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class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_mass_density" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_to_mass_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relationship to mass density</span> </div> </a> <ul id="toc-Relationship_to_mass_density-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Temporal_change" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Temporal_change"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Temporal change</span> </div> </a> <ul id="toc-Temporal_change-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spherical_harmonics_representation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spherical_harmonics_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Spherical harmonics representation</span> </div> </a> <ul id="toc-Spherical_harmonics_representation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Geoid</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 65 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-65" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">65 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AC%D8%B3%D9%85_%D8%A3%D8%B1%D8%B6%D9%8A" title="مجسم أرضي – Arabic" lang="ar" hreflang="ar" data-title="مجسم أرضي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Xeoide" title="Xeoide – Asturian" lang="ast" hreflang="ast" data-title="Xeoide" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Geoid" title="Geoid – Azerbaijani" lang="az" hreflang="az" data-title="Geoid" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AD%E0%A7%82%E0%A6%97%E0%A7%8B%E0%A6%B2%E0%A6%95" title="ভূগোলক – Bangla" lang="bn" hreflang="bn" data-title="ভূগোলক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D1%96%D0%B4" title="Геоід – Belarusian" lang="be" hreflang="be" data-title="Геоід" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D1%96%D0%B4" title="Геоід – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Геоід" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A5%85%E0%A4%87%E0%A4%A1" title="ज्याॅइड – Bhojpuri" lang="bh" hreflang="bh" data-title="ज्याॅइड" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B8%D0%B4" title="Геоид – Bulgarian" lang="bg" hreflang="bg" data-title="Геоид" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Geoide" title="Geoide – Catalan" lang="ca" hreflang="ca" data-title="Geoide" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Geoid" title="Geoid – Czech" lang="cs" hreflang="cs" data-title="Geoid" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Geoide" title="Geoide – Danish" lang="da" hreflang="da" data-title="Geoide" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geoid" title="Geoid – German" lang="de" hreflang="de" data-title="Geoid" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Geoid" title="Geoid – Estonian" lang="et" hreflang="et" data-title="Geoid" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CF%89%CE%B5%CE%B9%CE%B4%CE%AD%CF%82" title="Γεωειδές – Greek" lang="el" hreflang="el" data-title="Γεωειδές" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Geoide" title="Geoide – Spanish" lang="es" hreflang="es" data-title="Geoide" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Geoido" title="Geoido – Esperanto" lang="eo" hreflang="eo" data-title="Geoido" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Geoide" title="Geoide – Basque" lang="eu" hreflang="eu" data-title="Geoide" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%D9%85%DB%8C%D9%86%E2%80%8C%D9%88%D8%A7%D8%B1" title="زمین‌وار – Persian" lang="fa" hreflang="fa" data-title="زمین‌وار" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/G%C3%A9o%C3%AFde" title="Géoïde – French" lang="fr" hreflang="fr" data-title="Géoïde" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Xeoide" title="Xeoide – Galician" lang="gl" hreflang="gl" data-title="Xeoide" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%80%EC%98%A4%EC%9D%B4%EB%93%9C" title="지오이드 – Korean" lang="ko" hreflang="ko" data-title="지오이드" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D5%A5%D5%B8%D5%AB%D5%A4" title="Գեոիդ – Armenian" lang="hy" hreflang="hy" data-title="Գեոիդ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Geoid" title="Geoid – Croatian" lang="hr" hreflang="hr" data-title="Geoid" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Geoid" title="Geoid – Indonesian" lang="id" hreflang="id" data-title="Geoid" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geoide" title="Geoide – Italian" lang="it" hreflang="it" data-title="Geoide" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%90%D7%95%D7%90%D7%99%D7%93" title="גאואיד – Hebrew" lang="he" hreflang="he" data-title="גאואיד" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%94%E1%83%9D%E1%83%98%E1%83%93%E1%83%98" title="გეოიდი – Georgian" lang="ka" hreflang="ka" data-title="გეოიდი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B8%D0%B4" title="Геоид – Kazakh" lang="kk" hreflang="kk" data-title="Геоид" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B8%D0%B4" title="Геоид – Kyrgyz" lang="ky" hreflang="ky" data-title="Геоид" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%A2eo%C4%ABds" title="Ģeoīds – Latvian" lang="lv" hreflang="lv" data-title="Ģeoīds" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Geoid" title="Geoid – Luxembourgish" lang="lb" hreflang="lb" data-title="Geoid" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Geoid" title="Geoid – Hungarian" lang="hu" hreflang="hu" data-title="Geoid" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B8%D0%B4" title="Геоид – Macedonian" lang="mk" hreflang="mk" data-title="Геоид" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9C%E0%B4%BF%E0%B4%AF%E0%B5%8B%E0%B4%AF%E0%B4%BF%E0%B4%A1%E0%B5%8D" title="ജിയോയിഡ് – Malayalam" lang="ml" hreflang="ml" data-title="ജിയോയിഡ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Geoid" title="Geoid – Malay" lang="ms" hreflang="ms" data-title="Geoid" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geo%C3%AFde" title="Geoïde – Dutch" lang="nl" hreflang="nl" data-title="Geoïde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE_%E0%A4%97%E0%A5%81%E0%A4%B0%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%B5_%E0%A4%B8%E0%A4%A4%E0%A4%B9_(%E0%A4%9C%E0%A4%BF%E0%A4%AF%E0%A5%8B%E0%A4%87%E0%A4%A1)" title="सम गुरुत्व सतह (जियोइड) – Nepali" lang="ne" hreflang="ne" data-title="सम गुरुत्व सतह (जियोइड)" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B8%E3%82%AA%E3%82%A4%E3%83%89" title="ジオイド – Japanese" lang="ja" hreflang="ja" data-title="ジオイド" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Geoide" title="Geoide – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Geoide" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Geoide" title="Geoide – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Geoide" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Geoid" title="Geoid – Uzbek" lang="uz" hreflang="uz" data-title="Geoid" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Geoida" title="Geoida – Polish" lang="pl" hreflang="pl" data-title="Geoida" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Geoide" title="Geoide – Portuguese" lang="pt" hreflang="pt" data-title="Geoide" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Geoid" title="Geoid – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Geoid" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Geoid" title="Geoid – Romanian" lang="ro" hreflang="ro" data-title="Geoid" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B8%D0%B4" title="Геоид – Rusyn" lang="rue" hreflang="rue" data-title="Геоид" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B8%D0%B4" title="Геоид – Russian" lang="ru" hreflang="ru" data-title="Геоид" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gjeoidi" title="Gjeoidi – Albanian" lang="sq" hreflang="sq" data-title="Gjeoidi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Geoid" title="Geoid – Simple English" lang="en-simple" hreflang="en-simple" data-title="Geoid" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Geoid" title="Geoid – Slovak" lang="sk" hreflang="sk" data-title="Geoid" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Geoid" title="Geoid – Slovenian" lang="sl" hreflang="sl" data-title="Geoid" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B8%D0%B4" title="Геоид – Serbian" lang="sr" hreflang="sr" data-title="Геоид" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Geoid" title="Geoid – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Geoid" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Geoidi" title="Geoidi – Finnish" lang="fi" hreflang="fi" data-title="Geoidi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Geoid" title="Geoid – Swedish" lang="sv" hreflang="sv" data-title="Geoid" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%82%E0%AE%AE%E0%AE%BF%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%AE%E0%AF%8D" title="பூமிவடிவம் – Tamil" lang="ta" hreflang="ta" data-title="பூமிவடிவம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B5%E0%B8%AD%E0%B8%AD%E0%B8%A2%E0%B8%94%E0%B9%8C" title="จีออยด์ – Thai" lang="th" hreflang="th" data-title="จีออยด์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%97%D0%B0%D0%BC%D0%B8%D0%BD%D0%B2%D0%BE%D1%80%D0%B0" title="Заминвора – Tajik" lang="tg" hreflang="tg" data-title="Заминвора" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Geoit" title="Geoit – Turkish" lang="tr" hreflang="tr" data-title="Geoit" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D1%97%D0%B4" title="Геоїд – Ukrainian" lang="uk" hreflang="uk" data-title="Геоїд" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Geoid" title="Geoid – Vietnamese" lang="vi" hreflang="vi" data-title="Geoid" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A4%A7%E5%9C%B0%E6%B0%B4%E5%87%86%E9%9D%A2" title="大地水准面 – Wu" lang="wuu" hreflang="wuu" data-title="大地水准面" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%A7%E5%9C%B0%E6%B0%B4%E6%BA%96%E9%9D%A2" title="大地水準面 – Cantonese" lang="yue" hreflang="yue" data-title="大地水準面" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Geoid" title="Geoid – Zazaki" lang="diq" hreflang="diq" data-title="Geoid" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%A7%E5%9C%B0%E6%B0%B4%E5%87%86%E9%9D%A2" title="大地水准面 – Chinese" lang="zh" hreflang="zh" data-title="大地水准面" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q185969#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Ocean shape without winds and tides</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Geode" title="Geode">geode</a>, a crystal-filled rock.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/GEOID" class="mw-redirect" title="GEOID">GEOID</a>, a geocoding scheme.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Earth_Gravitational_Model_1996.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Earth_Gravitational_Model_1996.png/330px-Earth_Gravitational_Model_1996.png" decoding="async" width="330" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Earth_Gravitational_Model_1996.png/495px-Earth_Gravitational_Model_1996.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Earth_Gravitational_Model_1996.png/660px-Earth_Gravitational_Model_1996.png 2x" data-file-width="742" data-file-height="468" /></a><figcaption>Map of the undulation of the geoid in meters (based on the <a href="/wiki/EGM96" class="mw-redirect" title="EGM96">EGM96</a> gravity model and the <a href="/wiki/WGS84" class="mw-redirect" title="WGS84">WGS84</a> reference ellipsoid).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output 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.sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Globe_Atlantic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/120px-Globe_Atlantic.svg.png" decoding="async" width="120" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/180px-Globe_Atlantic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/240px-Globe_Atlantic.svg.png 2x" data-file-width="717" data-file-height="721" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></li> <li><a href="/wiki/Geodynamics" title="Geodynamics">Geodynamics</a></li> <li><a href="/wiki/Geomatics" title="Geomatics">Geomatics</a></li> <li><a href="/wiki/History_of_geodesy" title="History of geodesy">History</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Concepts</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Geographical_distance" title="Geographical distance">Geographical distance</a></li> <li><a class="mw-selflink selflink">Geoid</a></li> <li><a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">Figure of the Earth</a> <small>(<a href="/wiki/Earth_radius" title="Earth radius">radius</a> and <a href="/wiki/Earth%27s_circumference" title="Earth's circumference">circumference</a>)</small></li> <li><a href="/wiki/Geodetic_coordinates" title="Geodetic coordinates">Geodetic coordinates</a></li> <li><a href="/wiki/Geodetic_datum" title="Geodetic datum">Geodetic datum</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Horizontal_position_representation" title="Horizontal position representation">Horizontal position representation</a></li> <li><span class="nowrap"><a href="/wiki/Latitude" title="Latitude">Latitude</a> / <a href="/wiki/Longitude" title="Longitude">Longitude</a></span></li> <li><a href="/wiki/Map_projection" title="Map projection">Map projection</a></li> <li><a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">Reference ellipsoid</a></li> <li><a href="/wiki/Satellite_geodesy" title="Satellite geodesy">Satellite geodesy</a></li> <li><a href="/wiki/Spatial_reference_system" title="Spatial reference system">Spatial reference system</a></li> <li><a href="/wiki/Spatial_relation" title="Spatial relation">Spatial relations</a></li> <li><a href="/wiki/Vertical_position" title="Vertical position">Vertical positions</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Technologies</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Satellite_navigation" title="Satellite navigation">Global Nav. Sat. Systems (GNSSs)</a></li> <li><a href="/wiki/Global_Positioning_System" title="Global Positioning System">Global Pos. System (GPS)</a></li> <li><a href="/wiki/GLONASS" title="GLONASS">GLONASS <span style="font-size:85%;">(Russia)</span></a></li> <li><a href="/wiki/BeiDou" title="BeiDou">BeiDou (BDS) <span style="font-size:85%;">(China)</span></a></li> <li><a href="/wiki/Galileo_(satellite_navigation)" title="Galileo (satellite navigation)">Galileo <span style="font-size:85%;">(Europe)</span></a></li> <li><a href="/wiki/Indian_Regional_Navigation_Satellite_System" title="Indian Regional Navigation Satellite System">NAVIC <span style="font-size:85%;">(India)</span></a></li> <li><a href="/wiki/Quasi-Zenith_Satellite_System" title="Quasi-Zenith Satellite System">Quasi-Zenith Sat. Sys. (QZSS) <span style="font-size:85%;">(Japan)</span></a></li> <li><a href="/wiki/Discrete_Global_Grid" class="mw-redirect" title="Discrete Global Grid">Discrete Global Grid and Geocoding</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed plainlist"><div class="sidebar-list-title" style="color: var(--color-base)">Standards (history)</div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="nowrap" style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;line-height:1.2em;"><tbody><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Sea_Level_Datum_of_1929" class="mw-redirect" title="Sea Level Datum of 1929">NGVD 29</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Sea Level Datum 1929</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/OSGB36" class="mw-redirect" title="OSGB36">OSGB36</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Ordnance Survey Great Britain 1936</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/SK-42_reference_system" title="SK-42 reference system">SK-42</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Systema Koordinat 1942 goda</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/ED50" title="ED50">ED50</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> European Datum 1950</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/South_American_Datum#SAD69" title="South American Datum">SAD69</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> South American Datum 1969</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/GRS_80" class="mw-redirect" title="GRS 80">GRS 80</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Geodetic Reference System 1980</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/ISO_6709" title="ISO 6709">ISO 6709</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Geographic point coord. 