Rapid modeling method and device based on corrected spherical crown harmonic analysis and storage medium

Abstract

The invention relates to the field of earth science, and discloses a rapid modeling method and device based on corrected spherical crown harmonic analysis and a storage medium, and the method comprises the steps: carrying out the interpolation of data points, which are arranged according to a spherical coordinate latitude and longitude rule, of modeling data into data points which are arranged according to a spherical crown coordinate system rule; according to the data points regularly arranged in the spherical crown coordinate system, rapid modeling of an earth ultra-high-order geomagnetic field model is carried out based on corrected spherical crown harmonic analysis; and iteratively solving the ultrahigh-order geomagnetic field model of the earth, recalculating the kernel function of the current row or column when matrix and vector multiplication needs to be carried out in each iteration process instead of storing the kernel function in a computer memory, and only reserving a data point matrix and a special function matrix which is calculated in advance in the computer memory. The method has the beneficial effects that the superhigh-resolution regional lithosphere magnetic field correction spherical crown harmonic model can be constructed, and the forward modeling and inversion process speed is higher; and the occupied computer memory is smaller.

Description

Technical Field

[0001] The present invention relates to the field of earth science, and in particular to a rapid modeling method, device and storage medium based on modified spherical cap harmonic analysis. Background Art

[0002] The Geomagnetic Field is one of the Earth's inherent natural physical fields, and plays an important role in many fields including solid earth science, marine science, environmental science, resource and energy exploration, passive navigation, and military defense.

[0003] At present, geomagnetic field data in land areas are generally collected through aeromagnetic surveys, ground-based magnetic surveys, and geomagnetic stations, while geomagnetic field data in ocean areas mainly come from shipborne measurements. Aerial magnetic surveys and shipborne magnetic survey data are closer to the lithosphere, and the information on short-wavelength components in the data is relatively rich. However, due to differences in measurement time, observation instruments, resolution and accuracy, and data correction and processing methods among the sub-datasets in the compiled data set, splicing the sub-datasets may lead to unreliability of low-frequency components in near-surface magnetic survey data. In contrast, satellite magnetic surveys can achieve uniform coverage of the globe in a relatively short period of time.

[0004] At present, the reliable order of the global lithospheric magnetic field spherical harmonic model constructed solely from satellite data can reach spherical harmonic order 90. However, satellite data are generally distributed above 250 kilometers from the ground, resulting in sparse information on the medium and high frequency components of the lithospheric magnetic field.

[0005] In addition, satellite data are easily affected by external magnetic fields and random noise, and this effect is more serious in the polar regions. The geomagnetic field data obtained by different acquisition methods have their own advantages and disadvantages. How to combine multi-source and multi-component data to maximize the extraction of reliable signals in multi-source data sets, eliminate inconsistent external field signals and noise interference, and jointly construct a geomagnetic field model is one of the current hot topics in geomagnetism research.

[0006] When the research target is limited to a specific region rather than the entire world, the regional potential field modeling method has greater advantages than the global potential field modeling method. Under the premise of the same computing resources, regional modeling can construct a model with higher resolution. Revised Spherical Harmonic Analysis (R-SCHA) is a method that can jointly use multi-source and multi-component data distributed at different altitudes to model the geomagnetic field. It has the characteristics of high computational accuracy and good continuation effect. Therefore, this method has been widely used in lithospheric magnetic field modeling, main magnetic field modeling, and paleomagnetic field modeling.

[0007] The higher the resolution of the geomagnetic field model, the higher the order of basis function expansion is needed. However, in the past, there were two major difficulties in constructing ultra-high resolution models based on the modified spherical cap harmonic analysis method.

[0008] First, as the expansion order increases, the Legendre function and Mehler function in the modified spherical cap harmonic series may have numerical over / underflow problems, resulting in numerical errors in the forward and inversion processes of modeling. Second, in order to ensure the stability and high resolution of the model, more modeling data and model parameters are required, which will increase the inversion kernel function in two dimensions, thereby increasing memory requirements.

