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Keywords:

  • ionospheric ridge;
  • ELF scattering;
  • underground imaging

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References

[1] In the present work the variation in ionospheric height observed during the dawn or dusk hours is used to develop an electromagnetic imaging system for extremely low frequencies (ELF). The description concerns the mathematical principles of the proposed method rather than a realistic implementation of the technique. The scattering of ELF waves emanating from a point source on the Earth surface by a cylindrical underground formation is studied analytically with use of a Green's function method. The ionospheric discontinuity is treated with the help of mode matching technique and the subterranean scatterer by application of the method of auxiliary sources. Numerical results are presented and discussed for several cases.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References

[2] The study of electromagnetic scattering by buried obstacles has attracted much attention for many years. Especially when the scatterer is located beneath the ground, the theoretical analyses can have numerous practical applications. Howard [1972] determined the field developed by a subterranean cylinder under external excitation. Carin et al. [2002] implement an algorithm to reconstruct material profiles of dielectric objects buried in a lossy Earth using transmitters and receivers placed in the air. Also, Cui et al. [2001] present the main issues concerning microwave subsurface sensing and provide examples relevant to the sensing of land mines.

[3] Another popular topic that pertains more to geophysics than to electromagnetics is the effect of the ionosphere on wireless terrestrial communications. At low operating frequencies the ionosphere behaves like a good conductor and therefore a waveguide is formed between it and the conducting Earth's surface [Pappert, 1989]. The models describing the propagation of electromagnetic waves inside this structure have been reviewed by Cummer [2000], while the presence of an ionized column into the aforementioned waveguide is investigated in [Wait, 1991]. Moreover, the radiation of a dipole between Earth and ionosphere is treated by Barrick [1999] with use of simplified boundary conditions.

[4] The ionosphere possesses an interesting property not extensively examined in the above references: its height varies during the transition from day to night (dusk) and vice versa (dawn) [Uberall and Seaborn, 1982; Surana and Williams, 2003]. This height variation could be seen as a natural scanner for the substrate because the ridge of the ionosphere is moving as time goes by. In particular, an image of the ionospheric discontinuity is created inside the conducting ground and could provide information about subterranean inhomogeneities. The coupling between sliding ionosphere and subterranean formations is examined in this work. The motivation is similar to that of Nam et al. [2007] where the magnetotelluric method is investigated.

[5] In the present study a two-dimensional, double-layered plane model is considered and an abrupt variation in ionospheric altitude is assumed. A cylindrical formation is buried inside the lossy Earth and scatters the field produced by a dipole source located between the two layers: air, ground. As the longitudinal dimensions are infinite, a study for the two-layered, parallel-plate waveguide is carried out in the first place. After the derivation of the guiding condition, the propagation constants are determined and the orthogonality of the supported modes is proved (section 3).

[6] The Green's function of the problem (for subterranean sources) is essential to proceed further to the solution. The term indicating the influence of the horizontal layers is obtained with use of spectral integrals satisfying the Helmholtz equation. The other term of the Green's function (expressing the effect of the ridge) is found by implementing the mode matching technique along the step discontinuity plane [Mahmoud and Real, 1975]. The elements of the constant vector of the linear system are rapidly converging spectral integrals, while the singular part of the Green's function is taken to be common for both sides of the discontinuity. The mode matching technique is utilized for the incident field as well but in this case the direct evaluation of the integrals is not possible (due to the position of the source) and thus residue theorem is employed instead (section 4).

[7] The scattering by the buried cylinder is treated with use of the method of auxiliary sources (MAS) [Leviatan et al., 1983; Uberall et al., 1987] which is usually applied for two adjacent regions. The scatterer itself is considered as the one region and all the remaining area as the other. The Green's function for the ridged ionosphere accompanied by ground with finite conductivity, is essential to implement the method for the cylindrical scatterer formation. To this end, a point matching technique is applied for the manipulation of the boundary conditions (section 5).

[8] Numerical results are presented concerning the scattered field measured by a receiver stationed on the Earth's surface. The measurements are taken periodically and correspond to different positions of the ionospheric discontinuity. In many cases the detection of the obstacle is possible and its location is revealed by the local maxima or minima of the curves. Several conclusions are drawn related to the behavior of the scattering field distribution for different observation points, operating frequencies, radii and conductances of the scatterer (section 6).

