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An ideal laterally infinite horizontal slab, whose vertical gravity anomaly is proportional to its thickness $h$ and geophysical density contrast $\Delta \rho$ with replaced material. A negative geophysical density contrast gives a negative anomaly in a downward-positive convention. The slab's field above it does not specify burial depth; applying it to finite bodies is an approximation.

In the linear flat-elastic plate model, let downward deflection be $w$, applied downward pressure be $q$, flexural rigidity be $D$, and the restoring geophysical density contrast be $\Delta \rho >0$. The static balance isFor a sinusoidal Fourier mode of horizontal wavenumber magnitude $k$, applying the fourth spatial derivative multiplies its amplitude by $k^4$. HenceThe displayed dimensionless ratio compares bending resistance with buoyant restoration. Large $\chi$ means substantial elastic support and suppressed deflection. Small $\chi$ gives nearly local isostasy. The associated length $L_f=(D/\Delta \rho g{)}^{1/4}$ makes the load-width dependence explicit; some conventions define a flexural parameter with an additional factor $4^{1/4}$. At zero wavenumber, bending contributes nothing: total support remains buoyant even when the spatial distribution of subsidence changes.

Inference of subsurface geophysical density contrast from a gravity anomaly. Different geometries and density distributions can produce the same external field, so independent geological or seismic constraints are essential.

The difference $\Delta \mathbf{M}={\mathbf{M}}_{\mathrm{target}}-{\mathbf{M}}_{\mathrm{reference}}$ between a target region and the reference material it replaces. A weakly magnetic crater infill inside magnetic basalt can have a negative induced contrast, even though both materials have positive magnetic susceptibility. The sign and direction of this contrast, including remanent magnetization, determine a total-field magnetic anomaly together with geometry and the ambient field. Geophysical density contrast does not determine magnetic contrast.

The positive centers in the supplied maps are examples of lunar mascons, especially associated with large impact basins. An empty excavation by itself removes mass and would not explain a positive central anomaly. Basin formation can thin the low-density lunar crust and raise denser lunar mantle beneath the floor; impact melting, collapse and later cooling can also create dense material and lock in a mass excess. Where a basin was later flooded, dense mare basalt adds another contribution. Mare infill is therefore one possible contributor, not a necessary explanation of every mascon.^[ref]

The difference between the maps is the modeled topographic contribution. On a common reference surface, write schematicallyThe free-air gravity anomaly retains relief-related attraction. Rims, excavated floors and central peaks therefore contribute fine structure. The Bouguer gravity anomaly removes the attraction predicted from the measured topography at an assumed crustal density, making internal density and interface structure more apparent. A low basin floor has a negative topographic contribution; subtracting it can enhance an underlying positive central anomaly. Removing a positive rim contribution can weaken a rim high. The supplied lower map is correspondingly smoother at short wavelengths and shows prominent residual basin centers and surrounding rings. This is a change of gravity reduction, not a physical change of the craters.^[ref]

For scale, a Bouguer slab approximation gives $\Delta g\simeq 2\pi G\Delta \rho d$. A $5$ km layer with geophysical density contrast $300\mathrm{kg}{\mathrm{m}}^{-3}$ produces about $63$ mGal above an ideal infinite slab. Actual basin geometry, burial, porosity and mantle uplift require a finite three-dimensional model. Smaller craters can have central lows, damaged porous interiors or more complex rings; the maps do not imply that every crater has a positive anomaly or that positive anomalies measure basalt thickness uniquely.

Elasticity lets the lithosphere share a load with neighbouring regions through lithospheric flexure. Beneath the centre of a localized load, subsidence is generally smaller than the local isostasy estimate, so the deposited surface stands higher. The surrounding plate also bends down, and a more distant flexural forebulge may rise. The extra support is regional bending resistance, rather than removal of the load's mass.

The controlling comparison is load width against the lithosphere's flexural length, which increases with flexural rigidity and decreases with the relevant buoyant geophysical density contrast. Equivalently, the flexural support ratio becomes large for short-wavelength loading. A very broad nearly uniform load produces little curvature in its interior, so even a strong elastic lithosphere has a nearly local isostatic adjustment there. Bending matters most near its edges. A narrow load requires more curvature and can be supported appreciably by the surrounding elastic lithosphere.

The Deccan Traps form a broad continental large igneous province. The volcanic edifices of the Hawaiian Islands are more localized loads on an old oceanic lithosphere, for which regional bending and a flexural moat are prominent. Under the usual broad-Deccan versus narrow-Hawaiian-load comparison, elastic strength matters more for the Hawaiian Islands. The more coherent mantle-dominated strength of an old oceanic plate can reinforce this contrast. It cannot be deduced simply from “oceanic” versus “continental”: a cold Indian craton can be strong, and a site's effective elastic thickness depends on its thermal and loading history.

