Method and system for constructing a very low frequency ionospheric model based on a dynamic layered transmission matrix algorithm

Abstract

The application belongs to the technical field of very low frequency electromagnetic wave propagation, and discloses a method for constructing a very low frequency ionosphere model based on a dynamic layered transmission matrix algorithm. The dynamic layered design comprises the following steps: firstly, filtering and optimizing electron density; secondly, dynamically and adaptively adjusting the layers in the vertical direction; thirdly, transmission matrix hybrid regularization; fourthly, on the basis of the dynamic layered ionosphere, combining the transmission matrix hybrid regularization algorithm, performing SVD decomposition and dynamic truncation on the transmission matrix; and finally, obtaining the optimized parameter ionosphere model by adjusting the reference thickness and the adjustment factor parameter. The average relative error of the reflection coefficient of the dynamic layered transmission matrix algorithm of the application in the very low frequency range is 4.7%, and the calculation efficiency is improved by more than 50%, thereby providing a good reference for engineering calculation.

Description

Technical Field

[0001] This invention belongs to, but is not limited to, the field of very low frequency electromagnetic wave propagation technology, and particularly relates to a method and system for constructing a very low frequency ionospheric model based on a dynamic hierarchical transmission matrix algorithm. Background Technology

[0002] Very low frequency (VLF, 3-30 kHz) electromagnetic waves have long played an irreplaceable role in submarine communications, geological exploration, and space weather monitoring due to their unique propagation characteristics—the ability to transmit over long distances of thousands of kilometers through the ionosphere-surface waveguide. The interaction mechanism between VLF waves and the ionosphere is complex, and their propagation characteristics are significantly affected by the vertical gradient changes in the ionospheric electron density profile.

[0003] Since 1950, the ionospheric layered transmission matrix algorithm has been continuously developed and has become the core analytical method for dealing with the anisotropic propagation of magnetic ions. Its evolution can be divided into four stages: (1) Theoretical foundation period (first half of the 20th century). In the 1920s, British physicist Oliver Heaviside and American scientist Arthur Kennelly proposed the existence of the ionosphere and explained the phenomenon of long-distance radio propagation. In the 1950s and 1960s, Wait systematized the VLF wave propagation theory and proposed the waveguide mode theory, which regarded the Earth's surface and the ionosphere as waveguide boundaries, in which VLF waves propagate in a multimode form, providing theoretical support for the subsequent transmission matrix method. Early ionospheric models regarded the ionosphere as a single layer or two layers (D layer, E layer), ignoring gradient changes, and VLF reflection was mainly dominated by the low-altitude D layer (60-90 km). (2) Numerical methods and the introduction of transmission matrix (1960s-1980s). Budden and Barron proposed treating the ionosphere as a layered medium and solving the boundary matching problem of Maxwell's equations in each layer using the matrix method. Subsequently, Wait and Spies further developed a piecewise uniform layered model of the vertical profile of the ionosphere, connecting the solutions of each layer through a propagation matrix (or chain matrix). In 1971, Pappert et al. first used the matrix method to calculate VLF waveguide modes, improving computational efficiency and marking the transition of this method from theory to engineering application. (3) Refined Models and Algorithm Improvements (1980s-2000s). With the advancement of ionospheric detection technology and the introduction of high-performance computing methods, scholars proposed more refined ionospheric profile models and improved the computational efficiency and stability of numerical algorithms such as the transmission matrix. These included the introduction of the International Reference Ionosphere (IRI), numerical optimization of the transmission matrix algorithm (such as the impedance matrix method and adaptive layering), the application of hybrid algorithms (transmission matrix + waveguide mode theory) and parallel computing technology, satellite data assimilation and experimental verification, etc. These advances have significantly improved the accuracy of VLF propagation prediction and provided reliable theoretical tools for fields such as space weather and lightning detection. (4) Modern Development (2000s-present). With the increasing complexity of application scenarios, traditional models have limitations (such as ignoring abrupt changes in terrain, gradient effects in equatorial anomalies, and transient disturbances), making it difficult to handle rapidly changing events such as solar flares and geomagnetic storms. VLF research is developing towards multi-physics coupling and real-time prediction, studying the integration of high-performance computing and artificial intelligence algorithms in transmission matrix modeling, the refinement of dynamic ionospheric disturbance mechanisms, and the expansion of emerging application scenarios.

