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 GRAVITYABSOLUTE VS RELATIVE

• Absolute – Pendulum– Weight Drop– Rise and Fall

A-10 FG-5

Rise & Fall Weight 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 ~360Gals/6 hours = 1Gal/minute• Correction from recording gravimeter $, tidetables (obsolete),

computer program• Computer Tide Corrections (Examples)

SAGE 2004 TIDE CORRECTIONS

NOTE: MAXIMUMAMPLITUDE OF

~320GALS

SAGE 2010 TIDE CORRECTIONS

COMPUTING OBSERVED GRAVITYTIDE AND DRIFT CORRECTIONS

DRIFT CORRECTION CAUSED BY LONG TERM RELAXATION

ASSUMED TO BE SMOOTH, SLOW AND LINEARESTIMATE 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)

GEOIDELLIPSOID TOPO SURFACE

GEOIDHEIGHT

EARTH’SSURFACE

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)

– THEOG33 = 978049.0(1+0.0052884 sin²θ-0.0000059 sin² 2θ)

• 1967 – Correction for Potsdam (Geoid based)– THEOG67 = 978031.846(1+0.005278895 sin²θ-0.000023462 sin4θ)

• 1984 – Based on GRS 1980 – World Geodetic System (WGS84)– THEOG84 = 978032.67714 (1+0.00193185138639sin²θ)

– (1-0.00669437999013sin²θ)

– Requires correction for atmosphere (ATMCR).

– ATMCR = 0.87e-0.116h1.047 (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 = (0.094112-0.000134sinθ²-0.0000000134SELEVf) = ~0.09412SELEVf• FACm = (0.308768-0.000440sinθ²-0.0000001442SELEVm)• 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 = (2π6.6720.3048/1000.0)SELEVf = 0.03412SELEVf• BSCm = (2π6.672/1000.0)SELEVm = 0.04192SELEVm• 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 crustw=density of sea waters=density of substratum

h=density of crust –mountainso=density of crust-oceansr=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

- 32

-24-24

-

-32

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-di)/di)n

•

DOMAIN RADIUS (R)

• + + + + + + + + + + + + +• + + + + + + • +• + + +• + + + + + + +• + + + + + + +• + + + + + + + + + + + +

+ + + + ++++++• + + + + + +

• + + R

• + + + + +• + + + + +• +• + + + + + +• + + +

• + di

• +++ + + + + • + • + + + • + • + + + + + + +• • +• + + + + + + + + + • + +• + + + + + +• +• + + + + + + +• +• + + + + + + + + • + • + + + + + +

GRAVITY MODELING• DENSITY-DEPTH-

RELATIONSHIP.

GRAVITY MODELING

• VELOCITY-DENSITY RELATIONSHIP

• NAFE-DRAKE CURVE

VE

LOC

ITY

km

/sec

DENSITY gm/cm³

GRAVITY MODELING• VELOCITY-DENSITY RELATIONSHIP

GRAVITY MODELING

• EFFECTIVE DENSITY

LAYEREDMODEL

CONTINUOUSMODEL

Δρ(h) CAN BE CONSTANTLINEAR,EXPONENTIAL,OR HYPERBOLIC WITH

DEPTH

DENSITY-DEPTH RELATIONS

• EXPONENTIAL DENSITY-DEPTH = max +Δoe-bh

• Δ = -max = Δoe-bh

• Δ = Δo(1 - e-bH)/bh

• HYPERBOLIC DENSITY-DEPTH = Δo( β²/(h+β)²) + max

• Δ = Δo β²/(h+β)²

• Δ = Δo β/(H+β)

CALCULATING β

• From the infinite slab formula:

• Δg = 2πγΔoβH/(H + β)• Δg = 41.92 ΔoβH/(H + β)• H = - Δgβ/(Δg – 41.92Δoβ)• β = ΔgH/(41.92 ΔoH- Δg)

• If we know the residual anomaly (Δg) at a point and the depth of the basin (H) and the surface density contrast (Δo) 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

SPHERE CYLINDER

Z = X½Z=1.305X½

Gmax

Gmax/2

x½ x½

Gz= 4/3 π γR3(z/(x² + z²)3/2 Gz = 2πγR²(z/x² + z²)

REGIONAL – RESIDUAL SEPARATION