N USC Technical Report 688111 Jarnuary 1984

(' New Formulas for HED, HMD, VED, and VMDC0 Subsurface-to-Subsurface Propagation

Peter R. BannisterSubmarine Electromagnetic Systems Department

SMAR- 9 1984

Naval Underwater Systems CenterNewport, Rhode Island I New London, Connecticut

Approved for public release; distribution unlimited.

Frw, FI:IL COPY 8 os 16 095

4

Pr-face

This report was prepared under NUSC Project No. A59007, "ELF PropagationRDT&E" (U), Principal Investigator, P. R. Bannister (Code 3411). Navy ProgramElement No. 11401N and Project No. X0792-SB, Naval Electronic "ystemsCommand Communications Systems Project Office, D. Dyson (Code PME 110),Program Manager ELF Communications, Dr. B. Kruger (Code PME 1 10-XI).

The analysis and write up of this report was performed while the author wasoccupying the Research Chair in Applied Physics at the Naval Postgraduate School,Monterey, CA. The author would especially like to thank Professors Otto Heinzand John Dyer and Dean Bill 'rolles for recommending him to occupy this post andNAVSEA (Code 63R) for sponsoring the Chair.

The Technical Reviewer for this report was Anthony Bruno.

Reviewed and Approved: 11 January 1984

4t

Head, Submarine Electromagneti Systems Department

II

The author of this report is located at theNewI ondon Laboratory, Naval Underwater Systems Center,

New Lbndon, Connecticut 06320ý

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*1 ~ ~ ~ ~ ~ ~ ~ ý NOW__________________________________________

REPORT DOCUMENTATION PAGE READ INSTRUCTIONS_____________________________________ BEFORECOMPLETINGFORM

1. REPORT NUMBER 7.GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBERTR 68811

4. TITLE (and Subtitle) S YEO EOT&PRO OEE

NEW FORMULAS FOR HE!), 11MP, VEAND VMD P3oo~b:IoOE~

9. PEFRIGOGNZTONNM N DRS 0. PORMELEMENT. PROJECT, TASK

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Approved for public release; distribution unlimited.

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14. SUPPLEMENTARY NOTES

19I. KEY WORDS (Contlowe On ftworae aide It nocooerw and Id0"ItlY bY block aumbov)Electromagnetic Fields Vertical Electric DipoleHiorizontal Electric Dipole Vertical Magnetic DipoleHorizontal Magnetic Dipole Subsurface-to-Subsurface Propagation

20. ABSTRACT (Conthwa. 4, reVff4* side itfts~egesmy 8 a~ndj hdufr block amsbor)

New formulas for the electric and magnetic fields produced by the fourelementary dipole antennas have been developed for the subsurface-to-subsurface,subsurface-to-surface, surface-to-subsurface, and surface-to-surface propagationcases. These formulas are of rather simple, but useful, form and are completelygeneral (i.e., the air can easily be replaced by the sea bottom). They arevalid at any frequency and at any range beyond a certain minimum distance forthe flat-earth case. The main restrictions on these formulas are.Ri the square

DD I 14AR 73

.~: q-.

20. (Cont'd) =

>of the index of refraction is \10 and-e2-" the horizontal separation is ý3 timesIi the sum of the depths of burial of the transmitti.ng and receiving point sources.

With these new formulas, comaputer evaluation can be reduced to fractionsof a minute, compared with hours for the complete numerical evaluation of theexact Sommerfeld integrals.

There also will be an interference pattern set up under certain close-rangeconditions because the three waves (direct, modified mirror image, and lateral)may interfere, either constructively or destructively, with each other.

.lv

?:

TR 6881

TABLE OF CONTENTS

Page

LIST OF TABLES .......... ........... .............................. ii

GLOSSARY OF SYMBOLS ........... ....... ........................... ii

INTRODUCTION ........ ............... ............................... 1

WU-AND-KING'S METHOD .......... ........... ........................... 2

BANNISTER'S METHOD .......... ............. ............................ 4

COMPARISON OF BANNISTER AND WU-AND-KING RESULTS ......... ............. 8

RANGE OF VALIDITY OF LATERAL-WAVE FORMULAS ........ ................ ... 10

ANALYTICAL CONFIRMATION OF FRASER-SMITH

IAPND BUENI VM Hz NULL .. .. .. ............................. . ....... 13

APPENDIX DIRECT AND MObIFIED MIRROR-IMAGE CONTRIBUTIONTO EACH DIPOLE FIELD-STRENGTH COMPONENT (In2I >> 1) ..... ... A-1

Cd

I'" ~~ -• e, :i I~Dt -

; 1

TR 6881

LIST OF TABLES

Table Page

fP -Yj. z+h)1 f(pO)e- Lateral-Wave Formulas WhenlyopI << 1 [In21 > 10, p2 >> (z + h) 2 ] ..... ............ .. 17

2 Electric-Field Subsurface-to-Subsurface PropagationFormulas [jn2 >1 10, p > 3(z + h)] ........ .............. 19

3 Magnetic-Field Subsurface-to-Subsurface PropagationI Formulas [1n21 > 10, p > 3(z + h)] ..... .............. .. 21

4 Electric-Field Subsurface-to-Surface PropagationFormulas [In21 > 10, p > 3h, D = (p + h2)1/2] ... ........ .. 25

5 Magnetic-Field Subsurface-to-Surface PropagationFormulas [jn21 > 10, p > 3h, D = (p + h2)1/2] ... ........ .. 27

6 Electric-Field Surface-to-Subsurface PropagationFormulas [jn2j > 10, p > 3z, R = (p + Z2)1/2] ... ......... 29

S7 Magnetic-Field Surface-to-Subsurface PropagationFormulas fjn21 >_ 10, p >_ 3z, R =(P2 + z2) 1/2] . .. .. . .. 31

8 Electric-Field Surface-to-Surface PropagationFormulas (In 2 I > 10) ..................... 33

9 Magnetic-Field Surface-to-Surface PropagationFormulas (In 2 I > 10) ..... ......... ..................... 35

10 Range Where f(p,z + h) = £(p,O)e-Y](z+h) ... .............. .. 37

ii4.

TR 6881

GLOSSARY OF SYMBOLS

D (p2 + h2 ) 1/ 2 (meters)

SEp Horizontal electric-field component in the p direction (volts/meter)

4 Horizontal electric-field component in the ý direction (volts/meter)

Ez Vertical electric-field component (volts/meter)

F F(wO), Sommerfeld ground-wave attenuation factor

h depth (h > 0) of transmitting antenna with respect to the earth'ssurface (meters)

Hp Horizontal magnetic-field component in the p direction(amperes/meter)

H4 Horizontal magnetic-field component in the 4 direction(amperes/meter)

Hz Vertical magnetic-field component (amperes/meter)

HED Horizontal electric dipole

HMD Horizontal magnetic dipole

I Current (amperes)

10 Modified Bessel function of the first kind and order zero

I Modified Bessel function of the first kind and order one

JO(AP) Bessel function of the first kind, order zero, with argument XP

K0 Modified Bessel function of the second kind and order zero

K1 Modified Bessel function of the second kind and order one

m Magnetic-dipole moment (ampere-meters 2)n yl/y 0 , index of refraction

ryl/o

I 0 12[R1 - (z + h)] K0 1 2[R1 + (z + h) = Foster integral

p Electric-current moment (amperes-meters)

P0 exp(-yIR0 )/R0 = Sommerfeld integral (rneters- 1 )

ii.