[0004] Based on the above analysis, despite the progress made in the aforementioned research, the following unresolved issues remain regarding the VLF transfer matrix algorithm for ionospheric stratification: 1. Lack of hierarchical-transmission coupling: Dynamic hierarchical algorithms mostly focus on the geometric optimization of electron density profiles (such as adaptive mesh generation), but lack collaborative design with the numerical stability of the transmission matrix (such as condition number control and singular value decay); 2. Unclear parameter sensitivity: The influence mechanism of key parameters such as layer thickness and interpolation order on VLF propagation characteristics has not yet been quantitatively modeled; 3. Insufficient engineering applicability: Existing dynamic algorithms have high computational complexity, insufficient numerical stability, and slow iterative convergence speed due to the lack of preconditioning techniques. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides a method and system for constructing a very low frequency ionospheric model based on a dynamic layered transmission matrix algorithm. Simulation results show that, compared with fixed layering, this algorithm has an average relative error of 4.7% in the reflection coefficient range within the very low frequency range, and its computational efficiency is improved by more than 50%, providing a good reference for engineering calculations.

[0006] This invention is implemented as follows: a method for constructing a very low frequency ionospheric model based on a dynamic hierarchical transfer matrix algorithm, comprising the following steps: Step 1, Dynamic Layering Design: First, filter and optimize electron density, then dynamically and adaptively adjust the layering in the vertical direction. Step 2, Hybrid regularization of the transfer matrix; By establishing a nonlinear mapping relationship between the electron density gradient and the layer thickness, based on the dynamic layering of the ionosphere, the transfer matrix is ​​decomposed by SVD and dynamically truncated by combining the hybrid regularization algorithm of the transfer matrix. Step 3: By adjusting the baseline thickness and adjusting the factor parameters, the optimized parameter ionospheric model is obtained.

[0007] Furthermore, the dynamic stratification design first performs electron density gradient analysis, obtains the ionosphere within a certain range using the IRI model, and then performs fixed stratification in the vertical direction. i The center height of the layer is h i ( i= 1,2,3·····), the electron density at the corresponding height is N e ( i ).

[0008] Furthermore, the electron density is optimized using Savitzky-Golay filtering, as shown in the following formula:

[0009] in: For height h iThe filtered electron density value; c k The filter coefficients are determined by least squares fitting and reflect the contribution weights of neighboring points to the current point. m This is the half-window density (total window points = 2). m +1); The electron density gradient is obtained using the central difference method: .

[0010] Furthermore, the formula for calculating the layer thickness is as follows:

[0011] in: d 0 As the reference thickness, As an adjustment factor; physical parameter constraints are applied to the layer thickness, where the minimum allowable layer thickness is limited based on the Debye length, resulting in:

[0012] in: It is the vacuum permittivity; k B Boltzmann's constant; T e For electron temperature; N e Electron density; e Let be the electron charge. Then, based on numerical stability constraints, the maximum allowable delamination thickness is obtained:

[0013] in: c The speed of light in a vacuum; f max Maximum frequency; is the relative permittivity of the ionosphere.

[0014] Furthermore, the hybrid regularization of the transmission matrix first analyzes the single-layer transmission matrix, assuming the first layer... i The ionospheric transport matrix is T i (4×4 matrix), using Tikhonov preprocessing, the calculation formula is as follows:

[0015] Where: γ is the regularization intensity parameter; I It is the identity matrix. for T i The conjugate transpose operation.

[0016] Furthermore, SVD decomposition and dynamic truncation include: To solve the ionospheric transport matrix To address the ill-conditioned characteristics that are prone to occur in mutation layers such as EF, and to ensure numerical stability, Perform regularization:

[0017] U i ，V i It is a unitary matrix, from T i The singular value decomposition (SVD) satisfies ; It is a singular value matrix containing T i singular values (Sorted in descending order), set the singularity truncation threshold. :

[0018] in, Let be the maximum singular value. Let the singular value after filtering according to the threshold be . Gradient-weighted correction is applied to obtain ,have to

[0019] Therefore, the final reconstructed regularization matrix is:

[0020] Furthermore, the total transfer matrix is ​​obtained by multiplying all the regularized transfer matrices together. T total The ionospheric reflection and transmission characteristics are calculated using a transfer matrix algorithm; the formula for calculating the total transfer matrix is ​​as follows:

[0021] in, N This represents the total number of vertical strata in the ionosphere.

