Complex path asf prediction method based on parameter-optimized three-dimensional fdt

Abstract

The application belongs to the technical field of radio wave propagation, and specifically discloses a complex path ASF prediction method based on parameter optimization three-dimensional FDTD, which is aimed at the problem of ASF prediction error caused by the neglect of the lateral diffraction effect in the two-dimensional method, and the problem of poor precision caused by the cumulative phase error when the traditional three-dimensional FDTD method is applied to large-area ASF prediction, and the electromagnetic propagation characteristics of complex irregular terrain are accurately presented by means of three-dimensional terrain modeling, so as to significantly improve the accuracy of the complex path ASF prediction result. In addition, the complex path ASF prediction method based on parameter optimization three-dimensional FDTD introduced by the application also reduces the phase cumulative error by introducing dispersion optimization parameters, so as to improve the calculation efficiency while ensuring the precision. The method of the application can provide an efficient solution for ASF prediction in a large-scale complex terrain scene, and is more suitable for popularization and application in engineering.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of wave propagation, and particularly relates to a complex path ASF prediction method based on parameter optimization three-dimensional FDTD. BACKGROUND

[0002] Loran-C and its enhanced system have become an important part of the positioning, navigation and timing (PNT) system due to its small propagation loss, strong anti-interference ability and long propagation distance. At present, the accurate prediction of the additional secondary factor (ASF) of the traditional Loran-C system has become the key to improving the accuracy of the Loran-C and its enhanced system.

[0003] The modeling of the ASF of the traditional Loran-C system mainly adopts a two-dimensional method. Although this method reduces the computational complexity, it has obvious theoretical deficiencies and is difficult to fully reflect the interaction between low-frequency ground waves and complex terrain environments. When the terrain undulates greatly or there are complex ground objects, the two-dimensional method cannot accurately describe the multipath propagation and terrain diffraction in three-dimensional space, resulting in systematic errors between the theoretical prediction value and the actual observation value of the ASF.

[0004] With the rapid development of computer technology, it is feasible to use three-dimensional modeling to solve the ASF problem. However, due to the vulnerability of the algorithm itself, this method is difficult to extend to the three-dimensional field. The existing three-dimensional parabolic equation (3DPE) method mainly uses the virtual point method to handle the three-dimensional terrain boundary conditions, which introduces an irreducible error for the inclined surface. Compared with other methods, the three-dimensional finite-difference time-domain (3D-FDTD) method can accurately handle complex paths in a natural way. However, due to the cumulative phase error, there is a numerical dispersion error in the 3D-FDTD method, which limits its application in large-scale electrical problems. It is necessary to reduce the grid resolution to keep the memory requirement within a reasonable range in such scenarios, making it difficult to achieve accurate prediction of the ASF in a large area.

[0005] Therefore, it is necessary to provide a complex path ASF prediction method based on parameter optimization three-dimensional FDTD. SUMMARY

[0006] This invention provides a complex path ASF prediction method based on parameter-optimized 3D FDTD. This method uses a 3D approach to model complex terrain, enabling accurate prediction of ASF on long-distance complex and irregular terrain. At the same time, it introduces dispersion optimization parameters to reduce numerical dispersion error, thereby enabling the use of low-resolution grids and reducing the required computational resources.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A complex path ASF prediction method based on parameter-optimized 3D FDTD includes the following steps:

[0009] Step 1. Set and initialize the 3D terrain modeling parameters, which include the mesh parameters of the computational domain, the electrical parameters of the radio wave propagation path, the dispersion optimization parameters, the target optimization angle range, and the parameters of the absorbing boundary PML.

[0010] Step 2. Preprocess the 3D terrain modeling parameters;

[0011] Step 3. Solve the numerical dispersion relation by introducing the three-dimensional FDTD equation with dispersion optimization parameters;

[0012] Step 4. Calculate the dispersion optimization parameters using an optimization algorithm based on the numerical dispersion relation;

[0013] Step 5. Substitute the dispersion optimization parameters into the three-dimensional FDTD equation with PML fitting, and iteratively calculate the electric field components throughout the entire computational domain. , , ;

[0014] Step 6. Update the field source;

[0015] Step 7. Substitute the dispersion optimization parameters into the three-dimensional FDTD equation with PML fitting, and iteratively calculate the magnetic field components throughout the computational domain. , , ;

[0016] Step 8. Determine if the current running time step is equal to the preset running time step; if yes, proceed to step 9; otherwise, update the current running time step. The value is Then proceed to step 5;

[0017] Step 9. Analyze the electric field components throughout the entire computational domain. The ASF is calculated at the positive zero-crossing point of the third carrier period.

