Analytic Hierarchy Process (AHP)-Based Evaluation of Extremely-Low-Frequency Magnetic Field Contribution Rates

Abstract

The extremely-low-frequency (ELF) magnetic fields of submarines serve as key characteristics for target detection, with their formation mechanisms being complex and diverse. Effectively manipulating a submarine to reduce its magnetic signature is crucial for enhancing its magnetic stealth capabilities. However, current research on the impact of various causative factors is insufficient. This study proposes a contribution rate assessment method based on the Analytic Hierarchy Process (AHP) model, aiming to provide a theoretical basis for effective manipulation. Initially, a thorough analysis of the threat causes of a submarine’s ELF magnetic fields is conducted, and a corresponding hierarchical threat structure model is established. Subsequently, magnetic field signal characteristics generated by different causes are obtained through simulation, and threat matrices and characteristic matrices are constructed. Finally, the contribution rates of different causative magnetic fields to the total magnetic field are calculated, and the simulation results validate the effectiveness of the method. At the stern detection line, the contribution rate of the wake magnetic field is the highest, reaching 0.7649. Along the radial detection line, the contribution rate of the shaft frequency magnetic field is the highest and gradually decays, eventually falling below the wake magnetic field at 150 m and remaining at an approximately 0.5 contribution rate. This study calculates the contribution rates under different operational conditions and detection scenarios, laying a technical foundation for research on the comprehensive active control strategies of submarine ELF magnetic fields in different scenarios.

1. Introduction

In the context of the increasingly complex maritime strategic environment, submarines, as the core equipment of modern naval forces, play a crucial role in underwater warfare, with their safety being of paramount importance [1]. Enhancing the magnetic stealth capabilities of submarines is vital to ensure their concealment and survivability in underwater environments, which is a significant consideration in submarine design and operations [2].

The extremely-low-frequency (ELF) magnetic fields of submarines exhibit diverse characteristics, including a wide range of sources and multidimensional and multimodal features. The threats posed by these ELF fields to submarine safety are multifaceted and represent a comprehensive system engineering challenge. In the context of this study, ELF magnetic fields primarily refer to magnetic fields with frequencies below 30 Hz.

In recent years, major military powers around the world have begun to research various ELF magnetic fields, primarily focusing on the mechanisms of and control strategies for individual types of ELF magnetic fields [3]. For instance, Canada’s Department of National Defence and the UK’s Ultra Electronics have been concentrating on the corrosion-related shaft frequency magnetic fields of ships, designing software for Beasy boundary element simulation to model and analyze their distribution [4]. Russia and the US’s Cortana Corporation have been researching the Debye magnetic fields generated by the ion separation motion in submarine wakes and exploring the feasibility of using these fields for submarine detection [5]. Ship magnetic control mainly involves measures taken against individual magnetic sources and the conditions that allow for the existence of ship magnetic fields. The shaft frequency electric field control system developed jointly by Canada’s Davis Company and the Pacific Defense Research Institute of Canada’s Department of National Defence has been applied to newly built ships in multiple countries. Davis Company has provided shaft frequency electric field control systems for the US Navy’s “Seawolf” class SSN-21 and SSN-22 nuclear submarines [6]. Vickers Shipbuilding and Engineering Technology Limited has also developed a shaft frequency electric field control system, which has been equipped on the British Navy’s “Trafalgar” and “Turbulent” class nuclear submarines. The Polish Navy Technical Research and Development Center has tested shaft-based ELF electric fields and studied a short-circuit slip ring device, which can effectively reduce shaft-based ELF magnetic fields [7,8,9]. Russia has developed a ship magnetic field compensation system called “Kacкaд-э”. This system is designed to counteract the magnetic fields of ships. Ships equipped with the “Kacкaд-э” system have effective insulation. This insulation prevents any contact with dissimilar metals. The system uses magnetic modulation sensors to measure something important: the shaft frequency current characteristic parameters. To eliminate shaft frequency magnetic fields, reverse currents are applied. These currents are directed through the tail anode. [10].

In practical scenarios, the magnetic field as the main component changes dynamically with different frequency bands, distances, environments, and operating conditions. Consequently, it cannot be measured in isolation and is necessary to estimate the contribution rates based on the measurement signals of the overall ELF magnetic fields. However, comprehensive research on the various causes of ELF fields in submarines and the assessment of their threat contribution rates remain nascent. Therefore, it is imperative to clarify the relationships among various ELF fields and, in conjunction with detection scenarios, quantitatively assess the systemic contribution rates of each ELF field to inform effective stealth strategies and countermeasures [11].

In the field of military science, the contribution rate is a crucial indicator for measuring the impact of military equipment or operations on overall combat capability or operational effectiveness. In this study, it refers to the degree of influence of different types of ELF magnetic fields on the submarine’s total magnetic field when being detected [12]. Recent advancements in system engineering and big data have prompted many scholars to conduct systematic research on the contribution rates of multifactorial systems, providing new insights into the composition and control of the ELF magnetic fields of underwater platforms. Current research indicates that the understanding of system contribution rates and their assessment methods has matured, with extensive research conducted on the contribution rates of equipment to combat systems. Among these, the Analytic Hierarchy Process (AHP) has emerged as an effective tool for assessing system contribution rates, and it has been widely applied in various fields. The Analytic Hierarchy Process is a systematic analytical tool designed to address complex decision-making problems. Proposed by Professor Thomas L. Saaty in the 1970s, the AHP integrates both qualitative and quantitative analyses. The core concept of the AHP involves breaking down complex problems into more manageable components to simplify the decision-making process. This method constructs a hierarchical model that divides the decision-making problem into multiple levels, such as the target level, criterion level, and alternative level. Through pairwise comparisons and weight assignments, the AHP provides decision-makers with quantitative support [13]. The AHP has been widely applied in risk assessments [14], performance assessments [15], and evaluations of the contribution rates of weapon systems [16].