1983</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/North_American_Datum#North_American_Datum_of_1983" title="North American Datum">NAD 83</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> North American Datum 1983</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/World_Geodetic_System" title="World Geodetic System">WGS 84</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> World Geodetic System 1984</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/North_American_Vertical_Datum_of_1988" title="North American Vertical Datum of 1988">NAVD 88</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> N. American Vertical Datum 1988</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/European_Terrestrial_Reference_System_1989" title="European Terrestrial Reference System 1989">ETRS89</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> European Terrestrial Ref. Sys. 1989</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Restrictions_on_geographic_data_in_China" title="Restrictions on geographic data in China">GCJ-02</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Chinese obfuscated datum 2002</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Geo_URI_scheme" title="Geo URI scheme">Geo URI</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Internet link to a point 2010</td></tr></tbody></table> <ul><li><a href="/wiki/International_Terrestrial_Reference_System" class="mw-redirect" title="International Terrestrial Reference System">International Terrestrial Reference System</a></li> <li><a href="/wiki/SRID" class="mw-redirect" title="SRID">Spatial Reference System Identifier (SRID)</a></li> <li><a href="/wiki/Universal_Transverse_Mercator_coordinate_system" title="Universal Transverse Mercator coordinate system">Universal Transverse Mercator (UTM)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Geodesy" title="Template:Geodesy"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Geodesy" title="Template talk:Geodesy"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Geodesy" title="Special:EditPage/Template:Geodesy"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p class="mw-empty-elt"> </p><p>The <b>geoid</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="/dʒ/: 'j' in 'jam'">dʒ</span><span title="/iː/: 'ee' in 'fleece'">iː</span><span title="/./: syllable break">.</span><span title="/ɔɪ/: 'oi' in 'choice'">ɔɪ</span><span title="'d' in 'dye'">d</span></span>/</a></span></span> <a href="/wiki/Help:Pronunciation_respelling_key" title="Help:Pronunciation respelling key"><i title="English pronunciation respelling"><span style="font-size:90%">JEE</span>-oyd</i></a>) is the shape that the <a href="/wiki/Ocean" title="Ocean">ocean</a> surface would take under the influence of the <a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">gravity of Earth</a>, including <a href="/wiki/Gravitational_attraction" class="mw-redirect" title="Gravitational attraction">gravitational attraction</a> and <a href="/wiki/Earth%27s_rotation" title="Earth's rotation">Earth's rotation</a>, if other influences such as winds and <a href="/wiki/Tide" title="Tide">tides</a> were absent. This surface is extended through the <a href="/wiki/Continent" title="Continent">continents</a> (such as might be approximated with very narrow hypothetical <a href="/wiki/Canal" title="Canal">canals</a>). According to <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a>, who first described it, it is the "mathematical <a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">figure of the Earth</a>", a smooth but irregular <a href="/wiki/Surface" title="Surface">surface</a> whose shape results from the uneven distribution of mass within and on the surface of Earth.<sup id="cite_ref-Gauß1828_2-0" class="reference"><a href="#cite_note-Gauß1828-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It can be known only through extensive gravitational measurements and calculations. Despite being an important concept for almost 200 years in the history of <a href="/wiki/Geodesy" title="Geodesy">geodesy</a> and <a href="/wiki/Geophysics" title="Geophysics">geophysics</a>, it has been defined to high precision only since advances in <a href="/wiki/Satellite_geodesy" title="Satellite geodesy">satellite geodesy</a> in the late 20th century. </p><p>The geoid is often expressed as a <b>geoid undulation</b> or <b>geoidal height</b> above a given <a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>, which is a slightly flattened sphere whose <a href="/wiki/Equatorial_bulge" title="Equatorial bulge">equatorial bulge</a> is caused by the planet's rotation. Generally the geoidal height rises where the Earth's material is locally more dense and exerts greater gravitational force than the surrounding areas. The geoid in turn serves as a reference <a href="/wiki/Coordinate_surface" class="mw-redirect" title="Coordinate surface">coordinate surface</a> for various <a href="/wiki/Vertical_coordinate" class="mw-redirect" title="Vertical coordinate">vertical coordinates</a>, such as <i><a href="/wiki/Orthometric_height" title="Orthometric height">orthometric heights</a></i>, <i><a href="/wiki/Geopotential_height" title="Geopotential height">geopotential heights</a></i>, and <i><a href="/wiki/Dynamic_height" title="Dynamic height">dynamic heights</a></i> (see <a href="/wiki/Geodesy#Heights" title="Geodesy">Geodesy#Heights</a>). </p><p>All points on a geoid surface have the same <a href="/wiki/Geopotential" title="Geopotential">geopotential</a> (the sum of <a href="/wiki/Gravitational_energy" title="Gravitational energy">gravitational potential energy</a> and <a href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal</a> potential energy). At this surface, apart from temporary tidal fluctuations, the <a href="/wiki/Force_of_gravity" class="mw-redirect" title="Force of gravity">force of gravity</a> acts everywhere perpendicular to the geoid, meaning that <a href="/wiki/Plumb_bob" title="Plumb bob">plumb lines</a> point perpendicular and <a href="/wiki/Bubble_level" class="mw-redirect" title="Bubble level">bubble levels</a> are parallel to the geoid. Being an <a href="/wiki/Equigeopotential" class="mw-redirect" title="Equigeopotential">equigeopotential</a> means the geoid corresponds to the <a href="/wiki/Free_surface" title="Free surface">free surface</a> of water at rest (if only the Earth's gravity and rotational acceleration were at work); this is also a sufficient condition for a ball to remain at rest instead of rolling over the geoid. Earth's gravity acceleration (the <a href="/wiki/Vertical_derivative" class="mw-redirect" title="Vertical derivative">vertical derivative</a> of geopotential) is thus non-uniform over the geoid.<sup id="cite_ref-BookGeodesyConcepts_3-0" class="reference"><a href="#cite_note-BookGeodesyConcepts-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geoid_undulation_10k_scale.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Geoid_undulation_10k_scale.jpg/220px-Geoid_undulation_10k_scale.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Geoid_undulation_10k_scale.jpg/330px-Geoid_undulation_10k_scale.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Geoid_undulation_10k_scale.jpg/440px-Geoid_undulation_10k_scale.jpg 2x" data-file-width="1200" data-file-height="1200" /></a><figcaption>Geoid undulation in <a href="/wiki/Pseudocolor" class="mw-redirect" title="Pseudocolor">pseudocolor</a>, <a href="/wiki/Shaded_relief" class="mw-redirect" title="Shaded relief">shaded relief</a> and <a href="/wiki/Vertical_exaggeration" title="Vertical exaggeration">vertical exaggeration</a> (10000 vertical scaling factor).</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geoid_undulation_to_scale.