0.1257

0. 1 334

0.1012

0. 0591

0.0227

NORMALIZING FACTOR - -0.1538E-01
22 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISOTROPIC EARTH 

Example 19: Example 21:

0.4 0.7

REAL

3 305. 50.0 5. 

RESULTS ARE AS FOLLOWS... 

THETA FREQ REAL MAG HMAG QREAL QIMAG

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

0. 9996

0.9978

0. 9958

0. 9899

0.9578

0.9188

0.8214

0.5771

0.4707

0.4277

0.4175

0.4229

-0.0025

-0.0056

-0.0155

-0.0244

-0.0450

-0.1113

-0.1592

-0.2261

-0.2425

-0.1801

-0.1161

-0.0651

-0.0290

-0.0002

-0. 0005

-0.0017

-0.0030

-0.0071

-0.0311

-0.0635

-0.1541

-0.4042

-0.5108

-0.5531

-0.5632

-0.5578

-0.0021

-0.0046

-0.0127

-0.0204

-0.0383

-0.0996

-0.1476

-0.22H

-0.2443

-0.1799

-0.1158

-0.0651

-0.0290

1.0002

1.0000

0.9995

0. 9988

0. 9969

0.9890

0. 982 4

0.9755

0. 981 2

0. 981 5

0. 9808

0.9807

0. 9808

-0. 0005

-0.0010

-0.0027

-0.0041

-0.0067

-0.0117

-0.0116

-0.0050

0.0018

-0.0002

-0.0003

-0.0000

0. 0000

NORMALIZING FACTOR - -0.3398E-01

0.0

05

0.4

1.0

REAL

N

5.

RESULTS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

A

305

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

RHO-1

50.0

FOLLOWS. .

REAL

1.0000

0. 992 1

0. 9560

0.9164

0.8220

0.5845

0. 5005

0. 51 36

0.6739

0.7225

0.7645

0. 8034

0.8390

RHO-2

10.0

IMAG

-0.0218

-0.0459

-0.1079

-0.1495

-0.2049

-0.1825

-0.0821

0. 0893

0.1915

0.2367

0.2824

0.3219

0. 3568

DEPTH

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

61.

FREAL

.0036

.0114

.0473

.0869

.1825

.4408

.5581

.6297

.6484

.7018

.7469

.7856

.8209

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

W J

1. 1.

PIMAG QREAL

.0232 1.0036

.0492 1.0035

.1183 1.0033

.1672

.2414

.2889

.2423

.1514

.1815

.2404

.2838

.3216

.3556

.0033

.0045

.0252

.0586

.1433

.3223

.4244

.5114

.5890

.6599

M

2.

QIMAG

0.0014

0. 0034

0.0104

0.0177

0. 0365

0.1064

0. 1602

0.2407

0. 3730

0.4771

0.5662

0.6435

0.7124

NORMALIZING FACTOR - -0.191 IE-02

Example 20: 

-0.2

0.0

REAL

1.0

N

1

RESULTS

THETA

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

A

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

RHO-1

50.0

FOLLOWS. .

REAL

1.0000

0. 9998

0. 9984

0.9967

0. 991 4

0. 9649

0.9366

0.8711

0.7155

0.6654

0.6622

0.6801

0.7072

RHO-2

10.0

.

IMAG

-0.0017

-0.0039

-0.0109

-0.0172

-0.0310

-0.0700

-0.0947

-0.1214

-0.0665

0. 0353

0.1270

0.2030

0.2653

DEPTH

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

61.

FREAL

.0001

.0003

.0016

.0033

.0085

.0344

.0635

.1297

.3212

.4310

.5011

.5508

.5894

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

-0

W

1.

PIMAG

.0020

.0046

.0130

.0207

.0383

.0929

.1332

.2009

.3016

.3293

.3435

.3560

.3693

1

1

1

1

J

1.

