A multi-ship tracking method based on shaft frequency electromagnetic field

Abstract

The present invention discloses a multi-ship tracking method based on shaft-frequency electromagnetic fields, comprising: step S1, windowing the shaft-frequency electromagnetic field time series and extracting the amplitude and phase; step S2, constructing an objective function for multi-ship tracking based on the shaft-frequency electromagnetic field; step S3, setting an initial model for multi-ship tracking; step S4, obtaining the Jacobian matrix and Hessian matrix of the electromagnetic data with respect to ship inversion parameters; step S5, calculating the objective function fitting error of the inversion iterative model; step S6, determining whether the inversion requirements are met, and if so, proceeding to step S7; otherwise, returning to step S4; and step S7, outputting the final tracking result. The present invention can process the electromagnetic field amplitude and phase information at different times in real time, and the processed characteristic signals are used to simultaneously locate multiple ships with shaft-frequency electromagnetic fields. The initial phase of the shaft-frequency electromagnetic field is used as an inversion parameter for real-time inversion.

Description

Technical Field

[0001] The present invention relates to the technical field of tracking marine targets such as ships, and in particular to a multi-ship tracking method based on shaft-frequency electromagnetic fields. Background Art

[0002] As a ship navigates, the rotation of its propellers cuts through the Earth's magnetic field, generating induced electromotive force and induced current, which in turn forms a magnetic field. Simultaneously, the propeller's rotation disturbs the current distribution between the ship's surface and the seawater due to electrochemical corrosion, also causing electric field variations. These changes in electric and magnetic fields together constitute the shaft-frequency electromagnetic field. The shaft-frequency electromagnetic field has a wide range of applications in ship monitoring, early warning, positioning, and tracking.

[0003] The shaft-frequency electromagnetic field has a distinct spectral characteristic, making its frequency response commonly used for ship tracking. Both the amplitude and phase of the shaft-frequency electromagnetic field in the frequency domain contain information such as the ship's position, electric moment, and initial phase. It should be noted that the ship's initial phase significantly affects the phase of the electromagnetic signal received at all times. In particular, when multiple ships are within the detection range, the initial phases of all ships are aliased, resulting in a more complex phase of the recorded signal. While the principles of ship tracking are similar to those of marine controlled-source electromagnetic methods used in geophysical exploration, the application scenarios differ. In geophysical exploration, all information about the marine controlled-source electromagnetic transmitter, including the initial phase, is known, and its influence can be eliminated during data preprocessing. However, the tracked vessel is typically a non-cooperative target, and its parameters cannot be known in advance. Therefore, the phase of the acquired electromagnetic field signal cannot be directly used as the ship's true phase for inversion. Therefore, the initial phase should be considered as an inversion parameter along with information such as the ship's position and electric moment. However, existing research has primarily focused on tracking a single ship, with inversion parameters limited to the ship's position and electric moment, excluding the initial phase. Furthermore, in areas with high ship traffic, such as waterways and ports, multiple vessels may be within the sensor's detection range. Ship tracking methods that can only handle single-target scenarios are unable to handle the complexities of these areas. Furthermore, when conducting ship electromagnetic field observations, electromagnetic recorders record a time series of the electromagnetic field. Developing real-time positioning of multiple ships based on this time series is currently unresolved. Summary of the Invention

[0004] The present invention provides a multi-ship tracking method based on shaft-frequency electromagnetic fields, which can obtain electromagnetic field amplitude and phase information at different times in real time, and simultaneously locate multiple ships with shaft-frequency electromagnetic fields based on the processed characteristic signals, and perform real-time inversion using the initial phase of the shaft-frequency electromagnetic field as an inversion parameter, thereby effectively solving the problem that the phase of the collected electromagnetic field signal cannot be directly used as the true phase of the ship for inversion.

