Introduction
Satellite gravimetry (notably NASA’s GRACE mission) maps departures of Earth’s gravity field from an idealized smooth reference surface—the terrestrial ellipsoid—revealing positive anomalies (stronger gravity; commonly shown in red) and negative anomalies (weaker gravity; shown in blue). These spatial variations reflect heterogeneity in Earth’s internal mass distribution and its temporal changes.
The gravity of Earth is the net vector acceleration experienced by a test mass resulting from the sum of Newtonian gravitational attraction of the planet’s mass and the centrifugal acceleration arising from Earth’s rotation. Its direction is that of a plumb line and its scalar magnitude is g = ‖g‖. In SI units g is expressed as metres per second squared (m·s−2) or equivalently as newtons per kilogram (N·kg−1). Near the surface, g is approximately 9.8 m·s−2 (≈32 ft·s−2) to two significant figures; by international convention the standard gravity is defined exactly as 9.80665 m·s−2. The normal gravity at the equator is slightly smaller (9.7803267715 m·s−2). Symbols used in the literature include g, g0, gn and ge (often for equatorial normal gravity).
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An object’s weight is the downward force given by Newton’s second law, F = ma, where the local gravitational acceleration is the principal contribution to a. However, centrifugal acceleration due to rotation modifies the apparent weight, and the gravitational influences of the Moon and Sun are ordinarily treated as external tidal effects rather than components of local gravity.
Gravity studies sit within a broad, multidisciplinary geophysics. Core organizational domains include computational and exploration geophysics, historical and synthetic overviews, and discipline-specific branches. Electromagnetic geophysics addresses ionospheric processes, polar wind, thunderstorms and lightning and their coupling to magnetic and plasma environments. Fluid‑dynamical approaches cover atmospheric circulation, oceanography, turbulence and magnetohydrodynamics. Geodynamics integrates mantle processes, plate tectonics, volcanism, glaciology, climate and planetary evolution. The gravity‑focused subfields—geodesy, the geoid concept (equipotential surface approximating mean sea level) and physical geodesy—quantify Earth’s figure, gravity field and their temporal variability. Magnetic studies treat the geomagnetic field, magnetosphere, paleomagnetism and core dynamo processes, while wave phenomena (seismology, spectral analysis, vibration studies) probe internal structure and dynamics.
Historical and contemporary development of these topics is associated with numerous key figures in geophysics, including Aki, Alfven, Anderson, Benioff, Bowie, Dziewonski, Forbes, Eötvös, Gilbert, Gutenberg, Heiskanen, Hotine, von Humboldt, Jeffreys, Kanamori, Love, Matthews, McKenzie, Mercalli, Molodenskii, Munk, Press, Richter, Turcotte, Van Allen, Vaníček, Vening Meinesz, Wegener and Wilson.
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In a hypothetical, non-rotating planet with perfect spherical symmetry—whether of uniform density or with density varying only radially—the gravitational acceleration at the surface would be uniform because the mass distribution is the same in every direction and there are no inertial forces. Earth’s gravity, however, departs from this ideal because the planet both rotates and departs from spherical form: it is an oblate spheroid, flattened at the poles and bulging at the Equator. These two factors produce systematic spatial differences in surface gravity.
Quantitatively, measured gravitational acceleration on Earth’s surface varies by roughly 0.7%, with observed values ranging from about 9.7639 m s−2 (Nevado Huascarán, Peru) to about 9.8337 m s−2 (Arctic Ocean surface). Urban examples illustrate the same pattern on a smaller scale: cities near the Equator such as Kuala Lumpur, Mexico City and Singapore register about 9.7806 m s−2, whereas higher‑latitude cities like Oslo and Helsinki measure closer to 9.825 m s−2. The geographic explanation combines two effects: the oblate figure changes the radial distance from surface points to Earth’s centre (greater at the Equator, smaller at the poles, tending to weaken gravity at the Equator), and planetary rotation produces a centrifugal acceleration that reduces the apparent gravitational acceleration most strongly at low latitudes. Local variations in subsurface mass distribution then superimpose smaller anomalies on this global pattern.
