Gravity 2

Geoid, spheroid

Measurement of gravity Absolute measurements Relative measurements True gravitational Acceleration Difference in gravitational acceleration

Measurement of absolute gravity z – distance the object falls t – time to fall the distance z v o – initial velocity g – absolute gravity T – period of the pendulum L – length of the pendulum Expensive and time consuming measurements

Measurement of relative gravity F = mg g 1 < g 2  F 1 < F 2 g=F/m L ~ g  gravimeter

Worden gravimeter

Measurement of relative gravity F = mg g 1 < g 2  F 1 < F 2 g=F/m L ~ g Base station – absolute gravity – calibration of L (  L   g) Other stations – relative gravity – change in L Problems with surveying in the ocean

Eötvös correction (required for seaborne, airborne measurements) Apparent gravity is affected by motion of moving platforms Eastward ship (aircraft) travel: adds to the earth's rotation, increases centrifugal forces and decreases the gravity readings. Westward travel: increases the gravity reading. North-south travel: is independent of rotation, and decreases the gravity reading in either case. Correction required:

Isostasy

Deflection of a plumb bob Expected deflection due to the attraction of the mass of the mountain Actual deflection for Andes and Himalayas (less than expected due to a deficiency of mass beneath the mountains) Measurements: - in – Bouguer - Andes - mid -1800’s – Sir George Everest - Himalayas

Local Isostasy ( ) Block of different density The same pressure from all blocks at the depth of compensation (crust/mantle boundary) Blocks of the same density but different thickness The base of the crust is exaggerated, mirror image of the topography

Hydrostatic (lithostatic) pressure P =  gh P – pressure  – density z – thickness Archimedes – a floating body displaces its own weight of water “Floating” rigid surface layer Denser substratum The isostasy restores the equilibrium

Hydrostatic (lithostatic) pressure Pratt model P=  2 gh 2 =  3 gh 3 =  4 gh 4 =  5 gh 5 P/g=  2 h 2 =  3 h 3 =  4 h 4 =  5 h 5 Airy model P/g=  2 h 5 = (  2 h 4 +  1 h 4 ’) = (  2 h 3 +  1 h 3 ’)  1 – mantle density  2 – crustal density (constant)  1 >  2 P =  gh

Airy Isostatic Model 1) P/g=  a h a +  w h w +  c h c +  m h m = Constant  Total pressure 2) T = h a + h w + h c + h m = Constant  Total thickness a –air c – crust w – water m – mantle 5-8 times

Regional Isostasy Compensation directly below the load, no rigidity (like water) Take lithospheric strength into account, there is flexural rigidity

Regional Isostasy – Elastic Plate Elastic thickness  Flexural Rigidity (Bending)

Regional Isostasy – Elastic Plate (Turcotte and Schubert, 1982) Elastic thickness  Flexural Rigidity (Bending) D(d 4 w/d 4 x)+(  b -  a )gw=q(x) – differential equation D – flexural rigidity w – vertical deflection at x g – gravity x – horizontal distance from the load q(x) – load applied to the top of the plate at x Indexes: “a” – above, “b” - below

Regional Isostasy – Elastic Plate Elastic thickness  Flexural Rigidity (Bending) D(d 4 w/d 4 x)+(  b -  a )gw=q(x) Small amplitude of w, long wavelength Large w, smaller wavelength Peripheral bulge + depression Collapse into local equilibrium

Regional Isostasy – Examples Elastic thickness  Flexural Rigidity (Bending) ~ bending of diving plate Load – topography of accretionary plate wedge + volcanic arc ~ bending of elastic plate Load – mass of high mountains

Isostatic Rebound Historical levels of Lake Superior due to post-glacial rebound

Isostatic Rebound Present-day uplift rate, horizontal velocity, free air gravity anomaly, and rate of change in gravity (Wu, 2001)