[0005] The present invention solves the technical problem by adopting the following technical solutions:

[0006] A multi-ship tracking method based on shaft frequency electromagnetic field comprises the following steps:

[0007] Step S1, windowing the shaft frequency electromagnetic field time series and extracting the amplitude and phase;

[0008] Step S2, constructing an objective function for multi-ship tracking based on shaft frequency electromagnetic field;

[0009] Step S3, setting up an initial model for multi-vessel tracking;

[0010] Step S4, obtaining the Jacobian matrix and Hessian matrix of the electromagnetic data with respect to the ship inversion parameters;

[0011] Step S5, calculating the objective function fitting error of the inversion iterative model;

[0012] Step S6, judging whether the inversion requirements are met, if so, proceeding to step S7, otherwise returning to step S4;

[0013] Step S7: output the final tracking result.

[0014] Furthermore, in step S1, the method of windowing the shaft frequency electromagnetic field time series and extracting the amplitude and phase includes:

[0015] Step S11: Windowing

[0016] The time series is segmented according to the time interval required for tracking, and the window function is designed according to the time interval, and the time series and the window function are convolved;

[0017] Step S12: Frequency identification

[0018] Perform amplitude spectrum analysis on the windowed signal and select the strong energy frequency of the harmonic signal to identify the frequency of the time-harmonic electric dipole source;

[0019] Step S13, using slow Fourier transform to estimate Fourier coefficients

[0020] The slow Fourier transform has the form

[0021]

[0022] Where, f is the frequency identified in step S11, t is the time, is the center point of the time window, For the corresponding time The time series of the shaft frequency electromagnetic field; is the window function centered at time t, M is the number of data;

[0023] Multiply the shaft frequency electromagnetic field time series by the window function and perform linear least squares analysis on the time series within the specified time window to determine the optimal Fourier coefficients;

[0024] Step S14, amplitude and phase calculation

[0025] The amplitude and phase of the shaft-frequency electromagnetic field within a specified time window are calculated using the following formulas:

[0026]

[0027]

[0028] in and are the Fourier coefficients estimated by least squares, and are the amplitude and phase of the shaft frequency electromagnetic field, respectively.

[0029] Furthermore, in step S2, the method for constructing the objective function of multi-ship tracking based on shaft frequency electromagnetic field is as follows:

[0030] Step S21, converting the shaft speed electromagnetic data inversion problem of time-harmonic electric dipole positioning into an optimization problem of a multiplication objective function, the objective function used is:

[0031]

[0032] Among them, m is the model parameter vector, is the fitting difference between the frequency domain shaft frequency electromagnetic field response and the observed data at the current moment; is the normalized model constraint between the model parameters at the current moment and the model parameters at the previous moment;

[0033] Step S22: the fitting difference between the current frequency domain shaft frequency electromagnetic field response and the observed data Expressed as:

[0034]

[0035] in, is the standard deviation operator, is the observed data, M is the number of observed data, is the frequency domain forward response, W is the data weighting matrix;

[0036] Step S23: Normalize the model constraints between the current model parameters and the previous model parameters Has the following form:

[0037]

[0038] in, is the inversion model parameter at the previous moment, b is the normalization coefficient, and has the following form:

[0039]

[0040] in, is a minimum value, which is used to ensure that the term is differentiable, thereby constraining the continuity of the parameter at different times, that is, no mutation occurs at adjacent times.

[0041] Furthermore, in step S3, the model parameter vector m in the initial model of multi-vessel tracking contains 7 parameters and has the following form:

[0042]

[0043] Among them, [x s ,y s ,z s ] is the position of the ship, [I x ,I y ,I z ] is the electric moment of the ship in three orthogonal directions, and φ is the initial phase of the ship.

[0044] Furthermore, in step S3, the method for setting the initial model for multi-vessel tracking is as follows:

[0045] Step S31, performing amplitude spectrum analysis on the time series and determining the number of ships by evaluating the number of high-energy fundamental frequencies;

[0046] Step S32: for multiple ships with different axial rate frequencies, each axial rate signal is processed separately to obtain an amplitude-time curve and a phase-time curve, thereby achieving tracking of multiple single ships;

[0047] Step S33, when the ships have the same propeller speed, the shaft-frequency electromagnetic field only appears as one high-energy frequency on the amplitude spectrum; in this case, it is first assumed that there is only one ship, and an initial model with only one ship is set, and the inversion is started; when the inversion of the initial model with one ship converges, it means that there is only one ship in the current model; when the inversion does not converge, one ship is added to the inversion initial model, and the inversion is repeated until the inversion converges.