[0018] In addition, on the basis of the above-mentioned complex path ASF prediction method based on parameter optimization three-dimensional FDTD, the application further provides a computer device, which comprises a memory and one or more processors.

[0019] The memory stores executable code, and the processor executes the executable code to implement the steps of the above-mentioned complex path ASF prediction method based on parameter optimization three-dimensional FDTD.

[0020] The application has the following advantages:

[0021] As described above, the application provides a complex path ASF prediction method based on parameter optimization three-dimensional FDTD, which is aimed at the problem of ASF prediction error caused by the neglect of transverse diffraction effect in the two-dimensional method and the problem of poor precision caused by cumulative phase error when the traditional three-dimensional FDTD method is applied to large-area ASF prediction, and presents the electromagnetic propagation characteristics of complex irregular terrain accurately with the aid of three-dimensional terrain modeling, thereby significantly improving the accuracy of the complex path ASF prediction result. In addition, the complex path ASF prediction method based on parameter optimization three-dimensional FDTD provided by the application reduces the phase cumulative error by introducing dispersion optimization parameters, and can improve the calculation efficiency while ensuring the precision. The method of the application provides an efficient solution for ASF prediction in large-scale complex terrain scenarios, and is more suitable for popularization and application in engineering. BRIEF DESCRIPTION OF DRAWINGS

[0022] Figure 1 The figure is a flowchart of the complex path ASF prediction method based on parameter optimization three-dimensional FDTD in the embodiment of the application.

[0023] Figure 2 The figure is a principle diagram of the complex path ASF prediction method based on parameter optimization three-dimensional FDTD in the embodiment of the application.

[0024] Figure 3 The figure is an error diagram before dispersion optimization in the embodiment of the application.

[0025] Figure 4 The figure is an error diagram after dispersion optimization in the embodiment of the application.

[0026] Figure 5 The figure is a terrain modeling diagram on an actual terrain path in the embodiment of the application.

[0027] Figure 6 The figure is an ASF prediction result diagram on an actual terrain path in the embodiment of the application. DETAILED DESCRIPTION

[0028] The application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0029] Example 1

[0030] This invention provides a complex path ASF prediction method based on parameter-optimized 3D FDTD. This method solves the problem of dispersion accumulation error and difficulty in large-area calculation in traditional 3D FDTD methods. The overall process is as follows:

[0031] First, the model parameters are input and initialized. Then, parameter preprocessing is performed. Next, the numerical dispersion relation is solved, and the dispersion optimization parameters are calculated. Subsequently, the electromagnetic field components in the entire computational region are calculated. Finally, the calculation is completed, and the ASF (Automatic Field Signal) is output. This invention solves the ASF prediction error caused by neglecting transverse diffraction effects in two-dimensional methods, and overcomes the problem of poor accuracy caused by cumulative phase errors when applying traditional three-dimensional FDTD methods to ASF prediction in large areas. It provides theoretical support and technical solutions for improving the service accuracy of low-frequency radio navigation and timing systems.

[0032] Based on the above inventive concept, the following detailed description, in conjunction with the accompanying drawings, of the complex path ASF prediction method based on parameter-optimized 3D FDTD proposed in this invention. Figure 1 As shown, the method of the present invention includes the following steps:

[0033] Step 1. Set and initialize the 3D terrain modeling parameters. The 3D terrain modeling parameters include the mesh parameters of the computational domain (i.e., the simulation domain), the electrical parameters of the radio wave propagation path (i.e., the simulation path), the dispersion optimization parameters, the target optimization angle range, and the parameters of the absorbing boundary PML.

[0034] Setting 3D terrain modeling parameters includes:

[0035] Set the mesh parameters for the computation domain and define the computation domain as a three-dimensional Cartesian coordinate system. ,in , , The number of grids in each direction is set to , , The mesh partitioning step size is set to... , , In this embodiment, the calculation region is as follows: Figure 2 As shown, where This represents the origin of the coordinate system.

[0036] The electrical parameters that determine the propagation path of radio waves include the relative permittivity in free space. With conductivity relative permittivity of the stratigraphic region With conductivity Relative permittivity of seawater region with conductivity .

[0037] set the dispersion optimization parameters to , , .

[0038] set the target optimization angle range to where represents the polar angle sampling, represents the azimuthal angle sampling.

[0039] set the parameters of the absorbing boundary PML, which includes the number of absorbing boundary layers .

