Geoid construction method and system based on high-frequency terrain influence combination algorithm

Abstract

The application discloses a geodetic datum surface construction method and system based on a high-frequency terrain influence combination algorithm, relates to the field of geodetic datum surface measurement, and comprises the following steps: terrain quality compression is performed by using a Helmert second compression method; high-frequency terrain influence generated by terrain compression is calculated based on a high-frequency terrain influence combination algorithm; specifically, a combination method of a spherical harmonic algorithm of global terrain influence, a prism algorithm of near-zone terrain influence and a Molodenskii spectrum algorithm of truncation to the fourth power of elevation of far-zone terrain influence is adopted to calculate the high-frequency terrain influence; and finally, the high-frequency terrain influence is applied to the second or third geodetic boundary value theory based on a removal recovery technology to construct a geodetic datum surface or a quasi-geodetic datum surface. The application can calculate the high-frequency terrain influence, improve the calculation efficiency and calculation precision of the terrain influence, and simultaneously, the application of the high-frequency terrain influence combination algorithm helps to improve the accuracy of the construction of the geodetic datum surface or the quasi-geodetic datum surface.

Description

Technical Field

[0001] This invention relates to the field of (quasi)geoid measurement, and in particular to a method and system for constructing (quasi)geoids based on a high-frequency topographic influence combination algorithm. Background Technology

[0002] Currently, there are two types of elevation datums in use: the geoid and the quasi-geoid. Since the distance between the geoid and the reference ellipsoid is the geoid height, and the distance between the quasi-geoid and the reference ellipsoid is the geoid undulation, the geoid undulation and geoid height are usually used to represent the quasi-geoid and the geoid model, respectively. The International Association of Geodesy (IAG) has stated that determining a "geoid with 1 cm accuracy" is a global strategic goal for geodesy development in the 21st century. With the increasing demand for (quasi-)geoid accuracy and the deepening research on geodetic boundary value theory based on Helmert's second compression method, in-depth research on topographic influence algorithms is essential.

[0003] High-precision (quasi-)geoid models can be constructed using second or third geodetic boundary value theories (Heiskannen and Moritz, 1967; Hofmann-Wellenhof and Moritz, 2006). Geodetic boundary value theories require the absence of topographic mass outside the boundary surface; therefore, it is necessary to compress or transfer the topographic mass outside the boundary surface to the boundary surface or its interior space. The compression or transfer of topographic mass has both direct and secondary indirect effects on ground gravity values, and an indirect effect on the (quasi-)geoid (cf. Martinec, 1998). Among numerous topographic processing methods, Helmert's second compression method, which compresses the topographic mass outside the boundary surface into a thin layer of mass covering the boundary surface, causes relatively small indirect effects and is therefore widely used in the construction of (quasi-)geoids.

[0004] In the research of algorithms for direct and indirect terrain effects, international geodesists have achieved a series of representative research results. Currently, the most commonly used terrain effect algorithms are the integral algorithm and the spherical harmonic algorithm. Nahavandchi (1999) pointed out that there is a significant deviation in the terrain effect calculated by these two algorithms, due to the different error characteristics and frequency characteristics of the terrain effect calculated by the two algorithms. The integral formula for direct and indirect terrain effects exhibits singularity, which can be eliminated by introducing a Bouguer spherical shell (Martinec and...). 1994; Martinec, 1998). Nahavandchi (1999) and others derived the Taylor expansion of the indirect effects of topography truncated to the cube of elevation. (2000) derived the Taylor expansion of the direct and indirect effects of topography truncated to the square of elevation. Nahavandchi (2000) proposed a combined method to calculate the direct effects of topography using the traditional integral formula for the near area and the 360th order topographic spherical harmonic algorithm for the far area. (2001) et al. proposed an algorithm for calculating the far-field topographic impact in the Stokes-Helmert boundary value problem. The derivation of this algorithm uses a global topographic spherical harmonic model and the Molodenskii spectral method. Makhloof and Ilk (2008) further provided calculation formulas for the far-field topographic impact of topographic processing methods such as the Helmert first and second compression methods and the Airy-Heiskanen equilibrium algorithm. Wang (2011) proposed a method for calculating the direct and indirect topographic impact in the global domain, in which the near-field impact is calculated using an integral algorithm, and the far-field impact is calculated using a one-dimensional fast Fourier transform algorithm. Wang and Yang (2013) gave spherical and ellipsoidal harmonic expressions for the topographic gravity potential, which has important reference value for the research and application of spherical harmonic algorithms for residual topographic potential.

[0005] Currently, when constructing regional geoids and quasi-geoids using geodetic boundary value theory, the removal-and-recovery technique is widely adopted to avoid issues in global integration. In the removal-and-recovery model, the gravity data in the Hotine / Stokes integral is high-frequency gravity data; therefore, the topographic influence in the Hotine-Helmert / Stokes-Helmert integral based on the removal-and-recovery technique should be adjusted to high-frequency topographic influence. Existing topographic influence algorithms are mainly used to calculate global and distant-area topographic influence, and cannot obtain high-frequency information about topographic influence. Summary of the Invention

[0006] The purpose of this invention is to provide a geoid construction method and system based on a high-frequency topographic influence combination algorithm. The high-frequency topographic influence combination algorithm can be used to calculate the high-frequency topographic influence, which helps to improve the accuracy of geoid or quasi-geoid construction.

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

[0008] This invention provides a method for constructing a geoid based on a high-frequency topographic influence combination algorithm, the method comprising:

[0009] The Helmert second compression method is used to compress the terrain quality outside the boundary surface onto the boundary surface; the boundary surface refers to the boundary surface involved when constructing a geoid or quasi-geoid using the second or third geodetic boundary value theory;

[0010] The high-frequency terrain influence generated by terrain compression is calculated based on the high-frequency terrain influence combination algorithm; the high-frequency terrain influence combination algorithm is a combination of the spherical harmonic algorithm for global terrain influence, the prism algorithm for near-area terrain influence, and the Molodenskii spectrum algorithm truncated to the fourth power of elevation for far-area terrain influence.