[0048] Furthermore, in step S4, the method for obtaining the Jacobian matrix and Hessian matrix of the electromagnetic data with respect to the ship inversion parameters is as follows:

[0049] The Gauss-Newton method is used to solve the optimization problem of the product objective function. The n-th model update amount Expressed as:

[0050]

[0051] in, and are the Hessian matrix and gradient matrix of the objective function respectively; Has the following form:

[0052]

[0053] The gradient in the above formula has the following form:

[0054]

[0055]

[0056] Where J is the Jacobian matrix;

[0057] Hessian matrix Has the following form:

[0058]

[0059] in

[0060]

[0061] .

[0062] Furthermore, in step S5, during the inversion iteration process, the inversion iteration model gradually converges to the true model, and the objective function fitting error is shown as follows:

[0063]

[0064] Gradually converges to 1.0. In the above formula, d is the inversion data, F is the forward response of the inversion model, and N is the number of data. is the standard deviation of the kth data.

[0065] Furthermore, in step S6, there are two criteria for determining whether to exit the inversion: 1) whether the inversion target fitting error is met; 2) whether the number of inversion iterations reaches the maximum number of iterations; if not, return to step S4 and continue to iterate to solve the model update amount; if so, enter step S7.

[0066] Beneficial effects of the present invention:

[0067] This invention primarily addresses the issues of extracting electromagnetic field data applicable to real-time ship positioning from the time series of a ship's shaft-frequency electromagnetic field recorded by a receiving system, as well as achieving real-time tracking of multiple non-cooperative vessels. A multi-vessel tracking method based on the shaft-frequency electromagnetic field is proposed. This method extracts the characteristic response of the shaft-frequency electromagnetic field through three steps: windowing the recorded time series, extracting characteristic frequencies, extracting Fourier coefficients based on a slow Fourier transform, and calculating the amplitude and phase of the ship's shaft-frequency electromagnetic field time series. Tracking of multiple non-cooperative vessels is then performed based on a product objective function. Because traditional single-vessel positioning methods are insufficient in densely populated areas such as ports and waterways, the inversion method proposed in this invention provides an effective method for real-time positioning of multiple vessels in dense areas. Compared to traditional inversion algorithms, the data processing method proposed in this invention can obtain more accurate and robust shaft-frequency electromagnetic field amplitude and phase in real time. Furthermore, the multi-vessel tracking method proposed in this invention can achieve real-time positioning of at least two moving vessels. It is important to note that, compared to the inversion parameters used in existing research, the proposed method fully considers the unknown phase of the ship as a non-cooperative party, using the ship's initial phase as the inversion parameter. It also fully considers the continuity of ship parameters at different times, adding constraint functionals regarding the parameters at different times to the objective function, thereby improving the reliability and continuity of the inversion. This inversion approach and method are unprecedented in existing research. The proposed ship tracking inversion method has wider applicability and can be applied to the real-time monitoring and tracking of multiple non-cooperative ships. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 Flow chart of the method of the present invention.

[0069] Figure 2 Schematic diagram of layered geoelectric model;

[0070] Figure 3 The three-component electric field signal curves of two ships observed on the seabed;

[0071] Figure 4 The amplitude and time curves and phase and time curves of the three orthogonal electric field components of the two ships;

[0072] Figure 5 This is the positioning result diagram of the two ships;

[0073] Figure 6 is the probability density distribution diagram of the inverted electric moment vector of ship 1 and ship 2;

[0074] Figure 7 is the probability density distribution diagram of the ship's initial phase obtained by inversion. DETAILED DESCRIPTION

[0075] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts shall fall within the scope of protection of the present invention.

