A three-dimensional parabolic equation prediction method for low-frequency ground wave propagation over a relief terrain
By employing explicit finite difference discretization and unilateral approximation merging, the accuracy and efficiency issues of low-frequency ground wave propagation characteristic prediction under undulating terrain are resolved, achieving efficient and accurate low-frequency ground wave propagation characteristic prediction.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- UNIV OF JINAN
- Filing Date
- 2026-04-28
- Publication Date
- 2026-05-29
AI Technical Summary
Existing methods for predicting the propagation characteristics of low-frequency ground waves are not accurate enough in undulating terrain. Traditional 3D-PE models have low solution efficiency, high computational complexity, and large memory requirements, making them difficult to meet the needs of practical engineering applications.
By combining explicit finite difference discretization with 3D impedance boundary conditions and unilateral approximation merging, matrix inversion is avoided, and the low-frequency ground wave propagation characteristics under undulating terrain are solved by explicit recursive formulas.
It significantly reduces computational complexity and memory requirements, improves computational efficiency, accurately predicts the impact of irregular terrain on low-frequency ground waves, and enhances prediction accuracy.
Smart Images
Figure CN122115767A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of radio wave propagation technology, specifically relating to a three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain. Background Technology
[0002] Low-frequency ground waves, with their characteristics of low propagation loss, strong diffraction capability, wide coverage, and anti-interference, have become the core propagation method in fields such as long-distance radio communication, navigation and timing, and marine monitoring. Accurate prediction of their propagation characteristics is the key foundation for ensuring that low-frequency radio navigation and timing systems can achieve long-distance high-precision positioning, navigation, and timing functions.
[0003] In practical engineering applications, the propagation path of low-frequency ground waves includes undulating terrain such as mountains and hills. The irregular 3D undulations of the terrain cause physical effects such as reflection, diffraction, and multipath interference during the propagation of radio waves. In particular, the lateral undulations of the terrain can significantly affect key propagation parameters such as the magnetic field strength and propagation delay of low-frequency ground waves. Therefore, how to accurately predict the impact of 3D undulating terrain on the propagation of low-frequency ground waves has become the core issue for improving the accuracy of propagation characteristic prediction.
[0004] Due to the complexity of 3D terrain modeling and limitations in computer resources, the prediction of low-frequency ground wave propagation characteristics is currently dominated by 2D approximation models, which are difficult to accurately predict complex electromagnetic environments with significant 3D effects such as lateral reflection and scattering. To overcome the inherent limitations of 2D models, the 3D parabolic equation 3D-PE model has been gradually applied to the analysis of low-frequency ground wave propagation characteristics under undulating terrain. This model can accurately predict the interaction between terrain and electromagnetic waves, and is an effective tool for analyzing the low-frequency ground wave propagation characteristics under complex undulating terrain.
[0005] However, most of the mainstream solution methods for existing 3D-PE models use implicit algorithms. When dealing with 3D impedance boundary conditions, these algorithms involve complex matrix inversion operations. When facing complex 3D scenes with undulating terrain, the matrix dimension grows exponentially, which not only significantly extends the computation time but also requires extremely high memory, making it difficult to meet the needs of practical engineering applications and severely limiting the promotion and application of 3D-PE models.
[0006] In summary, given the insufficient accuracy of existing low-frequency ground wave propagation characteristic prediction methods in undulating terrain, and the technical pain points of traditional 3D-PE models such as low solution efficiency, high computational complexity, and large memory requirements, developing a 3D-PE model prediction method that can accurately characterize the impact of 3D undulating terrain on low-frequency ground wave propagation, and is computationally efficient and easy to implement numerically, has become an urgent need in the field of radio wave propagation technology. This method is of great practical significance for promoting the engineering application of low-frequency ground waves in various undulating terrain scenarios. Summary of the Invention
[0007] The purpose of this invention is to propose a three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain. This method avoids matrix inversion by combining explicit finite difference discretization with one-sided approximation merging of 3D impedance boundary conditions, which can significantly reduce computational complexity and memory consumption and improve computational efficiency.
[0008] To achieve the above objectives, the present invention adopts the following technical solution: A three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain includes the following steps: Step 1. Input the basic model parameter file. The input basic model parameter file includes the mesh parameters of the computational domain, the electrical parameters of the electromagnetic wave propagation path, and the excitation source parameters; Step 2. Solve for the initial field using the flat ground formula, and use the initial field as the source field of the 3D conformal mapping model - parabolic equation, i.e., the 3D-CMM-PE equation; Step 3. Derive the Taylor-type 3D-CMM-PE equations for undulating terrain conditions; Step 4. By implementing explicit differential discretization on the Taylor-type 3D-CMM-PE equation, an explicit iterative step scheme for the field quantities is derived, and absorbing boundary conditions are set to eliminate truncated reflections at the boundary of the computational domain. Step 5. Based on the one-sided approximation method, the 3D impedance boundary conditions are explicitly discretized and incorporated into the field value recursion relationship to obtain an explicit recursion formula with boundary constraints. Step 6. Using the source field as the initial value, solve the problem point by point using an explicit recursive formula with boundary constraints to obtain the field distribution results in the entire computational domain; Step 7. Based on the field distribution results within the entire calculation area, solve for the magnetic field strength and second time delay of low-frequency ground waves propagating on undulating terrain.
