A new method for gravity modeling using tesseroids and 2D Gauss-Legendre quadrature rule

Zhong, Yiyuan; Ren, Zhengyong; Chen, Chaojian; Huang, Chen; Yang, Zhi; Guo, Zhenwei

doi:10.1016/j.jappgeo.2019.03.003

Cited by 20 publications

(4 citation statements)

References 32 publications

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“…The resulting surface integral is then numerically solved by using the GLQ rule (e.g., Wild-Pfeiffer 2008), or by the split quadrature method using the double exponential quadrature (DEQ) rule, resulting in the so-called DEQ method (e.g., Fukushima 2017Fukushima , 2018. Most recently, Zhong et al (2019) proposed a new method for computing the gravity field of a tesseroid, in which the original volume integral is first transformed into two surface and four edge integrals, and then the GLQ rule is adapted to evaluate these integrals. In comparison with the TSE method, the quadrature method (e.g., GLQ method) is able to provide more accurate approximations on the cost of more computational time (e.g., Lin and Denker 2019).…”

Section: Introductionmentioning

confidence: 99%

“…Fukushima (2017Fukushima ( , 2018) developed a novel method, which first computes the GP of the tesseroids at an arbitrary point with very high precision by the powerful DEQ rule and then approximates the GV and GGT by numerical partial differentiation of the computed GP. In Li et al (2011), Grombein et al (2013), Uieda et al (2016), Lin and Denker (2019), Soler et al (2019), andZhong et al (2019), the improvement of the approximation is achieved by regularly or adaptively subdividing the tesseroids close to the computation point into smaller tesseroid elements along both horizontal and vertical dimensions or only in the horizontal dimension first, and then summing all effects of the subdivided tesseroid elements computed by the GLQ rule. Among the above-mentioned approaches, the last one is easy to implement and further requires no elementary body conversion, flat Earth approximation, coordinate transformation, tesseroid rotation, and numerical differentiation.…”

Section: Introductionmentioning

confidence: 99%

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Gravity Field Modeling Using Tesseroids with Variable Density in the Vertical Direction

2020

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We present an accurate method for the calculation of gravitational potential (GP), vector (GV), and gradient tensor (GGT) of a tesseroid, considering a density model in the form of a polynomial up to cubic order along the vertical direction. The method solves volume integral equations for the gravitational effects due to a tesseroid by the Gauss–Legendre quadrature rule. A two-dimensional adaptive subdivision technique, which automatically divides the tesseroids near the computation point into smaller elements, is applied to improve the computational accuracy. For those tesseroids having small vertical dimensions, an extension technique is additionally utilized to ensure acceptable accuracy, in particular for the evaluation of GV and GGT. Numerical experiments based on spherical shell models, for which analytical solutions exist, are implemented to test the accuracy of the method. The results demonstrate that the new method is capable of computing the gravitational effects of the tesseroids with various horizontal and vertical dimensions as well as density models, while the evaluation point can be on the surface of, near the surface of, outside the tesseroid, or even inside it (only suited for GP and GV). Thus, the method is attractive for many geodetic and geophysical applications on regional and global scales, including the computation of atmospheric effects for terrestrial and satellite usage. Finally, we apply this method for computing the topographic effects in the Himalaya region based on a given digital terrain model and the global atmospheric effects on the Earth’s surface by using three polynomial density models which are derived from the US Standard Atmosphere 1976.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gravity Field Modeling Using Tesseroids with Variable Density in the Vertical Direction

2020

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“…Only five components of the gravity gradient tensor are independent since the tensor components are symmetric and T rr + T θθ + T φφ = 0. As analytical solutions are unavailable, we use Gauss‐Legendre quadrature to evaluate the integrals in Equations 1–3 in the forward modeling (Asgharzadeh et al., 2007; Li et al., 2011; Lin et al., 2020; Uieda et al., 2016; Zhong et al., 2019).…”

