Abstract
An accurate series-form explicit expression is derived for the voltage induced in the small receiving loop of a central loop electromagnetic sounding system positioned above a homogeneous earth model. The solution is obtained by converting the semi-infinite integral representation for the vertical magnetic field induced at the center of the two loops into a series of simpler integrals, amenable to analytical evaluation. The developed formulation is based on replacing the exponential term in the integrand of the field integral with its power series expansion with respect to the difference of the squares of the wavenumbers in air and in the conducting medium. As a result, the vertical magnetic field and the induced voltage are finally expressed as sums of the spherical Hankel functions depending on geometrical parameters and the wavenumbers. The derived solution exhibits advantages in terms of both time cost and accuracy, when compared to the conventional numerical schemes for evaluating the Sommerfeld-type integrals and to the previously published quasistatic solution to the considered problem.
1. Introduction
Central loop electromagnetic sounding is a widespread technique for the exploration of earth structures, which consists of interpreting a set of measurements of the voltage induced in a small receiving loop placed at the center of a larger transmitting loop [1–17]. The underlying concept is that the presence of buried objects or other inhomogeneities in the subsurface structures may be revealed by the discrepancy between the recorded experimental data and the theoretical response curves associated with different earth models [8, 11–13, 18]. This implies that correct data interpretation requires the a priori knowledge of highly accurate theoretical response curves, at least for the standard homogeneous earth model. Yet, despite this necessity, only under the quasistatic field assumption can we derive an accurate analytical solution for the time-harmonic voltage induced in the receiver of a transmitter-receiver loop system located on a homogeneous ground. Thus, when the quasistatic assumption fails, the induced voltage is typically through time-consuming numerical approaches or 3D electromagnetic software tools [5, 13]. An attempt to overcome the restrictions implied by the quasistatic assumption and, at the same time, providing an efficient method to compute the voltage induced in the receiver has been made in [2], where a hybrid analytical-numerical solution valid for the general case of multilayer ground has been proposed. The solution has been shown to be robust and efficient but it suffers from the limitation that its usage requires the a priori computation of a set of numerical coefficients through an expensive iterative optimization algorithm.
The purpose of this work is to derive the exact series-form explicit solution describing the voltage response of a central-loop electromagnetic sounding system positioned on a homogeneous medium. The solution is obtained by converting the semi-infinite integral representation for the vertical magnetic field induced at the center of the two loops into a series of simpler integrals, amenable to analytical evaluation. The developed formulation is based on replacing the exponential term in the integrand of the field integral with its power series expansion with respect to the difference of the squares of the wavenumbers in air and in the conducting medium. As a result, the field and the voltage are expressed in terms of spherical Hankel functions of the wavenumbers and geometrical parameters. The obtained solution is not subject to the quasistatic assumption and is valid in a wide frequency range. Numerical tests are carried out to illustrate the advantages of the proposed approach. In particular, the comparison with the outcomes from the previously published quasistatic solution reveals the failure of the latter even in the low-frequency range. This evidence further confirms the usefulness of the derived expressions for the vertical magnetic field produced at the center of the two loops and the voltage induced in the small receiving loop.
2. Theory
Consider a current-carrying circular loop antenna and a co-axial and co-planar small receiving loop located at height above a homogeneous earth structure as sketched in Figure 1. The dielectric permittivity and electric conductivity of the ground are denoted by and , respectively, while the magnetic permeability is assumed to be constant everywhere and equal to that of free space .