[0009] Furthermore, based on the three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain described above, this invention also proposes a computer device, which includes a memory and one or more processors. The memory stores executable code, and when the processor executes the executable code, it implements the steps of the three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain, as described above.
[0010] Furthermore, based on the aforementioned method for predicting the propagation characteristics of low-frequency ground waves in undulating terrain using a three-dimensional parabolic equation, this invention also proposes a computer-readable storage medium storing a program thereon; when executed by a processor, this program is used to implement the steps of the aforementioned method for predicting the propagation characteristics of low-frequency ground waves in undulating terrain using a three-dimensional parabolic equation.
[0011] The present invention has the following advantages: As described above, this invention discloses a three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain. This method introduces the conformal mapping model (CMM) for diffraction propagation into terrain modeling, establishing a 3D-CMM-PE equation on undulating terrain. By employing an explicit finite difference algorithm to discretize the Taylor-type 3D-CMM-PE equation, and combining it with explicitly discretized 3D impedance boundary conditions based on a one-sided approximation method, the influence of dispersion error is reduced. Simultaneously, it can predict the lateral diffraction effect generated by the interaction between irregular terrain and low-frequency ground waves. This invention avoids the complex matrix inversion operations required by traditional implicit algorithms; all recursive calculations are explicit operations, significantly reducing computational complexity and memory requirements, greatly improving computational efficiency, and providing an effective technical means for the engineering prediction and application of low-frequency ground wave propagation. Attached Figure Description
[0012] Figure 1 This is a flowchart of a three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain, as described in an embodiment of the present invention.
[0013] Figure 2 The method of this invention and the conventional method are used to predict the magnetic field strength on a plane. The result is illustrated in the diagram.
[0014] Figure 3 To employ the method of this invention compared to conventional methods in Predicted quadratic delay on the plane The result is illustrated in the diagram.
[0015] Figure 4 In this embodiment of the invention, when an isolated Gaussian mountain exists in the propagation path, the 3D-FDTD method is used to predict the second time delay of low-frequency ground waves. The result is illustrated in the diagram.
[0016] Figure 5 In this embodiment of the invention, when an isolated Gaussian mountain exists in the propagation path, the 3D-CMM-PE method, i.e., the method of this invention, is used to predict the second time delay of low-frequency ground waves. The result is illustrated in the diagram.
[0017] Figure 6 In this embodiment of the invention, when an isolated Gaussian mountain exists in the propagation path, the 2D-PE method is used to predict the second time delay of low-frequency ground waves. The result is illustrated in the diagram. Detailed Implementation
[0018] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments: Example 1 This invention specifically provides a three-dimensional conformal mapping model-parabolic equation (3D-CMM-PE) method for predicting the propagation characteristics of low-frequency ground waves in undulating terrain based on an explicit finite difference (FD) algorithm. This method improves upon the 2D-PE method's inability to consider the influence of lateral irregular terrain on low-frequency ground waves, overcomes the shortcomings of traditional 3D-PE methods in the accuracy of predicting the propagation characteristics of low-frequency ground waves in undulating terrain, and avoids the complex matrix inversion operations required by traditional implicit algorithms. All recursive calculations are explicit operations, significantly reducing computational complexity and memory requirements, and greatly improving computational efficiency.
[0019] The overall process of this invention is as follows: First, the 3D-CMM-PE form under irregular terrain conditions is derived, and then an explicit difference scheme for 3D-CMM-PE is proposed. Based on this, by combining the one-sided approximation and the explicit difference scheme, the 3D impedance boundary conditions are effectively incorporated into the explicit finite difference algorithm. Subsequently, the field distribution within the computational region is solved point by point, and based on the full-field calculation results, the magnetic field strength for radio wave propagation is further solved. and secondary delay .
[0020] The significant advantage of this invention lies in its ability to efficiently and accurately predict the propagation characteristics of low-frequency ground waves in undulating terrain environments by introducing an explicit finite difference algorithm and integrating 3D impedance boundary processing. This method avoids the complex matrix inversion process of traditional implicit algorithms, significantly reducing computational complexity and memory requirements, thereby improving overall computational efficiency and providing effective technical support for the engineering prediction and practical application of low-frequency ground wave propagation characteristics.
[0021] Based on the above inventive concept, the following detailed description of the three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain proposed in this invention, with reference to the accompanying drawings, will be provided.
