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Geoid

The geoid is the equipotential surface of the Earth's gravity field that best fits, in a least squares sense, the mean sea level in the absence of external forces (except the Earth's gravity).^[1] It represents the shape the ocean surface would adopt under the influence of gravity and Earth's rotation alone, excluding effects like winds, tides, and currents, and serves as a model of global mean sea level.^[2] This surface extends continuously through the continents as an imaginary reference for elevation measurements. Unlike the smooth, oblate spheroid known as the reference ellipsoid used in geodesy, the geoid is highly irregular and undulating due to variations in the Earth's internal mass distribution, which cause local gravitational anomalies.^[2] These geoid undulations—the vertical deviations between the geoid and the ellipsoid—range globally from approximately -106 meters in the Indian Ocean to +84 meters near Iceland, reflecting denser or less dense subsurface materials that alter gravitational pull. The geoid is fundamental to physical geodesy, providing the zero-level reference for orthometric heights (elevations above sea level) in surveying, mapping, and navigation systems like GPS, where ellipsoidal heights must be corrected by subtracting geoid undulation values to obtain accurate terrain elevations.^[3] It also supports applications in oceanography, coastal management, and hydrology by enabling precise modeling of water levels, flooding risks, and sea-level changes.^[4] Geoid models are computed using a combination of terrestrial gravity surveys, satellite altimetry, and global gravimetric data processed via methods like the Stokes integral, with missions such as NASA's GRACE and its successor GRACE-FO providing high-resolution insights into gravitational variations.^[5]^[6]^[7] Refinements incorporating data from NOAA's GRAV-D project (2007–2023) and ongoing efforts like the National Spatial Reference System (NSRS) modernization aim to achieve centimeter-level accuracy for national and global height systems.^[2]^[8]

Fundamentals

Definition and Physical Concept

The geoid is the equipotential surface of the Earth's gravity potential that best fits, in a least-squares sense, the global mean sea level.^[9] This surface represents a level where the gravitational potential due to Earth's mass distribution and rotation is constant, meaning no net force acts to move water or other fluids across it. In the absence of disturbing influences such as winds, tides, or ocean currents, the geoid coincides precisely with the mean sea level in the oceans.^[4] Unlike the physical surface of the Earth, which features dramatic topographic variations from ocean trenches to mountain peaks spanning thousands of meters, the geoid exhibits a smoother, undulating form resulting from the uneven internal mass distribution of the planet. This irregularity causes the geoid to deviate from an idealized sphere or reference ellipsoid by up to approximately ±100 meters, creating subtle "hills" and "valleys" in the gravity field.^[9] While spheres and ellipsoids serve as simplified mathematical models for global calculations, the geoid provides a more accurate physical representation of Earth's shape under gravitational equilibrium. The geoid plays a crucial role in defining "true" elevations, as heights above sea level on land are measured orthometrically relative to this surface rather than to an ellipsoid or sphere. For instance, if ocean waters were entirely at rest without any dynamic effects, their surface would conform exactly to the geoid, illustrating how it acts as the natural zero-level reference for vertical measurements worldwide.^[10] The conceptual foundation of the geoid as an equipotential surface emerged from the work of 18th-century scientists, including Alexis-Claude Clairaut and Leonhard Euler, who advanced theories on the rotational figure of the Earth.^[11] This idea was formalized in 1828 by Carl Friedrich Gauss, who defined the mathematical figure of the Earth as the equipotential surface of the gravity field that most closely approximates mean sea level.^[12]

Historical Development

The concept of the geoid emerged from early efforts to understand Earth's shape through gravitational principles in the late 17th century. In 1687, Isaac Newton proposed in his Philosophiæ Naturalis Principia Mathematica that Earth is an oblate spheroid, flattened at the poles due to rotation, with polar flattening estimated between 1/230 and 1/579 based on Newtonian gravitation.^[13] This theoretical deduction laid the groundwork for gravity-based models of Earth's figure. Building on this, Christiaan Huygens in 1690 used pendulum clock observations to measure latitudinal variations in gravity, noting that pendulum lengths required adjustment with latitude to maintain consistent periods, further supporting the oblate spheroid idea and linking it to surface gravity.^[13] The formalization of the geoid occurred in the 19th century amid advances in geodesy. In 1828, Carl Friedrich Gauss defined the geoid as the "mathematical figure of the Earth," an equipotential surface where the plumb line is perpendicular to the gravity potential, distinguishing it from idealized ellipsoids.^[1] This conceptualization emphasized the geoid's role as a reference for gravitational equilibrium. Later, in 1872, Johann Benedict Listing coined the term "geoid" for this equipotential surface in his work Über unsere jetzige Kenntniss der Gestalt u. Grösse der Erde, refining astrogeodetic methods and establishing it as a standard in physical geodesy.^[14] Twentieth-century developments advanced geoid determination through theoretical and observational innovations. In the 1940s, Mikhail Molodensky developed a theory for normal heights, referencing a normal gravity field rather than actual gravity, which allowed height computations on Earth's physical surface without assuming a distant geoid, improving practical geodesy in irregular terrains.^[15] Post-World War II, global gravity models proliferated with improved instrumentation like spring gravimeters, enabling the International Gravity Standardization Net 1971 (IGSN 71) and early global geoid computations from block-averaged gravity data.^[16] The 1960s marked the integration of satellite geodesy, revolutionizing geoid understanding. Missions like Vanguard I (launched 1958) provided orbital data revealing Earth's gravitational irregularities, including a pear-shaped geoid with a northern stem and undulations up to 15 meters, confirming a third zonal harmonic (J3) and refining polar flattening estimates to 1/297.4.^[17] These observations, analyzed through dynamical methods, shifted geodesy from terrestrial-only approaches to space-based global models. The International Association of Geodesy (IAG), established in 1919, played a pivotal role in standardizing geoid-related frameworks. By 1967, it adopted the Geodetic Reference System 1967 (GRS 67), specifying ellipsoid parameters and gravitational constants for consistent geoid modeling; this evolved into GRS 80 in 1980, incorporating satellite-derived gravity insights.^[16] Non-European contributions, often underrepresented, included 19th-century efforts like India's Great Trigonometrical Survey (1802–1871), led by William Lambton and George Everest, which measured extensive baselines and triangulations across the subcontinent using the Everest Spheroid. Though lacking direct gravity data, these surveys informed early regional figure-of-Earth approximations and influenced subsequent gravity-integrated models in Asia.^[18]

Mathematical Description

Formulation and Equations

The total gravity potential W at a point in space is the sum of the gravitational potential V, generated by the Earth's mass distribution, and the centrifugal potential \Phi, resulting from the planet's rotation:

W = V + \Phi.