The current flowing in the larger transmitting loop is taken to be time-harmonic and equal to , with being the angular frequency. Because of the symmetry of the problem, currents in the material medium flow only horizontally. This implies that the generated electric field has no vertical component, and as a consequence, the EM field is transverse electric (TE) with respect to the vertical direction (termed as direction in the following) [5]. In this situation, it is well known that the magnetic field generated at the center of the two loops has only a vertical component, whose integral representation reads as [5].where denotes the radius of the emitter and is the th-order Bessel function, and
And with the time-harmonic factor suppressed for notational simplicity. The aim of this paper is to derive exact series representations for . To this goal, it is first convenient to use the differentiation properties of the Bessel functions and the identityso as to decompose (1) into three terms as follows:
The first two integrals on the right-hand side of (5) are special cases of the well-known tabulated Sommerfeld integral [19–23]and as a consequence, it yields the following:with . Next, to evaluate the last integral within the square brackets of (7), the numerator and denominator of the fraction in the integrand are first multiplied by . This leads to express (7) in the following form:with
The explicit expression of may be found straightforwardly, since it is easily recognized to be related to in (6) by the following differential relation:
On the other hand, to evaluate , it is convenient to rewrite it as follows:and then, change the variable of integration from to and express as follows:where and , being . This makes it possible to rewrite as follows:whereand being , , and . Next, since , the exponential term of the integral may be replaced with its power series expansion with respect to in Lommel’s sense, namely,where in the limit as , the quantity within the square brackets may be confused with . This makes it possible to turn (15) into the following:where is the th-order spherical Hankel function of the first kind, which is defined as follows:
Substitution of (16) into (14) provides the expressionwhose integrals on the right-hand side are known in the explicit form and tabulated in [19]. In particular, upon setting , we make use of the well-known result ([19] No. 13.47.2)with being the th-order modified Bessel function of the second kind and , in combination with the identities [24]
allows to obtain the expressionthat is, equivalently,
which leads to express (18) as follows:
with
Making use of (24) into (13) provides the following:and, upon introducing the differentiation properties of the spherical Bessel functions, namely ([24], No. 10.1.23, 10.1.24),it is straightforward to conclude thatwhere the argument of the spherical Hankel functions has been omitted for notational simplicity. Thus, after taking the limit as , it turns out that , , andwhereand with . Use of (10) in conjunction with (31)–(33) into (8) provides, after performing the derivatives of the spherical Hankel functions, the following expression for the field calculated at the center of the loopsbeing
Finally, since the radius of the receiver has been assumed to be sufficiently small, this implies that and ; it turns out that the voltage induced in the receiver is given by (34) multiplied by .
3. Numerical Results
In this section, the results arising from (34) are compared with those provided by the previously published quasistatic solution to the considered problem, namely [5],
which is obtained by assuming that both the emitter and the receiver lie on the top surface of the conducting medium to be explored (that is by assuming ). For the sake of validation of the proposed formulation, the results generated by (34) are also compared with the data arising from numerical integration of (1), performed by using an adaptive Gauss–Kronrod G7-K15 quadrature rule, resulting from the combination of a 7-point Gauss rule with a 15-point Kronrod rule. In particular, after mapping the semi-infinite integration interval onto the finite interval [0 : 1] through a suitable change of variable, the latter interval has been recursively decomposed until the estimates of the relative errors associated with all the sub-integrals, each one given by the relative difference between the Gaussian and Kronrod rule [25], do not exceed a fixed tolerance level.
First, the three different approaches are applied to the computation of the amplitude-frequency spectrum of the field produced at the center of a loop carrying 1 A of current and placed on a clay soil. The index at which the infinite sum in (36) is terminated is taken to be equal to 15, while two distinct values for the radius of the source loop are considered, that is, m and m. Finally, the electrical conductivity and dielectric permittivity of the clay soil are taken to be mS/m and [8], respectively. The obtained curves, illustrated in Figure 2, show how the numerical data agree well with the outcomes of (34) all over the considered frequency interval. Thus, the results from (34) may be referred to as exact curves. Conversely, discrepancy is observed between the quasistatic and exact trends even in the low-frequency range, to a greater extent the larger the radius of the source loop. Moreover, the quasistatic curves start to deviate from the exact ones at the frequencies of about 3 MHz when m, and 300 kHz for m. This means that the extension of the quasistatic frequency range decreases as is increased.