[0040] time setting includes the current running time step , time step and the preset running time step .

[0041] The initialized parameters include:

[0042] electric field components , , .

[0043] magnetic field components , , .

[0044] intermediate variables , , , where = , , , , , .

[0045] electromagnetic field component calculation coefficients , , , .

[0046] intermediate variable calculation coefficients , , .

[0047] current running time step .

[0048] dispersion optimization parameters , , .

[0049] Third carrier cycle zero-crossing time output quantity .

[0050] Step 2. Preprocess the calculation parameters in the three-dimensional terrain modeling parameters.

[0051] Preprocess the PML-related parameters , , :

[0052] (1)

[0053] (2)

[0054] (3)

[0055] wherein, , when , represents the coordinate position of the field source on the x coordinate axis, when , represents the coordinate position of the field source on the y coordinate axis, and when , represents the coordinate position of the field source on the z coordinate axis. represents the coordinate position of the position where the field source is located on the three-dimensional rectangular coordinate system . , represents the wavelength. is a parameter for adjusting the performance of PML. is the thickness of the absorbing boundary. is an integer, is a constant.

[0056] Preprocess the coefficients for calculating electromagnetic field components , , , :

[0057] (4)

[0058] (5)

[0059] (6)

[0060] (7)

[0061] wherein, , and respectively represent the relative permittivity and conductivity in the non-free space. The magnetic permeability of free space. The relative magnetic permeability.

[0062] When the calculation region is free space, the relative permittivity and conductivity are respectively taken as and When the calculation region is a stratum, the relative permittivity and conductivity are respectively taken as and When the calculation region is seawater, the relative permittivity and conductivity are respectively taken as and .

[0063] The calculation coefficient of the intermediate variable , , Preprocessing:

[0064] (8)

[0065] (9)

[0066] (10)

[0067] Step 3. Solve the numerical dispersion relation by introducing a three-dimensional FDTD equation of a dispersion optimization parameter.

[0068] In an electrically lossy medium, introduce a dispersion optimization parameter , , into the three-dimensional FDTD equation to obtain the update formula of the electric field component in the entire calculation region as:

[0069] (11)

[0070] wherein denotes time, and discretize formula (11) as:

[0071] (12)

[0072] wherein and respectively denote the values of and at , is a space index. and respectively denote the grid sizes in the y and z directions, , . ​​

[0073] for In time ,space The value at that location. Indicates time difference. , , For spatial labeling, , , They represent , , Spatial difference in direction.

[0074] Including electric field components , , Magnetic field components , , intermediate variables , , , .

[0075] Simplifying formula (12) yields:

[0076] (13)

[0077] in, express The value at position m. express The value at position m.

[0078] Next, the electric field components across the entire computational domain... and The calculations are performed using a three-dimensional FDTD algorithm in a three-dimensional Cartesian coordinate system. Since the wave propagation along the x, y, and z directions exhibits symmetry, the update formulas for different field components have the same structure. Therefore, the method incorporating dispersion optimization parameters is directly given. and The update formula is:

[0079] (14)

[0080] (15)

[0081] in, The grid size is in the x-direction. = .

[0082] Dispersion optimization parameters , 、 The update formula of the magnetic field component in the whole calculation region is obtained by introducing the three-dimensional FDTD equation

[0083] (16)

[0084] The difference form of formula (16) is

[0085] (17)

[0086] Wherein, represents the value at the point.

[0087] After simplifying formula (17), the following formula is obtained:

[0088] (18)

[0089] Wherein, represents the value at the point. represents the value at the point. Then, the magnetic field components in the whole calculation region, and

[0090] are calculated. Similarly, since it is the FDTD algorithm in the three-dimensional rectangular coordinate system, the wave propagation along the x, y and z directions has symmetry, and the update formula structures of different field components are the same, so the update formula of and introducing the dispersion optimization parameter is directly given as follows:

[0091] (19)

[0092] (20)

[0093] Suppose that the general solution of the plane wave of the electromagnetic field is

[0094] (21)

[0095] Wherein, is the imaginary symbol, is the angular frequency, 、 、 respectively represent the amplitudes of the electric field components along the x, y and z directions, 、 、 、 ,​​​​​​ respectively represent the amplitudes of the magnetic field components along the , , directions, , , respectively represent the wave numbers along the , , directions.

[0096] Substitute equation (21) into equation (13) to obtain:

[0097] (22)

[0098] After simplification, we obtain:

[0099] (23)

[0100] Substitute equation (21) into equations (14), (15), (18), (19), (20) to obtain:

[0101] (24)

[0102] (25)

[0103] (26)

[0104] (27)

[0105] (28)

[0106] Let , where .

