Articles | Volume 13, issue 5
https://doi.org/10.5194/se-13-849-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/se-13-849-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Method article
| 
04 May 2022
Method article |
| 04 May 2022
| 04 May 2022
Benchmark forward gravity schemes: the gravity field of a realistic lithosphere model WINTERC-G
Delft University of Technology, Faculty of Aerospace Engineering, Department of Space Engineering, Delft, the Netherlands
Dublin Institute for Advanced Studies, School of Cosmic Physics, Geophysics Section, Dublin, Ireland
Josef Sebera
Astronomical Institute of the Czech Academy of Sciences, Department of Galaxies and Planetary Systems, Prague, Czech Republic
Christian Albrechts University, Department of Geosciences, Kiel, Germany
Wolfgang Szwillus
Christian Albrechts University, Department of Geosciences, Kiel, Germany
Cedric Thieulot
Utrecht University, Faculty of Geosciences, Department of Earth Sciences, Utrecht, the Netherlands
Zdeněk Martinec
Dublin Institute for Advanced Studies, School of Cosmic Physics, Geophysics Section, Dublin, Ireland
Javier Fullea
Universidad Complutense de Madrid (UCM), Physics of the Earth and Astrophysics department, Madrid, Spain
Dublin Institute for Advanced Studies, School of Cosmic Physics, Geophysics Section, Dublin, Ireland
Related authors
Marc Rovira-Navarro, Wouter van der Wal, Valentina R. Barletta, Bart C. Root, and Louise Sandberg Sørensen
Solid Earth, 11, 379–395, https://doi.org/10.5194/se-11-379-2020, https://doi.org/10.5194/se-11-379-2020, 2020
Short summary
Short summary
The Barents Sea and Fennoscandia were home to large ice sheets around 20 000 years ago. After the melting of these ice sheets, the land slowly rebounded. The rebound speed is determined by the viscosity of the deep Earth. The rebound is ongoing and causes small changes in the Earth’s gravity field, which can be measured by the GRACE satellite mission. We use these measurements to obtain the viscosity of the upper mantle and find that it is 2 times higher in Fennoscandia than in the Barents Sea.
Cedric Thieulot and Wolfgang Bangerth
EGUsphere, https://doi.org/10.5194/egusphere-2024-1668, https://doi.org/10.5194/egusphere-2024-1668, 2024
Short summary
Short summary
One of the main numerical methods in geodynamics is the finite-element method. Many types of elements have been used in the past decades in hundreds of publications. They usually fall under two categories: quadrilaterals and triangles. For the first time we compare results obtained with the most used elements of each type on a series of geodynamical benchmarks and draw conclusions as to which are the best ones and which are to be preferably avoided.
Rene Gassmöller, Juliane Dannberg, Wolfgang Bangerth, Elbridge Gerry Puckett, and Cedric Thieulot
Geosci. Model Dev., 17, 4115–4134, https://doi.org/10.5194/gmd-17-4115-2024, https://doi.org/10.5194/gmd-17-4115-2024, 2024
Short summary
Short summary
Numerical models that use simulated particles are a powerful tool for investigating flow in the interior of the Earth, but the accuracy of these models is not fully understood. Here we present two new benchmarks that allow measurement of model accuracy. We then document that better accuracy matters for applications like convection beneath an oceanic plate. Our benchmarks and methods are freely available to help the community develop better models.
Judith Freienstein, Wolfgang Szwillus, Agnes Wansing, and Jörg Ebbing
Solid Earth, 15, 513–533, https://doi.org/10.5194/se-15-513-2024, https://doi.org/10.5194/se-15-513-2024, 2024
Short summary
Short summary
Geothermal heat flow influences ice sheet dynamics, making its investigation important for ice-covered regions. Here we evaluate the sparse measurements for their agreement with regional solid Earth models, as well as with a statistical approach. This shows that some points should be excluded from regional studies. In particular, the NGRIP point, which strongly influences heat flow maps and the distribution of high basal melts, should be statistically considered an outlier.