QREAL

.0001

.0001

.0001

.0000

0.9999

0.9993

0.9990

1

1

1

1

1

1

.0008

.0368

.0964

.1633

.2309

.2966

M

2.

QIMAG

0. 0003

0. 0007

0.0021

0. 0036

0.0073

0.0226

0.0385

0.0795

0.2352

0. 3645

0.4705

0.5590

0.6346

Example 22: 

-0.3

o.o

04

0.4

10. 305. 50.0 

RESULTS ARE AS FOLLOWS... 

THETA FREQ REAL PIMAG QREAL QIMAG

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

0.9620

0.8258

0.7150

0. 5467

0.4826

0.5548

0.6134

0.6822

0.7354

0.7797

0.8188

0. 8546

-0.0664

-0.1256

-0.2245

-0.2471

-0.1963

0.0461

0.1007

0.1075

0. 1838

0.2395

0.2854

0.3251

0.3607

-0.0195

-0.0574

-0.1944

-0.3081

-0.4893

-0.6202

-0.5936

-0.5962

-0.6767

-0.7299

-0.7742

-0.8134

-0.8492

-0.0692

-0.1323

-0.2457

-0.2827

-0.2636

-0.0884

-0.0612

-0.1047

-0. 1850

-0.2404

-0.2859

-0.3254

-0.3607

1.0195

1.0194

1.0202

1.0231

1. 0360

1.1028

1.1484

1.2096

1.3589

1.4653

1.5539

1.6322

1.7038

0. 0028

0.0068

0.0212

0.0356

0. 0672

0.1346

0.1619

0.2122

0.3688

0.4799

0.5713

0. 6505

0.7214

NORMALIZING FACTOR - -0. 7602E-01

NORMALIZING FACTOR - -0.2978E-03
0.0

0.8 1.0

CONCLUSIONS AND COMPUTER RESULTS 

Example 25: 

-0.5

0.0 

0.6

23

1.0

REAL

1. 305. 50.0 

RESULTS ARE AS FOLLOWS.•. 

THETA FREQ REAL QREAL QIMAG

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

1.0001

1.0001

1.0001

1.0000

0. 9995

0.9987

0. 9962

0. 9808

0. 9603

0.9370

0. 91 20

0.8859

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0001

0003

0011

0018

0035

0103

0169

0326

0884

1362

1778

2142

2462

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0000

0000

0000

0000

0001

0007

0016

0046

0236

0484

0757

1039

1322

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0001

0003

0010

0016

0031

0091

0148

0282

0729

1070

1327

1518

1651

1.0000

1.0001

1.0001

1.0001

1.0001

1.0002

1.0003

1.0009

1.0044

1.0086

1.0127

1.0159

1.0181

0. 0001

0. 0001

-0. 0001

-0. 0002

-0. 0004

-0.0013

-0.0021

-0.0044

-0.0155

-0.0292

-0.0450

-0.0624

-0. 081 1

NORMALIZING FACTOR - -0.1169E-t01

RHO-2 DEPTH

5 305. 50.0 5000.0 61. 1. 1. 2. 

RESULTS ARE AS FOLLOWS. .. 

THETA FREQ REAL IMAG PREAL PIMAG QREAL QIMAG

0.01

0.04

0.07

0.09

0.12

0.21

0.27

0.38

0.66

0.86

1.01

1.15

1.27

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

1.0012

1.0Q05

1.0003

1.0001

0. 9982

0.9953

0.9853

0.9228

0.8443

0.7618

0. 6808

0. 6042

0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0000

0006

0031

0048

0096

0260

0456

0866

2199

3158

3836

4300

4600

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0000

0000

0001

0002

0005

0032

0073

0217

1055

1969

2777

3420

3889

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0004

0009

0027

0045

0088

0252

0405

0743

1598

1886

1807

1501

1065

1.0000

1.0012

1.0006

1.0005

1.0007

1.0014

1.0027

1.0070

1.0283

1.0412

1.0395

1.0229

0.9931

0. 0012

0. 0003

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

-0.

