RADIO INTERFEROMETRY DEPTH SOUNDING: PART I-THEORETICAL Dlscusslont

GEOPHYSICS, VOL. 38, NO. 3 UUNE 1973), P. 557-580, 10 FIGS., 7 TABLES

RADIO INTERFEROMETRY DEPTH SOUNDING: PART I-THEORETICAL DlSCUSSlONt

A. P. ANNAN*

Radio interferometry is a technique for mea- layer earth. Approximate expressions for the fields suring in-situ electrical properties and for detect- have been found using both normal mode analysis ing subsurface changes in electrical properties of and the saddle-point method of integration. The geologic regions with very low electrical conduc- solutions yield a number of important results for tivity. Ice-covered terrestrial regions and the the radio interferometry depth-sounding method. lunar surface are typical environments where this The half-space solutions show that the interface method can be applied. The field strengths about modifies the directionality of the antenna. In a transmitting antenna placed on the surface of addition, a regular interference pattern is present such an environment exhibit interference maxima in the surface fields about the source. The intro- and minima which are characteristic of the sub- duction of a subsurface boundary modifies the surface electrical properties. surface fields with the interference pattern show- This paper (Part I) examines the theoretical ing a wide range of possible behaviors. These the- wave nature of the electromagnetic fields about oretical results provide a basis for interpreting various types of dipole sources placed on the sur- the experimental results described in Part II. face of a low-loss dielectric half-space and two-

INTRODUCTION lunar surface material has similar electrical prop- The stimulus for this work was the interest in erties. the measurement of lunar electrical properties in Since electromagnetic methods commonly used situ and the detection of subsurface layering, if in geophysics are designed for conductive earth any, by electromagnetic methods. Unlike most problems, a method of depth sounding in a dom- regions of the earth’s surface, which are conduc- inantly dielectric earth presented a very different tive largely due to the presence of water, the lunar problem. One possible method of detecting the surface is believed to be very dry and, therefore, presence of a boundary at depth in a dielectric is to have a very low electrical conductivity (Strang- the radio interferometry technique, first suggested way, 1969; Ward and Dey, 1971). Extensive ex- by Stern in 1927 (reported by Evans, 1963) as a perimental work on the electrical properties of method to measure the thickness of glaciers. The dry geologic materials by Saint-Amant and only reported application of the technique is the Strangway (1970) indicates that these materials work of El-Said (1956), who attempted to sound are low-loss dielectrics having dielectric constants the depth of the water table in the Egyptian des- in the range 3 to 15 and loss tangents considerably ert. Although he successfully measured some in- less than 1, in the Mhz frequency range. Analysis terference maxima and minima, his method of of the electrical properties of lunar samples by interpretation of the data is open to question in Katsube and Collett (1971) indicates that the light of the present work.

t Presented at the 39th Annual SEG International Meeting, September 18, 1969. Manuscript received by the Editor April 6, 1972; revised manuscript received September 22,1972. * University of Toronto, Toronto 181, Ontario, Canada. Downloaded 12/13/19 to 198.120.252.61. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ @ 1973 Society of Exploration Geophysicists. All right reserved.

557 558 Annan

The radio interferometry technique is concep- electrical properties of the earth and can be used tually quite simple. The essential features of the as a method of inferring the earth’s electrical method are illustrated in Figure 1. A radio-fre- properties at depth. quency source placed on the surface of a dielectric The problem chosen for study in the theoretical earth radiates energy both into the air (or free work was that of the wave nature of the fields space) above the earth and downward into the about various point-dipole sources placed on the earth. Any subsurface contrast in electrical prop- surface of a two-layer earth. The mathematical erties at depth will result in some energy being solution to this type of boundary-value problem reflected back to the surface. As a result, there is found in numerous references. The general will be interference maxima and minima in the problem of electromagnetic waves in stratified field strengths about the source due to waves media is extensively covered by Wait (1970), traveling different paths. The spatial positions of Brekhovskikh (1960), Budden (1961), Norton the maxima and minima are characteristic of the (1937), and Ott (1941, 1943). Although the solu-

