Quasi-Static Solution for ELF/SLF near-Zone Field of a Vertical Magnetic Dipole near the Surface of a Lossy Half-Space

Dipole antennas over the boundary between two different media have been widely used in the fields of geophysics exploration, oceanography, and submerged communication. In this paper, an analytical method is proposed to analyse the near-zone field at the extremely low frequency (ELF)/super low frequency (SLF) range due to a vertical magnetic dipole (VMD). For the lack of feasible analytical techniques to derive the components exactly, two reasonable assumptions are introduced depending on the quasi-static definition and the equivalent infinitesimal theory. Final expressions of the electromagnetic field components are in terms of exponential functions. By comparisons with direct numerical solutions and exact results in a special case, the correctness and effectiveness of the proposed quasi-static approximation are demonstrated. Simulations show that the smallest validity limit always occurs for component H2z, and the value of k2ρ should be no greater than 0.6 in order to keep a good consistency.


Introduction
e propagation properties of extremely low frequency (ELF: 3-30 Hz) and super low frequency (SLF: 30-300 Hz) waves over the boundary between two different media have been extensively investigated for many years because of their useful applications in geophysics exploration, oceanography, overwater or underwater communication, and so on. For example, the US system Seafarer, operated at the frequency of 78 Hz [1], while the Russian one, called ZEVS, operated at the frequency of 82 Hz [2]. ese systems have been used to communicate from a fixed station on the sea surface with a submarine traveling close to the ocean floor since 1990s [3,4]. e original expression of this problem can date back to 1909 by Sommerfeld in his classic literature [5]. Subsequently, Norton proposed simple formulas and graphs in 1936 [6]. Following their works, many other progresses were made on numerical [7,8] and analytical solutions [9][10][11][12][13][14][15] at a large propagation distance of kρ ≫ 1, where k represents the wave number and ρ is the propagation distance.
Evidently, in practical physical models such as a dipole source on the interface between two different media like air and Earth or sea water and rock, the electric propagation distance for the ELF/SLF ranges is usually very small, namely, kρ ≪ 1. Sommerfeld integrals are used for field components which can be expressed in terms of the derivatives of integrals including the Bessel function. Due to the divergence terms in Sommerfeld integrals, the investigation of ELF/SLF near-zone field is still not well developed yet. In the recent work by Xu et al. [16], a new quasi-static technique is used for analyzing the near-zone field radiated by a horizontal electric dipole near the sea-rock boundary.
is new development rekindled our interest in investigating the magnetic-dipole excitation problem.
Several investigations of simplifying formulas had been made by Banos and Wesley [17]. ey obtained approximation solutions suitable for both near and far fields in terms of Hertz potentials. Based on their efforts, Durrani [18,19] addressed the sufficiently accurate but relatively simple field components in the sea produced by vertical and horizontal dipoles (both electric and magnetic types) under reasonable limitations. Also of interest is the satisfactory communication range between air and sea. Later on, in a series of papers by Bannister [20,21], previous theoretical work like [22] was well summarized, and all field components for these four different types of dipoles located on the boundary were presented over the quasistatic range. In recent years, some near-field research has gradually been carried out both analytically [23][24][25] and numerically [26]. Particularly in 2010, Parise [27] established the exact closed-form expressions of the electromagnetic field components excited by a vertical magnetic dipole lying on the surface of a flat and homogeneous lossy half-space, but unfortunately H ρ is still in terms of Bessel functions, and these formulas are valid only when both of the dipole source and the observation point are embedded on the boundary. In many practical applications, both the dipole source and the receiving point may be located near the boundary. However, the corresponding study on this near-field radiation of a vertical magnetic dipole is still in the dark. Since it is a formidable task to calculate it accurately by existing analytical methods, in this paper, we will attempt to extend the work by Xu et al. [16] and evaluate the integrals in purpose of deriving new approximation formulas.
Note that the propagation distance of interest is up to dozens of kilometers, in this case, the propagation path via the reflection of the ionosphere is much longer than that along the Earth's surface. Hence, the effect of the ionosphere can be neglected and a simple half-space flat model is established. At ELF/SLF ranges, media such as sea water/ ground/Earth are always regarded as highly lossy ones, while the air is lossless. So, the ratio of the wave numbers is far smaller than 1, i.e., k air /|k media | ≪ 1. Furthermore, usually the heights of both the dipole source and the receiving point are much smaller than the horizontal distance, i.e., d ≪ ρ and z ≪ ρ. e following are some concrete evaluations and simplifications. Comparisons with direct numerical simulations and exact solutions are also carried out from both the amplitude and phase angle aspects eventually. e time dependence e iωt is assumed and suppressed throughout the analysis.