The conventional acceleration of gravity adopted for general use is g_n = 9.80665 m/s^2. This single numerical standard was formally established by the third General Conference on Weights and Measures in 1901 to provide a common reference value when more accurate local determinations are unavailable.
The magnitude of g_n derives from instrumental gravity measurements made at the Pavillon de Breteuil near Paris in 1888, together with a theoretical adjustment that transposes that signal to a set of defined geographic conditions: latitude 45° and sea level. Consequently, the adopted number represents gravity under those reference conditions rather than the gravity at the original measurement site.
The convention is explicitly pragmatic: g_n is a standardized reference, not a precise measurement for any particular location nor an exact global mean of terrestrial gravity. Its role in metrology reflects this practicality—g_n is used to convert mass to force in the definitions of units such as the kilogram-force and the pound-force. The combination of the 1888 measurements, the latitude–sea-level correction, and the 1901 formal adoption explains both the specific numerical value and its status as a conventional reference rather than a site-specific geophysical constant.
Latitude
Earth’s rotation makes the surface a non‑inertial frame, producing an outward centrifugal acceleration whose magnitude varies with latitude and attains its maximum at the Equator. That centrifugal component opposes true gravitational attraction and reduces the apparent downward acceleration experienced by bodies near the equator by up to about 0.3%. Rotation also deforms the planet into an oblate spheroid, so equatorial locations lie at greater radial distance from Earth’s center than polar locations; by the inverse‑square law this larger radius weakens the gravitational pull at the Equator. Further, lateral differences in subsurface mass distribution beneath equatorial versus polar regions produce additional local variations in the gravity field. Taken together, these effects yield the familiar latitude dependence of sea‑level gravity, from roughly 9.780 m·s−2 at the Equator to about 9.832 m·s−2 at the poles. Practically, an object will therefore weigh approximately 0.5% more at the poles than at the Equator, reflecting both the smaller centrifugal reduction and the shorter distance to Earth’s center at polar latitudes.
Altitude
Gravitational acceleration decreases with increasing altitude because greater distance from the Earth’s centre reduces the central attraction; for example, rising from sea level to 9 000 m (≈30 000 ft) lowers weight by about 0.29% if all other factors are held constant. A convenient closed-form approximation that captures this first-order effect is
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g(h) = g0 · (Re / (Re + h))^2,
where Re is the mean planetary radius and h is altitude measured from mean sea level (both in the same units). Using the conventional constants Re = 6 371.00877 km and g0 = 9.80665 m·s−2 gives g(h) = g0 at h = 0 and yields the nearly 90% surface gravity still present at typical low-orbit altitudes (for example, at ≈400 km the factor is ≈0.89–0.90). Apparent weightlessness in orbit, therefore, is not due to absence of gravity but to the spacecraft and its contents being in continuous free-fall.
Atmospheric effects modify the apparent weight at altitude: reduced air density diminishes buoyant force, producing a slight increase in apparent weight relative to the pure gravitational change (about +0.08% at 9 000 m). Local mass distribution and topography also matter—the gravity felt at a given geometric altitude depends on the density and mass beneath that altitude, so flying a fixed height above sea level over mountainous crust yields a marginally larger gravity than the same height above deep oceanic crust. Similarly, an observer standing on the surface will experience lower gravitational acceleration at higher elevations above mean sea level because of the greater radial distance from the centre.
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The spherical approximation embodied in the simple inverse-square formula is useful for order-of-magnitude and many engineering estimates, but it neglects the Earth’s oblateness, lateral density variations (local geology), detailed topography and the rotational (centrifugal) contribution to apparent gravity. For precise geophysical or geodetic work these factors must be incorporated; nonetheless, the same basic inverse-square principle and analogous formulae apply to other planetary bodies (Mars, Moon, etc.) when their mean radii and surface gravities are used.
Depth
The radial variation of gravity within Earth is governed by the mass enclosed at each radius; under spherical symmetry the gravitational acceleration at radius r depends only on M(r) via g(r) = −G M(r) / r^2, a consequence of the Shell theorem that allows the interior field to be treated as if the enclosed mass were concentrated at the center. Simple analytic end‑members illustrate how internal density controls g(r): for uniform density ρ, M(r) = (4/3)πρ r^3 and therefore g(r) ∝ r, so gravity decreases linearly with depth from the surface value g according to g′ = g(1 − d/R). If density declines linearly from a central value ρ0 to a surface value ρ1, the density law ρ(r) = ρ0 − (ρ0 − ρ1) r/R yields a gravity law with an additional quadratic term, g(r) = (4π/3)Gρ0 r − πG(ρ0 − ρ1) r^2/R, modifying the simple linear dependence.
