Methods, systems, media, and products for constructing elevation datums based on low-order cosine modified Stokes kernel functions.

Abstract

This invention discloses a method, system, medium, and product for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function, relating to the field of physical geodesy. The method includes: calculating model data using an Earth gravity field model, topographic data, and a high-frequency topographic influence combination algorithm; determining the direct and indirect effects of topographic compression on ground gravity anomalies and the elevation datum based on topographic compression; determining empirical and test values ​​of the correction order of the low-order cosine-corrected Stokes kernel function based on the cumulative error of the model gravity anomaly and the test frequency bandwidth; calculating an empirical and test elevation datum using the low-order cosine-corrected Stokes kernel function based on the empirical and test values ​​of the correction order; and determining the final elevation datum based on the empirical and test elevation datums. This invention can improve the accuracy of elevation datum construction.

Description

Technical Field

[0001] This invention relates to the field of physical geodesy technology, and in particular to a method, system, medium, and product for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function. Background Technology

[0002] Orthographic and normal height systems are widely used in different countries and regions, with the geoid and quasi-geoid serving as their respective elevation reference surfaces. Determining a geoid with 1 cm accuracy is a crucial goal of 21st-century geodesy. For decades, many geodesy researchers have continuously improved the accuracy of (quasi-)geoids by optimizing geoid construction methods. The third boundary value problem in physical geodesy is the theoretical foundation for regional (quasi-)geoid construction. The third boundary value theory uses the Stokes integral, with gravity anomaly data as its physical input. Gravity anomalies are the difference between actual and normal gravity values ​​caused by irregular distribution of Earth's mass, and they are of significant value in geodesy and geophysical research and applications.

[0003] Theoretically, the Stokes integral is a global integral, but due to the difficulty in obtaining global gravity data and the large computational cost of global integration, practical calculations typically use the removal-and-recovery technique, which only requires near-field integration. In this technique, the gravity field model serves as the reference gravity field in the boundary value calculation, meaning the Stokes integral actually includes two types of gravity data: measured gravity anomalies and reference model gravity anomalies. Errors in actual gravity measurements and gridding errors at discrete measurement points result in relatively low accuracy of measured gravity data over long wavelengths, while the accuracy of the gravity field model is very high in the low-frequency bands. As the spherical harmonic order increases, the accuracy of the reference model gravity data gradually decreases, while the relative accuracy of the measured gravity data gradually increases. An ideal Stokes kernel function should adjust the weights (or contribution rates) of the two types of gravity data in the generated elevation anomalies / geoid undulations based on their relative accuracy. When using the standard Stokes kernel function, the high conversion rate of measurement data and the low conversion rate of reference data in the low-frequency band result in large errors in the calculated low-order geoid undulations / geoid undulations. Conversely, the large far-field truncation error of gravity data in the high-frequency band leads to relatively large errors in the calculated high-order geoid undulations / geoid undulations. Therefore, some geodesists have conducted research on optimization methods for the Stokes kernel function. Modifying the Stokes kernel function can effectively improve the accuracy of the (pseudo)geoid solution under the removal-recovery mode, but past research on Stokes kernel function modification methods and spectral characteristic analysis is relatively limited. Summary of the Invention

[0004] The purpose of this invention is to provide a method, system, medium, and product for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function, which can improve the construction accuracy of geoids and quasi-geoids.

[0005] To achieve the above objectives, the present invention provides the following solution:

[0006] A method for constructing an elevation datum based on a low-order cosine modified Stokes kernel function includes:

[0007] The system acquires the Earth's gravity field model, ground gravity anomaly data, topographic data, GNSS leveling point data, the type of elevation datum, and the test frequency bandwidth. The elevation datum includes two types: geoid and quasi-geoid. The geoid is the datum for the orthogonal elevation system, and the quasi-geoid is the datum for the normal elevation system.

[0008] The model data, the direct impact of terrain compression on ground gravity anomaly, and the indirect impact of terrain compression on the elevation datum are calculated using a combination algorithm of Earth gravity field model, terrain data, and high-frequency terrain influence. The model data includes either the model geoid height or the model elevation anomaly, as well as the model gravity anomaly. The indirect impact of terrain compression on the elevation datum is either the indirect impact of terrain compression on the geoid or the indirect impact of terrain compression on the quasi-geoid.

[0009] The empirical value and the test value of the correction order of the low-order cosine-corrected Stokes kernel function are determined based on the cumulative error of the model gravity anomaly and the test frequency bandwidth; the correction order includes the initial correction order and the cutoff correction order.