[0011] The high-frequency topographic influence is applied to the geoid or quasi-geoid of the study area based on the second or third geodetic boundary value theory using removal and restoration techniques.

[0012] Optionally, the high-frequency terrain impacts generated by terrain compression are calculated based on a high-frequency terrain impact combination algorithm, specifically including:

[0013] The near-field full-frequency topographic impact of the study area is calculated using the aforementioned prism algorithm and near-field topographic data;

[0014] The far-field topographic impact of the study area was calculated using the Molodenskii spectrum algorithm truncated to the fourth power of elevation and the topographic spherical harmonic coefficient.

[0015] The global low-frequency topographic impacts of the study area are calculated using the spherical harmonic algorithm and topographic spherical harmonic coefficients.

[0016] The high-frequency topographic influence of the study area is obtained by summing the near-field full-frequency topographic influence and the far-field topographic influence, and then subtracting the global low-frequency topographic influence.

[0017] Optionally, the expression for the spherical harmonic algorithm for calculating the global low-frequency topographic influence is:

[0018]

[0019]

[0020]

[0021] in,

[0022] In the formula, δA sphere and δS sphere These represent the direct impact of global topographic compression on ground gravity and the secondary indirect impact of global topographic compression, respectively, calculated using the spherical harmonic algorithm; δζ sphere and δN sphere These represent the indirect effects of global topographic compression, calculated using the spherical harmonic algorithm, on geoid undulation and geoid height, respectively; r P γ is the geocentric radius at the calculation point; γ is the normal gravity at the normal elevation of the ground point; γ0 is the normal gravity on the reference ellipsoid; G is the Newtonian gravitational constant; and Ω0 represents the global integration region. ρ is the topographic spherical harmonic coefficient, ρ is the topographic density, R is the average radius of the Earth, N is the cutoff order, and h is the topographic elevation. Let Ω be a spherical harmonic function of order n and degree m, which is completely normalized. P Let P be the spherical coordinates of the calculation point on the ground, and Ω be the spherical coordinates of the integration point.

[0023] Optionally, the expression for the prism algorithm for calculating the near-field full-frequency terrain influence is as follows:

[0024]

[0025] in,

[0026]

[0027]

[0028] In the formula, δA near δS near ,δζ near δN near These represent the direct, secondary, and indirect effects of near-field topographic compression calculated using the prism algorithm; the indirect effects on elevation anomalies; and the indirect effects on geoid height, δA. i ,δζ i δN i These represent the direct impact of the compression of the i-th geographic grid in the near region, the indirect impact on the geoid undulation, and the indirect impact on the geoid height, calculated using the prism algorithm, respectively; G is the Newtonian gravitational constant; R is the Earth's average radius; h i x represents the elevation of the i-th discrete geographic grid; i x i+1 y i y i+1 Let ρ be the planar coordinates of the boundary of the i-th mass prism in a three-dimensional rectangular coordinate system. The three-dimensional rectangular coordinate system is established with the projection point of the calculation point on the boundary surface as the origin. The i-th mass prism is an approximate prism model of the i-th geographic grid in this three-dimensional rectangular coordinate system; i γ is the density of the i-th geographic grid; i γ represents the normal gravity at the center point of the i-th geographic grid on the elevation plane; 0i h represents the normal gravity at the projection point of the center point of the i-th mass prism onto the reference ellipsoid; P l0 represents the elevation of the ground calculation point P; l0 represents the horizontal distance between the center point of the mass prism and the calculation point.

[0029] Optionally, the expression for the Molodenskii spectral algorithm truncated to the fourth power of elevation for calculating the influence of the distant terrain is as follows:

[0030]

[0031]

[0032]

[0033]

[0034] In the formula, δA far δS far ,δζ far δN far The figures represent the direct, minor, and indirect effects of far-field topographic compression, calculated using the Molodenskii spectral algorithm truncated to the fourth power of elevation, as well as the indirect effects on geoid height. For kernel function J k The nth order Molodenskii cutoff coefficient, ψ represents the angular distance between the integration point and the calculation point; ψ0 represents the angular distance corresponding to the approximate spherical cap boundary of the near region; P is the topographic spherical harmonic coefficient; n Represents the nth-order Legendre function; For kernel function F k The nth order Molodenskii cutoff coefficient, For kernel function T k The nth order Molodenskii cutoff coefficient, Kernel function J k F k T k The expression is as follows:

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043]

[0044] This invention provides a geoid construction system based on a high-frequency topographic influence combination algorithm, the system comprising:

[0045] The terrain quality compression module is used to compress the terrain quality outside the boundary surface onto the boundary surface using the Helmert second compression method; the boundary surface refers to the boundary surface involved when constructing a geoid or quasi-geoid using the second or third geodetic boundary value theory.

[0046] The high-frequency terrain compression impact calculation module is used to calculate the high-frequency terrain impact generated by terrain compression based on the high-frequency terrain impact combination algorithm; the high-frequency terrain impact combination algorithm is a combination of the spherical harmonic algorithm for global terrain impact, the prism algorithm for near-area terrain impact, and the Molodenskii spectrum algorithm truncated to the fourth power of elevation for far-area terrain impact.

[0047] The geoid construction module is used to apply the high-frequency topographic influence to the geoid or the quasi-geoid of the research area based on the second or third geodetic boundary value theory using the removal and restoration technique.

[0048] Optionally, the high-frequency terrain compression effect calculation module specifically includes:

[0049] The near-field full-frequency topographic impact calculation unit is used to calculate the near-field full-frequency topographic impact of the study area using the prism algorithm and near-field topographic data.

[0050] The distant topographic impact calculation unit is used to calculate the distant topographic impact of the study area using the Molodenskii spectrum algorithm truncated to the fourth power of elevation and the topographic spherical harmonic coefficient.

[0051] A global low-frequency topographic impact calculation unit is used to calculate the global low-frequency topographic impact of the study area using the spherical harmonic algorithm and topographic spherical harmonic coefficients.