2. Problem Formulation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References

[9] The structure under study is shown is Figure 1 where the equivalent Cartesian (z, x, y) and cylindrical (ρ, equation image, y) coordinate systems are also defined. The ridged two-dimensional ionosphere is taken to be perfectly conducting (PEC) because of the supposed low frequencies. The conductivity of the ground medium is finite, while a PEC plane is placed at sufficiently large depth x = −L. These assumptions result into the discretization of the involved modes. The air medium (area 0) possesses a wave number k0 and an intrinsic impedance ζ0. The lower layer's (area 1) corresponding parameters are denoted as (k1 = k0equation image, ζ1 = ζ0/equation image), where ε1 is the dielectric constant of the Earth's ground (with relatively high losses). The height of the waveguide changes abruptly at z = d from (L + H) to (L + h) with H > h. The ridged PEC surface represents the day-night discontinuity of the ionosphere.

image

Figure 1. Geometry of the Earth-ionosphere model. Ionosphere is assumed to be a perfect conductor with an abrupt change in its height. The source is placed on the air-ground surface, and the developed field is scattered by the underground circular cylinder.

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[10] A cylindrical homogeneous scatterer with a dielectric constant equation image2 and radius a is located inside the Earth close to the plane z = d of the ionospheric discontinuity. The cross section of the scatterer is called area 2 with parameters (k2 = k0equation image, ζ2 = ζ0/equation image). The distance between its center and the Earth surface equals W with W > a. Both W and (La) are chosen having substantial values, namely the scatterer is not too close to the bounds of area 1. The discontinuous waveguide is excited by an infinite, parallel to y axis magnetic dipole of constant magnitude V Volts located slightly above the internal bound (inside vacuum area 0) at horizontal distance D from the x axis. A magnetic source is preferred against an electric one whose influence would be very weak due to high losses of area 1 and the developed opposite image inside it. Given the fact that both configuration and excitation are invariant with respect to y coordinate, the same should happen for all the investigated quantities and thus only three field components will be nonzero in the treated problem. The single magnetic component is the axial Hy and the nonvanishing electric components are the polar ones (Ez, Ex) or alternatively (Eρ, Eϕ). Throughout the analysis a time dependence of the form e+t is used and suppressed where ω = 2πf is the extremely low operating circular frequency. Because of the adopted time dependence, the propagation loss is indicated by the negative sign of the imaginary part of the dielectric constant, that is equation image1] < 0. The main purpose of this work is to observe the field produced by the cylindrical inhomogeneity on the upper side of the air-ground surface.

[11] Many quantities throughout the analysis can be written with a superscript time suggesting the side of the structure they are referred to. If time = night the quantity corresponds to the region z < d and in case time = day, to the region z > d. A subscript i separates horizontally the structure. When i = 0 the air region is observed and if i = 1, the quantity is related with ground's region. Absent superscript means that the quantity concerns the total problem and functions with no subscript represent field solutions for both areas.

3. Reduced Problem

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References

3.1. Supported Modes

[12] Consider a parallel-plate waveguide with infinite dimensions whose height is constant and equals (L + Y). It is a continuous structure as these of Figure 1 with Y = H, h. The approach is two-dimensional and thus y coordinate is suppressed and propagation across the z axis is assumed with a z-dependent factor ez. By supposing an x-dependent component which satisfies the homogeneous Helmholtz equation and imposing the suitable boundary conditions, one concludes to the following transcendental equation for the unknown β:

  • equation image

The radiation functions in each area are denoted by μ0(β) = equation image and μ1(β) = equation image. These functions are even with respect to β and the same happens for the complex roots of (1). Each one of them corresponds to a supported propagation constant by the investigated device.

