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1
Gravity Surveying
WS0506Dr. Laurent Marescot
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Introduction
Gravity surveying…
Investigation on the basis of relative variations in theEarth´gravitational field arising from difference of densitybetween subsurface rocks
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Application
• Exploration of fossil fuels (oil, gas, coal)• Exploration of bulk mineral deposit (mineral, sand, gravel)• Exploration of underground water supplies• Engineering/construction site investigation• Cavity detection• Glaciology• Regional and global tectonics• Geology, volcanology• Shape of the Earth, isostasy• Army
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Structure of the lecture
1. Density of rocks2. Equations in gravity surveying3. Gravity of the Earth4. Measurement of gravity and interpretation5. Microgravity: a case history6. Conclusions
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1. Density of rocks
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Rock densityRock density depends mainly on…• Mineral composition• Porosity (compaction, cementation)
Lab or field determination of density isuseful for anomaly interpretation and data reduction
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2. Equations in gravity surveying
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First Newton´s LawNewton´s Law of Gravitation
rr
mmGF rr2
21−=
( ) ( ) ( )2122
122
12 zzyyxxr −+−+−=
21311 skgm1067.6 −−−×=GGravitational constant
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Second Newton´s Law
amF rr= Ngr
RMGa rrr
=−= 2
kg10977.5 24×=M
km6371=R
gN: gravitational acceleration or „gravity“ for a spherical, non-rotating, homogeneous Earth, gN iseverywhere the same
mass of a homogeneous Earth
mean radius of Earth
2sm81.9≅Ng
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Units of gravity
• 1 gal = 10-2 m/s2
• 1 mgal = 10-3 gal = 10-5 m/s2
• 1 µgal = 10-6 gal = 10-8 m/s2 (precision of a gravimeter forgeotechnical surveys)
• Gravity Unit: 10 gu = 1 mgal
• Mean gravity around the Earth: 9.81 m/s2 or 981000 mgal
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Keep in mind…
…that in environmental geophysics,we are working with values about…
0.01-0.001 mgal ≈ 10-8 - 10-9 gN !!!
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Gravitational potential fieldThe gravitational potential field is conservative (i.e. the work to move a mass in this field is independent of the way)The first derivative of U in a direction gives the component of gravity in thatdirection
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withF U U UU g U i j km x y z
∂ ∂ ∂∇ = = ∇ = + +
∂ ∂ ∂
rrr rr
∫∫∞∞
=−=⋅=RR
RmG
rdrmGdrrgU 1
21rr
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The measured perturbations in gravity effectively correspond to the verticalcomponent of the attraction of the causative body
we can show that θ is usuallyinsignifiant since δgz<<gTherefore…
Measurement component
zgg δδ ≈
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Grav. anomaly: point mass
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2 3
from Newton's Law
( )cos
r
z
Gmgr
Gm Gm z zg gr r
θ
∆ =
′ −∆ = ∆ = =
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Grav. anomaly: irregular shape
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3
( )
for we derive:
( )
Gm z zgr
m x y z
G z zg x y zr
δ ρ δ δ δ
ρδ δ δ δ
′ −∆ =
′ ′ ′=
′ − ′ ′ ′=
( ) ( ) ( )2 2 2r x x y y z z′ ′ ′= − + − + −
with ρ the density (g/cm3)
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Grav. anomaly: irregular shape
3
3
for the whole body:( )
if , and approach zero:
( )
G z z x y zg r
x y z
G z zg dx dy dzr
ρ δ δ δ
δ δ δ
ρ
′ − ′ ′ ′∆ =
′ ′ ′
′ − ′ ′ ′∆ =
∑∑∑
∫∫∫
Conclusion: the gravitational anomaly can be efficiently computed! Thedirect problem in gravity is straightforward: ∆g is found by summing theeffects of all elements which make up the body
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3. Gravity of the Earth
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Shape of the Earth: spheroid• Spherical Earth with R=6371 km is an approximation!