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Geoid_undulation_to_scale.jpg/220px-Geoid_undulation_to_scale.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Geoid_undulation_to_scale.jpg/330px-Geoid_undulation_to_scale.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Geoid_undulation_to_scale.jpg/440px-Geoid_undulation_to_scale.jpg 2x" data-file-width="1200" data-file-height="1200" /></a><figcaption>Geoid undulation in pseudocolor, without vertical exaggeration.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The geoid surface is irregular, unlike the reference ellipsoid (which is a mathematical idealized representation of the physical Earth as an <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a>), but is considerably smoother than Earth's physical surface. Although the "ground" of the Earth has excursions on the order of +8,800 m (<a href="/wiki/Mount_Everest" title="Mount Everest">Mount Everest</a>) and −11,000 m (<a href="/wiki/Marianas_Trench" class="mw-redirect" title="Marianas Trench">Marianas Trench</a>), the geoid's deviation from an ellipsoid ranges from +85 m (Iceland) to −106 m (southern India), less than 200 m total.<sup id="cite_ref-TexasGeoid_4-0" class="reference"><a href="#cite_note-TexasGeoid-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>If the ocean were of constant density and undisturbed by tides, currents or weather, its surface would resemble the geoid. The permanent deviation between the geoid and <a href="/wiki/Mean_sea_level" class="mw-redirect" title="Mean sea level">mean sea level</a> is called <a href="/wiki/Ocean_surface_topography" title="Ocean surface topography">ocean surface topography</a>. If the continental land masses were crisscrossed by a series of tunnels or canals, the sea level in those canals would also very nearly coincide with the geoid. <a href="/wiki/Geodesist" class="mw-redirect" title="Geodesist">Geodesists</a> are able to derive the heights of continental points above the geoid by <a href="/wiki/Spirit_leveling" class="mw-redirect" title="Spirit leveling">spirit leveling</a>. </p><p>Being an <a href="/wiki/Equipotential_surface" class="mw-redirect" title="Equipotential surface">equipotential surface</a>, the geoid is, by definition, a surface upon which the force of gravity is perpendicular everywhere, apart from temporary tidal fluctuations. This means that when traveling by ship, one does not notice the <a href="#Undulation">undulation of the geoid</a>; neglecting tides, the local vertical (plumb line) is always perpendicular to the geoid and the local horizon <a href="/wiki/Tangential_component" class="mw-redirect" title="Tangential component">tangential</a> to it. Likewise, spirit levels will always be parallel to the geoid. </p> <div class="mw-heading mw-heading3"><h3 id="Simplified_example">Simplified example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=2" title="Edit section: Simplified example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geoida.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Geoida.svg/220px-Geoida.svg.png" decoding="async" width="220" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Geoida.svg/330px-Geoida.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Geoida.svg/440px-Geoida.svg.png 2x" data-file-width="730" data-file-height="338" /></a><figcaption><div><ol><li>Ocean</li><li>Ellipsoid</li><li>Local plumb line</li><li>Continent</li><li>Geoid</li></ol></div></figcaption></figure> <p>Earth's gravitational field is not uniform. An <a href="/wiki/Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">oblate spheroid</a> is typically used as the idealized Earth, but even if the Earth were spherical and did not rotate, the strength of gravity would not be the same everywhere because density varies throughout the planet. This is due to magma distributions, the density and weight of different <a href="/wiki/Geological" class="mw-redirect" title="Geological">geological</a> compositions in the <a href="/wiki/Earth%27s_crust" title="Earth's crust">Earth's crust</a>, mountain ranges, deep sea trenches, crust compaction due to glaciers, and so on. </p><p>If that sphere were then covered in water, the water would not be the same height everywhere. Instead, the water level would be higher or lower with respect to Earth's center, depending on the integral of the strength of gravity from the center of the Earth to that location. The geoid level coincides with where the water would be. Generally the geoid rises where the Earth's material is locally more dense, exerts greater gravitational force, and pulls more water from the surrounding area. </p> <div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation<span class="anchor" id="Shape"></span><span class="anchor" id="Undulation"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=3" title="Edit section: Formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geoundnsrp.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Geoundnsrp.png/220px-Geoundnsrp.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Geoundnsrp.png/330px-Geoundnsrp.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/78/Geoundnsrp.png 2x" data-file-width="396" data-file-height="381" /></a><figcaption>Meridional profile of geoid undulation (red) relative to the reference ellipsoid (black), greatly exaggerated; see also: <a href="/wiki/Earth%27s_pear_shape" class="mw-redirect" title="Earth's pear shape">Earth's pear shape</a>.</figcaption></figure> <p>The <b>geoid undulation</b> (also known as <b>geoid height</b> or <b>geoid anomaly</b>), <i>N</i>, is the height of the geoid relative to a given <a href="/wiki/Ellipsoid_of_reference" class="mw-redirect" title="Ellipsoid of reference">ellipsoid of reference</a>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=h-H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mi>h</mi> <mo>−</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=h-H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21cc74f889b44fd563a2bd82ac3ba3661041b106" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.405ex; height:2.343ex;" alt="{\displaystyle N=h-H}"></span> The undulation is not standardized, as different countries use different mean sea levels as reference, but most commonly refers to the <a href="/wiki/EGM96" class="mw-redirect" title="EGM96">EGM96</a> geoid. </p><p>In maps and common use, the height over the mean sea level (such as <a href="/wiki/Orthometric_height" title="Orthometric height">orthometric height</a>, <i>H</i>) is used to indicate the height of elevations while the <a href="/wiki/Ellipsoidal_height" class="mw-redirect" title="Ellipsoidal height">ellipsoidal height</a>, <i>h</i>, results from the <a href="/wiki/GPS" class="mw-redirect" title="GPS">GPS</a> system and similar <a href="/wiki/GNSS" class="mw-redirect" title="GNSS">GNSS</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=h-N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>h</mi> <mo>−</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=h-N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea6a84a3f012cad351f2b4361551b881657b6e7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.405ex; height:2.343ex;" alt="{\displaystyle H=h-N}"></span> (An analogous relationship exists between <a href="/wiki/Normal_height" title="Normal height">normal heights</a> and the <i><a href="/wiki/Quasigeoid" class="mw-redirect" title="Quasigeoid">quasigeoid</a></i>, which disregards local density variations.) In practice, many handheld GPS receivers <a href="/wiki/Spatial_interpolation" class="mw-redirect" title="Spatial interpolation">interpolate</a> <i>N</i> in a pre-computed <i>geoid map</i> (a <a href="/wiki/Lookup_table" title="Lookup table">lookup table</a>).