Formulations
e geometry and notations of the physical model are shown in Figure 1. It consists of a loop antenna (viewed as a vertical magnetic dipole) situated at a distance d, above a halfinfinite conducting space. Cartesian and cylindrical coordinates are introduced with z directed upwards through the dipole, and the radius ρ is taken horizontally in the x-y plane on the surface. e air characterized by the parameters ε 2 , μ 2 , and σ 2 occupies the upper space (z ≥ 0), and the lossy medium (like seawater or Earth) characterized by the parameters ε 1 , μ 1 , and σ 1 occupies the lower space (z ≤ 0). It is assumed that both regions are nonmagnetic so that μ 1 � μ 2 � μ 0 .

Integral Expressions for the Field Excited by an ELF/SLF
VMD. When its size is small enough compared with the propagation distance and free-space wavelength, it is a fairly common-sense idea to regard a small current loop as a vertical magnetic dipole (VMD) [28]. We identify IdS as the magnetic dipole moment, where I denotes the loop circulating current and dS represents the area of the loop. With a similar method by King [29], the integrals for the components of the electromagnetic field in Region 2 can be written as follows: where It is noted that the upper sign "+" in (2) corresponds to z > d, and the lower sign "− " corresponds to 0 < z < d, respectively. And the square root in (4) is taken in the first quadrant. e first two terms in formulas (1)-(3) stand for the direct wave and the ideal reflected wave. Meanwhile, the third term represents the lateral wave. us, these formulas can be rewritten in the following forms: In the above integrals, the direct-field and reflect-field terms can be derived via the method presented in the monograph by King et al. [15]. e direct-field components are e reflect-field components are where e rest lateral-wave terms are defined by Region 2 (air) Figure 1: Excitation of an ELF/SLF VMD above a lossy half-space.

International Journal of Antennas and Propagation
where M � c 1 + c 2 . It should be pointed out that these formulas contain Bessel functions, whose integrands are highly oscillatory in the near-zone. erefore, a novel approximation approach is required to be taken into consideration in the following subsection.

Simplifications of the Lateral-Wave Terms under the Equivalent Infinitesimal eory.
For the beginning of simplification, we should figure out the definition of "quasistatic" or "near-field" involved in this paper. It is defined as the situation where the distance from the source to the observation point is far less than a wavelength (kρ ≪ 1) [15,18]. Typically, if the frequency remains invariable, distances satisfying this condition contain the "near-field"; otherwise, if the distance is fixed, qualified frequencies form the "quasi-static" state. It is readily understood from the definition, when the "quasi-static" assumption is considered, ω ⟶ 0 and k 2 ρ ≪ 1 hold. Under this circumstance, contribution of the Sommerfeld integrals in (9)-(11) is mainly from the integration path where λ ≫ k 2 . erefore, the parameter c 2 can be expressed as follows: Normally, the medium in Region 1 is highly conductive when compared with air in Region 2, namely, k 2 ≪ k 1 . Also, d and z are usually far less than the horizontal distance ρ, i.e., d ≪ ρ and z ≪ ρ. us, we have where m � 1, 2. It is seen from (13) that the limit of i|c 1 − c 2 − k 1 |d m can be approximated as zero. e equivalent infinitesimal theory, a special case of Taylor's formula, could be subsequently applied here as follows: en, by some deformation, a useful alternative form of c 1 − c 2 is correspondingly obtained, so is the variable M. We write Hence, by substituting (16) into (9), we have Similarly, (10) is equal to (10) can be evaluated as follows:

International Journal of Antennas and Propagation
Finally, with the Appendix in [15], the final analytical results for F l 2φ , F l 2ρ , F l 2z can be obtained readily. We write So far, we have provided the process of calculating the field components for a VMD source over the near-field range in Region 2 in detail. e complete approximation solutions for the field components in Region 1 can be evaluated with a similar method as well.

