Evaluation of the fourth-order tesseroid formula and new combination approach to precisely determine gravitational potential

Shen, Wenbin; Deng, Xiao-Le

doi:10.1007/s11200-016-0402-y

Cited by 25 publications

(7 citation statements)

References 40 publications

(88 reference statements)

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“…There are two kinds of numerical approaches, which have been investigated extensively for the evaluation of tesseroids, namely the Taylor series expansion (TSE) method and the quadrature method. For the TSE method, Newton's integral is solved by expanding the integral kernel in a Taylor series expansion up to a certain degree, followed by an integration of the subsequent volume integrals (e.g., Heck and Seitz 2007;Wild-Pfeiffer 2008;Grombein et al 2013;Shen and Deng 2016). The TSE method is fast and can produce accurate results at low latitudes.…”

Section: Introductionmentioning

confidence: 99%

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%

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

2020

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“…The near-zone problem (or very-near-area problem) of the computation point for the tesseroid mass body (i.e., there are large approximation errors when the computation point approaches the surface of the tesseroid mass body) has been investigated by using tesseroids to discretize the spherical shell for the GP, GV, GGT (Uieda et al 2016;Shen and Deng 2016), GC Shen 2018a, b, 2019), and invariants of the gravity gradient tensor and their first-order derivatives (Deng et al 2021) with the constant density. Based on the same numerical situation to investigate the influence of the geocentric distance on the GC (Deng and Shen 2018a, Sect.3.3), we extend the constant density to the polynomial density up to fourth-order to reveal the density variation on the effects of the GC using the relative and absolute errors.…”

Section: Influence Of Height On the Gc With Polynomial Density Up To ...mentioning

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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“…Step02: To estimate next modification of the density of each Tesseroid cell, the gravity anomaly matrix g cal,0,q can be calculated from density matrix φ 0,(i,j) by (16). The deviation of gravity anomaly is δg 1 = g obs(q) − g cal,0,q , and the new modified density of each Tesseroid φ 1,(i,j) can be generated as:…”

Section: Apparent Density Mapping In Spherical Coordinatesmentioning

confidence: 99%

“…Step03: Renew the gravity anomaly matrix g cal,k,q at kth iteration of the density model matrix φ k,(i,j) by (16), and then renew the deviation of gravity anomaly δg k+1 = g obs(q) − g cal,k,q . We can obtain the density of each Tesseroid cell φ k+1,(i,j) by (20).…”

Section: Apparent Density Mapping In Spherical Coordinatesmentioning

confidence: 99%

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The Apparent Density Mapping Approach in Spherical Coordinates and the Crustal Density Distribution of Chinese Mainland

2019

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Apparent density mapping is a technique to obtain lateral density distribution of subsurface layer by inverting gravity anomaly. In general, the subsurface layer can be divided into vertical, juxtaposed prisms in Cartesian coordinates. However, for continental and global scales, the curvature of the earth should be taken into account, and the subsurface layer should be divided into Tesseroids in spherical coordinates instead of rectangular prisms. In this paper, we present a modified Gauss-Legendre algorithm to forward the gravity anomaly generated by an arbitrary Tesseroid, and the precision can be improved vastly when the observation points in the near surface of the Tesseroid. The shell tests show that the relative maximum error can be controlled in 0.0009343% when the Gauss-Legendre nodes are set as (4, 4) in the longitude and latitude directions and the subdivision parameter W is set as 1.5. Then, an apparent density mapping approach is presented in spherical coordinates to minimize the difference between the observed gravity anomalies and calculated gravity anomalies. The synthetic model verified the feasibility and precision of our approach. In real application, we have obtained the crustal apparent density distribution of Chinese mainland with 0.25 • ×0.25 • in longitude and latitude directions. The crustal apparent distribution of Chinese mainland is generally low in the western region and high in eastern region varying from 2.45-2.81 g/cm 3 and closely related to lithologic units and geological boundaries. INDEX TERMS Gravity inversion, apparent density mapping, spherical coordinates, Chinese mainland.

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Evaluation of the fourth-order tesseroid formula and new combination approach to precisely determine gravitational potential

Cited by 25 publications

References 40 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

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

The Apparent Density Mapping Approach in Spherical Coordinates and the Crustal Density Distribution of Chinese Mainland

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