Reformulation of Parker–Oldenburg's method for Earth's spherical approximation

SUMMARY Parker–Oldenburg's method is perhaps the most commonly used technique to estimate the depth of density interface from gravity data. To account for large density variations reported, for instance, at the Moho interface, between the ocean seawater density and marine sediments, or between sediments and the underlying bedrock, some authors extended this method for variable density models. Parker–Oldenburg's method is suitable for local studies, given that a functional relationship between gravity data and interface geometry is derived for Earth's planar approximation. The application of this method in (large-scale) regional, continental or global studies is, however, practically restricted by errors due to disregarding Earth's sphericity. Parker–Oldenburg's method was, therefore, reformulated also for Earth's spherical approximation, but assuming only a uniform density. The importance of taking into consideration density heterogeneities at the interface becomes even more relevant in the context of (large-scale) regional or global studies. To address this issue, we generalize Parker–Oldenburg's method (defined for a spherical coordinate system) for the depth of heterogeneous density interface. Furthermore, we extend our definitions for gravity gradient data of which use in geoscience applications increased considerably, especially after launching the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gravity-gradiometry satellite mission. For completeness, we also provide expressions for potential. The study provides the most complete review of Parker–Oldenburg's method in planar and spherical cases defined for potential, gravity and gravity gradient, while incorporating either uniform or heterogeneous density model at the interface. To improve a numerical efficiency of gravimetric forward modelling and inversion, described in terms of spherical harmonics of Earth's gravity field and interface geometry, we use the fast Fourier transform technique for spherical harmonic analysis and synthesis. The (newly derived) functional models are tested numerically. Our results over a (large-scale) regional study area confirm that the consideration of a global integration and Earth's sphericty improves results of a gravimetric forward modelling and inversion.

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A new look at Belgian aeromagnetic and gravity data through image-based display and integrated modelling techniques

Geological Magazine ◽

10.1017/s0016756800020884 ◽

1993 ◽

Vol 130 (5) ◽

pp. 583-591 ◽

Cited By ~ 22

Author(s):

B. C. Chacksfield ◽

W. De Vos ◽

L. D'Hooge ◽

M. Dusar ◽

M. K. Lee ◽

...

Keyword(s):

Large Scale ◽

Gravity Data ◽

Bouguer Anomaly ◽

Digital Processing ◽

Integrated Modelling ◽

Magnetic Data ◽

Gravity Gradient ◽

The North ◽

Anomaly Map ◽

Shaded Relief

AbstractDigital processing and image-based display techniques have been used to generate contour and shaded-relief maps of Belgian aeromagnetic data at a scale of 1:300000 for the whole of Belgium. These highlight the important anomalies and structural trends, particularly over the Brabant Massif. North and vertically illuminated shaded-relief plots, enhanced structural belts trending west–east to northwest–southeast in the Brabant Massif and west–east to southwest–northeast in the core of the Ardennes. The principal magnetic lineaments have been identified from the shaded-relief plots and tentatively correlated to basement structures. Most short lineaments are correlated with individual folds while the more extensive lineaments are correlated with large scale fault structures. Magnetic highs within the Brabant Massif are attributed to folded sediments of the Tubize Group. The magnetic basement in the east of Belgium is sinistrally displaced to the north by an inferred deep NNW–SSE crustal fracture. The Bouguer anomaly map of Belgium identifies the Ardennes as a negative area, and the Brabant Massif as a positive area, with the exception of a WNW–trending gravity low in its western part. The southern margin of the Brabant Massif is defined by a steep gravity gradient coincident with the Faille Bordiere (Border Fault). Trial modelling of the gravity and magnetic data, carried out along profiles across the Brabant and Stavelot massifs, has identified probable acid igneous intrusions in the western part of the Brabant Massif, and a deep magnetic lower density body underlying the whole Ardennes region, which is thought to be a distinctive Precambrian crustal block.