This study leverages the advantages of the Analytic Hierarchy Process in comprehensive effectiveness evaluation and multi-criteria decision-making to establish an AHP-based model for assessing the contribution rates of ELF magnetic fields [17]. From a system engineering perspective, this model is integrative. It combines factors from both the detection and detected entities. The model supports the acquisition of specific contributions. These contributions come from different types of ELF magnetic field characteristics. These characteristics are associated with submarines operating under various conditions. Additionally, they are relevant in different detection scenarios. The goal is to understand their impact on the overall ELF field. This provides a theoretical basis for the active control research of ELF magnetic fields in submarines [18].

2. Contribution Rate Modeling Framework

2.1. Analysis of the Causes and Threats

An analysis of the causes and threats of submarines’ ELF magnetic fields is crucial for research. Analyzing the causes of these magnetic fields provides a comprehensive understanding of their origins and composition. This understanding is vital for establishing a model. It is also essential for designing evaluation methods. ELF magnetic fields are particularly significant in the submarine domain, as they can be utilized for target detection and communication. Naval electromagnetic stealth refers to the use of specific designs, materials, and technical means to reduce the detectability of ships across the electromagnetic spectrum, which includes radio waves, microwaves, infrared, visible light, etc. This reduction in detectability aims to decrease the likelihood of ships being detected by enemy electromagnetic detection equipment, such as radar, infrared detectors, and photonic systems. The objective of this technology is to enhance the survivability and concealment of ships on the battlefield by minimizing their electromagnetic signal characteristics. The ELF magnetic fields generated by submarines during navigation can reveal their position. This poses a threat to their safety. In the current maritime battlefield, submarines face magnetic stealth threats. These threats are influenced by various causative magnetic fields.

Firstly, the magnetic fields generated by the electronic equipment, power systems, and communication systems of submarines can produce detectable magnetic signals. The equipment’s magnetic fields mainly comprise two forms: first, the submarine’s internal electrical system and powered equipment often have multiple grounding points, with resistances and poor grounding creating potential differences that result in leakage currents radiating magnetic fields through the hull; second, high-power equipment inside the submarine radiates magnetic waves, which propagate into the seawater. The submarine equipment’s magnetic fields are predominantly composed of extremely-low-frequency components, with frequencies related to equipment rotational speed. Changes in the magnetic field are related to equipment power and load, and the distribution of the magnetic field is closely related to the equipment’s position, with varying trends in different positions, exhibiting complex spatial distribution characteristics. The intensity of the equipment magnetic field is dependent on the equipment power, load, and distance.

Secondly, the rotation of the propeller causes periodic changes in the shaft system contact resistance, leading to corresponding periodic pulsations in the corrosion and anti-corrosion currents in the seawater. Naval corrosion is the process of gradual damage or destruction of a ship’s metal structures due to chemical or electrochemical reactions in seawater, atmospheric conditions, or other corrosive environments. This process typically manifests as oxidation, dissolution, or loss of metal, leading to decreased structural integrity and impaired functionality of the ship, and potentially causing safety incidents. Anti-corrosion currents are direct currents applied to the metal structures of a ship from an external power source to prevent or slow down the corrosion process. These currents are used in cathodic protection systems, where the metal structure becomes the cathode in an electrolytic solution, thereby reducing the corrosion rate of the metal. The application of anti-corrosion currents can be achieved through methods such as sacrificial anode or impressed current systems, and it represents a crucial technical means to protect the metal structures of ships from corrosion damage. This results in the generation of ELF alternating shaft frequency magnetic fields [19]. The movement of the submarine causes changes in the shaft frequency electric field. As the submarine’s speed or rotational speed increases, the intensity of the shaft frequency electric field also increases accordingly. When the submarine’s cathodic protection device is operational, it enhances the shaft frequency electric field. The shaft frequency electric field generated by the submarine can propagate over long distances and decays slowly, making it an important source of magnetic fields for submarine detection.

Thirdly, hydrodynamic disturbances caused by large underwater targets during movement lead to the separation of a significant number of sodium positive ions and chloride negative ions in high-salinity seawater, producing wake magnetic fields [20]. During maneuvering, the submarine’s speed, acceleration, and pitch angle affect the Debye effect magnetic field. When the submarine maneuvers, the changes in speed, acceleration, and pitch angle result in corresponding changes in the Debye effect magnetic field.

