Spherical Magnetic Vector Forwarding of Isoparametric DGGS Cells with Natural Superconvergent Points

Abstract

With the rapid advancement of satellite remote sensing technology, many scientists and organizations, including NASA, ESA, NAOC, and Roscosmos, observe and study significant changes in the geomagnetic field, which has greatly promoted research on the geomagnetic field and made it an important research direction in Earth system science. In traditional geomagnetic field research, tesseroid cells face degradation issues in high-latitude regions and accuracy limitations. To overcome these limitations, this paper introduces the Discrete Global Grid System (DGGS) to construct a geophysical model, achieving seamless global coverage through multi-level grid subdivision, significantly enhancing the processing capability of multi-source and multi-temporal spatial data. Addressing the challenges of the lack of analytical solutions and clear integration limits for DGGS cells, a method for constructing shape functions of arbitrary isoparametric elements is proposed based on the principle of isoparametric transformation, and the shape functions of isoparametric DGGS cells are successfully derived. In magnetic vector forwarding, considering the potential error amplification caused by Poisson’s formula, the DGGS grid is divided into six regular triangular sub-units. The triangular superconvergent point technique is adopted, and the positions of integration points and their weight coefficients are accurately determined according to symmetry rules, thereby significantly improving the calculation accuracy without increasing the computational complexity. Finally, through the forward modeling algorithm based on tiny tesseroid cells, this study comprehensively compares and analyzes the computational accuracy of the DGGS-based magnetic vector forwarding algorithm, verifying the effectiveness and superiority of the proposed method and providing new theoretical support and technical means for geophysical research.

1. Introduction

With the rapid development of Earth observation, satellite navigation and communication technologies in recent years, satellite remote sensing has been used by many scientists and organizations [1,2,3,4], such as NASA, ESA, NAOC, and Roscosmos, to observe and study obvious changes in the geomagnetic field, such as the movement of the magnetic north pole’s position, geomagnetic changes in the South Atlantic Anomaly region, and the potential phenomena of new global geomagnetic jerks [5,6].

Equipped with precision instruments such as magnetometers, accelerometers, and gyroscopes, remote sensing satellites provide high-precision, high spatio-temporal resolution, and large-scale geomagnetic data [7,8]. These models are primarily based on spherical harmonic representations of the main geomagnetic field and their long-term variations and are widely used in exploration geophysics, marine drilling and study of the Earth’s interior, ionosphere, and magnetosphere [9,10,11]. As the volume of Earth observation data exponentially increases, traditional spatial data models are no longer able to represent these vast amounts of global data at multiple resolutions [7,12]. Consequently, efficient utilization of these geomagnetic data obtained through satellite remote sensing and their potential application to satellite navigation, Earth science research, natural disaster warnings, and global changes have become pivotal research directions in the field of geophysics [13,14].

Just as physical properties are inverted in the Cartesian coordinate system, the spherical shell of the geophysical interpretation model is typically split into a set of discrete units by taking radius and latitude–longitude as parameters in a spherical coordinate system [15,16]. Tesseroid cells are simple and reliable volume discretization tools in geodetic or geocentric spherical coordinate systems that directly fit the curvature of the Earth through regular subdivision of the latitude and longitude grid and are performant for analyzing and interpreting gravity and magnetic field data, whether they are regional or global [17,18]. When the observation site is located outside the source tesseroid cell, the surface and edge integrals in Equation (19) of Zhong et al. [19] are regular, and their values can be calculated accurately using the standard Gauss–Legendre quadrature (GLQ) rule. Meanwhile, if the observation point is close to the source tesseroid cell, the integrand in these surface and edge integrals becomes singular [15,20,21,22,23]. As the tesseroid approaches the poles, it degenerates into a triangular prism instead of a rectangular prism [21,23]. In other words, with an increasing latitude ($\varphi$), the meridional scale of the tesseroid cell decreases while the latitudinal scale remains constant. This may lead to the resolution of the model becoming inconsistent in different directions [20,24,25]. Moreover, due to the singularity of the integral kernel near the poles [22,26], these singularities complicate the numerical integration process and introduce additional errors, especially when using low-order numerical methods or approximations [20,24]. Due to the combined effect of geometric distortions and singularities in the integral kernel, the accuracy of the tesseroid-based forward modeling is affected by the latitude of its location [24]. Therefore, although tesseroid cells are powerful for modeling regional and global geomagnetic phenomena, to prevent their accuracy from being impeded by shape degradation and numerical singularities [21,26,27], special considerations are required in high-latitude regions, especially polar regions ($\varphi$ > 85°).

With the goal of overcoming the limitations of the traditional latitude and longitude grid in mind, a Discrete Global Grid System (DGGS) has become the focus of research [28,29] since the 1980s. A DGGS is an innovative type of geospatial data management and analysis technology for constructing a spherical grid that discretizes the Earth’s surface into multi-scale polygon units of a consistent shape and size according to rule division. In this way, the DGGS is able to process massive, global, multi-resolution spatial data [30] and ensure that their consistency, hierarchy, and continuity are retained. Various DGGS variants are used across numerous fields, providing innovation within the existing Earth observation data models (further details are given in Section 6) [31,32,33,34,35,36,37]. As the main framework for processing satellite remote sensing data, the DGGS provides a spherical spatial model that can effectively minimize projection errors, efficiently process large-scale high-resolution remote sensing data, and support multi-scale analyses [38,39]. However, within geophysics applications, especially gravity and magnetic research, accurately obtaining the gravity/magnetic field corresponding to the DGGS cell is a key technical challenge that urgently needs to be addressed. In the forward modeling of gravity and magnetic fields based on polyhedron cells, the volume integration of a polyhedron is typically converted into an area integration by employing the Gauss/divergence theorem [40] and subsequently transformed into a line integral using various techniques, thereby obtaining the corresponding analytical/numerical solution [41]. While there is no analytical solution for a tesseroid cell, its integration limit is clearly defined. Therefore, numerical integration must be used to obtain the corresponding gravity and magnetic anomalies. However, there are no corresponding analytical solutions or fixed integration limits for DGGS cells. To overcome this problem, we need to use specific strategies to define the integration range and determine the appropriate numerical integration method [42].

Isoparametric transformation relies on the shape function of an element, with actual irregular elements mapped to regular elements (when there are as many shape functions as the number of element nodes, an element is called an isoparametric element) so that they can be analyzed or integrated efficiently. The shape functions must be adaptable and coordinated for stable and accurate analysis. Currently, there is no shape function or construction method that is directly applicable to DGGS cells. Since generating shape functions is more complicated, many scholars have proposed solutions for this purpose. For example, Barbier et al. [43] used REDUCE (a system that can accurately perform algebraic operations) to simplify algebraically derived shape functions, but the existing methods are mostly applicable to simple units, making it difficult for them to meet the requirements for complex DGGS cells.

The shape function is the key to calculating magnetic anomalies in irregular DGGS cells using isoparametric transformation. Therefore, based on isoparametric transformation, we propose a method for deriving the shape function, integral point position, and integral weight coefficient of arbitrary isoparametric elements such as DGGS cells as well as the calculation formula based on their volume integrals. However, based on Poisson’s formula [44,45], magnetic vector forwarding is vectorized by forward modeling of full-tensor gravity data, with the residuals in the former theoretically being about three times greater than those in the latter. Gaussian integration is still challenging when dealing with non-parallelograms and using linear, triangular, discrete grids in a DGGS and integration methods based on weighted residuals often lead to computational instability due to a lack of control over their accuracy [46].

Accuracy improvements in finite element analyses include increasing the number of integration points and optimizing and refining the grids, with the trade-off of a sharp increase in the computational costs. Meanwhile, superconvergence provides a way to balance accuracy and efficiency. By optimizing local grids or integration strategies, the accuracy can be significantly improved at specific points (called superconvergence points) or in specific regions [47] through direct evaluation or local post-processing [47]. Natural superconvergence points can exceed the global convergence rate without additional processing [48].