[0009] Therefore, as the model resolution increases, memory requirements will become a significant issue. In the process of building ultra-high-resolution geomagnetic field models, the amount of calculations in some extreme cases even exceeds the processing capacity of large computing clusters. So far, in the research results published in international journals, the modified spherical cap harmonic analysis method can only construct a spherical harmonic model of about one thousandth order at most. Therefore, the construction of ultra-high-resolution geomagnetic field models has been slow. Summary of the invention

[0010] The purpose of the present invention is to propose a rapid modeling method, device and storage medium based on modified spherical cap harmonic analysis, so as to solve the technical problems of basis function numerical overflow, slow forward and inversion calculation speed and large amount of computer memory occupation encountered when constructing ultra-high-order geomagnetic field model based on the existing modified spherical cap harmonic analysis method.

[0011] The present invention provides a rapid modeling method based on modified spherical cap harmonic analysis, comprising the following steps:

[0012] S1, interpolating the modeling data points arranged according to the spherical coordinate longitude and latitude rules into data points arranged according to the spherical cap coordinate system rules;

[0013] S2. Rapid modeling of the Earth's ultra-high-order magnetic field model based on modified spherical cap harmonic analysis according to the data points arranged in the spherical cap coordinate system;

[0014] S3. Iteratively solve the Earth's ultra-high-order geomagnetic field model. When matrix and vector multiplication is required during each iteration, the kernel function of the current row or column is recalculated instead of storing the kernel function in the computer memory. At the same time, only the data point matrix and the special function matrix calculated in advance are retained in the computer memory.

[0015] A storage medium stores instructions and data for implementing a rapid modeling method based on modified spherical cap harmonic analysis.

[0016] A rapid modeling device based on modified spherical cap harmonic analysis comprises: a processor and the storage medium; the processor loads and executes instructions and data in the storage medium to implement a rapid modeling method based on modified spherical cap harmonic analysis.

[0017] The beneficial effects provided by the present invention are:

[0018] (1) The application of an improved basis function calculation method greatly improves the numerical overflow / underflow problem in the basis function calculation process, thereby improving the stability of the calculation and being able to construct an ultra-high-resolution regional lithospheric magnetic field corrected spherical cap harmonic model;

[0019] (2) Since the idea of ​​interpolating the modeling data into data points arranged according to the rules of the spherical cap coordinate system is utilized, the amount of calculation of special functions is greatly reduced. Therefore, the forward modeling and inversion process of the modified spherical cap harmonic model based on the present invention is faster;

[0020] (3) Due to the calculation strategy of recalculating the kernel function of the current row or column when matrix and vector multiplication is required in each iteration, instead of storing the kernel function in the computer memory, the computer memory usage is greatly reduced, making it possible to build ultra-large data volumes and ultra-high resolution models. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 It is a schematic diagram of the modeling area of ​​the modified spherical cap harmonic analysis method;

[0022] Figure 2 It is a schematic flow chart of the method of the present invention;

[0023] Figure 3 It is a schematic diagram of the transformation relationship between the spherical coordinate system and the spherical cap coordinate system on the left;

[0024] Figure 4 It is the interpolation schematic diagram in the present invention;

[0025] Figure 5 It is a simulated schematic diagram of demonstration data of the present invention;

[0026] Figure 6 It is a working schematic diagram of the hardware device of the present invention. DETAILED DESCRIPTION

[0027] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0028] Before formally explaining the present invention, relevant technical terms in the present invention are explained. In addition, the scheme of the present invention is also generally explained first for easy understanding.

[0029] First, the modified spherical cap harmonic analysis method mentioned in the present invention was formally proposed by Thébault et al. in an article entitled "Revised spherical cap harmonic analysis (R-SCHA): validation and properties" published in the Journal of Geophysical Research in 2006. The modeling area Ω of this method is composed of a distance from the center of the earth to r a and r b The two spheres S 1 With S 2 and the vertex is the center of the earth and the half angle is θ 0 For details, please refer to Figure 1 , Figure 1 It is a schematic diagram of the modeling area of ​​the modified spherical cap harmonic analysis method.

[0030] Modified spherical cap harmonic analysis solves the Laplace equation in the modeling area and obtains the series expression of magnetic potential anomaly, as shown in formula (1):

[0031]

[0032] Among them, V is the magnetic potential anomaly, r, θ and λ are the geocentric distance, co-latitude and longitude in the spherical cap coordinate system respectively, a is the radius of the earth's reference sphere (generally taken as 6371.2km in geomagnetism), and the magnetic potential is composed of four function sequences: endogenous and exogenous Legendre functions, Maille functions and constant functions. is the Schmidt quasi-normalized Legendre function, is the normalized Maille function, is a normalized constant function, P max With M max is the maximum truncation order of each function sequence, To correct the spherical cap harmonic coefficient, R p With R 0 are the radius functions of the Mailer function and the constant function sequence respectively.