[0022] Another objective of this invention is to provide a very low frequency (VLF) ionospheric model construction system based on a dynamic hierarchical transfer matrix algorithm for constructing VLF ionospheric models, comprising: The dynamic stratification module first performs filtering to optimize the electron density, and then performs dynamic adaptive stratification adjustment in the vertical direction; The hybrid regularization module establishes a nonlinear mapping relationship between electron density gradient and layer thickness. Based on the dynamic layering of the ionosphere, it combines the hybrid regularization algorithm of the transfer matrix to perform SVD decomposition and dynamic truncation of the transfer matrix. The parameter adjustment module obtains an optimized ionospheric model by adjusting the baseline thickness and adjusting the factor parameters.

[0023] Another object of the present invention is to provide a computer device, the computer device including a memory and a processor, the memory storing a computer program, and when the computer program is executed by the processor, causing the processor to perform the steps of the method for constructing a very low frequency ionospheric model based on the dynamic hierarchical transfer matrix algorithm.

[0024] Another object of the present invention is to provide a computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the steps of the method for constructing a very low frequency ionospheric model based on the dynamic hierarchical transfer matrix algorithm.

[0025] Based on the above technical solutions and the technical problems solved, the advantages and positive effects of the technical solution to be protected by this invention are as follows: First, this invention proposes a transfer matrix optimization algorithm under dynamic stratification. By establishing a nonlinear mapping function between electron density gradient and stratification thickness, adaptive stratification (dynamic densification or sparseness) in the vertical direction is achieved. Subsequently, a transfer matrix synthesis method based on SVD regularization is designed [30, 31] to solve the numerical ill-conditioning problem in the abrupt change region of the EF layer. The optimal working range is determined through parameter sensitivity analysis (reference thickness, adjustment factor) to balance computational efficiency and accuracy, providing high-precision prediction of very low frequency propagation characteristics in complex ionospheric environments.

[0026] This invention optimizes the stratification method and improves the stability of the transmission matrix solution based on the classic fixed ionospheric stratification and transmission matrix algorithm, establishing a more accurate dynamic stratification transmission matrix algorithm ionospheric model. Taking a specific location as an example, the influence of reference thickness and adjustment factor on the propagation of radio waves in the ionosphere is analyzed, and the reflection coefficient of VLF band electromagnetic waves in the low ionosphere is calculated. Simulation results show that compared with fixed stratification, the algorithm has an average relative error of 4.7% in the reflection coefficient in the very low frequency range, and the computational efficiency is improved by more than 50%, providing a good reference for engineering calculations.

[0027] Second, by using Savitzky-Golay filtering and dynamic layering algorithms to model the ionosphere, higher-precision ionospheric gradient data can be provided for space weather research in the fields of scientific research and monitoring.

[0028] Co-design of dynamic layering and transfer matrix: For the first time, a nonlinear mapping between electron density gradient and layer thickness is achieved, solving the analytical insufficiency problem of fixed layering in abrupt change regions. Engineering treatment of ill-conditioned matrix: SVD dynamic truncation combined with Tikhonov preprocessing is used to improve the numerical stability of EF layer to an engineeringable level, which is beneficial for eliminating spurious fluctuations.

[0029] Traditional fixed-layer methods have inherent limitations in regions with abrupt changes in electron density gradients (such as the DE transition layer at 80-100 km), and the modeling accuracy of ionospheric abrupt changes in the VLF band is not high, leading to spurious fluctuations in the reflection coefficient calculation. This invention reduces the average relative error of the reflection coefficient through dynamic layer design and SVD dynamic truncation, which is superior to the fixed-layer model.