0004

0004

0009

0028

0050

0123

0601

1272

2029

2799

3535

NORMALIZING FACTOR • -0.1337E+00

W IMAGINARY , 

p 3 ta 

^W 5 

>le 24:

\

0.9

\1.0

REAL

N A

1. 305.

RESULTS ARE AS

THEXA FREQ

0. 01 0. 0

0. 04 0. 1

0.07 0.3

0. 09 0. 5

0.12 1.0

0.21 3.0

0.27 5.0

0.38 10.0

0.66 30.0

0.86 50.0

1.01 70.0

1.15 90.0

1.27 110.0

RHO-1

50.0

FOLLOWS. .

REAL

1. 0000

0. 9960

0.9949

1.0013

1.0114

1.0075

1.0054

1.0048

0. 9959

0.9826

0. 9664

0. 9483

0.9287

RHO-2

5000. 0

IMAG

-0.0040

-0.0030

0.0063

0.0100

0.0046

-0.0100

-0.0134

-0.0246

-0.0699

-0.1107

-0.1477

-0.1812

-0.2115

DEPTH

61.

REAL

-0.0000

-0.0000

-0.0000

-0.0000

-0.0001

-0. 0004

-0. 0009

-0. 0029

-0.0173

-0.0380

-0.0621

-0. 0882

-0.1151

W

1.

PIMAG

-0.0001

-0.0003

-0. 0008

-0.0013

-0.0026

-0.0078

-0.0129

-0.0250

-0. 0692

-0.1035

-0.1320

-0.1546

-0.1720

J

1.

qREAL

1.0000

0.9960

0.9949

.0013

.0114

.0079

.0063

.0077

.0132

.0206

.0286

.0365

1.0438

M

2.

QIMAG

-0.0039

-0. 0028

0.0071

0.0114

0.0072

-0.0021

-0.0005

0. 0005

-0.0017

-0.0072

-0.0156

-0.0265

-0.0395

Example 26:

0.2

N

3.

A

305.

RHO-1 RHO-2 DEPTH

50.0 10.0 61.

W

1.

J M

1. 2.

RHO-3

50.0

RESULTS ARE AS FOLLOWS...

THETA

ZFRSAPR

0.025

0.038

0.066

0.086

0.121

0.210

0.271

0.383

0.664

0.857

1.014

1.149

1.271

FREQ REAL IMAG

F10ATING UNDERFLOW PC0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

0. 9998

0. 9983

0.9958

0. 9866

0.9263

0.8518

0.6736

0.3645

0.3819

0.4664

0.5485

0.6143

-0.0033

-0.0078

-0.0228

-0.0372

-0.0707

-0.1746

-0.2397

-0.2969

-0.0880

0.1077

0.2023

0.2376

0.2433

PREAL

3646

-0. 0004

-0.0012

-0. 0052

-0.0101

-0.0236

-0.0746

-0.1135

-0.1707

-0.2172

-0.2264

-0.2378

-0.2500

-0.2616

PIMAG

-0.0035

-0.0076

-0.0200

-0.0301

-0.0498

-0.0883

-0.0997

-0.0956

-0.0694

-0.0789

-0.0933

-0. 1064

-0.1177

QREAL

1.0004

1.0010

1.0035

1.0059

1.0102

1.0008

0.9653

0.8443

0.5817

0. 6084

0. 7042

0. 7985

0.8760

QIMAG

0. 0001

-0. 0001

-0.0028

-0.0070

-0.0210

-0.0863

-0.1400

-0.2013

-0.0186

0.1866

0.2956

0. 3440

0.3610

NORMALIZING FACTOR - -0.2043E-01

NORMALIZING FACTOR - -0.1417E+01
55 H- I- H- <

*~WH-O»-1 OvOvWWi

Z h- \g *sj Ui W i-~ WH-00

•** ooooooooowwt—o

o o<k..PPPPPPPPf

S oooooioiooooi

» O 00 O -J O ^ I

H-H-H-H-OOOOOOOO

• f=5 » fe

v U> Ul * O

m p 

> 01

r 5.

