Consequently, research related to environmental and target electric field characteristics has garnered increasing attention in the field of marine environmental target detection and localization.
Sommerfeld pioneered the investigation into the electric field generated by an electric dipole on conductor surfaces beneath a planar interface18.
Building upon this foundation, Hoitham proposed an equivalent dipole pair model for the static electric field of vessels, thereby establishing the cornerstone for research on vessel electric field signatures19.
Lofgren and Weeks experimentally investigated the correlation mechanisms among sea ice salinity, ice growth rate, and environmental solute concentration38.
This paper proposes an analytical model of the electric field generated by horizontal static current sources in a four-layer medium (air-sea ice-seawater-seabed), aiming at the accurate modeling of the underwater electric field of vessels in sea areas with a sea ice layer.
The physical fields of vessels include electric fields, acoustic fields, magnetic fields, and hydrostatic pressure fields, which serve as crucial target signals for the identification and localization of underwater detection systems. At present, a variety of advanced underwater multi-physical field detection and measurement systems have been developed internationally, including the Multi-Influence Range System (MIRS) developed by SAES in Spain, the underwater multi-influence module developed by the Polish Naval Academy, and the TECOTEC Magnetic and Acoustic Measurement Range System1,2,3,4. These studies further demonstrate the crucial role of multi-field detection in target recognition. Among these detection modalities, electric field detection offers advantages such as being unaffected by hydrometeorological conditions in the marine environment, having stable and reliable detection performance, and possessing strong identification capability. Consequently, research related to environmental and target electric field characteristics has garnered increasing attention in the field of marine environmental target detection and localization. Furthermore, underwater electric fields can be applied to monitoring the corrosion status of vessel hulls and evaluating the effectiveness of anti-corrosion measures, conducting geophysical surveys to obtain seabed geological structure information, facilitating the exploration of seabed oil and natural gas deposits, and locating sunken vessels to support rescue and salvage operations. These diverse application scenarios validate the practical value of underwater electric fields and highlight their wide-ranging utility in the field of marine engineering.
Given that marine medium exhibits good electrical conductivity and corrosiveness, dissimilar metals submerged in it form corrosion currents, thereby generating corrosion-related static electric fields (CRSE)5. For vessels, corrosion currents form a loop through the hull, seawater, and propellers, which in turn gives rise to CRSE6,7,8. Among the existing modeling methods for CRSE, the finite element method (FEM), boundary element method (BEM), and equivalent source method (ESM) are widely applied9,10,11. Due to its concise mathematical form and clear physical meaning, the ESM has become an important theoretical basis for constructing CRSE models12,13,14. This method demonstrates that actual sources with arbitrarily complex distributions can be characterized by the linear superposition of a finite set of equivalent electric dipoles or current sources, as shown in Fig. 1. This significantly simplifies electric field calculations and enables adaptation to different seawater electrolyte environments15,16.
Fig. 1 Full size image Equivalent diagram of corrosion-related static electric field.
The equivalent source method (ESM) provides a crucial technical means for the inversion of electromagnetic fields from underwater targets. However, in practical underwater environments, when considering the electrostatic field distribution within underwater targets such as vessels and unmanned underwater vehicles (UUV), the issue of multiple layer media distribution should be taken into account17. Sommerfeld pioneered the investigation into the electric field generated by an electric dipole on conductor surfaces beneath a planar interface18. Building upon this foundation, Hoitham proposed an equivalent dipole pair model for the static electric field of vessels, thereby establishing the cornerstone for research on vessel electric field signatures19. Early research primarily concentrated on two-layer medium models. Bannister extended the theoretical framework towards practical scenarios by solving for the electromagnetic fields of horizontal electric dipoles in a two-layer medium, employing finitely conducting earth-image theory techniques20. Meanwhile, Schäfer demonstrated that the underwater potential is subject to the coupled effects of material polarization, measurement depth, and conductivity21. This finding underscores the necessity of accurately accounting for environmental factors in any predictive model.
