The Geopotential Surface
Research question: How can we define the physical shape of the Earth?
The Earth’s shape can be described in simple mathematical form as an ellipsoid, or an oblate spheroid, with a radius at the equator larger than at the poles. We consider the best fit ellipsoid for the Earth, or a given region, when half of the Earth’s topography is above the ellipsoid’s surface and half of the topography is below the surface. The benefit of a mathematical geometric system is that there are no uncertainties associated with its definition. The ellipsoid is thus a convenient shape for resolving coordinates, and as a reference to the geopotential surface.
In order to achieve a better fit to the Earth’s surface, we use a physical property to describe the Earth’s surface with more spatial details. Because the majority of the Earth’s surface is covered by water, it is possible to characterize its surface using a physical property, the Earth’s geopotential field. Water will flow as a function of height difference and/or changes in the gravity field. This combination is defined as geopotential. The gravity field itself is dependent on the Earth's rotation, the shape (morphology) and density (lithology) of the Earth's surface. Since ancient times, the definitions of “horizontal” and “vertical” were directly related to a physical motion related to the gravity field. Builders found the vertical by using a heavy object suspended on a string. In many cases, the heavy object was a lead weight (Amos, chapter 7, verses 7-8). As such, the vertical direction to the Earth’s geopotential surface was called “plumb line,” from the word “Plumb” from the Latin for lead (Pb) and the word line for rope or string. To define a horizontal line that is parallel to the Earth’s geopotential surface, builders and engineers used a “level” (liquid and gas bubble trapped in a glass) to establish a horizontal plane. Today, we define motions along the vertical that are associated with a gain or loss of energy, and we define horizontal motion, when there is no change in potential energy.
It is possible to generate an infinite number of geopotential surfaces using the water surface, such as the water levels in lakes or ponds, or even in a water reservoir. However, there is only one geopotential surface around the Earth that matches the global sea level on average (at rest). This surface is called a “geoid,” this term was coined by Listing (1873) and Gauss (1876) and is defined as: the Earth’s gravity potential field and the equipotential surface, W0, that best describes the earth's surface. It is also important to note that a certain geopotential value (also known as, geopotential constant) provides a better fit for a global coverage, but a different value is more suitable for a more local/regional application. The International Association of Geodesy (IAG) adopted the value of 62,636,853.4 m2 s-2 based on evaluation of global satellite altimetry. NOAA’s National Geodetic Survey (United States) and Canadian Geodetic Survey (Canada) empirically evaluated the best fitting geopotential constant that fits best suited for the North American – Pacific Geopotential Datum of 2022 (NAPGD2022). Averaged water level observations over a very long period (19-year tidal epoch from 1983 till 2001) were used to generate a water surface “at rest,” without any oceanographic processes, and directly compare it to different geopotential surfaces. Both the United States and Canada agreed to use the W0 value of 62,636,856.00 m2 s-2 for North America and the Pacific, a value that was also adopted by the International Astronomical Association (IAU) and the International Earth Rotation and Reference Systems Service (IERS).
As a result, a good physical indicator for a local fit is the height difference between the geoid surface (global mean sea level) and the averaged water level surface (local mean sea level), which is typically within 1 m difference. It is also important to note that over riverine environments the height difference will be greater than 1 m because of additional factors that are contributing to the water levels and are not related to oceanographic processes. There are multiple applications that use the geoid. From an operational standpoint, it is possible to use geopotential surfaces to supplement water observation in the short duration needed for observations (compared to the multi-year observation needed for water-level observations). In addition, it is possible to measure gravity over land, even when water is not present. As such, the geoid is commonly used to initialize water modeling and predict water levels extending to land for coastal management, flooding, and inundation. The geoid is also used for height relationships. We typically communicate the smoothly undulating shape that follows a certain constant value of Earth's gravity field using the ellipsoid — we call it geoid undulation.
Using the local mean sea level defined above, it is possible to relate the geometric (ellipsoid surface) and geopotential (geoid surface) to different tidal reference systems, such as NOAA’s National Tidal Datum Epoch that calculates tidal surfaces based on different 19-year statistical calculation (such as: Lowest Astronomical Tide, Mean Lower Low Water, and Mean High Water). The datum tutorial (VDatum) web page provides more discussions on the different height relationships.
References
Burša, M., J. Kouba, M. Kumar, A. Müller, K. Radej, S.A. True, V. Vatrt, and M. Vojtíšková. 1999. “Geoidal Geopotential and World Height System,” Studia Geophysica et Geodaetica, 43, pp. 327–337. https://doi.org/10.1023/A:1023273416512
Dayoub, N., S.J. Edwards, and P. Moore. 2012. “The Gauss-Listing potential value W0 and its rate from altimetric mean sea level and GRACE,” Journal of Geodesy, 86(9): 681–694. https://doi.org/10.1007/s00190-012-0547-6
Gauss, C.F. 1828: Bestimmung des Breitenunterscchiedes zwischen den Sternwarten von Gottingen und Altona, Gottingen: Bei Vandenhoeck und Ruprecht
Gauss, C.F. (1876) Trigonometrischen und polygonometrischen Rechnungen in der Feldmesskunst. Halle, a. S. Verlag von Eugen Strien. Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am ramsdenschen Zenithsektor. In: Carl Friedrich Gauß Werke, neunter Band. Königlichen Gesellschaft der Wissenschaften zu Göttingen (1903)
IAG Resolution No. 1, 2015: IAG Resolution (No. 1) for the Definition and Realization of an International Height Reference System (IHRS), IAG Resolutions adopted by the IAG Council at the XXVI th IUGG General Assembly, Prague, Czech Republic, June 22–July 2, https://iag.dgfi.tum.de/fileadmin/IAG-docs/IAG_Resolutions_2015.pdf.
Listing, J.B. 1873: Uber unsere jetzige Kenntnis der Gestalt und Grosse der Erde, Nachr. d. Kgl., Gesellsch. d. Wiss. und der Georg-August-Univ., 33-98, Gottingen.
Roman, D. R., and X. Li. 2020: Analysis of a Geopotential Datum at Tide Gauge Stations, Paper 10426, Proceedings of the FIG Working Week (virtual), Amsterdam, the Netherlands, ISBN 978- 87-92853-93-6, ISSN 2307-4086. https://www.fig.net/resources/proceedings/fig_proceedings/fig2020/papers/ts04f/TS04F_roman_li_10426.pdf
Sánchez, L., J. Ågren, J. Huang, Y.M.Wang, J. Mäkinen, R. Pail, R. Barzaghi, G.S.Vergos, K. Ahlgren, and Q. Liu, .2021 “Strategy for the realisation of the International Height Reference System (IHRS).” J Geod 95, 33. https://doi.org/10.1007/s00190-021-01481-0
Sevilla, M., M. Burša, D. Dušátko, S. Kenyon, J. Kouba, Z. Šíma, Z., V. Vatrt, and M. VotjiÍšková.2008. Determination of Geopotential W0,ALICANTE and its Connection to W0,NAVD88.
Stokes, G.G., 1849: “On the variation of gravity at the surface of the Earth,” Transactions of the Cambridge Philosophical Society, V. 8, p. 672.