These three types of magnetic fields are closely related to the submarine’s state of movement, with their propagation characteristics influenced by factors such as the submarine’s speed and rotational speed. Additionally, changes in the detection position also affect the magnitude of these three magnetic fields. For instance, when detecting in the direction of the submarine’s stern, the wake magnetic field is relatively larger, whereas when detecting radially to the submarine, the shaft frequency magnetic field is more prominent. Furthermore, with advancements in magnetic detection technology, numerous detection methods and modes have emerged, which are also significant influencing factors. Therefore, the analysis of magnetic stealth threats in submarines is a complex multi-factor problem that cannot be assessed solely from a single characteristic dimension. It requires a comprehensive consideration of factors such as the submarine’s own characteristics, signal characteristics, adversarial detection technology, and the marine environment to evaluate the different causes of ELF magnetic fields in submarines. The analysis of causes and threats serves as the foundation for the application of the contribution rate assessment, providing specific decision factors and a hierarchical structure for the application of the AHP, ensuring that this method can be effectively applied to this problem.

2.2. Hierarchical Modeling

Based on the analysis of the threat causes of the submarine’s ELF magnetic fields, a further step is taken to construct an assessment model for the contribution rates of these ELF fields. This model comprises three levels, namely, the target level, the feature level, and the object level, as illustrated in Figure 1. The primary objective is to establish the safety threat posed by the submarine’s ELF magnetic fields as the target level of the hierarchical structure model, which serves as a key indicator. The feature level encompasses various signal characteristics, which are critical criteria for evaluating the ELF magnetic fields. The object level selects the different causes of the submarine’s ELF magnetic fields to reflect their complex composition.

By analyzing and determining the detection methods and by using submarine state information, spatiotemporal information, and other parameters as inputs, a hierarchical structure model is constructed. On this basis, a contribution rate judgment matrix can be designed to assess the contribution of different features to the magnetic field, and a consistency test is conducted to ensure the rationality of the judgment matrix. Ultimately, based on the model’s calculation results, the contribution rates of ELF magnetic fields from different causes can be assessed, providing guidance for active control strategies of ELF magnetic fields in different combat scenarios. This lays the foundation for the comprehensive active control of submarine ELF magnetic fields oriented towards battlefield applications. The assessment process for the contribution rates of submarine ELF magnetic fields is shown in Figure 2.

3. Methods for Contribution Rate Assessment

The basic idea and hierarchical model for constructing a framework for the assessment of the contribution rates of ELF magnetic fields were introduced above. To provide a specific explanation of the calculation method for the contribution rates, the following section elaborates on the calculation process of the ELF magnetic field contribution rates based on the contribution rate hierarchical structure model.

The AHP deconstructs a complex problem into a hierarchical structure, decomposing the objective into multiple component factors. The process is as follows: first, the causes of the threat posed by the submarine’s ELF magnetic fields are analyzed; then, a hierarchical structure model and a judgment matrix are constructed based on the relationships among the factors, assigning appropriate weights to each level of indicators using the judgment matrix; finally, a bottom-up analysis is conducted to obtain a ranking of the importance of the lowest-level factors for the highest-level indicators. The advantage of the AHP lies in its ability to systematically divide the various factors involved in complex problems into an orderly hierarchy, making intricate issues more coherent and structured. This enables the establishment of a more comprehensive, objective, and reliable model for the contribution rates of different types of submarine ELF magnetic fields.

3.1. Threat Matrix Construction

In this paper, the judgment matrix between the target level and the feature level is defined as the threat matrix. Assuming that there are r features at the feature level, there exists an r × r judgment matrix between the target level and the feature level, referred to as the threat matrix T. The construction of the threat matrix requires an analysis of the detection methods, evaluating the sensitivity of different detection methods to the various characteristics of the submarine’s magnetic signals. The threat matrix is then constructed based on the analysis results.

Matrices are typically constructed using the 1~9 scale method or an improved scale method, as shown in

Table 1. Scale for pairwise comparison of importance levels and their assignments.

	Importance Levels	aij
1	Elements i and j are equally important.	1
2	Element i is slightly more important than element j.	3
3	Element i is clearly more important than element j.	5
4	Element i is strongly more important than element j.	7
5	Element i is extremely more important than element j.	9
6	Element i is slightly less important than element j.	1/3
7	Element i is clearly less important than element j.	1/5
8	Element i is strongly less important than element j.	1/7
9	Element i is extremely less important than element j.	1/9

. Here, 1 represents equal importance, and 9 represents extreme importance. The matrix constructed based on the pairwise comparison results is known as the judgment matrix, and it must satisfy the conditions a_ij > 0, a_ii = 1, and a_ij = 1/a_ji, which means that the judgment matrix is a symmetric matrix.

In this study, based on the characteristics of the submarine’s ELF magnetic fields, the judgment matrix between the feature level and the target level needs to be designed according to the threat methods. In Equation (1) shown in the threat matrix, element a_ij represents the ratio of the threat level of feature i to the magnetic stealth to the threat level of feature j to the magnetic stealth. According to the sensitivity of different detection methods to different signal features, the threat level of feature i and feature j is compared, and the value of a_ij is determined according to the scaling method. If the threat level of feature i is greater than the threat level of feature j, then a_ij > 1; otherwise, a_ij < 1.

$$T=\left[\begin{array}{cccc}1& a_{12}& \dots & a_{1r}\\ {\displaystyle \frac{1}{a_{12}}}& 1& \dots & a_{2r}\\ \dots & \dots & 1& \dots \\ {\displaystyle \frac{1}{a_{1r}}}& {\displaystyle \frac{1}{a_{2r}}}& \dots & 1\end{array}\right]\in {ℝ}_{r\times r}$$

(1)

The threat matrix reflects the degree of threat of different features to magnetic stealth, and the determination of its element values requires consideration of the impact of various features on the submarine’s magnetic stealth.

