Corrections to “A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling” (Heck and Seitz, 2007) and “Optimized formulas for the gravitational field of a tesseroid” (Grombein et al., 2013)

Deng, Xiao-Le; Grombein, Thomas; Shen, Wenbin; Heck, Bernhard; Seitz, Kurt

doi:10.1007/s00190-016-0907-8

Cited by 24 publications

(4 citation statements)

References 7 publications

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“…The tesseroid method for modeling the GP is surveyed in Section 2, and the series expansion of Heck and Seitz (2007) is also provided therein. We note that the general Taylor series expansion given by Heck and Seitz (2007) as well as that formula cited by Grombein et al (2013) from which the gravitational attraction of a tesseroid is derived, is incorrect; however the second-order formula is correct, as it was pointed out by Deng et al (2016). A correct general Taylor series expansion expression of tesseroid formula is provided in Section 2.…”

Section: Introductionmentioning

confidence: 88%

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

Shen

Deng

2016

Stud Geophys Geod

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Calculating topographic gravitational potential (GP) is a time-consuming process in terms of efficiency. Prism, mass-point, mass-line, and tesseroid formulas are generally used to calculate the topographic GP effect. In this study, we reformulate the higher-order formula of the tesseroid by Taylor series expansion and then evaluate the fourth-order formula by numerical tests. Different simulation computations show that the fourth-order formula is reliable. Using the conventional approach in numerical calculations, the approximation errors in the areas of the north and south poles are extremely large. Thus, in this study we propose an approach combining the precise numerical formula and tesseroid formulas, which can satisfactorily solve the calculation problem when the computation point is located in the polar areas or areas very near the surface. Furthermore, we suggest a "best matching choice" of new combination approach to calculate the GP precisely by conducting various experiments. Given the computation point at different positions, we may use different strategies. In the low latitude, we use a precise numerical formula, the fourth-order tesseroid formula, the second-order tesseroid formula, and the zero-order formula, in the 1° range (from the computation point), 1° to 15° range, 15° to 40° range, and the range outside 40°, respectively. The accuracy can reach 2 × 10 5 m 2 s 2 . For the high latitude, we use the precise numerical formula, fourth-order tesseroid, second-order tesseroid, and zero-order tesseroid formulas in the ranges of 0° to 1°, 1° to 10°, 10° to 30°, and the zones outside 30°, respectively. However, if an accuracy level of 2 × 10 5 m 2 s 2 is required, the zero-order tesseroid formulas should not be used and the second-order tesseroid formula should be used in the region outside 15° for the low latitude and in the region outside 10° for the high latitude. K e y w o r d s : Newton's integral, gravitational potential, tesseroid, new combination method W.B. Shen and X.L. Deng ii Stud. Geophys. Geod., 60 (2016)

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Section: Introductionmentioning

confidence: 88%

“…i j k . Fortunately, various international authors (e.g., Wild-Pfeiffer, 2008;Tsoulis et al, 2009;Chaves and Ussami, 2013;Shen and Han, 2013;Du et al, 2015) quoted only the second-order expression given by Heck and Seitz (2007), which is correct because of (Deng et al, 2016).…”

Section: Formulas For Gravitational Potential Generated By a Tesseroidmentioning

confidence: 95%

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

Shen

Deng

2016

Stud Geophys Geod

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“…However, the Newton's integral in spherical coordinates cannot be solved analytically except for the case of a homogeneous spherical shell and a spherical cap along the polar axis (Grombein et al, ; Mikuska et al, ). Numerical integration methods have been used to solve this problem, including the 2‐D, 3‐D Gauss‐Legendre quadrature (GLQ) (Asgharzadeh et al, ; Wild‐Pfeiffer, ), the Taylor series expansion (Deng et al, ; Grombein et al, ; Heck & Seitz, ), and approximation of tesseroids by other forms, which induce nearly the same gravity field but which effect can be easily computed (e.g., Kaban et al, ; Kaban, El Khrepy, & Al‐Arifi, ). Also, there are widely used direct methods, which operate in the spherical harmonic domain providing sufficient accuracy of the computed geoid, gravity field, and its derivatives (e.g., Root et al, ).…”

