Gravity Summary In first approximation we can chose for the shape of the Earth an ellipsoid of rotation defined essentially by the degree n=2 m=0 of the potential field plus the centrifugal potential. This is known as ellipsoid of reference. In 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 by g( ) = g e (1 +  sin 2 +  sin 4 )  and  are experimental constants  =  = g e =9.78 m s -2 From Fowler

Gravity Summary A better approximation of the shape of the Earth is given by the GEOID. The GEOID is an equipotential surface corresponding to the average sea level surface From Fowler

H elevation over Geoid h elevation over ellipsoid N=h-H Local Geoid anomaly

Geoid Anomaly gΔh=-ΔV

Geoid Anomaly gΔh=-ΔV Dynamic Geoid

Geoid Anomaly

Example of Gravity anomaly A buried sphere:  g z = 4G  π b 3 h (x 2 + h 2 ) 3/2 From Fowler

Gravity Correction: Latitude The reference gravity is in general expressed by g( ) = g e (1 +  sin 2 +  sin 4 )  and  are experimental constants  =  = g e =9.78 m s -2 The changes are related to flattening and centrifugal force. From Fowler

Change of Gravity with elevation g(h) = GM/(R+h) 2 = GM/R 2 ( R / (R+h)) 2 = g 0 ( R / (R+h)) 2 But R >> h => ( R / (R+h)) 2 ≈ (1 - 2h/R) This means that we can write g(h) ≈ g 0 (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  g FA = 2h/R g 0 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 correction Is known as FREE AIR GRAVITY ANOMALY g F = g obs - g ( ) +  g FA

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   g B = 2 π G  h The residual anomaly after we appy this correction is called BOUGUER GRAVITY ANOMALY g B = g obs - g ( ) +  g FA -  g B +  g T Where 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:  g z = 4G  π b 3 h (x 2 + h 2 ) 3/2 From Fowler

Example of Gravity anomaly

Isostasy In reality a mountain is not giving the full gravity anomaly! Airy Pratt From Fowler

Isostasy and Gravity Anomalies From Fowler