The Preliminary Reference Earth Model (PREM) provides a geophysically constrained radial profile of density and gravity that maps crustal and mantle layers and the core (labels: continental crust, oceanic crust, upper/lower mantle, core; crust–mantle boundary), and differs from the simple analytic models because it is derived from seismic data and the Adams–Williamson relation. In graphical comparisons, a constant‑density (average ρ) model gives a straight linear g(r) baseline, while a linear‑decrease model produces a gently curved profile; PREM’s curve departs from both because it captures layered structure inferred from observations. Practically, gravity inside the planet (r < R) is sensitive to the detailed radial density distribution, whereas outside (r ≥ R) the total mass M(R) alone controls g by the 1/r^2 law, so accurate ρ(r) models such as PREM are essential for predicting internal and external gravity.
Local contrasts in mass—produced by surface relief (including mountain masses), variations in near-surface rock density and deeper tectonic structures—give rise to gravity anomalies: measurable departures from the idealized gravitational field of the Earth. These anomalies are not purely theoretical; sufficiently large or extensive mass excesses or deficits can perturb the equilibrium shape of the sea surface, producing detectable sea‑level bulges, and can measurably alter the period of pendulum clocks, demonstrating tangible surface effects of subsurface mass distribution.
The discipline of gravitational geophysics quantifies these effects using highly sensitive gravimeters. Field and satellite measurements are corrected for known influences (for example, topographic loading, tidal forces and instrument drift) and the computed contributions of predictable sources are removed; the remaining residuals are then interpreted to infer the geometry and density contrasts of subsurface bodies. In applied exploration this same procedure is used to detect economically important targets: relatively dense intrusive or ore-bearing bodies generate positive gravity anomalies at the surface, whereas thick packages of low‑density sediments produce negative anomalies.
At regional to global scales, satellite gravimetry (notably the GRACE mission) produces maps of gravity that, when compared with geological observations, reveal systematic associations: regions showing gravity higher than simple theoretical models tend to coincide with recent volcanism and mid‑ocean ridge spreading. This spatial correspondence implies that tectonically active zones possess excess mass or distinctive density structures relative to background models, making them gravitationally distinct.
Operational gravimetric surveys—whether ground‑based or spaceborne—therefore follow a consistent workflow: precise measurement, rigorous correction for predictable surface and instrumental effects, and analysis of residual anomalies to constrain subsurface mass distribution. Such inferences inform interpretations of tectonic architecture, volcanic processes and the location of potential mineral or hydrocarbon resources.
Other factors that influence the apparent strength of Earth’s gravity at a site include buoyant forces from surrounding fluids and the time‑varying gravitational pull of external bodies. Both produce systematic deviations between true gravitational acceleration and the force measured by a scale or gravimeter and therefore require explicit correction in precise geophysical and geodetic work.
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Buoyancy arises because an object immersed in a fluid displaces a volume of that fluid and experiences an upward force equal to the weight of the displaced fluid (Archimedes’ principle). In practice the buoyant force is Fb = ρf V g, and the force registered by a scale is reduced to Wapp = mg − Fb. The magnitude of this reduction depends on the density ρf of the ambient medium: in air ρf is sensitive to pressure, temperature and altitude, while in water it additionally depends on salinity and hydrostatic pressure. Consequently, variable fluid‑density conditions produce a measurable buoyant correction that must be applied when high accuracy is required.
Superposed on these static effects are small temporal variations in local gravity produced by the tidal potential of the Moon and Sun. The tidal contribution depends on their relative geometry with respect to the observation point and typically has a diurnal amplitude of about 2 μm s−2 (≈0.2 mGal). Although small, this signal is comparable to the sensitivity of modern gravimetric and geodetic measurements and therefore must be removed from observational records.