[0107] Let the variable be:

[0108] (29)

[0109] Let:

[0110] (30)

[0111] (31)

[0112] Let , , then equation (30) and equation (31) are simplified to and , where represents the size of identity matrix.

[0113] Let:

[0114] (32)

[0115] (33)

[0116] matrix and matrix are multiplied to obtain:

[0117] (34)

[0118] wherein, , .

[0119] then formulas (23) to (28) are transformed into:

[0120] (35)

[0121] wherein, is expressed as:

[0122] (36)

[0123] the determinant of the matrix shown in formula (36) is:

[0124] (37)

[0125] wherein, denotes calculating the determinant of the matrix.

[0126] Let the determinant shown in formula (36) be equal to 0, and the numerical dispersion relation is obtained by solving the determinant:

[0127] (38)

[0128] wherein, denotes the speed of light, denotes the numerical wave number, is the polar angle, is the azimuthal angle.

[0129] Step 4. Calculate the dispersion optimization parameters by using an optimization algorithm through the numerical dispersion relation.

[0130] Step 4.1. Set the target optimization angle range.

[0131] In formula (38), when , , With When approaching 0, the numerical dispersion relation is converted to the analytical dispersion relation:

[0132] (39)

[0133] where, is the analytical wave number. Define the numerical dispersion error as:

[0134] (40)

[0135] In order to reduce the numerical dispersion error, the average dispersion error of a preset target optimization angle is defined as the objective function:

[0136] (41)

[0137] where, , , the preset target optimization angle range is:

[0138] (42)

[0139] where, represents the polar angle sampling step, represents the azimuthal angle sampling step.

[0140] In this embodiment, when the target angle range is , , , , , . In region I, = =54m, =18m, in region II, = = =54m.

[0141] In order to obtain the artificial anisotropy parameters , and , the average dispersion error ​Formula (41) is a typical multi-parameter nonlinear constrained optimization problem. The method of this invention determines the anisotropic parameters by combining the Nelder-Mead method (simplex search) and the sequential quadratic programming (SQP) algorithm. The method of combining Nelder-Mead and sequential quadratic programming effectively balances global search and local convergence, reducing the risk of getting trapped in local optima and providing high-precision solutions for multi-parameter constrained optimization problems. The specific solution steps are as follows:

[0142] Step 4.2. Set initial parameters The convergence criterion of the optimization algorithm is defined as the Lagrange function with respect to... The gradient is less than the preset gradient, where , , These represent the dispersion optimization parameters. , , The initial value.

[0143] Step 4.3. Optimize parameters based on dispersion Equation (28) is solved using the Nelder-Mead method, and the incident angle is scanned to obtain the optimal wavenumber for each propagation direction. ,in , , These represent the dispersion optimization parameters. , , No. The value after the next update This indicates the number of times the dispersion optimization parameters have been updated. Then, the average dispersion error within the target optimization angle range is calculated. The objective function of the SOP algorithm, i.e., the optimization algorithm.

[0144] Step 4.4. Construct a quadratic programming subproblem for the constrained optimization problem in formula (28) using a sequential quadratic programming algorithm, and update the dispersion optimization parameters by solving this quadratic programming subproblem. ,in , , These represent the dispersion optimization parameters. , , No. The value after the next update.

[0145] Step 4.5. [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require The dispersion optimization parameters after the latest update As The current value, and determine the current whether the convergence criterion of the optimization algorithm is satisfied. If the convergence criterion is satisfied, the current value of the dispersion optimization parameter is taken as the result of the calculation of the dispersion optimization parameter. Otherwise, return to step 4.3.

[0146] Step 5. Substitute the dispersion optimization parameter into the three-dimensional FDTD equation with PML, and iteratively calculate the electric field components in the entire calculation region

[0147] Substitute the dispersion optimization parameter calculated in step 4 into the three-dimensional FDTD equation with PML, and iteratively calculate the electric field components in the entire calculation region The specific calculation formula is:

[0148] (43)

[0149] The calculation formula of the intermediate variables and is:

[0150] (44)

[0151] (45)

[0152] wherein represents the value of at , and represents the value of at .