[13] An analytical solution for (1) is not available and hence an iterative method should be used to specify numerically the complex propagation constants. By taking into account that the Earth-ionosphere structure can be approximated by a parallel-plate waveguide with perfectly conducting sidewalls, it is sensible to suppose that β is close to k0 [Collin, 1991]. Since the magnitude of equation image1 is large compared to unity, a reasonable approximation is obtained:

  • equation image

With use of (2), the transcendental expression (1) yields to the following quadratic equation with respect to μ0(β):

  • equation image

for each integer n. Owing to the adopted time dependence, the propagation constants should lie on the second or forth quadrant of the complex plane. Therefore the radicand of μ0(β) possesses a negative imaginary part and μ0(β) itself belongs also to the second or forth quadrant of the β plane. The first solution set of (3) is rejected for not complying this demand. The second series of solutions is accepted and given (as function of the upper plane's height Y) by

  • equation image

where n is positive integer (or zero). The notation equation imagen(Y) is used for the initial guesses of the roots existing on the forth quadrant, namely equation image[equation imagen(Y)] > 0 and equation image[equation imagen(Y)] < 0. Note that for β = −equation imagen(Y) the characteristic equation (1) is also satisfied because of the symmetry of the continuous structure.

[14] Once the exact values of the supported propagating constants bn(Y) are found, the x-dependent components of the field modes can be derived. They satisfy the homogeneous Helmholtz equation in each area and for the case of the single axial magnetic field (Hy) are given by:

  • equation image

where μ0n(Y) = equation image, μ1n(Y) = equation image and n is positive integer (or zero). The normalization constant Qn(Y) is written as follows:

  • equation image

The modes of the electric component Ex, which is parallel to the discontinuity plane are defined as:

  • equation image

By means of direct integration, the orthogonality between f and g modes is verified:

  • equation image

where δnm is the Kronecker's delta. If the two sets of modes correspond to waveguides of different heights, i.e., y and Y with y < Y, then the new product is given by:

  • equation image

The derivation of (9) is achieved by taking into account that bn(Y) and bm(y) are roots of (1) for Y = Y, y.

3.2. Green's Function of Continuous Structure

[15] Because of the nature of the investigated problem, the scalar Green's function of magnetic type is necessary to be found. Such a function equals the axial magnetic field produced by an infinite dipole of constant current with magnitude 1/k1 in Volts [Tai, 1994]. This is the so-called “Green's source”, an elementary magnetic current positioned across the axis (z′, x′) inside area 1 where the scatterer exists. The upper height of the structure equals Y and it can take any value corresponding to the arbitrary superscript time. In addition, all the quantities in this section are written with a bar accent equation image indicating the continuous configuration of the problem. Given that area 1 contains the magnetic source, the local Green's function owns the form:

  • equation image

where equation image1,prim(z, x, z′, x′) is the primary excitation (independent of Y) and equation image1,sectime(z, x, z′, x) the secondary response.

[16] The primary field developed due to the elementary magnetic current source, possesses the following expression [Collin, 1991]:

  • equation image

with ℜ[μ1(β)] > 0 and in case it is zero equation image[μ1(β)] > 0. The symbol H0(2) is used for the Hankel function of second kind and zeroth order. The secondary quantities satisfy the homogeneous Helmholtz equation. These functions are written in terms of the well-known integral representation [Collin, 1991]:

  • equation image

The coefficients C1(β) and C2(β) are arbitrary complex functions and μ(β) denotes the radiation function of the area the quantity is referred to. The necessary boundary conditions are enforced to give the following forms:

  • equation image
  • equation image

The behavior and the numerical features of the integrands will be discussed later.

4. Complete Problem

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References

4.1. Green's Function of Discontinuous Structure

[17] In this section is examined the influence of the abrupt discontinuity of the upper wall on the field quantities. All the related functions are written with a hat accent equation image indicating the noncontinuous transition. The analysis aims at specifying that part of the Green's function owed to the vertical conducting edge at z = d:

  • equation image

for i = 0, 1 and time = night, day. The magnetic fields emanating from the discontinuity edge are written as infinite sums of modes with outgoing propagation directions from it:

  • equation image
  • equation image

The subscripts are not shown because the functions fn(x, Y) are defined for each x ∈ (−L, Y) and consequently the formulas above are valid for both horizontal regions.

