Gravity Surveying

WS0506 Dr. Laurent Marescot

1 Introduction

Gravity surveying…

Investigation on the basis of relative variations in the Earth´ arising from difference of density between subsurface rocks

2 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 3 Structure of the lecture

1. Density of rocks 2. Equations in gravity surveying 3. Gravity of the Earth 4. Measurement of gravity and interpretation 5. Microgravity: a case history 6. Conclusions

4 1. Density of rocks

5 Rock density

Rock density depends mainly on… • Mineral composition • Porosity (compaction, cementation)

Lab or field determination of density is useful for anomaly interpretation and data reduction

6 2. Equations in gravity surveying

7 First ´s Law

Newton´s Law of Gravitation

r G m m F = − 1 2 rr r 2

2 2 2 r = ()()()x2 − x1 + y2 − y1 + z2 − z1

Gravitational constant G = 6.67×10−11 m3kg−1s−2 8 Second Newton´s Law

r r r G M r r F = m a a = − r = g N R2 2 g N ≅ 9.81 m s gN: gravitational or „gravity“

for a spherical, non-rotating, homogeneous Earth, gN is everywhere the same

M = 5.977×1024 kg mass of a homogeneous Earth

R = 6371 km mean radius of Earth 9 Units of gravity

• 1 = 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 for geotechnical surveys)

• Gravity Unit: 10 gu = 1 mgal

• Mean gravity around the Earth: 9.81 m/s2 or 981000 mgal

10 Keep in mind…

…that in environmental , we are working with values about…

-8 -9 0.01-0.001 mgal ≈ 10 -10 gN !!!

11 Gravitational potential field

The gravitational potential field is conservative (i.e. the work to move a mass in this field is independent of the way) Thefirstderivativeof U in a direction gives the component of gravity in that direction

r FUUU∂∂∂rrr ∇==Ugr with ∇= U i + j + k mxyz2 ∂∂ ∂

R R dr G m U = gr ⋅rr dr = −G m = 1 ∫ 1 ∫ 2 ∞ ∞ r R

12 Measurement component

The measured perturbations in gravity effectively correspond to the vertical component of the attraction of the causative body

we can show that θ is usually insignifiant since δgz<

δg ≈ δg z 13 Grav. anomaly: point mass

Gm ∆=g from Newton's Law r r 2 Gm Gm() z′ − z ∆=∆=gg cosθ = z rr23 14 Grav. anomaly: irregular shape

Gm() z′ − z ∆=g r3

for δρδδδmxyz= ′′′ we derive:

Gzz()′ − gxyzδδδδ= ′′′ rρ3

with ρ the density (g/cm3)

222 rxxyyzz=−+−+−()()()′′′ 15 Grav. anomaly: irregular shape

for the whole body: Gzzρ()′ − δ xyz′δδ′′ ∆=g δδ∑∑∑ δ r3

if xy′′ , and z ′ approach zero: ρ Gzz()′ − ∆=g dxdydz′′′ ∫∫∫ r3

Conclusion: the gravitational anomaly can be efficiently computed! The direct problem in gravity is straightforward: ∆g is found by summing the 16 effects of all elements which make up the body 3. Gravity of the Earth

17 Shape of the Earth: spheroid

• Spherical Earth with R=6371 km is an approximation!

• Rotation creates an ellipsoid or a spheroid

R − R 1 e p = Re 298.247

Re − R = 7.2 km Deviation from a spherical model: R − R =14.3 km p 18 19 The Earth´s ellipsoidal shape, rotation, irregular surface relief and internal mass distribution cause gravity to vary over it´s surface 20 ω g = g + g = G (M − 2 R cosφ) n C R2

• From the equator to the pole: gn increases, gc decreases • Total amplitude in the value of g: 5.2 gal

21 Reference spheroid

• The reference spheroid is an oblate ellipsoid that approximates the mean sea-level surface () with the land above removed

• The reference spheroid is defined in the Gravity Formula 1967 and is the model used in

• Because of lateral density variations, the geoid and reference spheroid do not coincide

22 Shape of the Earth: geoid

• It is the sea level surface (equipotential surface U=constant)

• The geoid is everywhere perpendicular to the plumb line

23 24 25 Spheroid versus geoid

Geoid and spheroid usually do not coincide (India -105m, New Guinea +73 m)

26 4. Measurement of gravity and interpretation

27 Measurement of gravity

Absolute measurements Relative measurements

• Large •Gravimeters • Use spring techniques T = 2π L g • Precision: 0.01 to 0.001 mgal • Falling body techniques Relative measurements are used 1 z = g t 2 since absolute gravity 2 determination is complex and long! For a precision of 1 mgal Distance for measurement 1 to 2 m z known at 0.5 µm t known at 10-8 s 28 Gravimeters

Scintrex CG-5 LaCoste-Romberg mod. G

29 Source: P. Radogna, University of Lausanne Stable gravimeters

k ∆=gx ∆ Hook´s Law m 4π 2 m gx=∆with T = 2 Tk2 For one period π k is the elastic spring constant

Problem: low sensitivity since the spring serves to both support the mass and to measure the data. So this technique is no longer used… 30 LaCoste-Romberg gravimeter

This 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 the tension in the spring is proportional to the actual lenght of the spring.

• More precise than stable gravimeters (better than 0.01 mgal) • Less sensitive to horizontal vibrations • Requires a constant temperature environment

31 CG-5 Autograv

CG-5 electronic gravimeter:

CG-5 gravimeter uses a mass supported by a spring. The position of the mass is kept fixed using two capacitors. The dV used to keep the mass fixed is proportional to the gravity.