[0052] The high-frequency topographic impact calculation unit is used to sum the near-area full-frequency topographic impact and the far-area topographic impact, and then subtract the global low-frequency topographic impact to obtain the high-frequency topographic impact of the study area.

[0053] The present invention also provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the geoid construction method based on the high-frequency terrain influence combination algorithm.

[0054] According to specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0055] This invention proposes a method and system for constructing a geoid based on a high-frequency topographic influence combination algorithm. It utilizes the Helmert second compression method for topographic quality compression and calculates the high-frequency topographic influence generated by this compression based on the high-frequency topographic influence combination algorithm. Specifically, it employs a combination of a spherical harmonic algorithm for global topographic influence, a prism algorithm for near-field topographic influence, and a Molodenskii spectrum algorithm that truncates far-field topographic influence to the fourth power of elevation to calculate the high-frequency topographic influence. Finally, the high-frequency topographic influence is applied to the second or third geodetic boundary value theory based on removal and restoration techniques to construct a geoid or quasi-geoid. This invention, based on the high-frequency topographic influence combination algorithm, improves the calculation efficiency and accuracy of topographic influence. Furthermore, the application of this algorithm contributes to improving the accuracy of geoid or quasi-geoid construction. Attached Figure Description

[0056] 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.

[0057] Figure 1 This is a flowchart of a geoid construction method based on a high-frequency terrain influence combination algorithm provided in Embodiment 1 of the present invention;

[0058] Figure 2 This is a schematic diagram of the prism model of the i-th discrete grid of the integration region in a three-dimensional rectangular coordinate system, as provided in Embodiment 1 of the present invention. Detailed Implementation

[0059] 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.

[0060] When constructing a (quasi-)geoid model using the second or third geodetic boundary value theory, to meet the requirement that there is no terrain mass outside the boundary surface, the Helmert second compression method is usually used in practice to compress the terrain mass outside the boundary surface onto the boundary surface. Terrain compression has a direct and secondary indirect impact on gravity, and an indirect impact on the (quasi-)geoid. Currently, the removal-and-recovery model is commonly used for boundary value calculation. In this model, high-frequency terrain influences should be used for boundary value calculation. Traditional terrain influence algorithms cannot calculate high-frequency terrain influences; therefore, the purpose of this invention is to conduct in-depth research on high-frequency terrain influence algorithms. Based on the research of the prism algorithm, the Molodenskii spectrum algorithm truncated to the fourth power of elevation, and the spherical harmonic algorithm, a combined algorithm for high-frequency terrain influences that balances computational accuracy and speed is proposed: a combined algorithm of "prism algorithm + Molodenskii spectrum algorithm - spherical harmonic algorithm," which calculates high-frequency terrain influences and thus improves the accuracy of (quasi-)geoid model construction.

[0061] 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.

[0062] Example 1

[0063] like Figure 1 As shown, this invention provides a geoid construction method based on a high-frequency topographic influence combination algorithm, the method comprising:

[0064] S1: The Helmert second compression method is used to compress the terrain quality outside the boundary surface onto the boundary surface; the boundary surface refers to the boundary surface involved when constructing a geoid or quasi-geoid using the second or third geodetic boundary value theory.

[0065] S2: Calculate the high-frequency terrain impact generated by terrain compression based on the high-frequency terrain impact combination algorithm; the high-frequency terrain impact combination algorithm is a combination of the spherical harmonic algorithm for global terrain impact, the prism algorithm for near-area terrain impact, and the Molodenskii spectrum algorithm truncated to the fourth power of elevation for far-area terrain impact.

[0066] The expression for the spherical harmonic algorithm for calculating the global low-frequency topographic effects is as follows:

[0067]

[0068]

[0069]

[0070]

[0071] in,

[0072] In the formula, δA sphere and δS sphere These represent the direct impact of global topographic compression on ground gravity and the secondary indirect impact of global topographic compression, respectively, calculated using the spherical harmonic algorithm; δζ sphere and δN sphere These represent the indirect effects of global topographic compression, calculated using the spherical harmonic algorithm, on geoid undulation and geoid height, respectively; r P γ is the geocentric radius at the calculation point; γ is the normal gravity at the normal elevation of the ground point; γ0 is the normal gravity on the reference ellipsoid; G is the Newtonian gravitational constant; and Ω0 represents the global integration region. ρ is the topographic spherical harmonic coefficient, ρ is the topographic density, R is the average radius of the Earth, N is the cutoff order, and h is the topographic elevation. Let Ω be a spherical harmonic function of order n and degree m, which is completely normalized. P Let P be the spherical coordinates of the calculation point on the ground, and Ω be the spherical coordinates of the integration point.

[0073] The derivation of the prism algorithm for calculating the influence of near-field topography is given below:

[0074] Since topographic data is stored in a discrete grid format, and the surface integral formula is a continuous double integral, the integral formula needs to be discretized during calculation. By deriving the analytical expression of the double integral formula of topographic influence in a discrete grid, the discretization error introduced by the discretization of the integral formula can be avoided. The analytical algorithm for the double integral formula in a discrete grid can be simply called the prism algorithm. Geodesists have previously derived prism algorithms for topographic gravitational potential and gravity. Based on this, this invention further derives the prism algorithm for the function of residual topographic potential—topographic influence. A three-dimensional rectangular coordinate system is established with the projection point of the calculation point on the boundary surface as the origin. Figure 2 This is a schematic diagram of the prism model of the i-th discrete mesh in the integration region in the three-dimensional Cartesian coordinate system.

[0075] h in the figure i Let x be the elevation of the i-th discrete grid. i x i+1 y i y i+1 These are the boundary coordinates of the prism model of the mesh along the x and y axes, respectively. The terrain impact caused by terrain compression in the integration region is the sum of the terrain impacts generated by each mass prism within the region, i.e.