For accurate determination of Earth’s gravity field or of weight in precision experiments, corrections for both local fluid‑density (buoyancy) and the time‑varying lunar–solar tidal potential are therefore essential.
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A plumb bob aligns with the local gravity acceleration vector and thus indicates the true vertical used in leveling and astronomical observations. Gravity is a vector field with both magnitude and direction; it is the direction component at a point near Earth’s surface that orients objects and defines the local vertical.
Under an idealized, spherically symmetric Earth every gravity vector at the surface would be radial, pointing toward the geocentre. Because the real Earth is an oblate spheroid, however, the gravity vector at a given point generally does not coincide with the line to the centre. This systematic misalignment is reflected in the difference between geocentric latitude (the angle subtended at the centre of the Earth) and geodetic latitude (the angle between the normal to a chosen reference ellipsoid and the equatorial plane).
Vertical deflection denotes the angular offset between the true vertical (the gravity or astronomical vertical indicated by a plumb bob) and the normal to the reference ellipsoid (the geodetic vertical). In addition to the global effect of Earth’s oblateness, local mass anomalies—such as topography, near‑surface density contrasts, or subsurface structures—perturb the gravity vector and produce measurable local tilts of the plumb line. These deviations are critical in precise surveying and geodesy: they must be accounted for when converting astronomical observations to geodetic coordinates, when defining heights with respect to the geoid or ellipsoid, and when interpreting gravity anomalies that diagnose subsurface mass distribution.
Comparative values worldwide
Measured surface values of gravitational acceleration for many cities have been tabulated (values cited here are taken from T. M. Yarwood and F. Castle, Physical and Mathematical Tables, Macmillan, rev. ed. 1970). These observations, obtained by a combination of direct measurement and geographic calculation, display systematic spatial patterns that reflect the principal physical controls on g.
Latitude exerts a primary control: gravity increases from the equator toward the poles. Representative high‑latitude stations such as Anchorage and Helsinki record values near 9.825–9.826 m·s–2, roughly 0.5% larger than typical near‑equatorial coastal cities such as Kuala Lumpur (~9.776 m·s–2). This latitudinal trend arises from the combined effects of the Earth’s equatorial bulge (larger radial distance at low latitudes) and the latitude‑dependent centrifugal reduction of apparent weight.
Elevation provides the secondary control: gravitational acceleration decreases with increasing altitude above mean sea level. For example, Mexico City (≈2,240 m) registers ~9.776 m·s–2, while a controlled comparison at ~39°N shows Denver (≈1,616 m) at ~9.798 m·s–2 versus Washington, D.C. (≈30 m) at ~9.801 m·s–2, demonstrating the measurable decline of g with height even at similar latitudes.
Regional compilations conform to these controls while also showing local variability. Northern Europe and the British Isles (high–to–mid latitudes) generally exhibit the largest surface values in the tabulation (approximately 9.807–9.825 m·s–2). Continental and Mediterranean mid‑latitude locations lie somewhat lower (≈9.797–9.808 m·s–2). Across North America the range spans from high values in the subarctic (Anchorage ~9.826 m·s–2) to lower values in equatorial or high‑altitude locales (Mexico City ~9.776 m·s–2), with most temperate cities clustering near 9.79–9.81 m·s–2. Southern Hemisphere and southern mid‑latitude examples (e.g., Buenos Aires, Cape Town, Perth, Melbourne, Sydney, Auckland) typically occupy intermediate values around 9.794–9.803 m·s–2. In East, South and Southeast Asia, tropical lowland cities (e.g., Singapore, Kuala Lumpur) show some of the lowest listed values (~9.776–9.785 m·s–2), while temperate or higher‑altitude Asian stations reach slightly larger values.
Overall, the dataset illustrates that observed variations in surface gravity among world cities result mainly from latitude and elevation, modulated by local geophysical factors; highest values occur at high latitudes, lowest values where proximity to the equator and/or significant elevation reduce gravitational acceleration.