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Study of the accuracy of monthly time-variable satellites gravity field estimates

10.5194/egusphere-egu21-4136 ◽

2021 ◽

Author(s):

Hugo Lecomte ◽

Severine Rosat ◽

Mioara Mandea

Keyword(s):

Gravity Field ◽

Large Scale ◽

Gravity Data ◽

European Space Agency ◽

Field Measurements ◽

Time Variable ◽

Research Centre ◽

Data Products ◽

Level 2

The GRACE and GRACE Follow-On (GRACE-FO) missions have been providing monthly time-variable gravity field estimates since 2002 with a one-year gap between 2017 and 2018. The Level 2 data products are available through several processing centers with independent computation strategies. The Center of Space Research (CSR), the German Research Centre for Geosciences (GFZ) and the Jet Propulsion Laboratory (JPL) as part of the GRACE/GRACE-FO Science Data System (SDS) process gravity data with RL06 standards. The French National Centre for Space Studies (CNES) and the Graz University of Technology delivered GRACE gravity fields models respectively named CNES/GRGS RL05 and ITSG-GRACE2018. Besides GRACE data, the European Space Agency (ESA) delivers Level 2 data products for the Swarm mission. Swarm data enables the evaluation of gap-filling methods between the GRACE and GRACE-FO missions. These datasets are very valuable inputs in studying the Earth's deep interior and could open new windows into the study of core-mantle boundary processes and core dynamics.&#160;Earth's core dynamical processes inferred from geomagnetic field measurements are characterized by large-scale patterns. Studying them via gravity field observations involves the use of spherical harmonic coefficients up to degree and order 10. Particular attention needs to be dedicated to Stokes coefficients that are affected by problematic reconstruction effects such as C2,0 or C3,0. The comparison of time-series from various processing centers with Satellite-Laser Ranging (SLR) gravity products and hydrological loading models provides information on the consistency between different solutions and the accuracy of space gravity field measurements. The correction of hydrological and glacial isostatic adjustment (GIA) effects is an additional source of error in the determination of the gravity field. For example, the actual uncertainty of the GIA model over North America might lead to an error of 10% for some Stokes coefficients. Mismodelling in the seasonal loading could also affect the retrieved Stokes coefficients.&#160;This study firstly provides a comparison of existing gravity field solutions and their accuracy. Secondly, a detailed analysis of different error sources provides us with better estimates of the current limits in the determination of elusive signals coming from the deep Earth's interior. It also offers the possibility to better describe the external sources and then to minimize their contribution to the signal we are interested in.

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Evaluation of terrestrial and airborne gravity data over Antarctica – a generic approach

Journal of Geodetic Science ◽

10.1515/jogs-2019-0004 ◽

2019 ◽

Vol 9 (1) ◽

pp. 29-40

Author(s):

P. Zingerle ◽

R. Pail ◽

M. Scheinert ◽

T. Schaller

Keyword(s):

Gravity Field ◽

Large Scale ◽

Gravity Data ◽

Evaluation Procedure ◽

Airborne Gravity ◽

Small Scale ◽

German Research Foundation ◽

Worst Case ◽

Spectral Bands ◽

Topographic Model

Abstract The AntGrav project, funded by the German Research Foundation (DFG) has the main objective to homogenize and optimize Antarctic gravity field information. Within this project an evaluation procedure is needed to inspect all different kind of gravity field surveys available in Antarctica. In this paper a suitable methodology is proposed. We present an approach for fast 3D gravity point data reduction in different spectral bands. This is achieved through pre-calculating a fine 3D mesh of synthesized gravity functionals over the entirety of the Antarctic continent, for which two different global models are used: the combined satellite model GOCO05s for the long-wavelength part, and the topographic model Earth2014 for the shorter wavelengths. To maximize the applicability separate meshes are calculated for different spectral bands in order to specifically reduce a certain band or a selected combination. All meshes are calculated for gravity anomalies as well as gravity disturbances. Utilizing these meshes, synthesized gravity data at arbitrary positions is computed by conventional 3D interpolation methods (e.g. linear, cubic or spline). It is shown that the applied approach can reach a worst-case interpolation error of less than 1 mGal. Evaluation results are presented for the AntGG grid and exemplary for the in-situ measurements of the AGAP and BAS-LAND campaigns. While general properties, large-scale errors and systematic effects can usually be detected, small-scale errors (e.g. of single points) are mostly untraceable due to the uncertainties within the topographic model.