[0033] The second is the kernel function, which is a technical term in the inversion process in geophysics. The kernel function contains the location information of the observation system and the physical laws.

[0034] Please refer to Figure 2 , Figure 2 It is a schematic diagram of the process of the present invention.

[0035] The present invention provides a rapid modeling method based on modified spherical cap harmonic analysis, comprising the following steps:

[0036] S1, interpolating the modeling data points arranged according to the spherical coordinate longitude and latitude rules into data points arranged according to the spherical cap coordinate system rules;

[0037] It should be noted that before modeling with the modified spherical cap harmonic analysis method, the data needs to be converted from the spherical coordinate system to the spherical cap coordinate system, such as Figure 3 As shown, the conversion process has four steps:

[0038] First, transform the data from spherical coordinates to Figure 3 The rectangular coordinate system shown in (a);

[0039] The second step is, Figure 3 As shown in (b), the rectangular coordinate system is rotated along the z axis by λ 0 , so that the center of the spherical cap is located on the z'x' plane;

[0040] The third step is Figure 3 As shown in (c), the coordinate system rotated in the second step is rotated again along the y' axis by θ 0 , so that the center of the spherical cap (θ 0 ,λ 0 ) is located at the North Pole of the new coordinate system,

[0041] Finally, the coordinates and components in the rectangular coordinate system (x", y", z") are converted to the corresponding spherical coordinate system, thus completing the component conversion between the spherical coordinate system and the spherical cap coordinate system.

[0042] Generally, the data used in the modified spherical cap harmonic analysis method are arranged according to the longitude and latitude rules of the spherical coordinate system. After these coordinates are converted into the spherical cap coordinate system, the distribution of the coordinates is irregular, and the spherical cap co-latitude θ of each coordinate is different.

[0043] The special function in the modified spherical cap harmonic analysis method is only related to the spherical cap co-latitude θ.

[0044] It should be noted that the special functions include: Schmidt quasi-normalized Legendre function, normalized Mailer function, normalized constant function and radius function R of Mailer function and constant function sequence. p With R 0 .

[0045] As an example, the formula of the special function is as follows:

[0046] Schmidt quasi-normalized Legendre function The expression of is shown in formula (2):

[0047]

[0048] Where n and m are the order and degree of Schmidt's quasi-normalized associated Legendre function, respectively, Γ(x) is the gamma function with x as the variable, 2 F 1 (a, b; γ; x) is the Gaussian hypergeometric function, and its expression is:

[0049]

[0050] Among them, x (k) is the Pochhammer symbol, defined as:

[0051]

[0052] Normalized Mailer function The expression of is shown in formula (5):

[0053]

[0054] in:

[0055]

[0056] Normalized constant function The expression of is shown in formula (7):

[0057]

[0058] in:

[0059] The radius function R of the Mailer function and constant function sequence p With R 0 They are:

[0060]

[0061] Where s = ln(r b / r a ).

[0062] From formulas (2)-(9), we know that the calculation of special functions is related to the spherical cap co-latitude θ.

[0063] Therefore, in the present invention, the data points arranged according to the spherical coordinate longitude and latitude rules are interpolated into data points arranged according to the spherical cap coordinate system rules.

[0064] After doing this, while keeping the data point density the same, after coordinate transformation, multiple data points will be located on the same spherical cap co-latitude, forming data points evenly distributed on multiple concentric rings centered on the spherical cap center.

[0065] Since the special functions of data points distributed on the same spherical cap co-latitude of the spherical cap only need to be calculated once, the number of calculations of special functions in the modified spherical cap harmonic analysis method can be greatly reduced by reasonably designing the calculation program, thereby accelerating the forward and inverse processes of the modified spherical cap harmonic analysis method.

[0066] For details, please refer to Figure 4 , Figure 4 is a schematic diagram of interpolation in the present invention, which specifically adopts a bilinear interpolation method. Figure 4 (a) in the figure represents data points arranged according to the spherical coordinate longitude and latitude rules. Figure 4 (b) in the figure represents data points arranged according to the rules of the spherical cap coordinate system.

