IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 8 August 1981THE COMPLEX GROUND RETURN PLANE

A SIMPLIFIED MODEL FOR HOMOGENEOUS AND MULTI-LAYER EARTH RETURN

A. Deri G. TevanTechnical University of Budapest

A. Semlyen A. CastanheiraUniversity of Toronto

Abstract - For modelling current return in homo-geneous ground, the paper introduces the concept of anideal (superconducting) current return plane placed be-low the ground surface at a complex distance p equal tothe complex penetration depth for plane waves. This"complex"' plane appears as a mirroring surface, so thatconductor images can be used to derive very simpleformulae for self and mutual impedances under groundreturn conditions. Such equations, without proofs, wereoriginally proposed by Dubanton and published by Gary.1In this paper, plausibility arguments serve to initiallyjustify the procedure, then the equations are analyti-cally related to those of Carson and, finally, theerrors, which in most cases are less than a few percent,are numerically evaluated. The ideal return plane atcomplex depth can also be used for multi-layer earthreturn.

INTRODUCTION

Knowledge of ground return parameters of transmi-ssion lines has long been recognized as being importantin telecommunications theory. Later, applications forcalculation of transients on power lines and problemsrelated to harmonics have become equally important. Anumber of fundamental papers2 7 have developed.ex-pressions for line return parameters with ground return,Carson's approach2 being the most widely known andaccepted. It is valid for homogeneous earth and forfrequencies low enough so that the capacitive displace-ment currents in ground can be neglected and the wave-length is sufficiently large compared to transversalgeometrical dimensions.

The calculation of line impedances according toCarson is based on equations which contain infiniteintegrals with complex arguments. For their evaluationCarson has proposed infinite series and also some con-venient approximations for low and high frequencies(see also references [8], [9] and [10]). While theseapproximations are relatively simple they are valid eachfor a limited range of frequencies only, and medium fre-quencies are not covered.

In this paper it will be demonstrated and analyti-cally proved that simple and sufficiently accurate ex-pressions for line impedances, initially proposed byDubanton at Electricite de France and published byGary1,are valid for the whole range of frequencies.

Gary mentions in his paper1, that the Dubantonequations have probably been obtained by intuitive in-sight and a proof is still required. The present paperprovides a heuristic justification of the concept of aground-current return plane placed at a complex depth,suggested by Dubanton; then the resultant equations areanalytically related to those of Carson and the errors

are numerically evaluated. The method of the "complexground return plane" is, inaddition, extended for themodelling of multi-layer ground return.

THE COMPLEX GROUND RETURN PLANE

Fundamental Equations

The ideal conductor/ground return (self and mutual)impedances, according to Carsonand in a formadapted toour purpose, are:

Zs_2h + w

VJZs = jw - n--+ uJPo5 2iT r iT 5 =11

(hh+h)2 +d2Zn jw-0 i hk) kk

2i7r (hk-h') 2' + d2 Tr Jm s "Vo (2')

where the ground corrections terms J are the infiniteintegrals:

. -2hXJ = P + jQ =

j -dXs s s X +5 +j

o

( "t)

andco

.-(hk+hk) XJ = P + jQ = -je ~~ cosXd x dX (2")m m m x+v7 T 1 r

k=P+jQ~~Oand where:

h, hk, ht are the conductor height above ground

r is the conductor radiusd is the horizontal distance between conductors

k and Qa is the earth conductivity

The above equations were derived by Carson for theearthpermeability p equal to po and an approximation will begiven below for that case.

The complex ground return plane is a plane of per-fect conductibility, replacing the real earth andsituated at a complex distance p below the.earth surface,so that the ideal conductor/complex plane impedancesresult from the following equations:

Z = i °-O n 2(h + p)s i 2Tr r

WM 222-9 A paper recommended aiid approved by theIEEE Transmission & Distribution Committee of theIEEE Power Engineering Society for presentation atthe IEEE PES Winter Meeting, Atlanta, Georgia,February 1-6, 1981. M.anuscript submitted August 22,

1980; made available for printing December 8, 1980.

(hk+hg+2p)+dk2Z =j& Qn -

k

m 2r 2 2

(hk-hI) + dkQ

(4)

These are the Dubantonm equations. They provide simpleand remarkably accurate substitutes to Carson's equa-tions (1) and (2) over the whole range of frequencies

1981 IEEE

3686

(3)

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for which Carson's equations are valid.It should be mentioned that the convenience of

equations (3) and (4) results from the fact that theycan be evaluated by electronic hand held calculators,while the evaluation of Carson's equations requires acomputer program or reading of numerical values fromgraphs.