IMAGINARY

OOOOOOUiWl— OOOO

• ••••••••••••

oooooooooww^-o

OOOOOOOOOOOH-t—

poopppppppppp

S »°

OOOOOOOOOOOOO

£ncvui*>*>wroro^-H-ooO

iiiiiiiiiiiii .*....* *..**

ooooooooooooo

ooooooop ooooo

p p

1) N)

IMAG

* i-3 !

» g i * y'

H ppppppppppppp ° o

UJ OOOOOOOOOOOOO Q •

^ ---------,^oowf^^^-p »g ^*

S *°

ppppppippoppp w^ « 

ooooooooooooo £»~

____.,ppittppp ^ ooooooooooooo *g

oooooooooooop

t W ISS N3 I 

> *- v£) 00 •

' ro oo oo i

n vD NS *> I

„„ ._.- . J t— O O W *> W >— O

^ ^-^-H-pppppppppp*HP" 

£ i—'VD4j*^j*>W^-OH-O\CvOOUi ^2"

z; ^ \c -^

o poc

•^ OOOOOOOOOWiWH-O

1 C!

o ^ I

>iiiiiiiiiii 

iiiiii^>^>iiiii ^ ( ooooooooooooo *! 

ui*>*>wwroi-'»—ooooo * |

oiiiiiiiioiio ^

OOOOOOOOOOOOO *| 

LnooUi^slOOH-^CvvDhOO\Ui^J * (

OppppppOJ-'H-H-H-Jooooiioooooop S 

^-' oo^' IOH-OOOOO*^ o

IMAGINARY
CONCLUSIONS AND COMPUTER RESULTS 25

Example 31: Example 33: /-~^

0.6

N

3.

RESULTS

THE1A

0.025

0.038

0.066

0.086

0.121

0.210

0.271

0.383

0.664

0.857

1.014

1.149

1.271

0.7 0.8 0.9 1. 

REAL

A

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

RHO-1

50.0

FOLLOWS. .

REAL

1.0000

1.0039

1.0066

0. 9997

0. 9696

0. 8652

0. 8096

0. 7460

0. 6886

0. 6855

0.6848

0.6782

0. 6671

RHO-2

1.0

.

-0.2 >

(C

-0.1 -

o

0.0 - 

0

DEPTH W J

IMAG

0. 0004

-0. 0043

-0. 0288

-0.0518

-0.0918

-0.1332

-0.1295

-0.1073

-0.0544

-0. 0384

-0. 0397

-0.0454

-0. 0488

305.

PREAL

-0.0005

-0.0015

-0.0045

-0.0070

-0.0116

-0.0223

-0.0299

-0.0474

-0.1296

-0.2342

-0.3491

-0.4620

-0.5639

1. 1.

PIMAG

-0.0022

-0.0044

-0.0096

-0.0133

-0.0204

-0.0421

-0.0612

-0. 1047

-0.2453

-0.3431

-0.4003

-0.4212

-0.4127

M

2.

QREAL

1.0005

1.0053

1.0111

1.0067

0. 981 1

0.8875

0.8395

0. 7934

0.8182

0.9197

1.0338

1.1402

1.2311

RHO-3

50.0

QIMAG

0. 0026

0.0001

-0. 0192

-0.0385

-0.0714

-0.0912

-0. 0683

-0.0026

0. 1909

0.3046

0.3606

0.3759

0. 3639

NORMALIZING FACTOR - -0.2566E-01

0.0 0.2 0.3 0.4 0.5 0.6 

REAL

0.7 0.8 0.9 1.0

0.0

N

3.

RESULTS

THE1A

0.025

0.086

0.148

0.192

0.271

0.469

0.606

0.857

1.484

1.916

2.267

2.570

2.841

A

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

SO. 0

70.0

90.0

110.0

RRO-1

10.0

FOLLOWS.