[0022] like Figure 1 As shown, the three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain includes the following steps:
[0023] Step 1. Initialize model parameters: Input the basic model parameter file. The contents of the input basic model parameter file include the mesh parameters of the computational domain, the electrical parameters of the electromagnetic wave propagation path, and the excitation source parameters.
[0024] In this embodiment, step 1 specifically includes: In 3D Cartesian coordinate system In this context, the computational region size is defined as... ,in Horizontal Number of grid cells in each direction For vertical Number of grid cells in each direction, For height Number of grid cells in each direction; , , They are respectively , , Mesh subdivision step size in the direction.
[0025] The electrical parameters of the radio wave propagation path include the relative permittivity of free space. With conductivity and the relative permittivity of mountainous terrain regions With conductivity .
[0026] The excitation source uses a vertical electric dipole, and the parameters of the excitation source include current. , charge spacing and the height of the source above the ground .
[0027] Step 2. Solve for the initial field using the flat ground formula, and use the initial field as the source field of the 3D conformal mapping model - parabolic equation, i.e., the 3D-CMM-PE equation.
[0028] In this embodiment, step 2 specifically includes: In a rectangular coordinate system In the middle, the time harmonic factor is set to ,in The imaginary unit, Angular velocity, It is a time variable.
[0029] Solving the initial field and ground wave magnetic field components of 3D-CMM-PE using the flat ground formula. This can be expressed by formula (1): (1) in, The distance is the great circle distance. Vacuum wavenumber, Ground wavenumber; From source point to observation point The straight-line distance at the location; The straight-line distance from the mirror image of the source to the observation point; For Fresnel integration, It is an intermediate variable.
[0030] Great circle distance Defined as: .
[0031] Fresnel points Defined as: .
[0032] intermediate variables Defined as: .
[0033] in, Location of the observation point This represents the vertical height coordinate.
[0034] For the preset initial distance The initial field of the low-frequency ground wave is obtained from formula (1). As shown in formula (2): (2)
[0035] in, express The Earth wave magnetic field component at that location.
[0036] Initial field As the source field of the 3D-CMM-PE equation.
[0037] Step 3. Derive the Taylor-type 3D-CMM-PE equations for undulating terrain conditions, where the 3D-CMM-PE equations are the 3D-PE equations based on the conformal mapping model, and the undulating terrain conditions are irregular terrain conditions.
[0038] The Conform Map Model (CMM) is a classic terrain model proposed by Belilis and Tappett. Its core is based on coordinate transformation. Irregular terrain is mapped to flat ground, resulting in a 3D conformal mapping model-parabolic equation adapted to undulating terrain; Taylor-type 3D-CMM-PE equation refers to the 3D-PE equation after CMM transformation, which uses Taylor series expansion to approximate the lateral partial derivative terms and retains them to second-order accuracy.
[0039] In this embodiment, step 3 specifically includes: The equation for a 3D parabola on a flat surface is expressed in the form of formula (3): (3) in, The transverse Laplace operator is defined as follows: ; The target field to be solved. The refractive index of the medium, , The relative permittivity of the medium, such as when propagating in free space, i.e., air. . This refers to the relative permeability, such as when propagating in free space, i.e., air. , It represents the relative permeability when propagating in the air.
[0040] The conformal mapping model (CMM) is used to map the original undulating terrain to a flat ground. The transformation relationship between height and distance variables is shown in formula (4): (4) in, The terrain height function is used; after coordinate transformation, the spatial step size satisfies... , , .
[0041] Substituting equation (4) into equation (3) simplifies the equation, resulting in the Taylor-type 3D-CMM-PE equation for undulating terrain conditions, as shown in equation (5): (5) Among them, the terrain height function Simplified to .
[0042] The 3D-CMM-PE form constructed by formula (5) under irregular terrain is optimized for the scenario where the lateral terrain undulation is significant in 3D undulating terrain. By retaining the partial derivative terms of the terrain height function with respect to the y direction in the 3D-CMM-PE equation, the prediction accuracy of low-frequency ground wave propagation characteristics is effectively guaranteed.
[0043] Step 4. Propose an explicit difference scheme for 3D-CMM-PE: By implementing explicit difference discretization on the Taylor-type 3D-CMM-PE equation, an explicit iterative step scheme for the field quantities is derived. At the same time, absorbing boundary conditions are set to eliminate truncated reflections at the boundary of the computational domain.
[0044] In this embodiment, step 4 specifically includes: In the computational region, along the direction of propagation Discretize the field as If there are 1 point, then the waveguide is composed of a continuous arrangement of 1 point. It consists of several horizontal planes, among which Indicates the first The point is at The coordinates of the direction satisfy ; in the horizontal In the direction, the field is discretized into One point, Indicates the first The point is at The coordinates of the direction satisfy At a height In the direction, the field is discretized into One point, Indicates the first The point is at The coordinates of the direction satisfy .