The next section contains a plausibility argumentfor the validity of these equations.

Plausibility ArgumentationThe following concepts will be introduced in what

follows:.Equivalent return distance*Complex depth-Complex ground return plane

The equivalent return distance is defined with the aidof Fig. 1. The conductor is assumed to be a plane at

at any depth, and an ascending wire. For sucha contour:

f(Edz) = -jwf (5)

where 4 is the flux crossing the loop. The line-inte-gral in (5) is

f(EdQ) = -V + ZcI + IJ (6)

where Z is the impedance (per unit length) of the con-ductor, and J the current density at the depth of thereturn line of the contour.

From (5) and (6):

V = Z I + 1 J + jw4c ar (7)

This equation shows the voltage components of V. If weput Zc = 0, we have

V = 1 J + jw4a

(7')

Clearly, the breakdown of V into its two components in(7') is arbitrary since the return depth was selectedarbitrarily. It is therefore convenient to considerthe return line at infinite depth, where J = 0. Then,from (7'):

(711)V = jw

which is valid for a return line at infinite depth.Equation (7") shows that the impedance of the loop

is

(8)Z = V =

(a) (b)

Fig. 1 To the definition ofdistance

equivalent return

height h above ground. The magnetic field is parallelto the ground. The impedance of the conductor/groundreturn loop is Z = V/I, as shown in Fig. la. In Fig.lbthe earth is replaced by a perfectly conductive planeat distance D from the plane conductor. The equivalentreturn distance is defined by the requirement that theimpedance Z of the conductor/return plane loop of Fig. lbbe the same as of the loop in Fig. la.

Let D be the equivalent return distance. It turnsout to be a complex length since the inductance of theloop has to account for the resistance of the earth re-turn. Therefore p = D - h represents the complex depthof the ground return plane below the earth surface.

Because p is complex, this plane will be calied the com-

plex ground return plane. For the plane field of our

discussion the complex plane replaces effectively, andwithout loss of accuracy the actual ground return path.

The complex ground return plane is significant be-cause the whole current returns through it. It is re-

tained for the case of actual thin conductors, toconstruct image conductors. This isa heuristic approachto extend theusability of the concept of complex groundreturn plane for conditions different from those forwhich it has been defined. It is assumed that a planewhich serves as an ideal current return path is, by thisfact, an effective mirroring surface, which can be usedeven in the case of real conductors. It appears thusplausible that equations such as (3) and (4) will begood substitutes for Carson's formulae, (1) and (2).

Calculation of Complex Depth

Consider in Fig. la a closed loop formed by thesource, (a line in the) conductor plane, descendingwire (dashed), a straight horizontal line in the earth

Its calculation requires the determination of the fluxproduced by I.

Now we recall the definition of equivalent returndistance (see Fig. lb). Let us assume h = 0. Then we

obtain the complex depth p as the distance between thetwo planes of Fig. lb for which the flux between theplanes equals the total flux in earth, from its surfaceto infinity.

We will calculate the complex depth p for threecases, all with p =

- Homogeneous earth- Multi-layer earth- Continuously stratified earth (here, of

course, strata do not have the customarysignificance of discrete layers)

Homogeneous and Multi-Layer Earth

If I denotes the current per unit width (perpendi-cular in Fig. 1 to the plane of the paper) thenH = H(O) = I, where H(x) is the magnetic field inten-Qsity at depth x. The pertinent differential equationsare:

dE -jyp H

dx 0

dHdx

and their solutionFig. 2):

(9 1)

(9")

is, for a layer numbered by K (see

HEK7 rC BKDK Hr 1 (10)

conductor plane+

v

conductor plane

or, in the homogeneous earth case (n = 0):

-1 A

A D-6k 6K =coshKK2DKPK

-1

Co =-- = sinhK KK2 K PK

where

(11)

1

PK

= e PK (12

These equations can be obtained directly or usingwave concepts. With F indicating a forward and B a

backward wave, we have:

F B

F B

and

EF = K HF = EF e

pEB = KHB BBK

K-1

(14)

PKHF=HFK-IeH =H K

K-1

B BK e

KEC-1,

E H

EKHK~~~~~~~

Fig. 2 Notations for layer no. K

Note, that solution (10) is analogous to that of trans-mission lines connected in a chain.

If there are n + 1 layers, in the last one (ex-tending to infinity) there will be no backward wave.