[0010] A low-order cosine-corrected Stokes kernel function is used to calculate an empirical elevation datum and a test elevation datum based on ground gravity anomaly data, topographic data, model data, the direct impact of topographic compression on ground gravity anomaly, the indirect impact of topographic compression on the elevation datum, empirical values ​​of the correction order, and test values ​​of the correction order. The empirical elevation datum is an empirical geoid or an empirical quasi-geoid; the test elevation datum is a test geoid or a test quasi-geoid.

[0011] The final elevation datum is determined based on GNSS leveling point data, empirical elevation datum, and test elevation datum; the final elevation datum is the final geoid or the final quasi-geoid.

[0012] A computer system includes: a memory, a processor, and a computer program stored in the memory and executable on the processor, the processor executing the computer program to implement the steps of the elevation datum construction method based on the low-order cosine modified Stokes kernel function as described in any of the preceding claims.

[0013] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the elevation datum construction method based on the low-order cosine modified Stokes kernel function as described in any of the preceding claims.

[0014] A computer program product includes a computer program that, when executed by a processor, implements the steps of the elevation datum construction method based on the low-order cosine modified Stokes kernel function as described in any of the preceding claims.

[0015] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0016] This invention discloses a method, system, medium, and product for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function. The method includes acquiring an Earth gravity field model, ground gravity anomaly data, terrain data, GNSS leveling point data, the type of elevation datum, and the test frequency bandwidth; using the Earth gravity field model, terrain data, and a high-frequency terrain influence combination algorithm, calculating the model data, the direct impact of terrain compression on ground gravity anomalies, and the indirect impact of terrain compression on the elevation datum; determining the empirical value and test value of the correction order of the low-order cosine-corrected Stokes kernel function based on the cumulative error of the model gravity anomaly and the test frequency bandwidth; using the low-order cosine-corrected Stokes kernel function to calculate the empirical elevation datum and the test elevation datum based on the ground gravity anomaly data, terrain data, model data, the direct impact of terrain compression on ground gravity anomalies, the indirect impact of terrain compression on the elevation datum, the empirical value of the correction order, and the test value of the correction order; and determining the final elevation datum based on the GNSS leveling point data, the empirical elevation datum, and the test elevation datum. This invention can improve the accuracy of constructing elevation datum surfaces. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 A schematic diagram of the normalized far-field cutoff coefficients for the standard Stokes kernel function;

[0019] Figure 2 This is a diagram illustrating the conversion rate of gravity data when using the standard Stokes kernel function;

[0020] Figure 3 A schematic diagram of the normalized far-field cutoff coefficients for the low-order cosine-corrected Stokes kernel function;

[0021] Figure 4 A schematic diagram illustrating the conversion rate of gravity data when using a low-order cosine-corrected Stokes kernel function;

[0022] Figure 5 This is a schematic diagram of the process for constructing an elevation datum based on a low-order cosine modified Stokes kernel function. Detailed Implementation

[0023] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0024] The purpose of this invention is to provide a method, system, medium, and product for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function, aiming to improve the construction accuracy of geoids and quasi-geoids.

[0025] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0026] Example 1

[0027] like Figure 5 As shown, a method for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function in this embodiment includes:

[0028] Step 101: Obtain the Earth's gravity field model, ground gravity anomaly data, topographic data, GNSS leveling point data, the type of elevation datum, and the test frequency bandwidth. The elevation datum types include geoid and quasi-geoid. The geoid is the datum for the orthographic elevation system. The quasi-geoid is the datum for the normal elevation system.

[0029] Step 102: Using the Earth's gravity field model, topographic data, and a high-frequency topographic influence combined algorithm, calculate the model data, the direct impact of topographic compression on ground gravity anomalies, and the indirect impact of topographic compression on the elevation datum. The model data includes either the model geoid height or the model elevation anomaly, as well as the model gravity anomaly. The indirect impact of topographic compression on the elevation datum is either the indirect impact of topographic compression on the geoid or the indirect impact of topographic compression on the quasi-geoid.

[0030] If the elevation datum is a geoid, the geoid height and ground model gravity anomaly are calculated using the Earth's gravity field model and topographic data.

[0031] A high-frequency topographic influence combination algorithm was used to calculate the direct impact of topographic compression on ground gravity anomaly and the indirect impact of topographic compression on geoid based on topographic data.

[0032] If the elevation datum is a geoid, then the model elevation anomaly and the ground model gravity anomaly are calculated using the Earth gravity field model and topographic data.

[0033] A high-frequency topographic influence combination algorithm was used to calculate the direct impact of topographic compression on ground gravity anomaly and the indirect impact of topographic compression on the quasi-geoid based on topographic data.