This formulation accounts for both attractive and rotational forces acting on a test particle.^[19]^[9] The geoid is the particular equipotential surface of W on which W = W_0, a constant value chosen to approximate mean sea level. Orthometric heights H, which measure elevation above the geoid corrected for gravity variations along the plumb line, are defined relative to this surface. In contrast, ellipsoidal heights h are geometric distances from a reference ellipsoid, typically determined via satellite methods like GPS. The geoid undulation N, representing the vertical separation between the geoid and the ellipsoid, is thus given by

N = h - H.

This relation connects geometric and gravity-based height systems.^[19]^[9] To quantify deviations from the reference ellipsoid, the normal potential U is introduced as the potential generated by a idealized, rotating ellipsoid of revolution with the same total mass and angular velocity as the Earth. The disturbing potential T, which captures the irregularities due to the actual mass distribution, is defined as

T = W - U.

On the geoid, where W = W_0 and U \approx U_0, T simplifies to reflect these anomalies.^[19]^[20] Bruns' formula provides the direct link between the geoid undulation and the disturbing potential at points on or above the geoid:

N = \frac{T}{\gamma},

where \gamma is the magnitude of the normal gravity vector on the reference ellipsoid. This linear approximation holds under the assumption of small disturbances relative to the normal field and is fundamental for computing geoid heights from potential data.^[19]^[9]^[20] The mathematical foundation for V derives from Newton's law of gravitation. Inside the Earth, where mass density \rho is nonzero, V satisfies Poisson's equation:

\nabla^2 V = -4\pi G \rho,

with G as the gravitational constant. Outside the Earth, in mass-free space, it reduces to Laplace's equation:

\nabla^2 V = 0.

These partial differential equations are solved subject to boundary conditions at the Earth's surface, including continuity of V and its normal derivative across the interface, ensuring a unique harmonic exterior solution that decays at infinity. The centrifugal potential \Phi, being quadratic in coordinates, is analytically known and added post-solution. This framework yields W, from which T and N follow.^[19]^[21]

Relation to Reference Ellipsoid

The reference ellipsoid serves as a smooth mathematical approximation to the geoid, modeled as an oblate spheroid that aligns closely with the geoid at the equator and poles, reflecting the Earth's rotational flattening. A standard example is the WGS 84 ellipsoid, defined with a semi-major axis a = 6,378,137 m and flattening f = 1/298.257223563.^[22] The separation between the geoid and the reference ellipsoid is described by the undulation N, the vertical distance measured along the ellipsoidal normal from the ellipsoid to the geoid. Complementing this, the deflection of the vertical quantifies angular discrepancies, with components ξ (north-south) and η (east-west) denoting the differences between the local plumb line (gravity direction) and the ellipsoid normal.^[23] These deflections relate to spatial changes in undulation, approximated along the meridian by ξ ≈ - ∂N / ∂s, where s is the surface distance; this connects deflections to gravity anomalies influencing the geoid's slope.^[24] By construction in global models like EGM96, the mean undulation N is approximately 0 m, with a root-mean-square (RMS) value of about 30 m reflecting worldwide variations; regionally, undulations reach highs of +85 m near Iceland due to subsurface density variations.^[25] Unlike the smooth, rotationally symmetric reference ellipsoid, the geoid exhibits irregularities from heterogeneous mass distribution, necessitating undulation and deflection corrections for accurate height reference; for instance, the North American Vertical Datum of 1988 (NAVD88) employs geoid models to convert ellipsoidal heights to orthometric heights above the geoid.^[26]

Determination and Measurement

Gravimetry and Ground-Based Methods

Ground-based gravimetry provides essential measurements of the Earth's gravity field for determining local geoid undulations through the computation of gravity anomalies. These methods, predominant before the advent of satellite missions, rely on direct terrestrial observations to capture variations in gravitational acceleration that deviate from a reference model, enabling the integration of data into geoid models via techniques like Stokes' integral. Absolute gravimetry establishes precise reference values at benchmark stations, while relative gravimetry facilitates extensive surveys across varied terrains.^[5]^[27] Absolute gravimeters, such as the FG5 series developed by Micro-g LaCoste, measure gravity by tracking the free fall of a corner-cube retroreflector in a vacuum, achieving accuracies on the order of 2 microgals at established stations. These instruments serve as primary references for calibrating relative measurements and tying local surveys to global networks. Relative gravimetry, commonly employing spring-based instruments like the LaCoste-Romberg model, detects differences in gravity between stations by monitoring beam deflection or spring elongation, with typical survey accuracies reaching 0.1 mGal after calibration against absolute sites. Such relative surveys form the backbone of dense gravity networks on land, allowing for the mapping of regional anomalies essential to geoid computation.^[28]^[29] Gravity anomalies, defined as Δg = g_observed - g_normal where g_normal is the gravity predicted by a reference ellipsoid, quantify deviations attributable to mass irregularities and are integrated via Stokes' method to yield geoid heights N. On land, surveys using relative gravimeters build comprehensive networks adjusted for consistency through least-squares techniques that minimize discrepancies at crossovers and ties to absolute stations. Marine gravity measurements, conducted from ships with stabilized gravimeters, extend coverage over oceans where land-based data are unavailable, though they require corrections for platform motion to maintain 1-2 mGal precision. Airborne gravity campaigns, emerging prominently in the 1980s, employed aircraft-mounted LaCoste-Romberg systems to survey remote or rugged areas, such as early North American initiatives that filled gaps in continental data with resolutions down to 5-10 km.^[27]^[30]^[31] The foundations of global gravity mapping trace to pendulum-based absolute measurements from the 1950s to 1970s, which compiled the first worldwide datasets with accuracies around 0.1 mGal, culminating in networks like the International Gravity Standardization Net 1971 that integrated over 25,000 observations. These efforts produced initial global anomaly maps, highlighting major features like ocean trenches and continental shields. However, ground-based methods face inherent limitations, including sparse spatial coverage—particularly over vast oceanic expanses and polar regions—and the necessity for meticulous terrain corrections to account for local topography, which can introduce errors up to several mGals if not modeled accurately using prism or digital elevation methods. These challenges necessitated network adjustments to ensure datum consistency but restricted pre-satellite geoid resolutions to tens of kilometers in data-poor areas.^[32]^[33]^[34]^[35]

