{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T22:00:44Z","timestamp":1764194444880,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,3,26]],"date-time":"2020-03-26T00:00:00Z","timestamp":1585180800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>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\u2019s 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\u201d 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.<\/jats:p>","DOI":"10.3390\/rs12071063","type":"journal-article","created":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T03:44:13Z","timestamp":1585712653000},"page":"1063","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["TGF: A New MATLAB-based Software for Terrain-related Gravity Field Calculations"],"prefix":"10.3390","volume":"12","author":[{"given":"Meng","family":"Yang","sequence":"first","affiliation":[{"name":"Institute for Astronomical and Physical Geodesy, 85748 TU Munich, Germany"}]},{"given":"Christian","family":"Hirt","sequence":"additional","affiliation":[{"name":"Institute for Astronomical and Physical Geodesy, 85748 TU Munich, Germany"},{"name":"Institute for Advanced Study, 85748 TU Munich, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4364-4012","authenticated-orcid":false,"given":"Roland","family":"Pail","sequence":"additional","affiliation":[{"name":"Institute for Astronomical and Physical Geodesy, 85748 TU Munich, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4618","DOI":"10.1029\/2019GL082521","article-title":"SRTM2gravity: An Ultrahigh Resolution Global Model of Gravimetric Terrain Corrections","volume":"46","author":"Hirt","year":"2019","journal-title":"Geophys. Res. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"7843","DOI":"10.1029\/JB086iB09p07843","article-title":"The use of height data in gravity field approximation by collocation","volume":"86","author":"Forsberg","year":"1981","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Sans\u00f2, F., and Sideris, M.G. (2013). Topographic Reductions in Gravity and Geoid Modeling. Geoid Determination, Springer.","DOI":"10.1007\/978-3-540-74700-0"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Elhabiby, M., Sampietro, D., Sans\u00f2, F., and Sideris, M.G. (2009). BVP, Global Models and Residual Terrain Correction. Observing our Changing Earth, Springer.","DOI":"10.1007\/978-3-540-85426-5_25"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00190-007-0159-8","article-title":"Effects of topographic\u2013isostatic masses on gravitational functionals at the Earth\u2019s surface and at airborne and satellite altitudes","volume":"82","author":"Makhloof","year":"2007","journal-title":"J. Geod."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Forsberg, R. (1984). A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling, The Ohio State University.","DOI":"10.21236\/ADA150788"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1080\/01490419.2013.779334","article-title":"RTM Gravity Forward-Modeling Using Topography\/Bathymetry Data to Improve High-Degree Global Geopotential Models in the Coastal Zone","volume":"36","author":"Hirt","year":"2013","journal-title":"Mar. Geod."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6846","DOI":"10.1002\/2016JB013249","article-title":"Topographic gravity modeling for global Bouguer maps to degree 2160: Validation of spectral and spatial domain forward modeling techniques at the 10 microGal level","volume":"121","author":"Hirt","year":"2016","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_9","first-page":"570","article-title":"Vertical datum unification for the International Height Reference System (IHRS)","volume":"209","author":"Sideris","year":"2017","journal-title":"Geophys. J. Int."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s00190-011-0533-4","article-title":"Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies","volume":"86","author":"Balmino","year":"2011","journal-title":"J. Geod."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hirt, C., and Kuhn, K. (2012). Evaluation of high-degree series expansions of the topographic potential to higher-order powers. J. Geophys. Res. Solid Earth, 117.","DOI":"10.1029\/2012JB009492"},{"key":"ref_12","unstructured":"Rexer, M. (2017). Spectral Solutions to the Topographic Potential in the context of High-Resolution Global Gravity Field Modelling. [Doctoral Dissertation, Technical University of Munich]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s00190-018-1139-x","article-title":"Cap integration in spectral gravity forward modelling: Near- and far-zone gravity effects via Molodensky\u2019s truncation coefficients","volume":"93","author":"Bucha","year":"2008","journal-title":"J. Geod."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2574","DOI":"10.1029\/2018GC007529","article-title":"SHTools: Tools for Working with Spherical Harmonics","volume":"19","author":"Wieczorek","year":"2018","journal-title":"Geochem. Geophys. Geosyst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1007\/s10712-015-9345-z","article-title":"2015 Ultra-high-Degree SHA Gauss\u2013Legendre Driscoll\/Healy Quadrature Theorem Earth, Mars and Moon","volume":"36","author":"Rexer","year":"2015","journal-title":"Surv. Geophys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1035","DOI":"10.1007\/s10712-016-9382-2","article-title":"Layer-Based Modelling of the Earth\u2019s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation","volume":"37","author":"Rexer","year":"2016","journal-title":"Surv. Geophys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s10712-016-9376-0","article-title":"The Rock\u2013Water\u2013Ice Topographic Gravity Field Model RWI_TOPO_2015 and Its Comparison to a Conventional Rock-Equivalent Version","volume":"37","author":"Grombein","year":"2016","journal-title":"Surv. Geophys."},{"key":"ref_18","first-page":"7725","article-title":"High-resolution global forward modelling - A degree-5480 global ellipsoidal topographic potential model","volume":"19","author":"Rexer","year":"2017","journal-title":"Eur. Geosci. Union Gen. Assembl."