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>So a <a href="/wiki/GPS_receiver" class="mw-redirect" title="GPS receiver">GPS receiver</a> on a ship may, during the course of a long voyage, indicate height variations, even though the ship will always be at sea level (neglecting the effects of tides). That is because GPS <a href="/wiki/Satellite" title="Satellite">satellites</a>, orbiting about the center of gravity of the Earth, can measure heights only relative to a geocentric reference ellipsoid. To obtain one's <a href="/wiki/Orthometric_height" title="Orthometric height">orthometric height</a>, a raw GPS reading must be corrected. Conversely, height determined by spirit leveling from a <a href="/wiki/Tide_gauge" title="Tide gauge">tide gauge</a>, as in traditional land surveying, is closer to orthometric height. Modern GPS receivers have a grid implemented in their software by which they obtain, from the current position, the height of the geoid (e.g., the EGM96 geoid) over the <a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a> (WGS) ellipsoid. They are then able to correct the height above the WGS ellipsoid to the height above the EGM96 geoid. When height is not zero on a ship, the discrepancy is due to other factors such as ocean tides, <a href="/wiki/Atmospheric_pressure" title="Atmospheric pressure">atmospheric pressure</a> (meteorological effects), local <a href="/wiki/Sea_surface_topography" class="mw-redirect" title="Sea surface topography">sea surface topography</a>, and measurement uncertainties. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Geoundaequrp.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Geoundaequrp.png/220px-Geoundaequrp.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Geoundaequrp.png/330px-Geoundaequrp.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/41/Geoundaequrp.png 2x" data-file-width="396" data-file-height="381" /></a><figcaption>Equatorial profile of geoid undulation (red) relative to the reference ellipsoid (black), greatly exaggerated; see also: <a href="/wiki/Triaxial_Earth" class="mw-redirect" title="Triaxial Earth">triaxial Earth</a>.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Determination">Determination</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=4" title="Edit section: Determination"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Earth_ellipsoid#Determination" title="Earth ellipsoid">Earth ellipsoid § Determination</a></div> <p>The undulation of the geoid <i>N</i> is closely related to the <a href="/wiki/Disturbing_potential" class="mw-redirect" title="Disturbing potential">disturbing potential</a> <i>T</i> according to <b>Bruns' formula</b> (named after <a href="/wiki/Heinrich_Bruns" title="Heinrich Bruns">Heinrich Bruns</a>):<span class="anchor" id="Bruns_formula"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=T/\gamma \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>γ</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=T/\gamma \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9207c2abcdf973c872d7b3b9fc4084bc6136c4f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.257ex; height:2.843ex;" alt="{\displaystyle N=T/\gamma \,,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the force of <a href="/wiki/Normal_gravity" class="mw-redirect" title="Normal gravity">normal gravity</a>, computed from the normal field potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>. </p><p>Another way of determining <i>N</i> is using values of <i><a href="/wiki/Gravity_anomaly" title="Gravity anomaly">gravity anomaly</a></i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf020ee71d673ec00c0f5848bac6480fa0b75a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.052ex; height:2.509ex;" alt="{\displaystyle \Delta g}"></span>, differences between true and normal reference gravity, as per <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="Stokes_formula"></span><span class="vanchor-text">Stokes formula</span></span></b> (or <b>Stokes' integral</b>), published in 1849 by <a href="/wiki/George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">George Gabriel Stokes</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N={\frac {R}{4\pi \gamma _{0}}}\iint _{\sigma }\Delta g\,S(\psi )\,d\sigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>g</mi> <mspace width="thinmathspace" /> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>σ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N={\frac {R}{4\pi \gamma _{0}}}\iint _{\sigma }\Delta g\,S(\psi )\,d\sigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70975ce6281bb7fa3a4315f235b130edaad00399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.057ex; height:5.676ex;" alt="{\displaystyle N={\frac {R}{4\pi \gamma _{0}}}\iint _{\sigma }\Delta g\,S(\psi )\,d\sigma .}"></span></dd></dl> <p>The <a href="/wiki/Integral_kernel" class="mw-redirect" title="Integral kernel">integral kernel</a> <i>S</i>, called <i>Stokes function</i>, was derived by Stokes in closed analytical form.<sup id="cite_ref-Wang_2016_pp._1–8_6-0" class="reference"><a href="#cite_note-Wang_2016_pp._1–8-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Note that determining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> anywhere on Earth by this formula requires <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf020ee71d673ec00c0f5848bac6480fa0b75a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.052ex; height:2.509ex;" alt="{\displaystyle \Delta g}"></span> to be known <i>everywhere on Earth</i>, including oceans, polar areas, and deserts. For terrestrial gravimetric measurements this is a near-impossibility, in spite of close international co-operation within the <a href="/wiki/International_Association_of_Geodesy" title="International Association of Geodesy">International Association of Geodesy</a> (IAG), e.g., through the International Gravity Bureau (BGI, Bureau Gravimétrique International). </p><p>Another approach for geoid determination is to <i>combine</i> multiple information sources: not just terrestrial gravimetry, but also satellite geodetic data on the figure of the Earth, from analysis of satellite orbital perturbations, and lately from satellite gravity missions such as <a href="/wiki/GOCE" title="GOCE">GOCE</a> and <a href="/wiki/Gravity_Recovery_and_Climate_Experiment" class="mw-redirect" title="Gravity Recovery and Climate Experiment">GRACE</a>. In such combination solutions, the low-resolution part of the geoid solution is provided by the satellite data, while a 'tuned' version of the above Stokes equation is used to calculate the high-resolution part, from terrestrial gravimetric data from a neighbourhood of the evaluation point only. </p><p>Calculating the undulation is mathematically challenging.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The precise geoid solution by <a href="/wiki/Petr_Van%C3%AD%C4%8Dek" title="Petr Vaníček">Petr Vaníček</a> and co-workers improved on the <a href="/wiki/George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">Stokesian</a> approach to geoid computation.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Their solution enables millimetre-to-centimetre <a href="/wiki/Accuracy" class="mw-redirect" title="Accuracy">accuracy</a> in geoid <a href="/wiki/Computation" title="Computation">computation</a>, an <a href="/wiki/Order_of_magnitude" title="Order of magnitude">order-of-magnitude</a> improvement from previous classical solutions.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Geoid undulations display uncertainties which can be estimated by using several methods, e.g., <a href="/wiki/Least_squares" title="Least squares">least-squares</a> collocation (LSC), <a href="/wiki/Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a>, <a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">artificial neural networks</a>, <a href="/wiki/Radial_basis_function" title="Radial basis function">radial basis functions</a> (RBF), and <a href="/wiki/Geostatistics" title="Geostatistics">geostatistical</a> techniques. Geostatistical approach has been defined as the most-improved technique in prediction of geoid undulation.