[18] The coefficients equation imagentime(z′, x′) with “time” = “night/day” are unknown and can be found via the mode matching technique. This procedure concerns the projection of the boundary conditions on the set of the field eigenfunctions. In this case, the continuity of the tangential magnetic (Hy) and electric (Ex) components will be enforced across the discontinuity plane {z = d, −L < x < H}. In particular, the magnetic boundary condition is imposed just for −L < x < h as the vertical edge across H > x > h is perfectly conducting. For the same reason, the tangential electric field of the left side equals partially to zero. Furthermore, the electric field is written as a weighted series of gn(x, Y) functions. To manipulate numerically the expressions, the sums are truncated by keeping the first (N + 1) terms. In order to exploit the orthogonality property suggested by (8), the magnetic boundary condition should be projected on a gm(x, Y) set and the electric one on a fm(x, Y) set for m = 0, ., N. The suitable argument Y is found from the interval of x that each equation is referred to. With use of (9), the following (2N + 2) × (2N + 2) linear system is educed.

  • equation image
  • equation image

for m = 0, ., N. The behavior and the numerical features of the integrands will be discussed later.

4.2. Incident Field

[19] Apart from the Green's function, another quantity necessary for the manipulation of the problem is the incident field. The notations of the previous analysis are kept unchanged (subscripts i = 0, 1, superscript “time” = “night/day”, accents equation image, equation image). The excitation is the same with the Green's function problem (the cylindrical scatterer is again absent) and thus a similar method is followed. The source in this case is positioned into vacuum area 0 at (z′, x′) = (D, W) and the primary field is given by

  • equation image

With use of the integral representation (12) and by enforcing the proper boundary conditions, one receives the expressions of the single magnetic component for the incident field in each area.

  • equation image
  • equation image

[20] The direct evaluation (through numerical integration) of (21), (22) for x = W is not possible as w(β, Y) ∼ (1 + 1/ε1)∣β∣, β [RIGHTWARDS ARROW] ±∞. Therefore the residue theorem should be utilized instead. The integrands become singular only when (1) is satisfied, as the zeros of the other factors of the denominators are neutralized by the poles of w(β, Y). In addition, they have no branch cuts because of the finite size of the layers [Schevchenko, 1964]. The two alternative integration paths are depicted in Figure 2 according to the sign of the quantity (zD). The incident fields of the continuous problem in series form are written as follows:

  • equation image
  • equation image

where the prime in w′(β, y) represents the derivative with respect to β. Coincidentally, the two expressions above can be combined to one series of modes (because the dipole is located between the two horizontal layers):

  • equation image

where

  • equation image

On the point (z, x) = (D, W), the exponent is nullified, the series (25) diverges and the value of the field increases unboundedly to give the singularity of the excitation source.

image

Figure 2. Contour integration on the complex β plane followed for the computation of the incident field in series form. For the points on the right of the source the circumvention is made through the lower semicircle. For these on the left the integration path occupies the upper half plane. The two sets of poles (marked by X) are opposite as the guiding condition is even with respect to β.

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[21] In order to determine the incident field created because of the discontinuity edge, the mode matching method will be implemented once again. The response fields are denoted as equation imagey,inctime(z, x) and equal to sums of modes with coefficients equation imagentime. After the truncation of the series (25) (by keeping (M + 1) terms), another (2N + 2) × (2N + 2) linear system is educed.

  • equation image
  • equation image

where sgn(z) is the sign function and m = 0, ., N. The constant vector of the system is easily computable this time as it is expressed in terms of the product function Pmn(Y, y) of (9). Once the solution is found, the form of the following incident field owed to the discontinuity is derived.

  • equation image
  • equation image

4.3. Computational Issues

[22] The derived Green's function of the Earth-ridged ionosphere structure is defined by different formulas for z > d and z < d (ionospheric height Y = H, h) satisfying different Helmholtz equations for −L < x < W and W < x < Y (subscript i = 0, 1). On the contrary, the expression is uniform with respect to (z′, x′). As far as the linear system of (18), (19) is concerned, its matrix is numerically invertible because all matrices produced by similar projection procedures do so [Jones, 1964]. To this, stability also contributes the proper scaling of the elements of the matrix acquired from the shift of the modes' exponent by the quantity bn(Y)d.