[0034] Step 103: Based on the cumulative error of the model's gravity anomaly and the test frequency bandwidth, determine the empirical value and the test value of the correction order of the low-order cosine-corrected Stokes kernel function. The correction order includes the initial correction order and the cutoff correction order.

[0035] The cumulative error of the model gravity anomaly is calculated based on the formula for calculating the cumulative error of the model gravity anomaly.

[0036] The formula for calculating the cumulative error of the gravity anomaly in the model is as follows:

[0037]

[0038] Where L is the cutoff order of the reference gravity field model, and δΔg G (L) represents the cumulative error of the gravity anomaly truncated to order L. The error represents the gravity anomaly error of the nth-order model, γ represents normal gravity, and δC represents gravity anomaly error. nm C represents the spherical harmonic coefficient term in the gravity field model. nm The error, δS nm S represents the spherical harmonic coefficient term in the gravity field model. nm The error is σ, where σ represents the standard deviation, n represents the order, and m represents the degree.

[0039] The cumulative error of the model gravity anomaly is calculated based on the formula for calculating the cumulative error of the model gravity anomaly.

[0040] The lowest order corresponding to when the cumulative error of the model's gravity anomaly exceeds 0.01 mGal is set as the empirical value of the initial correction order.

[0041] The lowest order corresponding to when the cumulative error of the model's gravity anomaly exceeds 1 mGal is set as the empirical value of the cutoff correction order.

[0042] The empirical value of the initial correction order is set as the test value of the initial correction order by adding or subtracting multiples of 10 or 100 within the frequency band of the test frequency bandwidth.

[0043] The empirical value of the cutoff correction order is set as the test value by adding or subtracting multiples of 10 or 100 within the frequency band of the test frequency bandwidth.

[0044] Step 104: Using a low-order cosine-corrected Stokes kernel function, calculate the empirical elevation datum and the test elevation datum based on ground gravity anomaly data, topographic data, model data, the direct impact of topographic compression on ground gravity anomaly, the indirect impact of topographic compression on the elevation datum, empirical values ​​of the correction order, and test values ​​of the correction order. The empirical elevation datum is an empirical geoid or an empirical quasi-geoid. The test elevation datum is a test geoid or a test quasi-geoid.

[0045] The low-order cosine-corrected Stokes kernel function includes both non-extended and extended forms, specifically:

[0046]

[0047] Where S(ψ) represents the low-order cosine modified Stokes kernel function in its non-extended form; S(r P ,ψ) denotes the extended form of the low-order cosine-corrected Stokes kernel function; rp denotes the geocentric radial distance of the computation point; ψ denotes the angular distance between the computation point and the flow integral point; This represents the nth-order spectral weight of the low-order cosine-corrected Stokes kernel function; n represents the order; R represents the Earth's radius; P n (cosψ) represents an n-order Legendre polynomial; L0 is the initial correction order, and L1 is the cutoff correction order.

[0048] The difference between the ground gravity anomaly and the ground model gravity anomaly is obtained.

[0049] The remaining ground Helmert gravity anomaly is obtained by adding the gravity anomaly difference to the direct impact of terrain compression on the ground gravity anomaly.

[0050] The remaining ground Helmert gravity anomaly is extended to obtain the processed remaining boundary surface Helmert gravity anomaly.

[0051] If the elevation datum is a geoid, then the Stokes kernel function with a non-extended form, based on the empirical value and the test value of the correction order, is used to perform Stokes integration on the processed residual boundary surface Helmert gravity anomaly to obtain the empirical residual geoid height and the test residual geoid height.

[0052] The empirical geoid is obtained by adding the empirical residual geoid height, the indirect effect of topographic compression on the geoid, and the model geoid height.

[0053] The test geoid is obtained by adding the remaining geoid height, the indirect effect of topographic compression on the geoid, and the model geoid height.

[0054] If the type of the elevation datum is a geoid, then the extended form of the low-order cosine-corrected Stokes kernel function is used to perform Stokes integration on the processed residual boundary surface Helmert gravity anomaly based on the empirical value and the test value of the correction order, to obtain the empirical residual elevation anomaly and the test residual elevation anomaly.

[0055] The empirical residual elevation anomaly, the indirect effect of the topographic compression on the quasi-geoid, and the model elevation anomaly are added together to obtain the empirical quasi-geoid.

[0056] The test geoid is obtained by adding the test residual elevation anomaly, the indirect effect of the terrain compression on the quasi-geoid, and the model elevation anomaly.

[0057] Step 105: Determine the final elevation datum based on the empirical elevation datum and the test elevation datum; the final elevation datum is the final geoid or the final quasi-geoid.

