Several alternative gravity forward modelling methodologies and associated numerical codes with their own advantages and limitations are available for the solid Earth community. With 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: laterally varying density structures and a crust–mantle interface density. The differences between the four codes were all below 1.5 % of the modelled gravity signal suitable for reproducing satellite-acquired gravity data. TESS and GSH produced the most similar potential fields (

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 modelling approach. This spectral code was benchmarked against the GSH, and it was confirmed that both approaches produce a 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 produce 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 % 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 allow 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 with a laterally 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.

Dedicated gravimetric satellite missions such as NASA's GRACE and ESA's GOCE missions have generated unprecedented views of the Earth's gravity field

The recent global model WINTERC-G

Forward gravity field modelling discretization can be classified as space domain or spectral domain

We present a benchmark study comparing three space domain (triangles, tesseroids, hexahedra) and one spectral domain approach applied to the layered WINTERC-G 3-D density model, in order to assess the usability of the model including the uncertainty resulting from different forward modelling approaches. While the focus of this benchmark is the different parameterizations, we also need to address the approximations and inaccuracies of each individual method to better appraise the differences between the methods. Therefore, we have carried out different tests, ranging from simple shell tests to a more complex upper mantle model. We present the forward modelling scheme used in WINTERC-G and compare it to different forward modelling codes. Finally, the full 3-D upper mantle model WINTERC-G is used as an encompassing benchmark of the various forward modelling schemes in comparison to the XGM2016 gravity model that was used in the construction of WINTERC-G

The inversion code used to construct WINTERC-G relies on a spectral forward gravity modelling approach. The mathematical description of that method can be found in Appendix

On the sphere, there is a wide range of point distributions that can represent the geometry of the shell

The GSH code bench-marked here is based on the fast spectral method described in

The main difference between the GSH code and the spectral approach used in the development of WINTERC-G is the way the non-spherical boundary and laterally varying density are added. The GSH code adds these together before performing a spectral analysis on the combined function, whereas the WINTERC-based code performs the spectral analysis on the individual components of the boundaries (Eq.

Another requirement of the GSH code is that the boundaries and density should be on an equi-angular grid, similar to the WINTERC-G grid.

In the geoscientific community, the tesseroid algorithm of

A tesseroid is a spherical prism that is described by six values: the boundary coordinates in east, west, north, south, top, and bottom. Equi-angular tesseroids have a prismatic shape at low latitudes but degenerate into increasingly triangular shapes closer to the poles. In terms of Newton's integral (Eq.

Numerical integration of the triangular grids in this work also satisfies Eq. (

Sketch of the triangular grid on the sphere – grid or data points (red circle with black contour), triangle centre of mass (blue pentagon), and side midpoints (green circle).

Figure

Figure

In the first step WINTERC-G development uses a triangular grid to invert seismic tomography, surface heat flow, and isostasy data, whereas in the second step the gravity field is modelled based on a spherical equi-angular grid to accommodate the fast spectral code. The specific challenge is the calculation of gravity from a triangular grid because there is no triangular grid that would be perfectly uniform on the sphere. The nodes associated with larger triangles thus produce a larger signal so that the results are then systematically affected by the triangular patterns.

ASPECT (short for Advanced Solver for Problems in Earth's ConvecTion) is a code originally intended to solve the equations of conservation of mass, momentum, and energy in the context of convection in the Earth's mantle and lithosphere dynamics

Given a density field in the computational mesh, ASPECT can also compute the gravity acceleration vector, the gravitational potential, and the gravity gradients on any point in space. Since the integrand of the integral equations is not a polynomial the GLQ-based computed integral will not be exact. Nevertheless, we expect that an increase in the number of quadrature points inside the elements leads to a more accurate calculation. ASPECT relies on quadratic elements for velocity and temperature, and an array of

The (default) topology of the mesh in ASPECT is shown in Fig.

The most basic shell test is a spherical shell with finite thickness and a constant density. Here, we mainly assess the volumetric-based approaches and their resolution because spectral-type codes have an exact solution for a homogeneous density shell down to machine precision. Therefore, two other shell tests have been proposed: an equal-thickness shell with laterally varying density and a shell with a depth-varying density discontinuity. The WINTERC-G model is described by layers with laterally varying density as well as laterally varying density interfaces (e.g. surface topography, basement, Moho discontinuity).