[0076]

[0077] In the formula δA near δS near ,δζ near δNnear These represent the direct, secondary, and indirect effects of near-field topographic compression calculated using the prism algorithm; the indirect effects on elevation anomalies; and the indirect effects on geoid height; δA i ,δζ i δN i These represent the direct impact of compressing the i-th geographic grid in the near area, the indirect impact on elevation anomalies, and the indirect impact on geoid height, calculated using the prism algorithm. The discrete geographic grid is approximated as a mass prism; the specific formula for the topographic impact of a single mass prism is derived below.

[0078] According to the definition of direct influence, the triple integral formula for the direct influence of a mass prism in a three-dimensional rectangular coordinate system is:

[0079]

[0080] In the formula ρ i h is the density of the i-th geographic grid. P Let P be the elevation of the ground calculation point. The double integral of equation (2) is derived as follows:

[0081]

[0082] Using the relevant knowledge of indefinite integrals, we can obtain the single integral form of the above equation (3) as follows:

[0083]

[0084] Based on equation (4) and the relevant knowledge of indefinite and definite integrals, the formula for the directly affected prism algorithm can be further derived as follows:

[0085]

[0086] In the formula,

[0087]

[0088] The formula for the indirect effect of terrain compression on elevation anomalies in a rectangular coordinate system is:

[0089]

[0090] In the formula γ i Let δζ be the normal gravity at the center point of the i-th geographic grid on the elevation plane. Using equation (7) and related knowledge of indefinite integrals, we can obtain δζ. i The prism algorithm is as follows:

[0091]

[0092] In the formula See equation (6) for the definition.

[0093] Let h in equation (8) above P =0, γ i =γ 0i (γ 0i Let δN be the normal gravity at the projection point of the center point of the i-th mass prism onto the reference ellipsoid. From this, we can obtain the indirect effect δN of topographic compression on the geoid. i The prism algorithm is as follows:

[0094]

[0095] In the formula

[0096]

[0097] Since the Earth's surface is approximately spherical, the prismatic algorithm, which treats the near-field area as a plane due to the direct and indirect effects of terrain, suffers from planar approximation errors. This is why the prismatic algorithm is primarily used to calculate the impact of near-field terrain. To minimize planar approximation errors, the influence of the Earth's curvature can be further considered in the prismatic algorithm. According to Forsberg (1984), this can be achieved by simply changing the elevation of the calculation point to account for the Earth's curvature; the formula is as follows.

[0098]

[0099] In the formula To account for the influence of the Earth's curvature on the calculation point elevation, l0 is the horizontal distance between the center point of the mass prism and the calculation point. Replace h in equations (5) and (8) P Thus, δA, taking into account the influence of Earth's curvature, can be obtained separately. i With δζ i Since equation (9) is h in equation (8) P The simplified form is 0, and h in equation (8) is... P Replace with γ i Replace with γ 0i This allows us to obtain δN that takes into account the influence of Earth's curvature. i .

[0100] Therefore, the expression for the prism algorithm for calculating the near-field topographic influence, taking into account the effect of Earth's curvature, is as follows:

[0101]

[0102] in,

[0103]

[0104]

[0105]

[0106] In the formula, δA near δS near ,δζ near δN near The values ​​δA represent the direct, secondary, and indirect effects of near-field topographic compression calculated using a prism algorithm that takes into account the Earth's curvature; the indirect effects on elevation anomalies; and the indirect effects on geoid height. i ,δζ i δN i These represent the direct impact of the compression of the i-th geographic grid in the near region, calculated using a prism algorithm that takes into account the curvature of the Earth, the indirect impact on the elevation anomaly, and the indirect impact on the geoid height.

[0107] The following is the derivation of the Molodenskii spectral algorithm truncated to the fourth power of elevation for calculating the influence of distant terrain:

[0108] When r P >r、r P When >R, and The reciprocals of the Legendre polynomials are respectively

[0109]

[0110] In the formula l P To calculate the distance between the point and the integration point on the ground, l P0 This is the distance between the calculation point and the integration point on the boundary surface. For equation (12), 1 / l P0 The Legendre polynomial has two sides with respect to r. P Taking the first to fourth derivatives, we can obtain the following equations.

[0111]

[0112] Both the direct and indirect effects of topography are functionals of the residual topographic potential (the difference between the gravitational potential generated by the real topography outside the boundary and the gravitational potential generated by the compressed topography on the boundary). The formula for calculating the residual topographic potential in spherical coordinates is:

[0113]

[0114] Substituting equation (12) into equation (14), we obtain the Legendre polynomial for the topographic residual potential at point P on the ground.

[0115]

[0116] Furthermore, the fourth-order Taylor series expansion of the residual position of point P on the terrain surface in Legendre form is:

[0117]

[0118] Based on the functional relationship between the residual and the direct influence, the fourth-order Taylor series expansion of the direct influence δA in Legendre form is:

[0119]

[0120] From equations (13) and (17), we can obtain the Taylor series expansion that directly affects the rigorous form as follows:

[0121]

[0122] The formulas for calculating kernel functions J2, J3, and J4 are as follows:

[0123]

[0124] According to Bruns' theorem, the Legendre form of the fourth-order Taylor series expansion of the indirect effect of topographic compression on elevation anomaly δζ can be obtained from equation (16) as follows:

[0125]

[0126] From equations (13) and (20), the Taylor series expansion of δζ in rigorous form can be obtained as follows:

[0127]

[0128] In the formula, F2, F3, and F4 are the 2nd to 4th order kernel functions of δζ, and their formulas are as follows:

[0129]

[0130] In order (13) r P =R, we can get

[0131]

[0132] According to equation (23), the Legendre expression for the topographic residual potential at point P0 on the boundary surface r = R can be derived as follows:

[0133]

[0134] In the formula, l is the distance between the projection point of the calculation point on the boundary surface and the integration point on the ground, and its formula is: l0 is the distance between the projection point of the calculation point on the boundary surface and the integration point on the boundary surface, and its formula is l0 = 2Rsin(ψ / 2).