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Benchmark forward gravity schemes: the gravity field of a realistic lithosphere model WINTERC-G

10.5194/se-2021-145 ◽

2021 ◽

Author(s):

Bart Root ◽

Josef Sebera ◽

Wolfgang Szwillus ◽

Cedric Thieulot ◽

Zdenek Martinec ◽

...

Keyword(s):

Gravity Field ◽

Spherical Harmonic ◽

Short Wavelength ◽

Field Data ◽

The Other ◽

Potential Fields ◽

Spherical Harmonic Analysis ◽

Constant Density ◽

Forward Modelling ◽

Density Interface

Abstract. Several alternative gravity forward modelling methodologies and associated numerical codes with their own advantages and limitations are available for the Solid Earth community. With the upcoming state-of-the-art lithosphere density models and accurate global gravity field data sets it is vital to understand the opportunities and limitations of the various approaches. In this paper, we discuss the four widely used techniques: global spherical harmonics (GSH), tesseroid integration (TESS), triangle integration (TRI), and hexahedral integration (HEX). A constant density shell benchmark shows that all four codes can produce similar precise gravitational potential fields. Two additional shell tests were conducted with more complicated density structures: lateral varying density structures and a Moho density interface between crust and mantle. The differences between the four codes were all below 1.5 percent of the modeled gravity signal suitable for reproducing satellite-acquired gravity data. TESS and GSH produced the most similar potential fields (< 0.3 percent). To examine the usability of the forward modelling codes for realistic geological structures, we use the global lithosphere model WINTERC-G, that was constrained, among other data, by satellite gravity field data computed using a spectral forward modeling approach. This spectral code was benchmarked against the GSH and it was confirmed that both approaches produce similar gravity solution with negligible differences between them. In the comparison of the different WINTERC-G-based gravity solutions, again GSH and TESS performed best. Only short-wavelength noise is present between the spectral and tesseroid forward modelling approaches, likely related to the different way in which the spherical harmonic analysis of the varying boundaries of the mass layer is performed. The Spherical harmonic basis functions produces small differences compared to the tesseroid elements especially at sharp interfaces, which introduces mostly short-wavelength differences. Nevertheless, both approaches (GSH and TESS) result in accurate solutions of the potential field with reasonable computational resources. Differences below 0.5 percent are obtained, resulting in residuals of 0.076 mGal standard deviation at 250 km height. The biggest issue for TRI is the characteristic pattern in the residuals that is related to the grid layout. Increasing the resolution and filtering allows for the removal of most of this erroneous pattern, but at the expense of higher computational loads with respect to the other codes. The other spatial forward modelling scheme HEX has more difficulty in reproducing similar gravity field solutions compared to GSH and TESS. These particular approaches need to go to higher resolutions, resulting in enormous computation efforts. The hexahedron-based code performs less than optimal in the forward modelling of the gravity signature, especially of a lateral varying density interface. Care must be taken with any forward modelling software as the approximation of the geometry of the WINTERC-G model may deteriorate the gravity field solution.

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Three‐dimensional regularized focusing inversion of gravity gradient tensor component data

Geophysics ◽

10.1190/1.1778236 ◽

2004 ◽

Vol 69 (4) ◽

pp. 925-937 ◽

Cited By ~ 128

Author(s):

Michael S. Zhdanov ◽

Robert Ellis ◽

Souvik Mukherjee

Keyword(s):

Local Density ◽

Mineral Exploration ◽

Gravity Data ◽

Three Dimensional ◽

Tensor Component ◽

Airborne Gravity ◽

Gravity Gradient ◽

Gravity Method ◽

Density Anomaly ◽

And Inversion

We develop a new method for interpretation of tensor gravity field component data, based on regularized focusing inversion. The focusing inversion makes its possible to reconstruct a sharper image of the geological target than conventional maximum smoothness inversion. This new technique can be efficiently applied for the interpretation of gravity gradiometer data, which are sensitive to local density anomalies. The numerical modeling and inversion results show that the resolution of the gravity method can be improved significantly if we use tensor gravity data for interpretation. We also apply our method for inversion of the gradient gravity data collected by BHP Billiton over the Cannington Ag‐Pb‐Zn orebody in Queensland, Australia. The comparison with the drilling results demonstrates a remarkable correlation between the density anomaly reconstructed by the gravity gradient data and the true structure of the orebody. This result indicates that the emerging new geophysical technology of the airborne gravity gradient observations can improve significantly the practical effectiveness of the gravity method in mineral exploration.