Section: Methodsmentioning

confidence: 99%

Constrained Gravity Inversion With Adaptive Inversion Grid Refinement in Spherical Coordinates and Its Application to Mantle Structure Beneath Tibetan Plateau

et al. 2022

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We develop a novel gravity inversion algorithm in spherical coordinates based on adaptive inversion mesh refinement and multiphysical parameter constraints. The inversion mesh is discretized into tesseroids (spherical prisms) to take the curvature of the Earth into account. To reduce the number of unknowns and computational cost, the inversion mesh is adaptively refined according to the spatial variation of parameters at each iteration. Wavelet compression is used to further reduce the computational requirement. Besides, to alleviate the nonuniqueness of inversion, the algorithm is capable of incorporating a priori models from other geophysical methods via cross‐gradient coupling and/or direct parameter coupling. For cross‐gradient coupling, the vector product of spatial gradients of two model parameters is included in the objective function. For direct parameter coupling, an a priori model is converted to a density model using the relation between two physical properties, which is then used as the initial and reference model in the inversion. A synthetic example is presented to demonstrate the effectiveness of our inversion method. Finally, we invert the gravity data over the Tibetan Plateau to obtain the density variations in the upper mantle, with a global S wave tomographic model used as a constraint. The inversion model shows that the mantle lithosphere beneath the Himalayan collision zone varies from west to east. Low‐density anomalies are observed beneath the southern and the northern Tibetan Plateau, which are consistent with published low‐velocity anomalies and magmatism distributions, possibly indicating two asthenospheric upwellings.

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“…The relative errors in log 10 scale of the GGT and GC are large than 0 below the height of about 24 km and 50 km, respectively. The key to solving the near-zone problem of the GC of the tesseroid lies in the improvement of the numerical algorithm to calculate the triple or double integrals and the selection of the geometrical shape of the tesseroid mass body, e.g., the rotation method (Marotta and Barzaghi 2017;Marotta et al 2019), splitting line method using the double exponential quadrature (Fukushima 2018a), different types of the regular, adaptive and combined subdivision (Li et al 2011;Grombein et al 2013;Shen and Deng 2016;Uieda et al 2016;Deng and Shen 2019;Lin and Denker 2019;Soler et al 2019;Zhong et al 2019;Zhao et al 2019;Lin et al 2020;Chen 2020, 2021).…”

Section: Conclusion and Outlooksmentioning

confidence: 99%

Evaluation of gravitational curvatures for a tesseroid and spherical shell with arbitrary-order polynomial density

Deng

2023

J Geod

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In recent years, there are research trends from constant to variable density and low-order to high-order gravitational potential gradients in gravity field modeling. Under the research circumstances, this paper focuses on the variable density model for gravitational curvatures (or gravity curvatures, third-order derivatives of gravitational potential) of a tesseroid and spherical shell in the spatial domain of gravity field modeling. In this contribution, the general formula of the gravitational curvatures of a tesseroid with arbitrary-order polynomial density is derived. The general expressions for gravitational effects up to the gravitational curvatures of a spherical shell with arbitrary-order polynomial density are derived when the computation point is located above, inside, and below the spherical shell. When the computation point is located above the spherical shell, the general expressions for the mass of a spherical shell and the relation between the radial gravitational effects up to arbitrary-order and the mass of a spherical shell with arbitrary-order polynomial density are derived. The influence of the computation point’s height and latitude on gravitational curvatures with the polynomial density up to fourth-order is numerically investigated using tesseroids to discretize a spherical shell. Numerical results reveal that the near-zone problem exists for the fourth-order polynomial density of the gravitational curvatures, i.e., relative errors in $$\log _{10}$$ log 10 scale of gravitational curvatures are large than 0 below the height of about 50 km by a grid size of $$15'\times 15'$$ 15 ′ × 15 ′ . The polar-singularity problem does not occur for the gravitational curvatures with polynomial density up to fourth-order because of the Cartesian integral kernels of the tesseroid. The density variation can be revealed in the absolute errors as the superposition effects of Laplace parameters of gravitational curvatures other than the relative errors. The derived expressions are examples of the high-order gravitational potential gradients of the mass body with variable density in the spatial domain, which will provide the theoretical basis for future applications of gravity field modeling in geodesy and geophysics.

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A new method for gravity modeling using tesseroids and 2D Gauss-Legendre quadrature rule

Cited by 20 publications

References 32 publications

Gravity Field Modeling Using Tesseroids with Variable Density in the Vertical Direction

Gravity Field Modeling Using Tesseroids with Variable Density in the Vertical Direction

Constrained Gravity Inversion With Adaptive Inversion Grid Refinement in Spherical Coordinates and Its Application to Mantle Structure Beneath Tibetan Plateau

Evaluation of gravitational curvatures for a tesseroid and spherical shell with arbitrary-order polynomial density

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