[0135] According to equation (24), the fourth-order Taylor series expansion of the residual position of point P0 on the inner boundary surface of the terrain surface in Legendre form is:

[0136]

[0137] Combining equations (23) and (25), we can obtain the rigorous Taylor series expansion of the indirect effect of topographic compression on geoid height δN:

[0138]

[0139] In the formula, T2, T3, and T4 are the 2nd to 4th order kernel functions of δN, and their formulas are as follows:

[0140]

[0141] According to equation (18), the direct impact of the far-field region can be expressed as:

[0142]

[0143] In the formula δA far The kernel function represents the direct impact of the far-field region, where Ω0-C0 is the integration domain of the far-field region (Ω0 represents the global domain, and C0 represents the near-field region). The conversion relationship between the kernel functions J2, J3, and J4 (the specific calculation formulas for J2, J3, and J4 are shown in equation (19)) is as follows:

[0144]

[0145] In the formula, ψ0 represents the angular distance corresponding to the approximate spherical cap boundary of the near region. For a piecewise function, its Legendre expression is:

[0146]

[0147] In the formula For kernel function J k The nth order Molodenskii cutoff coefficients can be obtained from the spherical function expansion theory. The calculation formula is:

[0148]

[0149] Substitute equation (30) into equation (28) and use equation (28) The Molodenskii spectral algorithm for the direct influence of the far-field region can be obtained as follows:

[0150]

[0151] In the formula The topographic spherical harmonic coefficient is calculated using the following formula:

[0152]

[0153] According to equation (21), the indirect impact of remote topographic compression on elevation anomalies can be expressed as:

[0154]

[0155] In the formula δζ far To understand the indirect impact of far-field topographic compression on elevation anomalies, the kernel function... The conversion relationship between the kernel functions F2, F3, and F4 (the specific calculation formulas for F2, F3, and F4 are shown in equation (22)) is as follows:

[0156]

[0157] Piecewise continuous function Legendre's expression is

[0158]

[0159] In the formula For kernel function F k The nth order Molodenskii cutoff coefficient is calculated using the following formula:

[0160]

[0161] Substitute equation (35) into equation (33) and use equation (35) The indirect influence δζ in the far region can be obtained. far The Molodenskii spectral algorithm is

[0162]

[0163] Similarly, the Molodeskii spectral algorithm for the indirect influence of far-field topographic compression on geoid height can be obtained as follows:

[0164]

[0165] In the formula δN far The indirect impact of remote topographic compression on geoid height. (k takes values ​​of 2, 3, and 4 respectively) is the kernel function T k The nth order Molodenskii cutoff coefficient is calculated using the following formula:

[0166]

[0167] The specific calculation formulas for kernel functions T2, T3, and T4 are shown in equation (27).

[0168] For combined algorithms that address high-frequency terrain effects:

[0169] The direct and indirect effects of topography are mainly applied to solving for geoid undulations or geoid heights using second or third boundary value theories (i.e., Hotine-Helmert or Stoke-Helmert boundary value theories) based on the Helmert second compression method. The physical inputs to the Hotine-Helmert and Stoke-Helmert boundary value theories are Helmert gravity perturbations and Helmert gravity anomalies, respectively. Their conversion relationships with ordinary gravity perturbations and gravity anomalies are as follows:

[0170] δg+δA=δg H ,Δg+δA+δS=Δg H (40)

[0171] In the formula, δg represents the gravitational disturbance. H The helmert gravity disturbance is represented by Δg, which is the gravity anomaly. H This refers to the Helmert gravity anomaly. Performing an extended Hotine / Stokes integration on the Helmert gravity disturbance / Helmert gravity anomaly yields the Helmert geoid or Helmert geoid height. Using indirect effects, the Helmert geoid / Helmert geoid height can be restored to the true geoid / geoid height, i.e.

[0172] ζ H +δζ=ζ,N H +δN=N (41)

[0173] In the formula ζ H Here, ζ represents the Helmert geoid, and N represents the geoid. H Let N be the Helmert geoid height. The Hotine / Stokes integration should theoretically use a global integration, but due to the difficulty in obtaining global gravity data and the large computational cost of global integration, the removal-and-recovery technique is commonly used when applying boundary value theory. In the removal-and-recovery mode, the gravity data used by Hotine-Helmert or Stokes-Helmert boundary value theory are Helmert high-frequency gravity perturbations or Helmert high-frequency gravity anomalies, and the formula is:

[0174] (δg H ) high =δg high +δA high ,(Δg H ) high =Δg high +δA high +δShigh (42)

[0175] In the formula (δg H ) high For Helmert high-frequency gravitational perturbation, δg high For high-frequency gravitational disturbances, δA high For high-frequency direct influence, (Δg) H ) high For Helmert high-frequency gravity anomaly, Δg high This is a high-frequency gravity anomaly, δS high This is a secondary, indirect influence at high frequencies. Helmert high-frequency gravity disturbances / Helmert high-frequency gravity anomalies can be obtained through (extended) Hotine / Stokes integration, yielding Helmert high-frequency geoid anomalies or Helmert high-frequency geoid heights. The conversion relationships between these geoids and high-frequency geoid anomalies / geoid heights are as follows:

[0176] (ζ H ) high +δζ high =ζ high ,(N H ) high +δN high =N high (43)

[0177] In the formula (ζ) H ) high For Helmert high-frequency elevation anomaly, δζ high For the high-frequency indirect effects of topographic compression on elevation anomalies, ζ high This is a high-frequency elevation anomaly, (N) H ) high For Helmert high-frequency geoid height, δN high N represents the high-frequency indirect impact of topographic compression on the geoid height. high This represents the high-frequency geoid height. The high-frequency topographic influence used in the Hotine-Helmert or Stoke-Helmert boundary value theory based on removal restoration techniques can be obtained by subtracting the low-frequency topographic influence from the full-frequency topographic influence.

[0178]

[0179] In the formula δA full δA low The direct effects on full and low frequencies are respectively δS full δS low These represent the secondary indirect effects at full and low frequencies, respectively, δζ full ,δζ lowThese represent the full-frequency and low-frequency indirect effects of topographic compression on elevation anomalies, δN. full δN low These represent the full-frequency and low-frequency indirect effects of topographic compression on the geoid height.