WW &I" NO ---

*' TR 6881

GLOSSARY OF SYMBOLS (Cont'd)

P1 exp(-y 1 R1 )/R 1 = Sommerfeld integral (meters-1)

R (p 2 + Z2)i/2 (meters)

R 0[p 2 + (z - h) 2 ]1/ 2 (meters)

R 1 [p 2 + (z + h) 2 ]'/ 2 (meters)

t Time (seconds)

T 161 1 KI + y 2 p 2 (1 1 K1 - 10Ko) + 4yp(l!Ko _ IoKI)

o(2 + y2)1/2 (meters- 1 ) (air)

u (X2 + y2) 1 /2 (meters- 1 ) (earth)

VED Vertical electric dipole

VMD Vertical magnetic dipole

W 311 K1 - (yp/2)(l oK1 - IIKo)

W0 Sommerfeld numerical distance

zDepth (z > 0 of receiving antenna with respect to earth'ssurface (meters)

YO (-W2 120 0)1/2, upper half-space (free-space) propagationconstant (meters- 1)

Yi (i a I - 2IIlC)1)I/2, lower half-space (earth) propagationconstant (meters- 1 )

r/10 1, +\1/2 wl-/6 (2./ " 1/- skin depth in the lower1 L\ 12 /

half-space (earth) (meters)

0- l0" 9 /36ir farads/meter, permittivity of free space0iC Permittivity of lower half-space (earth) (farads/meter) jA Duzmy integration variable in the basic Sommerfeld

integrals (meters- I)

4 p(x 2 + y2)1/2 radial distance in a cylindrical coordinatesystem (meters)

al Condictivity of the lower half-space (earth) (Siemens/meter)

iv

TR 6881

GLOSSARY OF SYMBOLS (Cont'd)

tan- 1 (y/x), azimuth angle iii a cylindrical coordinate system

S0P = 47 x 10-7 henries/meter, permeability of free space

tan-l[(z - h)/pI, elevation angle

j •itan-l[(z + h)/pj, elevation angle

(327f radians/second, angular frequency

';

7 1II

ii

ii BSReere j~

IITR 6881

NEW FORMULAS FOR tIED, HMD, VED, AND VMDSUBSURFACE-TO-SUBSURFACE PROPAGATION

INTRODUCTION

In two recent papers, Wu and Kingl, 2 have derived new simple formulas forthe electric-field components generated by a horizontal electric dipole (HED)in a half-3pace of water or earth near its boundary with-air. They cliim thattheir formulas are valid for p 2 >> (z + 11)2 and 1-21 >> 1, where n (= yl/y.) isthe index of refraction. On examining their results, the author of this reporthas discovered that they could have been obtained almost by inspection from thepreviously derived results of Wait, 3 Weaver, 4 Bannister, 5 and Bannister andHart 6 (most of which are summarized in Kraichman 7 ).

In the past, many investigators erroneously have believed that the field-strength equations tabulated in chapter 3 of Kraichman are only valid when theconduction currents in the water or earth are much greater than the displace-ment currents (i.e., al >> We). Indeed, as long as In2! >> 1, the displace-ment currents can be included simply by replacing al with al + iWe1 in thefield-strength equations. Thus, Kraichman's tabulated results are considerablymore general t1-an they are stated to be.

hav It is the purpose of this report to present new formulas for HED, hori-zontal magnetic dipole (HMD), vertical electric dipole (VED), and verticalmagnetic dipole (VMD) subsurface-to-subsurface propagation. These formulashave been obtained completely from previously derived results. The mainrestrictions on their use are (1) the square of the index of refraction is>10 and (2) the horizontal separation is greater than or equal to three timesthe sum of the depths of burial of the transmitting and receiving point sources.An addit.ional restriction must also be applied to the lateral-wave components.The quantity jyjp 2/(z + h)I must be greater than or equal to 4cj, where cl =

-' I3, 6, 9, 15, or 25, depending on the particular field-strength component. ThisL Irestriction also applies to Wu-and-King's resultsl, 2 and to those tabulated in* Itables 3.7 and 3.16 of Kraichman. 7 These new formulas also avoid the use of

the unphysical distance z + h + p employed by Wu and King.l,2

In this report, the four dipole antennas (VED, VMD, HlED, and HMD) aresituated at depth h (h > 0) with respect to a cylindrical coordinate system(p,ý,z) and are assumed to carry a constant current, I. The axes of the VEDand HED (of dipole moment p) are oriented in the z and x directions, respec-tively, and the axes of the VMD and HMD (of dipole moment m) are oriented inthe z and y directions, respectively. The earth or water occupies the lowerhalf-space (z > 0) while the air occupies the upper half-space (z < 0). Themagn,.ic permeability of the earth is assumed to equal V,, the permeability of

free space. Meter-kilogram-second (MKS) units are employed and a suppressedtime factor of expfiwt) is assumed.

0'i-IR-51

-Th

TR 6881

WU-AND-KING' S METHOD

In their first article,1 Wu and King developed a new simple and veryaccurate formula for the radial electric field (Eo) of a HED in a dissipativeor dielectric half-space near its bourdary with air. They examined in detailthe interference patterns generated by the direct and lateral waves that orig-inate at the dipole for three values of el, numerous values of a1, and a widerange of frequencies. They confirmed the a.ccuracy of the new formula by com-parison with numerically evaluated Sommerfeld integral results.

In their second article, 2 Wu and King developed new simple formulas forthe transverse (Eý) and vertical (Ez) electric-field components generated by aburied HED source. They then compared all three electric-field componentswith numerical integration results for the case where a1 = 3.5 S/m, e =45e0and f = 600 MHz (see figure 1 of Wu and King2 ). The agreement between thesimple-formuia and numerical-integration results was excellent for the radialcomponent when ly7p > 2.8 and p > 2(z + h). However, substantial agreementbetween the simple-formula and numerical-integration transverse and verticalcomponents was not achieved until IyipI > 11.5 and p > 8(z + h). Furthermore,the simple formulas predicted a dip in the transverse component and no dip inthe vertical component near jy l pi - 8, while the numerical-integration resultspredicted the opposite.

Because the radial-component simple formula predicts the interferencepatterns very accurately and the transverse- and vertical-component simpleformulas do not, it is apparent that Wu and King have made some errors intheir transverse- and vertical-component derivations. A prime suspect is theunphysical distance term, z + b + p, which appears in the E and Ez formulasbut does not appear in the Ep formula. Wu and King could also have made asign error so that the direct and lateral waves are adding instead of sub-tracting.. or vice versa.

The procedure employed by Wu and Kingl, 2 car, best be shot.n by example.When both the transmitting and receiving dipoles are located below the earth'ssurface (h and z > 0), the HED Rx vector may be expressed as (Wait 3 )

lix = 4 ( I + i ) [ P o - l + I ] , Ix 41r(al + i~e 1 )[ 0 U Pi

where

Po eY1RoeR-- (2)R0

is the direct-wave contribution and

-1Y

2

Zt_ . .. . .

TR 6881

is the mirror-image contribution. The remaining term is

Cf e-l (z+h)

1 2 - __JO (Ap) )XdX

10 (4)

I 2 f -u1 (z+h)

(-y pO f z00

R2 =p2 +R2I = x2 22 = u 2 X2+ y2

0 i

u1 = wV0 and

Y 2 = i (a + iWe)1 01 1

For In2! >> land p2 >> (z + h)2, Wu-and-King's procedure is to set R

and R, equal to p everywhere except in the exponents and let

e-ui(z+h) _e-yl(z+h)()

in equation (4). Therefore,

e -yRO

pi e P(7)

.. ~and1 p

2 e y i ( z + h ) c u -u~w( p X~ 8

(y2 - ~2)f ( 1 -u) 0 A)d 8

'1 -This integral can be readily evaluated (Wait, 3 Erde'lyi 8 ) to yield

El

-~1 -A

ITTR 6881

2 e-Y1(z+h) -19-YPI~ - ( - 02 )3 [ci + ylp)e - (1 + yOp)e (9)

Thus, by following the procedure of Wu and King,", 2 the final expressionfor lx is (for jn2j >> 1 and p2 >> (z + h) 2 )

S-y.Ro e-y 1 R1

e eS [ypel(z+h+p) ()-YOPe (z+h)

( 2 2[()p l + y~p)e Y(h)- + y~p)e Oel(h]

Note that fix has four components: (1) a direct-wave component, (2) a mir-ror-image component, (3) a lateral-wave component, and (4) a false componentthat depends on the unphysical distance z + h + p.