[0076] Reference Attachment Figure 1 The present invention provides a multi-ship tracking method based on shaft frequency electromagnetic field, comprising the following steps:

[0077] Step S1: Window the shaft frequency electromagnetic field time series and extract the amplitude and phase as follows:

[0078] Step S11: Windowing

[0079] The time series is segmented according to the time interval required for tracking, and the window function is designed according to the time interval, and the time series and the window function are convolved;

[0080] Step S12: Frequency identification

[0081] According to the generation mechanism of the shaft speed electromagnetic field, the shaft speed electromagnetic field signal has a prominent line spectrum feature, which is consistent with the propeller rotation frequency and is accompanied by multiple harmonics. Therefore, the amplitude spectrum analysis of the signal after windowing is performed, and the strong energy frequency containing the harmonic signal is selected to identify the frequency of the time-harmonic electric dipole source;

[0082] Step S12, using slow Fourier transform to estimate Fourier coefficients

[0083] The slow Fourier transform has the form

[0084]

[0085] Where, f is the frequency identified in step S11, t is the time, is the center point of the time window, For the corresponding time The time series of the shaft frequency electromagnetic field; is the window function centered at time t, M is the number of data;

[0086] Multiply the shaft frequency electromagnetic field time series by the window function and perform linear least squares analysis on the time series within the specified time window to determine the optimal Fourier coefficients;

[0087] Step S14, amplitude and phase calculation

[0088] The amplitude and phase of the shaft-frequency electromagnetic field within a specified time window are calculated using the following formulas:

[0089]

[0090]

[0091] in and are the Fourier coefficients estimated by least squares, and are the amplitude and phase of the shaft frequency electromagnetic field, respectively.

[0092] Step S2: construct an objective function for multi-ship tracking based on shaft frequency electromagnetic field, as follows:

[0093] Step S21, converting the shaft speed electromagnetic data inversion problem of time-harmonic electric dipole positioning into an optimization problem of a multiplication objective function, the objective function used is:

[0094]

[0095] Among them, m is the model parameter vector, is the fitting difference between the frequency domain shaft frequency electromagnetic field response and the observed data at the current moment; is the normalized model constraint between the model parameters at the current moment and the model parameters at the previous moment; this objective function takes into account both the data fitting at the current moment and the continuity of the model parameters at adjacent moments.

[0096] Step S22: the fitting difference between the current frequency domain shaft frequency electromagnetic field response and the observed data Expressed as:

[0097]

[0098] in, is the standard deviation operator, is the observed data, M is the number of observed data, is the frequency domain forward response, W is the data weighting matrix;

[0099] Step S23: Normalize the model constraints between the current model parameters and the previous model parameters Has the following form:

[0100]

[0101] in, is the inversion model parameter at the previous moment, b is the normalization coefficient, and has the following form:

[0102]

[0103] in, is a minimum value, which is used to ensure that the term is differentiable, thereby constraining the continuity of the parameter at different times, that is, no mutation occurs at adjacent times.

[0104] Step S3, setting the initial model of multi-ship tracking. The model parameter vector m in the initial model of multi-ship tracking contains 7 parameters and has the following form:

[0105]

[0106] Among them, [x s ,y s ,z s ] is the position of the ship, [I x ,I y ,I z ] is the electric moment of the ship in three orthogonal directions, and φ is the initial phase of the ship.

[0107] In step S3, the method for setting the initial model for multi-vessel tracking is as follows:

[0108] Step S31, performing amplitude spectrum analysis on the time series and determining the number of ships by evaluating the number of high-energy fundamental frequencies;

[0109] Step S32: for multiple ships with different axial rate frequencies, each axial rate signal is processed separately to obtain an amplitude-time curve and a phase-time curve, thereby achieving tracking of multiple single ships;

[0110] Step S33, when the ships have the same propeller speed, the shaft-frequency electromagnetic field only appears as one high-energy frequency on the amplitude spectrum; in this case, it is first assumed that there is only one ship, and an initial model with only one ship is set, and the inversion is started; when the inversion of the initial model with one ship converges, it means that there is only one ship in the current model; when the inversion does not converge, one ship is added to the inversion initial model, and the inversion is repeated until the inversion converges.