[0030] Overcoming the technical bias of traditional inertia and practical tendency that "VLF modeling requires uniform layering," experiments have shown that dynamic layering automatically densifies in abrupt change regions, which is more in line with physical reality than fixed layering. Overcoming the bias of low efficiency in transfer matrix algorithms, efficiency is improved while maintaining accuracy through hybrid regularization and parallel matrix multiplication. Attached Figure Description

[0031] Figure 1 This is a flowchart of the method for constructing a very low frequency ionospheric model based on the dynamic hierarchical transfer matrix algorithm provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of ionosphere layering provided in an embodiment of the present invention; Figure 3 This is a block diagram of the very low frequency ionosphere model construction system provided in this embodiment of the invention; Figure 4 This is a schematic diagram of the dynamic hierarchical transfer matrix algorithm provided in an embodiment of the present invention; Figure 5 This is a graph showing the variation of electron density (a) and gradient (b) of the ionosphere at 60-120km in a certain region, provided by an embodiment of the present invention. Figure 6 This is a comparison chart of the results of the fixed-layer and dynamic-layer algorithms provided in this embodiment of the invention; (a) Reflection coefficient R; (b) Difference in reflection coefficients, where R1 represents the reflection coefficient under fixed layering and R2 represents the reflection coefficient under dynamic layering; (c) Reflection coefficient deviation rate. r =( R 2- R 1) / R 1; (d) Layer-by-layer thickness of the ionosphere; Figure 7 The embodiments of the present invention provide (a) a graph showing the effect of different thicknesses on the reflection coefficient; and (b) a graph showing the ionosphere layering effect under different thicknesses. Figure 8 This is a graph showing the effect of different adjustment factors on the reflection coefficient provided in the embodiments of the present invention; Figure 9 This is (a) provided in the embodiment of the present invention. d 0 =3, When the reflection coefficient R is 1; (b) d 0 =3, The ionospheric stratification effect when =1. Detailed Implementation

[0032] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0033] The application of ionospheric models in very low frequency (VLF) communication and radar detection often suffers from propagation path errors that significantly increase with temporal and spatial fluctuations because traditional fixed-layer models neglect the dynamic changes in ionospheric electron density. To address this industrialization need, this invention employs an adaptive layering strategy based on observation and physical constraints. This strategy isolates abrupt gradient changes in the electron density profile from conventional models, ensuring that the thickness of each sublayer accurately reflects the continuous changes in the atmospheric ionization structure, thus providing higher accuracy support at the forefront of modeling.

[0034] In the dynamic layering design, the elevation electron density data is smoothed using Savitzky-Golay filtering, and the local gradient is calculated using a data-driven central difference method to form an initial estimate of the layer thickness. This process incorporates the Debye length physical limit to automatically constrain the minimum layer scale; at the same time, a numerical stability criterion is introduced to limit the maximum thickness, ensuring that each layer has sufficient physical resolution in subsequent matrix operations without causing computational non-convergence due to over-refinement.

[0035] When constructing the transfer matrix, this invention proposes to integrate the electromagnetic response matrix of each layer with Tikhonov regularization, avoiding the failure of inversion caused by the surge in the matrix condition number under typical abrupt change layers (such as the boundary between E and F layers). This regularization adaptively balances the contradiction between data fitting and noise amplification, selecting the regularization strength through the least squares criterion, thus preserving the model's sensitivity to details while preventing unstable frequency components from disrupting the overall matrix operations.

[0036] For the singular value decomposition of matrices at each level, this method sets a dynamic truncation threshold for singular values ​​and applies gradient weighting correction to the retained singular values ​​after truncation, so as to balance the necessary contribution of small singular values ​​with numerical stability. This gradient weighting strategy not only maintains the physical consistency of the matrix spectral structure, but also significantly reduces error accumulation in the matrix reconstruction stage, enabling the overall transfer matrix to maintain high accuracy and high robustness during multi-level multiplication.

[0037] Finally, the regularization matrices of each sublayer are multiplied sequentially to construct the total transmission matrix, which is used to calculate the absorption, reflection, and transmission characteristics of the signal in the very low frequency band. This model, through comparison and correction with measured phase delay and amplitude attenuation data, forms a closed-loop parameter adjustment mechanism, achieving high-precision matching between industrial-grade ionospheric simulation results and field observation errors of less than 0.5° phase and 0.2 dB amplitude.

[0038] This method has been piloted in fields such as remote monitoring of power lines, detection of submarine communication pipelines, and long-distance navigation and positioning. It has fully demonstrated the reliability and applicability of the joint framework of dynamic layering and hybrid regularization in complex ionospheric environments, and provided effective model support for very low frequency industrial communication and radar systems.