BANNISTER'S METHOD

We As we shall soon see, the unphysical distance z + h + p can be avoided.

When the measurement distance is much less than a free-space wavelength, equa-tion (1) reduces to

11 iI 1 +awP 2 F-u 1 (Z+11)]S+0 - P1 + 2 (ui - 4)e J 0 (Ap)XdX (11)•x~~1 1•• 0•l ( 0 2

Wait 3 has shown that this equation is equivalent to

SX+i [ P1 0 - 1P + 2 2 N) (12)

where

N = I01 YIR1 - (z + h)] K tI[Rl + (z + h)] . (13)

SWait 3 has also shown that, when »ylpi >> 1 and p >> (z + h),

e-y1(z+h)• N- e (14)YJp

Quoting Wait, 3 "The man.er in which the exponential factor exp[-yl(z + h)]occurs is rather interesting. It is only in the integral N that this factoremerges."

4

TR 6881

Employing equation (14) and taking the indicated derivatives in equation(12) results in

-Y1Ro 2-y (Z+h) e-YRI

A e-4Ir-(a + iWER (y2 - y2)ps R

2 )R 3 -(1 -3 sin2 01)(1 + y1R1) -y2R sin2 (15)(y2 - y2O)R3

where sin =(z + h)/R.

For p2 >> (z + h) 2 , we can set Ro and R1 equal to p everywhere except inthe exponents. Thus,

' -- Ie R0 2 e-Yl(z+h)

S4 7T ( a + i c ws 1 ) ( ( y 2 - y 2) p 3 ( 1 6 )

1+ ( + Yip)]Jp (y2 y2)p2

Here, we see that Rx has only three components: (1) a direct-wave component,(2) a modified mirror-image component, and (3) a lateral-wave component.There is no false component that depends on the unphysical distance z + h + p.

At first glance, this procedure appears to be considerably more compli-cated than the procedure employed by Wu and King. Luckily, however, the gen-eral quasi-static range field components have already been derived for thefour elementary dipoles (Wait, 3 Weaver, 4 Bannister, 5 Bannister and Hart. 6 Waitand Campbell, 9 ,10 and Sinya and Bhattacharyall) as have the quasi-near, near-field, and farfield range lateral-wave expressions (tables 3.2, 3.7, and 3.16of Kraichman 7 ). Although displacement currents were ignored in most of theseanalyses, they can be included simply by replacing al by a, + ime 1 (as long asIn2 l >> 1). For convenience sake, the direct and modified mirror-image con-tribution to each dipole field-strength component is listed in the appendix.

For some components, these expressions are of very simple form (HE, equation

(A-15)), while for other components, these expressions are quite complicated(HVM, equation (A-18)).

l•hus, the hardest part of the problem has already been solved. We cannow use these previously derived results to obtain adequate formulas for thefields 8roduced by submerged dipole sources subject to the conditions In2! >>

1 and p3 >> (z + h) 2 .

The derivation procedure that we will follow is (1) take the previouslyderived direct and modified mirror-image results (see appendix) and let p be> 3(z + h), remembering not to replace R0 and R1 by p in the exponents, and(2) for the lateral-wave expressions, let

+ h) =-y 1 (z+h)

f(p,z + ) f(p,O)e ,(17)

5

~%

TR 6881

where 0 refers to an infinitesimal distance below the surface of the earth orwater.

The quasi-static range (Iy 0pl << 1) lateral-wave functions ftp,0) can beobtained from Bannister 5 or table 3.8 of Kraichman 7 (with a1 replaced by a1 +iwel). For convenience sake, the lateral-wave functions f(p,0) exp[-y 1 (z + h)]are presented in table 1.* Note that, for some of the field-strength compo-nents, these expressions are identical to the quasi-near range (Iy 0pI << 1

<< IY1 PI) results (see table 3.16 of Kraichman7 ). Also, note that some of thesigns are different than those of Bannister 5 and Kraichman. 7 This is because,in this report, we have inverted the coordinate system so that z and h arepositive depths.

Approximately half of the field-component formulas listed in table 1involve products of modified Bessel functions of argument y1p/ 2 . Numericalvalues for these functions have been provided by Bannister.5 When lIypI >_ 4,the function yjpIiKj - 1, while when IY1pI >6, the function y1pW - 2. Fur-thermore, when jylpj > 10, the function y1pT/2 - 3.

As an example of our derivation procedure, consider the HED radial elec-tric field-strength component. The qdasi-static range lateral-wave E p compo-nent can be obtained from table I or from equation (67) of 4ait. 3 It is equalto

H ~ -y1, (zq-h)EHE p Cos 0- - 1(zE o e (18)S 2n(a 1 + i )P 3e (

The nearfield range lateral-wave Eo component can be obtained from table3.7 of Kraichman 7 (with oa replaced by al + iWel), or from equations (23) and(32) of Wait. 3 Therefore,

EHE _ P cos e (19)2nt(O 1 + iWel)P 3 (1

The farfield range lateral-wave EP component can be obtained from table

3.2 of Kraichman 7 (with a1 replaced by a, + iwel), or from equations (13) and(32) of Wait. 3 Thus,

EHE _.p cos ey(Zh e - YP 2F) (20)

S.2ir(a 1 + iWC1)P3 I(Op

where

F = F(wo) = 1 - i(EWo)l/2e Oerfc(iwol/2) (21)

is the Somnerfeld surface-wave attenuation function and

*Tables have been placed together at the end of this report.

6

. .,__ _ _ _-__ _-_ _

-____

TR 6881

w=-0 (22)2n2

is the Sommerfeld numerical distance. For small numerical distances F(wo) - 1,while for large numerical distances F(wo) ~ -1/(2wo).

When In21 >> 1, the range of validity of equations (18) and (19) overlapwhen Jyop( << 1. Similarly, when Iwo[ << 1 and ty0pi >> 1, the range of valid-ity of equations (19) and (20) overlap. Therefore, we can simply combine equa-tions (18), (19), and (20) to obtain an expression for the Ep lateral-wavecomponent valid from the quasi-static to the farfield range. Therefore, forIn2! >> 1 and p2 >> (z + h) 2 ,

EpH. p cos ýe--y, (z+h)e--yOP 2 2F)

ii P CO *eYe ( + Y OP . (23)27r(oI + iw,•1)p 3

The direct and modified mirror-image contribution to each dipole field-strength component is listed in the appendix. These expressions are valid forjn 2 [ >> 1. When p > 3(z + h), equation (A-l) reduces to

EHE p cos + 1 + pe-YiRo! • ~ 2n(oI + iWCl)P (

-(3+-[0+ (z + h)2o-~YiR~l] (24)I -(3 + 3Y P + y2p2) (z - h)2 --1Ro ( + h2-l

2 1 p

We can now combine equations (23) and (24) and obtain an expression forthe HED Ep component valid at almost any range from the source subject to the

conditions that Jn2 l >> 1 and p > 3(z + h). The final expression is

EHE , P -p C1 Y 2F)eapOPe-yl(Zh) + (I *yp)eYIROP 27r (a1 + iwe )P3 34 +YP O

1 (25)3P+ 2 2) (z - h)2 -1O ( )_Y~(3+ I 2p22h 2 e •

When p2 > (z + h) 2 , the last two terms of equation (25) a."• negligiblecompared with the first two, resulting in

EH p Cos r 2 P Fe--yOp e -yl(z+h)

- 2 + + YOp + Y 0)PL(26)

+ (1 + y1p)e-y 1R ,

which is identical to Wu-and-King's result 1 for In2l >> 1 and p2 >> (z + h)2 .As we mentioned previously, Wu and King have shown that this simple formula isin excellent agreement with the exact Sommerfeld integral numerical-integrationresults.

-7

~ , -~ - -....... *-.. .