[0067] S2. Rapid modeling of the Earth's ultra-high-order magnetic field model based on modified spherical cap harmonic analysis according to the data points arranged in the spherical cap coordinate system;

[0068] It should be noted that the modeling and calculation in step S2 have been described in the previous steps, such as formulas (1)-(9). It should be noted that the gamma function appears many times in the calculation formula of the special function, and Γ(x) is equivalent to the factorial of (x-1). Therefore, the gamma function increases sharply with the increase of the parameter, and the range of double-precision floating-point numbers in computer programs can only be -2 1024 ~2 1024 As the order increases, some parameters in the formula will exceed this range, and the calculation will fail due to the appearance of nan (not a number) values. Even if it does not exceed this range, for a very large number, or a number very close to 0, addition, subtraction, multiplication and division of these numbers will lose precision, leading to actuarial loss of precision. This problem is called the overflow / underflow problem of numerical calculation.

[0069] In order to solve this problem in a targeted manner, the method for calculating the Legendre function in the present invention adopts the algorithm mentioned in the article "An algorithm for the numerical evaluation of the associated Legendre functions that runs in time independent of degree and order" published by James Bremer in the Journal of computational physics in 2018. This algorithm can effectively calculate the order of 10 6 The associated Legendre functions within can fully meet the requirements of the modified spherical cap harmonic analysis method.

[0070] The calculation method of the Mehler function applies the extended exponential method mentioned in the article "Extended-range arithmetic and normalized Legendre polynomials" published by Smith et al. in "ACM transactions on mathematical software" in 1981 to the calculation of the Mehler function, which solves the problem of numerical overflow leading to calculation failure when the spherical cap angle is small and the order of the Mehler function is large.

[0071] S3. Iteratively solve the Earth's ultra-high-order geomagnetic field model. When matrix and vector multiplication is required during each iteration, the kernel function of the current row or column is recalculated instead of storing the kernel function in the computer memory. At the same time, only the data point matrix and the special function matrix calculated in advance are retained in the computer memory.

[0072] It should be noted that by separating the modified spherical cap harmonic coefficients from the series expression, the positive problem of modified spherical cap harmonic modeling can be described as a linear system:

[0073] Gm=d(10)

[0074] Among them, G is the kernel function matrix containing the basis function, m is the modified spherical cap harmonic coefficient matrix, and d is the observation data matrix. Therefore, it can be seen that the larger the amount of data, the more rows of the kernel function, and the more model parameters, the more columns of the kernel function.

[0075] In the practical application of regional geomagnetic field modeling, the number of observation data is usually much larger than the number of coefficients that need to be solved. Therefore, the inverse problem of modified spherical cap harmonic modeling is to find the weighted least squares solution of the linear equations:

[0076]

[0077] Where superscript T represents the transpose of the matrix, and λ is the inversion damping factor. The size of the damping factor determines the trade-off between model complexity and data fitting. The smaller the damping factor, the better the model fits the data; conversely, the larger the damping factor, the smaller the model complexity and the worse the data fitting effect. In order to prevent the model from fitting the noise in the observed data, the complexity of the model cannot be too high, so it is necessary to add a damping factor for regularization. I is the unit matrix, W d This is the data weighting matrix:

[0078]

[0079] In the formula, σ i is the standard deviation of the ith observation.

[0080] It should be noted that as the dimension of the large linear equation system shown in formula (11) increases, the process of matrix multiplication and matrix inversion will become very time-consuming.

[0081] Therefore, large linear equations are usually solved by more efficient iterative methods, such as the steepest descent method, the conjugate gradient method, etc. In the iterative method, each iteration only involves a few matrix and vector multiplications, so it is more efficient.

[0082] At the same time, since the present invention adopts coordinate points arranged according to the rules of the spherical cap coordinate system, the forward modeling time of the kernel function is greatly shortened.

[0083] In order to reduce memory usage, the present invention adopts a new calculation strategy: whenever the iterative process requires matrix and vector multiplication, the kernel function of the current row or column is recalculated instead of storing the kernel function in the computer memory.

[0084] Therefore, only the data point matrix and the special function matrix calculated in advance need to be kept in the computer memory, which greatly reduces the computer memory usage during the forward and inversion process. This strategy makes it possible to build ultra-large data volumes and ultra-high resolution models.

[0085] To verify the method of the present invention, the present invention provides a specific embodiment as follows.

[0086] The simulation data demonstration example focuses on China and its consular district. The center of the modeled spherical cap is located at 105° east longitude and 27.5° north latitude, and the spherical cap angle of the data distribution is 30°.