Air Transmitter Receiver

Dielectric

Transmittei-receiver separation (b) FIG. 1. (a) Transmitter-receiver configurationfor radio interferometry, showinga direct wave and a reflected Downloaded 12/13/19 to 198.120.252.61. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ wave. (b) Schematic sketch of typical field-strength maxima and minima as the transmitter-receiverseparation increases. Radio Interferometry: Part I 559

tion to the boundary-value problem can be found The geometry and coordinate systems used in analytically, the integral expressions for fields the boundary-value problem are shown in Figure cannot be evaluated exactly. In the radiation 2. A point-dipole source is located at a height h zone, approximate solutions to the integrals can on the z-axis above a two-layer earth, where the be obtained by use of the theory of complex vari- earth’s surface is in the x-y-plane at z = 0, and the ables and special methods of contour integration. subsurface boundary is at z= -d. The region The preceding references, plus numerous others, z>O is taken as air or free space. The region discuss these techniques in detail. Since much of -d

2 = h--~----~---~--_____ Sourct I 1 Region0 KO=MO=l

/- Region1 Kl, Ml X Z = -d\ 7

Region2 K2, M2

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This choice of scaling parameter makes all the vm + k?rI = - ; 1 (1) following integral solutions dimensionless. As shown by Sommerfeld (1909) for a half- with the electric and magnetic fields defined by space earth, and extended to a multilayered earth by Wait (1970), the Hertz vectors for the vertical E = Kzn + vv .I& (2) dipole sources have only a z-component, while for the horizontal dipole sources, the Hertz vectors and have both x- and z-components. The vertical di- H = - iwKe,,V X n (3) pole sources have solutions of the form where k is the propagation constant. wd/KMeopo n; = -@OR and eo,~0 denote the permittivity and permeabil- R ity of free space throughout. P is the electric dipole moment density distribution. Similarly, for uo(X)e-Po(Z+h)H~(Xp)dX, (10) magnetic dipole sources, the results for the mag- m 0 netic Hertz vector are +kr ;

V2n + K2n = - M, (4) l-&L- * x [al@) ePIZ + a2(X)e-Prz] 2w s -_m PO and .e-PohZ3~(Xp)dX, (11) H = K2n + vv *II, (3) E = iwM/.o,v X II, (6) and

where M is the magnetic dipole moment density distribution. (12) The dipole sources are taken as “unit” point . eP*Z+(PZ-Pl)d-POh~~(~p)~~, dipole sources located in the region 220. The electric dipole moment density distribution is X is the separation constant of the differential equation, and pj= (X2- ky)12,’ with the sign of the P = 4xei$(R)ej, (7) root being chosen such that the solution satisfies where ej is a unit vector in the z-direction for the radiation condition. In the above form, after a vertical electric dipole source and in the scaling by W, X is a dimensionless parameter, and x-direction for the horizontal dipole source. 6(R) kj= 2s(KjMj)12‘ is the relative propagation con- is the three-dimensional delta function, and stant of each region. The aj(X) are unknown func- R= [~2+y~+(z-k)2]12.’ Similarly for the mag- tions of X which are found by satisfying the netic dipole sources, boundary conditions that tangential E and H be continuous at z = 0 and z = -d. M = 4&(R)ej. (8) For horizontal dipole sources, the solutions for the Hertz vector take the form In the following discussions, no distinction between the electric and magnetic Hertz vectors is eikoR made. When we refer to electric dipole sources, r( = - the electric Hertz vector is implied; for magnetic WR (13) dipole sources, the magnetic Hertz vector is implied. In addition, the free-space wavelength is bo(X)e-PO(Z+h)B~(Xp)dX, taken as the scaling parameter for all length mea- +iJ_-m G0 surements. In other words, a distance, denoted p, and is in free-space wavelengths, and the true length is Wp, where W is the free-space wavelength: cos+ - x2 n; = - COG9 2* 2w s-_m -PO (14) w= (9)

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for region 0; corresponds to setting both z and It equal to 0 in the preceding expressions for the Hertz vectors. nr, 2.2w _I; [bl(A)eP1z + b2(A)e-P1zl In the following discussions, h is always set equal s to 0, and, in most instances, z is assumed to be (15) close to 0. The solutions are discussed in two -e-PohH~(Xp)dX, parts; the half-space solutions and the two-layer and earth solutions. The half-space solutions for the Hertz vectors are obtained by setting KI= K2 and Ml= Mz in expressions (10) through (18). The half-space solution is of considerable interest since (16) the fields about the source show interference maxima and minima without a subsurface re- flector present. It also provides a base level for for region 1; and detection of reflections from depth.