[0039] like Figure 1 As shown, this embodiment of the invention provides a method for constructing a very low frequency ionospheric model based on a dynamic hierarchical transfer matrix algorithm, including the following steps: Step 1, Dynamic Layering Design: First, filter and optimize electron density, then dynamically and adaptively adjust the layering in the vertical direction. Step 2, Hybrid regularization of the transfer matrix; By establishing a nonlinear mapping relationship between the electron density gradient and the layer thickness, based on the dynamic layering of the ionosphere, the transfer matrix is ​​decomposed by SVD and dynamically truncated by combining the hybrid regularization algorithm of the transfer matrix. Step 3: By adjusting the baseline thickness and adjusting the factor parameters, the optimized parameter ionospheric model is obtained.

[0040] Furthermore, the dynamic stratification design first performs electron density gradient analysis, obtains the ionosphere within a certain range using the IRI model, and then performs fixed stratification in the vertical direction. i The center height of the layer is h i ( i= 1,2,3·····), the electron density at the corresponding height is N e ( i (See Figure 2).

[0041] Furthermore, the electron density is optimized using Savitzky-Golay filtering, as shown in the following formula:

[0042] in: For height h i The filtered electron density value; c k The filter coefficients are determined by least squares fitting and reflect the contribution weights of neighboring points to the current point. m Half-window density (total window points = 2) m +1); The electron density gradient is obtained using the central difference method: .

[0043] Furthermore, the formula for calculating the layer thickness is as follows:

[0044] in: d 0 As the reference thickness, As an adjustment factor; physical parameter constraints are applied to the layer thickness, where the minimum allowable layer thickness is limited based on the Debye length, resulting in:

[0045] in: It is the vacuum permittivity; k B Boltzmann's constant; T e For electron temperature; N e Electron density; e Let be the electron charge. Then, based on numerical stability constraints, the maximum allowable delamination thickness is obtained:

[0046] in: c The speed of light in a vacuum; f max Maximum frequency; is the relative permittivity of the ionosphere.

[0047] Furthermore, the hybrid regularization of the transmission matrix first analyzes the single-layer transmission matrix, assuming the first layer... i The ionospheric transport matrix is T i (4×4 matrix), using Tikhonov preprocessing, the calculation formula is as follows:

[0048] Where: γ is the regularization intensity parameter; IIt is the identity matrix. for T i The conjugate transpose operation.

[0049] Furthermore, SVD decomposition and dynamic truncation include: To solve the ionospheric transport matrix To ensure numerical stability, we must address the ill-conditioned characteristics that are prone to occur in mutation layers such as EF. Perform regularization:

[0050] U i ，V i It is a unitary matrix, from T i The singular value decomposition (SVD) satisfies ; It is a singular value matrix containing T i singular values (Sorted in descending order), set the singularity truncation threshold. :

[0051] in, Let be the maximum singular value. Let the singular value after filtering according to the threshold be . Gradient-weighted correction is applied to obtain ,have to

[0052] Therefore, the final reconstructed regularization matrix is:

[0053] Furthermore, the total transfer matrix is ​​obtained by multiplying all the regularized transfer matrices together. T total The ionospheric reflection and transmission characteristics are calculated using a transfer matrix algorithm; the formula for calculating the total transfer matrix is ​​as follows:

[0054] in, N This represents the total number of vertical strata in the ionosphere.

[0055] like Figure 2 As shown, this embodiment of the invention provides a very low frequency (VLF) ionospheric model construction system based on a dynamic hierarchical transfer matrix algorithm for constructing VLF ionospheric models, comprising: The dynamic stratification module first performs filtering to optimize the electron density, and then performs dynamic adaptive stratification adjustment in the vertical direction; The hybrid regularization module establishes a nonlinear mapping relationship between electron density gradient and layer thickness. Based on the dynamic layering of the ionosphere, it combines the hybrid regularization algorithm of the transfer matrix to perform SVD decomposition and dynamic truncation of the transfer matrix. The parameter adjustment module obtains an optimized ionospheric model by adjusting the baseline thickness and adjusting the factor parameters.

[0056] In the field of geophysical exploration and disaster monitoring, the dynamic layered transmission matrix algorithm provided by this invention can accurately capture the anomalous characteristics of VLF wave propagation, establish a correlation model between ionospheric disturbances and crustal activity, and provide highly timely auxiliary criterion data for earthquake early warning systems. Furthermore, based on the coupling mechanism between VLF waves and the ionosphere-surface waveguide, it can assist in the non-invasive detection of underground anomalous geological structures (including but not limited to oil and gas reservoirs and metal veins).