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Mathematical models
Normal gravity on a reference spheroid is represented by closed-form latitude-dependent formulas that approximate the combined effects of Earth’s rotation and its ellipsoidal shape. For terrain at mean sea level the Geodetic Reference System 1980 (GRS80) expression is commonly given in Helmert’s (International Gravity Formula 1967 / Clairaut) form, for example:
g(φ) = 9.780327 m·s⁻² (1 + 0.0053024 sin²φ − 0.0000058 sin²2φ)
,
an expression that is algebraically equivalent to alternative expansions in sin⁴φ
, cos²φ
or cos²2φ
and therefore describes the same latitude dependence of normal gravity on the GRS80 reference spheroid.
The World Geodetic System 1984 (WGS‑84) treats the ellipsoidal geometry explicitly and writes normal gravity on the reference ellipsoid as
g(φ) = G_e [ (1 + k sin²φ) / √(1 − e² sin²φ) ]
,
where a
and b
are the equatorial and polar semi‑axes, e² = 1 − (b/a)²
is the squared eccentricity, G_e
and G_p
are the defined equatorial and polar gravity values, and the dimensionless constant k = (b G_p − a G_e)/(a G_e)
encapsulates the pole–equator gravity difference. Instantiated with WGS‑84 constants (a = 6 378 137.0 m
, b = 6 356 752.314245 m
), the numeric form becomes
g(φ) = 9.7803253359 m·s⁻² [ (1 + 0.001931852652 sin²φ) / √(1 − 0.0066943799901 sin²φ) ]
,
with e² = 0.0066943799901
, G_e = 9.7803253359 m·s⁻²
, G_p = 9.8321849378 m·s⁻²
, and k ≈ 0.001931852652
.
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The WGS‑84 ellipsoidal formula and the GRS80 (Helmert) formula differ only by a few tenths of a microgal (less than 0.68 μm·s⁻²), so they are effectively interchangeable for most geodetic and geophysical work at the reference ellipsoid. Converting these normal‑gravity models to observed gravity requires additional, site‑specific corrections (topography, crustal density variations, tidal effects, etc.) to obtain gravity anomalies.
Applying Newton’s law of universal gravitation to a test mass near Earth and equating the resulting force to Newton’s second law yields the local gravitational acceleration
g = G M⊕ / r^2,
where r is the radial distance from the measurement point to the Earth’s centre, G the universal gravitational constant, and M⊕ the Earth’s mass. Substituting commonly used mean values (G ≈ 6.674×10^−11 m^3 kg^−1 s^−2, M⊕ ≈ 6×10^24 kg, r ≈ 6.4×10^6 m) gives g ≈ 9.78 m s^−2 at sea level.
That point‑mass form of the law is justified for a spherically symmetric Earth by Newton’s shell theorem, which states that the external gravitational field of a spherically distributed mass is identical to that of the same mass concentrated at the centre. However, the simple calculated value differs from precise measurements because the real Earth departs from the ideal assumptions: internal density varies with depth, the planet is oblate and has surface topography so the local radius differs from the mean, and the rotating reference frame introduces a centrifugal acceleration that reduces the apparent weight. Practical determination is further limited by experimental uncertainties (notably in G, and in the adopted values of r and M⊕); conversely, knowing g, r and G permits estimating M⊕ — an approach used historically (e.g., Cavendish) to infer the Earth’s mass.
Measurement (Gravimetry)
Gravimetry is the quantitative measurement and analysis of the Earth’s gravity field to determine the magnitude and spatial and temporal variations of gravitational acceleration for geodetic, geophysical and surveying applications. The conventional reference value is defined as g0 = 9.80665 m s^-2 (exact). Gravity is also expressed in Gals, where 1 Gal = 0.01 m s^-2 (so g0 = 980.665 Gal); routine gravimetric units are the milligal (1 mGal = 10^-5 m s^-2) and the microgal (1 μGal = 10^-8 m s^-2).
Systematic variations in gravity arise from planetary shape and rotation and from elevation. Normal gravity varies with latitude—approximately 9.780325 m s^-2 at the equator to about 9.832184 m s^-2 at the poles—and is represented by normal‑gravity formulae (e.g., Somigliana) that define a reference field. Gravity also decreases with height; the free‑air gradient is roughly −0.3086 mGal per metre, a correction routinely applied when reducing observations to a common datum.
