Comprehensive Guide to Gravity Anomalies and Corrections in Geodesy

1. Gravity Summary A 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 CM The 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

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

4. Geoid Anomaly gΔh=-ΔV

5. Geoid Anomaly gΔh=-ΔV Dynamic Geoid

6. Geoid Anomaly

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

10. Gravity Summary • 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() = ge (1 +  sin2 +sin4 ) • and  are experimental constants • = 5.27 10-3 =2.34 10-5 ge=9.78 m s-2 From Fowler

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

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

13. Change of Gravity with elevation g(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 write g(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 correction Is known as FREE AIR GRAVITY ANOMALY gF = gobs - g () + gFA

14. 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 π G  h The residual anomaly after we appy this correction is called BOUGUER GRAVITY ANOMALY gB = gobs - g () + gFA -gB + gT 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!)

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

16. Example of Gravity anomaly

17. Example of Gravity anomaly

18. Example of Gravity anomaly

19. Isostasy and Gravity Anomalies From Fowler