[0057] In the field of space weather and ionospheric monitoring applications: it provides research institutions with high-resolution electron density gradient data, supports space weather modeling, and facilitates ionospheric disturbance monitoring. It can also provide auxiliary positioning for satellite communications and calibrate ionospheric delay errors in low-frequency satellite signals, making it particularly suitable for complex environments such as equatorial anomalies.

[0058] Specialized software product development: VLF propagation prediction software can be designed and developed based on this invention: integrating dynamic layering algorithm and hybrid regularization module, providing standardized data output of key parameters such as reflection / transmission coefficient, optimal layer thickness distribution, and propagation loss, and supporting API integration and visualization.

[0059] Evidence related to the technical effects obtained by the embodiments of the present invention.

[0060] 1. Algorithm quality evaluation coefficient Set algorithm quality evaluation coefficients The approximation of physical reality by the dynamic hierarchical transfer matrix algorithm is measured by calculating the average relative error of the reflection coefficient.

[0061] in: N This represents the number of frequency sampling points. R i For the first i Reflection coefficient at each frequency point; R ref λ is the reference reflection coefficient; λ is the efficiency weighting coefficient. T dynamic This represents the total computation time of the dynamic hierarchical algorithm. Tfixed This represents the total computation time of the fixed-layer algorithm. In summary, the flowchart of the dynamic layered transfer matrix algorithm is as follows: Figure 4 As shown.

[0062] 2 Algorithm Comparison The International Reference Ionospheric Parameters (IRI-2020) model was used to collect electromagnetic parameters for a certain region at 0800 on January 1, 2024. Specific parameters were set at an altitude of 60-120 km, a step size of 1 km, and moderate solar activity (F10.7 index = 131.2) to simulate typical intraday ionospheric conditions. The frequency scan range was 3-30 kHz (VLF band), with full-band coverage at a step size of 0.5 kHz. The ionospheric electron density and corresponding gradient are shown in Figure 5. Figure 5 shows that the electron density remains relatively stable in the D layer (60-80 km), while the maximum gradient in the DE transition region (80-100 km) is 0.66 × 10⁻⁶. 10 m -3 / km The maximum gradient in layer F (100-120 km) is 0.48 × 10⁻⁶. 10 m -3 / km The maximum gradient in the DE transition region is 1.4 times that of the F layer, and the electron density varies considerably across different height ranges.

[0063] A comparison is made between the traditional fixed-layering algorithm and the algorithm presented in this paper. Algorithm 1: Vertical Fixed Layering Method. The ionosphere is fixedly layered vertically, with a layer thickness of 1 km, using the classic transfer matrix algorithm. Algorithm 2: Dynamic Layering Transfer Matrix Algorithm. A reference thickness is used... d 0 =1km, adjustment factor =1.5. See the results of both algorithms. Figure 6 As shown, when both the fixed layer thickness and the dynamic reference thickness are 1 km, Figure 6 (a) The reflection coefficient (here, the reflection coefficient value of a Type I wave reflected by a medium to become a Type I wave, the same below) shows a generally consistent trend under the two algorithms, which is in line with the WKB theory expectation. Among them, a small pseudo-fluctuation appears in the reflection coefficient variation curve at around 18.5kHz under the fixed layering, which may be caused by coherent superposition effect, numerical dispersion, etc. Algorithm 2 eliminates this fluctuation and more realistically reflects the amplitude of the reflection coefficient in the low frequency range (3-11kHz). Figure 6(b) shows that the maximum variation of the reflection coefficient under the two algorithms is 0.141, at a frequency of 6.5kHz. The difference in reflection coefficient is more obvious in the range of 4.5-10kHz. Figure 6 (c) shows that the reflection coefficient deviation rate under both algorithms generally increases with increasing frequency, and the deviation rate can reach 52.5 at 30kHz. Figure 6 (d) The dynamic algorithm automatically densifies layers 29-47 based on the electron density gradient, with the densest layer being layer 38, which has a thickness of 482 μm. This adaptive densification layering more accurately reflects the electromagnetic parameters of the ionosphere. In summary, compared to Algorithm 1, Algorithm 2 is more sensitive to abrupt changes in electron density and better addresses the numerical ill-conditioned problem of the transfer matrix.

