Optimization method of submarine concealment based on game theory and linear programming under complex terrain

Abstract

The application discloses a submarine concealment optimization method based on game theory and linear programming under complex terrain, considers the influence of complex terrain on submarine concealment in the researched sea area, carries out sound field calculation in the activity depth range of the submarine and a surface ship carrying a sonar, obtains a detection probability matrix based on a sonar equation, determines a payoff matrix of the submarine based on game theory, further solves the payoff matrix through a linear programming method, and quantitatively optimizes the submarine depth strategy. Compared with a traditional method, the application has significant advantages in concealment optimization, calculation efficiency and adaptability, is suitable for complex seabed environments, and provides a reliable, simple and efficient solution for submarine concealment optimization.

Description

Technical Field

[0001] This invention belongs to the field of submarine technology, specifically relating to a method for optimizing submarine stealth under complex terrain based on game theory and linear programming. Background Technology

[0002] In modern naval warfare, the stealth of submarines is directly related to the success or failure of their missions. With the rapid development of sonar detection technology, the resolution and detection capabilities of sonar have been significantly improved, posing a severe challenge to the survivability of submarines in complex terrain and dynamic marine environments. Therefore, how to formulate efficient and quantitative stealth optimization strategies to evade sonar detection has become a key research issue.

[0003] Existing technologies mainly employ strategy optimization methods based on empirical rules and numerical simulation. However, these methods have certain limitations when dealing with complex terrain and multivariate adversarial environments, as detailed below:

[0004] (1) Stealth optimization based on empirical rules. Traditional methods are mostly based on empirical rules, and submarine stealth optimization is carried out through qualitative analysis of sonar detection probability. Although basic avoidance strategies are achieved, the effects of multipath propagation of sound waves, terrain shielding effect and environmental noise cannot be fully considered in complex terrain, resulting in large deviations in stealth assessment and optimization results.

[0005] (2) Numerical simulation method. Numerical simulation method requires high computational resources, has low computational efficiency in complex environments, and mainly focuses on sound field analysis, lacking the ability to quantitatively assess stealth. Therefore, relying solely on numerical simulation method may not be sufficient to comprehensively evaluate the stealth of a submarine.

[0006] In summary, existing technologies have several shortcomings in optimizing submarine stealth under complex terrain conditions, including low computational efficiency, qualitative analysis of stealth only, and insufficient reliability of optimization results. Therefore, there is an urgent need for a game theory-based approach that constructs an adversarial model between the submarine and sonar, and utilizes linear programming techniques to solve for submarine stealth optimization strategies. This approach would provide an efficient and accurate technical means for optimizing submarine stealth in complex terrain, thereby overcoming the deficiencies of existing technologies. Summary of the Invention

[0007] To overcome the shortcomings of existing technologies, this invention provides a submarine stealth optimization method based on game theory and linear programming for complex terrain. In the studied sea area, the impact of complex terrain on submarine stealth is considered. Within the operational depth range of submarines and surface ships carrying sonar, sound field calculations are performed, and the detection probability matrix is ​​obtained based on the sonar equations. The submarine's payoff matrix is ​​determined based on game theory. Furthermore, the payoff matrix is ​​solved using linear programming, and the submarine's depth strategy is quantitatively optimized. Compared to traditional methods, this invention has significant advantages in stealth optimization, computational efficiency, and adaptability. It is applicable to complex seabed environments and provides a reliable, simple, and efficient solution for submarine stealth optimization.

[0008] The technical solution adopted by this invention to solve its technical problem is as follows:

[0009] Step 1: Detection probability modeling;

[0010] The depth range of submarine and sonar operations is determined, followed by the creation of the complex terrain required for the ray model, sound field calculations to obtain propagation loss values ​​at different sound source and receiving depths, and the establishment of a model based on the sonar equations, including the narrowband passive sonar equations under coherent processing and narrowband passive detection:

[0011] SE=SL-TL-NL+(AG-BW)-DT (1)

[0012] Where SE is the signal margin received by the passive sonar; SL is the narrowband source level of the submarine's radiated noise; TL is the sound propagation loss from the submarine's location to the sonar location; NL is the ocean background noise related to sea state; AG is the array gain; BW is the analysis bandwidth; and DT is the detection threshold of the passive sonar detection system.

[0013] The specific detection probability matrix is ​​obtained according to formula (1), and the specific formula is shown in formula (2):

[0014]

[0015] Step 2: Game equilibrium model;

[0016] Applying matrix game theory to submarines, a metric called payoff is defined for the submarine; the payoff is the negative of the expected probability of passive sonar detection within a certain radius; in a two-dimensional scenario, the payoff is represented by U:

[0017]

[0018] Where R represents the left and right calculation radius centered on the submarine in a two-dimensional scene; P is the probability of the sonar detecting the submarine, i.e. the probability of the submarine being detected; d represents the depth of the sonar, z represents the depth of the submarine, and r represents the distance between the submarine and the sonar.

