Spherical prism gravity effects by Gauss-Legendre quadrature integration

Asgharzadeh, M. F.; Frese, Ralph R. B. von; Kim, H. R.; Leftwich, T. E.; Kim, J. W.

doi:10.1111/j.1365-246x.2007.03214.x

Cited by 97 publications

(83 citation statements)

References 30 publications

(45 reference statements)

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“…While Heck and Seitz (2007) originally derived formulas for the tesseroid potential and the first radial derivative, Wild-Pfeiffer (2007, 2008 extended the approach to all components of first-and second-order derivatives. Furthermore, also Gauss-Legendre cubature can be applied as proposed by Asgharzadeh et al (2007) and WildPfeiffer (2007WildPfeiffer ( , 2008. For global computations, another alternative consists of analytically solving the one-dimensional integral with respect to the geocentric distance r and calculating the remaining two-dimensional surface integral numerically (cf.…”

Section: Introductionmentioning

confidence: 99%

Optimized formulas for the gravitational field of a tesseroid

Grombein

Seitz

Heck

2013

J Geod

138

114

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Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton's integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid's potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian inte-T. Grombein · K. Seitz · B. Heck Geodetic Institute, Karlsruhe Institute of Technology (KIT) Englerstr. 7, Karlsruhe, D-76128 Germany Tel.: +49-721-608-42306 Fax: +49-721-608-46808 E-mail: grombein@kit.edu gral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45% compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.

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Section: Introductionmentioning

confidence: 99%

Optimized formulas for the gravitational field of a tesseroid

Grombein

Seitz

Heck

2013

J Geod

138

114

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“…According to the GLQI decomposition (Stroud & Secrest 1966;Ku 1977;von Frese et al 1981a;Asgharzadeh et al 2007Asgharzadeh et al , 2008Wild-Pfeiffer 2008;Grombein et al 2010;Li et al 2011), the integrals in eqs (22)- (24) have the least squares numerical solutions as…”

Section: -D Gauss-legendre Quadrature Integrationmentioning

confidence: 99%

“…While Heck & Seitz (2007) originally derived formulas for the tesseroid potential and the first radial derivative, and Wild-Pfeiffer (2007, 2008 extended the approach to all components of first-and second-order derivatives. Furthermore, also Gauss-Legendre cubature is applied to gravity and its gradient fields proposed by Asgharzadeh et al (2007) and Wild-Pfeiffer (2007, 2008. For global computations, another alternative consists of analytically solving the 1-D integral with respect to the geocentric distance and calculating the remaining 2-D surface integral numerically (e.g.…”

Section: Introductionmentioning

confidence: 99%

Magnetic potential, vector and gradient tensor fields of a tesseroid in a geocentric spherical coordinate system

Du¹,

Chen²,

Lesur³

et al. 2015

Geophysical Journal International

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S U M M A R YWe examined the mathematical and computational aspects of the magnetic potential, vector and gradient tensor fields of a tesseroid in a geocentric spherical coordinate system (SCS). This work is relevant for 3-D modelling that is performed with lithospheric vertical scales and global, continent or large regional horizontal scales. The curvature of the Earth is significant at these scales and hence, a SCS is more appropriate than the usual Cartesian coordinate system (CCS). The 3-D arrays of spherical prisms (SP; 'tesseroids') can be used to model the response of volumes with variable magnetic properties. Analytical solutions do not exist for these model elements and numerical or mixed numerical and analytical solutions must be employed. We compared various methods for calculating the response in terms of accuracy and computational efficiency. The methods were (1) the spherical coordinate magnetic dipole method (MD), (2) variants of the 3-D Gauss-Legendre quadrature integration method (3-D GLQI) with (i) different numbers of nodes in each of the three directions, and (ii) models where we subdivided each SP into a number of smaller tesseroid volume elements, (3) a procedure that we term revised Gauss-Legendre quadrature integration (3-D RGLQI) where the magnetization direction which is constant in a SCS is assumed to be constant in a CCS and equal to the direction at the geometric centre of each tesseroid, (4) the Taylor's series expansion method (TSE) and (5) the rectangular prism method (RP). In any realistic application, both the accuracy and the computational efficiency factors must be considered to determine the optimum approach to employ. In all instances, accuracy improves with increasing distance from the source. It is higher in the percentage terms for potential than the vector or tensor response. The tensor errors are the largest, but they decrease more quickly with distance from the source. In our comparisons of relative computational efficiency, we found that the magnetic potential takes less time to compute than the vector response, which in turn takes less time to compute than the tensor gradient response. The MD method takes less time to compute than either the TSE or RP methods. The efficiency of the (GLQI and) RGLQI methods depends on the polynomial order, but the response typically takes longer to compute than it does for the other methods. The optimum method is a complex function of the desired accuracy, the size of the volume elements, the element latitude and the distance between the source and the observation. For a model of global extent with typical model element size (e.g. 1 degree horizontally and 10 km radially) and observations at altitudes of 10s to 100s of km, a mixture of methods based on the horizontal separation of the source and observation separation would be the optimum approach. To demonstrate the RGLQI method described within this paper, we applied it to the computation of the response for a global magnetization model for observations at 300 and 30 km altitude. I...

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“…Similarly, d 2 , m 2 , and G 2 denote the magnetic data, susceptibility model, and kernel matrix. There is no closed-form solution for the field due to a tesseroid cell, so we adopt the Gauss-Legendre quadrature integration to calculate the entries of the kernel matrices of both gravity gradient and magnetic anomaly (e.g., Asgharzadeh et al, 2007).…”

Section: Inversion In Spherical Coordinatesmentioning

confidence: 99%