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Spherical and ellipsoidal surface mass change from GRACE time-variable gravity data

10.5194/egusphere-egu2020-13258 ◽

2020 ◽

Author(s):

Michal Šprlák ◽

Khosro Ghobadi-Far ◽

Shin-Chan Han ◽

Pavel Novák

Keyword(s):

Gravity Field ◽

Gravitational Potential ◽

Gravity Data ◽

Time Variable ◽

Mass Change ◽

Spherical Approximation ◽

Surface Mass ◽

Time Variable Gravity ◽

The Earth ◽

Variable Gravity

The problem of estimating mass redistribution from temporal variations of the Earth&#8217;s gravity field, such as those observed by GRACE, is non-unique. By approximating the Earth&#8217;s surface by a sphere, surface mass change can be uniquely determined from time-variable gravity data. Conventionally, the spherical approach of Wahr et al. (1998) is employed for computing the surface mass change caused, for example, by terrestrial water and glaciers. The accuracy of the GRACE Level 2 time-variable gravity data has improved due to updated background geophysical models or enhanced data processing. Moreover, time series analysis of &#8764;15 years of GRACE observations allows for determining inter-annual and seasonal changes with a significantly higher accuracy than individual monthly fields. Thus, the improved time-variable gravity data might not tolerate the spherical approximation introduced by Wahr et al. (1998).A spheroid (an ellipsoid of revolution) represents a closer approximation of the Earth than a sphere, particularly in polar regions. Motivated by this fact, we develop a rigorous method for determining surface mass change on a spheroid. Our mathematical treatment is fully ellipsoidal as we concisely use Jacobi ellipsoidal coordinates and exploit the corresponding series expansions of the gravitational potential and of the surface mass. We provide a unique one-to-one relationship between the ellipsoidal spectrum of the surface mass and the ellipsoidal spectrum of the gravitational potential. This ellipsoidal spectral formula is more general and embeds the spherical approach by Wahr et al. (1998) as a special case. We also quantify the differences between the spherical and ellipsoidal approximations numerically by calculating the surface mass change rate in Antarctica and Greenland.&#160;References:Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth&#8217;s gravity field: Hydrological and oceanic effects and their possible detection using GRACE. Journal of Geophysical Research: Solid Earth, 103(B12), 30205-30229.

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TGF: A New MATLAB-based Software for Terrain-related Gravity Field Calculations

Remote Sensing ◽

10.3390/rs12071063 ◽

2020 ◽

Vol 12 (7) ◽

pp. 1063

Author(s):

Meng Yang ◽

Christian Hirt ◽

Roland Pail

Keyword(s):