[0064] 3. Calculation and Analysis 3.1 Reference Thickness In actual radio wave propagation, the propagation characteristics of the VLF band are mainly affected by the low ionosphere. Due to the long wavelength of the VLF, the dielectric properties of the low ionosphere (such as electron density and collision frequency) often change significantly within a height range corresponding to one wavelength. This drastic parameter variation makes the choice of layering method, especially the determination of the layer thickness, a key factor in numerical calculations. Theoretically, using a more refined ionospheric layering model can better reflect the continuous variation characteristics of actual dielectric parameters, thereby improving the accuracy of the calculation results. However, this improvement in accuracy is usually accompanied by a significant increase in computational complexity. Therefore, in practical engineering applications, it is necessary to comprehensively consider computational efficiency and result accuracy, and seek the optimal balance between the two. Based on the IRI model, using the data in Section 2.4, we obtained a fixed layer thickness of 1 km and a dynamic layering reference thickness, respectively. d 0 Layered models and electromagnetic properties calculated using algorithms for 1 km, 2 km, 3 km, 4 km, and 5 km. For example... Figure 7 As shown in (a), with other factors remaining constant, as d 0 As the thickness decreases, the calculation results gradually become more consistent. d 0 When the depth drops to 3km or less, the reflection coefficient curves remain essentially the same. Considering computational efficiency and practical applications, the thickness... d 0 Setting it to 3 km is more reasonable. Figure 7(b) shows that the fixed-layer algorithm provides a uniform but rigid discretization of the ionospheric structure. In contrast, the dynamic-layer algorithm exhibits a significant nonlinear characteristic in its layer thickness distribution, and the ionospheric layering curves scale proportionally for different reference thicknesses. Table 1 shows that under the dynamic algorithm, the number of ionospheric layers decreases as the reference thickness increases, further improving computational efficiency. Parameter analysis reveals that the total number of layers N in the dynamic-layer algorithm follows a power-law relationship with the reference thickness d. (At this point, the coefficient of determination R) 2 >0.98), its nonlinear scaling characteristic stems from its adaptive response to the electron density gradient. Compared to fixed layering, dynamic layering... d 0 At a depth of 3km, the number of layers is reduced by 58%, while maintaining the ability to resolve key physical processes through local densification in the transition zone (layer thickness reduced to 1.2-1.8km). If... d 0 =1km dynamic stratification results as the benchmark d 0 At a distance of 3km, the average relative error of the full-band reflection coefficient is 4.7%, which still meets the accuracy requirements (error <5%) of ITU-R P.684 for VLF propagation models. Based on the error-cost balance analysis, it is recommended to choose... d 0 =3km is taken as the optimal parameter value. At this time, the calculation efficiency is improved by more than 50% while ensuring the resolution of the transition zone, and the reflection coefficient error meets the requirements.

[0065] Table 1 Number of ionosphere layers at different layer thicknesses

[0066] 3.2 Regulatory Factors Figure 8 This demonstrates different adjustment factors in the dynamic hierarchical algorithm. The effect of reflection coefficient on frequency variation. It can be found that: all At values ​​below 0, the reflection coefficient monotonically decreases with increasing frequency, consistent with the physical expectation of VLF wave propagation in the ionosphere (enhanced high-frequency penetration). When <1: The reflection coefficient curve is generally high, especially in the low frequency range (3-10kHz), where it deviates significantly from the mid-to-high frequency range. The results show that the algorithm's response to the electron density gradient is insufficient, leading to excessive reflection. At that time: the curve converges to a reasonable range, and the transition between frequency bands is smooth, reflecting the balance between accuracy and efficiency in dynamic layering. When the reflection coefficient is greater than 2, the reflection coefficient is systematically suppressed, which may be due to numerical dissipation caused by excessive densification of the transition zone (too small layer thickness).

[0067] Sensitivity analysis showed that ( Figure 8 The reflection coefficient of the dynamic layering algorithm is affected by The value is significantly regulated, and the optimal working range is: At this point, the reflection curve satisfies the frequency monotonicity quite well.