Therefore

En (5

H n+1 (15)n

12')

12")

For this latter case (14) yields

x

H = H e Po:

and the total flux in the earth is

co

f = P H dx = P VIH0

(16)

(17)

The complex depth p is defined by means of Fig. lb as

resulting in the same flux 4. Consequently, for homo-geneous earth and =1=po p-is the complex penetrationdepth

pe h1

related to the real penetrat ion depth

(18)

(18')6 = 1VI 'f p2f") by

- = (1l+i)-P 6

In the multi-layer case equation (10) can beapplied sequentially to obtain

(19)

(13)

EZ [AK BKo

In particular, for K = n, (20) becomes

En- -A B En no no o

yielding ,A H

n no no

yedn,with the'aid of (15):

E A -B :

Ano no n+l

Aow-I no no-

or, eliminating H :n

n+1 no no

°n+l ~n+1 Cn

(20)

(20')

(20")

(21)

since H = I, E can be calculated from (21) and all

EK, HK will be obtained from (20). Consequently, the

flux in each layer K is obtained by integrating (9'):

AK

k = PO Hdx = (EKi - ER)0)

(22)

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In (10):

0

° = C = C0

(15')

The total flux is

(23)

This permits to calculate the complex depth of equation(17):

KP p H (24)0 0

Continuously Stratified Earth

Quite naturally, the multi-layer case can be re-duced by a limiting process to the continuous problemformulation. So, for instance, equation (10) reproduces(9), which is available in the first place:

(9)

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from (3) and (4) that the impedances will be (approx-imately) the same.

Complex Inductance forHomogeneous Earth and } $ po

For ' ipo the concept of complex plane can stillbe used for calculating the conductor/ground returnimpedances, but the pertinent equations will be differ-ent from (3) and (4). Instead complex inductances

L = R- Js 7r s

L = kE Jm It m

(28')

(28")

can be calculated for equations (1') and (2'). Indeed,it will be shown later (see equations (40) and (47))that

The integration of (9), where a is a function ofx, is not possible in closed analytical form 1912. Butthe numerical integration is straight forward, forspecified starting values: E(o) = Eo, H(O) = Ho. Since,however, Eo is not known, one has to use several trialvalues until the following objective is satisfied:at a sufficiently large value of x, denoted by x.,, thebackward wave

j Sf 1k5 2 h

J i(hkh+2p)2+dk2J a J kn Q k9nm 2 (h+h2 +d2

E = 2 (E(x.) - C(X.) H(xw) )B 2

must vanish. Then, from (9'):

x I

J| Hdx i4 (E - E ) 00 jw) 0 Xc, jw

o.~~~~~~~~~~~c0.

Finally, the complex depth can be obtained from:

PH000i

(25)for any 1, i.e. for

1

=vrjwtc (30)

Fig. 4 helps to interpret equations (29). For two(26) conductors, k and i, equation (29") becomes

(31)tkQm 2

(27) k

If we review the way in which the complex depth phas been calculated, we can see that p represents thecentre of gravity of the return current (or of H(x)).

01

(2)

(3)-equivalenthomogeneous x

earth: Heq-Hoe P

Fig. 3 The complex depth p as centre of gravityof H(x)

Fig. 3 illustrates this in the case of three layers,where the area under H(x) has been shaded. The totalarea is pHo or the integral under Heq(x) ofxan equiva-lent homogeneous earth for which Heq = Hoe p . Nomatter what is the shape of H(x), as long as the centreof gravity is given by the same depth p, itis expected

Fig. 4 To the calculation of complex inductances

and (4) becomes

110 hk1m i2T DM (32)

Equation (31) is valid for homogeneous earth andany -p (and so is the correspondent expression for Zs).

(29')

(291")

f = E¢K

d E 0 -jwpo EF'dx H -a 0 H_

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It has to be used in conjunction with (28). Equation(32) is valid only for p = po, but its applicability isplausible for stratified earth as well.

In conclusion, the conductor/ground return loopimpedance can be related to the geometrical data ofFig. 4 by two terms:

1. A first term for the ideal ground return,in which we have the logarithm of theratioDg /DU , and the permeability is p .

2. A second term for the ground correction,in which we have the logarithm of the ratioDg /Dv , and the permeability is V.