Erik van der Wiel, Cedric Thieulot, and Douwe van Hinsbergen
EGUsphere, https://doi.org/10.31223/X56971, https://doi.org/10.31223/X56971, 2023
Short summary
Short summary
Geodynamic models of mantle convection provide a powerful tool to study the structure and composition of the Earth’s mantle. Comparing such models with other datasets is difficult. We explore the use of 'configurational entropy', which allows to quantify mixing in models. The entropy may be used to analyze the mixed state of the mantle as a whole and may also be useful to validate numerical models against anomalies in the mantle that obtained from seismology and geochemistry.
Iris van Zelst, Cedric Thieulot, and Timothy J. Craig
Solid Earth, 14, 683–707, https://doi.org/10.5194/se-14-683-2023, https://doi.org/10.5194/se-14-683-2023, 2023
Short summary
Short summary
A common simplification in subduction zone models is the use of constant thermal parameters, while experiments have shown that they vary with temperature. We test various formulations of temperature-dependent thermal parameters and show that they change the thermal structure of the subducting slab. We recommend that modelling studies of the thermal structure of subduction zones take the temperature dependence of thermal parameters into account, especially when providing insights into seismicity.
Iris van Zelst, Fabio Crameri, Adina E. Pusok, Anne Glerum, Juliane Dannberg, and Cedric Thieulot
Solid Earth, 13, 583–637, https://doi.org/10.5194/se-13-583-2022, https://doi.org/10.5194/se-13-583-2022, 2022
Short summary
Short summary
Geodynamic modelling provides a powerful tool to investigate processes in the Earth’s crust, mantle, and core that are not directly observable. In this review, we present a comprehensive yet concise overview of the modelling process with an emphasis on best practices. We also highlight synergies with related fields, such as seismology and geology. Hence, this review is the perfect starting point for anyone wishing to (re)gain a solid understanding of geodynamic modelling as a whole.
Cedric Thieulot and Wolfgang Bangerth
Solid Earth, 13, 229–249, https://doi.org/10.5194/se-13-229-2022, https://doi.org/10.5194/se-13-229-2022, 2022
Short summary
Short summary
One of the main numerical methods to solve the mass, momentum, and energy conservation equations in geodynamics is the finite-element method. Four main types of elements have been used in the past decades in hundreds of publications. For the first time we compare results obtained with these four elements on a series of geodynamical benchmarks and applications and draw conclusions as to which are the best ones and which are to be preferably avoided.
Pavol Zahorec, Juraj Papčo, Roman Pašteka, Miroslav Bielik, Sylvain Bonvalot, Carla Braitenberg, Jörg Ebbing, Gerald Gabriel, Andrej Gosar, Adam Grand, Hans-Jürgen Götze, György Hetényi, Nils Holzrichter, Edi Kissling, Urs Marti, Bruno Meurers, Jan Mrlina, Ema Nogová, Alberto Pastorutti, Corinne Salaun, Matteo Scarponi, Josef Sebera, Lucia Seoane, Peter Skiba, Eszter Szűcs, and Matej Varga
Earth Syst. Sci. Data, 13, 2165–2209, https://doi.org/10.5194/essd-13-2165-2021, https://doi.org/10.5194/essd-13-2165-2021, 2021
Short summary
Short summary
The gravity field of the Earth expresses the overall effect of the distribution of different rocks at depth with their distinguishing densities. Our work is the first to present the high-resolution gravity map of the entire Alpine orogen, for which high-quality land and sea data were reprocessed with the exact same calculation procedures. The results reflect the local and regional structure of the Alpine lithosphere in great detail. The database is hereby openly shared to serve further research.
Melchior Schuh-Senlis, Cedric Thieulot, Paul Cupillard, and Guillaume Caumon
Solid Earth, 11, 1909–1930, https://doi.org/10.5194/se-11-1909-2020, https://doi.org/10.5194/se-11-1909-2020, 2020
Short summary
Short summary
This paper presents a numerical method for restoring models of the subsurface to a previous state in their deformation history, acting as a numerical time machine for geological structures. The method relies on the assumption that rock layers can be modeled as highly viscous fluids. It shows promising results on simple setups, including models with faults and non-flat topography. While issues still remain, this could open a way to add more physics to reverse time structural modeling.