REAL

1.0000

1.0057

1.0050

1.0049

1.0046

1.0023

0.9977

0. 9740

0. 7378

0. 4094

0.1165

-0.0928

-0.2170

RHO-2 DEPTH

1000. 0

IMAG

0. 0006

-0.0024

-0.0067

-0.0111

-0.0221

-0.0660

-0.1101

-0.2191

-0.5667

-0.7128

-0.6958

-0.5879

-0.4466

61.

PREAL

-0.0000

-0.0000

-0.0001

-0.0003

-0.0008

-0. 0048

-0.0105

-0.0276

-0.0867

-0.1070

-0.1007

-0.0828

-0.0625

W J

1. 1.

PIMAG

-0.0001

-0.0007

-0.0020

-0.0033

-0.0065

-0.0174

-0.0262

-0.0403

-0.0349

0.0024

0.0355

0.0562

0.0657

M

2.

OREAL

1.0000

.0057

.0051

.0052

.0055

.0072

.0081

.0017

0.8245

0.5164

0.2172

-0.0100

-0.1544

RHO-3

50.0

QIMAG

0.0007

-0.0017

-0.0046

-0.0077

-0.0156

-0.0486

-0.0840

-0.1788

-0.5319

-0.7153

-0.7313

-0.6440

-0.5123

NORMALIZING FACTOR - -0.1314E+01

Example 32: Example 34:

-0.5

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

RHO-1 RHO-2 DEPTH

3. 305. 500.0 1.0 

RESULTS ARE AS FOLLOWS... 

THE1A FREQ REAL IMAG OREAL QIMAG

0.025

0.012

0.021

0.027

0.038

0.066

0.086

0.12V

0.210

0.271

0.321

0.363

0.402

0.4

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

1.0000

1.0244

1.0210

0. 9841

0. 8386

0. 3875

0. 2292

0.1509

0.1158

0.1073

0.1043

0. 1037

0. 1046

-0.1934

-0.0318

-0.1323

-0.2247

-0.3769

-0.4225

-0.3092

-0.1632

-0.0537

-0.0182

0. 0052

0.0240

0. 0404

-0.0012

-0.0003

-0.0009

-0.0013

-0.0020

-0.0027

-0.0028

-0. 0032

-0.0043

-0.0051

-0.0057

-0.0062

-0.0067

-0.0012

-0.0006

-0.0011

-0.0012

-0.0013

-0.0011

-0.0014

-0.0021

-0.0042

-0. 0060

-0.0077

-0.0093

-0.0109

1.0012

1.0246

1.0219

0. 9854

0. 8406

0. 3902

0.2320

0. 1542

0.1201

0.1123

0.1100

0.1099

0.1113

-0.1922

-0.0312

-0.1313

-0.2234

-0.3757

-0.4214

-0.3079

-0.1612

-0.0496

-0.0122

0.0129

0.0333

0.0512

N

3.

RESULTS

THE1A

0.025

0.086

0.148

0.192

0.271

0.469

0.606

0.857

1.484

1.916

2.267

2.570

2.841

0.7 

A

305.

ARE AS

FREQ

0.0

0.1

0.3

0.5

1.0

3.0

5.0

10.0

30.0

50.0

70.0

90.0

110.0

\

0.8 

RE

RHO-1

10.0

FOLLOWS

REAL

1.0000

1.0001

1.0038

1.0041

1.0033

1.0030

1. 0028

1.0015

0. 9859

0. 9547

0.9117

0. 8601

0. 8026

0.9 1. 

AL

RHO-2

10000. 0

. ..

IMAG

-0.0006

0.0021

0.0005

-0.0020

-0.0051

-0.0149

-0.0248

-0.0497

-0.1487

-0.2413

-0.3248

-0. 3980

-0.4608

-0.4 

> 

-0.3 <

-0.2 1 

-0.1 

0.0

0

DEPTH

9.