The decrease in accuracy of the quasistatic solution as frequency grows up is further illustrated in Figure 3, which shows the relative error originating from substituting (37) in place of (34)–(36) with , which is again plotted against frequency. Here, the electromagnetic parameters of the material medium are given the same values as in the previous example. From the analysis of the plotted curves, it is found that the relative error increases almost linearly in the low-frequency range, up to the limits of about 100% for m and m, and for m. This implies that the quasistatic solution exhibits a higher accuracy if the radius of the source coil is reduced. Once the upper limit is reached, the error generated by (37) does not change any longer with increasing frequency.
One would ask how quickly (34) converges to the exact solution for different values of the radius of the source loop. This aspect is investigated by Figure 4, which depicts the relative truncation error that results from using (34) with a truncated sum in place of the infinite sum, plotted against with the truncation index taken as a parameter. The operating frequency is assumed to be equal to 100 kHz, and the estimate of the infinite sum is obtained by terminating the sequence of partial sums when the relative difference between the two last partial sums is smaller than . The same material medium as in the preceding examples is considered. A glance at Figure 4 reveals that convergence is in general slower for small but especially for large values of . For instance, if m it suffices to use a sum made up of terms to get an acceptable relative truncation error (), but as is further increased the accuracy of the result of the computation rapidly drops, and it can be enhanced only by increasing the truncation order (up to , as an example). However, the need to resort to longer sums does not imply that the proposed approach is more computationally expensive than numerical integration. In fact, with accuracy being equal, the time cost of the analytical approach is always smaller than that of numerical integration. The validity of this statement is confirmed by the entries of Table 1, which shows the time taken by (34) and the Gauss–Kronrod scheme, on a standard 1.8 GHz processor, to determine the field produced for m and m. The estimate of the relative error generated by Gauss-Kronrod rule and the relative truncation error associated with (34) are also indicated in the third column of Table 1. As can be observed, for m, the use of (34) with ensures approximately the same level of accuracy offered by numerical quadrature, but with significant timesaving with respect to the latter. On the other hand, when m, the relative error associated with is no longer small, and longer sums must be used in order to improve accuracy.
The entries of Table 1 confirm that increasing up to 30 permits breaking down the truncation error and, at the same time, tell us that the improvement in accuracy is not achieved at the cost of significantly worsening the computation time, which is still far smaller than that implied by a numerical scheme which offers the same degree of precision.
It has been shown how increasing the radius of the source loop reduces the frequency range of validity of the quasistatic assumption, and that at higher frequencies, the quasistatic solution fails regardless of the value of . The only exception to this rule occurs when the loops lie on a poorly conducting ground. This point is clarified by Figure 5, which illustrates profiles of the voltage induced in the small receiving loop against the conductivity , calculated by assuming and cm. The operating frequency is taken to be 100 MHz, and the same values for as in the previous example are considered.
As is seen, for larger source loops ( m), there is no chance to obtain accurate results by using the quasistatic solution for the voltage, and this is expected because of the relatively high operating frequency. Conversely, for m, the quasistatic assumption still holds at 100 MHz, provided that the ground conductivity does not exceed few mS/m.
4. Conclusion
The purpose of the present work has been to derive the exact explicit expression describing the time-harmonic voltage response of a central loop electromagnetic sounding system placed on a homogeneous earth structure. The solution has been obtained by turning the semi-infinite integral representation for the vertical magnetic field induced at the center of the emitter into a series of simpler integrals amenable to analytical evaluation. The series expansion of the field integral has been made possible by the replacement of the exponential term of the integrand with its power series expansion with respect to the difference of the squares of the wavenumbers in air and in the conducting medium. As a result, the vertical magnetic field and the induced voltage have been expressed in terms of the spherical Hankel functions of the wavenumbers and geometrical parameters. The results of numerical tests have shown the validity of the proposed approach. In particular, it has been seen how the derived solution exhibits advantages in terms of time cost and accuracy, when compared, respectively, to the numerical quadrature rules devoted to the evaluation of the Sommerfeld-type integrals, and to the previously published quasistatic solution to the problem.
Data Availability
No data are available.
Conflicts of Interest
The authors declare that they have no conflicts of interest.