gravity summary

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Gravity Summary. A general solution for the laplace problem can be written in spherical harmonics: V=(GM/r)  n=0 ∞  m=0 n (R/r) n (C nm cos m + S nm sin m) P nm (sin )  latitude,  longitude, R Earth Radius, r distance from CM - PowerPoint PPT Presentation

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  • 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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