[23] The constant vector of the system contains semi-infinite integrals of the functions equation imagemM(β, x′), equation imagemE(β, x′) multiplied by the harmonic components cos(β(dz′)), sin(β(dz′)), respectively. These integrals should be computed numerically due to the complexity of their integrand functions (with a sampling of B points for the integrands across the integration interval). The oscillations prescribed by the harmonic components are not very rapid as the cylindrical scatteter is posed close to the discontinuity plane and thus ∣z′ − d∣ is moderate. Furthermore, the integrands exhibit no numerical singularities on the real β axis. As the integrals are of infinite interval, they should be truncated by selecting a suitable upper limit βmax. This parameter is chosen by inspection of the asymptotic expressions of the integrand function as β [RIGHTWARDS ARROW] +∞. By replacing the hyperbolic harmonic functions with exponential components cosh((β)), sinh((β)) ∼ e/2, β [RIGHTWARDS ARROW] +∞ for A > 0, the required expressions are derived:

  • equation image
  • equation image

Given that −L < x′ < W (area 1), the integrands above converge exponentially at least as fast as e−β(HW), e−β(hW) do, respectively.

[24] In evaluating the total Green's function, one should not only find the field developed due to the vertical PEC edge. The excitation quantity of the continuous problem given by (13), (14) is also necessary. The functions are defined by infinite spectral integrals and their integrands should be expanded asymptotically as β [RIGHTWARDS ARROW] +∞ in the same way. The expression corresponding to the area 0 is simplified as follows:

  • equation image

The function is decaying exponentially for all the combinations of (x, x′) except for the case that both equal W. The respective expression for area 1 is given by:

  • equation image

Only two cases are not numerically effective: both source and observation point on the upper (x = x′ = W) or lower (x = x′ = −L) bound of the area. Even though the field on the internal bound x = W is of interest, the obstacle is not very surficial and usually the numerical evaluation is possible. Because of the harmonic component cos(β(zz′)), the integrands are oscillating rapidly for points too far from the scatterer but these cases do not concern this work.

5. Underground Formation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References

[25] The method of auxiliary sources is a numerical technique suitable for a variety of scattering problems [Uberall et al., 1987]. The boundary conditions are imposed only on a discrete number of points of the physical boundaries, called collocation points. The scattering fields in each area satisfy the Helmholtz equation because they are written as weighted finite sums of the local Green's functions. These functions correspond to auxiliary sources posed on a set of points. The sources are located outside the area whose field they generate so that the investigated quantities inside it are smooth as a result of scattering. Sometimes it is convenient to group adjacent regions and to employ the modified Green's functions which incorporate the additional boundary conditions [Leviatan et al., 1983]. If the forms of the boundary quantities from both sides of the physical surfaces coincide being converging with increasing number of auxiliary sources, then the approximate field is very close to the unique solution of the problem.

[26] The method of auxiliary sources will be applied by regarding the cylindrical scatterer as one region and the remaining area (stepped discontinuous waveguide with two layers) as another. That is why the expression of the Green's function G(z, x, z′, x′) determined in sections 2–4 is required. The single magnetic component in the remaining area (superscript G in parentheses) is written as:

  • equation image

where (zGu, xGu), u = 0, ., (U − 1) are the auxiliary points of the sources generating the field of the remaining area. They are not contained in it but are located into the cross section of the cylindrical scatterer (which has been conditionally removed). It is obvious from (35) that the computation of the field outside the scatterer requires repeated implementation (for each auxiliary source) of the spectral integration and the mode matching procedures described previously. In the same way, the single axial magnetic component inside the dielectric obstacle (superscript g in parentheses) is given by

  • equation image

where the quantity g(z, x, z′, x′) represents the Green's function of the bound-free area in case it is filled by the material of the scatterer.

  • equation image

The auxiliary sources generating the field inside the underground formation are posed on the points (zgu, xgu), u = 0, ., (U-1) located outside of the cross section of the scatterer. The coefficients CGu and Cgu can be determined through the enforcement of the boundary conditions.

[27] Owing to the circular shape of the cylinder, the auxiliary surfaces (lines in two dimensions) are chosen to be circles with radii ag > a for the field inside the formation and aG < a for the field in the remaining area. The auxiliary points are distributed uniformly on the circles as in Figure 3. For the external circle z2 + x2 = ag2 the points are defined by zgu = ag cosequation image, xgu = ag sinequation image. For the internal circle z2 + x2 = aG2 the auxiliary points are given by zGu = aG cosequation image, xGu = aG sinequation image with u = 0, .(U-1). The continuity of the tangential field components to the scatterer's bound z2 + x2 = a2 will be imposed on discrete equispaced points zu = a cosequation image, xu = a sinequation image. The azimuthal electric field (Eϕ) is given in terms of the single magnetic component (Hy) by:

