Gravitational attraction and potential of spherical shell with radially dependent density

Karcol, Roland

doi:10.1007/s11200-011-0002-9

Cited by 12 publications

(6 citation statements)

References 6 publications

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“…Atmospheric correction is usually calculated based on a simple approximation according to Wenzel (1985). By the term true atmosphere, we mean the model of the atmosphere derived from the effect of a spherical shell with radially dependent density using the US standard atmosphere 1976 (Karcol, 2011) with an irregularly shaped bottom surface formed by the Earth's surface, calculated globally (Mikuška et al, 2008). The difference between atmospheric correction calculated by both approaches for the AlpArray region (calculated for selected database points) is shown in Fig.…”

Section: A Short Remark On Future Treatment Of True Atmosphere and Distant Relief Effectsmentioning

confidence: 99%

“…This modern array has used over 628 sites for more than 39 months across the greater Alpine area such that no point on land was farther than 30 km from a broadband seismometer (Hetényi et al, 2018). While seismic imaging of the entire Alps in 3D became a reality following decades of active-and passivesource projects, imaging efforts in gravity reached 3D earlier thanks to the availability of national data sets of the Alpine neighboring countries with partly high-resolution and 3D modeling approaches among others (Ehrismann et al, 1976;Götze, 1978;Kissling, 1980;Götze and Lahmeyer, 1988;Götze et al, 1991;Ebbing, 2002;Ebbing et al, 2006;Marson and Klingelé, 1993;Kahle and Klingelé, 1979). However, these land data sets for historical reasons were acquired in national reference systems and were seldom shared, preventing high-resolution pan-Alpine gravity studies using homogeneously processed data.…”

Section: Introductionmentioning

confidence: 99%

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The first pan-Alpine surface-gravity database, a modern compilation that crosses frontiers

Zahorec

Papčo

Pašteka

et al. 2021

Earth Syst. Sci. Data

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Abstract. The AlpArray Gravity Research Group (AAGRG), as part of the European AlpArray program, focuses on the compilation of a homogeneous surface-based gravity data set across the Alpine area. In 2017 10 European countries in the Alpine realm agreed to contribute with gravity data for a new compilation of the Alpine gravity field in an area spanning from 2 to 23∘ E and from 41 to 51∘ N. This compilation relies on existing national gravity databases and, for the Ligurian and the Adriatic seas, on shipborne data of the Service Hydrographique et Océanographique de la Marine and of the Bureau Gravimétrique International. Furthermore, for the Ivrea zone in the Western Alps, recently acquired data were added to the database. This first pan-Alpine gravity data map is homogeneous regarding input data sets, applied methods and all corrections, as well as reference frames. Here, the AAGRG presents the data set of the recalculated gravity fields on a 4 km × 4 km grid for public release and a 2 km × 2 km grid for special request. The final products also include calculated values for mass and bathymetry corrections of the measured gravity at each grid point, as well as height. This allows users to use later customized densities for their own calculations of mass corrections. Correction densities used are 2670 kg m−3 for landmasses, 1030 kg m−3 for water masses above the ellipsoid and −1640 kg m−3 for those below the ellipsoid and 1000 kg m−3 for lake water masses. The correction radius was set to the Hayford zone O2 (167 km). The new Bouguer anomaly is station completed (CBA) and compiled according to the most modern criteria and reference frames (both positioning and gravity), including atmospheric corrections. Special emphasis was put on the gravity effect of the numerous lakes in the study area, which can have an effect of up to 5 mGal for gravity stations located at shorelines with steep slopes, e.g., for the rather deep reservoirs in the Alps. The results of an error statistic based on cross validations and/or “interpolation residuals” are provided for the entire database. As an example, the interpolation residuals of the Austrian data set range between about −8 and +8 mGal and the cross-validation residuals between −14 and +10 mGal; standard deviations are well below 1 mGal. The accuracy of the newly compiled gravity database is close to ±5 mGal for most areas. A first interpretation of the new map shows that the resolution of the gravity anomalies is suited for applications ranging from intra-crustal- to crustal-scale modeling to interdisciplinary studies on the regional and continental scales, as well as applications as joint inversion with other data sets. The data are published with the DOI https://doi.org/10.5880/fidgeo.2020.045 (Zahorec et al., 2021) via GFZ Data Services.

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Section: A Short Remark On Future Treatment Of True Atmosphere and Distant Relief Effectsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The first pan-Alpine surface-gravity database, a modern compilation that crosses frontiers

Zahorec

Papčo

Pašteka

et al. 2021

Earth Syst. Sci. Data

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“…In general, the precision can be affected by many parameters, such as W, size of the Tesseroid, and Gauss-Legendre nodes. We create a homogenous spherical shell, which has analytical solutions to calculate the gravitational acceleration [29], [30] and can be perfectly subdivided into Tesseroids as well. The true gravitational acceleration of computation point P caused by the spherical shell can be written as:…”

Section: The Synthetic Data Tests a Precision Evaluation On Thementioning

confidence: 99%

The Apparent Density Mapping Approach in Spherical Coordinates and the Crustal Density Distribution of Chinese Mainland

2019

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Apparent density mapping is a technique to obtain lateral density distribution of subsurface layer by inverting gravity anomaly. In general, the subsurface layer can be divided into vertical, juxtaposed prisms in Cartesian coordinates. However, for continental and global scales, the curvature of the earth should be taken into account, and the subsurface layer should be divided into Tesseroids in spherical coordinates instead of rectangular prisms. In this paper, we present a modified Gauss-Legendre algorithm to forward the gravity anomaly generated by an arbitrary Tesseroid, and the precision can be improved vastly when the observation points in the near surface of the Tesseroid. The shell tests show that the relative maximum error can be controlled in 0.0009343% when the Gauss-Legendre nodes are set as (4, 4) in the longitude and latitude directions and the subdivision parameter W is set as 1.5. Then, an apparent density mapping approach is presented in spherical coordinates to minimize the difference between the observed gravity anomalies and calculated gravity anomalies. The synthetic model verified the feasibility and precision of our approach. In real application, we have obtained the crustal apparent density distribution of Chinese mainland with 0.25 • ×0.25 • in longitude and latitude directions. The crustal apparent distribution of Chinese mainland is generally low in the western region and high in eastern region varying from 2.45-2.81 g/cm 3 and closely related to lithologic units and geological boundaries. INDEX TERMS Gravity inversion, apparent density mapping, spherical coordinates, Chinese mainland.

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“…The GP V true at the computation point P on and outside the spherical shell can be accurately calculated (Heiskanen and Moritz, 1967;Vaníček et al, 2001;Karcol, 2011) as follows:…”

Section: Numerical Tests For Formulasmentioning

confidence: 99%

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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Gravitational attraction and potential of spherical shell with radially dependent density

Cited by 12 publications

References 6 publications

The first pan-Alpine surface-gravity database, a modern compilation that crosses frontiers

The first pan-Alpine surface-gravity database, a modern compilation that crosses frontiers

The Apparent Density Mapping Approach in Spherical Coordinates and the Crustal Density Distribution of Chinese Mainland

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

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