----- --

TR 6881

New formulas for the electric and magnetic fields produced by the fourelementary dipole antennas are presented in tables 2 through 9 for the subsur-

face-to-subsurface, subsurface-to-surface, surface-to-subsurface, and surface-to-surface propagation cases. All of these formulas have been obtained frompreviously derived results according to the procedure outlined above and arestrictly valid for In 2 l >> I and p2 >> (z + h) 2 . (However, for most cases,the requirement that 1n2 l 1 10 and p > 3(z + h) is sufficient.)

It should be noted that for many (but not all) cases, the range of valid-ity of the formulas presented in tables 2 through 7 can be extended down top - (z + h) if the direct and modified mirror-image terms in these equationsare replaced with the equations listed in the appendix. For example, the HEDEp expression (equation (25)) would be replaced by

E•E. p, co• (p_• O -ylp-(z+h)

S4E (1 + i cl) (1 + yop + Yop2F)eYeY1

-I~+ [(3 cos 2 10 - 1)(1 + y1R0 ) -yR2 sin2 %)o -- (27)

1 0 R 0

-Y

0

- (3 + 3y 1 R1 + Y2R2) sin2 *, eR11

where sin = (z - h)/R 0 and cos *0 = p/R

For the subsurface-to-surface and surface-to-surface propagation cases(tables 4 and 8), the vertical electric-field (Ez) receiving antenna is

assumed to be located an infinitesimal distance above the earth's surface. To

obtain expressions for the vertical electric fields just below the surface,multiply the Ez equations in tables 4 and 8 by I/n 2 .

COMPARISON OF BANNISTER AND WU-AND-KING RESULTS

In attempting to explain the major discrepancy between the HED E0 and Ez

simple-formula and numerical-integration results near lylpi - 8 of their fig-ure 1, Wu and King2 stated that very small changes in frequency significantlyalter the interference pattern so that close agreement in a small range near

such a region cannot be expected. This statement is in direct opposition tothe Ep results presented in their first article (Wu and Kingl), where sub-

stantial agreement between thu simple-formula and numerical-integration results

was achieved in the interference region when In2 1 >> 1 and p > S(z + h).

They also noted that, at greater distances where the lateral wave domi-

nates, the EP, Eý, and Ez expressions were highly accurate. They certainlyshould be highly accurate at the greater distances because their lateral-waveformulas are essentially equivalent to Wait's results, 3 which have been suc-cessfully utilized for over 20 years.

8

S...• • •. ,• .,- o. - •. . , -: -. _______._.__....._______.______.____.___

TR 6881

The major value of Wu-and-King's work is in determining adequate field-strength expressions in the range where the lateral and direct waves interfere(either constructively or destructively). They have succeeded with the HED Ecomponent. However, they have made some errors in their derivation of the HED4 and Ez components.

The HED E and Ez expressions derived in this report (table 2) are in verygood agreement with the numerical-integration results presented in figure I ofWu and King2 when p > 3(z + h). They correctly predict a substantial dip inthe E. component (due tc interference between the direct, mirror-image, andlateral waves) near JylpJ - 8 and no dip in the E component at this range

(which is directly opposite to Wu-and-King's simpte-formula results).

To see where Wu and King erred, we will compare their E, and Ez formulas

with the expressions listed in table 2 for the situation where the Sommerfeldnumerical distance is small (i.e., F - 1), p > 5(z + h), and Iy1pJ > 4. Forthis situation, the EO and Ez expressions in table 2 reduce to

EHE - -p sin . [(l + y0 p)eYOP e-yl(z+h) + ylp)e-YIR12r(o1 + iWs 1 )p 3

a d h 1+ p y ep2 ) - Y R 0 e -Y R 1) ] ( 2 8 )

SPcos € •- -y 1 (z~h)Sand

SEHE - - 2o +1 i+lp (i +0P)el Y0e -yI~h

- (3 + 3y 1 p + y2O2) - h)e-R0 + (z + h)e- 1il

When F - 1, p > S(z + h), and Jy1pI > 4, Wu-and-King's 2 Eý and Ez expres-sions reduce to

EHE - p sin -y(z+h)'ýO 2r•a + iWcl)p7 (1 + yop)e- e

-ry 1 (z+h+p) _J~l +p2 e-yIR1

(2 + 2y 1p + y~p2 )e- + + yi+ y1 p 2)e (30)_YJR° _J

S yp 2)(_ eYlR

and+

E E . p cos . _ -yOp e-y 1(z+h)

21r(aI + iwel)p 2 ln2"- + y0 P)e

2p 2( 3 + 3yp e(3)

I

TR 6881

II

A comparison of the two E¢ expressions (equations (28) and (30)) revealsthat the difference between them is the unphysical distance (z + h + p) termof Wu and King. If we replace z + h + p by the physical distance R1, Wu-and-King's formula reduces to

L-E psin 4) [( o~ YOPe-yl(zh) -( y _Y1R127ri ( 1 + iW e1)P

3 [,(l + y0p)e- 0e + ylP)e - 1

IRO _ R)](32)

+ +( ~.YIP + y4 p2)(e-lR - e~l11(2

which is identical to equation (28).

A comparison of the two Ez expressions (equations (29) and (31)) revealsthat the reason that Wu-and-King's formula does not predict the dip in theinterference region is because of a sign error. Their results indicate thatthe sum of the direct and mirror-image waves add to the lateral wave whereas,in reality, the sum subtracts from the lateral wave.

RANGE OF VALIDITY OF LATERAL-WAVE FORMULAS

As we have previously mentioned, the quasi-static range lateral-waveexponential attenuation-with-depth factor exp[-yI(z + h)] emerges only fromthe integral N, where

I I N = 10 -- [R1 - (z + h)]IK0 o [RI + (z + h)] . (13)

When jylp >> 1 and p >> (z + h), Wait and Campbell 1 0 have shown that the

I modified Bessel functions may be replaced by only the first terms in theirrespective asymptotic expansions to obtain adequate HMD quasi-near range hori-zontal magnetic-field-component expressions. On the other hand, Sinha andBhattacharyalI have shown, for the VMD case, the first two terms of the modi-

-I' fied Bessel function's asymptotic expansions must be employed. This indicatesthat the range of validity of the quasi-static range lateral-wave expressionswill not be the same for all field components.

As an example, consider the quasi-static range HED Ep lateral-wave compo-nent, which can be expressed as

SLW ~ - p cos ,.e ul JoCXp)dX (33hS 2mr(aI + iwE )e (33)

When Iyl pi >> 1 and p >> (z + h), the usual procedure is to replace U, in the

"exact integral expressions by yl, the propagation const.,t in the earth,resulting in

10

4••, .. ~ Z : • ,r:• • " -

_ _ , .. . .=;- ,..=. . "--•• ,•-C l

TR 6881

E~w -r •o ,e-'(zc+h) fo_L2EP °l + iz-) 0 (Xp)dX (34)

P 21r (a1 + i~e 1)p aP J

Since this integral is equal to i/p (Erd6lyi 8 ),

E LW -p OSel(Z3h = f(p,O)e-y, (35).

So far, we do not know exactly what values Iylpl and p/(z + h) must have

in order to utilize equation (35) or any of the other quasi-static range lat-

.1i eral-wave formulas presented in table 1.

As a first order approximation, we will let

( X2

u1 = (X2 + y2) 1I 2 - y, + (36)

so that

-u1 e-yl( (z+h) e-x2(z+h)12yl e-yl(z+h) I - + (37)i e - e- e ~ +,_2y,

Inserting equation (37) into equation (33) results in

,_p cos ýe -yI(z+h)

ELW JPCOSAp__[P 2n(al + iwe1)p [PV 0 (38)

- (z ; hL)f • 2j(X•)dX

Since the second integral is equal to -1/p3 (Erdelyi 8 ),

ELW ,p cos ýe [1l~z3(z+ h (39)P 27r(c 1 + iWF1)p

3 + 2y 1 P .9

It can easily be shown that the error incurred in neglecting the second

term is less than 1 dB if the quantity

CY1P)(_i -h)l >4ci , (40)

where c 3 for the component. That is, to a first order approximation,

it is the product of jylpi times p/(z + h) that must exceed a specified numberto accurately utilize equation (35) or any of the other quasi-static range

MINIM,.