• Self levelling • Rapid measurement rate (6 meas/sec) • Filtering •Datastorage 32 Gravity surveying

33 Factors that influence gravity

The magnitude of gravity depends on 5 factors:

•Latitude • Elevation • Topography of the surrounding terrains • Earth • Density variations in the subsurface: this is the factor of interest in gravity exploration, but it is much smaller than latitude or elevation effects!

34 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) (use dGPS) • Frequently read gravity at a base station (looping) needed

35 36 Observed data corrections

gobs can be computed for the stations using ∆g only after the following corrections:

• Drift correction

• Tidal correction

• Distance ground/gravimeter („free air correction“ see below)

37 Drift correction on observed data

Gradual linear change in reading with time, due to imperfect elasticity of the spring (creep in the spring)

38 Tidal correction on observed data

Effect of the Moon: about 0.1 mgal Effect of the Sun: about 0.05 mgal 39 After drift and tidal corrections, gobs can be computed using ∆g, the calibration factor of the gravimeter and the value of gravity at the base

40 Gravity reduction: Bouguer anomaly

BA = gobs − gmodel

gmodel = gφ − FAC + BC −TC

• 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 41 Latitude correction β φ β 2 4 gφ = gequator (1+ 1 sin + 2 sin φ)

• β1 and β2 are constants dependent on the shape and speed of rotation of the Earth

• The values of β1, β2 and gequator are definded in the Gravity Formula 1967 (reference spheroid)

42 Free air correction

The FAC accounts for variation in the distance of the observation point from the centre of the Earth. This equation must also be used to account for the distance ground/gravimeter.

43 Free air correction

GM g = R2

dg GM g =−22 =− N dR R3 R

gdR ∆g ≈≈20.3mgalN ⋅dR Höhe R

FAC = 0.3086 h (h in meters) 44 Bouguer correction

•TheBC 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 BC= 2π Gρ h

45 Terrain correction

The TC accounts for the effect of topography.

The terrains in green and blue are taken into account in the TC correction in the same manner: why?

46 47 48 Residual

The regional field can be estimated by hand or using more elaborated methods (e.g. upward continuation methods)

49 Bouguer anomaly

50 Interpretation: the

Two ways of solving the inverse problem:

• „Direct“ interpretation • „Indirect“ interpretation and automatic inversion

Warning: „direct“ interpretation has nothing to do with „direct“ (forward) problem!

51 Direct interpretation

Assomption: a 3D anomaly is caused by a point mass (a 2D anomaly is caused by a line mass) at depth=z

x1/2 gives z

surface z

masse m 52 Direct interpretation

53 Indirect interpretation

(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)

54 55 Non-unicity of the solution

56 Automatic inversion

Automatic computer inversion with a priori information for more complex models (3D) using non-linear optimization algorithms. Minimize a cost (error) function F n F = ()∆g − ∆g ∑ obsi calci i=1 with n the number of data

Automatic inversion is used when the model is complex (3D)

57 Automatic inversion

58 Mining geophysics

59 5. Microgravity: a case history

60 A SUBWAY PROJECT IN LAUSANNE, SWITZERLAND, AS AN URBAN MICROGRAVIMETRY TEST SITE P. Radogna, R. Olivier, P. Logean and P. Chasseriau Institute of Geophysics, University of Lausanne

• 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 61 bedrock Source: P. Radogna et al. Zone II

Scintrex CG5 62 200 gravity stations Source: P. Radogna et al. Geological section, approximately A´´-A´-A

63 Source: P. Radogna et al. 64 Source: P. Radogna et al. Profile A´´-A´-A

65 Source: P. Radogna et al. Building and basement gravity effect

66 DEM for topographical corrections

BASEMENT FROM: •Cadastral plan •Building typology •GIS

67 Source: P. Radogna et al. Bouguer Anomaly

68 Source: P. Radogna et al. Regional Anomaly

69 Source: P. Radogna et al. Residual Anomaly

70 Source: P. Radogna et al. Result…

71 Source: P. Radogna et al. Complex building corrections

Painting of the valley and the bridge before 1874 and actual picture of the same zone

72 Source: P. Radogna et al. Rectangular prisms are used for modeling the bridge’s pillars

73 Source: P. Radogna et al. Gravity effect of the bridge

40 38 •Formulation of rectangular 36 prism (Nagy, 1966) 34 32 •Pillar’s density is fixed 30 3 28 to 2.00 g/cm

26

24

22 mGal 0.17 20 0.16 0.15 0.14 18 0.13 0.12 16 0.11 0.1 0.09 Y (m.) 14 0.08 0.07 0.06 12 0.05 0.04 0.03 10 0.02 0.01 8 0

6

4

2

0

-2

-4

-6

-8 Gravity station 74 -10 -10-8-6-4-202468101214161820 Source: P. Radogna et al. X (m.) Standard corrections gravity anomaly without topographical corrections. Reduction density : 2.40 g/cm3

120

100 402 300 4022 4019 403 80 4000 60 404 µGal

40 4005 405 20 4003 4012 4009 0

Gravity anomaly with topographical corrections 980

960

940

920 µGal 900

880

860

Gravity anomaly with bridge effect corrections 980

960

940

920 µGal 900

880

860

0 5 10 15 20 25 30 35 40 45 50 55 60 6575 70 distance (meters) Source: P. Radogna et al. 6. Conclusions

76 Advantages

• The only geophysical method that describes directly the density of the subsurface materials • No artificial source required • Useful in urban environment!

77 Drawbacks

• Expensive • Complex acquisition process • Complex data processing • Limited resolution • Very sensitive to non-unicity in the modeling solutions

78