APPROXIMATE SOLUTIONS

(17) Half-space earth . ePtZ+.(PtP1)&Poh~~(~~)~~, The solution of the half-space problem is treated by numerous authors, and the wave nature of the and fields is well defined. In the following discussions,

2 COSQ O” A2 the results of Ott (1941) and Brekhovskikh (1960) nz = - - c&d are followed quite closely, and detailed discussions 2w s -00 PO (18) of various aspects of the solutions can be found in these references. The wave nature of the fields about the source is illustrated in Figure 3. The for region 2. wavefronts denoted A and B are spherical waves The parameters X and Pi are the same as for the in the air and earth regions; wave C in the air is vertical dipole solutions, and the coefficients hi(X) an inhomogeneous wave, and wave D in the earth and ci(X) are found by satisfying the condition has numerous names, the most common being that tangential E and H be continuous at the head, flank, or lateral wave. Waves C and D exist boundaries. only in a limited spatial region, which is defined The boundary conditions for the Hertz vectors as those points whose position vectors make an and the resulting expressions for @j(X), bj(X), and angle greater than & with the z-axis. The angle QL~ Cj(X) are tabulated in Appendix A. The expres- is related to the critical angle of the boundary and sions for ai( bj(X), and Cj(X) are written in terms is defined in Appendix B. of the TE and TM Fresnel plane-wave reflection All the dipole sources exhibit the same wave and transmission coefficients. Using this notation, nature. To demonstrate how the waves are de- the similarity of all the solutions is clearly empha- rived from the integral expressions, the vertical sized and makes general discussion of the solu- magnetic dipole source is used for illustration. In tions possible rather than dealing with each source the air, the Hertz vector is given by separately. In discussing the approximate evaluation of the above integral expressions, extensive use is made II; = g + $J sin BoRoI(Bo) (19) of the plane-wave spectrum concept, since the e wave nature of the problem is most clearly under- .eikoz conB~H;(kop sin 8o)&o, stood using this approach. A brief outline of the .plane-wave spectrum notation used and approxi- and in the earth by mate evaluation of integrals by the saddle-point method is given in Appendix B. For radio interferometry applications, the fields at the earth’s surface for the source placed

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FIG. 3. Wavefronts about a dipole source on the surface of a half-space earth. A and B are spherical waves in the air and earth, respectively, wave C is an inhomogeneous wave in air, and D is the head wave in the earth.

With the aid of the Hankel transform identity and (Sommerfeld, 1949),

eikoR ik,, sin e. WR=iif s c (21) (24) .eikolzI coaeoI!Z~(kop sin &J&3,,, - GR[z%(a) + cot ar:a(41}, and the relation R,j= T,j- 1 for the Fresnel coeffi- 1 cients, equation (19) becomes where R= (p2+22)1/2, and a!= tan-l p/lZl . Ex- pressions (23) and (24) correspond to the spherical waves A and B in Figure 3. The bracketed .eikolzI coSeoZY~(kOp sin Oo)dOo. terms on the right may be interpreted as the modification to the directionality of the source Using the saddle-point method as discussed in due to the presence of the boundary. Appendix B, the approximate solutions of the The waves C and D are generated by crossing integral expressions (20) and (22) are the branch points of ToI and Tlo to obtain the saddle-point solutions (23) and (24) for angles cr>a”. As outlined in Appendix B, the contribu- II; = tion of the branch point can be approximately (23) evaluated by the method of steepest descent as long as (Y is not close to the branch point. For (Y>cyc the expressions

2 21 2iklqoI(l - cot LYtan $,I)-3’2eik1P-(k1-ko) ’ I; = , (25) (k: - k;) Wp2

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2 21 2iko4 1 - cot a! tan ~EO)-32eikop--i’ k1--kg’ ’ ’ I; = (26) (k; - Ki) wp2

