# Electromagnetic Field Variation of ELF Near-Region Excited by HED in a Homogeneous Half-Space Model
## 1 Introduction  
Due to the high salinity and complex temperature and current distribution characteristics of seawater, its conductivity and dielectric constant are higher than those of air. The higher the conductivity of water, the greater the attenuation of EM waves. Therefore, the propagation characteristics of EM waves in seawater are considerably different from those in air. Experiments have shown that the amplitude and phase characteristics of ELF (3-30 Hz) EM waves are stable when they propagate in seawater, with small propagation loss and strong seawater penetration ability. Therefore, ELF EM waves have broad application prospects in geological exploration, submarine exploration, seismic detection, submarine communication, and other forms of geophysical exploration [1; 2; 3; 4; 5; 6]. Many scholars have conducted in-depth studies on EM fields generated by low-frequency HED located in seawater or soil. In the representative monograph written by [PERSON] [7], a complete analytical solution of the EM field excited by a dipole in half-space is given. In the decades that followed, [PERSON] [8] described in detail the lateral EM waves generated by the excitation of horizontal and vertical electric dipoles (VEDs) in two-layer and three-layer media. It is worth mentioning that the lateral wave term of his solution can be expressedas an approximate expression containing a Fresnel integral. Several years later, [PERSON] derived the exact solution expression of an EM field excited by electric dipoles at the interface of half-space [9]. However, the transmitting point and receiving point can only be on the ground; off the ground is not accurate. Figure 1 demonstrates the EM wave propagation model in the half-space model in the two-layer medium in [PERSON]'s monograph, in which \(L\) is a lateral wave, \(L_{1}\) is a direct wave, and \(L_{2}\) is a reflected wave. Generally speaking, the long-term focus of EM research has been on the far-field characteristics of traditional problems such as EM propagation, radiation, and scattering, while the attention and research on the near-field characteristics are still relatively weak [10; 11; 12]. [PERSON] was the first to give the approximate integral expression of EM field for electric dipoles in homogeneous half-space (hereinafter referred to as the Sommerfeld integral) [13], but this expression is difficult to solve by simple numerical integration. The key reason is that the Fourier-Bessel expression in the Sommerfeld integral has non-integrable terms at \(k\rho\to 0\). To solve this problem, [PERSON] [14] gave the exact near-field expression of the fundamental oscillator in half-space under the condition of \(k\rho>>1\) by assuming that \(k_{2}/k_{1}\) (\(k_{2}\) and \(k_{1}\) represent the wave number in seawater and air, respectively) tends to infinity. However, the half-space model of \"air-seawater\" cannot satisfy the condition of \(|k_{2}/k_{1}|<<1\). Therefore, the exact formula is not suitable for solving the ELF EM field of the \"air-seawater\" half-space model. The quasistatic approximate method was first proposed by [PERSON] [15], but it uses Laplace and Poisson equations to calculate the field in the air, so it cannot be applied to the near-field in half-space under the condition of \(k_{2}<<k_{1}\). In recent years, this problem was also revisited by [PERSON] and [PERSON] [16], who addressed the ELF near-field wave propagation for the application of the marine controlled-source electromagnetics method as an extension of the above work. The computational scheme is developed from the approximated formulas in the book [8] by retaining fundamental terms. This motivates us to employ [PERSON]'s expansion near the poles of integrands from the Fourier-Bessel representations.