[0045] According to formula (5), for The first-order partial derivatives are approximated using the forward difference method, resulting in: (6) in, Represents field quantity At grid points The value at that location, , , .
[0046] about The first-order partial derivative is approximated using the forward difference method as follows: (7) about The second-order partial derivatives are approximated using the forward difference method as follows: (8) about The second-order partial derivatives are approximated using the forward difference method as follows: (9) about and The second-order mixed partial derivative is approximated using the forward difference method as follows: (10) Substituting equations (6) to (10) into equation (5), we get: (11) in, , , .
[0047] Simplifying formula (11), we get: (12) Simplifying formula (12), we get: (13) make , , , , The explicit iterative step scheme for the field quantity is obtained as follows: (14) It should be noted that when using the method of this invention to solve 3D-CMM-PE, it is necessary to... direction and The direction is set to establish an absorbing boundary, thereby simulating the propagation characteristics of electromagnetic waves into infinite space and eliminating non-physical reflections caused by the truncation of the computational domain boundary. direction and The absorbing boundaries are set for each direction, and the window function is defined as follows: for Direction, window function The expression is: (15) in, Along the coordinate system Maximum propagation distance in the positive direction of the axis.
[0048] for Direction, window function The expression is: (16) in, Represents the coordinate system along Maximum propagation distance in the positive direction of the axis.
[0049] Then 3D-CMM-PE in In-plane absorbing boundary attenuation function Represented as: (17) Starting from the initial distance, the field distribution calculated by 3D-CMM-PE needs to be multiplied in each calculation. .
[0050] Step 5. Based on the one-sided approximation and explicit difference scheme, the 3D impedance boundary is merged into the explicit FD algorithm: The 3D impedance boundary conditions are explicitly discretized based on the one-sided approximation method and incorporated into the field value recursion relationship to obtain an explicit recursion formula with boundary constraints.
[0051] In this embodiment, step 5 specifically includes: To adapt to the application of 3D boundary conditions, formula (5) needs to be transformed to a transformed coordinate system. Equation (18) is obtained; combined with the impedance boundary condition formula in the transformed coordinate system shown in formula (19), formula (18) is discretized and solved to obtain the explicit recursive formula with boundary constraints as shown in formula (36).
[0052] Transform equation (5) to the transformed coordinate system This yields the equation shown in formula (18): (18) in, , , It represents the surface impedance.
[0053] Represented as: .
[0054] The 3D impedance boundary condition is expressed as: (19) in, Represents field quantity At grid points The value at that location, Represents the normalized surface impedance. , This represents the vacuum wave impedance.
[0055] The design concept of the unilateral approximation merging scheme for 3D impedance boundary conditions in this invention is consistent with that of the unilateral approximation merging scheme for 2D impedance boundary conditions. Both schemes first characterize the second derivative on the boundary surface using a unilateral approximation method, then replace the second derivative with the impedance boundary conditions, and finally substitute it into the 3D-CMM-PE equation to obtain a fused equation that simultaneously satisfies the impedance boundary conditions and the 3D-CMM-PE equation.
[0056] make , , , ,in ,get: (20) Discretizing equation (20) yields: (twenty one) in, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at that location.
[0057] make .
[0058] At the boundary At this point, based on the one-sided approximation method, the finite difference expression of the second-order partial derivative of formula (19) is approximately expressed as: (twenty two) in, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at that location.
[0059] Discretizing equation (22) yields: (twenty three) in, Represents field quantity At grid points The value at that location.
[0060] make Then we get: (twenty four) Applying both sides of the equation (20) with respect to... Taking the partial derivatives and changing the order of the mixed partial derivatives, we get: (25) Discretizing equation (25) yields: (26) in, , , , , , They represent , , , , , At grid points The value at that location.
[0061] Partial derivatives At grid points The value at the location for: (27) Partial derivatives At grid points The value at the location for: (28) in, , They are the partial derivatives At grid points , The value at; For grid points place value, For grid points place value, For grid points place value, For grid points place value.
[0062] Partial derivatives At grid points The value at the location for: (29) Partial derivatives At grid points The value at the location for: (30) Substituting equations (27) to (30) into equation (26) yields: (31) make: .
[0063] Substituting into formula (5), we get: (32) Simplifying formula (32), we get: (33) in, Represents field quantity At grid points The second partial derivative at that point, and Represents field quantity At grid points Discrete approximation of the second-order partial derivative at point . Represents field quantity At grid points Discrete approximation of the first-order partial derivative at a given point.