When the DGGS grid is regarded as an isoparametric distortion element (i.e., it can be regarded as the integral sum of six triangles), although the definition of superconvergence becomes complex and challenging to apply directly, the properties of the “optimal” sampling points may still be preserved [49]. Superconvergence in linear triangular cells was first studied by Oganesyan and Rukhovets [50] and Chen [51]. Regardless of the grid’s consistency, asymptotic superconvergence of stress on the whole unit sheet or optimal superconvergence at symmetrical points on the unit can be achieved by selecting specific sample points and refining the local midpoint [52]. In quadratic triangular cells, superconvergence was first reported in Zhu [53], who obtained comprehensive pointwise superapproximation and superconvergence results. Zienkiewicz and Zhu [49] noted that the values recovered at the nodes exhibit “superconvergence” characteristics, achieving accuracy at the $O\left(h^4\right)$ level (with h being the size of the units). In demonstrating that quadratic triangular cells can possess $O\left(h^4\right)$ superconvergence, this finding has greatly encouraged researchers with a preference for triangular cells seeking high precision at a reasonable cost. For cubic triangular cells, while Lagrange interpolation does not necessarily lead to superconvergence [54], superconvergence has been observed at local symmetric points in the grid [52,55].

In terms of the symmetry of the superconvergence points, triangular symmetric quadrature rules are particularly valuable because they can be mapped directly to the integration domain, and the integration points are not concentrated near certain vertices [56,57]. In contrast to asymmetric rules, the vertex mappings must be determined, and the points are concentrated inconsistently [58]. Considering both efficiency and aesthetics, quadrature formulas that are fully symmetric with respect to the three vertices of the triangle are the primary choice [59,60].

In linear finite element analyses of uniform equilateral triangular grids, node errors exhibit superconvergence to the order of $O\left(h^4\right)$, significantly surpassing the optimal global convergence rate by two orders [61]. For equilateral or regular quadratic finite elements, the midpoints of element edges and vertices become superconvergence points for function values [62]. Although Gaussian and Lobatto points appear to be superconvergence points, the latest research by Zhang and Naga [63] indicates that they are not natural superconvergence points for equilateral triangular grids in the context of cubic and quartic finite elements. Within a symmetric theoretical framework, for equilateral cells, some superconvergence points are local grid symmetry points. Specifically, for odd-order units (linear, cubic, etc.), symmetry points are also superconvergence points for derivatives while for even-order units (quadratic, quartic, etc.), symmetry points are also superconvergence points for function values [61].

Based on the unit analysis process involved in the finite element method (FEM) [64,65,66], this paper proposes a method for constructing the shape function of an arbitrary isoparametric element, with formulas for the shape function of a DGGS cell and thus its successfully obtained volume integral. Considering the fact that Poisson’s formula [44,45] can exponentially amplify the error of the gravity gradient tensor forward results in magnetic vector forwarding, the use of the triangular superconvergence point aims to significantly improve the computational accuracy without increasing the computational complexity. To this end, the DGGS cell is laterally divided into six positive triangular sub-cells, and the symmetry rule is introduced to determine the exact locations of the integration points and their corresponding integration weight coefficients. In this paper, the computation accuracy of the DGGS-based magnetic vector forwarding algorithm is analyzed in depth using a tiny tesseroid-based forwarding algorithm and comprehensively compared with other techniques. This algorithm is an extended algorithm for adaptive tesseroid forwarding, which uses a very small tesseroid cell and fits a complex source; with a large ratio of the distance between the observation points and the size of the tesseroid cell, the accuracy of the calculations is ensured [23,67].

The following is a description of how this work is organized. The tesseroid-based magnetic vector forwarding algorithm is initially presented in Section 2, followed by an explanation of the DGGS-based magnetic vector forwarding algorithm in Section 3. Section 4 is dedicated to the verification of the aforementioned algorithms, including the verification of the tesseroid-based magnetic vector forwarding algorithms and the verification of the DGGS-based magnetic vector forwarding algorithm via the tesseroid-based forwarding algorithm utilizing tiny tesseroids. The conclusions of this paper are given in Section 5. In addition, various variants of DGGS are described in Section 6.

2. Tesseroid-Based Magnetic Vector Forwarding Algorithm

As shown in Figure 1, the magnetic potential of a uniformly magnetized object is

$$U={\displaystyle \frac{-1}{4\pi }}\mathit{M}·\nabla {\int }_v{\displaystyle \frac{1}{r}}dv$$

(1)

where r is the vector from the observation point to the anomaly source, $\mathit{M}$ is the total magnetization vector, $\mathit{M}={[M_x,M_y,M_z]}^{\mathrm{T}}$, $M_x=M\cos I\cos D$, $M_y=M\cos I\sin D$, and $M_z=M\sin I$. Here, M, I, and D are the magnitude, inclination, and declination of $\mathit{M}$, respectively.

There is a specific relationship between the analytical expressions of the magnetic and gravitational potential of an object with uniform magnetization M and uniform density $\rho$, i.e., the Poisson relationship [44,45]. By utilizing this relationship, the magnetic field of the magnetic body can be calculated more conveniently. Equation (1) can be re-expressed using the gravitational potential V:

$$U={\displaystyle \frac{-1}{4\pi G\rho }}\mathit{M}·\nabla V$$

(2)

where $V=G\rho {\int }_v{\displaystyle \frac{1}{r}}dv$, G = $6.674\times {10}^{-11}$${\mathrm{m}}^3·\mathrm{k}{\mathrm{g}}^{-1}·{\mathrm{s}}^{-2}$ is the gravitational constant.

According to the Poisson relation, the linear expression between the magnetic vector $B={[B_x,B_y,B_z]}^{\mathrm{T}}$ and the gravity gradient tensor T is as follows [44]:

$$B=-C_{m}T·\mathit{M}$$

(3)

where $C_m={\displaystyle \frac{{\mu }_0}{4\pi G\rho }}$, ${\mu }_0=4\pi \times {10}^{-7}H·m^{-1}$ is the permeability in a vacuum.

Expanding Equation (3),

$$\left\{\begin{array}{c}{B}_x={\displaystyle \frac{{\mu }_0}{4\pi G\rho }}\left[M_x{g}_{xx}+M_y{g}_{yx}+M_z{g}_{zx}\right]\\ B_y={\displaystyle \frac{{\mu }_0}{4\pi G\rho }}\left[M_x{g}_{xy}+M_y{g}_{yy}+M_z{g}_{zy}\right]\\ B_z={\displaystyle \frac{{\mu }_0}{4\pi G\rho }}\left[M_x{g}_{xz}+M_y{g}_{yz}+M_z{g}_{zz}\right]\end{array}\right.$$

(4)

For the tesseroid-based gravitational forwarding algorithm, component $g_{\alpha \beta }$ of gravity gradient tensor T at observation point P due to the tesseroid cell (see Figure 1) can be expressed as

$$g_{\alpha \beta }\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)=G\rho {\int }_{{\lambda }_1}^{{\lambda }_2}{\int }_{{\varphi }_1}^{{\varphi }_2}{\int }_{r_1}^{r_2}{\displaystyle \frac{1}{{\mathsf{\ell }}^3}}\left(3{\Delta }_{\alpha }{\Delta }_{\beta }{\mathsf{\ell }}^{-2}-{\delta }_{\alpha \beta }\right)\kappa d{r}^{\prime }d{\varphi }^{\prime }d{\lambda }^{\prime }$$

(5)

where $\alpha ,\beta \in \left\{x,y,z\right\}$, ${\Delta }_x=r^{\prime }\left(\cos \varphi \sin {\varphi }^{\prime }-\sin \varphi \cos {\varphi }^{\prime }\cos \left({\lambda }^{\prime }-\lambda \right)\right),$ ${\Delta }_y=r^{\prime }\cos {\varphi }^{\prime }\sin \left({\lambda }^{\prime }-\lambda \right),$

${\Delta }_z=r^{\prime }\cos \psi -r,$$\mathsf{\ell }=\sqrt{{r^{\prime }}^2+r^2-2r^{\prime }r\cos \psi }$, $\cos \psi =\sin \varphi \sin {\varphi }^{\prime }+\cos \varphi \cos {\varphi }^{\prime }\cos \left({\lambda }^{\prime }-\lambda \right)$, $\kappa =r^{\prime 2}\cos {\varphi }^{\prime }$, and ${\delta }_{\alpha \beta }$ is a Kronecker function.