3.2. Feature Matrix Construction

The judgment matrix between the object layer and the feature layer is referred to as the feature matrix, which is determined based on the features of the submarine’s ELF magnetic field signals. Submarines, under different frequency bands, distances, environments, operational conditions, and detection modes, exhibit dynamic changes in the main components of their magnetic fields. Additionally, different types of ELF magnetic fields display multidimensional and multimodal characteristics, necessitating the use of multimodal feature extraction methods to process the signals. To ensure that the magnetic field contribution rate indicators objectively reflect the contribution rate of magnetic fields, the construction of the feature matrix is divided into two steps: signal feature extraction and feature matrix construction.

Firstly, the feature extraction of the extremely-low-frequency magnetic field of the submarine is conducted. The preliminary extracted features include the line spectrum and its amplitude, centroid frequency, half-energy bandwidth, higher-order statistics, information entropy, and extrema, as detailed below.

Line spectrum and its amplitude. This term usually refers to the phenomenon in spectral analyses where energy is concentrated on discrete spectral lines. The ELF magnetic field signals of submarines often exhibit distinct line spectrum features. For the wake magnetic field and the magnetic field effects of the propeller shaft rotation, the line spectrum features are generally related to the propeller frequency and its harmonic components. As for the magnetic field radiation from internal equipment, the line spectrum features are typically associated with the operational frequencies of the electrical equipment and the power supply frequency.

Centroid frequency. The centroid frequency refers to the weighted average frequency of a signal spectrum, which is the inverse of the weighted average of the signal energy (or power) with respect to frequency. It can be used to describe the “center of mass” position of the signal spectrum and provides an indicator for measuring the distribution of the signal spectrum. The calculation of the centroid frequency employs a weighted average using the amplitude of the power spectrum as the weight, with the following formula:

$$f_c={\displaystyle \frac{{\displaystyle {\int }_0^{+\infty }f\cdot P(f)\mathrm{d}f}}{{\displaystyle {\int }_0^{+\infty }P(f)\mathrm{d}f}}}$$

(2)

Here, P(f) is the power spectrum of the magnetic field signal; thus, the centroid frequency will shift towards the positions with greater amplitude in the power spectrum.

Half-energy bandwidth. The half-energy bandwidth refers to the frequency range at which the signal power drops to half of its maximum value, serving as an indicator for measuring the selectivity or resolution of a filter. The spectrum of the submarine’s ELF magnetic field often includes low-frequency band spectra. Although not as prominent as the line spectrum features, for signals with low signal-to-noise ratios, the line spectrum features are often submerged in noise due to their low intensity. However, the energy of the band spectrum is distributed over a frequency band, making the spectral information contained within it less prone to being obscured.

Higher-order statistics, information entropy, and extrema. Owing to the complexity of magnetic noise in real-world environments and the low signal-to-noise ratio of the submarine’s ELF magnetic field, features are extracted from the signal to represent its characteristics. These include higher-order statistics such as the mean, standard deviation, skewness factor, and kurtosis factor, as well as information entropy H(x) and the maximum and minimum values. The calculation formulas for these features are presented in

Table 2. Feature calculation formula.

Feature	Formulas
Mean	${\displaystyle \sum T/N}$
Standard deviation	$\sqrt{{\displaystyle \sum {(T-\overline{T})}^2}/N}$
Skewness	${\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\left(x_i-\overline{x}\right)}^3/{({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\left(x_i-\overline{x}\right)}^2)}^{3/2}$
Kurtosis	$[{\displaystyle \sum _{i=1}^n}{\left(x_i-\overline{x}\right)}^4/{\displaystyle \frac{1}{n}}{({\displaystyle \sum _{i=1}^n}{\left(x_i-\overline{x}\right)}^2)}^2]-3$
Information entropy H(x)	$-{\displaystyle \sum _{i=1}^n{p}_i{\log }_2{p}_i}$

.

N is the number of signal samples within the signal processing window. The mean describes the average level, while the extrema simply describe the range of the data.

Next, the construction of the feature matrix is carried out based on the extracted features. There are a total of r features in the feature layer and n types of magnetic fields in the object layer. The judgment matrices between the object layer and the feature layer are denoted as C₁, C₂, …, C_r, where r is the total number of features. Thus, there are r n × n matrices between the object layer and the feature layer, referred to as the feature matrices C_i (i = 1, 2, …, r), as shown in Equation (3):

$$C_i=\left[\begin{array}{cccc}1& b_{12}& \dots & b_{1n}\\ {\displaystyle \frac{1}{b_{12}}}& 1& \dots & b_{2n}\\ \dots & \dots & 1& \dots \\ {\displaystyle \frac{1}{b_{1n}}}& {\displaystyle \frac{1}{b_{2n}}}& \dots & 1\end{array}\right]\in {ℝ}_{n\times n}(i=1,2,\dots ,r)$$

(3)

The feature matrix C_i in this study is obtained by calculating and comparing the eigenvalues of the corresponding simulated signals. Suppose that there is a signal feature vector F = [F₁, F₂, …, F_n], which contains the numerical values of the feature in all magnetic fields, and each magnetic field has a corresponding eigenvalue. Here, F_i represents the numerical value of the signal feature in the i-th magnetic field. Equation (4) can be used to compare the importance of different magnetic fields:

$$b_{ij}={\displaystyle \frac{F_i}{F_j}}$$

(4)

Here, b_ij is an element in the feature matrix, representing the importance of the i-th magnetic field relative to that of the j-th magnetic field with respect to a specific signal feature. If b_ij > 1, then it indicates that the j-th magnetic field is more important than the i-th magnetic field with regard to this feature, and vice versa if b_ij < 1, indicating that the i-th magnetic field is more important than the j-th magnetic field with respect to this feature.

