{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T21:48:43Z","timestamp":1775857723067,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T00:00:00Z","timestamp":1654905600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T00:00:00Z","timestamp":1654905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Earth Sci Inform"],"published-print":{"date-parts":[[2022,9]]},"DOI":"10.1007\/s12145-022-00822-7","type":"journal-article","created":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T13:05:08Z","timestamp":1654952708000},"page":"2017-2029","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A semi-vectorized and relationally-operated algorithm for fast geoid computation using Stokes\u2019s integration"],"prefix":"10.1007","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4305-8248","authenticated-orcid":false,"given":"Ahmed","family":"Abdalla","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2209-9921","authenticated-orcid":false,"given":"Vagner","family":"Ferreira","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,6,11]]},"reference":[{"issue":"1","key":"822_CR1","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/s12145-014-0149-3","volume":"8","author":"RA Abbak","year":"2015","unstructured":"Abbak RA, Ustun A (2015) A software package for computing a regional gravimetric geoid model by the KTH method. Earth Sci Inform 8(1):255\u2013265. https:\/\/doi.org\/10.1007\/s12145-014-0149-3","journal-title":"Earth Sci Inform"},{"issue":"1","key":"822_CR2","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/s12145-021-00713-3","volume":"15","author":"RA Abbak","year":"2022","unstructured":"Abbak RA, Ellmann A, Ustun A (2022) A practical software package for computing gravimetric geoid by the least squares modification of Hotine\u2019s formula. Earth Sci Inform 15 (1):713\u2013724. https:\/\/doi.org\/10.1007\/s12145-021-00713-3","journal-title":"Earth Sci Inform"},{"key":"822_CR3","volume-title":"The combined modelling of the regional quasigeoid of New Zealand using gravity and GPS-levelling data","author":"A Abdalla","year":"2013","unstructured":"Abdalla A (2013) The combined modelling of the regional quasigeoid of New Zealand using gravity and GPS-levelling data. University of Otago, PhD thesis"},{"issue":"20","key":"822_CR4","doi-asserted-by":"publisher","first-page":"2115","DOI":"10.1007\/s12517-021-08450-z","volume":"14","author":"A Abdalla","year":"2021","unstructured":"Abdalla A (2021) Development of Sudan gravity field by combining local and global data. Arab J Geosci 14(20):2115. https:\/\/doi.org\/10.1007\/s12517-021-08450-z","journal-title":"Arab J Geosci"},{"issue":"4","key":"822_CR5","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.jafrearsci.2011.02.012","volume":"60","author":"A Abdalla","year":"2011","unstructured":"Abdalla A, Fairhead D (2011) A new gravimetric geoid model for Sudan using the KTH method. J Afr Earth Sci 60(4):213\u2013221. https:\/\/doi.org\/10.1016\/j.jafrearsci.2011.02.012","journal-title":"J Afr Earth Sci"},{"issue":"10","key":"822_CR6","doi-asserted-by":"publisher","first-page":"823","DOI":"10.1139\/cjes-2014-0192","volume":"50","author":"A Abdalla","year":"2015","unstructured":"Abdalla A, Mogren S (2015) Implementation of a rigorous least-squares modification of Stokes formula to compute a gravimetric geoid model over Saudi Arabia (SAGEO13). Can J Earth Sci 50(10):823\u2013832. https:\/\/doi.org\/10.1139\/cjes-2014-0192","journal-title":"Can J Earth Sci"},{"key":"822_CR7","unstructured":"Abdalla A, Denker H, M\u00fcller J (2018) A combined optimisation of local and global gravimetric data to improve geoid modelling in Sudan, p 5241"},{"key":"822_CR8","unstructured":"\u00c5gren J (2004) Regional geoid determination methods for the era of satellite gravimetry: numerical investigations using synthetic earth gravity models. Doctoral dissertation Royal Institute of Technology (KTH), Stockholm"},{"key":"822_CR9","unstructured":"Albertella A, Sans\u00f3 F, Wenzel G, Pavlis NK, Forsberg R (2008) International School for the Determination and Use of the Geoid: Lecture Notes. International Geoid Service, DIIAR - Politecnico di Milano"},{"issue":"1","key":"822_CR10","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s00190-008-0232-y","volume":"83","author":"MJ Amos","year":"2009","unstructured":"Amos MJ, Featherstone WE (2009) Unification of new zealand\u2019s local vertical datums: iterative gravimetric quasigeoid computations. J Geod 83(1):57\u201368. https:\/\/doi.org\/10.1007\/s00190-008-0232-y","journal-title":"J Geod"},{"key":"822_CR11","unstructured":"Drepper U (2007) What every programmer should know about memory. https:\/\/doi.org\/10.1.1.91.957"},{"issue":"1\u20133","key":"822_CR12","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/s00190-005-0438-1","volume":"79","author":"A Ellmann","year":"2005","unstructured":"Ellmann A (2005) Two deterministic and three stochastic modifications of Stokes\u2019s formula: a case study for the Baltic countries. J Geod 79(1\u20133):11\u201323","journal-title":"J Geod"},{"issue":"3","key":"822_CR13","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/s12145-010-0062-3","volume":"3","author":"M Eshagh","year":"2010","unstructured":"Eshagh M, Abdollahzadeh M (2010) Semi-vectorization: an efficient technique for synthesis and analysis of gravity gradiometry data. Earth Sci Inform 3(3):149\u2013158. https:\/\/doi.org\/10.1007\/s12145-010-0062-3","journal-title":"Earth Sci Inform"},{"key":"822_CR14","unstructured":"Forsberg R, Tscherning CC (2008) An overview manual for the GRAVSOFT geodetic gravity field modelling programs. Contract report for JUPEM"},{"key":"822_CR15","doi-asserted-by":"publisher","unstructured":"Franchetti F, Voronenko Y, P\u00fcschel M (2006) A rewriting system for the vectorization of signal transforms. In: High performance computing for computational science (VECPAR), pp 363\u2013377. https:\/\/doi.org\/10.1007\/978-3-540-71351-7_28","DOI":"10.1007\/978-3-540-71351-7_28"},{"key":"822_CR16","volume-title":"Physical geodesy. Series of books in geology","author":"W Heiskanen","year":"1967","unstructured":"Heiskanen W, Moritz H (1967) Physical geodesy. Series of books in geology. W.H. Freeman, San Francisco"},{"key":"822_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-211-33545-1","volume-title":"Physical geodesy","author":"B Hofmann-Wellenhof","year":"2005","unstructured":"Hofmann-Wellenhof B, Moritz H (2005) Physical geodesy, 2nd edn. Wien, SpringerWienNewYork. https:\/\/doi.org\/10.1007\/978-3-211-33545-1","edition":"2nd edn."},{"issue":"3","key":"822_CR18","doi-asserted-by":"publisher","first-page":"374","DOI":"10.1023\/A:1022160504156","volume":"44","author":"J Huang","year":"2000","unstructured":"Huang J, Van\u00ed\u010dek P, Nov\u00e1k P (2000) An alternative algorithm to FFT for the numerical evaluation of Stokes\u2019s integral. Studia Geophysica et Geodaetica 44(3):374\u2013380. https:\/\/doi.org\/10.1023\/A:1022160504156","journal-title":"Studia Geophysica et Geodaetica"},{"key":"822_CR19","unstructured":"ICGES (2019) International Centre for Global Earth Models (ICGES). http:\/\/icgem.gfz-potsdam.de\/home, accessed: 2\/13\/2019"},{"issue":"8","key":"822_CR20","doi-asserted-by":"publisher","first-page":"6985","DOI":"10.1029\/JB086iB08p06985","volume":"86","author":"C Jekeli","year":"1981","unstructured":"Jekeli C (1981) Modifying stokes function to reduce the error of geoid undulation computations. J Geophys Res 86(8):6985\u20136990","journal-title":"J Geophys Res"},{"key":"822_CR21","unstructured":"Kiamehr R, Sj\u00f6berg EL (2010) KTH-GEOLAB scientific software for precise geoid determination based on the least-squares modification of Stokes\u2019 formula. Royal Institute of Technology (KTH) Division of geodesy, Sweden Technical manual, Stockholm"},{"issue":"19","key":"822_CR22","doi-asserted-by":"publisher","first-page":"2003","DOI":"10.1007\/s12517-021-08247-0","volume":"14","author":"MF Pa\u2019suya","year":"2021","unstructured":"Pa\u2019suya MF, Din AHM, Yusoff MYM, Abbak RA, Hamden MH (2021) Refinement of gravimetric geoid model by incorporating terrestrial, marine, and airborne gravity using KTH method. Arab J Geosci 14 (19):2003. https:\/\/doi.org\/10.1007\/s12517-021-08247-0","journal-title":"Arab