• Rotation creates an ellipsoid or a spheroid
247.2981
=−
e
pe
RRR
Deviation from a spherical model: km3.14km2.7
=−=−
p
e
RRRR
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The Earth´s ellipsoidal shape, rotation, irregular surface relief and internal mass distribution cause gravity to vary over it´s surface
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• From the equator to the pole: gn increases, gc decreases• Total amplitude in the value of g: 5.2 gal
( )φω cos22 RR
MGggg Cn −=+=
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Reference spheroid
• The reference spheroid is an oblate ellipsoid that approximates the mean sea-level surface (geoid) with the land above removed
• The reference spheroid is defined in the Gravity Formula1967 and is the model used in gravimetry
• Because of lateral density variations, the geoid and reference spheroid do not coincide
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Shape of the Earth: geoid
• It is the sea level surface (equipotential surface U=constant)
• The geoid is everywhere perpendicular to the plumb line
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Spheroid versus geoidGeoid and spheroid usually do not coincide (India -105m, New Guinea +73 m)
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4. Measurement of gravity and interpretation
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Measurement of gravity
• Large pendulums
• Falling body techniques
• Gravimeters• Use spring techniques• Precision: 0.01 to 0.001 mgal
Relative measurements are usedsince absolute gravitydetermination is complex and long!
2
21 tgz =
gLT π2=
For a precision of 1 mgalDistance for measurement 1 to 2 mz known at 0.5 µmt known at 10-8 s
Absolute measurements Relative measurements
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Scintrex CG-5LaCoste-Romberg mod. G
Source: P. Radogna, University of Lausanne
Gravimeters
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Stable gravimeters
2
2
4 with 2
kg xm
mg x TT kπ π
∆ = ∆
= ∆ =
For one period
k is the elastic spring constant
Problem: low sensitivity since thespring serves to both support themass and to measure the data. So thistechnique is no longer used…
Hook´s Law
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LaCoste-Romberg gravimeterThis meter consists in a hinged beam, carrying a mass, supported by a spring attached immediately above the hinge.
A „zero-lenght“ spring can be used, where thetension in the spring is proportional to theactual lenght of the spring.
• More precise than stable gravimeters (betterthan 0.01 mgal)
• Less sensitive to horizontal vibrations• Requires a constant temperature environment
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CG-5 Autograv
CG-5 electronic gravimeter:
CG-5 gravimeter uses a mass supported bya spring. The position of the mass is keptfixed using two capacitors. The dV used to keep the mass fixed is proportional to thegravity.
• Self levelling• Rapid measurement rate (6 meas/sec)• Filtering• Data storage
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Gravity surveying
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Factors that influence gravity
The magnitude of gravity depends on 5 factors:
• Latitude• Elevation• Topography of the surrounding terrains• Earth tides• Density variations in the subsurface:
this is the factor of interest in gravity exploration, but it ismuch smaller than latitude or elevation effects!
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Gravity surveying• Good location is required (about 10m)• Uncertainties in elevations of gravity stations account for the greatest
errors in reduced gravity values (precision required about 1 cm) (usedGPS)
• Frequently read gravity at a base station (looping) needed
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Observed data corrections
gobs can be computed for the stations using ∆g only after thefollowing corrections:
• Drift correction
• Tidal correction
• Distance ground/gravimeter („free air correction“ see below)
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Drift correction on observed dataGradual linear change in reading with time, due to imperfectelasticity of the spring (creep in the spring)
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Tidal correction on observed data
Effect of the Moon: about 0.1 mgalEffect of the Sun: about 0.05 mgal
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After drift and tidal corrections, gobs can be computed using ∆g, thecalibration factor of the gravimeter and the value of gravity at the base
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Gravity reduction: Bouguer anomaly
modelobs ggBA −=
TCBCFACggmodel −+−= φ
• gmodel model for an on-land gravity survey• gφ gravity at latitude φ (latitude correction)• FAC free air correction• BC Bouguer correction• TC terrain correction
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Latitude correction
( )φβφβφ4
22
1 sinsin1 ++= equatorgg
• β1 and β2 are constants dependent on the shape and speedof rotation of the Earth
• The values of β1, β2 and gequator are definded in the GravityFormula 1967 (reference spheroid)
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Free air correction
The FAC accounts for variation in the distance of theobservation point from the centre of the Earth.This equation must also be used to account for the distance ground/gravimeter.