must be added to (23) and (24) in order that the der Waerden, 1951; Brekhovskikh, 1960; and solutions be correct. Expression (25) is readily Wait, 1970) have considered the problem since identified as the inhomogeneous wave C, and (26) then using the modified saddle-point technique to corresponds to the lateral wave D. evaluate the integrals for a*7r/2. While a true The asymptotic solutions have the form of the surface wave is not excited, the pole enhances the geometrical optics solution plus second-order cor- fields near the source in such a manner that they rection terms. For a perfectly dielectric earth, fall off approximately as (kR)-I. At large dis- expressions (24) and (26) become infinite as (Y tances from the source, the fields are those deter- approaches &. The singular behavior arises from mined by the normal saddle-point method which the second-order terms which depend on the have a (kR)+ fall off. The transition between the derivatives of T&Q. The first-order term which ranges is determined by the proximity of 8~ to is the geometrical optics solution remains x/2. In the radio interferometry application, the bounded. In this particular case, the angle (Ye earth properties of interest are those of a low-loss is the critical angle of Tl&). As a result, the sad- dielectric which is assumed to have only moderate dle point and branch point coincide at a=&, and contrasts with the free-space properties. The pole the approximate methods used to evaluate the in this case is well away from x/2 and does not integrals are no longer valid. The conical surface affect the preceding solutions. about the z-axis, defined by (Y=o!, is the region In the particular situation of an earth where where the lateral and spherical waves merge to- MO= Ml, the Hertz vector for a vertical magnetic gether. In this region the two waves cannot be dipole can be evaluated exactly for z=h=O. The considered separately; the combined effect of the result is saddle point and branch point must be evaluated. Detailed analysis of this region for integrals sim- &&-- 2 [eik+ko-;) (27) ilar to expression (20) is given by Brekhovskikh (kf - ki) wp2 (1960), who obtains an asymptotic solution with 1 the geometrical optics solution, as the leading _eiklp &__ , term plus a connection term which falls off as ( P (k,R)+4 instead of (k1R)-2. This result indicates )I that the geometrical optics solution still describes as shown by Wait (1951). This provides a check the fields adequately for a=& when (klR)+14<

tions is to equate the radiation pattern of the source on the boundary to the first-order term in the preceding solutions. This technique demon- strates how the boundary modifies the directional- The Gi(a) for the various sources are tabulated in ity of the source. The radiation pattern is sharply Table 1. The electric and magnetic fields can be peaked in the direction of the critical angle into obtained by differentiation of the preceding solu- the earth. A sketch of the radiation pattern for a tions; the particular form of the integrals encoun- vertical dipole source is shown in Figure 4. This tered permits interchange of the integration and directionality of the source is important when re- differentiation steps. flections from a subsurface boundary are con- The half-space solutions demonstrated that sidered. interference patterns will be observed in the field strengths even when there is no subsurface re- Two-layer earfh flector. This is readily seen from equation (28). The fields at the earth’s surface are composed of The analysis of the integral expressions for the two propagating components with one having the two-layer eart?hproblem is carried out in two dif- phase velocity of the air, and the other the veloc- ferent ways. The depth of the subsurface boun- ity of the earth. Another important feature of the dary and the electromagnetic losses of the first half-space solutions is that the fields near the layer determine which approach is more useful. boundary fall off as the inverse square of the radial The primary method of analysis is to treat the distance from the source at distances greater than first layer of the earth as a leaky waveguide and two or three wavelengths from the source. use normal mode analysis. In certain cases the A convenient method of interpreting the solu- mode analysis is cumbersome, and these cases

Table 1. Coefficients G&Y) for half-space earth solutions for various dipole sources.

Hertz G2 Source Vector 1z G I I - ToI - i(G:(a)‘ + c:(a) cota) kf - k: - - Vertical Electric Sot(a) -i(G:'(a) + G:(a) cota) Dipole

Magnetic0 0 2ik&o, SOlb) -i(G:'(a) + G:(a) cota) Horizontal nx k; - k: Magnetic -- Dipole Magnetic0 1 (yo, - l)Tol(a)Solb) =&o‘ + mo) -i(G:‘(a) + 3G:(4 cot OJ- 2(&(m)) IIZ PPo(4 k;(kz0 - kz)“*I - Electric 0 0 -i(G:‘(a) + G&Y) cot a) Horizontal ax Electric - 2 2iko(mo + Em) -i(G:‘(a) + 3G:(a) cot a - 2G,(4) k;(kz0 - k:)lz

Definition : Downloaded 12/13/19 to 198.120.252.61. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ Radio Intetferometry: Part I 565

FIG. 4. Sketch showinghow a dielectric boundary modifiesthe directionality of a vertical dipole sourceplaced on the boundary. (a) No boundary present. (b) Boundary at Z=O.