Density Distribution ◽

Gravity Field ◽

Mass Density ◽

Computation Time ◽

Spherical Approximation ◽

Constant Density ◽

Forward Modelling ◽

Gravity Effects ◽

Residual Terrain Modelling ◽

Elevation Model

With knowledge of geometry and density-distribution of topography, the residual terrain modelling (RTM) technique has been broadly applied in geodesy and geophysics for the determination of the high-frequency gravity field signals. Depending on the size of investigation areas, challenges in computational efficiency are encountered when using an ultra-high-resolution digital elevation model (DEM) in the Newtonian integration. For efficient and accurate gravity forward modelling in the spatial domain, we developed a new MATLAB-based program called, terrain gravity field (TGF). Our new software is capable of calculating the gravity field generated by an arbitrary topographic mass-density distribution. Depending on the attenuation character of gravity field with distance, the adaptive algorithm divides the integration masses into four zones, and adaptively combines four types of geometries (i.e., polyhedron, prism, tesseroid and point-mass) and DEMs with different spatial resolutions. Compared to some publicly available algorithms depending on one type of geometric approximation, this enables accurate modelling of gravity field and greatly reduces the computation time. Besides, the TGF software allows to calculate ten independent gravity field functionals, supports two types of density inputs (constant density value and digital density map), and considers the curvature of the Earth by involving spherical approximation and ellipsoidal approximation. Further to this, the TGF software is also capable of delivering the gravity field of full-scale topographic gravity field implied by masses between the Earth’s surface and mean sea level. In this contribution, the TGF software is introduced to the geoscience community and its capabilities are explained. Results from internal and external numerical validation experiments of TGF confirmed its accuracy at the sub-mGal level. Based on TGF, the trade-off between accuracy and efficiency, values for the spatial resolution and extension of topography models are recommended. The TGF software has been extensively tested and recently been applied in the SRTM2gravity project to convert the global 3” SRTM topography to implied gravity effects at 28 billion computation points. This confirms the capability of TGF for dealing with large datasets. Together with this paper, the TGF software will be released in the public domain for free use in geodetic and geophysical forward modelling computations.

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3D Subsurface Characterization of Banded Iron Formation Mineralization using Large-Scale Gravity Data: A Case Study in Parts of Bharatpur, Dausa and Karauli Districts of Rajasthan, India

Natural Resources Research ◽

10.1007/s11053-021-09880-y ◽

2021 ◽

Author(s):

Saudamini Sahoo ◽

Anand Singh ◽

Santu Biswas ◽

S. P. Sharma

Keyword(s):

Large Scale ◽

Gravity Data ◽

Iron Formation ◽

Banded Iron Formation ◽

Subsurface Characterization

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Joint Gravity and Seismic Reflection Methods to Characterize the Deep Aquifers in Arid Ain El Beidha Plain (Central Tunisia, North Africa)

Water ◽

10.3390/w13091310 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1310

Author(s):

Hajer Azaiez ◽

Hakim Gabtni ◽

Mourad Bédir

Keyword(s):

Seismic Reflection ◽

Large Scale ◽

Seismic Analysis ◽

Gravity Data ◽

Geophysical Methods ◽

Gravity Anomalies ◽

Integrated Approach ◽

Central Tunisia ◽

Deep Aquifers ◽

Advanced Analysis

Electric resistivity sounding and tomography, as well as electromagnetic sounding, are the classical methods frequently used for hydrogeological studies. In this work, we propose the development and implementation of an original integrated approach using the unconventional hydro–geophysical methods of gravity and seismic reflection for the fast, large–scale characterization of hydrogeological potential using the Ain El Beidha plain (central Tunisia) as an analogue. Extending the values of vintage petroleum seismic reflection profiles and gravity data, in conjunction with available geological and hydrogeological information, we performed an advanced analysis to characterize the geometry of deep tertiary (Oligocene and Eocene) aquifers in this arid area. Residual and tilt angle gravity maps revealed that most gravity anomalies have a short wavelength. The study area was mainly composed of three major areas: the Oued Ben Zitoun and Ain El Beidha basins, which are both related to negative gravity trends corresponding to low–density subsiding depocenters. These basins are separated by an important NE–SW trend called “El Gonna–J. El Mguataa–Kroumet Zemla” gravity high. Evaluation of the superposition of detected lineaments and Euler deconvolution solutions’ maps showed several NE–SW and N–S relay system faults. The 3D density inversion model using a lateral and vertical cutting plane suggested the presence of two different tectonic styles (thin VS thick). Results from the gravity analysis were in concordance with the seismic analysis. The deep Oligocene and Eocene seismic horizons were calibrated to the hydraulic wells and surrounding outcrops. Oligocene and Eocene geological reservoirs appear very fractured and compartmented. The faulting network also plays an important role in enhancing groundwater recharge process of the Oligocene and Eocene aquifers. Finally, generated isochron maps provided an excellent opportunity to develop future comprehensive exploration surveys over smaller and more favorable areas’ sub–basins.

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