Section: Introductionmentioning

confidence: 99%

Efficient 3‐D Large‐Scale Forward Modeling and Inversion of Gravitational Fields in Spherical Coordinates With Application to Lunar Mascons

et al. 2019

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An efficient forward modeling algorithm for calculation of gravitational fields in spherical coordinates is developed for 3-D large-scale gravity inversion problems. The 3-D Gauss-Legendre quadrature (GLQ) is used to calculate the gravitational fields of mass distributions discretized into tesseroids. Equivalence relations in the kernel matrix of the forward modeling are exploited to decrease storage and computation time. The numerical tests demonstrate that the computation time of the proposed algorithm is reduced by approximately 2 orders of magnitude, and the memory requirement is reduced by N′ λ times compared with the traditional GLQ method, where N′ λ is the number of the model elements in the longitudinal direction. These significant improvements in computational efficiency and storage make it possible to calculate and store the dense Jacobian matrix in 3-D large-scale gravity inversions. The equivalence relations can be applied to the Taylor series method or combined with the adaptive discretization to ensure high accuracy. To further illustrate the capability of the algorithm, we present a regional synthetic example. The inverted results show density distributions consistent with the actual model. The computation took about 6.3 hr and 0.88 GB of memory compared with about a dozen days and 245.86 GB for the traditional 3-D GLQ method. Finally, the proposed algorithm is applied to the gravity field derived from the latest lunar gravity model GL1500E. Three-dimensional density distributions of the Imbrium and Serenitatis basins are obtained, and high-density bodies are found at the depths 10-60 km, likely indicating a significant uplift of the high-density mantle beneath the two mascon basins.

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“…However, the Newton's integral in spherical coordinates cannot be solved analytically except for the case of a homogeneous spherical shell and a spherical cap along the polar axis [Grombein et al, 2013;Mikuska et al, 2006]. Numerical integration methods have been used to solve this problem, including the 2D, 3D Gauss-Legendre quadrature (GLQ) [Asgharzadeh et al, 2007;Wild-Pfeiffer, 2008], the Taylor-series expansion [Deng et al, 2016;Grombein et al, 2013;Heck and Seitz, 2007] and approximation of tesseroids by other forms, which induce nearly the same gravity field, but which effect can be easily computed (e.g., Kaban et al, 2004Kaban et al, , 2016a. Also, there are widely used direct methods, which operate in the spherical harmonic domain providing sufficient accuracy of the computed geoid, gravity field and its derivatives (e.g., Root et al, 2016).…”

Section: Introductionmentioning

confidence: 99%

Efficient 3D large-scale forward-modeling and inversion of gravitational fields in spherical coordinates with application to lunar mascons

Zhao¹,

Chen²,

Uieda³

et al. 2019

Preprint

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An efficient forward modeling algorithm for calculation of gravitational fields in spherical coordinates is developed for 3D large‐scale gravity inversion problems. 3D Gauss‐Legendre quadrature (GLQ) is used to calculate the gravitational fields of mass distributions discretized into tesseroids. Equivalence relations in the kernel matrix of the forward‐modeling are exploited to decrease storage and computation time. The numerical tests demonstrate that the computation time of the proposed algorithm is reduced by approximately two orders of magnitude, and the memory requirement is reduced by N'λ times compared with the traditional GLQ method, where N'λ is the number of the model elements in the longitudinal direction. These significant improvements in computational efficiency and storage make it possible to calculate and store the dense Jacobian matrix in 3D large‐scale gravity inversions. The equivalence relations can be applied to the Taylor series method or combined with the adaptive discretization to ensure high accuracy. To further illustrate the capability of the algorithm, we present a regional synthetic example. The inverted results show density distributions consistent with the actual model. The computation took about 6.3 hours and 0.88 GB of memory compared with about a dozen days and 245.86 GB for the traditional 3D GLQ method. Finally, the proposed algorithm is applied to the gravity field derived from the latest lunar gravity model GL1500E. 3D density distributions of the Imbrium and Serenitatis basins are obtained, and high‐density bodies are found at the depths 10‐60 km, likely indicating a significant uplift of the high‐density mantle beneath the two mascon basins.

show abstract

Corrections to “A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling” (Heck and Seitz, 2007) and “Optimized formulas for the gravitational field of a tesseroid” (Grombein et al., 2013)

Cited by 24 publications

References 7 publications

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

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

Efficient 3‐D Large‐Scale Forward Modeling and Inversion of Gravitational Fields in Spherical Coordinates With Application to Lunar Mascons

Efficient 3D large-scale forward-modeling and inversion of gravitational fields in spherical coordinates with application to lunar mascons

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