  • equation image

where ε is the local dielectric constant and ϕ = atan2(x, z). The coefficients CGu and Cgu of (36), (35) will be determined by the following 2U × 2U linear system:

  • equation image
  • equation image

for v = 0, ., (U − 1). The circles with radii aG and ag are placed far enough from the actual surface to obtain smoother results and better fitting of the boundary quantities. Simultaneously, aG, ag are chosen close enough to a for the sake of stability of the linear system above. From the convergent formulas satisfying the boundary conditions, one can evaluate the produced field on the observation point at horizontal distance zo from x axis (on the ground surface x = W).

image

Figure 3. Circular surfaces for the method of auxiliary sources. The field inside the scatterer is produced by a set of the Green's sources (in the free space with dielectric constant equation image2) uniformly distributed across the circle of radius ag > a (not necessarily inside the bottom layer). The field outside the scatterer is given by a sum of the Green's functions (of the remaining area) whose sources belong to the circle of radius aG < a.

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6. Numerical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References

[28] The basic purpose of this section is to present and interpret the behavior of the scattering field as the time goes by, namely as the position of the vertical discontinuity d varies. In particular, the electric component normal to the air-ground boundary (Ex) is measured by a static observer at x = W. This is feasible if one places a short vertical dipole on the specific point of the Earth's surface when the transition between day and night occurs (during dawn and dusk hours). The heights of the ionosphere during daytime and nighttime have been found approximately equal to h = 60 km and H = 120 km, respectively [Wait, 1991]. Given the large size of the structure, two extremely low frequencies are considered f = 100, 1000 Hz, while the lower restricting PEC surface is posed at a distance L = 25 km where the field is in all cases negligible. The relative complex dielectric constant of the ground is written as equation image1 = 4 − jequation image with σ1 = 10−3 S/m [Cummer, 2000]. The conductivity of the scatterer can be either smaller σ2 = 10−4 S/m or larger σ2 = 10−2 S/m than this of the surrounding environment, while ℜ[ε2] = ℜ[ε1]. The formation is buried at a fixed depth W = 5 km and the source is located relatively close to it at D = −2 km. Two different radii for the cylinder are supposed a = 1, 2 km and the observation point can be placed either above (zo = 0) or away (zo = −4 km) from the obstacle.

[29] The choice of the truncation parameters used in the aforementioned analysis is of prime importance. When one develops a program implementing the described method, one should first determine the truncation limit βmax for the integrals (13), (14), (18), (19). In all the following examples, convergence of these integrals is achieved for the required values of geometrical parameters. Even when the obstacle is quite surficial, a choice of βmax = 50 k0 is adequate, accompanied by a number of points B (for numerical integration) equal to 100 per k0 of integration interval βmax. The dimension N of the mode matching systems (18), (19) and (27), (28) indicating how many modes represent the field radiated from the ionospheric ridge is another issue. In this case not only convergence of the waveforms but also coincidence of the boundary field quantities from both sides of the discontinuity is demanded. In the considered models N = 30 modes describe sufficiently the response of the ridge. As far as the truncation limit M for the incident field is concerned, it increases as the distance between the plane z = d and the source decreases. Of course, the choice d = D should be avoided because the singularity ruins the mode matching procedure. For all the other cases M = 100 terms at maximum are enough for evaluating (29), (30). Finally, the number of auxiliary sources U participating in (35), (36), (39), (40) is critical for the implementation of the method. In all the suggested examples U = 30 sources are adequate for excellent fitting between external and internal tangential field on the scatterer's surface. The values of aG, ag were varied as a check for the convergence of wave shapes.