PREAL

-0.0000

-0. 0000

-0.0000

-0.0000

-0.0000

-0.0003

-0.0006

-0.0019

-0.0101

-0.0202

-0.0303

-0.0396

-0.0477

if J

1. 5.

PIMAG

-0.0000

-0.0001

-0. 0003

-0.0005

-0.0010

-0. 0028

-0.0046

-0.0087

-0.0209

-0.0278

-0.0308

-0.0310

-0.0291

M

2.

OREAL

1.0000

1.0001

1.0038

1.0041

1.0034

1.0033

1.0034

1.0034

0.9960

0.9748

0.9419

0. 8997

0.8503

RHO-3

200.0

QIMAG

-0.0006

0.0022

0. 0008

-0.0015

-0. 0042

-0.0120

-0.02B2

-0.0410

-0.1278

-0.2135

-0. 2939

-0.3670

-0.4317

NORMALIZING FACTOR - -0.3237E402

NORMALIZING FACTOR - -0.1708E-01
26 ELECTROMAGNETIC COUPLING WITH A COLLINEAR ARRAY ON A TWO-LAYER ANISTROPIC EARTH

SELECTED REFERENCES

Abramowitz, Milton, and Stegun, I. A., 1964, Handbook of mathematical 

functions — with formulas, graphs, and mathematical tables: National 

Bureau of Standards Applied Mathematics Series 55, 1046 p.

Anderson, W. L., 1975, Improved digital filters for evaluating Fourier and 

Hankel transform integrals: Available only from National Technical 

Information Service, Springfield, Va. 22161, as NTIS PB-242 800,119 

PDey, Abhijit, and Morrison, H. F., 1973, Electromagnetic coupling in 

frequency and time-domain induced-polarization surveys over a 

multilayered earth: Geophysics, v. 38, no. 2, p. 380-405.

Hallof, P. G., 1974,The IP phase measurement and inductive coupling: 

Geophysics, v. 39, no. 5, p. 650-665.

Hohmann, G. W., 1973, Electromagnetic coupling between grounded wires 

at the surface of a two-layer earth: Geophysics, v. 38, no. 5, p. 854-863.

——————1975, Three-dimensional induced polarization and electromagnetic coupling: Geophysics, v. 40, no. 2, p. 309-324.

Jahnke, Eugene, and Emde, Fritz, 1945, Tables of functions: New York, 

Dover, 380 p.

Millett, F. B., Jr., 1967, Electromagnetic coupling of collinear dipoles on a

uniform half-space, in Mining Geophysics, Volume II, Theory: Soc.

Exploration Geophysicists, p. 401-419. 

Riordan, J., and Sunde, E. D., 1933, Mutual impedance of grounded wires

for stratified two-layer earth: Bell Syst. Tech. Jour., v. 12, p. 162. 

Sumner, J. S., 1976, Principles of induced polarization for geophysical

exploration: New York, Elsevier, Developments in economic geology,

5. 277 p. 

Sunde, E. D., 1968, Earth conduction effects in transmission systems: New

York, Dover, 370 p. 

Wait, J. R., 1966, Fields of a horizontal dipole over a stratified anisotropic

half-space: IEEE Trans. on Antennas and Propagation, v. AP-14, no.

6. p. 790-792. 

Wynn, J. C., 1974, Electromagnetic coupling in induced polarization:

Arizona University Ph.D. thesis, 137 p. 

Wynn, J. C., and Zonge, K. L., 1975, EM coupling, its intrinsic value, its

removal and the cultural coupling problem: Geophysics, v. 40, no. 5, p.

831-850. 

Zonge, K. L., and Wynn, J. C., 1975, Recent advances and applications in

complex resistivity measurements: Geophysics, v. 40, no. 5, p. 851-864. 

Zonge, K. L., Wynn, J. C., and Young, G. N., 1976, Complex resistivity case

histories [abs.]: Geophysics, v. 41, no. 2, p. 382-383.

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