The gravity gradient tensor T is expressed in matrix form:

$$T=\left[\begin{array}{ccc}{g}_{xx}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)\hfill & g_{xy}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)& g_{xz}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)\\ g_{yx}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)\hfill & g_{yy}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)& g_{yz}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)\\ g_{zx}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)\hfill & g_{zy}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)& g_{zz}\left(r_P,{\varphi }_P,{\lambda }_P,r_Q,{\varphi }_Q,{\lambda }_Q\right)\end{array}\right]$$

(6)

where the gravitational potential satisfies the Laplace equation outside the anomalous source, T is a symmetric matrix, and the sum of its diagonal components is equal to 0. Therefore, only five components of T are independent.

Equation (5) has no analytical solution and needs to be solved through approximate calculation using numerical methods, including Taylor series expansion [20] and GLQ [68], etc. Since Equation (6) involves multiple formulas that require integration, the integration kernel f [69] is introduced to simplify the analysis by transforming Equation (6) into the general form of a triple-Gaussian integral [70].

$$G\rho {\int }_{{\lambda }^{\prime }={\lambda }_1}^{{\lambda }_2}{\int }_{{\varphi }^{\prime }={\varphi }_1}^{{\varphi }_2}{\int }_{r^{\prime }=r_1}^{r_2}f(r_P,{\varphi }_P,{\lambda }_P,r^{\prime },{\varphi }^{\prime },{\lambda }^{\prime })d{r}^{\prime }d{\varphi }^{\prime }d{\lambda }^{\prime }$$

(7)

When the observation point P is not of concern, assuming that the integration interval of the tesseroid cell in the longitude, latitude, and radial directions is $[a,b]$, node $x_k$ can be selected appropriately to find the original function of the integration kernel $f(r^{\prime },{\varphi }^{\prime },{\lambda }^{\prime })$ on the integration interval $[a,b]$. Then, the height $f\left(\xi \right)$ can be approximated using weighted averaging with $f\left(x_k\right)$ by constructing the following equation [23]:

$${\int }_a^{b}f\left(x\right)dx={\displaystyle \frac{b-a}{2}}\sum _{k=1}^n{w}_{k}f\left(x_k\right)$$

(8)

where $x_k$ is the integration point with subscript $k=1,\cdots ,n$; n is the number of integration points, and $w_k$ is the corresponding integration weight and can be expressed as follows:

$$w_k={\displaystyle \frac{2}{\left(1-x_k^2\right){\left({P_n}^{\prime }\left(x_k\right)\right)}^2}}$$

(9)

where $P_n^{\prime }\left(x\right)$ denotes the derivative of the n-th order Legendre polynomial $P_n\left(x\right)$. The corresponding zero point of $P_n\left(x\right)$ in this interval corresponds to a Gaussian point when we scale the integration interval of Equation (8) to the interval $[-1,1]$. For higher-order Legendre polynomials $P_n\left(x\right)$, their roots can be found by utilizing the recurrence relation of their derivatives $P_n^{\prime }\left(x\right)$.

Utilizing Equation (8) and the Gauss–Legendre integration method, Equation (7) is rewritten as a unified representation of numerical integration [23,68]:

$$G\rho S\sum _{k=1}^{n_{\lambda }}\sum _{j=1}^{n_{\varphi }}\sum _{i=1}^{n_r}{W}_r^i{W}_{\varphi }^j{W}_{\lambda }^{k}f\left({r^{\prime }}_i,{{\varphi }^{\prime }}_j,{{\lambda }^{\prime }}_k\right)$$

(10)

where $\alpha \in \left\{\lambda ,\varphi ,r\right\}$, $n_{\alpha }$ is the number of integration points along the three axes of the spherical coordinate system, and $W_r^i$, $W_{\varphi }^j$, and $W_{\lambda }^k$ are the corresponding integration weights. In practice, similar to other integration methods, the computational accuracy of the integral to be solved can usually be controlled by adjusting the number of integration points. However, while increasing the number of integration points will improve the GLQ’s accuracy, it may also decrease the computational efficiency [68,71]. Therefore, for the three axes of the spherical coordinate system, $n_{\alpha }$ is equal to 2 and $S=\left(r_2-r_1\right)\left({\varphi }_2-{\varphi }_1\right)\left({\lambda }_1-{\lambda }_2\right)/8$.

3. DGGS-Based Magnetic Vector Forwarding Algorithm

In discretizing traditional spherical shell models, inconsistencies are often found between their triangular, rhombus prism, or tesseroid cells, especially in high-latitude regions, where significant degradation phenomena occur [72]. To address these inherent defects, this study adopts spherical pentagons and hexagons (collectively referred to as DGGS grids) and their constituent prisms (i.e., DGGS cells) under the framework of the DGGS to discretize the spherical surface and spherical shell, respectively. These DGGS cells have approximate shapes and continuous scales and provide seamless global coverage through a multi-level grid system, thus effectively eliminating the inconsistency and degradation found in high-latitude regions when using the traditional methods [73]. Specifically, the primary DGGS grid consists of 12 pentagons and 30 hexagons, which can be further refined to construct a higher-precision global grid system, as shown in Figure 2. It is worth noting that the number of pentagonal DGGS cells remains unchanged when the grid is refined, reflecting the stability and flexibility of its design.

Although tesseroid cells lack analytical or closed-form solutions, they have well-defined integration limits; therefore, there are numerical integration methods to obtain their corresponding gravity or geomagnetic field anomaly responses. In contrast, theoretically, DGGS cells have neither a direct analytical solution nor a fixed integration limit. In order to determine the integration limits for a DGGS cell, this study draws on the integration analysis within the finite element method, i.e., isoparametric transformation [74] on the basis of a unit’s shape functions and integration schemes. Considering that there is a lack of methods for constructing shape functions and high-performance integration schemes for DGGS cells, here arbitrary convex unit shape functions are constructed to find the shape functions for isoparametric DGGS cells. Subsequently, isoparametric transformation is utilized to map an irregular DGGS cell from its original system coordinate system to a local coordinate system to facilitate analysis.

3.1. The Isoparametric Transformation of DGGS Cells

In finite element analysis, to establish the equations to be solved, volume and surface integrals must be computed for each discretized unit, thereby discretizing the continuous physical field. Since forward modeling of the magnetic field in spherical coordinates in this paper only requires consideration of the volume effects, the volume integral of field function f over discretized unit $V_e$ (Figure 3b) is expressed as

$${\int }_{V_e}fd{V}_e={\int }_{V_e}f\left(x,y,z\right)dxdydz$$

(11)

However, the limits of integration for field function f are not well defined in the global coordinate system, while the integration limits are standardized in local or natural coordinates, as shown in Figure 3. Therefore, it is generally preferable to use normalized numerical integration methods with natural coordinates to compute integrals such as those above. To this end, the volume element formed by local coordinates $d\xi$, $d\eta$ and $d\zeta$ is expressed in global coordinates as follows:

$$d{V}_e=d\xi ·(d\eta \times d\zeta )$$

(12)

where · and × are the dot-product and cross-product operators, respectively, and

$$\begin{array}{c}d\xi ={\displaystyle \frac{\partial x}{\partial \xi }}d\xi i+{\displaystyle \frac{\partial y}{\partial \xi }}d\xi j+{\displaystyle \frac{\partial z}{\partial \xi }}d\xi k\hfill \\ d\eta ={\displaystyle \frac{\partial x}{\partial \eta }}d\eta i+{\displaystyle \frac{\partial y}{\partial \eta }}d\eta j+{\displaystyle \frac{\partial z}{\partial \eta }}d\eta k\hfill \\ d\zeta ={\displaystyle \frac{\partial x}{\partial \zeta }}d\zeta i+{\displaystyle \frac{\partial y}{\partial \zeta }}d\zeta j+{\displaystyle \frac{\partial z}{\partial \zeta }}d\zeta k\hfill \end{array}$$

(13)