3.3. Consistency Test and Contribution Rate Calculation

Firstly, according to the actual situation, the characteristic vector corresponding to the maximum eigenvalue of the judgment matrix is solved using different methods. After normalization, the weight vector of the single-level ranking is obtained, and a consistency test is conducted. If the test fails, the judgment matrix must be corrected until it meets the satisfactory consistency criteria.

The maximum eigenvalue of the judgment matrix and its corresponding characteristic vector are calculated, and a consistency test is performed on the maximum eigenvalue.

$$CI={\displaystyle \frac{-{\displaystyle \sum _{i=2}^n{\lambda }_i}}{n-1}}={\displaystyle \frac{{\lambda }_{\max }-n}{n-1}}$$

(5)

$$\mathrm{CR}={\displaystyle \frac{\mathrm{CI}}{\mathrm{RI}}}$$

(6)

In the provided context, λ represents the eigenvalue of a matrix. RI can be found in

Table 3. RI values for three matrix orders from 3 to 9.

Matrix Order	3	4	5	6	7	8	9
RI	0.58	0.90	1.12	1.24	1.32	1.41	1.45

.

The judgment matrix is generally considered acceptable when CR < 0.1.

According to the AHP method, the weight vector is calculated using the sum-product method. Firstly, each column of the judgment matrix is normalized. Let the matrix element be x_ij, the weight vector be w_i, and the normalized matrix element be y_ij as follows:

$$y_{ij}={\displaystyle \frac{x_{ij}}{{\displaystyle \sum _{k=1}^n{x}_{kj}}}}(i=1,2,3,\cdots ,n)$$

(7)

After normalization, the sum of the elements in each column is 1. The rows of the normalized judgment matrix are added together as follows:

$$U_i={\displaystyle \sum _{j=1}^n{y}_{ij}}(i=1,2,3,\cdots ,n)$$

(8)

Then, vector U is normalized as follows:

$$w_i={\displaystyle \frac{U_i}{{\displaystyle \sum _{j=1}^n{U}_j}}}(i=1,2,3,\cdots ,n)$$

(9)

By using the above equation, the weight vector corresponding to each judgment matrix can be obtained. That is, the weight vector W can be calculated from the threat matrix T, and the weight vector p_i can be obtained from the feature matrix C_i. The n × r matrix composed of the feature matrix weight vectors p_i (i = 1, 2, …, r) is P = [p₁, p₂, …, p_r] ∈ ℝ_n×r. Here, W is the characteristic vector of the judgment matrix between the target layer and the feature layer, and p_i (i = 1, 2, …, r) is the priority weight vector of each judgment matrix between the feature layer and the object layer. Therefore, the contribution rate W_c of each magnetic field in the object layer to the total ELF magnetic field of the submarine in the target layer is

$$\begin{array}{l}{W}_c=PW,\\ P=[p_1,p_2,\dots ,p_r]\in {ℝ}_{n\times r}\\ W={[c_1,c_2,\dots ,c_r]}^T\in {ℝ}_{r\times 1}\end{array}$$

(10)

4. Simulation and Result Analysis

4.1. Equivalent Modeling in a Unified Spatiotemporal Reference Frame

The calculation method for the contribution rate of the ELF magnetic field was established in the previous section. Next, an analysis and evaluation of the contribution rates are conducted using three typical ELF magnetic fields as examples: the wake magnetic field, shaft frequency magnetic field, and equipment magnetic field.

Under a unified spatiotemporal reference, simulations of the shaft frequency magnetic field and the equipment magnetic field are performed using arrays of electric dipoles and magnetic dipoles [21,22]. The arrays are uniformly distributed along the length of the submarine, with different operating conditions corresponding to the simulated source strengths. The Kelvin wake model is used to calculate the wake Debye magnetic field [23]. According to relevant sources, the cruising speed of a conventional submarine is approximately 3 m/s, with a rotational speed of 1 r/s [24].

The detection line at the stern selects detection points on a measuring line away from the stern of the submarine (with the stern of the submarine as the origin), taking 10 detection points along the longitudinal direction of the submarine on the measuring line, with 1 detection point placed every L meters (L is the length of the submarine) [24]. The radial detection line is chosen at a point 10 m from the middle point on the right side of the submarine, with one detection point placed every 50 m. A schematic diagram of the simulation scenario is shown in Figure 3, and the simulation parameter settings are presented in

Table 4. Simulation parameter settings.

Parameters	Numerical Value
Length L	100 m
Breadth D	16.67 m
Geomagnetic field strength F	5 × 104 nT
Magnetic inclination I	π/4
Seawater conductivity σ	σ = 4 s/m
Relative dielectric constant εr	εr = 81

. In Table 4, the parameters presented are fixed values utilized within the simulation, representing both the submarine model parameters and the ocean environmental parameters.