Consequently, if 1' = Po, and only then, the two termscan be combined to yield the expression of the totalimpedance in which we have the logarithm of the ratioDj /Dk, and the permeability is i'

ANALYTICAL DERIVATION

Self Impedance

We intend to derive equation (3) from (1'). Firstwe substitute (30) into (1"):

I -2hXJ= 1 Je dXs + /X2+ /p2

where the limits indicate integration along the realaxis. Next we introduce a new dimensionless complexintegration variable w, using

s ( q) 2 Qn 1+q + Ks 2 q

where K = 0, since J (X) = 0. Then, with (34"),equation (39) yields

h+p

(39)

(40)

Expression (40) can be used for any p as the correctionterm in (1'). If u = po then (11) yields preciselythe approximate expression (3) which we wanted toderive.

It should be mentioned that, for low frequencies,equation (3) yields the familiar approximation

Z2

= -7f e+ IO

D

Z fi fl + jw -,Zn-t [si/ml5 2vT r (41)

Indeed, using (19), equation (3) becomes, for |p|>>h:1Im(w)

(33)

,k V

1/117

Fig. 5 To the integration of (35)

(34')wX =_-p

and

hq=-p

We obtain

-2qwJ (q) = Je dws I 7

J w + VW'+-(L)

(34")

(35)

where (L) is the line, at angle -450, shown in Fig. 5Since the integrand in (35) has no poles in the half-plane at the right of the dashed line and is zeroaround the arc LCL' at infinity, the integration canbe conveniently performed -along the real axis L'. Itis convenient then to consider the derivative of (35)with respect to q:

J'(q) f ~-2qw00

J (j w2w=w_+W dw (36) or

The integrals (35) and (36) can not be calculated ana-lytically but if the approximation (see Fig. 6):

-12w 1-22w_ 1 - e~2w + 1/42+1

is used in (36) we obtain

J'(q)&1( 1

-I2 1+q q

Consequently:

(37)

(38)

Fig. 6 To substitution (37)

Zs T27° r 2¶( 4 r

2 -f 7 "Io 711. 8

By defining the equivalent distance

De = 711.8//f

(41 ')

(42)

equation (41) is obtained from (41').

Mutual Impedance

We have to calculate for Jm the integral of (2").It differs from the integral (1") or (33) for Js byhaving in the integrand (hk+hk)/2 instead of h andalso the additional factor cosXdkf. This time, then,we have instead of (34"):

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3691

h +htq =

2p

and we define the tangent of e of Fig. 4 by:

dkQhk+h9

With (43), (44) and (34'), equation (2") becomes:

J -2qwJ (q) = i - - cos(2Sqw) dw

(L)

compared numerically, using generalized parameters.A(43) two wire horizontal line with fixed conductor height

and variable conductor separation, has been used forthis purpose.

Equations (1) and (2) have been evaluated usingthe infinite series expansions for the P and Q terms,given by Carson2, and were taken as the basis for the

(44) comparison.For each value (44) of S corresponding todifferent

conductor separations, the value of a (see equations(48) and (48')) was varied over the range of applica-bility of Carson's expressions (up to a few MHz). Theresults are presented in form of curves. Frequency is

(45') measured by a parameter a, defined as:

have

oc = (48)

or

e-2(1+jS)qw + e-2(1-ji)qwJm(q) 2 Jww + /w2 +1

(L)

Again, we wish to move the integration path to the realaxis. This is possible if the integrand is zero on thearc LCL' at infinity on Fig. 5. For this, it isnecessary that the complex exponents of the exponentialfunctions of (45") have negative real parts. We notethat, on arc LCL':

arg q = 450

arg w = -45°...0°

Therefore arg (1 + j5) must not exceed 450, i.e. werequire that 5 < 1 (e < 45 ). Then the integration canbe performed along the real axis L'.

It appears from the above discussion that ourproof is restricted to the case S < 1. This is indeedthe case but we proceed with the derivation of thesimplified expression for Zm: its validity over awider range for S will then have to be established bynumerical testing. This is done in the next sectionwith the result that equation (4) for Zm is valid, withonly small errors, for the full range of S of anypractical significance.