[0111] Step S4, obtaining the Jacobian matrix and Hessian matrix of the electromagnetic data with respect to the ship inversion parameters, the method is as follows:

[0112] The Gauss-Newton method is used to solve the optimization problem of the product objective function. The n-th model update amount Expressed as:

[0113]

[0114] in, and are the Hessian matrix and gradient matrix of the objective function respectively; Has the following form:

[0115]

[0116] The gradient in the above formula has the following form:

[0117]

[0118]

[0119] Where J is the Jacobian matrix;

[0120] Hessian matrix Has the following form:

[0121]

[0122] in

[0123]

[0124] .

[0125] Step S5, calculating the objective function fitting error of the inversion iteration model. During the inversion iteration process, the inversion iteration model gradually converges to the true model. The objective function fitting error is shown in the following formula:

[0126]

[0127] Gradually converges to 1.0. In the above formula, d is the inversion data, F is the forward response of the inversion model, and N is the number of data. is the standard deviation of the kth data.

[0128] Step S6, judging whether the inversion requirements are met, if so, proceeding to step S7, otherwise returning to step S4;

[0129] There are two criteria for determining whether to exit inversion: 1) whether the inversion target fitting error is met; 2) whether the number of inversion iterations reaches the maximum number of iterations; if not, return to step S4 and continue to iterate to solve the model update amount; if so, go to step S7.

[0130] Step S7: output the final tracking result.

[0131] refer to Figure 2Three seafloor layers with increasing resistivity with depth are set up. The resistivities are 2Ωm, 4Ωm, and 100Ωm, respectively, from shallow to deep. The thicknesses of the two shallow sediment layers are 70m and 270m, respectively. Assume that Ship 1 has a draft of 8 meters and a propeller rotating at a constant speed of 210 rpm, i.e., a shaft frequency of 3.5Hz. A ship with an electric moment of [1,30,1]Am moves along a straight line from A[20, -3000, 8] to B[20, 4200, 8] over a period of 60 minutes. The initial phase of the electromagnetic field is set to 60°. The ship continuously excites the shaft-frequency electromagnetic field during navigation. The seafloor electromagnetic acquisition station is located at [0, 0, 30] on the seafloor with a sampling rate of 125Hz. Assume that Ship 2 has a draft of 5m and a propeller speed identical to Ship 1, i.e., a shaft frequency of 3.5Hz. Ship 2 sails from C[-1500,-1500,5] to D[5700,5700,5] at a constant speed of [2,2,0] m / s. The electric moment and initial phase are set to [20,1,1]Am and 45°, respectively.

[0132] Ship 1 and Ship 2 moved from A and C to B and D respectively within 60 minutes. The three-component electric field recorded by the electromagnetic acquisition station placed at [0,0,30] is as follows Figure 3 As shown in the figure, all components record the shaft frequency electric field signals of the two ships. Due to the different motion trajectories and electric moments of the ships, the shapes and amplitudes of the three components are also different. From the peak position of the electric field amplitude, it can be seen that ship 1 and ship 2 are closest to the electromagnetic acquisition station at 12.5 minutes and 25 minutes respectively. Similarly, the processing method proposed in this invention is applied to Figure 3 The time series shown in the figure is used to obtain the amplitude and time curve (such as Figure 4 (a) in the figure) and the phase and time curves (as shown in Figure 4 (b) in Figure 2). The dotted line is the result of time series processing, and the solid line is the result of time series processing. Figure 2 The model shown directly solves Maxwell's equations. As can be seen, the obtained amplitude and phase are consistent with the true values.