Here, i, j, and k are the unit vectors in the x, y, and z directions of the global coordinate system, respectively. Substituting Equation (13) into Equation (12) yields

$$d{V}_e=\left[\begin{array}{ccc}{\displaystyle \frac{\partial x}{\partial \xi }}& {\displaystyle \frac{\partial y}{\partial \xi }}& {\displaystyle \frac{\partial z}{\partial \xi }}\\ {\displaystyle \frac{\partial x}{\partial \eta }}& {\displaystyle \frac{\partial y}{\partial \eta }}& {\displaystyle \frac{\partial z}{\partial \eta }}\\ {\displaystyle \frac{\partial x}{\partial \zeta }}& {\displaystyle \frac{\partial y}{\partial \zeta }}& {\displaystyle \frac{\partial z}{\partial \zeta }}\end{array}\right]d\xi d\eta d\zeta$$

(14)

To analyze the transformation relationship of the derivative between the two coordinate systems and the volume element in Equation (14), with the help of the shape function $N_i$, the isoparametric transformation is rewritten into the form of the following interpolation function:

$$x=\sum _{i=1}^{n_v}{N}_i{x}_i,y=\sum _{i=1}^{n_v}{N}_i{y}_i,z=\sum _{i=1}^{n_v}{N}_i{z}_i,u=\sum _{i=1}^{n_v}{N}_i{u}_i$$

(15)

where $n_v$ denotes the number of element nodes used for the isoparametric transformation, and $x_i,y_i,z_i$ represent the coordinate values of these nodes in the global coordinate system (Cartesian or spherical coordinates). u is the displacement function, and $u_i$ is the function value of node i.

According to the chain rule of calculus, the partial derivatives of shape functions $N=\left\{N_i{|}_{i=1}^{n_v}\right\}$ with respect to ${\left(\xi ,\eta ,\zeta \right)}^{\mathrm{T}}$ are computed, and the results are expressed in matrix form.

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial N}{\partial \xi }}\\ {\displaystyle \frac{\partial N}{\partial \eta }}\\ {\displaystyle \frac{\partial N}{\partial \zeta }}\end{array}\right\}=\left[\begin{array}{ccc}{\displaystyle \frac{\partial x}{\partial \xi }}& {\displaystyle \frac{\partial y}{\partial \xi }}& {\displaystyle \frac{\partial z}{\partial \xi }}\\ {\displaystyle \frac{\partial x}{\partial \eta }}& {\displaystyle \frac{\partial y}{\partial \eta }}& {\displaystyle \frac{\partial z}{\partial \eta }}\\ {\displaystyle \frac{\partial x}{\partial \zeta }}& {\displaystyle \frac{\partial y}{\partial \zeta }}& {\displaystyle \frac{\partial z}{\partial \zeta }}\end{array}\right]\left\{\begin{array}{c}{\displaystyle \frac{\partial N}{\partial x}}\\ {\displaystyle \frac{\partial N}{\partial y}}\\ {\displaystyle \frac{\partial N}{\partial z}}\end{array}\right\}=J\left\{\begin{array}{c}{\displaystyle \frac{\partial N}{\partial x}}\\ {\displaystyle \frac{\partial N}{\partial y}}\\ {\displaystyle \frac{\partial N}{\partial z}}\end{array}\right\}$$

(16)

where J is called a Jacobi matrix, and its physical meaning is the volume of the unit to be integrated $v_e$.

Bringing Equation (15) into Equation (16) above, we obtain the following:

$$J=\left[\begin{array}{cccc}{\displaystyle \frac{\partial N_1}{\partial \xi }}& {\displaystyle \frac{\partial N_2}{\partial \xi }}& \cdots & {\displaystyle \frac{\partial N_{n_v}}{\partial \xi }}\\ {\displaystyle \frac{\partial N_1}{\partial \eta }}& {\displaystyle \frac{\partial N_2}{\partial \eta }}& \cdots & {\displaystyle \frac{\partial N_{n_v}}{\partial \eta }}\\ {\displaystyle \frac{\partial N_1}{\partial \zeta }}& {\displaystyle \frac{\partial N_2}{\partial \zeta }}& \cdots & {\displaystyle \frac{\partial N_{n_v}}{\partial \zeta }}\end{array}\right]\left[\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ ⋮& ⋮& ⋮\\ x_{n_v}& y_{n_v}& z_{n_v}\end{array}\right]$$

(17)

Thus, the partial derivatives of n for x, y, and z can be explicitly expressed in natural coordinates as

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial N}{\partial x}}\\ {\displaystyle \frac{\partial N}{\partial y}}\\ {\displaystyle \frac{\partial N}{\partial z}}\end{array}\right\}=J^{-1}\left\{\begin{array}{c}{\displaystyle \frac{\partial N}{\partial \xi }}\\ {\displaystyle \frac{\partial N}{\partial \eta }}\\ {\displaystyle \frac{\partial N}{\partial \zeta }}\end{array}\right\}$$

(18)

Through the isoparametric transformation from the global coordinate system to the local coordinate system, the integration domain $V_e$ for the field function f in the original system can be mapped to a region with well-defined integration limits in the local coordinate system. For simplicity, assume these limits are $\left[a_{\xi },b_{\xi }\right]\times \left[a_{\eta },b_{\eta }\right]\times \left[a_{\zeta },b_{\zeta }\right]$. Therefore, the integral of the field function f in the local coordinate system can be expressed as follows:

$${\int }_{a_{\zeta }}^{b_{\zeta }}{\int }_{a_{\eta }}^{b_{\eta }}{\int }_{a_{\epsilon }}^{b_{\epsilon }}{f}^{*}\left(\epsilon ,\eta ,\zeta \right)d\epsilon d\eta d\zeta$$

(19)

where

$$f^{*}\left(\xi ,\mu ,\eta \right)=f\left(x\left(\xi ,\mu ,\eta \right),y\left(\xi ,\mu ,\eta \right),z\left(\xi ,\mu ,\eta \right)\right)\left|J\right|$$

(20)

Using the triple-Gaussian integral [68,75], Equation (19) can be rewritten as follows:

$$\begin{array}{c}\begin{array}{c}{\int }_{V_e}fd{V}_e={\displaystyle \sum _{k=1}^{n_k}}{\displaystyle \sum _{j=1}^{n_j}}{\displaystyle \sum _{i=1}^{n_i}}{w}_i{w}_j{w}_{k}f\left({\epsilon }_i,{\eta }_j,{\zeta }_k\right)det\left(J\right)\end{array}\hfill \end{array}$$

(21)

where $n_i,n_j,n_k$ and $w_i,w_j,w_k$ are the number of integral points $\left({\epsilon }_i,{\eta }_j,{\zeta }_k\right)$ and the integral weight coefficients of the local coordinate system along the $x,y,z$-axis directions.

By establishing the transformation relationship between the two coordinate systems in the above derivation process, a regularly shaped unit in natural coordinates (also known as the parent unit) is transformed into a unit (sub-unit) with a distorted shape in the overall (Cartesian or spherical) coordinate system to obtain a clear integral limit. Then, the volume integration of the field function f with respect to unit $v_e$ is obtained using the triple-Gaussian integration method, which is applicable to most “convex” cells, e.g., tetrahedrons and triangular prisms.

3.2. The Shape Function of DGGS Cells

To compute the magnetic anomaly response of a DGGS cell within the system coordinate system at observation point P, we employ Equation (4) of Poisson’s formula, Equation (5) derived from the integral kernel formula based on the gravity gradient tensor, and Equation (10) of the triple-Gauss integral. This requires us to specify the regular pentagonal/hexagonal prism’s vertex coordinates in the global coordinate system, along with the shape functions, integration point coordinates, and integration weight coefficients required to define the prism’s geometry in the local coordinate system.