[0180] The spherical harmonic algorithm for topographic effects and the far-field Molodenskii spectral algorithm have the advantages of computational simplicity and high speed. Both algorithms require spherical harmonic coefficients generated through harmonic analysis of global topographic data, and these coefficients can be repeatedly applied once generated. According to the Nyquist sampling theorem, grid resolution determines the frequency domain contained in the grid data. Higher grid resolution allows for higher-order spherical harmonic coefficients, but also increases the demands on computer storage and processing power. Therefore, the spherical harmonic algorithm and the far-field Molodenskii spectral algorithm are suitable for calculating low-frequency topographic effects. In view of this, the spherical harmonic algorithm is used to calculate the low-frequency topographic effects in the combined algorithm.

[0181]

[0182] In the formula, N is the highest cutoff order of the topographic spherical harmonic coefficients.

[0183] Equation (44) includes both high-frequency and low-frequency topographic effects. Theoretically, both high-frequency and low-frequency topographic effects should be considered global topographic effects. Since high-frequency topographic effects are mainly generated by near-field topographic compression, and far-field high-frequency topographic effects are minimal and negligible, the total frequency topographic effect can be approximated as the sum of near-field total frequency topographic effects and far-field low-frequency topographic effects, i.e.

[0184]

[0185] In the formula These are respectively the direct impact and secondary indirect impact across the entire frequency range in the near-field area. These represent the full-frequency indirect effects of near-field topographic compression on elevation anomalies and geoid height. These are respectively the direct impact and secondary indirect impact of low-frequency frequencies in the far region. These represent the low-frequency indirect effects of far-field topographic compression on elevation anomalies and geoid height.

[0186] Surface integration algorithms can be used to calculate the topographic effects in the near, far, and global domains, simply by setting the integration domain to the corresponding area. However, because the integration algorithm is less efficient than the far-field Molodenskii spectral algorithm when calculating the far-field topographic effects, the Molodenskii spectral algorithm is used to calculate the low-frequency topographic effects in the far-field region in the combined algorithm.

[0187]

[0188] The near-field full-frequency terrain impact can be calculated from high-resolution near-field terrain data using either an integral algorithm or a prismatic algorithm. Since the prismatic algorithm, as an analytical form of the integral algorithm, can effectively reduce the discretization error inherent in integral algorithms, the near-field full-frequency terrain impact in the combined algorithm will be calculated using the prismatic algorithm.

[0189] In boundary value calculation based on removal recovery, the calculation scheme of the high-frequency terrain influence combination algorithm is as follows: First, the near-field full-frequency terrain influence is calculated using the prism algorithm and high-resolution terrain data, the far-field terrain influence is calculated using the Molodeskii spectrum algorithm, and the global low-frequency terrain influence is calculated using the spherical harmonic algorithm. Then, the remaining high-frequency terrain influence is obtained by summing the near-field full-frequency terrain influence and the far-field terrain influence and subtracting the global low-frequency terrain influence. Therefore, the high-frequency terrain influence combination algorithm can be abbreviated as "prism algorithm + Molodeskii spectrum algorithm - spherical harmonic algorithm".

[0190] Therefore, step S2 specifically includes:

[0191] S21: Calculate the near-field full-frequency topographic impact of the study area using the prism algorithm and near-field topographic data;

[0192] S22: Calculate the far-field topographic impact of the study area using the Molodenskii spectrum algorithm truncated to the fourth power of elevation and the topographic spherical harmonic coefficient;

[0193] S23: Calculate the global low-frequency topographic impact of the study area using the spherical harmonic algorithm and topographic spherical harmonic coefficients;

[0194] S24: Summing the near-field full-frequency topographic influence and the far-field topographic influence, and then subtracting the global low-frequency topographic influence, yields the high-frequency topographic influence of the study area.

[0195] S3: Apply the topographic influence to the geoid or quasi-geoid of the study area constructed by the geodetic boundary value theory based on the removal and restoration technique; the geodetic boundary value theory is the second or third geodetic boundary value theory.

[0196] In this embodiment, when constructing the geoid and quasi-geoid based on geodetic boundary value theory, it is required that there is no terrain mass outside the boundary surface. Therefore, during boundary value calculation, the terrain mass outside the boundary surface is usually compressed into a thin film of mass covering the boundary surface using the Helmert second compression method. Using the Helmert second compression method to compress the terrain outside the boundary surface will have a direct and secondary indirect impact on gravity, and an indirect impact on the (quasi)geoid. Currently, the boundary value calculation commonly uses the removal recovery mode, in which the terrain influence should be a high-frequency terrain influence. Traditional terrain influence algorithms include surface integration algorithms, Molodenskii spectrum algorithms, spherical harmonic algorithms, and combinations of "integral algorithm + Molodenskii spectrum algorithm", but none of them can be used to calculate high-frequency terrain influence. Therefore, this invention studies high-frequency terrain influence.

[0197] To improve the calculation accuracy of near-field topographic effects, this invention derives a prism algorithm for near-field topographic effects. By deriving the analytical form of the integral formula, the discretization error of the traditional integral formula is effectively controlled. To improve the calculation efficiency and accuracy of far-field topographic effects, a Molodengskii spectrum algorithm truncated to the cube of elevation is further derived based on the traditional Molodengskii spectrum algorithm truncated to the fourth power of elevation. Furthermore, this invention also derives a spherical harmonic algorithm for topographic effects truncated to the fourth power of elevation. Based on this, this invention proposes a combined algorithm of "prism algorithm + Molodengskii spectrum algorithm - spherical harmonic algorithm" that can balance the calculation accuracy and speed of high-frequency topographic effects.

