GRAVITY

EARTH’S GRAVITY FIELD 978 Gals 983 Gals 1 Gal = 1 cm/sec² ELLIPSOID North-South change ~1 mGals/km ~1.5 mGals/mile ~1  Gals/m ~.3  Gals/ft

MEASURING GRAVITY ABSOLUTE VS RELATIVE Absolute –Pendulum –Weight Drop –Rise and Fall A-10 FG-5 Rise & FallWeight Drop

GRAVIMETERS Relative –Stable – Astatic Worden La Coste Romberg Scintrix Auto Grav Worden Gravity Meter

La Coste & Romberg –Zero length spring –T proportional L

GRAVITY FIELD METHODS Planning a Survey –Previous data – quality and quantity – targer vs station density vs dollar$. –Instrumentation and field procedures –Acquiring permits, field preparations, low profile –Locations –Base ties, recoccupations, calibration, drift tares and tides –Special considerations in microgal surveys –Typical field procedures –Pitfalls and disasters (ignoring the above)

COMPUTING OBSERVED GRAVITY (MEASURED) CORRECT METER READINGS FOR TIDES. –Earth Tides. Caused by pull of sun and moon Maximum change ~360  Gals/6 hours = 1  Gal/minute Correction from recording gravimeter $, tidetables (obsolete), computer program Computer Tide Corrections (Examples)

SAGE 2004 TIDE CORRECTIONS NOTE: MAXIMUM AMPLITUDE OF ~320  GALS

SAGE 2010 TIDE CORRECTIONS

COMPUTING OBSERVED GRAVITY TIDE AND DRIFT CORRECTIONS DRIFT CORRECTION CAUSED BY LONG TERM RELAXATION ASSUMED TO BE SMOOTH, SLOW AND LINEAR ESTIMATE BY REOCCUPATION OF BASE CHECK FOR QUALITY CONTROL ON REOCC.

COMPUTING OBSERVED GRAVITY OBSG = (SCGR – BCGR)GRCAL + ABGV –Where: OBSG = Observed gravity SCGR = Station corrected meter reading BCGR = Base corrected gravity reading ABGV = Absolute base gravity value GRCAL= Gravimeter calibration

GRAVITY REDUCTION (MODEL) GEOID – Theoretical sea level surface. ELLIPSOID – Mathematical model of the earth –(from satellites) SPHEROID – Clark spheroid ~ 1866 –(from land surveys) GEOID ELLIPSOIDTOPO SURFACE GEOID HEIGHT EARTH’S SURFACE GEOID ELLIPSOID

THEORETICAL GRAVITY (MODEL) Geodetic Reference System (GRS) formulae refer to theoretical estimates of the Earth’s shape. From these GRS formulae we obtain International Gravity Formulae (IGF) Several different formulae have been adopted over the years 1930 – First internationally accepted IGF (Geoid based) –THEOG 33 = ( sin²θ sin² 2θ) 1967 – Correction for Potsdam (Geoid based) –THEOG 67 = ( sin²θ sin 4 θ) 1984 – Based on GRS 1980 – World Geodetic System (WGS84) –THEOG 84 = ( sin²θ) – (  sin²θ) –Requires correction for atmosphere (ATMCR). –ATMCR = 0.87e h (SL =0.87, 5 km =0.47, 10 km = 0.23 mGals)

GRAVITY ANOMALIES = MEASURED-MODEL Free Air Anomaly (FAAyy) –FAAyy = OBSG-THEOGyy+FACu x SELEVu –FACu = Free air correction in feet or meters –SELEVu = Station elevation in feet or meters FACf = ( sinθ² SELEVf) = ~ SELEVf FACm = ( sinθ² SELEVm) SELEVf = Station elevation in feet SELEVm = Station elevation in meters Simple Bouguer Anomaly (SBAyy) –SBAyy = FAAyy-BSCu –BSCu = Bouguer Slab Correction in feet or meters BSCf = (2π  /1000.0)SELEVf = SELEVf BSCm = (2π  /1000.0)SELEVm = SELEVm Note (FACu - BSCu) ≈ 0.06 mGals/ft ≈ 0.20 mGals/meter Complete Bouguer Anomaly (CBAyy) –CBAyy = SBAyy + TC TC = Terrain Correction (usually calculated in two parts)