Satellite Missions and Data

Satellite missions have revolutionized the determination of the global geoid by providing comprehensive gravity field data from space, enabling the mapping of Earth's irregular gravitational potential on a planetary scale. The Gravity Recovery and Climate Experiment (GRACE), launched in March 2002 and operational until October 2017, consisted of twin satellites that measured variations in Earth's gravity field through microwave ranging in the K-band between the satellites, detecting minute changes in their separation due to gravitational anomalies.^[36]^[37] This approach allowed GRACE to recover monthly gravity field models, revealing time-variable mass redistributions such as those from ice melt and groundwater depletion. The GRACE Follow-On (GRACE-FO) mission, launched in May 2018 and continuing operations, extends this capability with identical K-band ranging technology plus a laser ranging interferometer for enhanced precision, maintaining continuity in global gravity monitoring.^[7] Complementing these, the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE), launched in March 2009 and concluding in November 2013, employed electrostatic gravity gradiometry to measure subtle differences in gravitational acceleration, achieving high spatial resolution for the static gravity field.^[38]^[39] GOCE's low-Earth orbit at approximately 260 km altitude minimized non-gravitational disturbances, focusing on medium- to high-wavelength gravity signals essential for geoid undulations. Data from these missions are processed into Level-1 products (raw observations like orbits and accelerations) and Level-2 gravity field solutions (spherical harmonic coefficients), with post-2020 enhancements in GRACE-FO models, such as Release 6.3, incorporating improved error characterization and atmospheric corrections for better low-degree harmonic recovery.^[40]^[41] Processing satellite data for geoid determination involves removing non-gravitational forces, including atmospheric drag, solar radiation pressure, and tidal effects, using onboard accelerometers to isolate gravitational signals.^[42]^[38] These refined gravity fields are then combined with terrestrial gravimetry through the remove-compute-restore (RCR) technique, where long-wavelength signals from a reference model are subtracted, short-wavelength contributions computed from residual data, and the full signal restored to yield a high-resolution geoid.^[43] In the post-GOCE era, reprocessed datasets, such as ESA's Release 6 gravity field model derived from the entire mission's gradients with advanced filtering, have filled observational gaps and improved consistency with GRACE-FO outputs.^[44] Recent advancements from 2023 to 2025 include the integration of GRACE-FO data into NOAA's North American-Pacific Geopotential Datum of 2022 (NAPGD2022), enhancing vertical datum accuracy by incorporating satellite-derived gravity models for long-wavelength features, achieving sub-decimeter improvements in regional geoid heights.^[45] The European Space Agency (ESA) is planning the Next Generation Gravity Mission (NGGM) as part of the Mass Change and Geophysics International Constellation (MAGIC), aiming for launch around 2030 to provide higher temporal and spatial resolution through multi-pair satellite configurations, building on GRACE-FO and reprocessed GOCE data.^[46] Combined analyses from GRACE, GOCE, and GRACE-FO have yielded global geoid models with root-mean-square (RMS) accuracies of 1-2 cm, particularly for undulations at scales above 100 km, marking a significant leap in precision for geodetic applications.^[47]

Variations and Influences

Spatial Variations Due to Mass Distribution

The geoid's surface exhibits significant spatial undulations due to lateral variations in Earth's mass distribution, primarily arising from density contrasts within the crust and mantle. These undulations reflect deviations from the ideal equipotential surface, with geoid heights ranging from approximately -106 m to +90 m globally, driven by subsurface mass excesses or deficits. In regions of subduction, sinking oceanic slabs introduce dense material into the mantle, producing pronounced geoid lows; for instance, the Indian Ocean Geoid Low reaches -106 m, attributed to the gravitational pull of ancient subducted slabs interacting with large low-shear-velocity provinces at the core-mantle boundary. Conversely, geoid highs occur over areas of mass deficit, such as mid-ocean ridges where hot, buoyant mantle upwelling thins the lithosphere and reduces local density; the North Atlantic geoid high, associated with the Mid-Atlantic Ridge, exemplifies this with elevations up to +65 m linked to enhanced volcanism and lithospheric thinning.^[48]^[49]^[50] Isostatic compensation plays a key role in these variations, where crustal thickness fluctuations maintain equilibrium between the lithosphere and underlying mantle. Thicker crust in continental regions, such as orogenic belts, displaces denser mantle material, leading to positive geoid anomalies, while thinner oceanic crust allows greater mantle influence, often resulting in subtler undulations. Mantle convection further modulates these effects by driving long-wavelength density anomalies; upwellings of hot, low-density material contribute to geoid highs, whereas downwellings from subduction or cooling enhance lows. For example, the Himalayan geoid high, reaching up to +50 m in some models, stems from isostatic rebound and tectonic loading by the thickened Tibetan crust, which exceeds 70 km in places due to ongoing India-Asia collision. Similarly, the African Superswell features a relative geoid low (down to -4 m in parts), influenced by broad mantle upwelling that thins the lithosphere but is partially compensated by compositional variations in the upper mantle.^[51]^[52]^[53] These spatial patterns are quantitatively linked through Poisson's relation, which connects gravity anomalies (Δg) to geoid undulations (N) via the radial derivative of the disturbing potential. Approximated as