[0198] This invention conducts in-depth research on high-frequency terrain influence algorithms. To balance the accuracy and efficiency of terrain influence calculation, the integration region is divided into near-field and far-field regions. First, to improve the calculation accuracy of near-field terrain influence, this invention derives a prism algorithm for near-field terrain influence, effectively controlling the discretization error of traditional integral formulas by deriving the analytical form of the integral formula. Then, to improve the calculation efficiency and accuracy of far-field terrain influence, this invention derives a Molodenskii spectrum algorithm for far-field terrain influence based on the fourth-order Taylor series expansion of terrain influence. Subsequently, a spherical harmonic algorithm for global domain terrain influence truncated to the fourth power of elevation is presented. Finally, a high-frequency terrain influence combination algorithm of "prism algorithm + Molodenskii spectrum algorithm - spherical harmonic algorithm" is proposed, which is a combination method of subtracting the global domain low-frequency terrain influence calculated by the spherical harmonic algorithm from the sum of near-field and far-field terrain influence. Finally, a terrain influence experiment is conducted in a mountainous area. The effectiveness of the algorithm is verified by comparing the terrain influence calculated by different algorithms, and the approximate error of the terrain influence algorithm is quantitatively given based on the analysis of the error characteristics of different algorithms. The conclusions drawn from the numerical experiments are as follows: (1) When the near-field terrain resolution is high, the difference between the terrain influence calculated by the integral algorithm or the prism algorithm is very small, thus proving the correctness of the prism algorithm. When the near-field terrain resolution is not high, the prism algorithm can effectively improve the accuracy of the near-field terrain influence; (2) The difference between the far-field terrain influence calculated by the Molodenskii spectral method and the far-field influence calculated by the integral algorithm is very small, thus proving the correctness of the Molodenskii spectral method. Considering the simplicity and efficiency of the calculation, it is recommended to use the Molodenskii spectral algorithm to calculate the far-field terrain influence; (3) The global domain terrain influence calculated by the integral algorithm, the "prism + Molodenskii algorithm" and the spherical harmonic algorithm has a high degree of consistency, indicating that the spherical harmonic algorithm can effectively calculate the low-frequency information of the terrain influence; (4) Through the experiment of calculating the high-frequency terrain influence using the combined algorithm of "prism algorithm + Molodenskii spectral algorithm - spherical harmonic algorithm", it can be seen that the cutoff order of the spherical harmonic coefficient has a small impact on the direct high-frequency influence and a large impact on the indirect high-frequency influence. In boundary value calculation applications, the cutoff order of the topographic spherical harmonic coefficients should be consistent with the order of the reference gravity field used in the removal and restoration technique.

[0199] Example 2

[0200] This embodiment provides a geoid construction system based on a high-frequency topographic influence combination algorithm, the system comprising:

[0201] The terrain quality compression module is used to compress the terrain quality outside the boundary surface onto the boundary surface using the Helmert second compression method; the boundary surface refers to the boundary surface involved when constructing a geoid or quasi-geoid using the geodetic boundary value theory.

[0202] The high-frequency terrain compression impact calculation module is used to calculate the high-frequency terrain impact generated by terrain compression based on the high-frequency terrain impact combination algorithm; the high-frequency terrain impact combination algorithm is a combination of the spherical harmonic algorithm for global terrain impact, the prism algorithm for near-area terrain impact, and the Molodenskii spectrum algorithm for truncating far-area terrain impact to the fourth power of elevation.

[0203] The geoid construction module is used to apply the high-frequency topographic influence to the geoid or the quasi-geoid of the research area based on the geodetic boundary value theory using the removal and restoration technique; the geodetic boundary value theory is the second or third geodetic boundary value theory.

[0204] The high-frequency terrain compression effect calculation module specifically includes:

[0205] The near-field full-frequency topographic impact calculation unit is used to calculate the near-field full-frequency topographic impact of the study area using the prism algorithm and near-field topographic data.

[0206] The distant topographic impact calculation unit is used to calculate the distant topographic impact of the study area using the Molodenskii spectrum algorithm truncated to the fourth power of elevation and the topographic spherical harmonic coefficient.

[0207] A global low-frequency topographic impact calculation unit is used to calculate the global low-frequency topographic impact of the study area using the spherical harmonic algorithm and topographic spherical harmonic coefficients.

[0208] The high-frequency topographic impact calculation unit is used to sum the near-area full-frequency topographic impact and the far-area topographic impact, and then subtract the global low-frequency topographic impact to obtain the high-frequency topographic impact of the study area.

[0209] Example 3

[0210] This embodiment provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the geoid construction method based on a high-frequency terrain influence combination algorithm of Embodiment 1.

[0211] In addition, this embodiment of the invention also provides an electronic device, including a memory and a processor. The memory is used to store a computer program, and the processor runs the computer program to enable the electronic device to execute the geoid construction method based on the high-frequency terrain influence combination algorithm of Embodiment 1.

[0212] Alternatively, the aforementioned electronic device may be a server.

[0213] Embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0214] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0215] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0216] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0217] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the systems disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the descriptions are relatively simple; relevant parts can be referred to the method section.

[0218] Specific examples have been used to illustrate the principles and implementation methods of this invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of this invention. Furthermore, those skilled in the art will recognize that, based on the ideas of this 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 this invention.