Solutions for the near-Zone Field with
e complete quasi-static field components are derived readily with (9)- (11) and (20)- (22). For comparison purpose, in the following, we will attempt to reduce these formulas to the special case when both the dipole source and the receiving point are embedded on the boundary. For brevity and clarity, only H 2z 's derivation procedure is given as a demonstration. e magnetic component H 2z can be expressed as Since k 2 ρ ≪ 1 is assumed, with Maclaurin series, we have By substituting (24) into (23) and setting z � 0, d � 0, it converts to: Now, we have derived the analytical expressions when z � d � 0. To understand this method better, computations and discussions under several different conditions will be carried out in the following section.

Comparison with Numerical Solutions
When z ≠ 0 and d ≠ 0. It is shown in literature studies [7,8] that, by some proper numerical approaches, one can solve Sommerfeld integrals directly. erefore, direct numerical simulation is International Journal of Antennas and Propagation employed and then compared with the analytical result obtained in this paper. We assume that Region 1 is seawater characterized by the permeability μ 0 , relative permittivity ε r1 � 80, and conductivity σ 1 � 4 S/m. Region 2 is air characterized by the permeability μ 0 and the permittivity ε 0 . For a general case, the dipole source and receiving point are placed at the height d � 1 m and z � 5 m, respectively. e amplitudes and phase angles of the three nonzero field components with respect to the propagation distance ρ are evaluated in Figures 2-4 at operating frequencies f � 3 Hz and 300 Hz.
It is seen that both the amplitudes and phase angles obtained by analytical and direct numerical methods have a consistent trend in general. Figures 2(a) and 2(b) show that at the beginning there exists a relatively apparent difference between these two computed results. is is probably because when ρ is too small, d ≪ ρ and z ≪ ρ are not very restrictive. But when ρ > 10 m, two curves shed light on a good correspondence with each other. e magnitude of E 2φ and H 2ρ in Figures 3(a), 3(b), 4(a), and 4(b) also show similar trends with H 2z . Obviously, in most cases of practical interest, ρ ≫ z and ρ ≫ d, so the results confirm the high reliability and validity of the proposed method. Furthermore, it is also noted that the operating frequency has little influence to the magnitude of H 2z and H 2ρ , but a larger frequency will lead to a smaller amplitude of E 2φ .
As illustrated in Figures 2(c) and 2(d), initially the phase of H 2z ascend sharply until reaching the value of π. It is noted that a higher frequency leads to a closer mutation point to the source. Eventually, for a sudden change to − π, as the distance further increases, the phase starts rising smoothly and stabilizes around − (π/2). Higher frequency also causes a faster phase variation from -π to − (π/2), which results in a shorter distance for reaching the final state. We can see that phase angles of the two methods are almost alike except with some latency. By observing other two components, though trajectories are not in full accord, similar conclusions can still be drawn.
It should also be born in mind that with large distances, k 2 ρ is more close to "1," which then also leads to a larger difference between the numerical and analytical results. In this paper, attention is restricted to the near field, and k 2 ρ ∼ 1 is out of consideration. Computations show that different components have different validity limits of k 2 ρ for maintaining a consistency between the proposal and numerical solutions. For E 2φ and H 2ρ components, results of both the amplitude and phase obtained by the two methods remain the same even if k 2 ρ is approaching 1. However, for H 2z , when the distance reaches 100 km (i.e., k 2 ρ ≈ 0.6, as can be seen in Figure 2(b)), two curves start to deviate from each other.