With the advancement of marine detection requirements, the three-layer medium model of “air-seawater-seabed” in the shallow marine environment has progressively become the research focus due to its closer alignment with realistic environments. Cao and Tian utilized CST simulations to demonstrate that increasing the number of medium layers can extend electromagnetic propagation distances, thereby validating the necessity of the three-layer medium model22,23. Further research has refined this understanding. Wang identified low-frequency and interface effects as critical factors for long-distance transmission and quantified the influences of water depth and medium conductivity24,25. Feng extended investigations to complex seabed topography and its impact on field distribution26,27, while Wu systematically investigated the influence of the seabed medium in the stratified marine environment on the electromagnetic field in the air radiated from the extremely low-frequency horizontal electric dipole located in the seawater28. Significant methodological innovations have emerged for solving the three-layer medium model. These include Huang’s complex image theory with vector potential, Zhang’s conclusion on surface-wave-dominated long-distance propagation, Xu’s near-field approximation formula for ELF signals, Wang’s simplified lateral wave model, and Ren’s field decomposition method (separating secondary and background fields)29,30,31,32,33. Collectively, these approaches provide efficient pathways for analyzing the electric field distribution in the three-layer medium. It is noteworthy that a fundamental similarity exists between the propagation characteristics of extremely low-frequency electromagnetic waves and those of static current sources within a three-layer medium. The propagation behavior of both is predominantly governed by the stratified structure and interfacial effects of the three-layer medium. Key parameters, such as seawater depth and the electrical conductivity of each layer, exhibit a commonality in their modulating effects on the electric field distribution and attenuation patterns. Nevertheless, the three-layer medium model struggles to fully capture the complex stratified structures prevalent in actual marine environments. This limitation has spurred the development of four-layer medium models. Woloszyn proposed an analytical four-layer medium model comprising “air-seawater-seabed-non-conductive” layer, enhancing the accuracy of describing complex seabed structures34.
Affected by special temperature-salinity structures, dynamic distribution of ice-water mixtures, and periodic freezing-thawing processes of ice layers, the electromagnetic environment in sea areas with ice layers differs significantly from that in conventional sea areas35. The low temperature of ice layers inhibits the migration rate of ions in seawater, reduces the overall medium conductivity, and intensifies the electrochemical polarization effect at the interface between the vessel hull and seawater36. In addition, the freshwater input from sea ice melting and the deep high-salinity seawater form a vertical salinity gradient, driving the salt-gradient battery effect and generating an electrostatic field dominated by ion transport37. Transient salinity fluctuations caused by ice layer freezing-thawing also induce transient currents, which in turn interfere with the stability of vessel’s electric field. Lofgren and Weeks experimentally investigated the correlation mechanisms among sea ice salinity, ice growth rate, and environmental solute concentration38. Islam introduced deep learning techniques to achieve intelligent estimation of sea ice thickness based on dielectric constant inversion39. However, due to the pronounced heterogeneity and environmental sensitivity of sea ice in terms of its physical structure and chemical composition, it is seldom treated as an independent and critical layer in most current electromagnetic modeling studies of multilayered marine environments, thereby lacking dedicated modeling and systematic analysis.
This paper proposes an analytical model of the electric field generated by horizontal static current sources in a four-layer medium (air-sea ice-seawater-seabed), aiming at the accurate modeling of the underwater electric field of vessels in sea areas with a sea ice layer. Firstly, based on Laplace’s equation for steady electric fields, the electric potential and electric field intensity generated by a pair of horizontal static current sources in the seawater layer were derived for a two-layer medium model. The repeated image method was then introduced to satisfy the multilayer interface boundary conditions via successive mirror reflections, leading to an analytical expression for the electric field in the seawater layer. Subsequently, a finite-volume numerical model was established under experimental conditions and compared with the analytical solution under the same idealized four-layer scenario to verify its correctness. Finally, a scaled-down experiment was designed, and the measured electric field value was compared with the analytical results to evaluate the applicability of the analytical model in realistic environments.