Wolfgang Szwillus, Jörg Ebbing, and Bernhard Steinberger
Solid Earth, 11, 1551–1569, https://doi.org/10.5194/se-11-1551-2020, https://doi.org/10.5194/se-11-1551-2020, 2020
Short summary
Short summary
At the bottom of the mantle (2850 km depth) two large volumes of reduced seismic velocity exist underneath Africa and the Pacific. Their reduced velocity can be explained by an increased temperature or a different chemical composition. We use the gravity field to determine the density distribution inside the Earth's mantle and find that it favors a distinct chemical composition over a purely thermal cause.
Marc Rovira-Navarro, Wouter van der Wal, Valentina R. Barletta, Bart C. Root, and Louise Sandberg Sørensen
Solid Earth, 11, 379–395, https://doi.org/10.5194/se-11-379-2020, https://doi.org/10.5194/se-11-379-2020, 2020
Short summary
Short summary
The Barents Sea and Fennoscandia were home to large ice sheets around 20 000 years ago. After the melting of these ice sheets, the land slowly rebounded. The rebound speed is determined by the viscosity of the deep Earth. The rebound is ongoing and causes small changes in the Earth’s gravity field, which can be measured by the GRACE satellite mission. We use these measurements to obtain the viscosity of the upper mantle and find that it is 2 times higher in Fennoscandia than in the Barents Sea.
Menno Fraters, Cedric Thieulot, Arie van den Berg, and Wim Spakman
Solid Earth, 10, 1785–1807, https://doi.org/10.5194/se-10-1785-2019, https://doi.org/10.5194/se-10-1785-2019, 2019
Short summary
Short summary
Three-dimensional numerical modelling of geodynamic processes may benefit strongly from using realistic 3-D starting models that approximate, e.g. natural subduction settings in the geological past or at present. To this end, we developed the Geodynamic World Builder (GWB), which enables relatively straightforward parameterization of complex 3-D geometric structures associated with geodynamic processes. The GWB is an open-source community code designed to easily interface with geodynamic codes.
Cedric Thieulot
Solid Earth, 9, 1169–1177, https://doi.org/10.5194/se-9-1169-2018, https://doi.org/10.5194/se-9-1169-2018, 2018
Short summary
Short summary
I present the GHOST (Geoscientific Hollow Sphere Tessellation) software which allows for the fast generation of computational meshes in hollow sphere geometries counting up to a hundred million cells. Each mesh is composed of concentric spherical shells made of quadrilaterals or triangles. I focus here on three commonly used meshes used in the geodynamics/geophysics community and further benchmark the gravity and gravitational potential procedures in the simple case of a constant density.
Alexis Plunder, Cédric Thieulot, and Douwe J. J. van Hinsbergen
Solid Earth, 9, 759–776, https://doi.org/10.5194/se-9-759-2018, https://doi.org/10.5194/se-9-759-2018, 2018
Short summary
Short summary
The thermal state of the Earth's crust determines how it reacts to tectonic forces and to fluid flow responsible for ore formation. We hypothesize that the angle between plate motion and convergent boundaries determines the thermal regime of subduction zones (where a plate goes under another one). Computer models and a geological reconstruction of Turkey were used to validate this hypothesis.
This research was done to validate a hypothesis made on the basis of nonquantitative field data.
Anne Glerum, Cedric Thieulot, Menno Fraters, Constantijn Blom, and Wim Spakman
Solid Earth, 9, 267–294, https://doi.org/10.5194/se-9-267-2018, https://doi.org/10.5194/se-9-267-2018, 2018
Short summary
Short summary
A nonlinear viscoplastic rheology is implemented and benchmarked in the ASPECT software, allowing for the modeling of lithospheric deformation. We showcase the new functionality with a four-dimensional model of thermomechanically coupled subduction.
Cedric Thieulot
Solid Earth, 8, 1181–1191, https://doi.org/10.5194/se-8-1181-2017, https://doi.org/10.5194/se-8-1181-2017, 2017
Short summary
Short summary
I present a new family of analytical flow solutions to the incompressible Stokes equation in a spherical shell. The velocity is tangential to both inner and outer boundaries, the viscosity is radial, and the solution has been designed so that the expressions for velocity, pressure, and body force are simple to implement in (geodynamics) codes. This forms the basis of a numerical benchmark for convection codes, and I have implemented it in two finite-element codes.