[0019] Step 3: Solve the payoff matrix using linear programming;

[0020] The following are the guidelines for minimizing the maximum loss of a submarine:

[0021]

[0022] Where V is the maximum loss of the submarine, U(z) i ,d j Z is the payoff matrix. i This indicates the submarine's chosen depth z. i The probability, m represents the different depths z that the submarine can choose. i The number, j represents the different depths d that the sonar can select. j Quantity;

[0023] Since surface ships carry variable sonar, the minimum benefit maximization rule for variable sonar is as follows:

[0024]

[0025] In this context, W represents the minimum gain for sonar, and D... i Indicates the sonar selection depth z i The probability; n represents the different depths z that the sonar can select. i The number, j represents the different depths d that the submarine can choose. j Quantity;

[0026] The optimal depth of the submarine is obtained by solving the payoff matrix U(i,j) using linear programming. Based on this depth, the stealth of the submarine is analyzed.

[0027] Preferably, the ray model is the Bellhop model.

[0028] A computer program that causes a computer to execute the above-described submarine stealth optimization method.

[0029] An electronic device includes a processor and a memory; the memory is used to store a computer program, and the processor is used to execute the computer program stored in the memory to enable the electronic device to perform the above-described submarine stealth optimization method.

[0030] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described submarine stealth optimization method.

[0031] A chip includes a processor for retrieving and running a computer program from a memory, causing a device equipped with the chip to perform the aforementioned submarine stealth optimization method.

[0032] A computer program product includes a computer storage medium storing a computer program, the computer program including instructions executable by at least one processor, which, when executed by the at least one processor, implement the aforementioned submarine stealth optimization method.

[0033] The beneficial effects of this invention are as follows:

[0034] (1) This invention utilizes a matrix game model to model the interaction between submarines and sonar, comprehensively considering the mutual influence of their strategies. Through the zero-sum game theory of game theory, the optimal submarine strategy is found to avoid detection by enemy sonar. This method has greater adaptability compared to conventional static strategies.

[0035] (2) This invention utilizes linear programming to solve for the optimal strategies of submarines and sonar, which can handle various constraints and perform real-time optimization in complex environments. Compared with traditional nonlinear optimization methods, linear programming has higher computational efficiency and accuracy, and can quickly find the optimal solution in a large parameter space.

[0036] (3) This invention utilizes the Bellhop model for sound field calculation, taking into account changes in seabed topography and sound propagation loss, to accurately simulate the interaction between submarines and sonar. Conventional methods often neglect the impact of complex terrain on sonar detection performance, which may lead to inaccurate estimations of submarine stealth. By considering complex terrain such as seabed mountains, this invention can more accurately calculate submarine stealth and provide a more reliable stealth optimization scheme. Attached Figure Description

[0037] Figure 1 The results of passive sonar detection probability modeling at different sound source depths are based on the munk sound velocity curve and seamount topography. (a) Passive detection probability modeling results at a sound source depth of 300m, and (b) Passive detection probability modeling results at a sound source depth of 1000m.

[0038] Figure 2 This is a flowchart illustrating the method framework of the present invention;

[0039] Figure 3 For the propagation loss results obtained based on the sound velocity profile and topography, (a) the sound velocity profile used in the simulation, and (b) the seamount topography and propagation loss map;

[0040] Figure 4The optimal depth strategy probability distributions for submarines at different horizontal distances from the center of the seamount are: (a) the payoff matrix when the submarine is 0 km away from the seamount, (b) the payoff matrix when the submarine is 5 km away from the seamount, (c) the payoff matrix when the submarine is 10 km away from the seamount, and (d) the payoff matrix when the submarine is 15 km away from the seamount.

[0041] Figure 5 This represents the probability distribution of the optimal depth strategy for submarines at different horizontal distances from the center of the seamount. Detailed Implementation

[0042] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0043] This invention proposes a stealth strategy optimization method based on game theory. By constructing a game equilibrium model and using linear programming techniques to solve for the optimal hybrid strategy, the stealth strategy of submarines in complex terrain is optimized, providing theoretical support and technical guidance for the quantitative study of submarine stealth.

[0044] This invention considers the impact of complex terrain on submarine stealth in the studied sea area. Within the operational depth range of sonar carried by submarines and surface ships, sound field calculations are performed, and the detection probability matrix is ​​obtained based on the sonar equations. Then, the submarine's payoff matrix is ​​determined based on game theory. Furthermore, the payoff matrix is ​​solved using linear programming, and the submarine's depth strategy is quantitatively optimized. The process consists of the following three steps:

[0045] Step 1: Detection probability modeling.