[30] In Figure 4 the case of low-frequency f = 100 Hz for a scatterer with radius a = 2 km is examined. The independent variable is the horizontal position of the ridge and varies from d = −5 km to d = 5 km (as in all figures). The values of the field correspond to the measurements of a fixed receiver which collects data periodically (during dawn or dusk zone). The represented quantity is the normalized magnitude of the scattered electric field, namely each series of measurements is divided by a different normalization constant (the larger sample of it). Four such series matching to different combinations of observation points zo and conductivities σ2 are presented. An estimation for the incident field can be made through the well-known dimensions, the given materials' characteristics and the easily determinable position of the ridge (by inspection). Therefore the separation of the scattered fields from the total quantities is possible. One can observe that in all cases a maximum is exhibited at the position of the scatterer (d = 0), a result that can have practical sensing applications. A significant parameter for determining the location of the obstacle is the variation of the quantity (the normalized difference between the maximum and the minimum measured value) as d changes. For the case under investigation the variations are small and mainly kept below 8%. Only for the scatterer with large conductivity with zo = 0 can reach 20%. With increasing conductivity, the rod is natural to make its presence more effective. In the limit, the field developed from a PEC cylinder is powerful because of the singularity of the secondary sources.

image

Figure 4. Normalized magnitude of the vertical scattered electric field as function of the horizontal algebraic distance of the discontinuity edge for different observation points and different conductivities of the scatterer. The operating frequency equals f = 100 Hz and the radius of the cylinder is a = 2 km.

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[31] In Figure 5 the same quantities are presented for the same operating frequency but with a rod of half radius. The parameter d belongs to the same interval. For this smaller cylinder the variations are restricted even more (below 9% in any case). This is natural as the scattered field is proportional to the size of the scatterer. This hindrance in locating the target is compensated by the sharper change of the observed quantity which can reveal the position of the obstacle.

image

Figure 5. Same quantities as in Figure 4. The operating frequency equals f = 100 Hz and the radius of the cylinder is a = 1 km.

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[32] In Figure 6 the radius of the cylinder is a = 2 km but the frequency f is ten times larger. An apparent difference is that minimum instead of maximum field is recorded on the position of the cylinder when the observer is standing above it. This local behavior can be used similarly as the location is again indicated but in an inverse way. By inspecting Figure 6 a conclusion valid also for Figures 1–5 can be drawn. When the observation point is away from the scatterer, the variations in measured field are significantly diminished. For example in Figure 6, the fluctuations for zo = −4 km are kept below 20% and for zo = 0 possess values over 35%. This is a weak point of the proposed method as the detection is possible only if the receiver is positioned horizontally close to scatterer. The field behavior is determined by the excited mode amplitudes of the guided modes in the Earth-ionosphere waveguide and because of this strong mode competition, interference phenomena take place. Such an argument explains the complex behavior of the fields and their dependence to oscillation frequency.

image

Figure 6. Same quantities as in Figure 4. The operating frequency equals f = 1000 Hz and the radius of the cylinder is a = 2 km.

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[33] In Figure 7 the case of a cylinder with a = 1 km is considered for operating frequency f = 1000 Hz. Similar to Figure 6 the fluctuation of the electric field is larger compared to Figures 4 and 5 because of the higher frequency. The electrically larger the cylinder gets, the more significant is the developed scattering field. Especially for the obstacle with higher conductivity, a very large variation of 55% is observed which is easily recordable by common devices.

image

Figure 7. Same quantities as in Figure 4. The operating frequency equals f = 1000 Hz and the radius of the cylinder is a = 1 km.

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7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References

[34] The analysis and the obtained numerical results demonstrate the possibility of developing an imaging method for the underground environment based on the measurement of scattered extremely low frequency fields by underground formations. The “sweeping” phenomenon of the image of the ionospheric ridge during day-night and night-day transition affects to a significant degree the scattered field values providing the opportunity of formulating such a sensing method. The variation of the scattered fields during the transitionary period of ionosphere is the proposed basic mechanism that could be used to collect electromagnetic data from the underground formations. The adopted model is two-dimensional and aims to develop the proof of concept. The Green's function of the Earth-ridged ionosphere structure is computed by using analytical methods and the effect of the inhomogeneities inside the Earth is analyzed with use of the technique of auxiliary sources.

[35] The preceding method can be easily generalized to include investigation of cylinders with arbitrary cross section or with many layers. Also a more realistic model can be chosen for the discontinuity (e.g., ramp model) as the transition from day to night is not abrupt. Finally, a more complicated configuration taking into account the local curvature of the Earth could be assumed instead of using the planar structure.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Problem Formulation
  5. 3. Reduced Problem
  6. 4. Complete Problem
  7. 5. Underground Formation
  8. 6. Numerical Results
  9. 7. Conclusions
  10. References
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