J Geosci"},{"issue":"5","key":"822_CR23","first-page":"371","volume":"89","author":"LE Sj\u00f6berg","year":"1980","unstructured":"Sj\u00f6berg LE (1980) Least squares combination of satellite harmonics and integral formulas in physical geodesy. Gerlands Beitraege zur Geophysik 89(5):371\u2013377","journal-title":"Gerlands Beitraege zur Geophysik"},{"key":"822_CR24","volume-title":"Least squares modification of Stokes\u2019 and Vening Meinesz\u2019 formulas by accounting for errors of truncation, potential coefficients and gravity data. University of Uppsala","author":"LE Sj\u00f6berg","year":"1984","unstructured":"Sj\u00f6berg LE (1984) Least squares modification of Stokes\u2019 and Vening Meinesz\u2019 formulas by accounting for errors of truncation, potential coefficients and gravity data. University of Uppsala. Institute of Geophysics, Dept. of Geodesy"},{"key":"822_CR25","first-page":"367","volume":"16","author":"LE Sj\u00f6berg","year":"1991","unstructured":"Sj\u00f6berg LE (1991) Refined least-squares modification of Stokes formula. Manuscr Geodaet 16:367\u2013375","journal-title":"Manuscr Geodaet"},{"issue":"7\u20138","key":"822_CR26","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1007\/s00190-003-0346-1","volume":"77","author":"LE Sj\u00f6berg","year":"2003","unstructured":"Sj\u00f6berg L E (2003) A general model for modifying Stokes\u2019 formula and its least-squares solution. J Geod 77(7\u20138):459\u2013464. https:\/\/doi.org\/10.1007\/s00190-003-0346-1","journal-title":"J Geod"},{"key":"822_CR27","unstructured":"USGS (2022) What is a Geoid? Why do we use it and where does its shape come from? https:\/\/www.usgs.gov\/faqs\/what-geoid-why-do-we-use-it-and-where-does-its-shape-come, accessed: 03.15.2022"},{"issue":"12","key":"822_CR28","doi-asserted-by":"publisher","first-page":"684","DOI":"10.1007\/s001900050209","volume":"72","author":"P Van\u00ed\u010dek","year":"1998","unstructured":"Van\u00ed\u010dek P, Featherstone WE (1998) Performance of three types of Stokes\u2019s kernel in the combined solution for the geoid. J Geod 72(12):684\u2013697. https:\/\/doi.org\/10.1007\/s001900050209","journal-title":"J Geod"},{"issue":"2","key":"822_CR29","first-page":"86","volume":"12","author":"P Van\u00ed\u010dek","year":"1987","unstructured":"Van\u00ed\u010dek P, Kleusberg A (1987) The canadian geoid-stokesian approach. Manuscr Geodaet 12(2):86\u201398","journal-title":"Manuscr Geodaet"},{"key":"822_CR30","unstructured":"Wenzel H (1982) Geoid computation by least-squares spectral combination using integral kernels. Internation, Tokyo, Japan, pp 438\u2013453"},{"issue":"1","key":"822_CR31","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1111\/j.1365-246X.1969.tb00264.x","volume":"18","author":"L Wong","year":"1969","unstructured":"Wong L, Gore R (1969) Accuracy of geoid heights from modified Stokes kernels. Geophys J R Astron Soc 18(1):81\u201391. https:\/\/doi.org\/10.1111\/j.1365-246X.1969.tb00264.x","journal-title":"Geophys J R Astron Soc"}],"container-title":["Earth Science Informatics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12145-022-00822-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12145-022-00822-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12145-022-00822-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,20]],"date-time":"2022-08-20T10:16:26Z","timestamp":1660990586000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12145-022-00822-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,11]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,9]]}},"alternative-id":["822"],"URL":"https:\/\/doi.org\/10.1007\/s12145-022-00822-7","relation":{},"ISSN":["1865-0473","1865-0481"],"issn-type":[{"value":"1865-0473","type":"print"},{"value":"1865-0481","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,11]]},"assertion":[{"value":"18 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 May 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 June 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflicts of interest to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of Interests"}}]}}