[0046] First, the depth range of the submarine and sonar operations needs to be determined. Then, the complex terrain required for creating the ray model (Bellhop model) is constructed, and sound field calculations are performed to obtain propagation loss values ​​at different sound source and receiver depths. Based on the sonar equations, a model is established, including the narrowband passive sonar equations under coherent processing and narrowband passive detection:

[0047] SE=SL-TL-NL+(AG-BW)-DT (1)

[0048] Where SE is the signal margin received by the passive sonar; SL is the narrowband source level of the submarine's radiated noise; TL is the sound propagation loss from the submarine's location to the sonar's location; NL is the ocean background noise related to sea state; AG is the array gain; BW is the analysis bandwidth; and DT is the detection threshold of the passive sonar detection system.

[0049] Based on the above formula, the specific detection probability matrix can be obtained as follows, and the detection probability P... d It is closely related to the signal margin (SE); the larger the signal margin, the greater the detection probability.

[0050]

[0051] Step 2: Game equilibrium model.

[0052] Next, we apply matrix game theory to our submarine, defining a success metric called payoff. Since the submarine must avoid passive detection by anti-submarine vessels, its payoff is the negative of the expected probability of passive sonar detection within a certain radius. In a two-dimensional scenario, payoff is represented by U:

[0053]

[0054] R represents the left and right computational radius centered on the submarine in a two-dimensional scene; P is the probability of sonar detection of the submarine, i.e., the probability of the submarine being detected. Therefore, for the discrete distribution of d and z values, U(z i ,d j This can generate a profit matrix U. ij Set the sonar gain to -U(z) i ,d j This would turn the situation into a zero-sum game between two people. This ensures that neither side can change their strategy in a way that benefits both sides, thus leaving no room for cooperation.

[0055] Step 3: Solve the payoff matrix using linear programming.

[0056] Finally, after obtaining the above profit matrix, the optimal depth of the submarine is solved using linear programming, as follows:

[0057] The following are the guidelines for minimizing the maximum loss of a submarine:

[0058]

[0059] Where V is the maximum loss of the submarine, U(z) i ,d j Z is the payoff matrix. i This indicates the submarine's chosen depth z. i The probability, m represents the different depths z that the submarine can choose. i The number, j represents the different depths d that the sonar can select. j Quantity;

[0060] Since surface ships carry variable sonar, the minimum benefit maximization rule for variable sonar is as follows:

[0061]

[0062] Where W is the minimum gain of the sonar, and D i Indicates the sonar selection depth z iThe probability; n represents the different depths z that the sonar can select. i The number, j represents the different depths d that the submarine can choose. j Quantity;

[0063] The optimal depth for the submarine can be obtained by solving the payoff matrix using linear programming. Based on this depth, the submarine's stealth capabilities can be analyzed.

[0064] This invention calculates the signal margin (SE) of a submarine at different depths and sonar depths using a passive sonar equation model, and further derives a detection probability matrix. This detection probability matrix describes the probability of a submarine being detected by sonar; the larger the signal margin, the higher the detection probability. This approach provides a mathematical basis for subsequent game theory analysis regarding detection probabilities.

[0065] This invention defines the submarine's payoff (i.e., the negative expected value of the probability of avoiding sonar detection) in a two-dimensional scenario and applies it to the game between the submarine and the sonar by constructing a payoff matrix. By setting the sonar's payoff to be opposite to the submarine's payoff, the game is transformed into a zero-sum game, ensuring that the strategy optimization between the submarine and the sonar can reach the optimal equilibrium.

[0066] This invention employs linear programming to solve for the optimal strategy of a submarine, aiming to minimize the submarine's maximum losses and maximize the minimum sonar gains. After obtaining the optimized depth strategy, a quantitative analysis of the submarine's stealth is performed to assess the probability of detection at that depth, thereby further improving the submarine's survivability and stealth.

[0067] Example:

[0068] Figure 1 The results of passive sonar detection probability modeling at different sound source depths are presented based on the Munk sound velocity curve and seamount topography. Figure (a) shows the passive detection results at a sound source depth of 300m, and Figure (b) shows the results at a sound source depth of 1000m. Modeling the passive sonar detection probability can further provide theoretical support for optimizing the depth of submarines under complex conditions.

[0069] Figure 2 The paper presents a flowchart of a submarine stealth optimization method based on game theory and linear programming under complex terrain. The implementation process consists of three steps: (1) Calculating the propagation loss value by inputting the depth range, sound speed profile, and terrain of the submarine and surface ship sonars into Bellhop. (2) Combining the calculated propagation loss, sonar equations, and detection probability functions to obtain the payoff matrix. (3) Solving the payoff matrix using linear programming to obtain the submarine's optimization strategy.