In accordance with the arrangement of the magnetic detection points mentioned above, this study analyzed the magnitude and signal characteristics of the magnetic field at the detection points. Through the simulation studies of the three typical ELF magnetic fields described, the simulated signals were obtained, and actual measured ocean noise from an experiment was added to the simulation signals. The experiment was conducted in Hainan, China, in August 2023, using a CS-VL magnetometer to measure real-world environmental noise.

Figure 4 illustrates the time-domain simulation signal diagram at a measurement point 100 m from the submarine’s stern.

4.2. Contribution Rate Analysis

Based on the signals of the three magnetic fields obtained from the aforementioned simulations, feature extraction is conducted according to the contribution rate evaluation process. The three components of the submarine’s ELF magnetic field—the wake magnetic field, propeller shaft rotation effect magnetic field, and internal equipment radiation magnetic field—typically exhibit a superposition of distinct periodic line spectra and low-frequency band spectra when measured using magnetic sensors. For periodic signals, the line spectrum and the amplitude at the corresponding frequency points serve as a type of frequency-domain feature. Additionally, the centroid frequency and half-energy bandwidth of the low-frequency band spectrum also possess characteristics related to underwater platforms. Due to the complexity of magnetic noise in actual environments and the low signal-to-noise ratio of underwater platform ELF magnetic fields, higher-order statistical features are extracted from the time-domain signals. Therefore, the line spectrum and its amplitude, centroid frequency, half-energy bandwidth, and higher-order statistical features are extracted as the frequency-domain features of the underwater platform’s ELF magnetic field.

According to the calculation method for the threat matrix described previously, different detection systems and scenarios result in varying signal sensitivities and processing methods, leading to different threat matrices. Therefore, when determining the detection mode and scenario, the threat level of different signal features can be established. The submarine’s fixed detection array is taken as an example.

In the fixed mode, the magnetic detection system remains stationary, while the submarine’s relatively slow speed (3 m/s) allows the system and the submarine to be considered relatively fixed within a certain signal acquisition time. This fixed mode typically operates at a distance from the target, resulting in weaker signals. However, as the magnetic detection system does not move, it can employ various techniques to better eliminate background noise. Research into the generation mechanism and propagation patterns of extremely-low-frequency magnetic fields from underwater platforms reveals that the magnetic field signals consist of a direct current (DC) component superimposed with fluctuating components, with the DC component being predominant. In the absence of relative spatial changes, the measured signal represents a time series with minimal amplitude variations, indicating that the DC component remains constant for the measurement system while only the fluctuating components change [25]. Due to the presence of clear periodic frequency characteristics in the fluctuating terms, it is possible to remove the constant DC component and utilize the frequency-domain characteristics of the signal for detection. This approach is suitable for monitoring critical waterways and defending against intrusions in important straits, such as by deploying surface/underwater magnetic detection arrays. Therefore, for fixed detection, the band spectrum information (centroid frequency and half-energy bandwidth) in the frequency-domain characteristics is considered the primary indicator, followed by the time–frequency characteristics and time-domain characteristics [16].

Hence, the designed threat matrix is denoted as T, and the corresponding feature weight vector is denoted as w. The maximum eigenvalues of the threat matrix and the feature matrix, along with their corresponding characteristic vectors, are calculated, and a consistency test is performed on the maximum eigenvalues. The elements within the Threat Matrix T correspond to the following six characteristics: line spectrum and its amplitude, half-energy bandwidth, centroid frequency, standard deviation, skewness factor, and kurtosis factor. The CR of the threat matrix ≈ 0.0376 < 0.1, which meets the consistency test.

$$\begin{array}{l}T=\left[\begin{array}{cccccc}1& 2& 6& 7& 9& 8\\ \frac{1}{2}& 1& 4& 5& 7& 6\\ \frac{1}{6}& \frac{1}{4}& 1& \frac{1}{2}& 1& \frac{1}{3}\\ \frac{1}{7}& \frac{1}{5}& 2& 1& 2& 1\\ \frac{1}{9}& \frac{1}{7}& 1& \frac{1}{2}& 1& \frac{1}{2}\\ \frac{1}{8}& \frac{1}{6}& 3& 1& 2& 1\end{array}\right],\\ w={\left[\begin{array}{cccccc}0.443& 0.284& 0.073& 0.075& 0.043& 0.082\end{array}\right]}^T\end{array}$$

Further feature extraction is conducted on the signals when the submarine passes through the stern detection line, yielding the detailed data shown in

Table 5. Signal feature extraction results at 200 m point of the stern detection line.

Feature Name	Shaft Frequency Magnetic Field	Wake Magnetic Field	Equipment Magnetic Field
Line spectrum frequency (Hz)	2.004	2.002	3.031
Line spectrum amplitude (nT)	0.1964	161.861	0.1912
Half-energy bandwidth	24.4257	33.0105	5.7716
Centroid frequency (Hz)	12.7852	15.8890	3.4438
Standard deviation	0.1420	144.0268	0.2461
Skewness factor	0.0083	0.3563	0.3150
Kurtosis factor	1.6274	2.4961	1.7593

and

Table 6. Signal feature extraction results at 400 m point of the stern detection line.