[0064] Simplifying formula (33), we get: (34) Simplifying formula (34), we get: (35) Simplifying formula (35), we obtain the explicit recursive formula containing boundary constraints: (36) In step 5 of the method of the present invention, the second derivative on the boundary surface is first characterized by a one-sided approximation method, and then the second derivative is replaced by the impedance boundary condition. Subsequently, it is substituted into the 3D-CMM-PE equation, and finally a fusion equation that satisfies both the impedance boundary condition and the 3D-CMM-PE equation is obtained. This method effectively simplifies the amount of calculation while ensuring the calculation accuracy.
[0065] Step 6. Solve the field distribution point by point: Using the source field as the initial value, solve the field distribution point by point using an explicit recursive formula with boundary constraints to obtain the field distribution results in the entire computational domain.
[0066] In this embodiment, step 6 specifically includes: The initial field calculated in step 2 based on the flat ground formula will be used as the initial field. Discretize and assign values to each grid point on the starting surface of the 3D computational grid. The starting face is The corresponding cross section yields the initial field value set. .
[0067] Initial field value set This is the source field for the numerical solution of the 3D-CMM-PE equation.
[0068] Starting from the initial field, along the direction of electromagnetic wave propagation The solution is solved step by step.
[0069] The process of advancement Beginning, until Ultimately, it covers the entire computing area.
[0070] For each recursive step from the nth section to the (n+1)th section, calculate the new section, i.e. Field values at all grid points on the corresponding cross section Different explicit recursive formulas are used depending on the location of the grid points: For points within the calculation region, i.e., all height coordinates grid points The calculation is performed using the formula obtained in step 4 by explicit differential discretization of the Taylor-type 3D-CMM-PE equation. Specifically, the field values from the previous integer step (the nth step) are used. The surrounding field value of the previous half step (n+1 / 2 step) is calculated directly by the explicit formula shown in formula (14).
[0071] For ground boundary points, i.e., all height coordinates grid points The calculation is performed using the explicit recursive formula with boundary constraints obtained in step 5. This formula combines the impedance conditions satisfied at the boundary points with the parabolic equation, forming an explicit relationship regarding the field values at the boundary points. For ground boundary points... Its recursive formula is shown in formula (36). Since formula (36) simultaneously includes , , Three unknowns, along the horizontal direction, i.e. When solving the direction recursively, the field values of all ground boundary points on the current section are obtained at once by combining the lateral boundary conditions and solving the tridiagonal equation system sequentially.
[0072] After calculating a complete After sectioning, the calculation area needs to be... Both sides and The field near the directional boundary is absorbed to simulate open space and suppress non-physical reflections caused by the truncated boundary. Specifically, after each recursive step, the field values of the grid points located within the absorption layer are processed. Multiply by the absorption boundary attenuation function defined in step 5 The grid points located within the absorption layer are located in or Grid points in the region.
[0073] Through the above process, from the initial surface Begin by recursively calculating until... Ultimately, the entire 3D computing area is obtained. Field distribution results within .
[0074] Step 7. Solve for the magnetic field strength and second time delay of the radio wave propagation: Based on the field distribution results across the entire computational region, solve for the magnetic field strength of the low-frequency ground wave propagating on undulating terrain. With secondary delay .
[0075] In this embodiment, step 7 specifically includes: The field distribution results calculated in step 6 Right now Substituting the discrete form into formula (2), we can obtain all the magnetic fields within the computational region. : .
[0076] in, , , These represent the propagation directions along the lower edge of the rectangular coordinate system. Horizontal direction Altitude direction Discrete grid coordinates, corresponding to grid indices are ; For field quantity At grid points The value at that location.
[0077] For the magnetic field at each location within the calculation area The real and imaginary parts are extracted separately, and then the phase delay accumulated along the propagation path of the electromagnetic wave is obtained. The calculation formula is as follows: (37) in, This indicates the operation of taking the real part. This indicates the operation of taking the imaginary part.
[0078] When the radio wave propagation path is an undulating terrain path, the relative permittivity and conductivity are respectively and The propagation phase of low-frequency ground waves along the undulating terrain path is calculated using formula (37). .
[0079] When the electromagnetic wave propagation path is a good conductor path, the relative permittivity and conductivity are respectively... and The propagation phase of low-frequency ground waves along a good conductor path is calculated using formula (37). .
[0080] Secondary time delay caused by low-frequency ground waves propagating along undulating terrain paths The calculation formula is: (38) In addition, to verify the effectiveness of the method proposed in this invention, the following specific experiments are also provided: The experiment involved the magnetic field strength measurements using the 3D-CMM-PE method (the method described in this invention), based on a one-sided approximation and explicit difference scheme, as well as the three-dimensional finite-difference time-domain (3D-FDTD) method and the two-dimensional parabolic equation (2D-PE) method. and secondary delay Comparison of predictions.
[0081] In the 3D-CMM-PE computational model, the computational region is set as follows: .
[0082] Grid step size is .
[0083] In the 3D-FDTD computational model, the computational domain is: .
[0084] Grid step size is Time step It is 22.2222ns.