B. Hillebrand, C. Thieulot, T. Geenen, A. P. van den Berg, and W. Spakman
Solid Earth, 5, 1087–1098, https://doi.org/10.5194/se-5-1087-2014, https://doi.org/10.5194/se-5-1087-2014, 2014
Short summary
Short summary
Our paper demonstrates that the level set method is a viable method for material tracking in multi-material flow models. The different benchmarks illustate several advantages that the level set method provides over tracer-based methods. We therefore conclude that the level set method is well suited for geodynamical modeling.
C. Thieulot
Solid Earth Discuss., https://doi.org/10.5194/sed-6-1949-2014, https://doi.org/10.5194/sed-6-1949-2014, 2014
Revised manuscript has not been submitted
I. Sasgen, H. Konrad, E. R. Ivins, M. R. Van den Broeke, J. L. Bamber, Z. Martinec, and V. Klemann
The Cryosphere, 7, 1499–1512, https://doi.org/10.5194/tc-7-1499-2013, https://doi.org/10.5194/tc-7-1499-2013, 2013
Related subject area
Subject area: Core and mantle structure and dynamics | Editorial team: Geodesy, gravity, and geomagnetism | Discipline: Geodesy
Increased density of large low-velocity provinces recovered by seismologically constrained gravity inversion
GRACE constraints on Earth rheology of the Barents Sea and Fennoscandia
Wolfgang Szwillus, Jörg Ebbing, and Bernhard Steinberger
Solid Earth, 11, 1551–1569, https://doi.org/10.5194/se-11-1551-2020, https://doi.org/10.5194/se-11-1551-2020, 2020
Short summary
Short summary
At the bottom of the mantle (2850 km depth) two large volumes of reduced seismic velocity exist underneath Africa and the Pacific. Their reduced velocity can be explained by an increased temperature or a different chemical composition. We use the gravity field to determine the density distribution inside the Earth's mantle and find that it favors a distinct chemical composition over a purely thermal cause.
Marc Rovira-Navarro, Wouter van der Wal, Valentina R. Barletta, Bart C. Root, and Louise Sandberg Sørensen
Solid Earth, 11, 379–395, https://doi.org/10.5194/se-11-379-2020, https://doi.org/10.5194/se-11-379-2020, 2020
Short summary
Short summary
The Barents Sea and Fennoscandia were home to large ice sheets around 20 000 years ago. After the melting of these ice sheets, the land slowly rebounded. The rebound speed is determined by the viscosity of the deep Earth. The rebound is ongoing and causes small changes in the Earth’s gravity field, which can be measured by the GRACE satellite mission. We use these measurements to obtain the viscosity of the upper mantle and find that it is 2 times higher in Fennoscandia than in the Barents Sea.
Cited articles
Abramowitz, M. and Stegun, I. (Eds.): Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 13 9780486612720, 1972.
Afonso, J. C., Fernández, M., Ranalli, G., Griffin, W. L., and Connolly, J. A. D.: Integrated geophysical-petrological modeling of the lithosphere and sublithospheric upper mantle: Methodology and applications, Geochem. Geophy. Geosy., 9, 1–36, 2008. a
Afonso, J. C., Fullea, J., Griffin W. L. , Yang, Y., Jones, A. G., Connolly, J. A. D., and O'Reilly, S. Y.: 3-D multi-observable probabilistic inversion for the compositional and thermal structure of the lithosphere and upper mantle, I: a priori petrological information and geophysical observables, J. Geophys. Res.-Sol. Ea., 118, 2586–2617, https://doi.org/10.1029/2007GC001834, 2013. a
Afonso, J. C., Fullea, J., Yang, Y., Connolly, J. A. D., and Jones, A. G.: 3-D multi-observable probabilistic inversion for the compositional and thermal structure of the lithosphere and upper mantle, II: General methodology and resolution analysis, J. Geophys. Res.-Sol. Ea., 118, 1650–1676, https://doi.org/10.1002/jgrb.50123, 2013. a
Afonso, J. C., Moorkamp, M., and Fullea, J.: Imaging the lithosphere and upper mantle: Where we are at and where we are going,
Integrated Imaging of the Earth: Theory and Applications, First Edition, edited by: Moorkamp, M., Lelièvre, P. G., Linde, N., Khan, A., John Wiley & Sons, Hoboken, N. J., 191–218, ISBN 9781118929063, 2016. a
Afonso, J. C., Salajegheh F., Szwillus, W., Ebbing, J., and Gaina, C.: A global reference model of the lithosphere and upper mantle from joint inversion and analysis of multiple data sets, Geophys. J. Int., 217, 1602–1628, https://doi.org/10.1093/gji/ggz094, 2019.