Feature Name	Shaft Frequency Magnetic Field	Wake Magnetic Field	Equipment Magnetic Field
Line spectrum frequency (Hz)	2.009	2.004	3.010
Line spectrum amplitude (nT)	0.0414	106.454	0.0091
Half-energy bandwidth	22.4313	29.8627	6.1274
Centroid frequency (Hz)	11.7879	15.3541	3.6701
Standard deviation	0.0423	95.8806	0.0098
Skewness factor	0.1144	0.3662	0.2455
Kurtosis factor	1.5950	2.5386	1.6464

. An example of the feature matrix at 100 m of the stern detection line is as follows.

$$\begin{gathered} C_1=\left[\begin{array}{ccc}1& 33.7539& 119.6961\\ 0.0296& 1& 3.5761\\ 0.0084& 0.2820& 1\end{array}\right];C_2=\left[\begin{array}{ccc}1& 1.2562& 5.4945\\ 0.7961& 1& 4.3739\\ 0.1820& 0.2286& 1\end{array}\right];C_3=\left[\begin{array}{ccc}1& 1.0157& 3.9644\\ 0.9846& 1& 3.9032\\ 0.2522& 0.2562& 1\end{array}\right]; \\ {C}_4=\left[\begin{array}{ccc}1& 6.2845& 102.5436\\ 0.1591& 1& 16.3170\\ 0.0098& 0.0613& 1\end{array}\right];C_5=\left[\begin{array}{ccc}1& 1.1977& 17.6078\\ 0.8349& 1& 14.7010\\ 0.0568& 0.0680& 1\end{array}\right];C_6=\left[\begin{array}{ccc}1& 1.3201& 1.2532\\ 0.7575& 1& 0.9493\\ 0.7980& 1.0534& 1\end{array}\right] \end{gathered}$$

Further feature extraction is conducted on the signals when the submarine passes through the radial detection line under the fixed detection mode, yielding the detailed data shown in

Table 7. Signal feature extraction results at 100 m point of the radial detection line.

Feature Name	Shaft Frequency Magnetic Field	Wake Magnetic Field	Equipment Magnetic Field
Line spectrum frequency (Hz)	2.007	2.000	3.005
Line spectrum amplitude (nT)	51.9841	16.4464	0.0865
Half-energy bandwidth	27.4869	32.6948	5.0312
Centroid frequency (Hz)	14.3182	15.1561	3.4111
Standard deviation	46.8661	17.1459	0.0496
Skewness factor	0.1744	0.2517	0.2955
Kurtosis factor	1.9255	2.6699	1.6473

and

Table 8. Signal feature extraction results at 150 m point of the radial detection line.

Feature Name	Shaft Frequency Magnetic Field	Wake Magnetic Field	Equipment Magnetic Field
Line spectrum frequency (Hz)	2.003	2.021	3.004
Line spectrum amplitude (nT)	1.8056	2.2731	0.0095
Half-energy bandwidth	25.4485	32.4229	5.1274
Centroid frequency (Hz)	13.3201	15.7780	3.5731
Standard deviation	1.3063	2.0210	0.0081
Skewness factor	0.0809	0.3629	0.2398
Kurtosis factor	1.7464	2.5185	1.6451

. An example of the feature matrix at 100 m of the radial detection line is as follows.

$$\begin{gathered} C_1=\left[\begin{array}{ccc}1& 0.3164& 190.1318\\ 3.1608& 1& 600.9723\\ 0.0053& 0.0017& 1\end{array}\right];C_2=\left[\begin{array}{ccc}1& 1.1895& 6.4984\\ 0.8407& 1& 5.4633\\ 0.1539& 0.1830& 1\end{array}\right];C_3=\left[\begin{array}{ccc}1& 1.0585& 4.4432\\ 0.9447& 1& 4.1975\\ 0.2251& 0.2382& 1\end{array}\right]; \\ {C}_4=\left[\begin{array}{ccc}1& 0.3658& 345.6835\\ 2.7334& 1& 944.8810\\ 0.0029& 0.0011& 1\end{array}\right];C_5=\left[\begin{array}{ccc}1& 1.4432& 0.8518\\ 0.6929& 1& 0.5902\\ 1.1740& 1.6944& 1\end{array}\right];C_6=\left[\begin{array}{ccc}1& 1.3866& 1.6208\\ 0.7212& 1& 1.1689\\ 0.6170& 0.8555& 1\end{array}\right] \end{gathered}$$

Due to space constraints, this article only presents partial results of the signal extraction for the stern detection line and the radial detection line. These data cover multiple parameters of the magnetic field signals, including the line spectrum feature, half-energy bandwidth, and centroid frequency.

The final calculations yield the contribution rates of the different causes of ELF magnetic fields at the two detection positions of the stern detection line and the radial detection line, as shown in

Table 9. Contribution rates of the stern detection line.

Distance	Magnetic Field Category	Contribution Rate
100 m	Wake magnetic field	0.7265
	Shaft frequency magnetic field	0.2083
	Equipment magnetic field	0.0652
200 m	Wake magnetic field	0.7653
	Shaft frequency magnetic field	0.1571
	Equipment magnetic field	0.0776
300 m	Wake magnetic field	0.7549
	Shaft frequency magnetic field	0.1726
	Equipment magnetic field	0.0725
400 m	Wake magnetic field	0.7630
	Shaft frequency magnetic field	0.1615
	Equipment magnetic field	0.0756
500 m	Wake magnetic field	0.7591
	Shaft frequency magnetic field	0.1648
	Equipment magnetic field	0.0761

and

Table 10. Contribution rates of the radial detection line.