Now we differentiate (45") with respect to q andthen substitute approximation (37). We have thusonly exponential functions in the integrand so,that wecan easily perform the integration to obtain J'(q). Itsintegration yields:

J (q) - 1 J-9,nm 2 /

or, taking into account (43) and (44):

(hk+ht+2p)2 + d 2

m 2n h )2 + dk

where have is the average height of the conductors and6 is the real penetration depth of equation (18').Therefore

(45")

h +hia = - vwll2

2/iT(48')

We note that a is the real part of q defined in (43).The parameters a and S are related to Carson's

r and e , as follows:

a -r cose2V'2

and (49)

S = tane

Figure 7 shows that if the separation betweenconductors is not too large (5 < 2), the absolute valueof the error in the total impedance, incurred by usingequations(3) and (4) instead of (1) and (2), does notexceed 3%. For large S (0 close to 900), there is afrequency range (approximately from 100 Hz up tolO kHz)

1

10

6

(46)4

2

0

(47)

Expression (47) can be used for any p as the correctionterm in (2'). If p = po then (2') yields precisely theapproximate expression (4) which we wanted to derive.

ERROR EVALUATION

Expressions (1) and (3) for the self impedance,and expressions (2) and (4) for the mutual impedanceof an ideal conductor/ground return loop have been

- 2

- 4 ---3 -2 -1 ogO

log1

lHz lOOHz lOkHz f IMHz-I

Fig. 7 Relative error in the absolute value of thetotal impedance, including fluxes in air(p=1OOQm, hk=hk=20m)

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in which the errors become appreciable, although stillacceptable.For geometries encountered at almost alltransmission lines (S < 0.5) the error is smaller thanabout 0.5 percent and tendsto 0 for smaller values ofa.All curves in this figure display an asymptotic beha-viour, the errors approaching zero, for both low andhigh frequencies.

The comparison has been extended to the resistiveand reactive components of the ground return impedancetaken separately. Equations (3) and (4) have beenevaluated and the impedance component related to linegeometry subtracted from them. The results were thencompared with similar quantities derived from equations(l't) and (2"). Figures 8aand b show the actual errorsfound in such comparison, for the resistive andreactivecomponents, respectively. As before, the error in-creases as the distance between conductors, expressedby S, becomes larger, and is negligibly small atextreme values of a, corresponding to low and highfrequencies.

6

31=

-6 - _ + (a)

-6

- 3 -2 21 0 ,loga

%l15 g =10

10 _=(b

Fig. 8 Relative error in the (a) resistive and(b) reactive components of the groundreturn impedances

It should be mentioned that the infinite seriesexpansion proposed by Carson became instable for rgreater than 8. This fact is responsible for the dis-continuities found in all error curves. Indeed, toavoid the instability referred to above, the asymptoticexpression for r > 5 (proposed by Carson2) was usedinstead of the infinite series after the limit of rwas reached-.

A comparison test has also been made between thesimplified equations based on the complex return planeand Carson's asymptotic expressions for other rangesof r. It can be stated that these asymptotic expres-sions are very accurate, but only in the range of rfor which they are defined. In this range they comparefavorably with the method given in this paper.

The equivalent distance, De, often found in theliterature, is by 9% smaller than that given byequation (41), but the latter proved to be marginallymore accurate in the frequency range of its applica-bility (50 to 200 Hz).

CONCLUSIONS

This paper has presented a formal proof of thecomplex ground return plane approach for calculation oftransmission line impedances. The main advantageoffered by this method consistsin its simplicity andthe ease of obtaining accurate results by means ofelectronic calculators, while Carson's expressions re-quire a digital computer for their evaluation.

Although asymptotic formulations of Carson'sequations are also easily evaluated, they have thedrawback of being valid for specific frequency ranges.On the other hand, the complex plane approach resultsin simple formulae which are valid throughout, fromvery low frequencies up to several MHz.

Discontinuities obtained by Carson's formulae,when using asymptotic forms for some frequency ranges,may be annoying when approximating functions- ( e.g.rational polynomials) have to be evaluated.

Finally the physical interpretation of the simpli-fied formulae makes them easy to be retained in memoryand also attractive for teaching purposes.

For calculation of transmission line impedances,for which a is usually less than 0.5, very good resultscan be obtained by using equations (3) and (4). Ifcouplings between transmission lines and other lines(for instance, communication lines), are of interest,such that the values of $ are large, the complex planeequations can still be used. In this case, however,larger errors would be introduced, mainly in the audiofrequency range.

REFERENCES

[1] C. Gary, "Approche Complete de la PropagationMultifilaire en Haute Frequence par Utilisationdes Matrices Complexes"f, EDF Bulletin de laDirection des Etudes et Recherches-Serie B, No.3/4, 1976, pp. 5-20.

[2] J.R. Carson, "Wave Propagation in Overhead Wires,with Ground Return", Bell Syst. Techn. J., 1926Vol. 5, pp. 539-554.