[0133] by Figure 2 Taking the model shown as an example, a tracking experiment is conducted on two ships with the same propeller speed. In target tracking, it is assumed that the seabed resistivity is known, while the ship parameters are unknown. The trajectory estimated by the inversion of this patent is as follows Figure 5 As shown in . The RMS value of the inversion result converges to about 1. Figure 5 It can be seen that the position positioning accuracy of the two ships is the highest, the inversion results are basically consistent with the true value, and the single positioning error gradually increases with the increase of the distance between the ship and the receiving station. The probability density distribution of the electric moment of the two ships obtained by inversion is as follows Figure 6 It should be noted that the present invention selects the value with the maximum probability density as the final ship electric moment. Figure 6 (a) and Figure 6 As can be seen from (b), the error between the inverted ship's first electric moment [Ix, Iy, Iz] and the true value is [0.3, 1.24, 0.21]Am, and the error between the inverted ship's second electric moment [Ix, Iy, Iz] and the true value is [0.82, 0.1, 1.03]Am. Similarly, Figure 7 is the probability density distribution of the inverted initial phase. The inverted initial phase distribution for ship one is between 45° and 100°, with the maximum probability density initial phase at 61.20°, and an error of 1.20° from the true value. The inverted initial phase distribution for ship two is between -20° and 80°, with the maximum probability density initial phase at 45.30°, and an error of 0.30° from the true value. The errors between the inverted results and the true results are all within an acceptable range, verifying the effectiveness of this method in the multi-ship scenario.

[0134] This invention primarily addresses the issues of extracting electromagnetic field data applicable to real-time ship positioning from the ship's shaft-frequency electromagnetic field time series recorded by the receiving system, as well as achieving real-time tracking of multiple non-cooperative ships. A multi-ship tracking method based on the shaft-frequency electromagnetic field is proposed. This method extracts the characteristic response of the shaft-frequency electromagnetic field through three steps: characteristic frequency extraction, Fourier coefficient extraction based on a slow Fourier transform, and calculation of the amplitude and phase of the ship's shaft-frequency electromagnetic field time series. The method then tracks multiple non-cooperative ships based on a product objective function. Because traditional single-ship positioning methods are insufficient in densely populated areas such as ports and waterways, the inversion method proposed in this invention provides an effective method for real-time positioning of multiple ships in dense areas. Compared to traditional inversion algorithms, the data processing method proposed in this invention can obtain more accurate and robust shaft-frequency electromagnetic field amplitude and phase. Furthermore, the multi-ship tracking method proposed in this invention can achieve real-time positioning of no fewer than two moving ships. It is important to note that, compared to the inversion parameters used in existing research, the proposed method fully considers the unknown phase of the ship as a non-cooperative party, using the ship's initial phase as the inversion parameter. It also fully considers the continuity of ship parameters at different times, adding constraint functionals regarding the parameters at different times to the objective function, thereby improving the reliability and continuity of the inversion. This inversion approach and method are unprecedented in existing research. The proposed ship tracking inversion method has wider applicability and can be applied to the real-time monitoring and tracking of multiple non-cooperative ships.

[0135] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. A multi-ship tracking method based on shaft frequency electromagnetic field, characterized in that: The steps include: Step S1, windowing the shaft frequency electromagnetic field time series and extracting the amplitude and phase; Step S2, constructing an objective function for multi-ship tracking based on shaft frequency electromagnetic field; Step S3, setting up an initial model for multi-vessel tracking; Step S4, obtaining the Jacobian matrix and Hessian matrix of the electromagnetic data with respect to the ship inversion parameters, the method is as follows: The Gauss-Newton method is used to solve the optimization problem of the product objective function, and the n-th model update Δp n Expressed as: Among them, H n and g n are the Hessian matrix and gradient matrix of the objective function respectively; g n Has the following form: The gradient in the above formula has the following form: Where J is the Jacobian matrix; The Hessian matrix H n Has the following form: in Step S5, calculating the objective function fitting error of the inversion iterative model; During the inversion iteration process, the inversion iteration model gradually converges to the true model, and its objective function fitting error is shown as follows: ψ n Gradually converges to 1.

0. In the above formula, d is the inversion data, F is the forward response of the inversion model, N is the number of data, δ k is the standard deviation of the kth data; Step S6, judging whether the inversion requirements are met, if so, proceeding to step S7, otherwise returning to step S4; Step S7: output the final tracking result.