To improve the computational accuracy, we propose an alternative method for constructing the shape function of the regular hexagonal prism/isoparametric DGGS cell, with its purpose being more consistent assignment of the integral weights under isoparametric transformation. First, as shown in Figure 4, assume that a regular hexagon with $n_{\overline{v}}$ nodes and its i-th vertex coordinates are denoted as $\left(x_i,y_i\right)$. Then, the regular hexagonal is extended from the $z=-1$ plane to the $z=1$ plane to construct a regular 3D hexagonal prism with $n_v=2\times n_{\overline{v}}$ nodes. Since constructing the shape functions requires specific interpolating polynomials, a list of polynomials to be used, $\overline{v}$, is determined using Pascal’s triangle, as shown in Figure 5.

$$\overline{v}=\left\{1,x,y,z,xy,\cdots ,x^5,\cdots ,y^5,\cdots ,z^5,\cdots \right\}$$

(22)

For a 3D linear shape function, it is assumed that function $\psi =\psi \left(x,y,z\right)$ varies linearly along all axes. The $\left\{v_i{|}_{i=1}^{n_v}\right\}$ are selected from the list of polynomials $\overline{v}$ and the corresponding displacement function $\psi$ is constructed:

$$\psi =a_1{v}_1+a_2{v}_2+a_3{v}_3+\cdots +a_{n_v}{v}_{n_v}$$

(23)

Alternatively, it can be written in vector form as

$$\psi =\left[\begin{array}{ccccc}{v}_1& v_2& v_3& \cdots & v_{n_v}\end{array}\right]{\left[\begin{array}{ccccc}{a}_1& a_2& a_3& \cdots & a_{n_v}\end{array}\right]}^{\mathrm{T}}$$

(24)

where $\left\{{\left.a_i\right|}_{i=1}^{n_v}\right\}$ are the coefficients to determine.

To determine the shape functions $\left\{{\left.N_i\right|}_{i=1}^{n_v}\right\}$ and their coefficients $\left\{{\left.a_i\right|}_{i=1}^{n_v}\right\}$, the following equation system in matrix form is constructed by substituting the vertices coordinates $\left(x_j,y_j,z_j\right)$ of the regular hexagonal prisms into the interpolating function $\psi$. Then,

$$\begin{array}{c}\mathit{\psi }=\mathit{Va}\hfill \\ \left\{\begin{array}{c}{\psi }_1\\ {\psi }_2\\ ⋮\\ {\psi }_i\\ ⋮\\ {\psi }_{n_v}\end{array}\right\}=\left[\begin{array}{cccccc}1& v_1\left(x_1,y_1,z_1\right)& \cdots & v_i\left(x_1,y_1,z_1\right)& \cdots & v_{n_v}\left(x_1,y_1,z_1\right)\\ 1& v_1\left(x_2,y_2,z_2\right)& \cdots & v_i\left(x_2,y_2,z_2\right)& \cdots & v_{n_v}\left(x_2,y_2,z_2\right)\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& v_1\left(x_j,y_j,z_j\right)& \cdots & v_i\left(x_j,y_j,z_j\right)& \cdots & v_{n_v}\left(x_j,y_j,z_j\right)\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& v_1\left(x_{n_v},y_{n_v},z_{n_v}\right)& \cdots & v_i\left(x_{n_v},y_{n_v},z_{n_v}\right)& \cdots & v_{n_v}\left(x_{n_v},y_{n_v},z_{n_v}\right)\end{array}\right]\left\{\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_i\\ ⋮\\ a_{n_v}\end{array}\right\}\hfill \end{array}$$

(25)

where $\mathit{\psi }={\left\{{\psi }_1,{\psi }_2,\cdots ,{\psi }_i,\cdots ,{\psi }_{n_v}\right\}}^{\mathrm{T}}$ and $\mathit{a}={\left\{a_1,a_2,\cdots ,a_i,\cdots ,a_{n_v}\right\}}^{\mathrm{T}}$.

Then,

$$\mathit{a}={\mathit{V}}^{-\mathit{1}}\mathit{\psi }$$

(26)

When $\mathrm{rank}\left(\mathit{V}\right)<n_v$, the inverse matrix of $\mathit{V}$ is unavailable, or its inverse matrix is meaningless.

Here, $\mathit{v}={\left[v_1,v_2,\cdots ,v_i,\cdots ,v_{n_v}\right]}^{\mathrm{T}}$ is re-selected from the list of polynomials $\overline{v}$ to ensure $\mathrm{rank}\left(\mathit{V}\right)=n_v$, which indicates that all the equations are not linearly correlated. Then, substituting Equation (26) into Equation (24) gives the following:

$$\psi ={\mathit{v}}^{\mathrm{T}}{\mathit{V}}^{-1}\mathit{\psi }$$

(27)

According to Equations (15), (24), and (27), the shape functions are defined as $\left[N\right]$ according to the conventions of finite element analysis:

$$\left[N\right]={\left[\begin{array}{cccccc}{N}_1& N_2& \cdots & N_i& \cdots & N_{n_v}\end{array}\right]}^{\mathrm{T}}={\mathit{v}}^{\mathrm{T}}{\mathit{V}}^{-1}$$

(28)

where $N_i$ is the shape function corresponding to the i-th vertex of the hexagonal prism.

The following list of polynomials is chosen artificially:

$$v={\left[\begin{array}{cccccccccccc}1& x& y& z& xy& xz& yz& x{y}^2& y^{2}z& y^2& x{y}^{2}z& xyz\end{array}\right]}^{\mathrm{T}}$$

(29)

Then, the following shape function scheme was obtained using a derivation process similar to the above:

$$\begin{array}{ccc}{N}_1=-{\displaystyle \frac{1}{12}}\left(z+1\right)\left(-4x^3-4x^2+x+1\right)\hfill & & N_7={\displaystyle \frac{1}{12}}\left(z-1\right)\left(-4x^3-4x^2+x+1\right)\hfill \\ N_2={\displaystyle \frac{1}{12}}\left(z+1\right)\left(2x+1\right)\left(-2x^2+\sqrt{3}y+2\right)\hfill & & N_8=-{\displaystyle \frac{1}{12}}\left(z-1\right)\left(2x+1\right)\left(-2x^2+\sqrt{3}y+2\right)\hfill \\ N_3=-{\displaystyle \frac{1}{12}}\left(z+1\right)\left(2x-1\right)\left(-2x^2+\sqrt{3}y+2\right)\hfill & & N_9={\displaystyle \frac{1}{12}}\left(z-1\right)\left(2x-1\right)\left(-2x^2+\sqrt{3}y+2\right)\hfill \\ N_4={\displaystyle \frac{1}{12}}\left(z+1\right)\left(-4x^3+4x^2+x-1\right)\hfill & & N_{10}=-{\displaystyle \frac{1}{12}}\left(z-1\right)\left(-4x^3+4x^2+x-1\right)\hfill \\ N_5={\displaystyle \frac{1}{12}}\left(z+1\right)\left(2x-1\right)\left(2x^2+\sqrt{3}y-2\right)\hfill & & N_{11}=-{\displaystyle \frac{1}{12}}\left(z-1\right)\left(2x-1\right)\left(2x^2+\sqrt{3}y-2\right)\hfill \\ N_6=-{\displaystyle \frac{1}{12}}\left(z+1\right)\left(2x+1\right)\left(2x^2+\sqrt{3}y-2\right)\hfill & & N_{12}={\displaystyle \frac{1}{12}}\left(z-1\right)\left(2x+1\right)\left(2x^2+\sqrt{3}y-2\right)\hfill \end{array}$$

(30)

