Gravity SummaryA general solution for the laplace problem can be written in spherical harmonics:V=(GM/r) n=0 m=0n (R/r)n (Cnm cos m + Snm sin m) Pnm (sin ) latitude, longitude, R Earth Radius, r distance from CMThe coefficient Cnm and Snm are called Stokes Coefficients. Pnm(sin ) are associated Legendre functions

Ynm (,) = (Cnm cos m + Snm sin m) Pnm (sin )Is called spherical harmonics of degree n order m

H elevation over Geoidh elevation over ellipsoid

N=h-HLocal Geoid anomaly

Geoid Anomalygh=-V

Geoid Anomalygh=-VDynamic Geoid

Geoid Anomaly

IsostasyIn reality a mountain is not giving the full gravity anomaly! AiryPrattFrom Fowler

Gravity SummaryIn general all the measure of gravity acceleration and geoid are referenced to this surface. The gravity acceleration change with the latitutde essentially for 2 reasons: the distance from the rotation axis and the flattening of the planet.The reference gravity is in general expressed byg() = ge (1 + sin2 +sin4 )and are experimental constants = 5.27 10-3 =2.34 10-5 ge=9.78 m s-2From Fowler

Example of Gravity anomaly A buried sphere: gz= 4G b3 h --------------- 3(x2 + h2)3/2From Fowler

Gravity Correction: LatitudeThe reference gravity is in general expressed byg() = ge (1 + sin2 +sin4 )and are experimental constants = 5.27 10-3 =2.34 10-5 ge=9.78 m s-2The changes are related to flattening and centrifugal force.From Fowler

Change of Gravity with elevationg(h) = GM/(R+h)2 = GM/R2 ( R / (R+h))2 = g0 ( R / (R+h))2

But R >> h => ( R / (R+h))2 (1 - 2h/R)This means that we can writeg(h) g0 (1 - 2h/R) The gravity decrease with the elevation above the reference Aproximately in a linear way, 0.3 mgal per metre of elevation

The correction gFA= 2h/R g0 is known as Free air correction(a more precise formula can be obtain using a spheroid instead of a sphere but this formula is the most commonly used)

The residual of observed gravity- latitude correction + FA correctionIs known as FREE AIR GRAVITY ANOMALYgF = gobs - g () + gFA

Change of Gravity for presence of mass (Mountain)The previous correction is working if undernit us there is only air if there is a mountain we must do another correction. A typical one is the Bouguer correction assuming the presence of an infinite slab of thickness h and density gB = 2 G hThe residual anomaly after we appy this correction is called BOUGUER GRAVITY ANOMALY gB = gobs - g () + gFA - gB + gTWhere I added also the terrain correction to account for the complex shape of the mountain below (but this correction can not be do analytically!)

Example of Gravity anomaly A buried sphere: gz= 4G b3 h --------------- 3(x2 + h2)3/2From Fowler

Example of Gravity anomaly

Example of Gravity anomaly

Example of Gravity anomaly

Isostasy and Gravity AnomaliesFrom Fowler

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