The gravity field of a homogeneous spherical shell can easily be calculated analytically because due to symmetry the relationship only depends on the radial distance of the computation point:

As expected, the GSH code produces a solution similar to the analytical value within machine precision, with a standard deviation of roughly

For the tesseroids, a small bias was found at the fifth significant digit, indicating a very good overall performance across all the latitudes. There is only a negligible variation close to the poles as seen in Fig.

Deviation of the tesseroid integration from the analytical shell result as a function of latitude. This is for the case of the 2 km thick shell with a 1

The triangle-based approach seems to perform slightly better than the tesseroid approach, but it does show different behaviour depending on the lateral resolution of the triangle grid. At a roughly 2

The ASPECT code outperforms the triangle and tesseroid approach concerning precision and lateral resolution. It achieves

So, all four codes are able to obtain the gravity signal up to

Summary of the homogeneous density shell test at the target height

The following shell test examines the capability of the different forward modelling schemes to handle lateral density variations. We will first show that the GSH code is capable of producing similar results compared to the WINTERC-G-based code. Then, we discuss the difference between the four modelling approaches in this benchmark.

The mass shell in this scenario is described by the outer radius, located at 56 km depth, and the inner radius, located at 80 km depth. The density values within the layer correspond to typical lithospheric-scale lateral density variations in WINTERC-G model, as presented in Fig.

The densities range between 3286 and 3419 kg m

The forward-modelled geoid differences of this layer between the WINTERC-G code and GSH code are shown in Fig.

This same shell is processed with the other forward modelling approaches. The radial component of the gravity field is computed at 250 km height above the mean sphere, as this was the height at which the satellite gravity data for the development of WINTERC-G were used. Shell test 1 showed that the mean gravity uncertainty between the different numerical codes, which is linked to the zero-degree coefficients, was insignificant. Therefore, the spherical harmonic coefficients 2–179 were used to focus more on the anomalies of the gravity solution. This meant that the solutions had to be post-processed by the GSH code to ensure that a similar spectral signature is used in the comparison. This introduced some errors at machine precision level.

Radial gravity component comparisons at 250 km height for shell test 2, wherein a 24 km thick shell is modelled with laterally varying density structure. A grid resolution of

Figure

Table

Statistical results from shell test 2: a density shell of equal thickness is modelled with laterally varying density structure.

High-resolution lithosphere–upper mantle models combined with increasing computing capabilities offer new possibilities in relation to dynamic studies like mantle convection, glacial isostatic adjustment (GIA), and geo-hazards. Geometric boundaries within the crust and mantle that vary in depth and thickness are difficult to represent in numerical models. Nevertheless, such discontinuities produce large gravitational signals, and hence density boundaries need to be represented as perfectly as possible as slight changes to their depths could have noticeable effects in the full lithospheric gravitational signal. In particular, the top and bottom boundaries of the crystalline crust (basement and Moho boundaries, respectively) are of importance. In this test, we model a single density interface, representing the crust–mantle interface taken from the CRUST1.0 crustal model

Radial gravity component comparisons at 250 km height for shell test 3, wherein a density shell of equal thickness is modelled with a density contrast at the CRUST1.0 Moho boundary.

Shell test 3 assesses the precision of the different codes in modelling a geometrically varying density interface. The results are depicted in Fig.

Statistical results from shell test 3: a density shell of equal thickness is modelled with a density contrast at the CRUST1.0 Moho boundary.

The results in Table

The choice of forward modelling scheme and parameterization of similar density models could lead to non-negligible local differences in the modelled gravitational signal. This could, if not properly understood, lead to erroneous interpretation of geological structures. With increasingly high-resolution gravity data sets and their associated density models this becomes an important technical modelling issue. In this section, we study the forward gravity modelling of the whole WINTERC-G model, and the data set can be found in

WINTERC-G goes through different vertical and horizontal parameterization during its two-stage inversion process

Hence, from a gravitational point of view, the WINTERC-G density model consists of 13 layers (see Table

Here, we compute the WINTERC-G-associated gravity signal by means of an independent gravity approach in order to assess the reproducibility. The calculated signal should match the gravity data inverted to build WINTERC-G, which is XGM2016. We use the GSH software for this, as it resembles the spectral code used for WINTERC-G the most.