[0153] represents the central difference of y at time , space , and . represents the central difference of z at time , space , and

[0154] . The calculation formula of the intermediate variables and is:

[0155] (46)

[0156] (47)

[0157] The calculation formula of the intermediate variables and is:

[0158] ​​​ (48)

[0159] (49)

[0160] where, represents the value at, represents the value at, represents the value at. represents the value at. represents the value at, represents the value at, represents the value at. represents the central difference of y at,

[0161] represents the central difference of z at,

[0162] The electric field components and in the calculation region are calculated, and the update formulae of and are:

[0163] (50)

[0164] (51)

[0165] where, , , , and the calculation modes of formula (44) and formula (45) are the same.

[0166] Step 6. Update the field source.

[0167] The added field source is the Loran-C signal, and the field source is excited by the electric field component , and the current waveform of the field source is represented as:

[0168] (52)

[0169] where, .

[0170] When ,​​​​​​​​​​​ When , .

[0171] In this embodiment, the signal frequency of the field source is updated , the Gaussian pulse width , the Gaussian pulse delay , respectively: , =0, . And according to the number of field sources, the position of the field source and the type of the field source set in the three-dimensional terrain modeling parameters, the field source is updated. For example, the number of field sources is set to . The field source position , , , , , is set to: , , , , , . The field source type is set to soft source.

[0172] Step 7. Substitute the dispersion optimization parameters into the three-dimensional FDTD equation with PML, and iteratively calculate the magnetic field components , , in the entire calculation area.

[0173] Substitute the dispersion optimization parameters into the three-dimensional FDTD equation with PML, and iteratively calculate the magnetic field components in the entire calculation area. The calculation formula is:

[0174] (53)

[0175] Wherein, the calculation formula of the intermediate variable and is:

[0176] (54)

[0177] (55)

[0178] Wherein, represents the central difference of y at time , space , represents the central difference of y at time , space , ​The central difference of z.

[0179] Intermediate variable With The calculation formula is:

[0180] (56)

[0181] (57)

[0182] Wherein, Indicates the time , space Place The central difference of y, Indicates the time , space Place The central difference of z.

[0183] The electric field components in the calculation region And The update formula of And Is obtained:

[0184] (58)

[0185] (59)

[0186] Wherein, , , , And formula (54), formula (55) calculation method.

[0187] Step 8. Determine whether the current running time step is equal to the preset running time step. If yes, go to step 9. Otherwise, update the value of the current running time step To And go to step 5.

[0188] After the magnetic field component calculation is completed, it is judged whether the current running time step is equal to the preset running time step. That is, it is judged whether the current running time step n satisfies , if not, continue to repeat steps 5 to 8, and update the value of the current running time step Until it reaches the preset value, that is, the preset running time step. If , end the loop.

[0189] Step 9. Calculate the ASF through the third carrier cycle positive zero point moment of the electric field component In the whole calculation region.

[0190] Output attenuation factor phase The calculation formula is:

[0191] (60)

[0192] Wherein, is the electric field component is the third carrier cycle positive zero moment, is the great circle distance of wave propagation.

[0193] When the three-dimensional terrain modeling is set to the actual complex propagation path, that is, the calculation area is set to the complex terrain containing free space and stratum area, the attenuation factor phase of Loran-C signal on the actual complex propagation path is obtained by formula (60) .

[0194] When the three-dimensional terrain modeling is set to the pure seawater propagation path, that is, the calculation area is set to the seawater area, the attenuation factor phase of Loran-C signal on the pure seawater propagation path is obtained by formula (60) .

[0195] Then the ASF is calculated, and the following is obtained:

[0196] (61)

[0197] Wherein, is the of Loran-C signal on the actual propagation path, is the of Loran-C signal on the pure seawater propagation path.

[0198] In addition, in order to verify the effectiveness of the method of the application, the following specific experiments are also given.

[0199] Experiment 1 is the error comparison before and after dispersion optimization.

[0200] Figure 2 The grid size in the middle grid is set to: region I is = =54m, =18m, region II is = = =54m. The time step = .

[0201] The dispersion optimization parameters obtained are: region I is , , . Region II is , , .

[0202] Figure 3 and Figure 4 The two methods are: the complex path ASF prediction method proposed in this invention, namely the parameter-optimized-3DFDTD method (PO-3D-FDTD), and the traditional 3D-FDTD method. At that time, the dispersion error increases with... The changing ASF prediction results. (From...) Figure 3 and Figure 4 It can be seen that, compared with the traditional 3D-FDTD method, the dispersion error of the method of the present invention is smaller, with the maximum dispersion error reduced from 0.052% and 0.040% to 0.011% and 0.016%, respectively.

[0203] Experiment 2 shows the ASF prediction results along the propagation path of a Gaussian mountain range.