\Delta g \approx -\frac{\partial (\gamma N)}{\partial r},

where γ is the normal gravity and r is the radial distance, this relation highlights how vertical changes in the gravitational potential from density heterogeneities produce observable gravity and geoid signals. Negative Δg over geoid highs indicate mass deficits, while positive Δg align with lows from mass excesses, enabling inference of subsurface structure from surface measurements.^[54] Geophysical modeling elucidates these mass distributions through forward and inverse techniques. Forward modeling simulates geoid undulations (N) by integrating density perturbations (ρ) into the gravitational potential, using parameterized mantle structures to predict observed anomalies; for subduction zones, this involves assigning higher densities to slabs (e.g., +1-2% relative to ambient mantle) to replicate lows like the Indian Ocean feature. Inverse problems reverse this process, constraining mantle density and viscosity from geoid data via tomographic inversions, often revealing deep-seated heterogeneities such as slab graveyards or upwellings that explain regional highs and lows. These approaches, grounded in spherical harmonic expansions of the potential, prioritize long-wavelength signals (>1000 km) to isolate convective influences from crustal effects.^[55]^[56]

Temporal Changes

Temporal variations in the geoid arise primarily from dynamic mass redistributions within the Earth system, including post-glacial rebound, ongoing ice sheet melting, and fluctuations in continental water storage. Post-glacial rebound, or glacial isostatic adjustment (GIA), results from the viscoelastic response of Earth's mantle to the melting of Pleistocene ice sheets, leading to uplift in formerly glaciated regions and corresponding positive geoid height changes. For instance, in the Hudson Bay region, GIA induces a geoid height increase of approximately +1 mm/yr. Conversely, mass loss from major ice sheets alters the geoid negatively over polar regions; combined contributions from Greenland and Antarctic ice melt equate to a sea-level rise of about 0.5 mm/yr in earlier assessments, though recent observations indicate higher rates of 1.2 mm/yr equivalent due to accelerated discharge. Hydrological cycles, involving seasonal storage changes in soil moisture, groundwater, and surface waters, produce oscillatory geoid signals with amplitudes reaching ±10 cm in regions like the Amazon basin and Eurasian river systems.^[57]^[58]^[59] Satellite gravimetry missions such as GRACE (2002–2017), GRACE-FO (2018–present), and GOCE (2009–2013) have enabled precise monitoring of these temporal geoid changes through time-series of gravity field variations. GRACE/GRACE-FO data reveal global geoid trends from 2002 to 2025 influenced by climate-driven mass shifts, with an average rise of 0.3 mm/yr attributed to net ocean mass gain from land ice melt and thermal expansion effects. Over Antarctica, mass loss averaged 150 Gt/yr from 2002 to 2023, contributing 0.4 mm/yr to sea-level rise and inducing localized geoid depression, while Greenland's 270 Gt/yr loss adds 0.8 mm/yr equivalent. These missions capture both secular and periodic signals, with GOCE providing complementary high-resolution static baselines adjusted for temporal components.^[60]^[61]^[62] To quantify these variations, geoid time-series are modeled using least-squares estimation for secular (long-term linear) trends and harmonic functions for annual and semi-annual periodicities, accounting for hydrological, atmospheric, and oceanic loading effects. This approach isolates climate signals from GIA, revealing accelerations such as the doubling of Antarctic mass loss rate from -74 Gt/yr (2002–2010) to -142 Gt/yr (2011–2020) during the 2010s, driven by enhanced glacier discharge in West Antarctica. Recent GRACE-FO observations from 2023 to mid-2025 indicate variable mass balance, with sustained average losses around 135 Gt/yr up to 2023, but a record mass gain in 2024-2025 due to high precipitation, particularly in East Antarctica, leading to temporary geoid stabilization over the continent. In 2024-2025, Antarctica experienced a net mass gain for the first time in decades, driven by exceptional snowfall, resulting in a temporary uplift in geoid heights over the ice sheet and altering short-term sea-level contributions.^[63]^[64]^[65]^[66]^[67]

Modeling and Representation

Spherical Harmonic Models

Spherical harmonic models represent the global geoid through the expansion of the Earth's disturbing potential T, which quantifies the deviation of the actual gravitational potential from a reference ellipsoidal potential. The disturbing potential outside the Earth is expressed in spherical coordinates ([r](/page/R), [\theta](/page/Theta), [\phi](/page/Phi)), where [r](/page/R) is the radial distance, [\theta](/page/Theta) the colatitude, and [\phi](/page/Phi) the longitude, as

T(r, \theta, \phi) = \sum_{l=1}^{\infty} \sum_{m=0}^{l} \left( \frac{R}{r} \right)^{l+1} \left[ \bar{C}_{lm} \cos(m\phi) + \bar{S}_{lm} \sin(m\phi) \right] \bar{P}_{lm}(\cos \theta),