[0087] The simulated data include simulated near-surface aeromagnetic scalar data and simulated satellite magnetic vector data. The simulated near-surface data comes from the second-generation WDMAM global lithosphere aeromagnetic anomaly compilation grid data, which compiles aeromagnetic data from more than 40 countries and regions and a large amount of shipborne marine magnetic survey line data. However, data collection for some countries and regions has not yet been completed, and the GRIMM_L120 global lithosphere magnetic anomaly field model is used to fill in the gaps without data. In areas where shipborne marine magnetic survey data are relatively sparse, the ocean crust single-layer magnetization intensity vector distribution model is used, the magnetization intensity of the model is adjusted based on the existing marine magnetic survey data to fit the actual observation data, and then the data in the vacant area is forward predicted, thereby filling in the ocean area with relatively sparse data. In addition, the compiled aeromagnetic data are affected by the measuring instruments, measuring time, different resolutions, and the time-varying changes of the Earth's main magnetic field. The low-frequency or long-wave components in the data are often unreliable (eg, Fu Aoyun, 1984; Ravat et al., 2003). Therefore, Lesur et al. used the spherical harmonics of the 16th to 100th order in the GRIMM_L120 global lithospheric magnetic field model to replace the corresponding low-frequency components in the WDMAMv2 grid data, thereby improving the accuracy of the low-frequency components of the compiled grid. The compiled grid data has an altitude of 5 km in the land area and 0 km in the ocean area. The coordinate system is the WGS84 ellipsoid coordinate system, the grid spacing of the data is 0.05°, and there are 1304210 data points in total. The simulated near-surface aeromagnetic data used in the simulation data demonstration example are as follows: Figure 5 As shown in (a) in .

[0088] The satellite magnetic data were simulated using the global lithospheric magnetic field model SH-WDMAMv2 derived from WDMAMv2 to calculate the B distribution at an altitude of 300 km. x , B y and B z The grid spacing of the data is 0.2°, with a total of 81,520 data points. The simulated satellite magnetic survey data used in the simulation data demonstration example are as follows: Figure 5 As shown in (a) to (c) in the figure. Figure 5 (a) represents the simulated near-surface aeromagnetic scalar data, and (b) to (c) represent the simulated satellite magnetic vector data.

[0089] The above data are used to model the modified spherical cap harmonic analysis. The resolution of the modified spherical cap harmonic analysis method is mainly determined by the size of the truncation order of the internal source field. The stage order of the external source field is set to 18, and the truncation technique of the Mehler function and the constant function is set to 3. The truncation order of the internal source field sequence is continuously increased to compare the computer memory usage and calculation time of the ordinary algorithm and the FastRSCHA algorithm. The iterative algorithm adopts the bistable linear conjugate gradient method (Bi-CGStab), and the number of iterations is set to 5 times.

[0090] The workstation computer used in the comparative experiment uses the Ubantu 22.04 system and is equipped with two CPU E5-2699v4, 256G computer memory, both the common algorithm and FastRCHA algorithm are programmed in Fortran, the program uses openMP parallelization, and the compiler used is ifx Fortran Compiler, the main memory usage of the two algorithms and the time required to complete the calculation are shown in Table 1:

[0091] Table 1 Comparison of computing resource usage and computing efficiency of common algorithms and FastRSCHA algorithms

[0092]

[0093] According to the data in Table 1, the FastRSCHA algorithm occupies less computing memory and has a faster computing speed than the common algorithm. This makes the construction of a regional lithospheric magnetic field model with ultra-large data volume and ultra-high resolution no longer constrained by computing resources. Therefore, the FastRSCHA algorithm (the abbreviation of the algorithm mentioned in this invention) has practical application value and is worthy of further promotion and use.

[0094] See also Figure 6 , Figure 6 4 is a schematic diagram of the working of the hardware device of an embodiment of the present invention, wherein the hardware device specifically comprises: a rapid modeling device 401 based on modified spherical cap harmonic analysis, a processor 402 and a storage medium 403.

[0095] A rapid modeling device 401 based on modified spherical cap harmonic analysis: The rapid modeling device 401 based on modified spherical cap harmonic analysis implements the rapid modeling method based on modified spherical cap harmonic analysis.