COMPLETE BOUGUER ANOMALIES OF THE UNITED STATES

ISOSTATIC ANOMALIES (PRATT – AIRY)  c=density of crust  w=density of sea water  s=density of substratum  h=density of crust –mountains  o=density of crust-oceans  r=density of crust-ridge

100% COMPENSATION

75% COMPENSATION

0% COMPENSATION

GEOLOGICAL CORRECTED ANOMALY EXAMPLES –IMPERIAL VALLEY –RIO GRANDE RIFT –LOS ANGELES BASIN

REGIONAL- RESIDUAL GRAVITY ANOMALIES DEFINITION: RESIDUAL = REGIONAL – COMPLETE BOUGUER REGIONAL ANOMALY IS DETERMINE BY SCALE OF THE TARGET. (NON UNIQUE) SEPARATION METHODS: LINEAR SEPARATION (PROFILE METHOD 1D) MAP SEPARATION (2D) LEAST SQUARES FIT OF GRAVITY ANOMALIES

LINEAR SEPARATION

MAP SEPARATION COMPLETE BOUGUER ANOMALY REGIONAL ANOMALY

RESIDUAL BOUGUER ANOMALY 0 5

LEAST SQUARES FIT OF STATION GRAVITY PROBLEM: PRODUCE A REGULAR GRID OF GRAVITY VALUES FROM A RANDOMNLY DISTRIBUTED DATA SET.

LEAST SQUARES FIT OF STATION GRAVITY General quadric function of form: F(x,y) = Ax² +By² +Cxy +Dx + Ey +F Weighting function of form: W = ((R-d i )/d i ) n

DOMAIN RADIUS (R) R di

GRAVITY MODELING DENSITY-DEPTH- RELATIONSHIP.

GRAVITY MODELING VELOCITY-DENSITY RELATIONSHIP NAFE-DRAKE CURVE VELOCITY km/sec DENSITY gm/cm³

GRAVITY MODELING VELOCITY-DENSITY RELATIONSHIP

GRAVITY MODELING EFFECTIVE DENSITY LAYERED MODEL CONTINUOUS MODEL Δρ(h) CAN BE CONSTANT LINEAR,EXPONENTIAL, OR HYPERBOLIC WITH DEPTH

DENSITY-DEPTH RELATIONS EXPONENTIAL DENSITY-DEPTH  =  max +Δ  o e -bh Δ  =  -  max = Δ  o e -bh Δ  = Δ  o (1 - e -bH )/bh HYPERBOLIC DENSITY-DEPTH  = Δ  o ( β²/(h+β)²) +  max Δ  = Δ  o β²/(h+β)² Δ  = Δ  o β/(H+β)

CALCULATING β From the infinite slab formula: Δg = 2πγΔ  o βH/(H + β) Δg = Δ  o βH/(H + β) H = - Δgβ/(Δg – 41.92Δ  o β) β = ΔgH/(41.92 Δ  o H- Δg) If we know the residual anomaly (Δg) at a point and the depth of the basin (H) and the surface density contrast (Δ  o ) we can calculate β.

GRAVITY MODELING FORWARD INVERSE MODELING USING RESIDUAL SIMPLE SHAPES –SLAB –SPHERE –HORIZONTAL CYLINDER TALWANI - BOTT (2D) CADY (2 ½D) TALWANI – CORDELL – BIEHLER (3D)

GRAVITATIONAL FIELD OF A SPHERE AND CYLINDER SPHERECYLINDER Z = X½Z=1.305X½ Gmax Gmax/2 x½x½x½x½ G z = 4/3 π γR 3  (z/(x² + z²) 3/2 G z = 2πγR²  (z/x² + z²)

REGIONAL – RESIDUAL SEPARATION