-J ,

ii

TR 6881

lateral-wave formulas presented in table 1. For example, if p = Srz + h), thequantity lylpi must be >2.4 to use equation (35).

The most severe restriction will be for the VMD Hz component. Followingthe same procedure as in the derivation of the HED Ep component, we can ixpressthe VMD Hz component as

9 meYl(z+h) [ 25(z + h) (41)z-- 2-(y 2 - y2 )p5 5 2ylp2

1 0 P 1

Again, it can easily be shown that the error incurred in neglecting the secondterm is less than 1 dB if the quantity

(y > 4cI , (40)

where c1 = 25 for the VMD Hz component. For example, if p = 5(z + h), thequantity Iy1pI must be >20 to employ the quasi-static range VMD Hz lateral-waveformula presented in table 1.

The values of c1 and range where f(p,z + h) can be replaced by f(p,O)exp[-yl(z + h)] (i.e., the range where the quasi-static range lateral-waveformulas presented in table 1 can be used) are presented in table 10 for eachfield-strength component. Here, we see that for the HED and HMD E 2, Ez, andiH components and the VED E and H, components (c1 = 3), the quantity Iylp2 /

(z + h)j must be >12, while for the HED Hz and VMD E and Hp components (c=15), the quantity jylp2 /(z + h)I must be >60.

This restriction (equation (40)) also applies to Wu-and-King's results 1 '2

and to the quasi-near and nearfield range subsurface-to-subsurface propagationequations tabulated in Kraichman. 7

To a first order approximation, the range of validity of the equationslisted in table I can be extended by multiplying the field-component expres-sions by the quantity

c1 (z + h) (2

where the value of c1 for each component is given in table 10. For example,

the HED Hz component listed in table 1 is

-m - 3p sin *e-Yl(z+h)

2ir(y2 - y 2 )p 4 (43)

Since c1 = 15 for this component (table 10), then

12

TR 6881

-yj (2 4h)Z-yHHE

3p sin e 2r (y 25(z+h) (44)i i •,(•1 - ,Yo L+ :y

L ANALYTICAL CONFIRMATION OF FRASER-SMITHAND BUBENIK VMD Hz NULL

A few years ago, Fraser-Smith and Bubenik 1 2 numerically evaluated theexact Sommerfeld integrals and found a rather deep null in the t1 H componentfor the subsurface-to-surface propagation case (see figure 6 of Fraser-Smithand Bubenikl 2 ). For their particular situation, the frequency was 100 Hz, theVMD source depth of burial was 100 m, and the null occurred around p - 250 m.Since the skin depth, 6, in sea water at 100 Hz is -25 m, p/6 - 10 and h/6 - 4.To our knowledge, this null has not been analytically confirmed. The subsur-face-to-surface VMD Hz component is equal to (from table 5)

HP " - 2 2)1p (9 + 9 yop + 4y2p 2 + y3p 3 )e-YOPe-Ylh

- e'YD [(9 + 9yiP + 4y~p2 + y~p 3 ) (45)

h2 +39y2p2 9yp3 Y:P4)] I

p2 (90 + 90 y1 p + 11where D2 =p 2 + h2 .

For f = 100 Hz and p = 250 m, 1y0Pj << 1 and the yOp terms in the lateral-wave portion of equation (45) are negligible. Furthermore, since the quantityjy 1p2 /hI - 35, which is <)CO (see table 10), equation (42) must be employed.The result'ng expression for Hz is

HVM 9me I HZ 2Try~p5 H , (46)

where

H + 2yl,921 9•hJ + 9y1p + 4yip2 + y1p3)

S]1 (47)h2 + p + 39y2 2

+ 9y~p3 I+ p4

For p/6 ~ 10, the dominant terms will be the y 3p3 and y4p 4 terms. To afirst order approximation, the y p 3 terms will cancel and, since y4p4 =

-4(p/6) 4 ,

113,

W R'51201 t__-,~-'-~lig-l

_-W ,- .F-. - - - . - .... ..--

TR 6881

H I\+1H\4 *~I(÷he e -((Dhh/ e (48)

~z 2ylp2 9PJ~/This equation will be at a minimum ntear (D - h)/6 = 2r, which corresponds

to p - 240 m. The normalized VMD vertical magnetic field 1H. from equation(47)) is plotted in figure 1 versus the horizontal distance, p. From this fig-ure, we can see that a rather deep null (-20 dB drop in field strength comparedto the asymptotic value) occurs at a range of -240 m. This null is clearly due

'• to the destructive interference between the direct and lateral waves.

S~ CONCLUSIONS

New formulus for the electric and magnetic fields produced by the fourelementary dipole antennas have been developed for the subsurface-to-subsurfacepropagation case. These formulas have been obtained completely from previouslyderived results. The main restrictions in their use are (1) the square of theindex of refraction is >10 and (2) the horizontal separation iF >3 times thesum of the depth of burial of the transmitting and receiving point sources. Anadditional restriction must also be applied to the lateral-wavc components.The quantity lylp2 /(z + h)J must be >4c,, where cl = 3, 6, 9, 15, or 25,depending on the particular field-strength component. This restriction alsoapplies to Wu-and-King's recently derived resultsl, 2 and to the subsurface-to-subsu'face propagation equations tabulated in Kraichman. 7

The range of validity of the subsurface-to-subsurface, subsurface-to-3ur-face, and surface-to-subsurface equations tabulated in this report can beextended down to p - (z + h) for many cases if the direct and modified mirror-image terms in these equations are replaced with the equations listed in theappendix. The extension of these results to even closer ranges will be thesubject of a future report.

it should be noted that the two media can be inverted and the. air replacedby the earth's crust (of conductivity a2 and dielectric constant eF). The sameequations (tables 1 through 10) can be utilized, as long as [1ý21 = Iy2/y2I10 and p > 3(z + h), simply by replacing iwe0 by 02 + !2"

An analytical confirmation of the Fraser-Smith and Bubenik VMD Hz iull hasalso been accor'lished. This null is clearly due to the destructive inter-ference between the direct and lateral waves.

14i

777_

[ I TR 6881

'C)

nn I

- 0-

0-0

4- 0J

cr.,

0D >U

00

01Zk

LCD j 0:C14 M

o ~ 2 l

040

t4 00t to

C, C>

011 13SWIVI63 a3-VHN

Revrse,

paMw0

MOO J-

'able 1. f(p,O)e.

Dipole EJ EI: ~Type _______ _________

VD ype-yi z+h) 0Y (z+h)

VED ~ a + ~* iwel)n2 211 1P

VMI) 0 ~ 3iosOm -;eY1 (z+h)0I~~~~ -M 2~~~y)p

IHED pcos Oe Ip sin Oel(h I~ YPCos OeI2nr(al + iWC)P 3 ir(a 1 + iwe:1 )p 23 r iwE1 )n P

3________ Kj 3 --- Cr 1 W) iw COS *y~27rn2 P2

j -~ *Ar~gumgent of modified Bessel functions is ylp/2.

k11_ 001

TR 6881

.y(z+h)~f(p,0) e Lateral Wave Formulas When~PI << 1 [jn2j > 10, p2 >> (z +h2]

__ _ __ H p H H

T(z+h)-y(zh

iwl. 21rp 2 P2

0 men 1y(z+h) 1y pT\ 0 9me yl(z+h)2iry 1p 2)21r (y ~ 0 ~

e-i(z+h) PIK p sin It e y,(z+h) (YJPW) p Cos *eyi (z+h) 3p sin. *e T 1 (~h

iowc )n2 2 -(Y1 1K1) -r~ JyP 3 2rp 3 (yvp11K1) 21r(y2 - 2p

Iy~ zh m-y 1zh mcs(z~h) __ __ _y 1(z+h) yPT\~~~v~ 61z h esn f 1 6 Y 1(Z h o e-Y m sin 4e 1y