[0058] If the elevation datum is a geoid, the final elevation datum is determined using GNSS leveling point data based on empirical geoids and test geoids.

[0059] If the type of the elevation datum is a geoid, the final elevation datum is determined using GNSS leveling point data based on empirical geoids and test geoids.

[0060] Standard Stokes kernel function

[0061] The Stokes integral is a theoretical tool for calculating geoid height or geoid undulation using gravity anomaly data in third boundary value problems. The Stokes integral includes two forms: non-extended and extended. The non-extended Stokes integral is used to calculate geoid height, while the extended Stokes integral is used to calculate geoid undulation. The standard Stokes kernel function for the extended form is...

[0062]

[0063] In the formula S ST (r P (,ψ) is the extended form of the Stokes standard kernel function, where R is the average radius of the Earth, and r P Let P be the geocentric radius vector of the point being calculated in spherical coordinates. n (cosψ) is an nth-order Legendre polynomial, ψ is the angular distance between the center of the sphere and the calculation point, and l P0 The formula for calculating the distance from a point to the center of a sphere in a spherical flow is as follows:

[0064] The standard Stokes kernel function in its non-extended form is

[0065]

[0066] In the formula S ST (ψ) is the standard Stokes kernel function in its non-extended form, and the expression for symbol t is sin(ψ / 2). Since S ST (ψ) is S ST (r P In ψ), the parameter r P Since this is a special case when R is taken, this invention only describes the method for modifying the extended kernel function.

[0067] The Stokes kernel function can be represented in the following general spectral form.

[0068]

[0069] In the formula S(r P ,ψ) represents the Stokes kernel function, ω n This represents the nth-order spectral weight (or correction factor). The difference between different types of Stokes kernel functions lies in the different values ​​of the spectral weights. When S(r P When ψ) specifically represents the standard Stokes kernel function, the spectral weights of each order in equation (10) are all taken as 1. When S(r P The spectral weights (ψ) typically range from [0,1] when modifying the Stokes kernel function. The formula for calculating the spectral weights of a low-order cosine modified Stokes kernel function is:

[0070]

[0071] in, This represents the n-th order spectral weight of a low-order cosine-corrected Stokes kernel function; n represents the order; L0 is the initial low-order correction order, and L1 is the cutoff low-order correction order. Cosine-corrected kernel functions include two forms: low-order correction only and correction of both high and low orders. The spectral weights of high and low order cosine-corrected Stokes kernel functions are:

[0072]

[0073] In the formula L2 represents the nth-order spectral weights of the cosine high- and low-order modified Stokes kernel function, L3 represents the high-order initial modification order, and L4 represents the high-order cutoff modification order.

[0074] Introducing a piecewise function Its relationship with the Stokes kernel function S(r P The relationship between ,ψ) is

[0075]

[0076] Piecewise function Legendre series form is

[0077]

[0078] In the formula Q n The nth-order far-field cutoff error coefficient of the Stokes kernel function is calculated as follows:

[0079]

[0080] According to the theory of spherical function expansion, we have

[0081]

[0082] In the formula Δg n Let be the nth-order gravity anomaly of the sphere, and Δg be the gravity anomaly of the sphere. The near-field Stokes integral can be expressed in the following general spectral form.

[0083]

[0084] In the formula, ζ represents the geoid, which is the vertical distance between the quasi-geoid and the reference ellipsoid. The quasi-geoid of the region is usually represented by a geoid data grid. C1 represents the near-field range centered at the calculation point with a radius of ψ1. γ is the normal gravity at the ground calculation point. The normalized nth-order far-field cutoff error coefficient is calculated as follows:

[0085]

[0086] The spectral relationship between elevation anomaly and gravity anomaly is:

[0087]

[0088] In the formula ζ n It is an nth-order elevation anomaly.

[0089] Using equations (15) and (17), the general spectral decomposition of the Stokes integral in the removal recovery mode can be derived as follows:

[0090]

[0091] In the formula Δg T To measure gravity anomalies on a spherical surface, Its nth-order gravity anomaly value, Δg G This is the spherical reference gravity anomaly value. Its nth-order gravity anomaly value, ζ G For the reference geoid, M is the highest order of the measured gravity anomaly (usually determined based on the resolution of the ground gravity grid according to the Nyquist sampling metric), and L is the cutoff order of the reference gravity field model. Typically, L2 ≥ M, L1 ≤ L, and the spectral weights above L are all 1. Since both the reference and measured gravity data contain errors, in equation (18)... This represents the conversion rate of nth-order measured gravity data and its errors into geoid anomalies. This represents the conversion rate of nth-order reference gravity data and its errors into geoid anomalies. For different types of Stokes kernel functions, it is only necessary to normalize the far-field cutoff coefficient in equation (18). Spectral weight factor ω n Simply adjust the coefficients and spectral weights to match the corresponding Stokes kernel function.