with R as the reference radius (typically 6,371 km), \bar{C}_{lm} and \bar{S}_{lm} the fully normalized spherical harmonic coefficients of degree l and order m, and \bar{P}_{lm} the associated Legendre functions.^[68] This series expansion allows the geoid height N to be derived via Bruns' formula, N = T / \gamma, where \gamma is the normal gravity at the ellipsoid surface, evaluated at the geoid.^[69] These models are constructed by determining the harmonic coefficients through least-squares adjustment to a combination of gravity data sources, including satellite altimetry, gravimetry, and satellite gravimetry. A seminal example is the Earth Gravitational Model 1996 (EGM96), complete to degree and order 360, providing a global resolution of approximately 55 km and incorporating about 130,000 coefficients from satellite tracking, surface gravity, and altimetry data.^[70] Subsequent advancements led to EGM2008, complete to degree and order 2159 with additional terms up to degree 2190, achieving a finer resolution of about 9 km (5 arcminutes) through integration of GRACE satellite data and enhanced terrestrial measurements.^[71] The coefficients in these models capture the geoid undulations with global root-mean-square accuracies improving from around 0.5 m for EGM96 to 0.15 m for EGM2008 in well-observed regions.^[72] Geoid heights from spherical harmonic models are computed by synthesizing the potential at desired points on or above the surface, often using fast Fourier transform (FFT) techniques for efficient global evaluation on a grid. Alternatively, for high-frequency components or local refinements, point-mass integration methods approximate the mass distribution to supplement the harmonic expansion. Error estimates are derived from the variance-covariance matrix of the coefficients, quantifying uncertainties in the potential and propagated to geoid heights, typically at the centimeter level for low degrees and increasing for higher ones due to data sparsity.^[73] Efforts toward the Earth Gravitational Model 2020 (EGM2020) are ongoing as of 2025, aiming to incorporate data from the GOCE and GRACE-FO satellite missions to enhance low-degree terms and reduce long-wavelength errors, while addressing omission errors from unmodeled high-degree signals through forward modeling of terrain effects. Omission errors, arising from truncating the series beyond the maximum degree, can contribute up to several decimeters in mountainous areas but are expected to be mitigated in future releases by blending with higher-resolution gravity datasets. Validation of current models like EGM2008 against independent GPS/leveling networks confirms global accuracies of 20-30 cm, with improvements to 15 cm in regions with dense control points.^[74]^[75]

Applications in Geodesy

In geodesy, the geoid plays a central role in height modernization efforts, enabling the transformation of Global Navigation Satellite System (GNSS) ellipsoidal heights to orthometric heights, which are referenced to mean sea level. This process relies on the geoid undulation N, where orthometric height H = h - N (with h as the ellipsoidal height), allowing for more accurate vertical datums without extensive traditional leveling. In the United States, the National Geodetic Survey (NGS) utilizes the GEOID2022 model as part of the North American Terrestrial Reference Frame of 2022 (NATRF2022) and North American-Pacific Geopotential Datum of 2022 (NAPGD2022), which shifted from the legacy North American Vertical Datum of 1988 (NAVD88) to a gravimetrically determined geopotential datum, achieving centimeter-level accuracy in continental regions. This modernization supports infrastructure projects, flood risk assessment, and coastal management by providing consistent, high-precision elevations across the nation.^[76]^[77]^[78]^[79] The geoid is integral to navigation and Geographic Information Systems (GIS), where it corrects for gravitational irregularities to ensure precise positioning and mapping. In marine charting, shipborne GNSS measurements of sea surface heights are adjusted using geoid models to derive accurate bathymetry and navigation aids, reducing errors in coastal surveys to the decimeter level. For aviation, geoid undulation data embedded in GNSS receivers convert ellipsoidal heights to orthometric ones, supporting altimetry corrections that enhance flight safety and terrain avoidance systems. In GIS applications, integration of geoid models with GNSS enables the creation of vertical-consistent digital elevation models for urban planning and environmental monitoring, as seen in tools that align satellite-derived elevations with local datums for seamless data fusion.^[80]^[81]^[82] In Earth sciences, geoid models facilitate monitoring of dynamic processes such as sea-level rise and tectonic activity. Satellite missions like the Gravity Recovery and Climate Experiment (GRACE) use geoid variations to quantify continental water storage changes, revealing groundwater depletion in regions like California's Central Valley, where mass loss contributes to subsidence and sea-level contributions on the order of millimeters per year globally. For sea-level monitoring, geoid adjustments in tide gauge and altimetry data isolate steric and mass-driven changes, improving estimates of global mean sea-level rise to within 0.4 mm/year. In tectonics, geoid anomalies help map crustal strain and post-glacial rebound, such as in Scandinavia, where viscoelastic models informed by geoid data explain uplift rates of up to 1 cm/year. These applications underscore the geoid's role in linking gravitational signals to climate and solid-Earth dynamics.^[83]^[84]^[85] Regional geoid models enhance local accuracy for specialized applications, often achieving centimeter-level precision through integration of dense gravity and GNSS/levelling data. The European Gravity Geoid Model 1997 (EGG97), developed under the European Subcommission for the European Geoid, combined over 2.7 million terrestrial gravity points with satellite altimetry and terrain corrections to produce a quasigeoid model with an estimated accuracy of 15-20 cm across Europe, enabling precise height networks for infrastructure like the Channel Tunnel. Such models support national cadastral systems and engineering surveys by minimizing undulation errors in orthometric height computations.^[86]^[87] Looking ahead, geoid applications are poised for advancement through emerging technologies like quantum gravimeters, which promise sub-milligal gravity measurements for refining high-resolution models post-2025. NASA's planned space-based quantum gravity gradiometer mission aims to detect subtle mass redistributions, enhancing geoid undulations for disaster response and resource management. In the context of United Nations Sustainable Development Goals, particularly Goal 13 on climate action, geoid-informed elevation data will bolster flood mapping tools, integrating GNSS and satellite observations to predict inundation risks in vulnerable coastal areas with improved vertical accuracy.^[88]^[89]^[90]