Asgharzadeh, M. F., von Frese, R. R. B., Kim, H. R., Leftwich, T. E., and Kim, J. W.: Spherical prism gravity effects by Gauss-Legendre quadrature integration, Geophys. J. Int., 169, 1–11, https://doi.org/10.1111/j.1365-246X.2007.03214.x, 2007. a
Bangerth, W., Dannberg, J., Gassmoeller, R., and Heister, T.: ASPECT v2.2.0, (version v2.2.0), Zenodo [code], https://doi.org/10.5281/ZENODO.3924604, 2020. a
Becker, T. W. and Boschi, L.: A comparison of tomographic and geodynamic mantle models, Geochem. Geophy. Geosy., 3, 2001GC000168, https://doi.org/10.1029/2001GC000168, 2002.
Burstedde, C., Wilcox, L. C., and Ghattas, O.: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees, SIAM J. Sci. Comp., 33, 1103–1133, https://doi.org/10.1137/100791634, 2011. a, b
Cammarano, F., Tackley, P., and Boschi, L.: Seismic, petrological and geodynamical constraints on thermal and compositional structure of the upper mantle: global thermochemical models, Geophys. J. Int., 187, 1301–1318, https://doi.org/10.1111/j.1365-246X.2011.05223.x, 2011.
D'Urso, M. G.: Analytical computation of gravity effects for polyhedral bodies, J. Geodesy, 88, 13–29, https://doi.org/10.1007/s00190-013-0664-x, 2014. a
Forte, A. M.: Constraints on seismic models from other disciplines - Implications for mantle dynamics and composition,
in: Treatise of Geophysics, 1, 805–857, Elsevier, https://doi.org/10.1016/B978-044452748-6.00027-4, 2007.
Fullea, J., Afonso, J. C., Connolly, J. A. D., Fernàndez, M., García-Castellanos, D., and Zeyen, H.: LitMod3-D: an interactive 3-D software to model the thermal, compositional, density, seismological, and rheological structure of the lithosphere and sublithospheric upper mantle, Geochem. Geophy. Geosy., 10, Q08019, https://doi.org/10.1029/2009GC002391, 2009. a
Fullea, J., Rodríguez-González, J., Charco, M., Martinec, Z., Negredo, A., and Villaseñor, A.: Perturbing effects of sub-lithospheric mass anomalies in GOCE gravity gradient and other gravity data modeling: Application to the Atlantic-Mediterranean transition zone, Int. J. Appl. Earth Obs., 35, 54–69, https://doi.org/10.1016/j.jag.2014.02.003, 2015. a
Fullea, J., Lebedev, S., Martinec, Z., and Celli, N.: WINTERC-G: mapping the upper mantle thermochemical heterogeneity from coupled geophysical-petrological inversion of seismic waveforms, heat flow, surface elevation and
gravity satellite data, Geophys. J. Int., 226, 146–191, https://doi.org/10.1093/gji/ggab094, 2020. a, b, c, d, e, f, g, h, i, j, k, l, m, n
Fullea, J., Lebedev, S., Martinec, Z., and Celli, N.: WINTERC-G: a global upper mantle thermochemical model from coupled geophysical-petrological inversion of seismic waveforms, heat flow, surface elevation and gravity satellite data [data set], Geophys. J. Int. (version 5.4, 226, 146–191). Zenodo, https://doi.org/10.5281/zenodo.5730195, 2021. a, b
Grombein, T., Seitz, K., and Heck, B.: Optimized formulas for the gravitational field of a tesseroid, J. Geodesy, 87, 645–660, https://doi.org/10.1007/s00190-013-0636-1, 2013. a
Grombein, T., Luo, X., Seitz, K., and Heck, B.: A wavelet-based assessment of topographic-isostatic reductions for GOCE gravity gradients, Surv. Geophys., 35, 959–982, https://doi.org/10.1007/s10712-014-9283-1, . a
Heck, B. and Seitz, K.: A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modeling, J. Geodesy, 81, 121–136, https://doi.org/10.1007/s00190-006-0094-0, 2007. a, b, c
Heiskanen, W. A. and Moritz, H.: Physical Geodesy, Reprint, Institute of Physical Geodesy, Technical University Graz, Austria, https://doi.org/10.1007/b139113, 1984.