[0092] In the removal-recovery mode, the kernel function affects the accuracy of low- and high-frequency elevation anomalies differently. In the low-frequency band of removal-recovery (2 ≤ n < L), both measured and reference gravity data contribute to the elevation anomaly. The sum of the conversion rates of each order of the measured and reference gravity data in the low-frequency band is always 1, indicating that there is no far-field truncation error in the low-frequency band. Since the errors of the two gravity data are also converted into the elevation anomaly with the Stokes integral, the error of the low-frequency elevation anomaly originates from the low-frequency errors of the measured and reference gravity data. Different Stokes kernel functions have different far-field truncation error coefficients, therefore, the proportion of the errors of the two gravity data in the conversion into low-order elevation anomaly errors also differs when using different kernel functions. In the high-frequency band of removal-recovery (L+1~M), the conversion rate of the measured data is... Reference data was not involved in the transformation. Since the high-order elevation anomaly is contributed only by the measurement data and the high-order transformation rate is not 1, the error sources of the high-order elevation anomaly are two types: high-frequency errors in the measured gravity data and far-field truncation errors.

[0093] In equation (18), r P By replacing R and γ with the normal gravity of the boundary surface γ0 and the geoid anomaly ζ with the geoid height N, we can obtain the corresponding spectral decomposition formula for calculating the geoid height in the removal and restoration mode.

[0094] The spectral characteristics of the Stokes integral using standard and cosine-corrected Stokes kernel functions are analyzed below based on the spectral decomposition of the Stokes integral in the removed recovery mode. In the spectral characteristic analysis of this invention, the elevation of the calculation point is taken as 1000m, the integration radius is taken as 1°, and the highest order of the reference model is taken as 2160.

[0095] Depend on Figure 1 It can be seen that the normalized far-field cutoff coefficient of the standard kernel function exhibits relatively large amplitude variations. Specifically, the maximum amplitude of the normalized far-field cutoff coefficient of the standard kernel function in the mid-to-high frequency band is approximately 0.2. According to... Figure 2 At lower frequencies, the conversion rate of measurement data is relatively high. However, due to the presence of many long-wavelength errors in the measurement data, the long-wavelength errors of the elevation anomaly calculated using the standard Stokes kernel function are relatively large. At higher frequencies, the amplitude of the conversion rate of measurement data when using the standard Stokes kernel function is approximately 1 ± 0.2, which deviates relatively significantly from the ideal value of 1.

[0096] Modified Stokes kernel functions include two types: low-order and high-low order. Since the difference in their spectral characteristics lies only in the final frequency band of the measurement data, and this band contains very little elevation anomaly information, the spectral analysis in this invention focuses only on the low-order modified Stokes kernel function. To illustrate the spectral characteristics of the cosine-modified Stokes kernel function under different modification frequency bands, the normalized far-field cutoff coefficient and conversion factor of the cosine-modified Stokes kernel function in the 100–300, 200–400, 2–720, and 2–2160 frequency bands are shown below. Figures 3 to 4 .

[0097] Figure 3 The results show that under four correction conditions, the normalized far-field cutoff coefficient of the Stokes kernel function in the 100–300 GHz band with low-order cosine correction is the largest, followed by the coefficient in the 200–400 GHz band, while the far-field cutoff coefficients in the 2–720 GHz and 2–2160 GHz bands are very small. Figure 4In the uncorrected low-frequency band, the conversion rate of the reference gravity data is approximately 1, while the conversion rate of the measured gravity data is approximately 0, indicating that the low-frequency elevation anomaly information mainly comes from the contribution of the reference model's gravity data. In the corrected frequency band, as the spherical harmonic order increases, the conversion rate of the reference data decreases while the conversion rate of the measured data increases. At the intermediate corrected order, the conversion rates of the reference and measured data approach 0.5. At the corrected cutoff frequency, the conversion rate of the reference data is approximately 1, while the conversion rate of the measured data is approximately 0.