At the first, fourth, seventh, and tenth nodes, the shape functions do not consider the information on the field function f along the y-axis. According to the definition of isoparametric transformation, the following shape function scheme for isoparametric DGGS cells is constructed using a similar derivation process to that above:

$$\begin{array}{cccc}{N}_1=-{\displaystyle \frac{1}{12}}\left(z+1\right)\left(x+1\right)\left(4y^2-3\right)\hfill & & N_7={\displaystyle \frac{1}{12}}\left(z-1\right)\left(x+1\right)\left(4y^2-3\right)\hfill & \\ N_2={\displaystyle \frac{1}{12}}\left(z+1\right)\left(2x+1\right)\left(2y+\sqrt{3}\right)y\hfill & & N_8=-{\displaystyle \frac{1}{12}}\left(z-1\right)\left(2x+1\right)\left(2y+\sqrt{3}\right)y\hfill & \\ N_3=-{\displaystyle \frac{1}{12}}\left(z+1\right)\left(2x-1\right)\left(2y+\sqrt{3}\right)y\hfill & & N_9={\displaystyle \frac{1}{12}}\left(z-1\right)\left(2x-1\right)\left(2y+\sqrt{3}\right)y\hfill & \\ N_4={\displaystyle \frac{1}{12}}\left(z+1\right)\left(x-1\right)\left(4y^2-3\right)\hfill & & N_{10}=-{\displaystyle \frac{1}{12}}\left(z-1\right)\left(x-1\right)\left(4y^2-3\right)\hfill & \\ N_5=-{\displaystyle \frac{1}{12}}\left(z+1\right)\left(2x-1\right)\left(2y-\sqrt{3}\right)y\hfill & & N_{11}={\displaystyle \frac{1}{12}}\left(z-1\right)\left(2x-1\right)\left(2y-\sqrt{3}\right)y\hfill & \\ N_6={\displaystyle \frac{1}{12}}\left(z+1\right)\left(2x+1\right)\left(2y-\sqrt{3}\right)y\hfill & & N_{12}=-{\displaystyle \frac{1}{12}}\left(z-1\right)\left(2x+1\right)\left(2y-\sqrt{3}\right)y\hfill \end{array}$$

(31)

3.3. The Integration Points of DGGS Cells

In conventional finite element analysis, the integration points are usually set at locations such as the Gaussian points of the units, equidistant points to its edges, or the midpoints of its faces, as shown in Figure 6a; however, the computational accuracy at these locations is significantly lower than when using the superconvergence point of the cell as the integration point, as shown in Figure 6b.

To set the superconvergence point in a regular hexagonal prism as the integration point for a section of the DGGS prism (i.e., the DGGS grid, see Figure 6b), it can be considered as a combination of six equilateral triangles. For linear triangles (see Figure 7), the integration points are located at the centroids of the units [76]. For quadratic triangles, the integration points (sampling points) are not uniquely defined [76]. When using Poisson’s formula on triangular grids, symmetry points of the grid are almost always superconvergent points [77]. Therefore, within the framework of symmetry theory [55], integration points must appear in groups of six, three, or a single point (with multiplicity), and each group of integration points must have equal corresponding integration weights [78]. As illustrated in Figure 7, for a group of six points, there is one point in each of the six sub-triangles defined by the meridians, which is the centroid of the equilateral triangles; for a group of three points ($\sum 3A$ and $\sum 3B$), the points lie on each meridian.

The symmetry rules for triangles remain unchanged when equilateral triangles are rotated or reflected around their medians and can be isoparametrically transformed into any other arbitrary triangle. As illustrated in Figure 8, the symmetrical integration rules for triangles primarily encompass designed combinations of type-0 (centroid), type-1 (midline point or high line), and type-2 (non-midline point). Type-0 contains only one point, namely the centroid, whose position in the barycenter coordinate system is fixed at (1/3, 1/3, 1/3). The type-1 orbit is composed of three points located in the middle, and their coordinates consist of three different arrangements of $(\alpha ,1-\alpha /2,1-\alpha /2)$0. The type-2 orbit contains six points not in the middle position, and their coordinates are determined by six unique arrangements of $(\alpha ,\beta ,1-\alpha -\beta )$. Each orbit type contributes a different number of points. The total number of points n is determined by the number of orbits ($n=n_0+3n_1+6n_2$), where $n_0=$ 0 or 1; $n_1$ and $n_2$ can be arbitrary.

With each type of integration point combination, it is assumed that the weights $w_i$ of all points may be preliminarily regarded as equal to simplify the calculation process. However, in practical application, the weights and coordinates of each point (especially the points in type-1 and type-2 orbits, as shown in Figure 8b,c) need to be finely optimized to accurately approximate the function integration in the triangular domain. The symmetrical integration rule for triangles ensures that their rotation and reflection do not vary for any triangle shape, making the symmetry rule especially effective for dealing with problems with rotational symmetry.

3.4. The Integration Weights of DGGS Cells

To compute the integral weights in integral domain $V_e$ for any function $\overline{f}$, in particular integral kernel f, as shown in Figure 6a, a cross-sectional view of a regular hexagonal prism (i.e., a horizontal hexagonal cell, as shown in Figure 4 in greater detail) is taken as the object of study, and $n_{xy}$ integration points are arranged in the horizontal hexagonal cell in Figure 6b. While the original Gaussian integration scheme is employed along the vertical direction for a regular hexagonal prism in Figure 6a, setting the $n_r$ (i.e., the quantity of 1D Gaussian integration points) layer of the integration points and the corresponding weight coefficients $w_r^k$ for the kth layer of integration points. Then, according to the definition of triple-Gaussian integration, Equation (19) is rewritten as

$${\int }_{V_e}\overline{f}d{V}_e=\sum _{j=1}^{n_w}{\overline{w}}_j\overline{f}\left({\epsilon }_j,{\eta }_j,{\zeta }_j\right)det\left(J\right)$$

(32)

where $\overline{f}\left({\epsilon }_j,{\eta }_j,{\zeta }_j\right)$ is the function to be integrated; the triple-Gaussian integration weight coefficient ${\overline{w}}_j=w_{xy}^i{w}_r^k$, with subscripts $j\equiv (k-1)\times n_{xy}+i$ and $1\le j\le n_w$, $1\le n_r\le n_r$; $w_{xy}^i$ are the weights of the ith integration points of the regular hexagonal cell; and the number of integration points $n_w=n_{xy}\times n_r$; and ${\epsilon }_j,{\eta }_j,{\zeta }_j$ are the coordinates of the jth integration point.

The combination of integration points and weights establishes an orthogonal integration rule whose accuracy is measured by the maximum order d of the polynomials that can be integrated accurately over the integration domain $V_e$. Given that Equation (32) exhibits linearity with respect to function $\overline{f}$, achieving a rule of (at least) degree d necessitates ensuring its exactness for a polynomial basis of degree d. In this article, a 1D Gaussian integration scheme is adopted in the vertical direction for an upright hexagonal prism, thereby focusing the research on a 2D space to ensure that the basis functions selected can match or exceed accuracy to the order of d. To this end, monomials ${\xi }^m{\eta }^n$ satisfying condition $m+n\le d$ are selected as candidate basis functions. However, considering the potential impact of rounding errors and Vandermonde matrices, directly applying these monomials may not be the optimal strategy for describing the polynomial space ${\mathit{P}}_d$. To improve the accuracy and efficiency of the description, this article uses orthogonal Koornwinder–Dubiner polynomials as a basis [79,80] in order to obtain a better numerical performance.

$$\begin{array}{c}{\mathit{P}}_d=\left\{p_i\left(x\right)|i=1\cdots M\right\}\hfill \\ =\mathrm{span}\left\{g_{m,n}|m+n\le d\right\}\hfill \end{array}$$

(33)

where

$$g_{m,n}(\xi ,\eta )=P_m^{(0,2n+1)}\left[2(\xi +\eta )-1\right]\left[{(\xi +\eta )}^n{P}_n^{(0,0)}\left({\displaystyle \frac{\eta -\xi }{\eta +\xi }}\right)\right]$$

(34)

Here, $P_n^{(\alpha ,\beta )}$ are the Jacobi polynomials with weights $(\alpha ,\beta )$ and a degree n.

