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

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.

GRV_D_inv: A graphical user interface for 3D forward and inverse modeling of gravity data

This paper presents a new gravity inversion tool GRV_D_inv, specifically a GUI-based Matlab code developed to determine the three-dimensional depth structure of a density interface. The algorithm used performs iteratively in the frequency-domain based on a relationship between the Fourier transforms of the gravity data and the sum of the Fourier transforms of the powers of the depth to the interface. In this context, the proposed code is time-efficient in computations, and thus, it is capable of handling large arrays of data. The GUI-enabled interactive control functions of the code enable the user with easy control in setting the parameters for the inversion strategy prior the operation, and allow optional choice for displaying and recording of the outputs data without requiring coding expertise. We validated the code by applying it to both noise-free and noisy synthetic gravity data produced by a density interface; we obtained good correlation between the calculated ones and the actual relief even in the presence of noise. We also applied the code to a real gravity data from Brittany (France) for determining the 3D Moho interface as a practical example. The recovered depths from the code compare well with the published Moho structures of this study area.

Performance comparison of the wavenumber and spatial domain techniques for mapping basement reliefs from gravity data

Estimating the density interface depth is an important task when interpreting gravity data. A range of techniques can be applied for this. Here we compare the effectiveness of the wavenumber and spatial domain techniques for inverting gravity data with respect to basement reliefs. These techniques were tested with two synthetic gravity models, and then applied to a real case: the gravity data of the Magura basin (East Slovakian Outer Carpathian). The findings show that the spatial domain technique can precisely estimate the structures, but the computation speed is slow, while the wavenumber domain technique can perform faster computations with less precision.

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.