Scattering of electromagnetic plane waves by a buried vertical dike

The complete and exact solution of the scattering of a TE mode frequency domain electromagnetic plane wave by a vertical dike under a conductive overburden has been established. An integral representation composed of one-sided Fourier transforms describes the scattered electric field components in each one of the five media: air, overburden, dike, and the country rocks on both sides of the dike. The determination of the terms of the series that represents the spectral components of the Fourier integrals requires the numerical inversion of a sparse matrix, and the method of successive approaches. The zero-order term of the series representation for the spectral components of the overburden, for given values of the electrical and geometrical parameters of the model, has been computed. This result allowed to determine an approximate value of the variation of the electric field on the top of the overburden in the direction perpendicular to the strike of the dike. The results demonstrate the efficiency of this forward electromagnetic modeling, and are fundamental for the interpretation of VLF and Magnetotelluric data.


INTRODUCTION
Electromagnetic (EM) wave propagation and EM geophysical methods are employed in mineral, groundwater and petroleum exploration, in shallow geotechnical investigation, or in related subjects such as Global Positioning System (GPS). They are based on the distribution of the EM field components in the ground induced by natural or man-made EM source fields. The electromagnetic description of a medium similar to the earth's crust consists in the determination of the electromagnetic parameters of the medium from the knowledge of the distribution of the components of the 190 LURIMAR S. BATISTA and EDSON E. S. SAMPAIO EM field inside it. Maxwell's and the constitutive equations provide the means for relating those field components and the electromagnetic parameters. Among the electromagnetic parameters the conductivity is the most diagnostic for the rocks of the crust. The analysis of the overall distribution of the charges, the electric currents, and the field components is in general very difficult because of complex geological structures. Therefore there are many unsolved problems related to Maxwell's equation demanding advanced research.
The investigation of the scattering of EM plane waves caused by lateral variation of the properties of the rocks is fundamental for the success of geophysical exploration. (Sommerfeld 1896) and (Wiener and Hopf 1931) employed different techniques to solve the problem of the scattering of EM plane waves in a perfectly conductive half-plane. The need for investigating three-dimensional problems, with a relatively small computation time, led to the development of several algorithms of numerical modelling (Gupta et al. 1987), (Livelybrooks 1993), (Avdeev et al. 1997), and(Zhdanov et al. 1997). Some are flexible, and either compute the wave field at non-uniform sampling intervals, or evaluate the derivatives with unequal deggrees of precision. Others excite the system from different functions, or restrict the information at the boundaries. All of them compromise the accuracy of the final result.
Presently there are very few options available in the literature of EM models obtained analytically. Generally the analytical solutions are related to one-dimensional models of the earth or simple structures as hemispheres or outcropping faults. The exact analytical solution of the scattering of a TE mode EM plane wave by an outcropping vertical fault in the frequency domain was done by (Sampaio and Fokkema 1992). Subsequently (Sampaio and Popov 1997) developed the correspondent analytical solution in the time domain for the zero-order term.
We present the basic formulation and the analytical solution of the scattering of a TE mode EM plane wave in the frequency domain by the following earth model: a vertical dike between two quarter-spaces and covered by an overburden. A preliminary version of this investigation has been presented by (Batista and Sampaio 1999). The analytical tools developed here will also be useful as a check of the efficiency of the numerical techniques of forward and inverse modelling.

The Wave Equation
The application of the concept of an EM plane wave to geophysical problems was proposed originally by (Tikhonov 1950) and by (Cagniard 1953) for the study of the magnetotelluric method. We will follow this concept for the configuration of the primary field.
Let a TE mode EM plane wave, which propagates in the positive z direction, be scattered by a geological structure consisting of a buried vertical dike as depicted in Figure 1. The horizontal layer thickness is h, its electrical conductivity is σ 1 , and it is in contact along the plane z = 0 with an infinitely resistive half-space (z < 0). The vertical dike presents an electrical conductivity σ 3 and it is laterally limited by two vertical half-planes x = −a and x = +a. The two quarter-spaces have, respectively, conductivity σ 2 for x < −a and conductivity σ 4 para x > +a. The horizontal SCATTERING OF EM WAVES BY VERTICAL DIKES 191 layer is in contact with both the dike and the two quarter-spaces along the horizontal plane z = +h. For the TE mode the electric vector is always along the y direction, and for the described model the total electrical conductivity presents a constant and finite value in each medium. So the problem consists in finding the solution for the two-dimensional homogeneous Helmholtz wave equation in each medium: where: • k n = √ −iωµ 0 (σ n + iωε 0 ), is the propagation constant in each medium; ω is the angular frequency; µ 0 and ε 0 are the free-space values of, respectively, the magnetic permeability and the dielectric permittivity; and σ n is the conductivity.
• E n represents the y component of the electric field vector.