Distance	Magnetic Field Category	Contribution Rate
10 m	Wake magnetic field	0.2335
	Shaft frequency magnetic field	0.7052
	Equipment magnetic field	0.0613
50 m	Wake magnetic field	0.2707
	Shaft frequency magnetic field	0.6573
	Equipment magnetic field	0.0720
100 m	Wake magnetic field	0.3488
	Shaft frequency magnetic field	0.5831
	Equipment magnetic field	0.0681
150 m	Wake magnetic field	0.5317
	Shaft frequency magnetic field	0.3992
	Equipment magnetic field	0.0691
200 m	Wake magnetic field	0.4901
	Shaft frequency magnetic field	0.4360
	Equipment magnetic field	0.0740

and Figure 5 It can be observed that, on the stern detection line, the contribution rate of the wake Debye magnetic field remains at a high level throughout, while the contribution rates of the shaft frequency magnetic field and the equipment magnetic field are lower, with overall variations being relatively stable. On the submarine’s radial detection line, the contribution rate of the shaft frequency magnetic field is the highest before the 100 m detection point and gradually decreases with an increasing distance. The contribution rate of the wake magnetic field gradually increases and exceeds that of the shaft frequency magnetic field at the 150 m detection point, remaining at about 0.5, while the contribution rate of the equipment magnetic field remains at a low level throughout.

5. Discussion

The generation of extremely-low-frequency magnetic fields in underwater platforms is complex, with diverse distribution patterns. The magnetic fields, which serve as the primary components, dynamically change with varying frequency bands, distances, environmental conditions, and operational states. To effectively implement active control of ELF magnetic fields, it is necessary to analyze the contribution rates of different types of ELF fields to the overall ELF magnetic environment, thereby providing a theoretical basis for the design of active control strategies. This study begins with magnetic field signals that share the same spatiotemporal reference, and extracts multi-modal features of the signals from aspects such as frequency-domain characteristics, time-domain characteristics, and time-frequency domain characteristics. Based on this, the Analytic Hierarchy Process (AHP) is utilized to establish a contribution rate model of ELF magnetic fields in underwater platforms based on multi-modal signal features. Finite element simulation methods are employed to analyze the contribution rates of ELF magnetic fields with different causes in typical scenarios. Due to the complexity of the causes of ELF magnetic fields in underwater platforms, there is currently a lack of systematic control strategies targeting different causes. To enhance the effectiveness of active control over ELF magnetic fields, this paper proposes a calculation model for the contribution rates of different types of ELF magnetic fields to the overall ELF magnetic environment, providing a theoretical foundation for future research and the design of active control strategies.

This study establishes a multidimensional feature extraction method for ELF magnetic fields based on time-domain, frequency-domain, and time-frequency domain characteristics. Signals from ELF magnetic fields in underwater platforms, which arise from different causes, exhibit multidimensional and multimodal features, making it impractical to directly utilize the raw signals for contribution rate analysis. Consequently, the project focuses on extracting the multimodal features of the signals, enabling the magnetic field contribution rate indicators to objectively reflect the contribution rates. However, the study employs an equivalent simulation model based on real experimental tests as the data source, which can reflect some signal characteristics to a certain extent but lacks completeness. In the next phase of research, a broader range of actual measurement signals will be incorporated into the analysis, and the equivalent simulation will be further refined to better represent the complete signal characteristics.

In this study, the judgment matrix between the target layer and the feature layer is defined as the Threat Matrix. Assuming there are r features in the feature layer, there exists an r × r judgment matrix between the target layer and the feature layer, referred to as the Threat Matrix T. The construction of the Threat Matrix must be based on the enemy’s detection means, analyzing the sensitivity of different detection methods to the magnetic signal characteristics of the submarine. The Threat Matrix is constructed according to the analysis results. Regarding the detection equipment and parameters, the data available in this study is not yet sufficient. In the next phase of research, it is hoped that more parameters of detection equipment under different detection modes can be obtained. By comparison and analysis, a more effective Threat Matrix can be constructed, which can fully reflect the threat level of different signal characteristics under different detection modes.

6. Conclusions

This study analyzes the causes and threats of submarine extremely-low-frequency (ELF) magnetic fields, revealing the complexity of their causes and the diversity of their distribution patterns. From a system engineering perspective, a contribution rate model based on the Analytic Hierarchy Process (AHP) is proposed. The first step in this model involves equivalent modeling under a unified spatiotemporal reference and the extraction of multimodal features in the frequency domain, time domain, and time–frequency domain. The second step involves using the AHP to establish a contribution rate model with a three-tier system consisting of the target layer, feature layer, and object layer. The judgment matrix between the target layer and feature layer is determined by the enemy’s detection systems, while the judgments between the feature layer and object layer depend on the magnetic characteristics of our submarines under different operating conditions. The third step involves assessing the combined effect of these factors to obtain the contribution rates of different causes of ELF magnetic fields. Finally, the study validates the effectiveness of the contribution rate assessment model and establishes a contribution rate analysis framework system that can adapt to the dynamic needs of both detection and detection by the opposing sides.