A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling

Heck, Bernhard; Seitz, Kurt

doi:10.1007/s00190-006-0094-0

Cited by 230 publications

(200 citation statements)

References 25 publications

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“…Alternatively, in the geocentric SCS, the SP or so-called the tesseroid (e.g. Anderson 1976;Heck & Seitz 2007;Fig. 1), is chosen as basic element for forward modelling, which is bounded by (i) a pair of surfaces with the constant spherical radius r 1 = const, r 2 = const, that are 'parallel' to the reference sphere; (ii) a pair of meridional planes with λ 1 = const, λ 2 = const and (iii) a pair of coaxial circular cones (ϕ 1 = const, ϕ 2 = const).…”

Section: Tesseroid and Coordinate Systemsmentioning

confidence: 99%

“…If considering the Earth's ellipticity, the radii r 1 and r 2 of the lower surface and the upper surface can be replaced, roughly by the surfaces with geocentric latitudedependent Earth's radii of the reference ellipsoid in the geocentric SCS, or more preferably by the surfaces with geographic latitude-dependent Earth's radii in the geographic SCS (e.g. Heck & Seitz 2007) where some extra transformations for the coordinates and the magnetic effects are required. Providing the magnetic effects are expressed in the ellipsoidal local CCS, the magnetic vector and its tensor should be rotated on the y P axis further because of the difference between the geocentric latitude and the geographic latitude.…”

Section: Basic Formulasmentioning

confidence: 99%

“…Equivalently to Heck & Seitz (2007) and Grombein et al (2013), we integrate the TSE of the Cartesian integral kernels in eqs (13), (15) and (16) at the tesseroid's geometrical centre point Q 0 (r 0 , ϕ 0 , λ 0 ), where the magnetization vector is treated as a constant vector. Here because the terms of odd order and mixed terms in the Taylor's series vanish after integration, the corresponding integrals can be expanded as:…”

Section: Taylor's Series Expansionmentioning

confidence: 99%

“…in the Appendix B is the Euclidean distance between computing point and integrating point; a, b and c are the length of sides in x Q , y Q and z Q directions of the RP and the lengths can be calculated by the 'equivalent volume' RP defined by Heck & Seitz (2007) as:…”

Section: Approximations By Cartesian Elementsmentioning

confidence: 99%

“…Heck & Seitz (2007) and Wild-Pfeiffer (2008) derived the fourth-order approximations which applied the Taylor's series expansion (TSE) of the integral kernel. 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.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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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Section: Tesseroid and Coordinate Systemsmentioning

confidence: 99%

Section: Basic Formulasmentioning

confidence: 99%

Section: Taylor's Series Expansionmentioning

confidence: 99%

Section: Approximations By Cartesian Elementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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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Modeling 3‐D density distribution in the mantle from inversion of geoid anomalies: Application to the Yellowstone Province

Chaves

Ussami

2013

JGR Solid Earth

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[1] We developed a three-dimensional scheme to invert geoid anomalies aiming to map density variations in the mantle. Using an ellipsoidal-Earth approximation, the model space is represented by tesseroids. To assess the quality of the density models, the resolution and covariance matrices were computed. From a synthetic geoid anomaly caused by a plume tail with Gaussian noise added, the inversion code was able to recover a plausible solution about the density contrast and geometry when it is compared to the synthetic model. To test the inversion algorithm in a natural case study, geoid anomalies from the Yellowstone Province (YP) were inverted. From the Earth Gravitational Model 2008 expanded up to degree 2160, lower crust-and mantle-related negative geoid anomalies with amplitude of approximately 70 m were obtained after removing long-wavelength components (>5400 km) and crustal effects. We estimated three density models for the YP. The first model, the EDM-1 (estimated density model), uses a starting model with density contrast equal to 0. The other two models, the EDM-2 and EDM-3, use an initial density derived from two S-velocity models for the western United States, the Dynamic North America Models of S Waves by Obrebsky et al. (2011) and the Northwestern United States Teleseismic Tomography of S Waves (NWUS11-S) by James et al. (2011). In these three models, a lower and an upper bound for the density solution was also imposed as a priori information. Regardless of the initial constraints, the inversion of the residual geoid indicates that the lower crust and the upper mantle of the YP have a predominantly negative density contrast (~À50 kg/m 3 ) relative to the surrounding mantle. This solution reveals that the density contrast extends at least to 660 km depth. Regional correlation analysis between the EDM-1 and NWUS11-S indicates an anticorrelation (coefficient of~À0.7) at 400 km depth. Our study suggests that the mantle density derived from the inversion of geoid could be integrated with seismic velocity models to image mantle anomalous features beyond the depth limit of investigation achieved combining gravity and seismic tomography.

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Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid

Claessens

Hirt

2013

JGR Solid Earth

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[1] Forward gravity modeling in the spectral domain traditionally relies on spherical approximation. However, this level of approximation is insufficient for some present day high-accuracy applications. Here we present two solutions that avoid the traditional spherical approximation in spectral forward gravity modeling. The first solution (the extended integration method) applies integration over masses from a reference sphere to the topography and applies a correction for the masses between ellipsoid and sphere. The second solution (the harmonic combination method) computes topographic potential coefficients from a combination of surface spherical harmonic coefficients of topographic heights above the ellipsoid, based on a relation among spherical harmonic functions introduced by Claessens (2005). Using a degree 2160 spherical harmonic model of the topographic masses, both methods are applied to derive the Earth's ellipsoidal topographic potential in spherical harmonics. The harmonic combination method converges fastest and-akin to the EGM2008 geopotential model-generates additional spherical harmonic coefficients in spectral band 2161 to 2190 which are found crucial for accurate evaluation of the ellipsoidal topographic potential at high degrees. Therefore, we recommend use of the harmonic combination method to model ellipticity in spectral-domain forward modeling. The method yields ellipsoidal topographic potential coefficients which are "compatible" with global Earth geopotential models constructed in ellipsoidal approximation, such as EGM2008. It shows that the spherical approximation significantly underestimates degree correlation coefficients among geopotential and topographic potential. The topographic potential model is, for example, of immediate value for the calculation of Bouguer gravity anomalies in fully ellipsoidal approximation.Citation: Claessens, S. J., and C. Hirt (2013), Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid,

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A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling

Cited by 230 publications

References 25 publications

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

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

Modeling 3‐D density distribution in the mantle from inversion of geoid anomalies: Application to the Yellowstone Province

Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid

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