References

[1]
What is the geoid? - National Geodetic Survey - NOAA
Jan 30, 2001 · Here is the one currently adopted at NGS: geoid: The equipotential surface of the Earth's gravity field which best fits, in a least squares ...
[2]
What is the geoid? - NOAA's National Ocean Service
Jun 16, 2024 · The geoid is a model of global mean sea level that is used to measure precise surface elevations.
[3]
[PDF] Datums, Heights and Geodesy - National Geodetic Survey
-geoid is a vertical datum tied to MSL. -geoid height is ellipsoid height from specific ellipsoid to geoid.
[4]
The Geopotential Surface - National Geodetic Survey - NOAA
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.”
[5]
Geoid modeling calculations | Geopotential Datums | Research
The geoid products are used in many NOAA operational applications, such as initializing coastal and ocean modeling (flooding, inundation, surge, and tides), ...Missing: importance | Show results with:importance
[6]
Earth's Gravity Field - NASA Earth Observatory
Jul 29, 2003 · GRACE is accomplishing that goal by providing a more precise definition of Earth's geoid—an imaginary surface defined only by Earth's ...
[7]
[PDF] Geoid Surfaces and Theory - National Geodetic Survey
Note that the geoid is a vertical datum surface. • A geoid height is the ellipsoidal height from an ellipsoidal datum to a geoid. • Hence, geoid height models ...
[8]
A tutorial on datums - VDatum - NOAA
GEOID. The geoid is a specified equipotential surface, defined in the Earth's gravity field, which best fits, in a least squares sense, global mean sea level ...
[9]
[2208.12757] Clairaut, Euler and the figure of the Earth - arXiv
Aug 17, 2022 · In this paper, we review the work of the eighteenth-century French mathematician, astronomer and geographer Alexis-Claude Clairaut related to the question of ...
[10]
[PDF] Facilities–Freytag - The University of Chicago Press
Figure of the Earth. In 1828 Carl Friedrich Gauss de- fined the mathematical figure of the earth as that equi- potential surface of the earth's gravity ...
[11]
random and systematic error in physical geodesy, c. 1800–1910
Starting with Isaac Newton's and Christian Huygens's influential work in 1687 and 1690, geodesists had modelled the earth as an oblate ellipsoid of ...
[12]
Earliest Known Uses of Some of the Words of Mathematics (G)
Mar 20, 2000 · GEOID was first used in German (geoide) in 1872 by Johann Benedict Listing (1808-1882) in Über unsere jetzige Kenntniss der Gestalt u. Grösze ...
[13]
Ellipsoid, geoid, gravity, geodesy, and geophysics - GeoScienceWorld
Mar 3, 2017 · Distinguished from Stokes' formula, Molodensky's theory says that the physical surface of the Earth can be determined without using the density ...
[14]
The International Association of Geodesy: from an ideal sphere to an ...
Gauss' student Johann Benedict Listing (1808–1882) called the corresponding surface the “geoid”. Geoid height differences (ΔN) result from multiplying the ...<|separator|>
[15]
[PDF] Satellite Geodesy 1958-1964 - NASA Technical Reports Server
Much progress has been made in satellite geodesy during the years. 1958 to 1964, although it is likely that sub- stantial improvements will follow the launch of ...
[16]
Geodetic infrastructure in India - Coordinates
The Great Trigonometrical Survey of India was completed in 19th century under leadership of the great surveyors- Lambton and Sir George Everest.
[17]
[PDF] Physical geodesy
Sep 25, 2022 · Abstract. Physical geodesy studies the large-scale figure and gravity field of the Earth, which are closely related.<|control11|><|separator|>
[18]
[PDF] PHYSICAL GEODESY
disturbing potential (as computed from an assumed normal gravity field). This formula is known as generalized Bruns' formula and lt relates the ou = yQ,NP ...
[19]
[PDF] 3.02 Potential Theory and Static Gravity Field of the Earth - Elsevier
This formula holds for T anywhere on or above the geoid and thus can also be used to determine the geoid undulation according to Bruns' formula [3]. ... Moritz H ...
[20]
World Geodetic System 1984 (WGS 84) - NGA - Office of Geomatics
These are the semi-major axis of the WGS 84 ellipsoid, the flattening factor of the Earth, the nominal mean angular velocity of the Earth, and the geocentric ...
[21]
Geoid Evaluation | Geopotential Datums | Research
Once a geoid model has been developed, it is important to evaluate the model ... geoid that may be compared with future geoid profiles to reveal geoid change.Missing: importance | Show results with:importance
[22]
[PDF] THE USE AND ABUSE OF VERTICAL DEFLECTIONS - Earth Survey
and the component of the deflection of the vertical can be resolved along a geodetic azimuth (α) by ε = ξ cos α + η sin α. (2). It is important to distinguish ...
[23]
11. the egm96 geoid undulation with respect to the wgs84 ellipsoid
The geoid is an equipotential surface of the Earth's gravity field that is closely associated with the mean ocean surface. "Closely associated" can be defined ...
[24]
[PDF] converting gps height into navd88 elevation with the geoid96 geoid ...
The conversion from ellipsoid to orthometric height requires a geoid height model. Geoid heights in the conterminous United States range from about -8 meters to ...
[25]
[PDF] Definition of Functionals of the Geopotential and Their Calculation ...
If geodesists nowadays speak about gravity anomalies, they usually have in mind this definition with h = ht, i.e. on the Earth's surface. 2.4.3 The Topography- ...
[26]
[PDF] NOAA Technical Memorandum NOS NGS 93 Absolute Gravity ...
Sep 20, 2023 · instruments: The Micro-G LaCoste FG5 and A10 absolute gravimeters. The FG5 gravimeter is the current state-of-the art laboratory instrument ...
[27]
chapter v physical geodesy - National Geodetic Survey
In using the earth's gravity field to determine the shape of the geoid, the acceleration of gravity is measured at or near the surface of the earth. It might be ...<|control11|><|separator|>
[28]
[PDF] BEST PRACTICE IN GRAVITY SURVEYING
Network adjustment - using crossovers. Network adjustment for marine (and airborne) gravity surveys uses interpolated values where the ship tracks cross as ...
[29]
Analysis and geological interpretation of gravity data from GEOS-3 ...
A number of detailed gravimetric geoids of portions of the world's oceans from marine gravity measurements were constructed. The geoids were constructed by ...
[30]