[0204] The computational region size is: {-2.16km≦} ≤102.6km, -4.86km≦ ≤4.86km, -1.44km≦ The grid partition size is ≤9.72km. When z ≤ 2.16km, = =54m, =18m. In other areas... = = =54m. Time step = Relative permittivity of the stratigraphic region electrical conductivity Relative permittivity of seawater region electrical conductivity Number of absorbing boundary layers .

[0205] like Figure 5 As shown, the terrain model is set to the actual propagation path from the Pucheng Loran launch station to the Qinling Mountains, and the ASF is predicted using the method proposed in this invention.

[0206] Figure 6 The PO-3D-FDTD method proposed in this invention is different from the traditional uniform fine grid 3D FDTD method (3D-FDTD: Fine grids), the traditional non-uniform grid 3D-FDTD method, and the fine grid cylindrical coordinate system 2D-FDTD method (2D-FDTD). ): Fine grids)in x direction from y=0 position, where and are the coordinates in the two-dimensional cylindrical coordinate system.

[0207] are the coordinates in the two-dimensional cylindrical coordinate system. Figure 5 and Figure 6 It can be seen that the accuracy of the cylindrical coordinate system 2D-FDTD method of ignoring the lateral terrain effect fine grid is the lowest. The ASF prediction result of the PO-3D-FDTD method of the method of the application is very consistent with the ASF prediction result of using the conventional uniform fine grid three-dimensional FDTD method, and the conventional non-uniform grid 3D-FDTD method gradually loses accuracy when the propagation range increases, which is caused by the cumulative numerical dispersion effect.

[0208] The method of the application overcomes the deficiency of the conventional two-dimensional modeling method in complex propagation path ASF prediction. Compared with the 2D-FDTD method, the method of the application can accurately represent the lateral diffraction effect of the radio wave in the irregular and discontinuous area of the terrain through three-dimensional terrain modeling, and obtain more accurate ASF prediction result. In addition, the method of the application also reduces the phase accumulation error by introducing the dispersion optimization parameter, improves the calculation efficiency, and does not lose the accuracy.

[0209] Embodiment 2

[0210] This embodiment 2 describes a computer device, which comprises a memory and one or more processors.

[0211] The executable code is stored in the memory, and when the processor executes the executable code, the steps of the complex path ASF prediction method based on parameter optimization three-dimensional FDTD in the above-mentioned embodiment 1 are implemented.

[0212] The computer device in this embodiment is any device or apparatus with data processing capability, which will not be described here.

[0213] Of course, the above description is only for the preferred embodiments of the application, and the application is not limited to the above-mentioned embodiments. It should be noted that any person skilled in the art can make all equivalent substitutions, obvious modifications under the teaching of the present application, which are within the scope of the present application, and should be protected by the present application.