[0085] The ground electrical parameters, namely the relative permittivity and conductivity of mountainous terrain regions, are: , S / m.
[0086] The mountain range has a Gaussian shape, and its height function in a rectangular coordinate system is... The format is as follows: .
[0087] in, Let the height of Mount Gauss be a value. ; and They represent Mount Gauss in and The center of the direction, with a value of , ; and To control the mountain peaks and The parameters for directional width are as follows: .
[0088] Figure 2 and Figure 3 For the 3D-CMM-PE method, 3D-FDTD method and 2D-PE method in Magnetic field strength at the Earth's surface on a plane and secondary delay The prediction results.
[0089] from Figure 2 and Figure 3 It can be seen that there are certain differences between the 3D-CMM-PE method and the 3D-FDTD method in the back mountain region, but they gradually become consistent as the propagation distance increases. The main reason for this error is that the 3D-CMM-PE method itself has a paraxial approximation, and the Taylor-type 3D-CMM-PE method is limited by the propagation elevation angle. The 2D-PE method cannot take into account the influence of lateral terrain on the propagation of low-frequency ground waves, which leads to its prediction results in the back mountain region being significantly different from the other two 3D methods.
[0090] Figure 4 , Figure 5 , Figure 6 The images show the quadratic time delay distribution of an isolated Gaussian mountain scene using the 3D-FDTD, 3D-CMM-PE, and 2D-PE methods, respectively, with the mountain height fixed. The mountain width control parameters are as follows: The influence of the Gaussian Mountains on the propagation characteristics of low-frequency ground wave transverse waves.
[0091] It can be seen that the second delay Significant numerical variations are observed in the region where Gaussian mountains exist, with the most pronounced variation occurring near the center of the mountains, decreasing with increasing distance from the mountain center. The changes gradually stabilized. In terms of direction, the symmetrical perturbation of the transverse wave propagation path by the Gaussian mountain results in a symmetrical distribution of time delay.
[0092] from Figure 4 and Figure 5 It can be seen that the time delay distribution and perturbation range of the 3D-CMM-PE method are highly consistent with those of the 3D-FDTD method, proving that the method of this invention can effectively predict the 3D terrain diffraction effect; however, in the isolated Gaussian mountain scene, lateral diffraction has a decisive influence on the secondary time delay distribution. Because the 2D-PE method ignores the lateral influence, its distribution deviates significantly from the 3D scene and cannot accurately predict the time delay perturbation on the side and rear of the mountain.
[0093] The method of this invention effectively solves the problem that the 2D-PE method cannot consider the influence of lateral irregular terrain on low-frequency ground waves, and at the same time makes up for the lack of accuracy of the traditional 3D-PE method in predicting the propagation characteristics of low-frequency ground waves under undulating terrain. It uses the finite difference algorithm to solve 3D-CMM-PE, which not only reduces dispersion error, but also effectively captures the lateral diffraction effect generated by the interaction between irregular terrain and low-frequency ground waves, thereby significantly improving the prediction accuracy of low-frequency ground wave propagation characteristics under undulating terrain.
[0094] Example 2 This embodiment 2 describes a computer device that includes a memory and one or more processors.
[0095] The memory stores executable code, which, when executed by the processor, is used to implement the steps of the three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics of undulating terrain in Embodiment 1 above.
[0096] In this embodiment, the computer device can be any device or apparatus with data processing capabilities, and will not be described in detail here.
[0097] Example 3 This embodiment 3 describes a computer-readable storage medium storing a program that, when executed by a processor, implements the steps of a three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain.
[0098] The computer-readable storage medium can be an internal storage unit of any device or apparatus with data processing capabilities, such as a hard disk or memory, or an external storage device of any device with data processing capabilities, such as a plug-in hard disk, smart media card (SMC), SD card, flash card, etc.
[0099] Of course, the above description is only a preferred embodiment of the present invention. The present invention is not limited to the above-described embodiments. It should be noted that any equivalent substitutions or obvious modifications made by those skilled in the art under the guidance of this specification fall within the scope of this specification and should be protected by the present invention.
Claims
1. A three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain, characterized in that, Includes the following steps: Step 1. Input the basic model parameter file. The input basic model parameter file includes the mesh parameters of the computational domain, the electrical parameters of the electromagnetic wave propagation path, and the excitation source parameters; Step 2. Solve for the initial field using the flat ground formula, and use the initial field as the source field of the 3D conformal mapping model - parabolic equation, i.e., the 3D-CMM-PE equation; Step 3. Derive the Taylor-type 3D-CMM-PE equations for undulating terrain conditions; Step 4. By implementing explicit differential discretization on the Taylor-type 3D-CMM-PE equation, an explicit iterative step scheme for the field quantities is derived, and absorbing boundary conditions are set to eliminate truncated reflections at the boundary of the computational domain. Step 5. Based on the one-sided approximation method, the 3D impedance boundary conditions are explicitly discretized and incorporated into the field value recursion relationship to obtain an explicit recursion formula with boundary constraints. Step 6. Using the source field as the initial value, solve the problem point by point using an explicit recursive formula with boundary constraints to obtain the field distribution results in the entire computational domain; Step 7. Based on the field distribution results in the entire calculation area, solve for the magnetic field strength and second time delay of low-frequency ground waves propagating on undulating terrain.