[PDF] World Gravity Standards - NASA Technical Reports Server
This network contains more than 1800 stations around the world. The data used in the adjustment included more than. 25,000 gravimeter, pendulum and absolute.
[31]
Historical development of the gravity method in exploration - Available
Jul 13, 2017 · By the early 1970s, the best measurements were in the range of 0.01 to 0.05 mGal, and by about 1980, free-fall gravimeters had replaced ...
[32]
[PDF] Facts and an Approach to Collecting Gravity Data Using Near-Real ...
Because the resolution of a gravity model is limited by the spacing of the gravity measurements, regions of sparse data coverage are also regions of ...Missing: ground- | Show results with:ground-
[33]
[PDF] Gravity surveys - UBC EOAS
The following points provide an outline for how data are acquired for common ground-based surveys. ... Terrain corrections are hard to get accurate to ...
[34]
The gravity recovery and climate experiment: Mission overview and ...
May 8, 2004 · The GRACE mission is designed to track changes in the Earth's gravity field for a period of five years. Launched in March 2002, ...
[35]
GRACE mission comes to an end
Oct 27, 2017 · "The GRACE mission was planned to last five years but has been working very successfully for more than 15 years – three times longer than ...
[36]
Overview | Mission - GRACE-FO - NASA
GRACE-FO, which launched May 22, 2018, will continue the work of tracking Earth's water movement to monitor changes in underground water storage, the amount of ...
[37]
GOCE (Gravity field and steady-state Ocean Circulation Explorer)
GOCE was a geodynamics and geodetics mission with the objective of determining the stationary gravitational field. The mission ended in October 2013 after ...
[38]
ESA - Instruments - European Space Agency
GOCE's main instrument was the Electrostatic Gravity Gradiometer (EGG), a set of six 3-axis accelerometers mounted in a diamond configuration in an ultra-stable ...
[39]
GRACE-FO Data
Launched on May-22, 2018, the GRACE Follow-On (GRACE-FO) mission continues the legacy of GRACE, tracking Earth's water movement and surface mass changes across ...
[40]
Data Updates & Announcements - GRACE Tellus - NASA
GRACE-FO: new data release RL06.3 now available. The updated RL06.3 Level-2 gravity fields improve the low-degree harmonics time series, and the ...
[41]
Advancements in the GRACE and GRACE‐FO Gradiometer Mode
May 12, 2025 · Nominally, GRACE accelerometer measurements are used to remove the non-gravitational contributions from precise orbit determination (POD) ...
[42]
A discussion on the approximations made in the practical ...
Apr 26, 2005 · The remove-compute-restore (RCR) technique is the most well known method for regional gravimetric geoid determination today.
[43]
ESA's Release 6 GOCE gravity field model by means of the direct ...
ESA's Release 6 GOCE gravity field model by means of the direct approach based on improved filtering of the reprocessed gradients of the entire mission ( ...Missing: post- era<|control11|><|separator|>
[44]
[PDF] NOAA Technical Report NOS NGS 69 - National Geodetic Survey
1.3 The importance of a static geoid model ... This highlights the need and importance of including high resolution geoid changes in specific regions.
[45]
Citation for Next Generation Gravity Mission Mission Requirements ...
NGGM will extend and improve time series of satellite gravity missions by providing enhanced spatial and temporal resolution time-varying gravity field ...Missing: plans | Show results with:plans
[46]
[PDF] Evaluation of GOCE/GRACE and combined global geopotential ...
achieving an accuracy of approximately 1-2 cm for geoid undulations at a spatial resolution of approximately 100 km. Consequently, numerous GGMs have been ...
[47]
Seismologists Search for the Indian Ocean's “Missing Mass” - Eos.org
Apr 18, 2019 · This low-gravity region, which shows up as a 106-meter “dent” in the geoid, is referred to as the Indian Ocean Geoid Low (IOGL) [Sreejith et al.
[48]
How the Indian Ocean Geoid Low Was Formed - Pal - AGU Journals
May 5, 2023 · Lower mantle slabs are indirectly important in generating this geoid low as these subducting slabs perturb the African LLSVP and generate ...Missing: magnitude | Show results with:magnitude
[49]
North Atlantic geoid high, volcanism and glaciations - AGU Journals
Feb 3, 2010 · Geoid high and intense volcanism in the northern Mid Atlantic Ridge are interpreted as enhanced by the loading on the adjacent continents by ice caps during ...Introduction · Model Description and Results · Discussion and Conclusions
[50]
Mantle dynamics, isostasy, and the support of high terrain - Molnar
Feb 11, 2015 · Although estimates of crustal thickness show variations of 10 km or more, there is at best a poor correlation of crustal thickness and ...
[51]
New insights into the crust and lithospheric mantle structure of Africa ...
Jun 24, 2016 · In contrast, the African Superswell is characterized by moderate geoid anomaly values (−4 to +4 m), except in the southwestern border of the ...
[52]
[PDF] Flexure of the Indian Plate and Intraplate Earthquakes
Aug 21, 1988 · The Indian plate is flexed downward by the load of the Himalaya and by bending moments related to plate collision. The curvature of the ...
[53]
[PDF] magsat investigation of crustal magnetic anomalies in the eastern ...
Feb 22, 1983 · Poisson's relation, an analytic relationship between the poten- tials of the gravity and magnetic anomaly fields, is used to relate the gravity ...<|separator|>
[54]
Modeling 3‐D density distribution in the mantle from inversion of ...
Nov 16, 2013 · [1] We developed a three-dimensional scheme to invert geoid anomalies aiming to map density variations in the mantle.
[55]
On the resolution of density anomalies in the Earth's mantle using ...
Contamination of seismic velocity structure into the density model space produces patterns that resemble those obtained from inversions of real data. The ...
[56]
Numerical models of the rates of change of the geoid ... - SpringerLink
The predicted geoid rate has a peak value of 1.3–1.4 mm/yr over Hudson Bay and agrees within the computational accuracy, i.e. ±0.1 mm/yr, with the postglacial ...
[57]
7 SEA-LEVEL CHANGE | Satellite Gravity and the Geosphere
... ice, equivalent to 0.6 mm/yr of sea-level change, for Antarctica, and 70 mm of ice (0.3 mm/yr sea-level equivalent) for Greenland. An error of this ...Thermal Expansion Of The... · Ice Mass Balance · Isostatic Rebound
