Analytic Expressions for the Gravity Gradient Tensor of 3D Prisms with Depth-Dependent Density

Jiang, Li; Liu, Jie; Zhang, Jianzhong; Feng, Ziqi

doi:10.1007/s10712-017-9455-x

Cited by 18 publications

(3 citation statements)

References 87 publications

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“…These almost negligible residuals verify the high accuracy of our new formula for the case of varying density contrasts. However, happening to almost all analytical formulae [Holstein, 2003;Ren et al, 2018a;Jiang et al, 2018;Chen et al, 2018], phenomenon of numerical instability still exist in our new formula when the distance between the observation site and the polyhedral mass body beyond a critical level. This phenomenon is caused by the limited machine precision to present the real number in the calculation, which can be resolved by using longer bits such as 128 bits to represent real numbers or equivalent real number representation methods.…”

Section: Theory and New Analytic Formulaementioning

confidence: 95%

Exact gravity field for polyhedrons with polynomial density contrasts of arbitrary orders

Ren¹,

Chen²,

Zhong³

et al. 2018

Preprint

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Section: Theory and New Analytic Formulaementioning

confidence: 95%

Exact gravity field for polyhedrons with polynomial density contrasts of arbitrary orders

Ren¹,

Chen²,

Zhong³

et al. 2018

Preprint

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“…For example, the density from the surface to the inner core of the Earth varied from 1020.0 to 13088.5 kg m −3 in the preliminary reference Earth model (Dziewonski and Anderson 1981) and from 2651.0 to 13012.2 kg m −3 in the radial ek137 model (Kennett 2020). The density hypothesis and digital models (e.g., CRUST1.0 (Laske et al 2013) and UNB_TopoDens (Sheng et al 2019)) arbitrary-order polynomial density (Zhang and Jiang 2017), GV and GGT in the Fourier domain with the depth-dependent polynomial density (Wu and Chen 2016;Wu 2018), GGT with the depth-dependent density (Jiang et al 2018), and GP, GV, and GGT with the depth-dependent nth-order polynomial density (Karcol 2018;Fukushima 2018b). Similarly, the gravitational effects of a spherical shell were derived in the form of the polynomial density, i.e., the GP and GV with the radial fifth-order polynomial density (Karcol 2011) and the GP, GV, and GGT with the linear, quadratic, and cubic order polynomial density (Lin et al 2020).…”

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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“…To derive closed-form solutions of gravity signals, only the underground mass anomaly needs to be discretized into disjoint elements. These elements can have different geometric shapes, such as circular disks and cylinders (Singh 1977;Krogh et al 1982;Damiata & Lee 2002;Rim & Li 2016;Asgharzadeh et al 2018), prisms and polyhedra (Li & Chouteau 1998;Wu & Chen 2016;Wu 2016Wu , 2018Jiang et al 2018;Ren et al 2018b). The prismatic body has been the focus of interest for an extended period of time due to its simplicity and high effectiveness.…”

Section: Introductionmentioning

confidence: 99%

Exact solutions of the vertical gravitational anomaly for a polyhedral prism with vertical polynomial density contrast of arbitrary orders

Chen

Ren

Pan

et al. 2018

Geophysical Journal International

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We present general closed-form solutions for the vertical gravitational anomaly caused by a polyhedral prism with mass density contrast varying with depth. Our equations are the first ones to implement a polynomial vertical mass density contrast of arbitrary order. Singularities in the gravity field which arise when the observation site is close to or in the anomalous polyhedral prism are removed in our analytic expressions. Therefore, the observation site can be located outside, on the faces of or inside the anomalous mass bodies. A simple prismatic body of anomalous density is adopted to test the accuracy of our newly developed closedform solution. Cases of constant, linear, quadratic, cubic and quartic polynomial orders of mass density contrast are tested. For cases of constant, linear, quadratic and cubic polynomial orders, the relative errors between our results and other published exact solutions are less than 10 −11 %. For the case of quartic polynomial order, relative errors less than 10 −10 % are obtained between our solutions and those computed by a high-order Gaussian quadrature rule (512 × 512 ×512 = 134 217 728 quadrature points), where our new analytic solution needs significantly less computational time (0.0009 versus 31.106 s). These numerical experiments not only verified the accuracy of our new formula but also demonstrated their potential in computing exact gravity anomalies for complicated mass density distributions in the Earth.

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Analytic Expressions for the Gravity Gradient Tensor of 3D Prisms with Depth-Dependent Density

Cited by 18 publications

References 87 publications

Exact gravity field for polyhedrons with polynomial density contrasts of arbitrary orders

Exact gravity field for polyhedrons with polynomial density contrasts of arbitrary orders

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

Exact solutions of the vertical gravitational anomaly for a polyhedral prism with vertical polynomial density contrast of arbitrary orders

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