Comparison with Exact Solutions
When z � 0 and d � 0. When both the dipole source and receiving point are located at the interface of a half-space, the exact field components excited by a vertical magnetic dipole have been addressed by Parise already [27]. e exact solutions of H 2z , E 2φ , and H 2ρ can be expressed in terms of where Q n (ρ) � ik 3 n ρ 3 − 4k 2 n ρ 2 − 9ik n ρ + 9 e ik n ρ ρ 5 , Note that I n and K n are the nth order modified Bessel functions of the first and second kind, respectively, where n � 1, 2.
Since it is a special case of the dipole radiation problem discussed in this paper, we can actually better validate the manner if we plot how that field changes as a function of the distance ρ by our approximation method, the direct numerical calculation and this exact analytical solution. Comparisons are depicted in Figures 5-7. All the parameters involved are the same with those in Figures 2-4

except with different d and z.
Apparently, our analytical results not only coincide with the numerical simulations but also have an excellent agreement with the exact solutions, and curves in Figures 5-6 bear a strong resemblance to those in Figures 2-3. It indicates that all methods are of good accuracy. Meanwhile, comparisons of H 2ρ are not satisfactory as can be seen from Figure 7. On the whole, curves of the analytical and numerical solution are relatively consistent, but do not match up to the exact solution as the propagation distance ρ increases. Take f � 3 Hz as an example. If ρ reaches several hundred meters, the gap in magnitude between exact solutions and the other two methods will become larger. And the phase difference is always about π. is is probably because the exact solution in [27] is expressed in terms of the modified Bessel functions rather than concrete analytical ones.

Evaluations for ese Analytical Formulas.
For further investigation about the features of the proposed method, several other evaluations are performed. As a natural extension, just to make it clear and to see how the locations of the dipole source and receiving point influence the propagation strength of electromagnetic fields, we consider the following simulations presented in Figure 8, with the relative permittivity ε r1 � 80 and conductivity σ 1 � 4 S/m in Region 1. e operating frequency is taken as f � 10 Hz.      respectively. It is seen that, with the increasing of the radial distance, |H 2z | will exhibit quite different characteristics. When ρ is relatively small (see Figures 8(a) and 8(b)), the amplitude decreases at first and then increases gradually as the height of the observing point becomes higher. Typically, the minimum point of the curve will have a shift to a farther position from the origin if ρ is larger. However, for sufficiently large values of ρ (Figures 8(c) and 8(d)), the amplitude increases monotonically as z increases. We also know that, when the dipole source is closer to the interface, the attenuation will be larger. Perhaps it is due to the different degrees of the effect by the lossy medium in Region 1 at different heights. When the receiving point is at an arbitrary place on the x-z plane in Region 2, and the dipole source is located at d � 0 m, 50 m, 150 m, respectively, with the operating frequency taken as f � 10 Hz, the strength distributions of the electric and magnetic components are revealed in Figures 9-10. We can actually see that both the electric and magnetic fields decay rapidly in all directions. As the position of the source is higher, the attenuation of the electromagnetic components will become slower. It is also noted that the distribution in      9(c) and 10(c) is very likely to that in free space, which implies that when the height of the dipole is high enough away from the interface, the radiation result is nearly not affected by the sea water.

Conclusions
is paper deals with the integrals for the electromagnetic field of a vertical magnetic dipole lying above the boundary of a lossy half-space in order to derive analytical formulas for the near-zone field. On the basis of the feature of "quasi-static" state and the equivalent infinitesimal theory, we can induce two rational assumptions. One is c 2 ≈ iλ, and the other is about the substitution of c 1 − c 2 . Comparisons with numerical solutions show that both the amplitude and phase are quite consistent with each other. Curves for contrast with the exact formulas are also drawn when the dipole source and observing point are both on the boundary, which is a further proof of the high accuracy and reliability of our manner. Simulations also show that different components have different validity limits of k 2 ρ for maintaining a consistency between the proposal and numerical solutions. e  International Journal of Antennas and Propagation smallest validity limit always occurs for the component H 2z , and the value of k 2 ρ should be no greater than 0.6 in order to keep a good consistency. Subsequent computations, which indicate that operating frequencies and the positions of the dipole source and the receiving point will all affect the field components, are also performed. In addition, it should be pointed out that these formulas are also suitable for other cases where the condition k 1 ≪ k 2 is satisfied. And the media in two regions are interchangeable. Under this circumstance, c 1 ≈ iλ is acquired, and the approximation in (13) changes to |c 2 − c 1 − k 2 |. en, by a similar method presented in this paper, the final field expression can be obtained.

Data Availability
All data are available as MATLAB code.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.