For an initially submerged Hawaiian load, replacing seawater changes both the load and buoyant restoring contrast. At fixed deposited thickness, the added load relative to the initial water column uses the rock-minus-water density, and the restoring term uses the mantle-minus-water density. An emerged island requires separate treatment of its subaerial and submerged portions. These corrections change the local isostasy reference but do not remove the importance of load width for lithospheric flexure. The broad Hawaiian swell should also be distinguished from the narrower volcanic load: deeper thermal or dynamic support of the Hawaiian swell is not the same mechanism as elastic support of the volcanic ridge.

Finally, over long periods material creep, yielding and thermal modification can reduce elastic support. Without measured load geometries, ages and effective elastic thicknesses, the ranking is a justified qualitative expectation rather than a theorem valid for every continental and oceanic load.

A gravity anomaly is measured gravity minus a specified reference gravity after the stated reductions. A Bouguer gravity anomaly aims to remove the predictable effects of latitude, observation elevation and surrounding topographic mass, so residual variations can be related to subsurface geophysical density contrasts.

Measure relative gravity with a calibrated gravimeter at stations across and beyond the crater, and survey station positions and elevations accurately, preferably with GNSS tied to a reference datum. Repeat a base station through the day to estimate instrument drift and tie the traverse to a gravity datum. Record station times, because tidal and drift corrections are time dependent.

Apply instrument scale and drift corrections, solid-Earth and relevant ocean-loading tidal corrections, latitude/reference-ellipsoid gravity subtraction, the free-air correction, the Bouguer slab correction for material between station and datum, and a terrain correction for departures from a flat slab. The crater’s strong relief makes the last correction important. A local regional-field/background estimate is also needed if the target is the crater’s residual anomaly. State the reduction density; it is not the inferred breccia density.^[ref] For a fixed land traverse there is no moving-platform Eötvös correction to add.

Treat the quoted anomaly as the response of a slab of thickness $h$ replacing host rock. The Bouguer slab formula applies to the geophysical density contrast, not the absolute density:With $\Delta g=-2.5\times 10^{-5}\mathrm{m}{\mathrm{s}}^{-2}$, $h=400$ m and ${\rho }_h=2750\mathrm{kg}{\mathrm{m}}^{-3}$,Thus ${\rho }_b\simeq 2601\mathrm{kg}{\mathrm{m}}^{-3}$ in the slab approximation. The negative anomaly requires less dense infill. A finite bowl has a different geometric response, and water, post-impact sediment and the regional baseline can matter; the stated calculation is the requested slab estimate. The plotted minimum appears closer to $-2$ mGal, but the numerical instruction explicitly supplies $-2.5$ mGal and controls this calculation.

A gravity anomaly is measured gravitational acceleration, reduced to a stated reference surface and compared with a reference gravity field. A local geological interpretation further removes or models the regional background. An absolute negative Bouguer gravity anomaly is therefore not itself a local negative geophysical density contrast.

Measure relative gravity with a calibrated gravimeter at stations across the profile, tie it to a base station, repeat base readings to estimate instrumental drift, and measure station coordinates and accurate elevations, for example by GNSS and levelling. Record observation times for tide corrections. Subtract the latitude-dependent reference gravity, correct instrument drift and Earth tides, add the free-air correction for station height, and subtract the attraction of the reference-density Bouguer slab. A terrain correction accounts for relief missing from or exceeding that slab. Use a consistent density such as $2670\mathrm{kg}{\mathrm{m}}^{-3}$ for the topographic correction.^[ref]

The question's reference to the anomaly in panel b is a panel-label slip: the upper plot contains gravity, while the lower plot supplies geometry and elevations. A geological slab thickness is the vertical thickness of the anomalous body, not simply its height above sea level.

For a laterally extensive Bouguer slab of thickness $h$ and geophysical density contrast $\Delta \rho$,The greenstone belt reaches from roughly $+1.0$ to $-0.5\mathrm{km}$ near its deepest central part, so use $h\simeq 1500\mathrm{m}$. The gravity maximum is approximately $-100\mathrm{mGal}$; the exposed host-rock background in the south is approximately $-115\mathrm{mGal}$. Taking this as a constant regional reference gives a local excess $\Delta g\simeq 15\mathrm{mGal}=1.5\times 10^{-4}\mathrm{m}{\mathrm{s}}^{-2}$. ThusThis is an approximate density under a constant-background, infinite-slab model. The finite width, variable thickness and neighboring pluton require a fuller gravity inversion for a precise value. In particular, the absolute value $-100\mathrm{mGal}$ must not be inserted as the local excess.

A gravity anomaly is the difference between observed gravitational acceleration and a specified reference prediction after defined reductions. A free-air gravity anomaly, a Bouguer gravity anomaly, and a residual local gravity anomaly differ because they remove different reference contributions. State the reference ellipsoid, datum, reference density, and regional trend before interpreting an anomaly as a subsurface geophysical density contrast.