Claims

1. A method for complex path ASF prediction based on parameter-optimized 3D FDTD, characterized in that, Includes the following steps: Step 1. Set and initialize the 3D terrain modeling parameters, which include the mesh parameters of the computational domain, the electrical parameters of the radio wave propagation path, the dispersion optimization parameters, the target optimization angle range, and the parameters of the absorbing boundary PML. Step 2. Preprocess the 3D terrain modeling parameters; Step 3. Solve the numerical dispersion relation by introducing the three-dimensional FDTD equation with dispersion optimization parameters; Step 4. Calculate the dispersion optimization parameters using an optimization algorithm based on the numerical dispersion relation; Step 5. Substitute the dispersion-optimized parameters into the three-dimensional FDTD equation with PML, and iteratively calculate the electric field components in the entire calculation region , , ; Step 6. Update the field source; Step 7. Substitute the dispersion-optimized parameters into the three-dimensional FDTD equation with PML, and iteratively calculate the magnetic field components throughout the computational region , , ; Step 8. Determine whether the current running time step is equal to the preset running time step; if yes, go to step 9; otherwise, update the current running time step the value of and go to step 5; Step 9. Calculate the ASF at the instant of the third carrier cycle positive going zero crossing by the electric field component throughout the computational area of the third carrier cycle positive going zero crossing; Step 1 specifically involves: Setting 3D terrain modeling parameters includes: The grid parameters of the calculation region are set, and the calculation region is defined as a three-dimensional rectangular coordinate system wherein , , The grid numbers in the x, y and z directions are respectively set as , , The grid division steps in the x, y and z directions are respectively set as , , ; Setting the electrical parameters of the wave propagation path, including the relative permittivity of free space with electrical conductivity , the relative permittivity of the formation region with electrical conductivity , the relative permittivity of the sea water region with electrical conductivity ; The dispersion optimization parameter is set as , , ; The target optimization angle range is set as wherein denotes polar angle sampling, denotes azimuth angle sampling; Parameters of the absorbing boundary PML are set, which include the number of absorbing boundary layers ; The time setting includes a current running time step , a time step length , and a preset running time step ; The initialization parameters include: electric field component , , ; magnetic field component , , ; intermediate variable , , , wherein = , , , , , ; Electromagnetic field component calculation coefficient , , , ; Intermediate variable calculation coefficient , , ; current run time step ; Dispersion-optimized parameters , , ; Third carrier cycle zero crossing time output quantity ; Step 9 specifically involves: Output attenuation factor phase The formula for calculating the output attenuation factor phase is: (60) wherein is the electric field component is the third carrier cycle positive going zero crossing time, is the great circle distance of wave propagation, denotes the speed of light; When the three-dimensional terrain modeling is set to the actual complex propagation path, i.e. the calculation region is set to the complex terrain containing the free space and the stratum region, the phase of the attenuation factor of the Loran-C signal on the actual complex propagation path is obtained by formula (60) ; When the three-dimensional terrain modeling setting is set to a pure seawater propagation path, i.e., the calculation region is set to a seawater region, the phase of the attenuation factor of the Loran-C signal on the pure seawater propagation path is obtained by formula (60) ; Then calculate ASF and get: (61)。 2. The method of claim 1, wherein, Step 2 specifically involves: PML-related parameters , , Preprocessing: (1) (2) (3) wherein, represents the coordinate position of the field source on the x-coordinate axis when represents the coordinate position of the field source on the y-coordinate axis when represents the coordinate position of the field source on the z-coordinate axis when represents the coordinate position of the field source on the x-coordinate axis when represents the coordinate position of the field source on the y-coordinate axis when represents the coordinate position of the field source on the z-coordinate axis when , represents the wavelength; is a parameter for adjusting the PML performance; is the thickness of the absorbing boundary; is an integer, is a constant; Calculating coefficients for electromagnetic field components 、 、 、 Preprocessing: (4) (5) (6) (7) wherein, , with denote the relative permittivity and conductivity, respectively, in a non-free space; denotes the permeability of free space; denotes the relative permeability; When the calculation region is free space, the relative permittivity and conductivity are taken as and respectively; when the calculation region is a stratum, the relative permittivity and conductivity are taken as and respectively; when the calculation region is seawater, the relative permittivity and conductivity are taken as and respectively; Calculating coefficients for intermediate variables , , Preprocessing: (8) (9) (10)。 3. The method of claim 2, wherein, Step 3 specifically involves: In electrically lossy media, the dispersion optimization parameters are... , , By introducing the three-dimensional FDTD equation, the electric field components over the entire computational domain are obtained. The update formula is: (11) wherein denotes time, formula (11) is discretized as: (12) wherein, and respectively represent and the value at , is a space index; and respectively represent the grid size in the y, z direction, = , = ; For The value at time , space ; Indicates the time difference; , , Space label, , , Respectively , , Directional space difference; including an electric field component , , , a magnetic field component , , , an intermediate variable , , , ; Simplifying formula (12) yields: (13) wherein represents the value at m; represents the value at m; Optimizing parameters of dispersion 、 、 The three-dimensional FDTD equation is introduced to obtain the update formula of the electric field components in the entire calculation region and ​ (14) (15) wherein is the grid size in the x-direction, = 0.5 ; Dispersion optimization parameters , , By introducing the three-dimensional FDTD equations, the magnetic field components over the entire computational domain are obtained. The update formula is: (16) The difference form of formula (16) is: (17) wherein represents at the value at Simplifying formula (17) yields: (18) wherein represents the value at m; represents the value at m; Optimizing parameters of dispersion 、 、 The three-dimensional FDTD equation is introduced to obtain the update formula of the magnetic field components and in the entire calculation region as follows: (19) (20) Assume the universal solution for plane waves in an electromagnetic field is: (21) wherein is the imaginary unit, is the angular frequency, , , denote the amplitudes of the electric field components along , , directions, , , denote the amplitudes of the magnetic field components along , , directions, , , denote the wave numbers along , , directions; Substituting formula (21) into formula (13) yields: (22) After simplification, we get: (23) Substituting formula (21) into formulas (14), (15), (18), (19), and (20), we get: (24) (25) (26) (27) (28) Let where ; Let the variable be: (29) make: (30) (31) Let , , then formula (30) and formula (31) are simplified as and , where denotes a unit matrix of size . make: (32) (33) matrix and the matrix results in: (34) wherein , ; Then formula (23) to formula (28) are transformed into: (35) in, Represented as: (36) The determinant of the matrix shown in formula (36) for: (37) in, This indicates finding the determinant of a matrix; Setting the determinant shown in formula (36) to 0, solving the determinant yields the numerical dispersion relation as follows: (38) in, Represents the numerical wavenumber. Polar angle, It is the azimuth angle.