Claims

1. A method for constructing a geoid based on a high-frequency topographic influence combination algorithm, characterized in that, The method includes: The Helmert second compression method is used to compress the terrain quality outside the boundary surface onto the boundary surface; the boundary surface refers to the boundary surface involved when constructing a geoid or quasi-geoid using the second or third geodetic boundary value theory; The high-frequency terrain influence generated by terrain compression is calculated based on the high-frequency terrain influence combination algorithm; the high-frequency terrain influence combination algorithm is a combination of the spherical harmonic algorithm for global terrain influence, the prism algorithm for near-area terrain influence, and the Molodenskii spectrum algorithm truncated to the fourth power of elevation for far-area terrain influence. The high-frequency topographic effects are applied to the geoid or quasi-geoid of the study area based on the second or third geodetic boundary value theory using removal and restoration techniques; Among them, the high-frequency terrain impact calculated by the high-frequency terrain impact combination algorithm for terrain compression specifically includes: The near-field full-frequency topographic impact of the study area is calculated using the aforementioned prism algorithm and near-field topographic data; The far-field topographic impact of the study area was calculated using the Molodenskii spectrum algorithm truncated to the fourth power of elevation and the topographic spherical harmonic coefficient. The global low-frequency topographic impacts of the study area are calculated using the spherical harmonic algorithm and topographic spherical harmonic coefficients. The high-frequency topographic influence of the study area is obtained by summing the near-field full-frequency topographic influence and the far-field topographic influence, and then subtracting the global low-frequency topographic influence. The expression for the spherical harmonic algorithm used to calculate the global low-frequency topographic influence is as follows: ； in, , ; In the formula, and These represent the direct impact of global topographic compression on ground gravity and the secondary indirect impact of global topographic compression, calculated using the spherical harmonic algorithm, respectively. and These represent the indirect effects of global topographic compression, calculated using the spherical harmonic algorithm, on geoid undulation and geoid height, respectively. The geocentric radius of the calculation point; This represents normal gravity at a normal elevation on the ground. For reference, normal gravity on the ellipsoid; Newton's gravitational constant, Represents the global points region; The topographic spherical harmonic coefficient, For terrain density, The average radius of the Earth To truncate the order, This refers to the terrain elevation; for Step The spherical harmonic function that is completely normalized. For ground calculation points spherical coordinates, Let be the spherical coordinates of the integration point.

2. The method according to claim 1, characterized in that, The expression for the prism algorithm for calculating the near-field full-frequency terrain influence is as follows: ； in, ; ； ； ； ； ； ； In the formula, , , , These are the direct, secondary, and indirect effects of near-field topographic compression calculated using the prism algorithm; the indirect effects on elevation anomalies; and the indirect effects on geoid height. , , The near-zone number calculated using the prism algorithm is respectively the 1st... The direct impact of geographic grid compression, the indirect impact on elevation anomalies, and the indirect impact on geoid height; It is Newton's gravitational constant; The average radius of the Earth; For the first Elevation of a discrete geographic grid; , , , For the first The planar coordinates of the boundary of a mass prism in a three-dimensional rectangular coordinate system, where the origin is the projection of the calculation point onto the boundary surface. The mass prism is the first... A prism approximation model of a geographic grid in a three-dimensional rectangular coordinate system; For the first Density of geographic grids; For the first Normal gravity at the center point of the grid on the elevation surface of a geographic grid; For the first Normal gravity at the projection point of the center point of a mass prism onto the reference ellipsoid; For ground calculation points Elevation of the location; The distance between the center point of the mass prism and the calculation point is denoted as .

3. The method according to claim 1, characterized in that, The expression for the Molodenskii spectral algorithm, which truncates to the fourth power of elevation, to calculate the influence of the distant terrain is as follows: ； ； ； ； In the formula, , , , The figures represent the direct, minor, and indirect effects of far-field topographic compression, calculated using the Molodenskii spectral algorithm truncated to the fourth power of elevation, as well as the indirect effects on geoid height. For kernel function of Molodenskii cutoff coefficients of order 1 , This represents the angular distance between the integration point and the calculation point; This represents the angular distance corresponding to the approximate spherical cap boundary of the near zone; The topographic spherical harmonic coefficient; Represents the nth order Legendre function; For kernel function of Molodenskii cutoff coefficients of order 1 ; For kernel function of Molodenskii cutoff coefficients of order 1 Kernel function , , The expression is as follows: ； ； ； ； ； ； ； ； ； 。 4. A geoid construction system based on a high-frequency topographic influence combination algorithm, characterized in that, The system includes: The terrain quality compression module is used to compress the terrain quality outside the boundary surface onto the boundary surface using the Helmert second compression method; the boundary surface refers to the boundary surface involved when constructing a geoid or quasi-geoid using the second or third geodetic boundary value theory. The high-frequency terrain compression impact calculation module is used to calculate the high-frequency terrain impact generated by terrain compression based on the high-frequency terrain impact combination algorithm; the high-frequency terrain impact combination algorithm is a combination of the spherical harmonic algorithm for global terrain impact, the prism algorithm for near-area terrain impact, and the Molodenskii spectrum algorithm truncated to the fourth power of elevation for far-area terrain impact. The geoid construction module is used to apply the high-frequency topographic influence to the geoid or the quasi-geoid of the research area based on the second or third geodetic boundary value theory using removal and restoration technology; The high-frequency terrain compression effect calculation module specifically includes: The near-field full-frequency topographic impact calculation unit is used to calculate the near-field full-frequency topographic impact of the study area using the prism algorithm and near-field topographic data. The distant topographic impact calculation unit is used to calculate the distant topographic impact of the study area using the Molodenskii spectrum algorithm truncated to the fourth power of elevation and the topographic spherical harmonic coefficient. A global low-frequency topographic impact calculation unit is used to calculate the global low-frequency topographic impact of the study area using the spherical harmonic algorithm and topographic spherical harmonic coefficients. The high-frequency topographic impact calculation unit is used to sum the near-area full-frequency topographic impact and the far-area topographic impact, and then subtract the global low-frequency topographic impact to obtain the high-frequency topographic impact of the study area. The expression for the spherical harmonic algorithm used to calculate the global low-frequency topographic influence is as follows: ； in, , ; In the formula, and These represent the direct impact of global topographic compression on ground gravity and the secondary indirect impact of global topographic compression, calculated using the spherical harmonic algorithm, respectively. and These represent the indirect effects of global topographic compression, calculated using the spherical harmonic algorithm, on geoid undulation and geoid height, respectively. The geocentric radius of the calculation point; This represents normal gravity at a normal elevation on the ground. For reference, normal gravity on the ellipsoid; Newton's gravitational constant, Represents the global points region; The topographic spherical harmonic coefficient, For terrain density, The average radius of the Earth To truncate the order, This refers to the terrain elevation; for Step The spherical harmonic function that is completely normalized. For ground calculation points spherical coordinates, Let be the spherical coordinates of the integration point.

5. A computer-readable storage medium storing a computer program, characterized in that, When executed by a processor, the computer program implements the geoid construction method based on the high-frequency terrain influence combination algorithm as described in any one of claims 1 to 3.

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