Substitute the bases $P_n^{(\alpha ,\beta )}$ into Equation (32) to construct the following system of linear equations in order to obtain the integral weights for each integration point:

$$\left[\begin{array}{ccc}{p}_1\left({\epsilon }_1,{\eta }_1,{\zeta }_1\right)det\left(J\right)& \cdots & p_1\left({\epsilon }_{n_w},{\eta }_{n_w},{\zeta }_{n_w}\right)det\left(J\right)\\ p_2\left({\epsilon }_1,{\eta }_1,{\zeta }_1\right)det\left(J\right)& \cdots & p_2\left({\epsilon }_{n_w},{\eta }_{n_w},{\zeta }_{n_w}\right)det\left(J\right)\\ ⋮& & ⋮\\ p_M\left({\epsilon }_1,{\eta }_1,{\zeta }_1\right)det\left(J\right)& \cdots & p_M\left({\epsilon }_{n_w},{\eta }_{n_w},{\zeta }_{n_w}\right)det\left(J\right)\end{array}\right]\left\{\begin{array}{c}{\overline{w}}_1\\ {\overline{w}}_2\\ ⋮\\ {\overline{w}}_{n_w}\end{array}\right\}=\left\{\begin{array}{c}{\int }_{V_e}{p}_1\left(\epsilon ,\eta ,\zeta \right)d{V}_e\\ {\int }_{V_e}{p}_2\left(\epsilon ,\eta ,\zeta \right)d{V}_e\\ ⋮\\ {\int }_{V_e}{p}_M\left(\epsilon ,\eta ,\zeta \right)d{V}_e\end{array}\right\}$$

(35)

This is expanded into matrix form as

$$\begin{array}{c}\mathit{P}\overline{\mathit{w}}=\overline{\mathit{F}}\hfill \\ \left[\begin{array}{ccccc}{p}_{1,1}& \cdots & p_{1,j}& \cdots & p_{1,n_w}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ p_{i,1}& \cdots & p_{i,j}& \cdots & p_{i,n_w}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ p_{M,1}& \cdots & p_{M,j}& \cdots & p_{M,n_w}\end{array}\right]\left[\begin{array}{c}{\overline{w}}_1\\ ⋮\\ {\overline{w}}_j\\ ⋮\\ {\overline{w}}_{n_w}\end{array}\right]=\left[\begin{array}{c}\overline{F}\left(p_1\right)\\ ⋮\\ \overline{F}\left(p_i\right)\\ ⋮\\ \overline{F}\left(p_M\right)\end{array}\right]\hfill \end{array}$$

(36)

where, according to the definition of a volume integral as shown in Figure 4, the volume integral of field function f for a regular hexagonal prism is defined as follows:

$$\begin{array}{cc}\hfill \overline{F}\left(f\right)& ={\int }_{-1}^1{\int }_{-\sqrt{3}\left(x+1\right)}^{\sqrt{3}\left(x+1\right)}{\int }_{-1}^{-{\displaystyle \frac{1}{2}}}fdxdydz+{\int }_{-1}^1{\int }_{-{\displaystyle \frac{\sqrt{3}}{2}}}^{{\displaystyle \frac{\sqrt{3}}{2}}}{\int }_{-{\displaystyle \frac{1}{2}}}^{{\displaystyle \frac{1}{2}}}fdxdydz+{\int }_{-1}^1{\int }_{-\sqrt{3}\left(x-1\right)}^{\sqrt{3}\left(x-1\right)}{\int }_{{\displaystyle \frac{1}{2}}}^{1}fdxdydz\hfill \end{array}$$

(37)

Then, the integral point weighting coefficient $\overline{\mathit{w}}=\left\{{\left.{\overline{w}}_i\right|}_{i=1}^{n_w}\right\}$:

$$\overline{\mathit{w}}={\mathit{P}}^{-1}\overline{\mathit{F}}$$

(38)

When using six equilateral triangles to form a DGGS grid, the integration weight coefficients are related to the area of the units [49,81]. Considering the unit’s symmetry and the need for subsequent comparative analyses, four integration schemes ($\sum 1^u$, $\sum 1$, $\sum 3A$, and $\sum 3A+\sum 1$) are used, as shown in Figure 9. $\sum 1^u$ and $\sum 1$ correspond to linear integration schemes without and with unit symmetry, respectively, and $\sum 3A$ and $\sum 3A+\sum 1$ are both quadratic integration schemes with unit symmetry.

4. Algorithm Verification

4.1. Verification of the Tesseroid-Based Magnetic Vector Forwarding Algorithms

In a spherical coordinate system, there are no analytical solutions or explicit integration limits for DGGS cells, so the widely used tesseroid cell is used as a reference to verify the validity and reliability of the DGGS-based magnetic vector forwarding algorithm proposed in this paper. For this purpose, we first validated the tesseroid-based magnetic vector forwarding algorithm using a composite model of two tesseroids with the same depth according to Zhang et al. [82]. Inclination (45°), declination (45°), and total magnetization intensity (10 $\mathrm{A}·{\mathrm{m}}^{-1}$) were used for ease of comparison, as shown in Figure 3(a–c) of Zhang et al. [82]. One of the tesseroids occupied a 114°E to 115°E longitude, a 30.1°N to 30.9°N latitude, and a depth of 30 to 100 km, while the other occupied a 116°E to 117°E longitude, a 30.1°N to 30.9°N latitude, and a depth of 30 to 100 km. A total of 62,500 data points were computed 4 km above the lunar surface at intervals of 0.025° × 0.025° in the latitude and longitude directions. Then, we computed the corresponding magnetic field response using Equation (4) of Poisson’s formula, Equation (5) of the integral kernel formula based on the gravity gradient tensor, and Equation (10) of the triple-Gauss integral, as demonstrated in Figure 10. Comparing them with the results shown in Figure 3(a–c) of Zhang et al. [82], it is evident that the forward modeling results derived using the tesseroid-based magnetic vector forwarding algorithm are accurate.

4.2. Validating DGGS-Based Magnetic Vector Forwarding with the Tiny Tesseroid-Based Algorithm

Subsequently, the DGGS grid was generated using the software DGGRID (version 7.0) [83], which supports the Icosahedral Snyder Equal Area (ISEA) and regular icosahedral projection and can generate triangular, rhombic, pentagonal, or/and hexagonal grids [28]. Triangular and rhombic grids use a fixed aperture of 4, while pentagonal and hexagonal grids can utilize fixed apertures of 3, 4, 7, or a mixed sequence of these apertures. For simplicity, a DGGS grid with a resolution of 7 and a serial number of 2136 was selected, with its upper and lower boundaries set to 40 and 60 km, respectively, to obtain the corresponding DGGS cell, as illustrated in Figure 11, in spherical coordinates. As reference cells, the accuracy of calculations for tesseroids is influenced by the number of integration points and the KU criterion [84]. It is key to determine whether the tesseroid cell involved in the forward modeling must be segmented to improve the accuracy of the calculations, especially when the size of a tesseroid is relatively large compared to its distance from an observation point, as in this case, direct calculations may introduce significant errors. Therefore, Uieda et al. [23] proposed the following inequality:

$${\displaystyle \frac{d}{L_i}}\ge D$$

(39)

where $L_i,i\in \left(r,\varphi ,\lambda \right)$ represents the size of the “side lengths” of the tesseroid along the longitude, latitude, and radius directions, respectively [72]; d is the distance from the tesseroid to the observation point; and D is a preset threshold used to control the size of the error, which represents the “distance–size ratio” (also known as the feature size) and is a positive scalar. The choice of D depends on the desired level of accuracy. If the above equation holds, i.e., the feature size of the tesseroid relative to its distance to the observation point is sufficiently small, then it can be considered that the influence of the tesseroid on the observation point can be represented by its center point or a given approximation without the need for further segmentation. If this inequality does not hold, the tesseroid may need to be segmented into smaller parts in order to calculate its influence on the observation point more accurately.

However, irregular DGGS cells have no clear horizontal boundaries in the longitude and latitude directions. Therefore, based on the adaptive forward modeling algorithm for tesseroids proposed by Uieda et al. [23] and the tiny tesseroid forward modeling method devised by Cao et al. [67], the 3D space defined by the maximum and minimum values of the longitude, latitude, and radial extent of a DGGS cell was averaged and divided into identically sized tiny tesseroids: $2,097,152=n_x\times n_y\times n_z=128\times 128\times 128$. Tiny tesseroids whose centers were located outside the DGGS cells were discarded, while those with their centers within the DGGS cells were retained. The tesseroid-based forwarding algorithm was then used to compute the magnetic vector anomalies for the retained tiny tesseroids, and the corresponding result is shown in Figure 12. The results of the DGGS-based forwarding algorithm were computed using the shape function from Equations (31) and (32), with integral schemes $\sum 1^u$, $\sum 1$, $\sum 3A$, and $\sum 3A+\sum 1$. All the performance evaluations were carried out on a server equipped with an Intel^® Xeon^® Gold 5117 CPU and 64 GB of memory.