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_mass_density">Relationship to mass density</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=5" title="Edit section: Relationship to mass density"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Gravity_anomaly" title="Gravity anomaly">Gravity anomaly</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravity,_geoid_anomaly_synthetic_cases_with_local_isostasy_2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Gravity%2C_geoid_anomaly_synthetic_cases_with_local_isostasy_2.gif/220px-Gravity%2C_geoid_anomaly_synthetic_cases_with_local_isostasy_2.gif" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Gravity%2C_geoid_anomaly_synthetic_cases_with_local_isostasy_2.gif/330px-Gravity%2C_geoid_anomaly_synthetic_cases_with_local_isostasy_2.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Gravity%2C_geoid_anomaly_synthetic_cases_with_local_isostasy_2.gif/440px-Gravity%2C_geoid_anomaly_synthetic_cases_with_local_isostasy_2.gif 2x" data-file-width="2202" data-file-height="1132" /></a><figcaption><a href="/wiki/Gravity_anomaly" title="Gravity anomaly">Gravity</a> and Geoid anomalies caused by various crustal and lithospheric thickness changes relative to a reference configuration. All settings are under local <a href="/wiki/Isostasy" title="Isostasy">isostatic</a> compensation.</figcaption></figure> <p>Variations in the height of the geoidal surface are related to anomalous density distributions within the Earth. Geoid measures thus help understanding the internal structure of the planet. Synthetic calculations show that the geoidal signature of a thickened crust (for example, in <a href="/wiki/Orogen" class="mw-redirect" title="Orogen">orogenic belts</a> produced by <a href="/wiki/Continental_collision" title="Continental collision">continental collision</a>) is positive, opposite to what should be expected if the thickening affects the entire <a href="/wiki/Lithosphere" title="Lithosphere">lithosphere</a>. Mantle convection also changes the shape of the geoid over time.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gravity_anomalies_on_Earth.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Gravity_anomalies_on_Earth.jpg/220px-Gravity_anomalies_on_Earth.jpg" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Gravity_anomalies_on_Earth.jpg/330px-Gravity_anomalies_on_Earth.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Gravity_anomalies_on_Earth.jpg/440px-Gravity_anomalies_on_Earth.jpg 2x" data-file-width="540" data-file-height="334" /></a><figcaption>Three-dimensional visualization of <a href="/wiki/Gravity_anomaly" title="Gravity anomaly">gravity anomalies</a> in <a href="/wiki/Gal_(unit)" title="Gal (unit)">units of Gal.</a>, using <a href="/wiki/Pseudo_color" class="mw-redirect" title="Pseudo color">pseudo color</a> and <a href="/wiki/Shaded_relief" class="mw-redirect" title="Shaded relief">shaded relief</a>.</figcaption></figure> <p>The surface of the geoid is higher than the <a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a> wherever there is a positive <a href="/wiki/Gravity_anomaly" title="Gravity anomaly">gravity anomaly</a> or negative <a href="/wiki/Disturbing_potential" class="mw-redirect" title="Disturbing potential">disturbing potential</a> (mass excess) and lower than the reference ellipsoid wherever there is a negative gravity anomaly or positive disturbing potential (mass deficit).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>This relationship can be understood by recalling that gravity potential is defined so that it has negative values and is inversely proportional to distance from the body. So, while a mass excess will strengthen the gravity acceleration, it will decrease the gravity potential. As a consequence, the geoid's defining equipotential surface will be found displaced away from the mass excess. Analogously, a mass deficit will weaken the gravity pull but will increase the geopotential at a given distance, causing the geoid to move towards the mass deficit. </p><p>The presence of a localized inclusion in the background medium will rotate the gravity acceleration vectors slightly towards or away from a denser or lighter body, respectively, causing a bump or dimple in the equipotential surface.<sup id="cite_ref-Lowrie_1997_p._50_17-0" class="reference"><a href="#cite_note-Lowrie_1997_p._50-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The largest absolute deviation can be found in the <a href="/wiki/Indian_Ocean_Geoid_Low" title="Indian Ocean Geoid Low">Indian Ocean Geoid Low</a>, 106 meters below the average sea level.<sup id="cite_ref-Raman_2017_18-0" class="reference"><a href="#cite_note-Raman_2017-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Another large feature is the North Atlantic Geoid High (or North Atlantic Geoid Swell), caused in part by the weight of ice cover over North America and northern Europe in the <a href="/wiki/Late_Cenozoic_Ice_Age" title="Late Cenozoic Ice Age">Late Cenozoic Ice Age</a>.<sup id="cite_ref-x153_19-0" class="reference"><a href="#cite_note-x153-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Temporal_change">Temporal change</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=6" title="Edit section: Temporal change"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Geodesy#Temporal_change" title="Geodesy">Geodesy § Temporal change</a></div> <p>Recent satellite missions, such as the <a href="/wiki/Gravity_Field_and_Steady-State_Ocean_Circulation_Explorer" class="mw-redirect" title="Gravity Field and Steady-State Ocean Circulation Explorer">Gravity Field and Steady-State Ocean Circulation Explorer</a> (GOCE) and <a href="/wiki/GRACE_(satellite)" class="mw-redirect" title="GRACE (satellite)">GRACE</a>, have enabled the study of time-variable geoid signals. The first products based on GOCE satellite data became available online in June 2010, through the European Space Agency.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> ESA launched the satellite in March 2009 on a mission to map Earth's gravity with unprecedented accuracy and spatial resolution. On 31 March 2011, a new geoid model was unveiled at the Fourth International GOCE User Workshop hosted at the <a href="/wiki/Technical_University_of_Munich" title="Technical University of Munich">Technical University of Munich</a>, Germany.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Studies using the time-variable geoid computed from GRACE data have provided information on global hydrologic cycles,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> mass balances of <a href="/wiki/Ice_sheet" title="Ice sheet">ice sheets</a>,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Postglacial_rebound" class="mw-redirect" title="Postglacial rebound">postglacial rebound</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> From postglacial rebound measurements, time-variable GRACE data can be used to deduce the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> of <a href="/wiki/Earth%27s_mantle" title="Earth's mantle">Earth's mantle</a>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Spherical_harmonics_representation">Spherical harmonics representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=7" title="Edit section: Spherical harmonics representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Geopotential_spherical_harmonic_model" title="Geopotential spherical harmonic model">Geopotential spherical harmonic model</a></div> <p><a href="/wiki/Spherical_harmonic" class="mw-redirect" title="Spherical harmonic">Spherical harmonics</a> are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is <a href="/wiki/EGM2020" class="mw-redirect" title="EGM2020">EGM2020</a> (Earth Gravitational Model 2020), determined in an international collaborative project led by the National Imagery and Mapping Agency (now the <a href="/wiki/National_Geospatial-Intelligence_Agency" title="National Geospatial-Intelligence Agency">National Geospatial-Intelligence Agency</a>, or NGA). The mathematical description of the non-rotating part of the potential function in this model is:<sup id="cite_ref-noaa.gov_27-0" class="reference"><a href="#cite_note-noaa.gov-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {GM}{r}}\left(1+{\sum _{n=2}^{n_{\text{max}}}}\left({\frac {a}{r}}\right)^{n}{\sum _{m=0}^{n}}{\overline {P}}_{nm}(\sin \phi )\left[{\overline {C}}_{nm}\cos m\lambda +{\overline {S}}_{nm}\sin m\lambda \right]\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <mi>M</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mrow> </munderover> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>m</mi> <mi>λ</mi> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>m</mi> <mi>λ</mi> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {GM}{r}}\left(1+{\sum _{n=2}^{n_{\text{max}}}}\left({\frac {a}{r}}\right)^{n}{\sum _{m=0}^{n}}{\overline {P}}_{nm}(\sin \phi )\left[{\overline {C}}_{nm}\cos m\lambda +{\overline {S}}_{nm}\sin m\lambda \right]\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b89c24db57cda0eefb0ebfbca3f5bd031e193fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:72.12ex; height:7.509ex;" alt="{\displaystyle V={\frac {GM}{r}}\left(1+{\sum _{n=2}^{n_{\text{max}}}}\left({\frac {a}{r}}\right)^{n}{\sum _{m=0}^{n}}{\overline {P}}_{nm}(\sin \phi )\left[{\overline {C}}_{nm}\cos m\lambda +{\overline {S}}_{nm}\sin m\lambda \right]\right),}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c7f7052f131b0858274cdd86e38a4119fbb28b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.966ex; height:2.509ex;" alt="{\displaystyle \phi \ }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7098afc76a9c0a8bc5079e98cf4aad3a4f9face7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \lambda \ }"></span> are <i>geocentric</i> (spherical) latitude and longitude respectively, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}_{nm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}_{nm}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fab56f1b314a3393bb6d12d3145bba66c57a9488" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.686ex; height:3.343ex;" alt="{\displaystyle {\overline {P}}_{nm}}"></span> are the fully normalized <a href="/wiki/Associated_Legendre_polynomials" title="Associated Legendre polynomials">associated Legendre polynomials</a> of degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81a2f2e76ab2d87f5ed37aeb7018991a8d82b04b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle n\ }"></span> and order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f753d46b8449abfe4aab6f5f1058188e46492f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.621ex; height:1.676ex;" alt="{\displaystyle m\ }"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {C}}_{nm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {C}}_{nm}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e97cffbddbb11fcc71b857809f1eea8b0dfe75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.611ex; height:3.343ex;" alt="{\displaystyle {\overline {C}}_{nm}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {S}}_{nm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {S}}_{nm}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb32df6e72ff0d4c063f42c3688b20af1eee32b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.324ex; height:3.343ex;" alt="{\displaystyle {\overline {S}}_{nm}}"></span> are the numerical coefficients of the model based on measured data. The above equation describes the Earth's gravitational <a href="/wiki/Potential" title="Potential">potential</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, not the geoid itself, at location <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ,\;\lambda ,\;r,\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>λ</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>r</mi> <mo>,</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ,\;\lambda ,\;r,\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d5a99834a0ef84431850132bd1148acd81c476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.762ex; height:2.509ex;" alt="{\displaystyle \phi ,\;\lambda ,\;r,\ }"></span> the co-ordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb18f8bc1b4c5e03654f8e8cfc73581142a80742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.629ex; height:1.676ex;" alt="{\displaystyle r\ }"></span> being the <i>geocentric radius</i>, i.e., distance from the Earth's centre. The geoid is a particular <a href="/wiki/Equipotential" title="Equipotential">equipotential</a> surface,<sup id="cite_ref-noaa.gov_27-1" class="reference"><a href="#cite_note-noaa.gov-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and is somewhat involved to compute. The gradient of this potential also provides a model of the gravitational acceleration. The most commonly used EGM96 contains a full set of coefficients to degree and order 360 (i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\text{max}}=360}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mo>=</mo> <mn>360</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\text{max}}=360}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40aba2a8c360abb340dc0a310c949cb744bfee23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.272ex; height:2.509ex;" alt="{\displaystyle n_{\text{max}}=360}"></span>), describing details in the global geoid as small as 55 km (or 110 km, depending on the definition of resolution). The number of coefficients, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {C}}_{nm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>C</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {C}}_{nm}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e97cffbddbb11fcc71b857809f1eea8b0dfe75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.611ex; height:3.343ex;" alt="{\displaystyle {\overline {C}}_{nm}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {S}}_{nm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {S}}_{nm}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb32df6e72ff0d4c063f42c3688b20af1eee32b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.324ex; height:3.343ex;" alt="{\displaystyle {\overline {S}}_{nm}}"></span>, can be determined by first observing in the equation for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> that for a specific value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> there are two coefficients for every value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> except for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}"></span>. There is only one coefficient when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}"></span> since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(0\lambda )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(0\lambda )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870b18c951bf822a93c8d058c431668c7507235b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.444ex; height:2.843ex;" alt="{\displaystyle \sin(0\lambda )=0}"></span>. There are thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6128190b8d3e08afed5437adaa79d4367440bd6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.369ex; height:2.843ex;" alt="{\displaystyle (2n+1)}"></span> coefficients for every value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. Using these facts and the formula, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{I=1}^{L}I={\frac {1}{2}}L(L+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </munderover> <mi>I</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{I=1}^{L}I={\frac {1}{2}}L(L+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc4fac7f988308e5a5c3d0af07bcebbd5f4df3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.908ex; height:3.676ex;" alt="{\textstyle \sum _{I=1}^{L}I={\frac {1}{2}}L(L+1)}"></span>, it follows that the total number of coefficients is given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=2}^{n_{\text{max}}}(2n+1)=n_{\text{max}}(n_{\text{max}}+1)+n_{\text{max}}-3=130317}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> </mrow> </munderover> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>130317</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=2}^{n_{\text{max}}}(2n+1)=n_{\text{max}}(n_{\text{max}}+1)+n_{\text{max}}-3=130317}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8bbb511103012a32dea9bb20a7bd54d5f52294" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.888ex; height:7.009ex;" alt="{\displaystyle \sum _{n=2}^{n_{\text{max}}}(2n+1)=n_{\text{max}}(n_{\text{max}}+1)+n_{\text{max}}-3=130317}"></span> using the EGM96 value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\text{max}}=360}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </msub> <mo>=</mo> <mn>360</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\text{max}}=360}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40aba2a8c360abb340dc0a310c949cb744bfee23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.272ex; height:2.509ex;" alt="{\displaystyle n_{\text{max}}=360}"></span>. </p><p>For many applications, the complete series is unnecessarily complex and is truncated after a few (perhaps several dozen) terms. </p><p>Still, even higher resolution models have been developed. Many of the authors of EGM96 have published EGM2008. It incorporates much of the new satellite gravity data (e.g., the <a href="/wiki/Gravity_Recovery_and_Climate_Experiment" class="mw-redirect" title="Gravity Recovery and Climate Experiment">Gravity Recovery and Climate Experiment</a>), and supports up to degree and order 2160 (1/6 of a degree, requiring over 4 million coefficients),<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> with additional coefficients extending to degree 2190 and order 2159.