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Free air correction
)metersin(3086.0 hhFAC =
2
32 2
2 0.3 mgal
N
NHöhe
GMgR
gdg GMdR R R
g dRg dRR
=
= − = −
∆ ≈ ≈ ⋅
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Bouguer correction
2BC G hπ ρ=
• The BC accounts for the gravitational effect of the rocks present between the observation point and the datum
• Typical reduction density for the crust is ρ =2.67 g/cm3
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Terrain correction
The TC accounts for the effect of topography.
The terrains in green and blue are taken into account in the TCcorrection in the same manner: why?
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Residual gravity anomaly
The regional field can beestimated by hand orusing more elaboratedmethods (e.g. upwardcontinuation methods)
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Bouguer anomaly
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Interpretation: the inverse problem
Two ways of solvingthe inverse problem:
• „Direct“ interpretation• „Indirect“ interpretation
and automatic inversion
Warning: „direct“ interpretation has nothing to do with„direct“ (forward) problem!
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Direct interpretationAssomption: a 3D anomaly iscaused by a point mass (a 2D anomaly is caused by a linemass) at depth=z
x1/2 gives z
zmasse m
surface
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Direct interpretation
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(1) Construction of a reasonable model(2) Computation of its gravity anomaly(3) Comparison of computed with observed anomaly(4) Alteration of the model to improve correspondence of
observed and calculated anomalies and return to step (2)
Indirect interpretation
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Non-unicity of the solution
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Automatic inversion
Automatic computer inversion with a priori information formore complex models (3D) using non-linear optimizationalgorithms. Minimize a cost (error) function F
with n the number of data
Automatic inversion is used when the model is complex (3D)
( )∑=
∆−∆=n
icalcobs ii
ggF1
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Automatic inversion
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Mining geophysics
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5. Microgravity: a case history
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• length:6 km• difference in altitude: 323 m • geology: alpine molassic bedrock (tertiary sandstone) and
an overlaying quaternary glacial fill • depth of bedrock: varying from 1.5 m to 25 m • The choice of the corridor had to consider the depth of the
bedrock Source: P. Radogna et al.
A SUBWAY PROJECT IN LAUSANNE, SWITZERLAND, AS AN URBAN MICROGRAVIMETRY TEST SITE
P. Radogna, R. Olivier, P. Logean and P. ChasseriauInstitute of Geophysics, University of Lausanne
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Zone II
Scintrex CG5
200 gravity stations Source: P. Radogna et al.
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Geological section, approximately A´´-A´-A
Source: P. Radogna et al.
64Source: P. Radogna et al.
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Profile A´´-A´-A
Source: P. Radogna et al.
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Building and basement gravity effect
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DEM for topographical corrections
BASEMENT FROM:•Cadastral plan
•Building typology•GIS
Source: P. Radogna et al.
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Bouguer Anomaly
Source: P. Radogna et al.
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Regional Anomaly
Source: P. Radogna et al.
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Residual Anomaly
Source: P. Radogna et al.
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Result…
Source: P. Radogna et al.
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Painting of the valley and the bridge before 1874 and actual picture of the same zone
Complex building corrections
Source: P. Radogna et al.
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Rectangular prisms are used for modeling the bridge’s pillars
Source: P. Radogna et al.
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Gravity effect of the bridge
-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20X (m.)
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
18
20
22
24
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30
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36
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Y (m
.)
00.010.020.030.040.050.060.070.080.090.10.110.120.130.140.150.160.17
mGal
Gravity station
•Formulation of rectangularprism (Nagy, 1966)•Pillar’s density is fixedto 2.00 g/cm3
Source: P. Radogna et al.
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Standard corrections gravity anomaly without topographical corrections.
Reduction density : 2.40 g/cm3
860
880
900
920
940
960
980
µGal
300 4022 4019
4012 4009
4005
4003
4000
402403
404
405
0
20
40
60
80
100
120
µGal
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
µGal
distance (meters)
860
880
900
920
940
960
980
Gravity anomaly with bridge effect corrections
Gravity anomaly with topographical corrections
Source: P. Radogna et al.
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6. Conclusions
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Advantages
• The only geophysical method that describes directly the density of the subsurface materials
• No artificial source required• Useful in urban environment!
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Drawbacks
• Expensive• Complex acquisition process• Complex data processing• Limited resolution• Very sensitive to non-unicity in the modeling solutions
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