Local departures from the normal field reflect near‑surface and subsurface mass heterogeneity and are characterized by standard anomaly types and associated corrections. The free‑air anomaly corrects for elevation; the Bouguer anomaly applies a slab correction (≈ 0.1119 mGal per metre of slab thickness for a rock density of ≈2670 kg m^-3); and terrain corrections account for irregular topography surrounding the measurement site. Interpretation of anomalies yields information on mass excesses or deficits, isostatic compensation and geological structure.
Gravimetric theory rests on fundamental constants and field solutions: the Newtonian constant G ≈ 6.67430×10^-11 m^3 kg^-1 s^-2 and the point‑mass relation g = GM/r^2, while global gravity fields and geopotentials are expressed using spherical harmonics and geopotential models.
Instruments divide into absolute and relative gravimeters. Absolute devices (falling‑corner‑cube optical interferometers, atom‑interferometry systems) determine g directly with accuracies at the μGal level. Relative gravimeters (spring‑based instruments such as LaCoste & Romberg) measure changes with respect to a reference and, after calibration, achieve resolutions from sub‑mGal to μGal. Satellite gravimetry complements ground and airborne observations: GRACE (launched 17 March 2002, operations through 2017) and GRACE‑FO (launched 22 May 2018) detect time‑variable mass redistribution (e.g., hydrology, cryosphere); ESA’s GOCE (17 March 2009–11 November 2013) mapped the static gravity gradient at high spatial resolution.
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Gravimetric outputs include high‑resolution geoid and global geopotential models used to define vertical datums and to convert between ellipsoidal and orthometric heights, to monitor sea‑level and ice‑mass changes, and to detect subsurface density anomalies for mineral, hydrocarbon and tectonic studies. Spatial coverage ranges from centimetre‑to‑kilometre scales in local surveys to continental and global scales captured by airborne and satellite missions; temporal sensitivity spans seasonal to long‑term tectonic and isostatic signals.
A practical gravimetric workflow comprises field measurement (absolute and/or relative observations), standard reductions and corrections (tidal, instrument drift, free‑air, Bouguer, terrain), computation of anomalies relative to a chosen normal field, and integration with other geodetic and geophysical data to infer mass distribution, establish vertical datums, or monitor temporal mass changes.
Satellite measurements
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Modern determinations of both the static and time-varying components of the Earth’s gravity field are dominated by dedicated satellite missions that provide complementary observational modes and temporal coverage. Missions such as GOCE, CHAMP and Swarm furnish high‑resolution measurements of the static field and its spatial gradients, while GRACE and GRACE‑FO employ a two‑satellite inter‑satellite ranging technique to deliver time series of gravity change; together these datasets enable global gravity recovery across a range of spatial and temporal scales. The lowest‑degree global parameters (for example the degree‑2 zonal term that quantifies Earth’s oblateness, and the geocenter motion that describes the offset of the planet’s mass center relative to a reference frame) are most robustly constrained by satellite laser ranging rather than by the higher‑resolution gravimetric missions, so different techniques are used in a complementary fashion.
Gravity field solutions from these missions are conventionally expressed as spherical‑harmonic expansions of the gravitational potential, a multi‑scale mathematical representation whose coefficients underpin most derived products and permit explicit separation of spatial wavelengths. From those harmonic solutions researchers routinely compute physically interpretable maps such as geoid undulations (equipotential heights relative to a reference ellipsoid) and gravity anomalies (deviations from a reference gravity model), which are essential inputs to geodesy, oceanography and solid‑Earth studies. Large‑scale anomalies and long‑wavelength geoid features—reflecting broad crustal and mantle mass distributions—are detectable from space; for example, GOCE’s gravity‑gradient measurements substantially improved mapping of these deep‑seated structures.
GRACE’s twin‑satellite approach produces direct, space‑based measurements of mass redistribution by tracking temporal changes in the gravity field. The resulting gravity‑anomaly time series have been used to quantify seasonal and long‑term variations in terrestrial water storage, ice‑sheet mass balance and basin‑scale ocean mass changes; GRACE‑FO continues this essential record. A related demonstration of the twin‑satellite method was NASA’s GRAIL mission, which mapped the Moon’s gravity at high resolution using two lunar orbiters, yielding improved spherical‑harmonic models and new constraints on crustal thickness and internal structure before its intentional deorbit in 2015.