Representation of the Fields
The solutions of the differential equation in each medium are as follows ( Figure 2): 1. For −∞ < z < 0: f 0,2 e u 0 x cos(αz) + g 0,2 e u 0 z cos(αx) dα; (2) 2. For 0 < z < h: 3. For h < z < ∞: f 2 e u 2 x cos(αz) + g 2 e −u 2 z cos(αx) dα; Where: E I = e −ik 0 z represents the incident field for z < 0; E R 0,j = R 0,j e ik 0 z , R 0,j are the free-space reflection coefficients; E T 0,j = T 0,j e −ik 1 z , T 0,j are the transmission coefficients from the free-space into the horizontal layer; E R 1,j = R 1,j e ik 1 (z−h) , R 1,j are the reflection coefficients for the horizontal layer relative to medium 2, 3 and 4 respectively; E T j = T 1,j e −ik j (z−h) , T 1,j are the transmission coefficients from the horizontal layer into medium 2, 3 e 4 respectively; and u n = α 2 − k 2 n , represents the wave number in the transformed space.
From Equations (36) and (37) we obtain, Multiplying Equation (38) by W −1 (α), we obtain, Notice that W −1 (α) represents the inverse matrix, obtained by a straightforward operation; ψ(α) = W −1 (α) · y(α) represents the vector of the independent constants; and K(α; ξ) = W −1 (α) ·M(α; ξ) is a sparse square matrix, denominated nucleus matrix. The integral equations of the system (39) are classified as Fredholm singular integral equations of the second kind. The regular part, ∞ 0 K(α; ξ)φ(ξ)dξ is a Riemann improper integral, possessing a finite value. So, the solution of the equation (39) can be obtained employing the method of successive approximations (Kondo 1991): and Where: , for n > 1, and o K 1 (α; ξ) = K(α; ξ). The elements of the vector φ n (α) are the spectral components used to calculate the scattered electric field in the nine domains of the model under study.
It is sufficient the series (40) be convergent in order for A necessary but not sufficient condition for the convergence of the series is that lim m→∞ φ m (α) = 0. (Sampaio and Fokkema 1992) verified numerically that the terms of the series decrease and the series remains bounded for up to five terms for the model of a vertical fault, and we expect the model of the buried dike to present a similar behavior. However the rigorous proof of the convergence of the series is a difficult mathematical problem and we lack such a proof.

Basic Concepts
The values of frequency and the values of the earth's crust conductivity employed in geophysics are such that the modulus of the constant of propagation of the subsurface is always much larger

Computation and Representation of the Normalized Electric Field
After the determination of the spectral components and the check of the convergence of the integrals we computed the normalized electric field, E |k 3 | H 0 ωµ 0 , at the surface of the earth for the proposed dike model, employing Equations (5), (6) and (7). We employed different geoelectric parameters to simulate distinct models.  Figure (4) shows that as (p 1 → ∞) the contacts between the dike and the surrounding rocks are very well defined at |x| = a. In the limit the model becomes that of an outcropping dike. On the other hand, as p 1 → 0, which in the limit means either an infinite thickness of the overburden or a zero width of the dike, the same contacts become, as expected, progressively not well defined. Figure (5) displays the variation of the real and the imaginary parts of the normalized electric field on z = 0 as a function of |k 3 |x, for the vertical dike model with the following parameters: variable conductivity of the surrounding rocks and the dike; σ 1 = 0.05 S/m; overburden thickness such that h = 0.02|k 3 |; and width of the dike such that 2a = 0.4|k 3 |. The electric field has been computed for the following values of p 2 = σ 3 /σ 2 : 0.1; 1; 10; 100; and 1000. We observe in Figure (5) that, as expected, the behaviour of the curves for p 2 = 0.1 is the opposite to the curves for p 2 > 1. The real part curves for p 2 > 1 reach a maximum value at x = 0 and have an inflection point at |x| = a. For p 2 = 0.1 these curves reach a minimum at x = 0 and they also have an inflection point at |x| = a. The behaviour of the imaginary curves is exactly the opposed. Figure (6) displays the variation of the real and the imaginary parts of the normalized electric field on z = 0 as a function of |k 3 |x, for the vertical dike model with the following parameters: variable conductivity of the overburden and the dike; overburden thickness such that h = 0.02|k 3 |; dike width such that 2a = 0.4|k 3 |; and σ 2 = σ 4 = 0.0005 S/m. The electric field has been computed for the following values of p 3 = σ 3 /σ 1 : 0.1; 0.4; 1; 10; and 100. We observe that in the real part curves the contacts between the dike and the surrounding rocks are well defined for p 3 > 1, whereas for an overburden more conductive than the dike they are not.
However this distinction does not happen in the imaginary part curves. For them the contacts are always well defined. The maximum value is observed at x = 0 for p 3 ≥ 1, whereas for p 3 < 0.1 the position of the maximum is not well defined.

CONCLUSION
The complete and exact algebraic solution of the scattering of a monochromatic EM plane wave, for the case of a vertical dike immersed in two quarter-spaces and overlaid by a horizontal layer, was determined. As a first step, zero-order terms of the series representation of the spectral components were selected to compute an approximate value of the electric field above the vertical dike.
The results of the modeling show that: 1) for the outcropping dike, h → 0, the contacts between the dike and the surrounding rocks are very well defined; 2) for a large thickness of the overburden, h → ∞, the contacts between the dike and the quarter-spaces are not defined; 3) when the conductivity of the overburden is much larger than the conductivity of the dike, σ 1 >> σ 3 , it is not possible to define the contacts between the dike and the surrounding rocks employing the real component of the electric field, but the contacts are defined through the imaginary component.
The results show the precision of the zero-order terms in the calculation of the secondary electric field. So they can be used in the computation to substitute other techniques with advantage. Therefore the expressions of the analytical solution of the electric field are fundamental for the interpretation of magnetotelluric, or VLF (Very Low Frequency) data, associated to the geophysical exploration.

ACKNOWLEDGMENTS
We acknowledge our fellowships and the grant from CNPq (Brazilian National Science Foundation). We thank the fruitful comments from Prof. M. Popov. We also thank the help from Mr. J. Lago in the preparation of the manuscript.