yield more useful results when the integrals are ik? solved approximately using the saddle-point II; = cos f#JG sin2 ~:m~~(Bdc~(fh) s c (31) method of integration. 1 The various dipole sources may all be treated in the same manner. For the purpose of illustrat- where the integration variable is &, as defined in ing the method of analysis, the horizontal electric Appendix B. The singularities of the coefficients dipole source solutions are used as an example. bo and COdetermine the nature of the solutions to The X- and z-components of the electric Hertz equations (30) and (31). vector in the air for this source are given by equa- The expressions tions (13) and (14) where the coefficients bo and GO are listed in Table A-2. rllO~lO(l+ R12P) For normal mode analysis, equation (13) is re- rnlO(l+ bo)= , (32) written using the integral identity [equation (21)]. 1 - R,oJW The components of the Hertz vector are given by and

1 d = 2 J sin4mlo(e1) [1 + bo(eI)](30) mlOcO= - C 2PO 1 . e&o2 coSeOHo(kIp sin OI)dO1, . (YOI- l)rlloTlo(l+R12~)So1(1+X1~B) Downloaded 12/13/19 to 198.120.252.61. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ and (1--~10X128)(1-R10R12B) 566 Annan

(1 - R1oRd3) = 2 (R1oRd3>“. (36) (l-xXlOX12P)(l-R1OR12P) _I --I‘ IL-0

have branch points at the two critical angles of The positions of the poles and branch points in the two boundaries in question. In addition, the complex plane determine the wave nature of equation (32) has an infinite set of simple poles, the solutions to equations (30) and (31). The nor- and (33) has a doubly infinite set of poles on each mal mode solutions are obtained by deforming the Riemann surface. These poles are determined by contour of integration C to C’, as schematically the normal mode equations, illustrated in Figure 5. The integral from -(r/2) +ia, tor/2+im, andfromr/2+im to 7r/2--im 1 - RroRizP = 0, (34) is identically zero. Along the first part of C’ the integral is zero since the integrand is zero. Along and the second part of the contour C’ the integral is 1 - x,ox,zp = 0. (35) zero due to the asymmetry of the integrand about &=x/2. This result is common in mode analysis Equations (34) and (35) are the TE and TM nor- and has a wide range of applications as discussed mal mode equations, respectively. Both are trans- in detail by Brekhovskikh (1960). The integral cendental equations with infinite sets of roots. The along C is equal to the residues of the poles crossed relation between the normal modes and multiple in deforming the contour to C’, plus the integrals reflections is readily obtained by expanding the along contours Cr and Cz which run from +im denominators of equations (32) and (33) into around the branch points and back to +im. The infinite geoi ric series. For example, components of the Hertz vector are given by

‘2 In i n/2 _- 1

I2 Ia3

FIG. 5. Complex&plane showinghow integration contour C is modified to C’ in order to obtain normal mode solutions.Branch points denoted by solid circlesand the poles by squares. Downloaded 12/13/19 to 198.120.252.61. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ Radio Interferometry: Part I 567

Table 2. Residues of Roles for normal mode analysis for horizontal electric dipole.

TE Mode Residue

0 IIZ TM Mode Residue

where t(&) = s o : [d-lo(eP)[i+ R,~cePh9CeP)]sdl+ xIr(ef)8(e3)l

lTi = 2zri c (TE pole residues) The solutions (37) and (38) are completely gen- (37) eral and valid provided a branch point and a pole, + 11 + 12, or a TE and TM pole, do not coincide. In the first and situation, the pole and branch-point contributions II: = 2ai c (TE pole residues) must be considered together rather than separately, as indicated. In the other situation, (38) + 27ri c (TM pole residues) (38) must include the residue of a second-order pole rather than the residues of two simple poles, as + Ia + 14, indicated. Since these situations rarely occur, they where are not discussed further here. The expressions for the residues in (37) and (38) are tabulated in Table 2. Approximate solutions to the branch cut integrals may be obtained by steepest-descent integration as discussed in Appendix B. The solutions are the second-order lateral and inhomogeneous waves generated at the I2 2$ sinem10(f4>[l + hml boundaries. The approximate solutions are listed . c2 (40) s in Table 3. . eaoz O”ne ”H&p sin el)del, The behavior of the fields at the earth’s surface . 2 is very dependent on the position of the singular points in the complex & plane, which points are, la = cos f#J 2 sin2el~lo(fh)431) (41) in turn, determined by the material properties of S Cl the earth and the layer thickness d. The important features of the solutions are the radial depen- dences and the initial amplitudes of the various . 2 terms in the solutions. All the residues contain 14 = cos#j”k’ Hankel functions, which, for radial distances 2w Sc* sin2 ~l~lo(~l)dv (42) 1 greater than one or two wavelengths, have the . eikoZcon eoHI(klp sin el)d81. form Downloaded 12/13/19 to 198.120.252.61. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ 568 Annan

Table 3. Approximate steepest-descent solutions to branch-cut integrals for normal mode analysis of the horizontal electric dipole.