Hirt, C. and Kuhn, M.: Band-limited topographic mass distribution generates full-spectrum gravity field: Gravity forward modeling in the spectral and spatial domains revisited, J. Geophys. Res. Sol.-Ea., 119, 3646–3661, https://doi.org/10.1002/2013JB010900, 2014. a, b
Holzrichter, N. and Ebbing, J.: A regional background model for the Arabian Peninsula from modeling
satellite gravity gradients and their invariants, Tectonophysics, 692, 86–94, https://doi.org/10.1016/j.tecto.2016.06.002, 2016. a
Heister, T., Dannberg, J., Gassmöller, R., and Bangerth, W.: High Accuracy Mantle Convection Simulation through Modern Numerical Methods, II: Realistic Models and Problems, Geophys. J. Int., 210, 833–851, https://doi.org/10.1093/gji/ggx195, 2017. a, b, c
Kaban, M. K., Tesauro, M., and Cloetingh, S.: An integrated gravity model for Europe's crust and upper mantle, Earth Planet Sc. Lett., 296, 195–206, https://doi.org/10.1016/j.epsl.2010.04.041, 2010. a
Kaban, M. K., Tesauro, M., Mooney, W. D., and Cloetingh, S.: Density, temperature, and composition
of the North American lithosphere – New insights from a joint analysis of seismic gravity, and mineral physics data:
1. Density structure of the crust and upper mantle, Geochem. Geophy. Geosy., 15, 4781–4807, https://doi.org/10.1002/2014GC005483, 2014. a
Kimerling, J. A., Sahr, K., White, D., and Song, L.: Comparing geometrical properties of global grids, Cartogr. Geogr. Inf. Sc., 26, 271–288, https://doi.org/10.1559/152304099782294186, 1999. a
Kronbichler, M., Heister, T., and Bangerth, W.: High accuracy mantle convection simulation through modern numerical methods,
Geophys. J. Int., 191, 12–29, https://doi.org/10.1111/j.1365-246X.2012.05609.x, 2012. a, b, c
Kuhn, M., Featherstone, W. E., and Kirby, J. F.: Complete spherical Bouguer gravity anomalies over Australia, Austr. J. Earth Sci., 56, 213–223, https://doi.org/10.1080/08120090802547041, 2009. a
Lachapelle, G.: A Spherical Harmonic Expansion of the Isostatic Reduction Potential, Bollettino di Geodesia E Scienze Affini, 3, 1976. a
Laske, G., Masters, G., Ma, Z., and Pasyanos, M.: Update on CRUST1.0 – A 1-degree global model of Earth's crust, Geophysical Research Abstract, 15, EGU2013-2658, http://igppweb.ucsd.edu/gabi/rem.html, 2013. a
Martinec, Z., Pěč, K., and Burša, M.: The Phobos gravitational field modeled on the basis of its topography, Earth Moon Planet, 145, 219–235, https://doi.org/10.1007/BF00057745, 1989. a
Moritz, H.: Geodetic reference system 1980, J. Geodesy, 54, 395–405, https://doi.org/10.1007/s001900050278, 1980. a
Nagy, D., Papp, G., and Benedek, J.: The gravitational potential and its derivatives for the prism, J. Geodesy, 74, 552–560, https://doi.org/10.1007/s001900000116, 2000. a
Novák, P. and Grafarend, E.: The effect of topographical and atmospheric masses on space borne gravimetric and gradiometric data, Studia Geophysica et Geodaetica, 50, 549–582, https://doi.org/10.1007/s11200-006-0035-7, 2006. a
Pail, R., Bingham, R., Braitenberg, C., Dobslaw, H., Eicker, A., Güntner, A., Horwath, N., Ivins, E., Longuevergne, L., Panet, I., and Wouter, B.: Science and User Needs for Observing Global Mass Transport to Understand Global Change and to Benefit Society, Surv. Geophys., 36, 743–-772, https://doi.org/10.1007/s10712-015-9348-9, 2015. a
Pail, R., Fecher, T., Barnes, D., Factor, J. F., Holmes, S. A., Gruber, T., and Zingerle, P.: Short note: the experimental geopotential model XGM2016, J. Geodesy, 92, 443–451, https://doi.org/10.1007/s00190-017-1070-6, 2018. a