However, as shown in Figure 13, it can be observed that the results of the DGGS-based magnetic vector accuracy for forward modeling when the symmetry condition is not used, even with integration scheme $\sum 1^u$, are much smaller than those of the theoretical residuals (3%).

As shown in Figure 13 and Figure 14, compared to the linear triangle integration scheme $\sum 1^u$ without symmetry conditions, the calculation accuracy for the linear triangle integration scheme $\sum 1$ with symmetry conditions can be further improved because the integration point of $\sum 1$ is a superconvergence point. Furthermore, when using the quadratic triangular integration scheme $\sum 3A$ the computational accuracy is further improved, demonstrating that higher-order triangular superconvergent point integration schemes can effectively enhance the precision of the DGGS-based magnetic vector forwarding algorithm, as shown in Figure 15. However, as revealed in Figure 16, using the integration scheme $\sum 3A+\sum 1$ with more integration points did not yield better integration results, suggesting that a superconvergent integration scheme can achieve high-precision computational results using fewer integration points. Regarding the computational time, the influence of the superconvergent characteristics means that the computation times for schemes $\sum 1^u$, $\sum 1$, $\sum 3A$, and $\sum 3A+\sum 1$ do not significantly increase.

On the basis of efficiency and aesthetics, formulas that are completely symmetrical with respect to the three vertices of a triangle are preferred [59,60]. Regarding Poisson’s formula in the context of triangular grids, the grid symmetry points are almost superconvergent [77]. Therefore, two triangular integration schemes $\sum 1$ and $\sum 3A$ are used to calculate the results of the DGGS-based magnetic vector forwarding algorithm. Considering the integration schemes’ dependency on order, the number of integration points, and the point characteristics—without overemphasizing the integration accuracy—the default integration scheme $\sum 1$ is recommended.

Referring to Figures 4 and 5 in Uieda et al. [23], when the observation altitude is 10 km, the accuracy of the tesseroid-based forwarding algorithm exceeds 10% at $D\le 3$, while at an observation altitude of 260 km, the accuracy of the tesseroid-based forwarding algorithm is about 0.1%. Therefore, for Earth and lunar satellites, the observation points are located 260 and 10 km above the planetary surface, respectively. Referring to the forward modeling parameter settings in Uieda et al. [23], at different observation altitudes (260 km), we recalculated the results of the DGGS-based magnetic vector forwarding algorithm due to the DGGS cell in Figure 11 with the integral schemes $\sum 1^u$, $\sum 1$, $\sum 3A$, and $\sum 3A+\sum 1$.

As shown in Figure 17, Figure 18, Figure 19 and Figure 20, increasing the observation height—thus indirectly increasing the distance–size ratio—significantly reduces the residuals between the results of the DGGS-based (see Figure 18, Figure 19 and Figure 20) and tiny tesseroid-based forwarding algorithms, with an error percentage of ∼${\displaystyle \frac{5\times {10}^{-3}}{3}}\times 100\%=0.1667\%$.

5. Conclusions

Although tesseroid cells lack analytical solutions for gravity, their integral limits are clearly defined. Meanwhile, both of these attributes are lacking in DGGS cells. By applying isoparametric transformation, irregular DGGS cells in a global coordinate system can be converted into regular hexagonal prisms in a local coordinate system, facilitating the implementation of a DGGS-based forwarding algorithm in spherical coordinates. The integration process in this algorithm is similar to that in the tesseroid-based algorithm, which uses spherical gravity integral kernels in spherical coordinates. Consequently, the DGGS-based forwarding algorithm can also accurately fit to curvature. The tiny tesseroid-based forwarding algorithm is highly accurate, has strong adaptability, and allows the size of the tesseroid cells to be adjusted [67]. However, a large number of cells are introduced when this algorithm is used to fit complex structures, leading to a sharp increase in the computation time, so it should be used solely as a verification tool [23].

At superconvergence points, even without using symmetry conditions to determine the corresponding integration weights, the integration results maintain good accuracy, demonstrating the superconvergence characteristics at these points. Moreover, when symmetry conditions are applied to determine the integration weights at these superconvergence points, the integration accuracy is improved even further. Furthermore, employing high-order triangular superconvergence points can further enhance the precision of the integration.

The percentage error of the DGGS-based gravity forwarding algorithm is about 1% [67]. Magnetic vector forwarding uses the DGGS-based gravity algorithm, and the theoretical percentage error is expected to be 3%. In this paper, the integration characteristics of natural superconvergence points and symmetry conditions are innovatively used to reduce the residual error of the magnetic vector forwarding algorithm to less than 1%, which provides solid support for the geomagnetic field research of DGGS cells.

For numerical integration calculations, a greater number of integration points corresponds to a higher calculation accuracy, but this also results in a decrease in calculation efficiency. In this article, since the proposed two integration schemes $\sum 1$ ($n_{xy}=6$, $n_r=2$) and $\sum 3A$ ($n_{xy}=18$, $n_r=2$) have consistent integration weights, Equation (21) has the same $w_i{w}_j{w}_k$; and when the unit is not distorted, the corresponding value of $det\left(J\right)$ (which physically represents the volume of the integration unit to be solved) is the same. This greatly reduces the complexity of the integration formula, accelerating the computational capabilities of Equation (21).

Following similar principles to those for the construction of regular hexagonal prisms, regular pentagonal prisms are constructed vertically from $z=-1$ to $z=1$. Considering regular pentagonal cells with $n_{xy}$ integral points, two, three, or five layers of integral points [78,85] are arranged vertically in the direction to obtain an isoparametric element scheme for a regular pentagonal prism. Because it involves trigonometric function expansions of ${18}^{\circ }$ and ${36}^{\circ }$, the shape function expression is lengthy. Therefore, only the list of polynomials required to construct the regular pentaprism shape function is provided:

$$v={\left[\begin{array}{cccccccccc}1& x& y& z& x^2& xy& xz& yz& x^{2}z& xyz\end{array}\right]}^{\mathrm{T}}$$

(40)

The list of polynomials given above can be used by interested readers to generate the relevant shape functions for a regular pentagonal prism.

6. NOTES: Variants of the DGGS

Several variants of the DGGS are used across various fields, adding innovation to these Earth observation data models. The details are as follows:

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The Hierarchical Hexagonal Hierarchical Spatial Indexing System (H3) is a DGGS developed by Uber that is used to index geographic data into a hexagonal grid. It guarantees the uniqueness of each location and that it will not appear repeatedly in the same grid [31,86];

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Hierarchical Equal Area iso-Latitude Pixelization (HEALPix) is a program that was developed to map the cosmic microwave background. Its main purpose is to quickly and accurately count and analyze massive astrophysical data [34];

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Regularized Hierarchical Equal Area iso-Latitude Pixelization (rHEALPix) is a program that converts HEALPix into a square projection on the basis of HEALPix, which is sufficient to meet the geometric requirements of the proposed OGC DGGS standard and is helpful in the design of an index system for effectively converting latitude and longitude coordinates [35];

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ISEA is a type of azimuthal projection that was proposed by Snyder [28] which takes the 12 centers of an icosahedron as the projection centers and offers simpler operations and a faster speed for establishing the corresponding relationship between a plane and a sphere [37];

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Open Equal Area Global GRid (OpenEAGGR) models the Earth’s surface as an equal-area unit grid with a maximum resolution of 1 square centimeter. When using the icosahedron as the Earth model, it can cooperate with ISEA to meet the needs of a DGGS [32];

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DGGRID is a program developed by Sahr [83] for generating and using icosahedral discrete global grids. It includes a variety of grid types and topologies, e.g., triangular, rhombic, pentagonal, and hexagonal grids, and the subdivision and shape of the grid can be controlled by setting different topologies and apertures [33];

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Science Collaboration Environment for New Zealand Grid (SCENZ-Grid) is a program developed by the governments of Australia and New Zealand to solve problems within spatial analysis and data collection more quickly and power land protection research [36];

HEALPix and rHEALPix include both forward and inverse settings and spherical and ellipsoidal projections, and ISEA, OpenEAGGR and DGGRID are all icosahedral-based modeling methods, with DGGRID based on a DGGS running on the same discrete global mesh as the H3 system [31,34,35,86].