[0098] To improve the accuracy of geoid height or geoid height calculations using Stokes integrals, the conversion rate of the reference model gravity data should approach 1 in the ultra-low frequency band (with very high accuracy) and approach 0 in the high frequency band (with significantly lower accuracy than measured gravity data). This invention sets the corresponding reference gravity data error thresholds to 0.01 mGal and 1 mGal, respectively. That is, the conversion rate of the reference data should be 1 when the reference model gravity data error is less than 0.01 mGal, and 0 when the reference model gravity data error is greater than 1 mGal. Therefore, the lowest order corresponding to the cumulative error of the model gravity anomaly exceeding 0.01 mGal can be set as the empirical value of the initial correction order L0, and the lowest order corresponding to the cumulative error of the model gravity anomaly exceeding 1 mGal can be set as the empirical value of the cutoff correction order L1. Calculations show that when using the EIGEN-6C4 reference model, the empirical value for L0 is 104, and the empirical value for L1 is 305.

[0099] To test the practical application effects of different Stokes kernel functions, a geoid construction experiment (i.e., calculation of elevation anomaly grid data) was conducted in a test area in central China. This test area is a mountainous region with latitude and longitude ranging from 106° to 116°E and 26° to 34°N. The topographic data used in this experiment was SRTM data. At 2′ resolution, the highest elevation in the test area was 3423.56m, the lowest was 11.08m, the average was 537.40m, and the root mean square was 714.51m. The regional geoid was constructed using the Stokes-Helmert boundary value theory method, with analytical continuation used for downward extension. Both the continuation integration radius and the Stokes integration radius were set to 1°. The constructed regional gravity geoid ranged from 108° to 114°E and 28° to 32°N, using the 2190th order EIGEN-6C4 gravity field model as the reference model. The accuracy of the gravity-based geoid was verified using GNSS leveling data. The 2′ resolution gravity-based geoid solutions obtained based on different Stokes kernel functions are shown in Table 1. The parameters of the Stokes kernel functions with high and low order cosine corrections were set to 104 for L0, 305 for L1, 5400 for L2, and 5600 for L3.

[0100] Table 1. Accuracy (m) and calculation time (s) of the quasi-geoid calculated using different Stokes kernel functions.

[0101]

[0102] According to Table 1, among several statistical measures, the standard deviation (SD) index is typically used to characterize the accuracy of the quasi-geoid without systematic bias. Besides kernel function correction methods and model gravity field errors, the accuracy of the quasi-geoid construction is also affected by factors such as local gravity measurement data errors and GNSS leveling point errors. Table 1 shows that the quasi-geoid calculated using the standard Stokes kernel function has the largest error and is therefore unsuitable for solving (quasi-)geoids. The accuracy of the quasi-geoid calculated using a low-order cosine-corrected kernel function is improved by 47.61% compared to the traditional standard kernel function. Using high- and low-order cosine-corrected Stokes kernel functions, compared to the corresponding low-order corrected kernel function, the change in the accuracy of the quasi-geoid in the experimental area is very small (0.01 cm), but the calculation time is significantly increased. This experiment demonstrates that high-order correction of the Stokes kernel function is unnecessary.

[0103] Finally, to illustrate the impact of the corrected frequency band on the accuracy of the quasi-geoid construction, a gravity quasi-geoid was calculated based on test values ​​of multiple sets of low-order cosine-corrected Stokes kernel function correction orders. Table 2 shows the accuracy statistics of the gravity quasi-geoid.

[0104] Table 2. Accuracy of the pseudo-geoid based on test values ​​with different correction orders (unit: m)

[0105] Corrected frequency band Maximum value Minimum value average value Root Mean Square Standard deviation 2～300 0.1844 -0.1339 0.0878 0.1055 0.0590 2～400 0.1907 -0.0911 0.0881 0.1042 0.0561 100～300 0.1797 -0.1009 0.0872 0.1034 0.0561 200～400 0.2142 -0.0603 0.0890 0.1091 0.0636

[0106] Table 2 shows that the correction frequency band has a significant impact on the accuracy of the quasi-geoid. The conversion rates of measured and reference gravity data differ under different correction frequency bands. Under the correction frequency bands of 2–300, 2–400, 100–300, and 200–400, the orders corresponding to a conversion rate of 0.5 for both reference and measured gravity data are approximately 151, 200, 200, and 300, respectively. Table 3 shows that the solution accuracy for the 2–400 and 100–300 correction frequency bands is consistent (5.61 cm), which is better than the solution accuracy for the 2–300 and 200–400 correction frequency bands (5.90 cm and 6.36 cm, respectively). This indicates that when the spherical harmonic order increases to around 200, reducing the conversion rate of the reference gravity data to approximately 0.5 and increasing the conversion rate of the measured gravity data to approximately 0.5 is more reasonable. Theoretically, the intermediate order of the corrected frequency band (i.e., the average of the initial and cutoff corrected orders) should be close to the order corresponding to the equality of the order errors of the measured gravity anomaly and the model gravity anomaly. In practical applications of the corrected kernel function, based on empirically selected correction parameters, and considering the order of the reference gravity field model used, as well as the accuracy of the measured and reference gravity data in the test area, multiple sets of corrected order test values ​​can be selected for repeated experiments to determine the Stokes kernel function corrected frequency band most suitable for the local (quasi-)geoid.