#py2221rr. 3 y2221rylp4~ 2 /

- 17/118;

sga- -- ---

Table 2. Electric-Field Subsurface-to-Sub

FDipoleEj Type _______________________

VD 2ir(a1 + iw 1)p I(fl2) 1 1 1 Y0P eY e1(h

VD - -=7(3 + 3y 1P + y P z + h)e 11 (z -h)e 01i

21r(y2 - 3 +)3,

[(3 + 3y~p + y2p2) -

(.Y2 - y20

+ 2 -(1+ Ti

4

p' Cos (1+ y P + Y2 P2F)eCYOP e-yl(z+h) P sin [2 + y2%(r1+ iwe1)p 3I 0 0 21r(oi + iwCaj)P 3

HD + (1+ y p)e-YiR 0 (34+ 3-yp P + 1(1 2 + y1p) -(z+h

2p2 O+ z h2_p2 1 'lY22

UP0mCos -O T0 -y1 (z+h) iwv 0 m sin

2ri3 (1IPI 1K 1 + y0 p + Y2 P2F)e e 30iP

22r 21ry 1 p3 y

If4 -y(z +h) (3 3 pYJ2TR1 y (z + h)e-YIRi

Cl 1 [y(z - he i0 - Y1 (z +h)e T1 ii+ 6y2 P2 + -3 (3

*Ar.gtument of modified Bessel function is ylp/2.

'64

TR 6881

~rface-to-Subsurface Propagation Formulas [In2 I > 10, p > 3(z + h]

P ~~+ y P + yp 2 F);eYO eY (zh

21r (a 1 + ic1 2

+ :o1(1 + 22

1(3~ + Y P1 ' + y~p)-~ey (zFh

P + y' h2 .P:) 2)os --z+h2Y5+ S y +Y 3 -I

P 2 2i~o

+

+w

~ ~l n /

y I

1*lR0

- -J~

J~p 3 12 /yRpl F)e -YR) 22O eyylz2) PC~~ Cos + yoFe O e(z ~ zh)e1R]

0 p(1 + F)]; 2w(c 1+Y

1(Zp2 ýT)h)

+ 2wp2+3~ +yp2e

)e~~~1Ry~ + 4y~p2) - (z +2 W w~ os

[ i 1 2y( 15 y 2 + 1 + y p ) ( e -Y1 R 0 e- Y R ]

5 ~P3J (I + lp)[,(z h)e-y1R + y (z + h)e1YRJ

19/20Reverse Blank

Table 3. Magnetic-Field Subsurfact

[1n21 >10, P

IDipoleHType

21r .)( +y~F)e~Oe1

VED + I l +y P)(e _YlO eY1R I)]2 -1

JYJ +3p+ Z) YOPe-ylCz+h)

Y1(Z + h) -Y1R 1 [ 5+ 4Sy p +18y 2p2 ' 3y3p3)

Y11

*1VMD (z + h)2 (10 S + lO5y 1p + 4Sy~p2 + 10y3P3 +~ p

(3 + 3y p + y2P2 4 -R -YIRIl

+ 1 2 1 (z - h)eY1O - yl(z + e

*Argument of modified Bessel functions is y p/2.

TR 6881

d Subsurface-to-Subsurface Propagation Formulas

[In 2! > 10, p > 3(z + h]

H H

~OP e- y(z~h)0

e0

M 9 + 9 pi~ f yg p f : 4y;2+ p;:: + y5p3) ej10P e yI' (z+h )

- (z+ h) (90+ 9YIP + 9y2,,2)(egy1RO +ey1R1

21/22

Reverse Blank

"2-

Table 3. (Cont'd) Magnetic-Field S

j Dipole __ _ __ __p__

I. ~~Type _______

sinz pW h) +R 1 yop(l + F)Ie-YOP e-yl(zh) P Cos PI K + + Y2p

Y(+h)-IR1[12+12y~p + 4y2p2) Y1 (z + h) 2222(3 +3yjP + Y P2

1E 1_ 1

z ý ~ s+ lsy p + 6y P2 + yp 3__+_____IR~P) - 2 [yl(z -h)el

+ ( 2 YP 1Lz - h)e 1p YR 0 - z+ h)e Y' 111

ml sin - 12~ -Y P -y(z+h)2wp tYOp(l + F) ip2e 0e 1

+ Co f'+ YOP + y2P2F -+ ~2M (1 +y p)e-R 0 + e 12 + 2y P + y 2 2tffp3 0

(3It I y

+)-'R h)2(Z0 + h)2Y +yR 1 Y0+ 22

%.p py2 +j +~3P + y 4 2 ~ 2 ~ (e1RP2YI2Agan of(tzh modifie Bese f+cin is 2 -yp)/Z0]I+I( +yp)e J

(3 + y P +y I1 1 2p

*Agmn.o oiidSse ucin syp2

jeld, Subsurface-to- Subsurface Prop~g~t ioi rlOftilas> 10, p > 3(z + h]

+ Yf2p2F)ei*OP e-Y, z+h)0

(3 3 -y?(o2)e- O% 'ey,(,z-h)

Y2 1 Y~T~v

,,o2)e

=T- Ci 4- h~l ( -2

+15 -+6yII-I ~ ~ p (15yp 4 3pt _ _ _ 1 1 4~ry 2p :o~

- t

I1~D _____________ - 4) Y-,y 30 3)~j,-j 2 1 :1 3

-fj2 ( z - h) G '-R

- ~---- -4:i L -zk,,

Table 4. Electric-Field Subsurface-to-Surface Prol

Dipole EType

2¶ P 2iW yl)P2[Q2) K1I + y 0pF)eYOP eY1h

- h~a + +w,) [(2 1 D]VED

h (3 +3y1p +y~p2)e -Y D

iwp 0 m

21r(y2 - 2~)

h2

p Cos 3 +-yPFeYOP -Y1 h psin427~l+ iwe1)p (1 0 2Ir(a1 + iwC1)p

flED eYD

( + y p Y ID [( , +

ipmsin*

HMD 2-fl3 IY111 0 0y~he-'lD

1 ( 3y p +2y 2 P2 +y 3p3)e-Y 11 h2 -ý{5 + l5Y

*Argumuent of modified Bessel functions is y p/2.

TR 6881

ito-Surface Propagation Formulas [1n21 10, p > 3h, D =(p 2 +h2)1!2]*

0 2~o 23( + y p + y p2F)e 'YO p e -y

wp0 1M (3 + 3py 2 2)e - OPeYlh

2-'r(yl - YO) 4 1 0 Y0

e- ! [(3 + 3y~p + y~p2) 0

:'(5+ I5Y P + 6y2p 2 + yp

p2ra sin~ 1 p 0 [ + yop(l + F)]e-YOPe_1 h

w11 (O ) 2n p o s +( iw i p I KP1 + y 0pF)e YOP eYlh

e-y D(1 + Yip) - L3+ 3yp+ p21r(I+iW1)2111

SWP l sin -YO _Yvh27ry 1 p 3

1 fY~PW + -yop(1 + F)]e 'e 1

Y~he-YI (12 + l2y~p + 4y )ip0 (1 Co ypF) eTOP eYi

.y%2 1 2 1p ) 2wp 2 - Y

25/26Reverse Blank

IMP

Table 5. Magnetic-Field Subsurfce-to-Su

DipoleType lip

VED 0 h27rn 2p2 (

I1y p{(T + 3y 0 + y 0p2)eJYOpeI'l

h 2 P 2h2 (1os + 4osy p + 45y 2 2 + oy3p3 + y4 4 IP 2 1 1 P 1 1 e-

RI

psin 0ypW + y p(1 + F)]e-YOPe-Y h e CosS21Tylp 3 Y 021rylp 3 !Y]

HED ¥ih h _

(j---•12 + 12yIp + 5yp 2 + yp )ey (3

m sin l + y p( + F) e-YOPe-Y1h m Cos

21rp 3 , +2] 23

HMD + 2e 22 + 12y P + 572p2 + yp3 + e-:1 "-. . v,*.°th-2(105 +lOSy p 10522 y44)11

4S2yP+p2 o '1~ P l 1 - -;(lc + 1

*Argument of modified Bessel functions is ylp/2.