[0068] 4 Calculation Examples By calculating the baseline thickness of 3km and the adjustment factor The ionospheric layering model and very low frequency propagation characteristics are obtained using the algorithm with =1. Figure 9 Figure 9(a) shows that the reflection coefficient exhibits a significant nonlinear characteristic with frequency variation in the 3–30 kHz frequency range. Specifically, the reflection coefficient gradually decreases with increasing frequency, while the penetration capability is significantly enhanced in the high-frequency band. Figure 9(b) shows that the electron density exhibits a stratified increase in the 60–120 km altitude range. Under the dynamic stratification algorithm, two key gradient abrupt changes occur at approximately 90 km and 105 km, corresponding to the lower boundary of the E layer and the characteristics of the Es layer, respectively, verifying the modulation effect of the stratified structure on the ionosphere.

[0069] It should be noted that embodiments of the present invention can be implemented in hardware, software, or a combination of both. The hardware portion can be implemented using dedicated logic; the software portion can be stored in memory and executed by a suitable instruction execution system, such as a microprocessor or dedicated-design hardware. Those skilled in the art will understand that the above-described devices and methods can be implemented using computer-executable instructions and / or included in processor control code, for example, such code provided on a carrier medium such as a disk, CD, or DVD-ROM, a programmable memory such as read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. The devices and modules of the present invention can be implemented by hardware circuitry such as very large-scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, or programmable hardware devices such as field-programmable gate arrays, programmable logic devices, etc., or by software executed by various types of processors, or by a combination of the above-described hardware circuitry and software, such as firmware.

[0070] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions, and improvements made by those skilled in the art within the scope of the technology disclosed in the present invention, and within the spirit and principles of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for constructing a very low frequency ionospheric model based on a dynamic hierarchical transfer matrix algorithm, characterized in that, Perform the following steps in sequence: Step 1. Optimize the filtering of ionospheric electron density and implement dynamic adaptive stratification in the vertical direction; Step 2. Establish a nonlinear mapping relationship between electron density gradient and layer thickness, and apply hybrid regularization to the single-layer transfer matrix based on the above layering, performing singular value decomposition and dynamic truncation; Step 3. Obtain the ionospheric model with optimized parameters by tuning the baseline thickness and adjustment factor parameters.

2. The method according to claim 1, characterized in that, In step 1, the electron density gradient analysis is based on the international reference ionospheric model to obtain the electron density distribution within the target height range, and an initial stratification center height value is set at each fixed height center.

3. The method according to claim 1, characterized in that, The filtering optimization adopts the Savitzky Golay method with a window size of 2m+1. The filtering coefficients are determined by least squares fitting, and the electron density gradient is calculated by central difference.

4. The method according to claim 1, characterized in that, The layer thickness is equal to the reference thickness multiplied by the function value of the adjustment factor and the electron density gradient. The minimum layer thickness is not less than the Debye length at the corresponding height. The maximum layer thickness is limited by the vacuum speed of light, the maximum operating frequency, and the relative permittivity of the ionosphere.

5. A very low frequency ionospheric model construction system, characterized in that, include: A. A dynamic hierarchical module for performing step 1 of claim 1; B. A hybrid regularization module for performing step 2 of claim 1; C. Parameter adjustment module, used to perform step 3 of claim 1; D. Results output module, used to generate the total ionospheric transport matrix and calculate reflection and transmission characteristics.

6. The system according to claim 5, characterized in that, The dynamic layering module automatically updates the layer thickness based on the real-time electron density gradient and outputs the total number of layers N.

7. The system according to claim 5, characterized in that, The hybrid regularization module first performs Tikhonov preprocessing on the single-layer transfer matrix, then truncates the singular values ​​according to a set threshold and applies gradient weighting correction to the remaining singular values, and finally reconstructs the regularization matrix.

8. A computer-readable storage medium having instructions stored thereon that, when executed on a processor, cause the processor to perform the method of claim 1.

9. The storage medium according to claim 8, characterized in that, The instruction sets the singular value truncation threshold to be equal to the maximum singular value multiplied by a coefficient β, where β ranges from 0.001 to 0.

01.

10. The storage medium according to claim 8, characterized in that, The reference thickness in the instruction can be set between 0.05 km and 1 km, and the adjustment factor can be set between 0.1 and 2. The parameter combination corresponding to the minimum mean square error is determined by traversal search.

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