<sup id="cite_ref-nga.mil_29-0" class="reference"><a href="#cite_note-nga.mil-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> EGM2020 is the international follow-up that was originally scheduled for 2020 (still unreleased in 2024), containing the same number of harmonics generated with better data.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Deflection_of_the_vertical" class="mw-redirect" title="Deflection of the vertical">Deflection of the vertical</a></li> <li><a href="/wiki/Geodetic_datum" title="Geodetic datum">Geodetic datum</a></li> <li><a href="/wiki/Geopotential" title="Geopotential">Geopotential</a></li> <li><a href="/wiki/International_Terrestrial_Reference_Frame" class="mw-redirect" title="International Terrestrial Reference Frame">International Terrestrial Reference Frame</a></li> <li><a href="/wiki/Physical_geodesy" title="Physical geodesy">Physical geodesy</a></li> <li><a href="/wiki/Planetary_geoid" class="mw-redirect" title="Planetary geoid">Planetary geoid</a> <ul><li><a href="/wiki/Areoid" class="mw-redirect" title="Areoid">Areoid</a> (Mars's geoid)</li> <li><a href="/wiki/Selenoid" class="mw-redirect" title="Selenoid">Selenoid</a> (Moon's geoid)</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output 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.citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20200808025531/https://earth-info.nga.mil/GandG/wgs84/gravitymod/wgs84_180/wgs84_180.html">"WGS 84, N=M=180 Earth Gravitational Model"</a>. <i>NGA: Office of Geomatics</i>. 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Porto, Portugal, 2004.</span> </li> <li id="cite_note-nga.mil-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-nga.mil_29-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100508002312/http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/index.html">"Earth Gravitational Model 2008 (EGM2008)"</a>. <a href="/wiki/National_Geospatial-Intelligence_Agency" title="National Geospatial-Intelligence Agency">National Geospatial-Intelligence Agency</a>. Archived from <a rel="nofollow" class="external text" href="http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/index.html">the original</a> on 8 May 2010<span class="reference-accessdate">. Retrieved <span class="nowrap">9 September</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Earth+Gravitational+Model+2008+%28EGM2008%29&rft.pub=National+Geospatial-Intelligence+Agency&rft_id=http%3A%2F%2Fearth-info.nga.mil%2FGandG%2Fwgs84%2Fgravitymod%2Fegm2008%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeoid" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarnesFactorHolmesIngalls2015" class="citation journal cs1">Barnes, D.; Factor, J. K.; Holmes, S. A.; Ingalls, S.; Presicci, M. R.; Beale, J.; Fecher, T. (2015). <a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015AGUFM.G34A..03B/abstract">"Earth Gravitational Model 2020"</a>. <i>AGU Fall Meeting Abstracts</i>. <b>2015</b>: G34A–03. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015AGUFM.G34A..03B">2015AGUFM.G34A..03B</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=AGU+Fall+Meeting+Abstracts&rft.atitle=Earth+Gravitational+Model+2020&rft.volume=2015&rft.pages=G34A-03&rft.date=2015&rft_id=info%3Abibcode%2F2015AGUFM.G34A..03B&rft.aulast=Barnes&rft.aufirst=D.&rft.au=Factor%2C+J.+K.&rft.au=Holmes%2C+S.+A.&rft.au=Ingalls%2C+S.&rft.au=Presicci%2C+M.+R.&rft.au=Beale%2C+J.&rft.au=Fecher%2C+T.&rft_id=https%3A%2F%2Fui.adsabs.harvard.edu%2Fabs%2F2015AGUFM.G34A..03B%2Fabstract&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeoid" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=10" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiGötze2001" class="citation journal cs1">Li, Xiong; Götze, Hans-Jürgen (November 2001). <a rel="nofollow" class="external text" href="https://geored2.sgc.gov.co/Articulos%20y%20documentacion/Li_G_Tut.pdf">"Ellipsoid, geoid, gravity, geodesy, and geophysics"</a> <span class="cs1-format">(PDF)</span>. <i>Geophysics</i>. <b>66</b> (6): 1660–1668. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001Geop...66.1660L">2001Geop...66.1660L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1190%2F1.1487109">10.1190/1.1487109</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geophysics&rft.atitle=Ellipsoid%2C+geoid%2C+gravity%2C+geodesy%2C+and+geophysics&rft.volume=66&rft.issue=6&rft.pages=1660-1668&rft.date=2001-11&rft_id=info%3Adoi%2F10.1190%2F1.1487109&rft_id=info%3Abibcode%2F2001Geop...66.1660L&rft.aulast=Li&rft.aufirst=Xiong&rft.au=G%C3%B6tze%2C+Hans-J%C3%BCrgen&rft_id=https%3A%2F%2Fgeored2.sgc.gov.co%2FArticulos%2520y%2520documentacion%2FLi_G_Tut.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeoid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoritz2011" class="citation journal cs1">Moritz, H. (March 2011). <a rel="nofollow" class="external text" href="https://doi.org/10.2478%2Fv10156-010-0010-7">"A contemporary perspective of geoid structure"</a>. <i>Journal of Geodetic Science</i>. <b>1</b> (1): 82–87. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011JGeoS...1...82M">2011JGeoS...1...82M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2478%2Fv10156-010-0010-7">10.2478/v10156-010-0010-7</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geodetic+Science&rft.atitle=A+contemporary+perspective+of+geoid+structure&rft.volume=1&rft.issue=1&rft.pages=82-87&rft.date=2011-03&rft_id=info%3Adoi%2F10.2478%2Fv10156-010-0010-7&rft_id=info%3Abibcode%2F2011JGeoS...1...82M&rft.aulast=Moritz&rft.aufirst=H.&rft_id=https%3A%2F%2Fdoi.org%2F10.2478%252Fv10156-010-0010-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeoid" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003C.HTM">"Physical Geodesy"</a>. <i>Geodesy for the Layman</i>. NOAA. 1984.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Physical+Geodesy&rft.btitle=Geodesy+for+the+Layman&rft.pub=NOAA&rft.date=1984&rft_id=https%3A%2F%2Fwww.ngs.noaa.gov%2FPUBS_LIB%2FGeodesy4Layman%2FTR80003C.HTM&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeoid" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geoid&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/geoid" class="extiw" title="wiktionary:Special:Search/geoid">geoid</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://earth-info.nga.mil/">NGA webpage on Earth Gravitational Models</a></li> <li><a rel="nofollow" class="external text" href="https://cddis.nasa.gov/926/egm96/egm96.html">NASA webpage on EGM96</a></li> <li><a rel="nofollow" class="external text" href="https://www.ngs.noaa.gov/GEOID/">NOAA webpage on Geoid Models</a></li> <li><a rel="nofollow" class="external text" href="https://icgem.gfz-potsdam.de/home">International Centre for Global Earth Models (ICGEM)</a></li> <li><a rel="nofollow" class="external text" href="https://www.isgeoid.polimi.it/">International Service for the Geoid (ISG)</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output 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