Pasyanos, M. E., Masters, T. G., Laske, G., and Ma, Z.: LITHO1. 0: An updated crust and lithospheric model of the Earth, J. Geophys. Res.-Sol. Ea., 119, 2153–2173, https://doi.org/10.1002/2013JB010626, 2014. a, b
Pavlis, N. K. and Rapp, R. H.: The development of an isostatic gravitational model to degree 360 and its use in global gravity modeling, Geophys. J. Int., 100, 369–378, https://doi.org/10.1111/j.1365-246X.1990.tb00691.x, 1990. a
Rapp, R. H.: Degree variances of the Earth's potential, topography and its isostatic compensation, Bull. Geodesique, 56, 84–94, https://doi.org/10.1007/BF02525594, 1982. a
Root, B. C.: GSH is a MATLAB package to do Global Spherical Harmonic Analyses (GSHA) and Synthesis (GSHS) for Crust1.0., 4TU, Research Data, Software [code], https://doi.org/10.4121/16764238.v1, 2021. a
Schaeffer, A. J. and Lebedev, S.: Global shear speed structure of the upper mantle and transition zone, Geophys. J. Int., 194, 417–449, https://doi.org/10.1093/gji/ggt095, 2013. a
Sebera, J., Haagmans, R., Floberghagen, R., and Ebbing J., Gravity spectra from the density distribution of Earth's uppermost 435 km, Surv. Geophys., 39, 227–244, https://doi.org/10.1007/s10712-017-9445-z, 2018. a, b, c, d
Simmons, N. A., Forte, A. M., Boschi, L., and Grand, S. P.: GyPSuM: A joint tomographic model of mantle density and seismic wave speeds,
J. Geophys. Res., 115, B12310, https://doi.org/10.1029/2010JB007631, 2010.
Sneeuw, N.: Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective, Geophys. J. Int., 118, 707–716, https://doi.org/10.1111/j.1365-246X.1994.tb03995.x, 1994. a
Szwillus, W. and Götze, H.-J.: Efficient Mass Correction Using an Adaptive Method, in: Understanding the Bouguer Anomaly, edited by: Pašteka, R., Mikuška, J. and Meurers, B., Elsevier, 93–112, https://doi.org/10.1016/B978-0-12-812913-5.00005-1, 2017. a, b
Thieulot, C.: GHOST: Geoscientific Hollow Sphere Tessellation, Solid Earth, 9, 1169–1177, https://doi.org/10.5194/se-9-1169-2018, 2018. a, b
Uieda, L.: Tesseroids v1.1.1: Forward modeling of gravitational fields in spherical coordinates, Zenodo [code], https://doi.org/10.5281/zenodo.15800, 2015. a
Uieda, L., Barbosa, V. C. F., and Braitenberg, C.: Tesseroids: Forward-modeling gravitational fields in spherical coordinates, Geophysics, 81, 41–48, https://doi.org/10.1190/geo2015-0204.1, 2016. a, b, c, d
Varshalovich, D. A., Moskalev, A. N., and Khersonskii, V. K.: Quantum Theory of Angular Momentum, World Scientific Publication, Singapore, https://doi.org/10.1142/0270, 1989. a, b
Wang, Z. and Dahlen, F. A.: Spherical-spline parameterization of three-dimensional Earth models, Geophys. Res. Lett., 22, 3099–-3102, https://doi.org/10.1029/95GL03080, 1995. a, b, c, d
Werner, R. A. and Scheeres, D. J.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia, Cell, 65, 313–344, https://doi.org/10.1007/BF00053511, 1996. a
Wild-Pfeiffer, F.: A comparison of different mass elements for use in gravity gradiometry, J. Geodesy, 82, 637–653, https://doi.org/10.1007/s00190-008-0219-8, 2008. a
Short summary
Several alternative gravity modelling techniques 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 differences of the various approaches. In this paper, we discuss the four widely used techniques: spherical harmonics, tesseroid integration, triangle integration, and hexahedral integration.
Several alternative gravity modelling techniques and associated numerical codes with their own...