2. The three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain according to claim 1, characterized in that, Step 1 specifically involves: In 3D Cartesian coordinate system In this context, the computational region size is defined as... ,in Horizontal Number of grid cells in each direction, For vertical Number of grid cells in each direction, For height Number of grid cells in each direction; , , They are respectively , , Mesh subdivision step size in the direction; The electrical parameters of the radio wave propagation path include the relative permittivity of free space. With conductivity and the relative permittivity of mountainous terrain regions With conductivity ; The excitation source uses a vertical electric dipole, and the parameters of the excitation source include current. , charge spacing and the height of the source above the ground .
3. The three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain according to claim 2, characterized in that, Step 2 specifically involves: In a rectangular coordinate system In the middle, the time harmonic factor is set to ,in The imaginary unit, Angular velocity, It is a time variable; Solving the initial field and ground wave magnetic field components of 3D-CMM-PE using the flat ground formula. This can be expressed by formula (1): (1) in, The distance is the great circle distance; Vacuum wavenumber, Ground wavenumber; From source point to observation point The straight-line distance at the location; is the straight-line distance from the mirror image of the source to the observation point; For Fresnel integration, As an intermediate variable; Great circle distance Defined as: ； Fresnel points Defined as: ； intermediate variables Defined as: ； in, Location of the observation point The vertical height coordinate; For the preset initial distance The initial field of the low-frequency ground wave is obtained from formula (1). As shown in formula (2): (2) in, express The Earth wave magnetic field component at that location; Initial field As the source field of the 3D-CMM-PE equation.
4. The three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain according to claim 3, characterized in that, Step 3 specifically involves: The equation for a 3D parabola on a flat surface is expressed in the form of formula (3): (3) in, The transverse Laplace operator is defined as follows: ; The target field to be solved. The refractive index of the medium; The conformal mapping model (CMM) is used to map the original undulating terrain to a flat ground. The transformation relationship between height and distance variables is shown in formula (4): (4) in, The terrain height function is given; after coordinate transformation, the spatial step size satisfies... , , ; Substituting equation (4) into equation (3) simplifies the equation, resulting in the Taylor-type 3D-CMM-PE equation for undulating terrain conditions, as shown in equation (5): (5) Among them, the terrain height function Simplified to .
5. The three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain according to claim 4, characterized in that, Step 4 specifically involves: In the computational region, along the direction of propagation Discretize the field as If there are 1 point, then the waveguide is composed of a continuous arrangement of 1 point. It consists of several horizontal planes, among which Indicates the first The point is at The coordinates of the direction satisfy ; in the horizontal In the direction, the field is discretized into One point, Indicates the first The point is at The coordinates of the direction satisfy At a height In the direction, the field is discretized into One point, Indicates the first The point is at The coordinates of the direction satisfy ; According to formula (5), for The first-order partial derivatives are approximated using the forward difference method, resulting in: (6) in, Represents field quantity At grid points The value at that location, , , ; about The first-order partial derivative is approximated using the forward difference method as follows: (7) about The second-order partial derivatives are approximated using the forward difference method as follows: (8) about The second-order partial derivatives are approximated using the forward difference method as follows: (9) about and The second-order mixed partial derivative is approximated using the forward difference method as follows: (10) Substituting equations (6) to (10) into equation (5), we get: (11) in, , , ; Simplifying formula (11), we get: (12) Simplifying formula (12), we get: (13) make , , , , The explicit iterative step scheme for the field quantity is obtained as follows: (14) exist direction and The absorbing boundaries are set for each direction, and the window function is defined as follows: for Direction, window function The expression is: (15) in, Along the coordinate system Maximum propagation distance in the positive direction of the axis; for Direction, window function The expression is: (16) in, Represents the coordinate system along Maximum propagation distance in the positive direction of the axis; Then 3D-CMM-PE in In-plane absorbing boundary attenuation function Represented as: (17)。 6. The three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain according to claim 5, characterized in that, Step 5 specifically involves: Transform equation (5) to the transformed coordinate system This yields the equation shown in formula (18): (18) in, , , It is the surface impedance; Represented as: ； The 3D impedance boundary condition is expressed as: (19) in, Represents field quantity At grid points The value at that location, Represents the normalized surface impedance. , Indicates vacuum wave impedance; make , , , ,in ,get: (20) Discretizing equation (20) yields: (21) in, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at; make ; At the boundary At this point, based on the one-sided approximation method, the finite difference expression of the second-order partial derivative of formula (19) is approximately expressed as: (22) in, Represents field quantity At grid points The value at that location, Represents field quantity At grid points The value at; Discretizing equation (22) yields: (23) in, Represents field quantity At grid points The value at; make Then we get: (24) Applying both sides of the equation (20) with respect to... Taking the partial derivatives and changing the order of the mixed partial derivatives, we get: (25) Discretizing equation (25) yields: (26) in, , , , , , They represent , , , , , At grid points The value at; partial derivatives At grid points The value at the location for: (27) partial derivatives At grid points The value at the location for: (28) in, , They are the partial derivatives At grid points , The value at; For grid points place value, For grid points place value, For grid points place value, For grid points place value; partial derivatives At grid points The value at the location for: (29) partial derivatives At grid points The value at the location for: (30) Substituting equations (27) to (30) into equation (26) yields: (31) make: ； Substituting into formula (5), we get: (32) Simplifying formula (32), we get: (33) in, Represents field quantity At grid points The second partial derivative at that point, and Represents field quantity At grid points Discrete approximation of the second-order partial derivative at point . Represents field quantity At grid points Discrete approximation of the first-order partial derivative at point ; Simplifying formula (33), we get: (34) Simplifying formula (34), we get: (35) Simplifying formula (35), we obtain the explicit recursive formula containing boundary constraints: (36)。 7. The three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain according to claim 6, characterized in that, Step 6 specifically involves: The initial field calculated in step 2 based on the flat ground formula will be used as the initial field. Discretize and assign values to each grid point on the starting surface of the 3D computational grid. The starting face is The corresponding cross section yields the initial field value set. ; Initial field value set This is the source field for the numerical solution of the 3D-CMM-PE equation; Starting from the initial field, along the direction of electromagnetic wave propagation Proceed step by step to solve the problem; The process of advancement Beginning, until Ultimately, this covers the entire computing area; For each recursive step from the nth section to the (n+1)th section, calculate the new section, i.e. Field values at all grid points on the corresponding cross section Different explicit recursive formulas are used depending on the location of the grid points: For points within the calculation region, i.e., all height coordinates grid points The calculation is performed using the formula obtained by explicit difference discretization of the Taylor-type 3D-CMM-PE equation in step 4, that is, by direct calculation using the explicit formula shown in formula (14). For ground boundary points, i.e., all height coordinates grid points The calculation is performed using the explicit recursive formula with boundary constraints obtained in step 5. The recursive formula for the ground boundary points is shown in formula (36); formula (36) also includes , , Three unknowns, along the horizontal direction, i.e. When solving the direction recursively, the field values of all ground boundary points on the current section are obtained at once by combining the transverse boundary conditions and solving the tridiagonal equation system in sequence or by solving the tridiagonal equation system. After calculating a complete After sectioning, the calculation region Both sides and The field near the directional boundary is absorbed. Specifically, after each recursive step, the field values of the grid points located within the absorption layer are processed. Multiply by the absorption boundary attenuation function defined in step 5 The grid points located within the absorption layer are located in or Grid points in the region; From the initial surface Begin by recursively calculating until... Ultimately, the entire 3D computing area is obtained. Field distribution results within .
8. The three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics in undulating terrain according to claim 7, characterized in that, Step 7 specifically involves: The field distribution results calculated in step 6 Right now Substituting the discrete form into formula (2), we can obtain all the magnetic fields within the computational region. : ； in, , , These represent the propagation directions along the lower edge of the rectangular coordinate system. Horizontal direction Altitude direction Discrete grid coordinates, corresponding to grid indices are ; For field quantity At grid points The value at; For the magnetic field at each location within the calculation area The real and imaginary parts are extracted separately, and then the phase delay accumulated along the propagation path of the electromagnetic wave is obtained. The calculation formula is as follows: (37) in, This indicates the operation of taking the real part. This indicates the operation of taking the imaginary part; When the radio wave propagation path is an undulating terrain path, the relative permittivity and conductivity are respectively and The propagation phase of low-frequency ground waves along the undulating terrain path is calculated using formula (37). ; When the electromagnetic wave propagation path is a good conductor path, the relative permittivity and conductivity are respectively... and The propagation phase of low-frequency ground waves along a good conductor path is calculated using formula (37). ; Secondary time delay caused by low-frequency ground waves propagating along undulating terrain paths The calculation formula is: (38)。 9. 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 three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics of undulating terrain as described in any one of claims 1 to 8.
10. A computer-readable storage medium having a program stored thereon, characterized in that, When the program is executed by the processor, it implements the steps of the three-dimensional parabolic equation prediction method for low-frequency ground wave propagation characteristics of undulating terrain as described in any one of claims 1 to 8.