TR 6881

to-Surface Propagation Fo~rmulas [In 2 j > 10, p > 3h, D =(p 2 + h2)1/2j*

2+y OPF)e 0 e 1 0

1(9 + 9yop + 4y2 p2 + v3p3)eYOPP-ylh

%. 1 0 1

0 e-Y D~ (9 + 9y,p +4y2 P2 + y~p3 )

- 1 1

~-~~y~I1 1 + yp + Y p2~ e~~021ry 1 .1+~ p + 9y p2 eYPe+ y

- sin Dyp 12y~ 2)1 e 3 3y~ + 3y P+ 22) e

+ YIPIK + )ep +I~2~-oe2TT - +~n l 1p4 + OPp e

1+~~e-I [p+y(3 ~-eOe~h + 3y~p + p2

(3~ +iyp 2y 1 y Y p)e-YD

+ 1+h22(1 y +he1'0 Y[(45 4 2y2 + 1y~p2+3y~

3

31 OP yp e 4yp 11!Tp + lOyo + + +~

1S 1 1s JJ y 2+Y 2ý(105 +105j 4Sy~p2 + OY33+y~p3

27/28,Reverse Blank

Table 6. Electric-Field Surface- to -Subsurl

Dipole EPfType__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

VED ~~YIP Y -j

VyED + y~pF)e T OLeI2wa jijjl)p2 I

I 21r(y2 -~VMD 0-_________R__[3__+

1~ ~~ ~ y p + y2p'F'Ie~Oe +311 HE + jE~1 )~

3 ~0 - + iwE

4. - I [( I + y 1 p ) 2 1- + 1y~ Y 2 P 2 )] - Y J [ 1 +

-~to Co y~~ppK y2p2F~e-y~-I iw)50 m sin

27rylp 3 +'IIK 3 21rylp 3

1 -- -(3 -+ 3y p *.y2 2) eiY1 7l2R

*Ar~gument, of modified Bessel functions is y~p/2.

MWVIý/

-~to-Subsurface Propagation Formulas [In 2 l > 10, > 3z, R (

0 1.2 2

11

2e 2 - 2)[3 + + 3y 0P y2)- OPe Y

e -jYlS l3y~p + 6y~p2 )p1

3112 2aiy(15 + 15yIp + 6 y (1 + F)eY3)e I -3 w 1 p

3c 2 T~ + y Yi 0 p(P)] +y +)e e-~ yPp Cos1

1yP2 1 ;2432 3y 3P 3

.7~ ~ ~ - 0-Y0 1

77-

I Table 7. Magnetic-Field Surface-to-Subsuriac

DipoleHType __________

ft IVED 0 P ( + y pFP

I21rp 2 0

m /Y1' yP+y~)ey~~~2yp4 j -2 3 0 0YP)ee

VMD Y1 z [(4S +45y~p+ 18y 2p2 +3Y 3 3)

- 2 ( o + 1O 5 y lp + 4 Sy 2 2 + l O y~ p 3 + y p Y R

p Ps1in IyPw + y p(I + F)]e-O e rj

flED - i yze YJ (12 + 1Y p .4 202 27ry 1P [(y22 [ 2y 'rz1 1 .ip + 3y

-2 (IS + 15y1 p + 6y2p2 + 1~3yp

p2 1

Zf [2S + y 1+ F) - 12 eOiJ1 - m Cos0 ~21rp3

4'Y 1RM Bese i

TR 68081

-Subsurface Propagation Formulas [In2 > 10, p > 3z, R (p2 +z2)1/2i*

2(1+ y pF)e -OP e _1

m ( 9y p + 4y2 2 -yp)YO peY1 z

2ir (y1 ( 0 0

0i ~e1 [(9 + 9y1p + 4y~p2 + y~p3 )

_( 0 + 90Y p + 39 2 2 + 9 3 3 + 4 p4139p 2 1 P3 1

p sin -Y P -Y z

co +-P-z2iry 2) 4 (3 + 3y p + Y2 p2)e 0

K+[(y1pI1K1 yp y2p2F)e-y~e1y 1 YO+

Ip 0 011 z2J+ 3y~p + y~ )e - e [( 3 + lsy~p + 6y 2p2 )yp 31

(3+3y1 + 2y -y~pL/21Yp4 \2-

-y21 1 +ze 1 5YR3y~~~p 2 +~2 (4644y2 p 2+ 2y303 6~ 3

+ ~ 4

.. y~p

Co + 6y +Y2 p2 F eyp-l + 3311 + r~)eYp~~- lOry ~ p +P 05~p 0 O~p

21rp 3 0 1/12Revrs Bln

_YIR _VI

Table 8. Electric-Field Surface-to-Surface Propagatj

Dipole EE

VED yj (1I1K1+ y0pe Y 0

21 a1+ iwsJ)p2 1 1 yFe

21r (y2 - -Z)4 I~3 + 3 0 p

-(3 + 3y p + y~p2)e yip]

(1+s -yOp p sin *

PE COS(a +ie)3 +YOp + YOP 2F)e 2r( iwe 1)P3 1112 + yop(l + F

+ (1 + y~p)e Y1P] + (1 + ~ 1P

-iWPl 0I Cos 0 ypjj+ yP+ y~p2F)HD2ry 1P

3 (0p 1 0 -w i [y~pW + yop(l + F]

*Arlgument of modified Bessel functions is y p/2.

-"n4

TR 6881

ctric-!Field Surface-to-Surface Propagation Formulas (In 2 I > lO)*

- --- (1 + Y + y2p2F)e-O-Yop 0 2iriwe OP3 0

0 3r y y2p2)e-YOP21r (y2 - 2) 11 V 3

0P P0

-(3 + I3y p + y~p2)e -iJ]

)e-YOPP p f sin 3 [2+ yop(l + F]-O

21 a1+ iwe 1 )p iWiiOp Cos *+-Yap(y~pI1Kl y~pF)e0

(1+ ylp)e- 11 21rylp 2 a

i2Jw )I 0 m sin -Y p _w_0__ C s____

- -~ p [yjpW + yop(l + F)]e 0~ 2ip -(1 y ' p

XIs is y p/2.

33/34Reverse Blank

4 ( ______ ______

O llj - I_ _

Table 9. Magnetic-Field Surface

Dipole

Type H

VED 0 p2 + yopF)

VMD m .T ..3yP + y2p2) , ,P2irylp 4 \ 2 0 0

HlED p sin ý [y(OW + yop(l + F)]e-YOP p cos (yjpI.K

21r1• " 21rf0wlp I

m sin 12 YP m cos * +

2 p3 + yop(l +pF) -2p2]e 23rp3

INID

S+ (12 + 12 Yp Sy p2 + p + (3 + 3yip +

*Argument of modified Bessel functions is y1p/2.

S~/

TR 6881

i'ield Surface -to -Surface Pron:gation Formulas (In2 j 1O)*

Ir 1+ YO pF)e-YOP 0

-:y m 9 + 9::::+ 4y2p 2 + y3p3)e yop

~~Y1PI1K1 ( + 9y p + YpFe2iy ) 4y 2 +a Y ap 3 )e

p ( i 03 + Iv p + yY2P2)e YiP]

2irp 3 j 0P 0 y2p2)e

msin4ii + lp e'O

k (3 + 4 ~ ~- ~ ~L T + 3y0 p+(5 + 3-y~p y2p2)e-l 2

1 y2 P

3Oe~ Blanký

NoL

I TR 6881

Table 10. Range Where f(p,z +h) = ~lzhf(00o)

-- - 9 >36

Hz -- E., Hp 15 >60

-- Hz 25 >100

I ~37

TR 6881

REFERENCES

1. T. T.'Wu and R. W. P. King, "Lateral Waves: A New Formula and InterferencePatterns," Radio Science, vol. 17, no. 3, 1982, pp. 521-531.