2. The multi-ship tracking method based on shaft frequency electromagnetic field according to claim 1, characterized in that: In step S1, the method of windowing the shaft frequency electromagnetic field time series and extracting the amplitude and phase includes: Step S11: Windowing The time series is segmented according to the time interval required for tracking, and the window function is designed according to the time interval, and the time series and the window function are convolved; Step S12: Frequency identification Perform amplitude spectrum analysis on the windowed signal and select the strong energy frequency of the harmonic signal to identify the frequency of the time-harmonic electric dipole source; Step S13, using slow Fourier transform to estimate Fourier coefficients The slow Fourier transform has the form Where f is the frequency identified in step S11, t is the time, τ is the center point of the time window, T = [T1, T2, ..., T n ] is the corresponding time t=[t1,t2,...,t n ] shaft frequency electromagnetic field time series; w(t) is the window function centered at time t, and M is the number of data; a linear least squares analysis is performed on the time series within the specified time window to determine the optimal Fourier coefficients; Step S14, amplitude and phase calculation The amplitude and phase of the shaft-frequency electromagnetic field within a specified time window are calculated using the following formulas: Among them A j and B j are the Fourier coefficients estimated by least squares, and are the amplitude and phase of the shaft frequency electromagnetic field, respectively.

3. The multi-ship tracking method based on shaft frequency electromagnetic field according to claim 2, characterized in that: In step S2, the method for constructing the objective function of multi-ship tracking based on shaft frequency electromagnetic field is as follows: Step S21, converting the shaft speed electromagnetic data inversion problem of time-harmonic electric dipole positioning into an optimization problem of a multiplication objective function, the objective function used is: Among them, m is the model parameter vector, is the fitting difference between the frequency domain shaft frequency electromagnetic field response and the observed data at the current moment; is the normalized model constraint between the model parameters at the current moment and the model parameters at the previous moment; Step S22: the fitting difference between the current frequency domain shaft frequency electromagnetic field response and the observed data Expressed as: Among them, ||*|| is the standard deviation operator, d=[d1,d2,...,d M ] is the observation data, M is the number of observation data, F(m) is the frequency domain forward response, and W is the data weighting matrix; Step S23: Normalize the model constraints between the current model parameters and the previous model parameters Has the following form: Among them, m p is the inversion model parameter at the previous moment, b is the normalization coefficient, and has the following form: Among them, δ 2 is a minimum value, which is used to ensure that the term is differentiable, thereby constraining the continuity of the parameter at different times, that is, no mutation occurs at adjacent times.

4. The multi-ship tracking method based on shaft frequency electromagnetic field according to claim 3 is characterized in that: In step S3, the model parameter vector m in the initial model of multi-vessel tracking contains 7 parameters and has the following form: Among them, [x s ,y s ,z s ] is the position of the ship, [I x ,I y ,I z ] is the electric moment of the ship in three orthogonal directions, is the initial phase of the ship.

5. The multi-ship tracking method based on shaft frequency electromagnetic field according to claim 4, characterized in that: In step S3, the method for setting the initial model for multi-vessel tracking is as follows: Step S31, performing amplitude spectrum analysis on the time series and determining the number of ships by evaluating the number of high-energy fundamental frequencies; Step S32: for multiple ships with different axial rate frequencies, each axial rate signal is processed separately to obtain an amplitude-time curve and a phase-time curve, thereby achieving tracking of multiple single ships; Step S33, when the ships have the same propeller speed, the shaft-frequency electromagnetic field only appears as one high-energy frequency on the amplitude spectrum; in this case, it is first assumed that there is only one ship, and an initial model with only one ship is set, and the inversion is started; when the inversion of the initial model with one ship converges, it means that there is only one ship in the current model; when the inversion does not converge, one ship is added to the inversion initial model, and the inversion is repeated until the inversion converges.

6. The multi-ship tracking method based on shaft frequency electromagnetic field according to claim 5, characterized in that: In step S6, there are two criteria for determining whether to exit the inversion: 1) whether the inversion target fitting error is met; 2) whether the number of inversion iterations reaches the maximum number of iterations; if not, return to step S4 and continue to iterate to solve the model update amount; if so, enter step S7.

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