[0107] The spectral analysis and boundary value calculation experiments in this embodiment are both conducted for the extended Stokes kernel function. The non-extended Stokes kernel function is a special case of the extended Stokes kernel function. Therefore, the analysis and conclusions of this invention are also applicable to the non-extended Stokes kernel function for calculating geoid undulation.

[0108] Example 2

[0109] A computer system includes: a memory, a processor, and a computer program stored in the memory and executable on the processor. The processor executes the computer program to implement the steps of the elevation datum construction method based on the low-order cosine modified Stokes kernel function in Embodiment 1.

[0110] Example 3

[0111] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the elevation datum construction method based on a low-order cosine modified Stokes kernel function in Embodiment 1.

[0112] Example 4

[0113] A computer program product includes a computer program that, when executed by a processor, implements the steps of the elevation datum construction method based on a low-order cosine modified Stokes kernel function in Embodiment 1.

[0114] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0115] This document uses specific examples to illustrate the principles and implementation methods of the present invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of the present invention. Furthermore, those skilled in the art will recognize that, based on the ideas of the present invention, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function, characterized in that, The method includes: The system acquires the Earth's gravity field model, ground gravity anomaly data, topographic data, GNSS leveling point data, the type of elevation datum, and the test frequency bandwidth. The elevation datum includes two types: geoid and quasi-geoid. The geoid is the datum for the orthogonal elevation system, and the quasi-geoid is the datum for the normal elevation system. The model data, the direct impact of terrain compression on ground gravity anomaly, and the indirect impact of terrain compression on the elevation datum are calculated using a combination algorithm of Earth gravity field model, terrain data, and high-frequency terrain influence. The model data includes either the model geoid height or the model elevation anomaly, as well as the model gravity anomaly. The indirect impact of terrain compression on the elevation datum is either the indirect impact of terrain compression on the geoid or the indirect impact of terrain compression on the quasi-geoid. The empirical and test values ​​of the correction order of the low-order cosine-corrected Stokes kernel function are determined based on the cumulative error of the model gravity anomaly and the test frequency bandwidth; the correction order includes the initial correction order and the cutoff correction order. A low-order cosine-corrected Stokes kernel function is used to calculate an empirical elevation datum and a test elevation datum based on ground gravity anomaly data, topographic data, model data, the direct impact of topographic compression on ground gravity anomaly, the indirect impact of topographic compression on the elevation datum, empirical values ​​of the correction order, and test values ​​of the correction order. The empirical elevation datum is an empirical geoid or an empirical quasi-geoid; the test elevation datum is a test geoid or a test quasi-geoid. The final elevation datum is determined based on GNSS leveling point data, empirical elevation datum, and test elevation datum; the final elevation datum is the final geoid or the final quasi-geoid.

2. The method for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function according to claim 1, characterized in that, The model data, topographic data, and high-frequency topographic influence combined algorithm were used to calculate the direct impact of topographic compression on ground gravity anomalies and the indirect impact of topographic compression on the elevation datum. Specifically, these include: If the type of the elevation datum is a geoid, then the geoid height and ground model gravity anomaly are calculated using the Earth gravity field model and topographic data. A high-frequency topographic influence combination algorithm was used to calculate the direct impact of topographic compression on ground gravity anomaly and the indirect impact of topographic compression on geoid based on topographic data. If the type of the elevation datum is a geoid, then the model elevation anomaly and the ground model gravity anomaly are calculated using the Earth gravity field model and topographic data. A high-frequency topographic influence combination algorithm was used to calculate the direct impact of topographic compression on ground gravity anomaly and the indirect impact of topographic compression on the quasi-geoid based on topographic data.

3. The method for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function according to claim 1, characterized in that, The low-order cosine-corrected Stokes kernel function includes both non-extended and extended forms, specifically: Where S(ψ) represents the low-order cosine modified Stokes kernel function in its non-extended form; S(r P ,ψ) denotes the extended form of the low-order cosine-corrected Stokes kernel function; rp denotes the geocentric radial distance of the computation point; ψ denotes the angular distance between the computation point and the flow integral point; This represents the nth-order spectral weight of the low-order cosine-corrected Stokes kernel function; n represents the order; R represents the Earth's radius; P n (cosψ) represents an n-order Legendre polynomial; L0 is the initial correction order, and L1 is the cutoff correction order.

