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. 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
aij > 0,
aii = 1, and
aij = 1/
aji, 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
aij 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
aij is determined according to the scaling method. If the threat level of feature
i is greater than the threat level of feature
j, then
aij > 1; otherwise,
aij < 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:
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.
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, …,
Cr, 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
Ci (
i = 1, 2, …,
r), as shown in Equation (3):
The feature matrix
Ci 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, …,
Fn], which contains the numerical values of the feature in all magnetic fields, and each magnetic field has a corresponding eigenvalue. Here,
Fi 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:
Here, bij 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 bij > 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 bij < 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.
In the provided context,
λ represents the eigenvalue of a matrix. RI can be found in
Table 3.
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
xij, the weight vector be
wi, and the normalized matrix element be
yij as follows:
After normalization, the sum of the elements in each column is 1. The rows of the normalized judgment matrix are added together as follows:
Then, vector
U is normalized as follows:
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
pi can be obtained from the feature matrix
Ci. The
n ×
r matrix composed of the feature matrix weight vectors
pi (
i = 1, 2, …,
r) is
P = [
p1,
p2, …,
pr] ∈ ℝ
n×r. Here,
W is the characteristic vector of the judgment matrix between the target layer and the feature layer, and
pi (
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
Wc of each magnetic field in the object layer to the total ELF magnetic field of the submarine in the target layer is