On land, tie gravimeter observations to a gravity base, calibrate them, correct instrumental drift and Earth tides, and determine station latitude and elevation accurately. Subtract latitude-dependent normal gravity $\gamma (\phi )$. Reduce to the reference elevation using a positive free-air correction for a station above that datum: about $+0.3086h\mathrm{mGal}$ for $h$ in metres. A simple Bouguer gravity anomaly then subtracts attraction of the intervening rock slab and adds the conventional terrain correction:with all acceleration terms expressed consistently. In SI units $1\mathrm{mGal}=10^{-5}\mathrm{m}{\mathrm{s}}^{-2}$.

At sea, also correct vessel-motion effects, especially the Eötvös correction from east–west vessel motion and Earth rotation, and vertical ship acceleration; average or filter the moving-platform data appropriately. GPS navigation and tide information specify position and measurement height. Sea-surface measurements give a marine free-air gravity anomaly after motion, latitude, tide and height reductions. A marine Bouguer gravity anomaly additionally allows for measured bathymetry and the replacement of low-density water by reference rock; the simple water-column replacement correction is positive, $2\pi G({\rho }_{\mathrm{ref}}-{\rho }_w)H$, relative to the sea-surface free-air value. A consistent three-dimensional bathymetric correction is preferable near irregular coasts. Remove a justified broad regional field if the intended quantity is a residual anomaly of the local body.

For a wide horizontal Bouguer slab replacing surrounding basement, use the density contrast $\Delta \rho ={\rho }_{\mathrm{sed}}-{\rho }_{\mathrm{base}}$, not the absolute sediment density:Take downward attraction as positive. The negative anomaly therefore requires $\Delta \rho <0$. With $\Delta g=-40\mathrm{mGal}=-4.0\times 10^{-4}\mathrm{m}{\mathrm{s}}^{-2}$, $h=3000\mathrm{m}$, and $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$,soThis is the requested Bouguer slab approximation: a flat observation surface, laterally infinite constant-thickness material, uniform average geophysical density contrast, and a regional anomaly already removed. The actual basin in Figure 1 narrows downwards and is finite. For a fixed maximum thickness, finite width reduces the central anomaly magnitude, so this calculation can underestimate the true magnitude of the average geophysical density contrast. It does not establish a unique basement density without the geometrical assumption.

Use the same surrounding basement density inferred in solution, carrying its unrounded value through the arithmetic. The intrusion's geophysical density contrast isThe Bouguer slab thickness needed to produce the same negative peak is thereforeThe smaller absolute geophysical density contrast requires a thicker body. This is an approximate vertical thickness and, if the top is close to the surface as drawn in Figure 1, an approximate base depth below that top. It is not an independent estimate of the intrusion's burial depth. For an infinite slab the anomaly does not depend on top depth; a buried top would require additional geometry. Finite width and the downward-widening outline in Figure 1 make 8 km a slab-equivalent estimate rather than a uniquely determined geological depth.

For the two negative gravity anomalies in the actual Figure 1, examine where the steepest flanks occur within the total fall into the trough. The solid upper intrusion profile has maximum slope relatively low on its flanks, close to the flat trough; its body widens downwards. The lower basin profile has maximum slope relatively high on its flanks, near the shoulders; its body narrows downwards. The dashed gradient traces and arrows mark these transitions. The signed derivative changes sign on opposite sides of a symmetric trough; a plot of gradient magnitude instead gives two positive peaks.

A shallow edge controls a relatively abrupt local change, whereas deeper geophysical density contrast spreads its effect over greater horizontal distance. Downward-narrowing geometry leaves relatively more of the anomaly associated with the shallow broad part, placing the strong flank change nearer the outer shoulders. Downward-widening geometry adds a broader deep contribution beneath the narrow outcrop, shifting the inflection towards the lower part of the trough. Thus the location of a flank's maximum slope relative to its shoulder and minimum is more useful here than the shared $-40\mathrm{mGal}$ peak alone.

A symmetric constant-density body with vertical sides gives a symmetric negative trough with paired rounded flanks; for a broad shallow body the main slope changes occur approximately halfway down those flanks. It is intermediate between the two end members, rather than a triangular anomaly or a rectangular discontinuity. Exact fractional inflection heights depend on width, depth, burial and rounding. The following independently calculated infinite-strike models illustrate the distinction without copying Figure 1.

This qualitative inference assumes approximately comparable simple bodies, uniform geophysical density contrast, and a properly removed regional field. Gravity inversion is non-unique: lateral density changes, multiple bodies, and a buried top can imitate changes of wall slope. Use the seismic reflection constraints to test the interpretation.

A body of igneous rock crystallized from magma emplaced beneath the surface. Granite is a common plutonic composition. A pluton's gravity anomaly depends on its geophysical density contrast and geometry, rather than on the rock name alone.

A region of long-term accommodation and accumulation of sediments, commonly associated with subsidence. Its gravity anomaly depends on geometry and the geophysical density contrast between fill and surrounding rock.