4. The method for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function according to claim 1, characterized in that, Based on the cumulative error of the model gravity anomaly and the test frequency bandwidth, empirical values ​​and test values ​​of the correction order of the low-order cosine-corrected Stokes kernel function are determined, specifically including: The cumulative error of the model gravity anomaly is calculated based on the formula for calculating the cumulative error of the model gravity anomaly. The lowest order corresponding to when the cumulative error of the model's gravity anomaly exceeds 0.01 mGal is set as the empirical value of the initial correction order; The lowest order corresponding to when the cumulative error of the model's gravity anomaly exceeds 1 mGal is set as the empirical value of the cutoff correction order. The empirical value of the initial correction order is added to or subtracted from a multiple of 10 or 100 within the frequency band of the test frequency bandwidth to set the test value of the initial correction order. The empirical value of the cutoff correction order is set as the test value by adding or subtracting multiples of 10 or 100 within the frequency band of the test frequency bandwidth.

5. The method for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function according to claim 4, characterized in that, The formula for calculating the cumulative error of the gravity anomaly in the model is as follows: Where L is the cutoff order of the reference gravity field model, and δΔg G (L) represents the cumulative error of the gravity anomaly truncated to order L. The error represents the gravity anomaly error of the nth-order model, γ represents normal gravity, and δC represents gravity anomaly error. nm C represents the spherical harmonic coefficient term in the gravity field model. nm The error, δS nm S represents the spherical harmonic coefficient term in the gravity field model. nm The error is σ, where σ represents the standard deviation, n represents the order, and m represents the degree.

6. The method for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function according to claim 1, characterized in that, Using a low-order cosine-corrected Stokes kernel function, based on ground gravity anomaly data, topographic data, model data, the direct impact of topographic compression on ground gravity anomaly, the indirect impact of topographic compression on the elevation datum, empirical values ​​of the correction order, and test values ​​of the correction order, the empirical and test elevation datums are calculated. Specifically, this includes: The difference between the ground gravity anomaly and the gravity anomaly of the ground model is obtained; The difference in gravity anomaly is added to the direct effect of terrain compression on ground gravity anomaly to obtain the remaining ground Helmert gravity anomaly. The remaining ground Helmert gravity anomaly is extended to obtain the processed remaining boundary surface Helmert gravity anomaly. If the type of the elevation datum is a geoid, then the Stokes kernel function with non-extended form, based on the empirical value of the correction order and the test value of the correction order, is used to perform Stokes integration on the processed residual boundary surface Helmert gravity anomaly to obtain the empirical residual geoid height and the test residual geoid height. The empirical geoid is obtained by adding the empirical residual geoid height, the indirect effect of topographic compression on the geoid, and the model geoid height. The test geoid is obtained by adding the remaining geoid height, the indirect effect of topographic compression on the geoid, and the model geoid height. If the type of the elevation datum is a geoid, then the extended form of the low-order cosine-corrected Stokes kernel function is used to perform Stokes integration on the processed residual boundary surface Helmert gravity anomaly based on the empirical value and the test value of the correction order, to obtain the empirical residual elevation anomaly and the test residual elevation anomaly. The empirical residual elevation anomaly, the indirect effect of the topographic compression on the quasi-geoid, and the model elevation anomaly are added together to obtain the empirical quasi-geoid. The test geoid is obtained by adding the test residual elevation anomaly, the indirect effect of the terrain compression on the quasi-geoid, and the model elevation anomaly.

7. The method for constructing an elevation datum based on a low-order cosine-corrected Stokes kernel function according to claim 1, characterized in that, The final elevation datum is determined based on empirical and test elevation datums, specifically including: If the type of the elevation datum is a geoid, the final elevation datum is determined using GNSS leveling point data based on empirical geoids and test geoids. If the elevation datum is a geoid, the final elevation datum is determined using GNSS leveling point data based on empirical and test geoids.

8. A computer system, comprising: The memory and processor contain a computer program stored in the memory and executable on the processor, characterized in that the processor executes the computer program to implement the steps of the elevation datum construction method based on the low-order cosine modified Stokes kernel function as described in any one of claims 1-7.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by a processor, the computer program implements the steps of the elevation datum construction method based on the low-order cosine modified Stokes kernel function as described in any one of claims 1-7.

10. A computer program product, comprising a computer program, characterized in that, When executed by a processor, the computer program implements the steps of the elevation datum construction method based on the low-order cosine modified Stokes kernel function as described in any one of claims 1-7.

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