The GSH software produces a geoid solution by computing the potential field from WINTERC-G and then divides this by 9.81 m s

Geoidal differences comparing the solutions made by the WINTERC-G-based code, the solution from GSH, and the observed XGM2016 gravity field on which WINTERC-G is based. Spherical harmonic coefficients 4–359

The total signal of the WINTERC-G model gives

In order for WINTERC-G to be useful for independent gravity-based research, the gravity field computed by our different forward modelling approaches (covering most of the commonly used techniques in solid Earth modelling) should be below the differences between WINTERC-G and XGM2016. In this section, the full WINTERC-G model is forward-modelled into the gravitational field by the selected methodologies. The approaches were kept free in selecting the best parameters (e.q. resolution, meshing) for the forward modelling result. The resulting radial gravity vector component would be examined at 250 km height above the reference sphere of 6371 km radius, and only SH degrees 2 to 179 were taken into account. The reduced spectral resolution of 179 degrees instead of 359 degrees was chosen because of reduction in computation time. The signal above 179 degrees has limited strength at 250 km altitude. This would result in differences of WINTERC-G and XGM2016 with a standard deviation of around 2 mGal, so all codes should be below these values.

Figure

The triangle integration, similarly to the results from previous tests, yields a radial gravity component very close to those from GSH and tesseroid codes: no apparent differences are visible except for the characteristic triangle features also seen in the shell tests. The residuals with GSH and tesseroids show that the triangle integration probably got close to the limit of this technique because we can see triangular artefacts only (see the mid-latitudes in Fig.

This is also the case for the ASPECT code, which has the largest differences (0.8 to 1 mGal standard deviation, with outliers up to 10 mGal). Unless a high resolution is used, the ASPECT code has difficulties in obtaining the correct gravity signal, mostly related to representing the various boundaries: surface, Moho, ice–bedrock, and other boundaries. This is mainly attributed to a lack of adequate radial resolution, as explained in Sect.

Gravity radial component at 250 km height comparison of the forward-modelled WINTERC-G model by the different forward modelling approaches. Spherical harmonic coefficients 2–179

Table

Statistical results from the WINTERC-G-grav benchmark.

The gravity field is extremely sensitive to the volume of the modelled masses and therefore the exact representation of the boundaries of individual mass layers. Different gravity signatures can be computed when you are not aware of this. The WINTERC-G lithosphere model is constructed with a spectral gravity forward modelling approach. This has consequences for the inverted densities and other physical parameters when the model is used as prior information in independent studies using different codes and gravity forward modelling and/or inversion approaches. This benchmark study was performed to (i) assess the differences arising from using different available gravity forward modelling approaches on a realistic global 3-D density distribution from the surface down to the base of the upper mantle (WINTERC-G model) and (ii) to independently assess the reproducibility of WINTERC-G from a gravity field point of view. The different tests devised in this study are summarized in Table

The GSH code is able to forward-model the WINTERC-G gravity signal to similar precision as the WINTERC-G model is intended. The variations with the XGM2016 gravity field data have similar variations as the WINTERC-G dedicated solution. The difference of

As a first “sanity check” we used a homogeneous spherical shell to show that all codes reproduce the gravity effect of such a simple model well with an exact analytical solution (Shell test 1). The largest errors are seen with the tesseroid code, but even here the relative accuracy achieved is still on the order of

A summary of the various benchmark tests described in this paper.

The result for the complete integration of the WINTERC-G model can now be interpreted due to the shell test results. Tesseroids and the spherical harmonic approach again agree very well and consistently achieve a relative agreement of 0.3 % with each other. This corresponds to the accuracy achieved with the laterally variable density structure, so this seems to be the limit in precision between these two approaches. One caveat is the observation height of 250 km, which suppresses the short-wavelength differences. If WINTERC-G were to be used as a starting model for a more regional model, integrating airborne and ground gravity data would be an important step. However, we have not compared the two methods at or near ground level, since this is computationally unfeasible for tesseroids on a global scale due to the needed increase in resolution to get similar precision. Triangles could be a viable choice to model WINTERC-G. However, at the resolution level 8 of the triangular refinement, the relative differences are still

GSH's ability to represent the laterally varying densities as much as possible makes this approach most suitable for forward modelling of global lithosphere density models. The GSH software is built for global models with laterally varying parameters, e.g. boundaries and density, but is less suitable for regional models. It is most suitable for WINTERC-G-like models based on spherical harmonic basis functions to represent the gravity field. The GSH software would be less suitable for models using a spatial forward modelling approach in the inversion, like LITHO1.0 (i.e. triangles). The resolution issue is mostly related to the Nyquist criteria. So, if the information is distributed