2 2 ikg 2ikos,a(l + R12(&)8(dO)) e (k: - ki)(l - R1*(@$8(~iJ)*WP*

I (1 _ COt (ydtan e;,)- Ji2Ca(eEe)ek0p+i2d(k:--k~)1a’ cos +ko Cl(e;O)elkop -_ 2(k:-_ 0 WP2 WP2 1

* * “22d-(k;-k;)l~z’ cos qkr(l - cot LYEtan 0&)-a/2C3(8~;)k ’zP--(kZdL) 2(k: - k;)“Wp”‘

7jOl(l - I&)(1‘ + XI& ’ + Eo,(l - x:*8*)(1 + iWqz cl(eL) - (1 - xl*s)*(l - RlZS) ’ 1

c,(ekd

2iqloTloS01&T{ (701 - l)[.&S~o(l + To@ - Ru@) + t112T10(1+ SoIS - X~@)l 2(1 - y&o1y&z1(l + X&(l - RIO@)+ vz1(1 - x~~6)(1 + R&l) cde2 - - kr,(l - yz~)l”(l - yz~)“~(l - X&)(l‘ - &S)*

ko kz sin eEo= - sin eEz= - h k1

--Illeiklp sin BP1 with Kt< K1 in case (c). In all cases MO= Ml= Mt P , (43) is assumed since permeabilities of bulk earth ma- where @ is a pole defined by equations (34) and terials vary little from the free-space value. The (35). The branch cut contributions, II and 13, poles lie on the lines 1I&&@[ = 1 and 1XIOX&I fall off as pe2 with no exponential attenuation, = 1, with the density of distribution of the poles while 12 and Z4 fall off as pe2and are exponentially on these lines controlled by the slab thickness d. attenuated when the earth is lossy. The ampli- For d small, the poles are widely spaced with most tudes of the modes are largest for those poles in of them lying high up the lines near the imaginary the vicinity of f?& This effect is related to the O1axis. For increasing d, the poles move down the modified directionality of the source as discussed lines toward &=r/2 and are more closely packed. for the half-space fields. When the earth has a Unattenuated modes occur only when the Oy lie finite loss all terms in the solution except II and on or close to the real O1axis. In case (a), the pole the first term of 13 have amplitudes which decay contours lie on the real axis for &, IO1

Downloaded 12/13/19 to 198.120.252.61. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ with Kz>K, in case (b), and a perfect dielectric damped due to energy leaking out of the slab. In Ko= 1 FREE SPACE

K* PERFECT DIELECTRIC

PERFECT CONDUCTOR

(4 Is 0.

Kg = 1 FREE SPACE

PERFECT DIELECTRIC -Re e1 K, n/2

K2 ’ K1 PERFECT DIELECTRIC

04 Im 8,

K, = 1 FREE SPACE

PERFECT DIELECTRLC /2Re 81 KI K2< K1 PERFECT DIELECTRIC

(4 FIG. 6. Sketches of complex Or-plane showing positions of branch points and pole lines for 3 simple two-layer earth cases; case (a), dielectric slab over a perfect conductor; case (b), dielectric slab over dielectric half-space with Kz>Kl; case (c) dielectric slab over a dielectric half-space with K*

case (b), no unattenuated mode can exist, since The Hertz vector components can then be writ- energy is continually flowing out of the slab into ten as the lower half-space. The effect of a finite loss ‘in the slab moves the pole contours up away from II: = JG+ Lx, (49) the real axis, and all modes have a finite attenua- and tion. These models demonstrate the general position and behavior of the singular points in the IC = La + L, (50) complex plane. More detailed studies are given by Wait (1970), Brekhovskikh (1960), and Bud- where L1 and La are the half-space earth solutions den (1961). discussed earlier, and L2 and L4, given by The normal mode approach is most useful when the distance from the source is greater than the depth of the boundary d and when the earth has a very low loss. In a lossy dielectric, the complex (51) dielectric constant is R’(l+i tan S), where K’ is 1 the real dielectric constant and tan 6 is the loss tangent; and the attenuation distance in free- and space wavelengths for a plane wave is