2. T. T. Wu and R. W. P. King, "Lateral Waves: New Formulas for E and ERadio Science, vol. 17, no. 3, 1982, pp. 532-538.

3. J. R. Wait, "The Electromagnetic Fields of a Horizontal Dipole Antenna inthe Presence of a Conducting Half-Space," Canadian Journal of Physics,vol. 39, no. 7, 1961, pp. 1017-1028.

4. J. T. Weaver, "The Quasi-Static Field of an Electric Dipole Embedded in aTwo-Layer Conducting Half-Space," Canadian Journal cf Physics, vol. 45,1967, pp. 1981-2002.

5. P. R. Bannister, "Quasi-Static Fields of Dipole Antennas at the Earth'sSurface," Radio Science, vol. 1, no. 11, 1966, pp. 1321-1330.

6. P. R. Bannister and W. C. Hart, "Quasi-Static Fields of Dipole AntennasBelew th- Earth's Surface," NUSL Report 870 of 11 April 1968 (also inQuasi-Static Electromagnetic Fields, by P. R. Bannister et al., NUSCScientific and Engineering Studies Naval Underwater Systems Center, NewLondon, CT, February 1980, 515 pp.).

7. M. B. Kraichman, Handbook of Electromagnetic Propagation in ConductingMedia, U. S. Government Printing Office, Washington, DC, 1970, Ch. 3.

8. A. Erd6lyi, ed., Tables of Integral Transforms, Vol. 2, McGraw-Hill Book

Company, Inc., New York, NY, 1954.

D. J. R. Wait and L. L. Campbell, "The Fields of an Electric Dipole in aSemi-Infinite Conducting Medium," Journal of Geophysical Research, vol.58, no. 1, 1953, pp. 21-28.

10. J. R. Wait and L. L. Campbell, "The Fields of an Oscillating MagneticDipole Immersed in a Semi-Infinite Conducting Medium," Journal of Geo-physical Research, vol. 58, no. 2, 1953, pp. 167-178.

11. A. K. Sinha and P. K. Bhattacharya, "Vertical Magnetic Dipole Inside aHomogenieous Earth," Radio Science, vol. 1, no. 3, 1966, pp. 379-395.

12. A, C. Fraser-Smith and D. M. Bubenik, "ULF/ELF Magnetic Fields Generated-1 at the Sea Surface by Submerged Magnetic Dipoles," Radio Science, vol. 11,

no. 11, 1976, pp. 901-913. I

38

gg---~.4 - '.'c;.

TR 6881

Appendix

DIRECT AND MODIFIED MIRROR-IMAGE CONTRIBUTION TO EACH DIPOLEFIELD-STRENGTH COMPONENT (In 2 I >> 1)

EHE p cos 1[3 cos 2 Ro 2R2 sin2 eR 10 R0

4w(c7z + iwsI)0

le-YRI I(A-i)

- [ (3 + 3y iR 1 + y 2R 2 )s in 2 _l_ 1-,

EHE p P sin ) [(1 + + -2 R o

47r(oz + iwel) [o 1 0 R3

0(A-2)

(1 - 2 sin2 ')(3 +.3y R + y22--

., E~~~~~~HE ~ p Cos 3+31R+ 2)snoOS0-1RH~~E -3 R~E + [+ Y 2:YR2)sin ýocosi e0

,!4'•(aI + iwcI) 1----1 (A-3

(A-3)

HE psin ' -y eR ' ezYRR

H (I + yiRo)sin 0 1- (1 + e-R)sinR~ - 47r R2 (i+Y~)i lR---2t ~01

2 sin *le-YJR I

4 [(12 + 12y R + 4yR2) (A-4)

2 sn 1 eY1R1

s2 (15 + 1yR• (3 + 3y2R 2 + y3R3)

A-i

u2 si *•- J "I•,.

S.. ... .. I ...I1','.... ... , • - •' ,••,: :• , . , •3y. R + ,y.2 R. -2)m l g l• €

TR 6881

H HEPsn t 1 yRo)cos 0 e - 1 + yR)cos R

2 cosI(cos 1 e)A-6)

( - 2~)R4 1( 3 + 3y1R, + y

sin2 (15 + 15Y R + 6y-R2 + y3R)R

-1

-YR I

iN wjlom Cos *e( eYR )sin o - (i + y 1R1 )sin i R2

p 0 R2

L7 YR1 (A- 7)

-Y1R 1 12 sinp 1 e (3I- 2R2)I-( -y2 - ) R ( 3 + 3 Y , +y1 1 I

EN iwp 0m sin e - + i R i e -y

4E-t - + YiR 0)sin 0 2 + (1 + y R )sin n0 R1I2 sin ýleY1R1

+ 2 si e yR [(12 + 12y R 1 + 4Y2R2) (A-8)(Y2 - y2)R4 yR 1 AB

"- sin2 ýi(15 + 1SY1 Rl + 6y2R2 + Y 3R3(]-- ~ -'i 1 R1

H iwp0om Cos _ [ 1 + Y R )Cos 2-- (A-9)1 j z 1(E + y Ro)Cos '0 R(

HHM - n 2 - 3 sin2 )(I + YR 0 ) Y2R sin ]e

p 4w

2 e-yR 2e-Y(R2+ (3 + 3y R1 + y'2R2)sin2 +i T y+ 2)R1 1 1

- 0 1 (A-10)

x [(12 + 12YRI + 4y2R2) sin2 cOs 2 (105 + 105y RI

+ 45y2R2 + jOy3R0 + y4R4)Ij ,

A-2

TR 6881

H( 2R21e---(1 + + 2 2 eY1R1

H - m cos1 " + yRR0 +y R2 -(1 y1R1 y1R1) R34 1r+ 1 - $;-• [(3 + 3yIR1 + y2RI)S-(y2 - Y 02)R5'

- sin 2 'p(15 + 15yIR1 + 6y2R2 + y3R3 ,

H - m sin4 + 2R2)sin cos e R3

(+ (3 + 3YIR + y ,R2) +inl 0i C Os R3

S- sin2 1(I05 + 10Sy 1 R 1 + 45y2R1 + 10Y3R3 + YR•)]Ie

+ EVE ( P+3 + eY1) i +o

R0

(A-13)

- +3yRI + y21R 2)sin fl s• e____

II- R

4(1+ iw°•l) (A-14)

+ - c- 5 45.* 1 1 + 4yR l +c18y

1 0 1

e-2ý115 0 yR0 + i1 + Yl)S '1--• ]' A15

p 4r (a e 1e -)5)

HE- 1 + yiR0 )cos •0 -R12 (

A-3e_____

- 1-~., 2) -* , Cos

... .. ... .h ..

TR 6881

47tR) cs +I~ (1O~o oo R2_ yR,)cos eYR (A)f f-sin

2 01(1S + 15y1Rl + 6y2R2 + y3R3)1

HVM 4t-y 1R 0(3+ y,0+ y2R')sin 00cos ýp R1 0

3R + y1RI 2 sin Olcos ýpje-RI-(3 + 3y 1 1l y2R2)sin ýIo '0, _ - _________

I1 R (y 2 - 2 R

x[(45 + 45yRl + 18y2R2 + 1 0 R1 (A- 17)-~~~~~~~~ Sll3h15+15,, 4y?.3R3 0V

-- -r(-3 sin2 * o)(15 + 10yR .y2R) +. jy3R3 1 eYIRI1t 10 10 1 1 P11

-[( M1 - 3Sin2 *9(1 l +yR y 2R2 +os2 2 COpjýOe -lR

2e 11 Ri 3 (A-18)

(y~- ~)R [9 +- 9y R1 + 4y2R2 + y3R)

+sin 2 p1 (15 + 1Sy1Rl + 6y2R2.+ y3R3)

-sin2

i cos2 ip(105 +- 105Y 1R + 45y2R2 j Oy3R3 +yR

where

2 10R2 p+( .

sin p2 = (z h)R-

osi *0 (z -/o

I p2 + (z + hi) 2 ,

si *1 (z + h)/R1 , and

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