[0096] Processor 402: The processor 402 loads and executes the instructions and data in the storage medium 403 to implement the rapid modeling method based on modified spherical cap harmonic analysis.

[0097] Storage medium 403: The storage medium 403 stores instructions and data; the storage medium 403 is used to implement the rapid modeling method based on modified spherical cap harmonic analysis.

[0098] The beneficial effects of the present invention are:

[0099] (1) The application of an improved basis function calculation method greatly improves the numerical overflow / underflow problem in the basis function calculation process, thereby improving the stability of the calculation and being able to construct an ultra-high-resolution regional lithospheric magnetic field corrected spherical cap harmonic model;

[0100] (2) Since the idea of ​​interpolating the modeling data into data points arranged according to the rules of the spherical cap coordinate system is utilized, the amount of calculation of special functions is greatly reduced. Therefore, the forward modeling and inversion process of the modified spherical cap harmonic model based on the present invention is faster;

[0101] (3) Due to the calculation strategy of recalculating the kernel function of the current row or column when matrix and vector multiplication is required in each iteration, instead of storing the kernel function in the computer memory, the computer memory usage is greatly reduced, making it possible to build ultra-large data volumes and ultra-high resolution models.

[0102] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A rapid modeling method based on modified spherical cap harmonic analysis, characterized in that: The method comprises the following steps: S1, interpolating the modeling data points arranged according to the spherical coordinate longitude and latitude rules into data points arranged according to the spherical cap coordinate system rules; S2. Rapid modeling of the Earth's ultra-high-order magnetic field model based on modified spherical cap harmonic analysis according to the data points arranged in the spherical cap coordinate system; S3. Iteratively solve the Earth's ultra-high-order geomagnetic field model. When matrix and vector multiplication is required during each iteration, the kernel function of the current row or column is recalculated instead of storing the kernel function in the computer memory. At the same time, only the data point matrix and the special function matrix calculated in advance are retained in the computer memory.

2. A rapid modeling method based on modified spherical cap harmonic analysis as claimed in claim 1, characterized in that: The interpolation in step S1 adopts bilinear interpolation.

3. A rapid modeling method based on modified spherical cap harmonic analysis as claimed in claim 1, characterized in that: In step S2, a rapid model is built and the Laplace equation is solved in the modeling area to obtain the series expression of the magnetic potential anomaly as follows: Among them, V is the magnetic potential anomaly, r, θ and λ are the geocentric distance, co-latitude and longitude in the spherical cap coordinate system respectively, a is the radius of the earth's reference sphere, and the magnetic potential is composed of four function sequences: endogenous and exogenous Legendre functions, Maille functions and constant functions. is the Schmidt quasi-normalized Legendre function, is the normalized Maille function, is a normalized constant function, P max With M max is the maximum truncation order of each function sequence, To correct the spherical cap harmonic coefficient, R p and R0 are the radius functions of the Mailer function and the constant function sequence respectively; the special functions in step S3 include: Schmidt quasi-normalized Legendre function, normalized Mailer function, normalized constant function and the radius function R of the Mailer function and the constant function sequence. p With R0.

4. A rapid modeling method based on modified spherical cap harmonic analysis as claimed in claim 3, characterized in that: The Legendre function is calculated using the James Bremer method.

5. A rapid modeling method based on modified spherical cap harmonic analysis as claimed in claim 3, characterized in that: The Mailer function is calculated using the extended exponential method.

6. A rapid modeling method based on modified spherical cap harmonic analysis as claimed in claim 3, characterized in that: When performing iterative solution in step S3, the modified spherical cap harmonic coefficients are separated from the series expression to obtain a linear system Gm=d, where G is the kernel function matrix containing the basis function, m is the modified spherical cap harmonic coefficient matrix, and d is the observation data matrix; the linear system is inversely solved to obtain Where, the superscript T represents the transpose of the matrix, λ is the inversion damping factor, I is the unit matrix, and W d is the data weighting matrix, In the formula, σ i is the standard deviation of the ith observation.

7. A storage medium, characterized in that: The storage medium stores instructions and data for implementing a rapid modeling method based on modified spherical cap harmonic analysis as described in any one of claims 1 to 6.

8. A rapid modeling device based on modified spherical cap harmonic analysis, characterized in that: include: Processor and storage medium; the processor loads and executes instructions and data in the storage medium to implement a rapid modeling method based on modified spherical cap harmonic analysis as described in any one of claims 1 to 6.

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