[3] F. Pollaczek, "fUber das Feld einer unendlichlangen wechselstromdurchflossenen Einfachleitung",Elektr. Nachr. Tech., 1926, Vol. 3, pp. 339-359.

[4] W.H. Wise, "Effect of Ground Permeability onGround Return Circuits", Bell Syst. -Techn. J.,1931, Vol. 10, pp. 472-484.

[5] W.H. Wise, "Propagation of High Frequency Currentsin Ground Return Circuits", Proc. Inst. RadioEngrs., 1934, Vol. 22, pp. 522-527.

[6] L.M. Wedepohl, and A.E. Efthymiadis, "Wave Propa-gation in Transmission Lines over Lossy Ground:a New, Complete Field Solution", Proc. Inst.Electr. Engnrs., Vol. 125, No. 6, June, 1978, pp.505-5 10.

[71 L..M. Wedepohl, and A.E. Efthymiadis, "PropagationCharacteristics of Infinitely-Long Single-Conduc-tor Lines by the Complete Field Solution Method",Proc. Inst. Electr. Engnrs., Vol. 125, No. 6,June 1978, pp. 511-517.

[8] "Transmission LineReference Book, 345 kVand Above",Electric Power Research Inst., Fred Weidnerand SonPrinters, Inc., 1975.

[9]

[10]

E. Clarks,,"Circuit Analysis of A.C. Power Systems",Vol. 1, John Wiley, New York, 1943.

G.O. Calabrese, "Symmetrical Components", RonaldPress, New York, 1959.

[11] B.C. Kuo,"Automatic Control Systems", Prentice-Hall,1975, 3rd Edition (book).

[12] H.D'Angelo,"Linear Time-VaryingSystems:Analysis &Synthesis",Allyn and Bacon, Boston, 1970 (book).

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Discussion

M. E. El-Hawary (Faculty of Engineering and Appled Science, Memor-ial University of Newfoundland, St. John's, Newfoundland, Canada):The authors should be commended for an interesting paper providingfurther insight into the problem of modeling the effects of earth returnin transmission line parameter evaluation. The author's analytical ap-proach using Dubanton equations is sound and provides logical yetsimple procedures for evaluating ground return impedances.

The evaluation of the complex return distance p in the case of dis-crete layered earth requires complex matrix multiplications as indicatedby equations (20). It is possible to derive sequentially recursive relation-ships to arrive at the elements required by Eq. (21). Have the authorsstudied this? This should provide efficient means for calculating thecomplex equivalent depth.

As an aside, this discusser would like to point out that many con-tributions in the area of developing synthesis seismograms in explora-tion geophysics utilize formulations partially similar to those developedin this paper [A]. This is to be expected since acoustic wave propaga-tion is modeled in the plane-wave case by equations analogous to (9)and (10) of the paper.

The authors should be commended for a very clear presentation.

REFERENCE

[A] Wuenschel, P.C., "Seismogram Synthesis Including Multiples andTransmission Coefficients," Geophysics, Vol. 25, No. 1, pp. 106-129, 1960.

Manuscript received February 23, 1981.

A. Semlyen: I would like to thank Dr. El-Hawary for his comments onour paper. We agree that for the calculation of the complex equivalentdepth, in the case of a multi-layer earth representation, a simpler ap-proach could be developed. This could be achieved (according to Dr.

Tevan's suggestion) by replacing equations (9) by a single differentialequation in Z = E/H:

dZ j + ¢z2dx j~a (50)

This equation should be integrated backwards from x = where Ztn+l is known, as shown for equation (15). Within a layer (see Fig. 2),a and ,u are constant and the integrated form of (50) will relate Z fromthe upper boundary to Z of the lower boundary. On the transition sur-faces between layers Z has no discontinuity because E and H vary con-tinuously. The fmal result is ZO = EO/Ho at the surface of the earth.This is the same as the ratio

Eo = n+iAno - BnoZo H0 Ano - ?n+lCno

obtained from equation (21).It is interesting to note that the equivalent depth p can be calcu-

lated directly from ZO using equation (24) where

iW2¢K o0

Therefore

4K Eo zon = -- =

(51r P OHO ijw"oHo jwp"o )

The most important advantage of equation (50) is perhaps that itcan be easily integrated numerically for obtaining ZO, even in the caseof continuous stratification, starting at a large enough value of x. Thentoo, (51) gives directly the value of the complex equivalent depth.

Finally, I would also like to thank Dr. El-Hawary for the informa-tion related to a procedure similar to ours in exploration geophysics.

Manuscript received April 13. 1981.