[58]
[PDF] GRACE Measurements of Mass Variability in the Earth System
The annual cycle in the geoid variations, up to 10 millimeters in some regions, peaked predomi- nantly in the spring and fall seasons. Geoid variations observed ...
[59]
GRACE and GRACE-FO polar ice mass loss
These images, created from GRACE and GRACE-FO data, show changes in polar land ice mass since 2002. Orange and red shades indicate areas that lost ice mass, ...Missing: geoid acceleration 2023-2025
[60]
Antarctic Ice Mass Loss 2002-2023 - NASA SVS
Between 2002 and 2023, Antarctica shed approximately 150 gigatons of ice per year, causing global sea level to rise by 0.4 millimeters per year.Missing: geoid signal acceleration
[61]
(PDF) Antarctic mass change rates from GRACE - ResearchGate
Aug 10, 2025 · 1] We estimate mass trends over Antarctica using gravity variations observed by the Gravity Recovery and Climate Experiment (GRACE) satellite ...
[62]
[PDF] Significance of secular trends of mass variations determined ... - HAL
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of ...
[63]
Regional acceleration in ice mass loss from Greenland and ...
Sep 17, 2014 · In Antarctica (Figure 3), the AP experiences a mass loss of 31±4 Gt/yr, with an acceleration in loss of 3.2±0.6 Gt/yr2. The northern tip of AP, ...
[64]
Ice Sheets - Earth Indicator - NASA Science
Sep 25, 2025 · Antarctica is losing ice mass at an average rate of about 135 billion tons per year, and Greenland is losing about 266 billion tons per year.<|control11|><|separator|>
[65]
[PDF] 6 Geopotential (01 February 2018) - IERS Conventions Centre
Feb 1, 2018 · The EGM2008 model (Pavlis et al., 2008) is complete to degree and order 2159, and contains addi- tional spherical harmonic coefficients up to ...<|separator|>
[66]
[PDF] Spherical harmonic expansions of geo-fields
Spherical harmonic expansions use coefficients to represent the global structure of geo-fields, like the Earth's gravity, in the spectral domain.
[67]
[PDF] NASA/TP—1998–206861
A related goal for an improved geoid was the establishment of a globally defined geoid that could form the reference surface for a global vertical datum. At ...
[68]
The development and evaluation of the Earth Gravitational Model ...
Apr 19, 2012 · EGM2008 is a spherical harmonic model of the Earth's gravitational potential, developed by a least squares combination of the ITG-GRACE03S gravitational model.Solution Development and... · Error Assessment · EGM2008 Model Products
[69]
NGA - Office of Geomatics
### Summary of EGM2008 Details
[70]
Spherical harmonic analysis and synthesis using FFT
Functionals of the Earth's gravity field such as gravity anomaly and geoidal height can also be computed via synthesis. Thermal-corrected sea level anomaly from ...
[71]
[PDF] EGM2020: Updates - Sirgas
Sep 18, 2019 · EGM2008 with improvements for regions where new data has been incorporated. Approved for Public Release #19-987. Page 5. 5.
[72]
[PDF] Evaluation of the EGM2008 Gravity Field by Means of GPS
The new EGM2008 global gravity field model is evaluated by comparisons of geoid heights computed from the model with those available at GPS levelling ...Missing: undulation | Show results with:undulation
[73]
GEOID18 Technical Details - National Geodetic Survey
The purpose of GEOID18 is to return a geoid height that is consistent with the published NAVD 88 orthometric heights on NGS Data Sheets when used with GPS ...
[74]
About - Height Modernization - National Geodetic Survey - NOAA
Height Modernization began by using orthometric heights from traditional leveling and ellipsoid heights from GNSS to create a geoid height model referenced to ...
[75]
Prepare your data for the NSRS modernization of 2022 - Esri
Sep 23, 2025 · When converting ellipsoidal heights to corresponding orthometric heights, the NGS recommends the use of the GEOID18 model for the contiguous ...
[76]
Shipborne GNSS-Determined Sea Surface Heights Using Geoid ...
This study developed a complete methodology for jointly deriving and validating shipborne GNSS-determined SSH, using a geoid model and realistic dynamic ...Missing: GIS aviation
[77]
3D GNSS data and the GEOID - GPS World
Jan 19, 2017 · The easiest way to check the accuracy of your vertical data is to use a high-precision GNSS receiver and collect a point on the survey mark.
[78]
Mean Sea Level, GPS, and the Geoid | Summer 2003 | ArcUser - Esri
The geoid approximates mean sea level. The shape of the ellipsoid was calculated based on the hypothetical equipotential gravitational surface.
[79]
Water Storage | Science - GRACE-FO - NASA
GRACE-FO detects groundwater changes, measures water storage, and shows human consumption depleting groundwater basins. It also measures land sinking from ...
[80]
Sea-level fingerprints emergent from GRACE mission data
May 9, 2019 · Here we perform a systematic calculation of sea-level fingerprints of on-land water mass changes using monthly Release-06 GRACE Level-2 Stokes coefficients.
[81]
Temporal Variations of the Marine Geoid - AGU Journals - Wiley
Oct 31, 2020 · Our study considers geoid variations arising from (i) water mass cycling between the Earth system components, including terrestrial water ...
[82]
[PDF] Evaluation and Improvement of the EGG97 Quasigeoid Model for ...
In the course of the European Geoid Project, about 2.7 million gravity data and about 700 million topographical height data were included in the project data ...
[83]
The European Gravimetric Quasigeoid EGG97 - An IAG Supported ...
The requirements for accuracy and resolution were derived from the potential of GPS heighting and satellite altimetry, and defined at the “cm” to “dm” level ...
[84]
Quantum gravity gradiometry for future mass change science
Mar 14, 2025 · A quantum gravity gradiometer in a low Earth orbit, operating in a cross-track configuration, could be a viable single-spacecraft measurement instrument.
[85]
NASA to Launch First Space-Based Quantum Gravity Sensor to Map ...
Apr 16, 2025 · NASA's Jet Propulsion Laboratory is developing the first space-based quantum gravity gradiometer to detect gravitational anomalies.
[86]
UN-SPIDER and ZFL Present Flood Mapping Innovations at LPS 2025
Jul 3, 2025 · Lastly, the latest project of UN-SPIDER was introduced, an AI-driven Digital Twin technology, illustrating future flood and sea level rise ...