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RADIO INTERFEROMETRY DEPTH SOUNDING: PART I-THEORETICAL Dlscusslont

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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 (4 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 <z <0 is a low-loss dielectric slab, and the the detailed work in the mathematical develop- region z < -d is a half-space of arbitrary electrical ment of these solutions is contained in the above properties. These regions are denoted 0, 1, and 2, references, the discussion of the solutions that fol- respectively. K, and M, are the complex dielectric low will be primarily aimed at the radio inter- constant and relative permeability of each region, ferometry application rather than the mathemati- respectively. For consideration of vertical dipole cal manipulations required to obtain them. sources, the dipole moments are taken aligned with the z-axis; for the horizontal dipole sources, THEORETICAL BOUNDARY-VALUE PROBLEM the dipole moments are taken aligned parallel to Although the various solutions of the boundary- the x-axis. value problem for horizontal and vertical electric The solutions are most conveniently written and magnetic dipole sources over a two-layer and discussed using the electric and magnetic earth appear in the literature, a complete and Hertz vector potential notation. On the assump- consistent tabulation of the solutions does not. tion of a time dependence ePiwtand linear consti- Therefore, the boundary-value problem is out- tutive equations in Maxwell’s equations, one ob- lined here, and a unified notation is used to ex- tains the following expressions for the electric and press the solutions. This consistent notation is of magnetic fields in terms of the Hertz vectors. For considerable help in later discussions of the solu- electric dipole sources, the electric Hertz vector tions. satisfies the Helmholtz equation, L 2 = h--~----~---~--_____ Sourct I 1 Region0 KO=MO=l , /- Region1 Kl, Ml X Z = -d\ 7 Region2 K2, M2 Downloaded 12/13/19 to 198.120.252.61. Redistribution subject SEG license or copyright; see Terms of Use at http://library.seg.org/ FIG. 2. Geometry of the boundary-valueproblem for a two-layer earth, showingnotation used. 560 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 di- pole 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.
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