[0070] Figure 3The results of propagation loss based on the given sound speed profile and terrain are presented.

[0071] Figure 4 The figure presents the payoff matrix for submarines at different distances from seamounts. As can be seen from the figure, the optimal depth varies depending on the distance between the submarine and the seamount. When the distance is 0 km from the seamount (Figure a), the submarine's optimal depth is 140-160 m, resulting in the highest payoff. When the distance is 15 km from the seamount (Figure d), the submarine's optimal depths are 30 m and 400 m, resulting in the highest payoff.

[0072] Figure 5 The optimal depth strategy probability distribution for submarines at different horizontal distances from the center of a seamount is presented. The figure shows that the probability distribution is concentrated in areas far from the seamount. Near the center of the seamount, the submarine's stealth capabilities change significantly. In the 0-1km region, the submarine's depth is concentrated between 100-160m; in the 2-3km region, the depth is concentrated between 350-400m, utilizing the seamount's shielding effect for detection. In areas 4km and beyond, the shielding effect weakens, and the submarine's depth strategy tends towards shallower (30-50m) and deeper water layers (350-400m), with a lower probability for strategies targeting intermediate water layers.

[0073] This invention has achieved significant results in seamount terrain conditions, effectively improving the stealth of submarines in complex environments. By modeling the game-theoretic relationship between the submarine and sonar, and combining this with complex ocean topography, the method achieves optimal submarine strategy. Compared to traditional methods, this invention has significant advantages in stealth optimization, computational efficiency, and adaptability, and is suitable for complex seabed environments, providing a reliable, simple, and efficient solution for submarine stealth optimization.

Claims

1. A method for optimizing the stealth of a submarine in complex terrain based on game theory and linear programming, characterized in that, The method comprises the following steps: Step 1: detection probability modeling; A depth range in which the submarine and the sonar operate is determined, and then a complex terrain required for a ray model is created, a sound field calculation is performed to obtain a propagation loss value at different sound source depths and receiving depths, a model is established according to a sonar equation, and a narrowband passive sonar equation under coherent processing and narrowband passive detection is as follows: SE = SL - TL - NL + (AG - BW) - DT (1) Wherein, SE is a signal excess received by the passive sonar; SL is a narrowband sound source level of submarine radiated noise; TL is a sound propagation loss value from a position of the submarine to a position of the sonar; NL is a sea state related ocean background noise; AG is an array gain; BW is an analysis bandwidth; and DT is a detection threshold of a passive sonar detection system; A specific detection probability matrix is obtained according to formula (1), and a specific formula is as formula (2): Step 2: game equilibrium model; The matrix game is applied to the submarine, and a measurement standard for the submarine is defined as a benefit; the benefit is an expected negative value of the passive detection probability of the sonar within a certain radius; in a two-dimensional scene, the benefit is represented by U: Wherein, R represents a left and right calculation radius centered on the submarine in the two-dimensional scene; P is a detection probability of the sonar on the submarine, that is, a detection probability of the submarine; d represents a depth of the sonar, z represents a depth of the submarine, and r represents a distance of the submarine from the sonar; Step 3: linear programming solution of the benefit matrix; The maximum loss minimization of the submarine is as follows: where V is the maximum loss of the submarine, U(z i ,d j ) is the payoff matrix, Z i represents the probability of the submarine choosing depth z i , m represents the number of different depths z i that the submarine can choose, and j represents the number of different depths d j that the sonar can choose. The minimum benefit maximization of the variable sonar carried by the surface ship is as follows: In this case, W is the minimum payoff of the sonar, D i represents the probability of the sonar selecting depth z i ; n represents the number of different depths z i that the sonar can select, and j represents the number of different depths d j that the submarine can select. The optimized depth of the submarine is obtained by solving the benefit matrix U(i,j) through linear programming, and on the basis of the depth, the concealment of the submarine is analyzed.

2. The method according to claim 1, wherein, The ray model is a Bellhop model.

3. A computer program, characterized in that, The computer program enables a computer to execute the method in any one of claims 1 to 2.

4. An electronic device, comprising: It comprises: a processor and a memory; The memory is used to store a computer program, and the processor is used to execute the computer program stored in the memory, so that the electronic device executes the method in any one of claims 1 to 2.

5. A computer-readable storage medium having stored thereon a computer program, characterized in that, The computer program is executed by the processor to implement the method in any one of claims 1 to 2.

6. A chip, characterized by It comprises: a processor, which is used to call and run a computer program from a memory, so that a device installed with the chip executes the method in any one of claims 1 to 2.

7. A computer program product, characterised in that, The computer program product comprises a computer storage medium, the computer storage medium stores a computer program, the computer program comprises instructions executable by at least one processor, and when the instructions are executed by the at least one processor, the method in any one of claims 1 to 2 is implemented.

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