},{"key":"ref_19","unstructured":"Abrykosov, O., Ince, E.S., Foerste, C., and Flechtner, F. (2020, January 13). Rock-Ocean-Lake-Ice Topographic Gravity Field model (ROLI Model) Expanded up to Degree 3660. Available online: http:\/\/doi.org\/10.5880\/ICGEM.2019.011."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"B02404","DOI":"10.1029\/2008JB005639","article-title":"Evaluation of precise terrain effects using high-resolution digital elevation models","volume":"114","author":"Tsoulis","year":"2009","journal-title":"J. Geophys. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1016\/j.cageo.2003.08.003","article-title":"Terrain correction computation using Gaussian quadrature","volume":"29","author":"Hwang","year":"2003","journal-title":"Comput. Geosci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"F41","DOI":"10.1190\/geo2015-0204.1","article-title":"Tesseroids: Forward-modeling gravitational fields in spherical coordinates","volume":"81","author":"Uieda","year":"2016","journal-title":"Geophysics"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"F1","DOI":"10.1190\/geo2010-0334.1","article-title":"Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals","volume":"77","author":"Tsoulis","year":"2012","journal-title":"Geophysics"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s00190-008-0219-8","article-title":"A comparison of different mass elements for use in gravity gradiometry","volume":"82","year":"2008","journal-title":"J. Geod."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.cageo.2015.07.015","article-title":"GTeC\u2014A versatile MATLAB\u00ae tool for a detailed computation of the terrain correction and Bouguer gravity anomalies","volume":"84","author":"Cella","year":"2015","journal-title":"Comput. Geosci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4279","DOI":"10.1002\/grl.50838","article-title":"New ultrahigh-resolution picture of Earth\u2019s gravity field","volume":"40","author":"Hirt","year":"2013","journal-title":"Geophys. Res. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1029\/2018GC008033","article-title":"3-D Density, Thermal, and Compositional Model of the Antarctic Lithosphere and Implications for Its Evolution","volume":"20","author":"Haeger","year":"2019","journal-title":"Geochem. Geophys. Geosyst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1007\/s11200-015-1258-2","article-title":"Gravimetric geoid for Egypt implementing Moho depths and optimal geoid fitting approach","volume":"61","year":"2017","journal-title":"Stud. Geophys. Geod."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"552","DOI":"10.1007\/s001900000116","article-title":"The gravitational potential and its derivatives for the prism","volume":"74","author":"Nagy","year":"2000","journal-title":"J. Geod."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/s00190-002-0264-7","article-title":"Corrections to \u201cThe gravitational potential and its derivatives for the prism\u201d","volume":"76","author":"Nagy","year":"2002","journal-title":"J. Geod."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s00190-006-0094-0","article-title":"A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling","volume":"81","author":"Heck","year":"2006","journal-title":"J. Geod."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1007\/s00190-013-0636-1","article-title":"Optimized formulas for the gravitational field of a tesseroid","volume":"87","author":"Grombein","year":"2013","journal-title":"J. Geod."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1007\/s11200-017-0656-z","article-title":"Experiences with the use of mass-density maps in residual gravity forward modelling","volume":"62","author":"Yang","year":"2018","journal-title":"Stud. Geophys. Geod."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1469","DOI":"10.1007\/s00190-019-01261-x","article-title":"A numerical study of residual terrain modelling (RTM) techniques and the harmonic correction using ultra-high-degree spectral gravity modelling","volume":"93","author":"Hirt","year":"2019","journal-title":"J. Geod."},{"key":"ref_35","unstructured":"Laske, G., Masters, G., Ma, Z.T., and Pasyanos, M. (2013, January 23\u201328). Update on CRUST1.0 A 1-degree Global Model of Earth\u2019s Crust. Proceedings of the EGU General Assembly, Vienna, Austria."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1016\/j.cageo.2010.07.010","article-title":"A digital rock density map of New Zealand","volume":"37","author":"Tenzer","year":"2011","journal-title":"Comput. Geosci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1007\/s00190-016-0917-6","article-title":"Topographic gravitational potential up to second-order derivatives: An examination of approximation errors caused by rock-equivalent topography (RET)","volume":"90","author":"Kuhn","year":"2016","journal-title":"J. Geod."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"3646","DOI":"10.1002\/2013JB010900","article-title":"Band-limited topographic mass distribution generates full-spectrum gravity field: Gravity forward modeling in the spectral and spatial domains revisited","volume":"119","author":"Hirt","year":"2014","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"5844","DOI":"10.1002\/2017GL072874","article-title":"A high-accuracy map of global terrain elevations","volume":"44","author":"Yamazaki","year":"2017","journal-title":"Geophys. Res. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1093\/gji\/ggz264","article-title":"The tree-canopy effect in gravity forward modelling","volume":"219","author":"Yang","year":"2019","journal-title":"Geophys. J. Int."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/12\/7\/1063\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:11:47Z","timestamp":1760173907000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/12\/7\/1063"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,26]]},"references-count":40,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["rs12071063"],"URL":"https:\/\/doi.org\/10.3390\/rs12071063","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2020,3,26]]}}}