Spherical integral transforms of second-order gravitational tensor components onto third-order gravitational tensor components

Šprlák, Michal; Novák, Pavel

doi:10.1007/s00190-016-0951-4

Cited by 13 publications

(3 citation statements)

References 46 publications

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“…With the first measurement of radial gravitational curvatures (or gravity curvatures, GC, third-order derivatives of the gravitational potential) using the atom interferometry sensors in the laboratory condition (Rosi et al 2015), the subsequent study of the GC was focused on their theory and simulation applications in geodesy and geophysics (Šprlák and Novák 2015(Šprlák and Novák , 2017Hamáčková et al 2016;Sharifi et al 2017;Novák et al 2017;Pitoňák et al 2017Pitoňák et al , 2018Pitoňák et al , 2020Shen 2018a, b, 2019;Novák et al 2019;Du and Qiu 2019;Romeshkani et al 2020Romeshkani et al , 2021. The GC were more sensitive to field sources than the loworder gravitational effects (i.e., GP, GV, and GGT) (Heck 1984;Du and Qiu 2019).…”

Section: Introductionmentioning

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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Section: Introductionmentioning

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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“…Some previous efforts have solved the singularity problem of SHEs of gravitational field elements (Balmino et al., 1990; Bettadpur, 1995; Casotto & Fantino, 2007; Eshagh, 2008, 2009; Eshagh & Sjöberg, 2009; Hotine & Morrison, 1969; Ilk, 1983; Liu et al., 2010, 2013; Petrovskaya & Vershkov, 2006, 2007, 2008; Wan, 2011; Zhu et al., 2017), whereas the same problem still exists in GVs and GGTs. The reason is that the gravitational field elements are expressed by fully normalized associated Legendre function (FNALF) (Chen et al., 2006; Fantino & Casotto, 2009; Fukushima, 2012a, 2012b; Hirt et al., 2010; Jekeli & Lee, 2007; Liu et al., 2012; Pail et al., 2011; Pavlis et al., 2012; Rummel et al., 2011; Wan & Yu, 2013; Šprlák & Novák, 2017), while the GVs and GGTs are expressed by Schmidt semi‐normalized associated Legendre function (SNALF) (Barraclough, 1974; Benton et al., 1982; Blakely, 1995; Chambodut et al., 2005; Du et al., 2015; Hemant & Maus, 2005; Huang et al., 2011; Kim et al., 2007; Kotsiaros & Olsen, 2012; Langel, 1987; Liu et al., 2019; Malin & Pocock, 1969; Quinn et al., 1986; Ravat et al., 1995; Shao et al., 1999; Wardinski & Holme, 2006), whose recursive formulae have a constant difference from those of FNALF. Although the calculation of GVs can be transformed into the non‐singular formulae developed in gravimetry, the formulae of GGTs contain the linear combinations of Legendre functions and their first‐ or second‐order derivatives, which cannot be converted simply by multiplying a constant.…”

Section: Introductionmentioning

confidence: 99%

“…Some previous efforts have solved the singularity problem of SHEs of gravitational field elements (Balmino et al, 1990;Bettadpur, 1995;Casotto & Fantino, 2007;Eshagh, 2008Eshagh, , 2009Eshagh & Sjöberg, 2009;Hotine & Morrison, 1969;Ilk, 1983;Liu et al, 2010Liu et al, , 2013Petrovskaya & Vershkov, 2006, 2008Wan, 2011;Zhu et al, 2017), whereas the same problem still exists in GVs and GGTs. The reason is that the gravitational field elements are expressed by fully normalized associated Legendre function (FNALF) (Chen et al, 2006;Fantino & Casotto, 2009;Fukushima, 2012aFukushima, , 2012bHirt et al, 2010;Jekeli & Lee, 2007;Liu et al, 2012;Pail et al, 2011;Pavlis et al, 2012;Rummel et al, 2011;Wan & Yu, 2013;Šprlák & Novák, 2017), while the GVs and GGTs are expressed by Schmidt semi-normalized associated Legendre function (SNALF) (Barraclough, 1974;Benton et al, 1982;Blakely, 1995;Chambodut et al, 2005;Du et al, 2015;Hemant & Maus, 2005;Huang et al, 2011;Kim et al, 2007;Kotsiaros & Olsen, 2012;Langel, 1987;Liu et al, 2019;Malin & Pocock, 1969;Quinn et ...…”

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confidence: 99%

Non‐Singular Calculation of Geomagnetic Vectors and Geomagnetic Gradient Tensors

Liu

Sun

et al. 2021

JGR Solid Earth

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Some spherical harmonic expressions of gravitational and geomagnetic field elements will become infinite when computation point approaches polar regions, as the sine function of the geocentric co‐latitude contained in the denominator tends to be zero. Currently, this singularity problem has been solved for gravitational field case, however, it remains unsolved for geomagnetic vectors (GVs) and geomagnetic gradient tensors (GGTs). Because the latter use Schmidt semi‐normalized associated Legendre function (SNALF), which is different from fully‐normalized associated Legendre function used in the former. To overcome this singularity problem, we derive new non‐singular expressions of the first‐ and second‐order derivatives of Schmidt SNALF (order m equals to 0 or 1), and the corresponding two kinds of spherical harmonic polynomials. When the non‐singular expressions are applied to the traditional formulae of GVs and GGTs, more practical expressions of GVs and GGTs with non‐singularity are formulated by refining the cases that the order m equals to 0, 1, 2 and other values. Furthermore, to provide flexible calculation strategies for Schmidt SNALF, we derive four kinds of recursive formulae, including the standard forward row recursion, the standard forward column recursion, the cross degree and order recursion, and the Belikov recursion. The standard formula to check the calculation accuracy of various recursive formulae of SNALF are also given. Besides, we demonstrate the effectiveness and reliability of the new derived non‐singular expressions of GVs and GGTs and analyze the computation speed and stability of the four recursive formulae by extensive numerical experiments.

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Non-singular calculation of geomagnetic vectors and geomagnetic gradient tensors

Liu¹,

Sun³

et al. 2021

Preprint

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Spherical integral transforms of second-order gravitational tensor components onto third-order gravitational tensor components

Cited by 13 publications

References 46 publications

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

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

Non‐Singular Calculation of Geomagnetic Vectors and Geomagnetic Gradient Tensors

Non-singular calculation of geomagnetic vectors and geomagnetic gradient tensors

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