4. The complex path ASF prediction method based on parameter-optimized 3D FDTD according to claim 3, characterized in that, Step 4 specifically involves: Step 4.

1. Set the target optimization angle range; In formula (38), when , , and As the value approaches 0, the numerical dispersion relation transforms into the analytical dispersion relation: (39) in, For the analytical wavenumber; define the numerical dispersion error. for: (40) Will Average dispersion error at a preset target optimization angle Define the objective function as: (41) in, , The preset target optimization angle range is: (42) in, Indicates the polar angle sampling step size. Indicates the azimuth sampling step size; Step 4.

2. Set initial parameters The convergence criterion of the optimization algorithm is defined as the Lagrange function with respect to... The gradient is less than the preset gradient, where , , These represent the dispersion optimization parameters. , , The initial value; Step 4.

3. Optimize parameters based on dispersion Equation (28) is solved using the Nelder-Mead method, i.e., the simplex search method, and the incident angle is scanned to obtain the optimal wavenumber for each propagation direction. ,in , , These represent the dispersion optimization parameters. , , No. The value after the next update This indicates the number of times the dispersion optimization parameters have been updated; then, the average dispersion error within the target optimization angle range is calculated. As the objective function of the SOP algorithm, i.e., the optimization algorithm; Step 4.

4. Construct a quadratic programming subproblem for the constrained optimization problem in formula (28) using the Sequential Quadratic Programming (SQP) algorithm, and update the dispersion optimization parameters by solving this quadratic programming subproblem. ,in , , These represent the dispersion optimization parameters. , , No. The value after the next update; Step 4.

5. [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require The dispersion optimization parameters after the latest update As The current value, and determine the current Does it meet the convergence criterion of the optimization algorithm? If it meets the convergence criterion, then... The current value is used as the calculation result of the dispersion optimization parameter; otherwise, return to step 4.

3.

5. The complex path ASF prediction method based on parameter-optimized 3D FDTD according to claim 4, characterized in that, Step 5 specifically involves: Substitute the dispersion optimization parameters calculated in step 4 into the three-dimensional FDTD equation of the PML fit, and iteratively calculate the electric field components throughout the entire computational domain. The specific calculation formula is as follows: (43) Among them, intermediate variables and The calculation formula is: (44) (45) in, express exist The value at that location, express exist The value at; Indicates time ,space Place Central difference of y Indicates time ,space Place Central difference with respect to z; and The calculation formula is: (46) (47) intermediate variables and The calculation formula is: (48) (49) in, express exist The value at that location, express exist The value at; express exist The value at that location, express exist The value at; Indicates time ,space Place Central difference of y Indicates time ,space Place Central difference with respect to z; For the electric field components within the computational region and Calculations were performed to obtain and The update formula is: (50) (51)。 6. The complex path ASF prediction method based on parameter-optimized 3D FDTD according to claim 5, characterized in that, Step 6 specifically involves: The added field source is a Loran-C signal, and the field source consists of an electric field component. Excitation, current waveform of the field source Represented as: (52) in, ; when hour, ;when hour, .

7. The complex path ASF prediction method based on parameter-optimized 3D FDTD according to claim 6, characterized in that, Step 7 specifically involves: Substituting the dispersion optimization parameters into the three-dimensional FDTD equations with PML, the magnetic field components throughout the computational domain are iteratively calculated. The calculation formula is: (53) Among them, intermediate variables and The calculation formula is: (54) (55) in, Indicates time ,space Place Central difference of y Indicates time ,space Place Central difference with respect to z; intermediate variables and The calculation formula is: (56) (57) in, Indicates time ,space Place Central difference of y Indicates time ,space Place Central difference with respect to z; For the electric field components within the computational region and Calculations were performed to obtain and Update formula: (58) (59)。 8. A computer device comprising a memory and one or more processors, wherein the memory stores executable code, characterized in that, When the processor executes the executable code, it implements the steps of the complex path ASF prediction method based on parameter optimization three-dimensional FDTD as described in any one of claims 1 to 7.

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