The tesseroid parameterization offers a great deal of flexibility, since each volume element is described explicitly. This makes tesseroids ideally suited to represent more regional geological models, which are more complicated than a simple layered structure. However, this flexibility comes at the expense of computation time because the gravity kernel needs to be evaluated for each tesseroid–station pair. This leads to a computation time scaling behaviour of

Integration of triangular grids is affected by grid irregularities so that there are multiple ways to define a volume element for each node. Furthermore, there are possibly no analytical expressions for the kernel functions (here, these functions are said to relate the point of calculation to the centre of mass of each triangle as a pointwise function). In all tests except the one for the homogeneous shell, the density structure needed to be interpolated from the native equi-angular grid into the triangular grid. Besides the differences in the mass due to the triangulation itself, the differences thus include the effect of the interpolation. The best result (compared with other integration schemes) was achieved by using the spline interpolation and the 0.5 arcdeg spatial resolution, which reduced the largest triangular artefacts significantly. What appeared very important was a vertical refinement of the data. All 13 layers spanning 400 km of mass from the surface downward were refined to thin slices with a thickness of 2 km maximally. Handling the triangular grids rather corresponds to the scattered data representation, while the integration can easily be done in parallel (here the integration was performed on an ordinary PC) and the data indexing allows for a multi-resolution approach (i.e. where possible the triangles can be divided into smaller surface and/or volume elements).

The ASPECT code is first and foremost a geodynamic code designed to solve the mass, momentum, and energy conservation equations on massively parallel architectures. Forward gravity calculations based on Gauss–Legendre quadrature were added to it as a post-processor. Despite its relying on octree-based mesh refinement

The biggest issue with the ASPECT approach is the inability to accurately model a variable density interface. Codes that cannot account for variable thickness in mass layers will find the WINTERC-G model difficult to implement. This was best seen in the ASPECT results. To investigate this more, we have examined the effect of constant layer-based codes and codes using variable geometry layers with respect to their resulting gravity field solutions.

Representing the model with varying boundary between the crust and the mantle, approximated by the spherical harmonics functions.

Representing the model in equal-thickness layers by changing the density laterally.

The difference between the two solutions is largest for radial resolution of 10 km thick layers, which is already a high radial resolution for fully global numerical models in mantle convection studies. For example, the 400 km deep WINTERC-G model is only represented by 13 layers. Differences in geoid undulations of 55 m can occur, which is more than 50 % of the observed geoid on Earth. Even with a layer thickness of 1 km, the two approaches differ significantly. The 100 m radial resolution produces sub-metre differences. At 10 m thick layers the differences between the two models become insignificant (

Currently, the ASPECT code needs an equal-thickness layer grid as an input file. Therefore, the WINTERC-G model needs to be converted to a grid-cube file. This is done by a Python parser script (attached to the paper). The grid mass elements will be calculated by taking into account the different volumes and densities in the layers of WINTERC-G. The thickness of the mass cubes can be chosen in the parser file by cutting up the WINTERC-G model in several equi-thickness layers. For this test, we have investigated the gravity difference for a cube grid of 50, 100, 200, 400, and 800 layers, equivalent to layer thicknesses of 8.109, 4.054, 2.027, 1.013, and 0.506 km. These mass cube grid models are compared to the GSH solution of the WINTERC-G layered model. The results are plotted in Fig.

Overall, the discussed approaches show similar attainable precision but have several differences with respect to handling the density models. But what about the practicality of the algorithms? What are their demands on the RAM and CPU time? How well can they be made suitable for parallel computing, and are the algorithms able to have local enhancement? The figures discussed here differ by orders of magnitude as the approaches have been implemented on different type of machines. Therefore, the exact figures for CPU usage differ due to the different hardware setup. However, the orders of magnitude already give an indication of the performance and practicality of the different approaches.

The runtime of the GSH approach for the full WINTERC-G model on a standard laptop is 3 min for the analysis of the WINTERC-G coefficients and 7 min for the synthesis, in total 10 min. The runtimes for the several shell tests in the case of GSH are negligible. The runtime of the tesseroid approach (TESS) for the complete WINTERC-G model was approximately 10 h, whereas for the simple shell tests a calculation took within 20 min. For the triangle code (TRI) orders of magnitude of the runtime for a full WINTERC-G model around are 0.5 d for modelling 204 layers and 192 000 nodes (the model was vertically and laterally refined). The shell tests took about 40 min to compute approximately 10 layers and 192 000 nodes. If the L7 resolution had been used, which is the native WINTERC-G resolution, the shell tests would have been about 5 to 10 times faster. The runtime for the WINTERC-G model in ASPECT for the

For GSH, the memory usage scales with resolution, but for the global

The GSH computation can be performed in parallel with respect to the number of layers. The layers are independent from each other and the corresponding coefficients are added to obtain the total SH coefficients of the model. However, laterally selected or regional modelling is not possible as the GSH needs global information on the layer's density distribution and its geometry. The least-squares fitting could be performed in parallel with proper numerical toolboxes. Local enhancement is not possible for GSH, which is one of the biggest drawbacks of the GSH code. Only an increase in the number of spherical harmonic coefficients would improve the resolution of the gravity output. For regional studies with high resolution, spatial forward modelling approaches are then advised. Parallelization of the tesseroid code would be straightforward. Increasing the resolution adaptable is also straightforward when tesseroids are combined with hierarchical subdivision methods like quadtrees (e.g.

This benchmark study is focused on the computation of the gravitational potential field associated with the crustal and upper mantle model WINTERC-G

Simple shell tests show that all four codes can produce similar gravitational potential fields suitable for modelling of satellite-acquired gravity data. The differences between the forward modelling schemes are all below 1.5 % of the modelled signal, and the tesseroid and GSH codes produced the most similar results (

The GSH code shows that it can produce almost similar potential fields as the internal spectral code that was used in the development of the WINTERC-G model. Mainly short-wavelength noise is seen between the two forward modelling codes that can be attributed to the different way the spherical harmonic analysis of the varying boundaries of the mass layer is performed. This produces small differences, especially at high gradient values of the boundary variations, introducing mostly short-wavelength differences. The spatial forward modelling schemes still have difficulty in reproducing similar gravity field solutions and would have to go to unrealistically high resolutions, resulting in enormous computation efforts. 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 if the density parameterization of this model is not taken into account.

The inversion code used to construct the WINTERC-G model relies on spherical harmonic forward gravity modelling code

Let the mass density

Let us now compute the gravitational potential

In view of the last expression, it is convenient to express the boundary topographies

In a particular case, when both bounding topographies are spherical,

Since there is no spherical triangular grid with constant-area surface elements, the spherical triangles necessarily differ in the sides and angles. When assigning an area to a node according to Fig.

Constant size – each node is given the same area that is proportional to a number of points on the sphere.

Local simple average – a node is given an average area value estimated with the neighbouring triangles (see Fig.

Weighted local average - similar to previous but the neighbouring triangles are weighted depending on the magnitude of the inner angle or its sine.

Sum of thirds – each node is given an area equal to a sum of thirds from surrounding triangles.

Centre of mass – each node is given the area according to Fig.

Surface element options in terms of the total spherical area and gravity residuals. For option 1 the total area is exact since the triangle area is calculated directly from

Relative volume differences per layer (from 400 km) of WINTERC-G with respect to the tesseroids.

The differences in the volume with respect to tesseroids are provided in Fig.

The layering of WINTERC-G can be viewed in Table

The WINTERC-G layering structure used in this study.

The software and model that are used in the study are all open-source. They can be found at the following locations.
The GSH approach is available at

The supplement related to this article is available online at:

BCR is the main contributor to the paper, who drafted the initial version, was responsible for the GSH results and discussions, and performed the comparison of all approaches and tests. JS was initiator of the benchmark study and was responsible for the triangle approach results and discussions, as well as reviewing the full paper. WS was the co-initiator of the benchmark study and was responsible for the tesseroid results and discussions, writing the Python parser, and reviewing the paper. CT was involved in finalizing the benchmark study and was responsible for the ASPECT results and discussions, as well as substantially reviewing the paper. ZM was involved in the initial benchmark study and developed the gravity code of the WINTERC-G model, wrote Appendix A, and reviewed the paper. JF is the developer of the WINTERC-G model and helped in the benchmark of all the forward modelling codes, benchmarked the parallel version of the WINTERC-G spectral code, wrote Appendix C, and reviewed the paper.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We would like to thank Jörg Ebbing and Roger Haagmans for their vital discussions on the topic. Furthermore, we were grateful for the efforts by Mikhail Kaban, an anonymous reviewer, and the topical editor Juliane Dannberg. Perceptually uniform colour maps were used in this study to prevent visual distortion of the data (

This research has been supported by the European Space Agency (grant no. 3-D Earth – A Dynamic Living Planet).

This paper was edited by Juliane Dannberg and reviewed by Mikhail Kaban and one anonymous referee.