1
Introduction to Inversion in
Geophysics
Dr. Laurent Marescot
Course given at the University of Fribourg (19 April 2010)
Laurent Marescot, 2010
Learning Objective and Agenda
Learning objective: get the basic understanding of
inversion processes to be able to use
geophysical software in a meaningful
way
Agenda:
• Some basic definitions
• Inversion concepts
• Linear inversion: temperature example
• Non-linear inversion: seismic and geoelectric examples
2
Laurent Marescot, 2010
Agenda
• Some basic definitions
• Inversion concepts
• Linear inversion: temperature example
• Non-linear inversion: seismic and geoelectric examples
3
Laurent Marescot, 2010
5
Definition: Contrast
To characterize different material using geophysics, a
contrast must exist (i.e. a difference in the physical
properties)
Laurent Marescot, 2010
Agenda
• Some basic definitions
• Inversion concepts
• Linear inversion: temperature example
• Non-linear inversion: seismic and geoelectric examples
9
Laurent Marescot, 2010
Agenda
• Some basic definitions
• Inversion concepts
• Linear inversion: temperature example
• Non-linear inversion: seismic and geoelectric examples
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Laurent Marescot, 2010
Direct and Inverse Problems
Inverse ProblemDirect Problem
Note: equation of a straight line: y=bx+aLaurent Marescot, 2010
Linear Inverse Problem:Even-determined Problem
b
a
Z
Z
T
T
2
1
2
1
1
1
22
11
bZaT
bZaT
- Nb of data = Nb of model parameters
- Suppose noise-free data!
dGm1
mGd
2
1
1
2
1
1
1
T
T
Z
Z
b
aor
or
Laurent Marescot, 2010
b
a
81
21
22
19
mbaC
mbaC
8*22
2*19
dGm1
mGd
22
19
11
28
6
1
b
aor
re) temperatu(surface18
(slope)5.0
Cb
a
22
11
bZaT
bZaT
Linear Inverse Problem:Even-determined Problem
or
Laurent Marescot, 2010
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Are two data points
enough?
Linear Inverse Problem:Even-determined Problem
Laurent Marescot, 2010
Linear Inverse ProblemOver-determined Problem
b
a
Z
Z
Z
T
T
T
NN 1
1
1
2
1
2
1
dGmg
NN bZaT
bZaT
bZaT
22
11
- Nb of data > Nb of model parameters
- G is no longer a square matrix: cannot be inverted!
mGdor
Laurent Marescot, 2010
2
1
2
22 :i
ienormL e
pre
i
obs
ii dde
)()( GmdGmdeeTTE
dGGGmT1Test ][Solution of inverse problem:
Min
0m
E
dGm gestor
Linear Inverse ProblemOver-determined Problem
Laurent Marescot, 2010
b
a
Z
Z
Z
T
T
T
NN 1
1
1
2
1
2
1
NN bZaT
bZaT
bZaT
22
11
mGd
dGGGmT1Test ][
Linear Inverse ProblemOver-determined Problem
or
Laurent Marescot, 2010
dGGGmT1Test ][
2
1
21
1
2
1
21
111
1
1
1
111
T
T
ZZZ
Z
Z
Z
ZZZb
a
N
N
N
Linear Inverse ProblemOver-determined Problem
Laurent Marescot, 2010
Over-determinedEven-determined Under-determined
dGGGmT1Test ][dGm
1
Linear Inverse ProblemEffect of Number of Data
Laurent Marescot, 2010
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dGGGmT1Test ][
b
a
Z
Z
T
T
1
1
2
1
bZaT
bZaT
2
1 mGd
2
1
1
11
1
111
T
T
ZZZ
Z
ZZb
aor
1)(2
12
22Z
ZZ
ZZ 10
12
Z
ZZ
Linear Inverse ProblemA Mixed Problem: Issue
or
Laurent Marescot, 2010
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dGGGmT1Test ][
2
1
1
11
0
0
1
111
T
T
ZZZ
Z
ZZb
a
dGIGGmT1Test ][
22
22
)22(
12
22Z
ZZ
Zε small, e.g. 0.0001
Least-squares inversion withMarquardt-Levenberg modification
Linear Inverse ProblemA Mixed Problem: Solution
Laurent Marescot, 2010
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or
For example, add a priori information on the surface temperature (e.g. b = 18°C)
In this case, the solution for a (slope) should reflect:
- information on the measured data
- a priori information on b
2
1
21
1
2
1
21
11
0
0
1
111
T
T
ZZZ
Z
ZZb
a
Linear Inverse ProblemIncluding a priori Information
dGIGGmT1Test ][
Laurent Marescot, 2010
Linear Inverse ProblemEffect of Sampling on Solution Variance
Because of measurement errors, inversion solution is NOT UNIQUE!
Laurent Marescot, 2010
dWGWεGWGm e
T1
me
Test ][
Mixed (Marquardt-Levenberg)
W are weighting matrices
Linear Inverse ProblemIncluding a priori Information
Laurent Marescot, 2010
More information in this paper:Un algorithme d’inversion par moindres carrés pondérés: application aux données géophysiques par méthodes électromagnétiques en
domaine fréquence: Marescot, 2003, Bull. Soc. vaud. Sc. nat. 88.3: 277-300. www. tomoquest.com
• Direct vs. Inverse problems
• A priori Information
• G can be pre-evaluated
• Even-, Over-, Under-determined inverse problems
• Mixed-determined inverse problem
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Linear Inverse ProblemKey Concepts
dGIGGmT1Test ][
Laurent Marescot, 2010
Agenda
• Some basic definitions
• Inversion concepts
• Linear inversion: temperature example
• Non-linear inversion: seismic and geoelectric examples
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Laurent Marescot, 2010
Linear / Non-linear Problems
In fact, seismic tomography is
NON-LINEAR, since the ray
paths depend on the
UNKNOWN velocities in the
model:
- No straight-line rays
- Iterative solution required
Laurent Marescot, 2010
E(m)=c1
E(m)=c2 < c1 E(m)=c3 < c2
m0
m4
m3
m2
m1
Non-linear Inverse ProblemDescent Methods
Laurent Marescot, 2010
Non-linear Inversion
Make the problem linear using a Taylor series around an
estimated solution:
)m(mG)g(m)mg(m)g(mg(m)est
nn
est
n
est
n
est
n
)g(mdΔmGest
n1nn
mGddΔdcalcobs
j
i
ijm
mgG
)(
Laurent Marescot, 2010
Non-linear Inversion
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mGddΔdcalcobs
dGIGGmT1T ][
Compare with the linear inverse solution:
dGIGGmT1Test ][
mGd
j
i
ijm
mgG
)(
Laurent Marescot, 2010
Non-linear Inversion
1) Donner une valeur initiale pour le modèle m
2) Calculer la réponse de ce modèle (dcalc). C’est le problème direct.
3) Evaluer la correction m à apporter au modèle et corriger le modèle
4) Recommencer le processus au point 2) jusqu’à convergence
dGIGGmT1T ][
Laurent Marescot, 2010
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Initial Traveltimes
50
40
30
20
10
00 50 100 150 200 250
T-D
ist/
20
00 [
ms]
Distance [m]
Laurent Marescot, 2010
Source: ETHZ
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Final Traveltimes
50
40
30
20
10
00 50 100 150 200 250
T-D
ist/
20
00 [
ms]
Distance [m]
Laurent Marescot, 2010
Source: ETHZ
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Final Traveltimes504030201000 50 100 150 200 250T-Dist/2000 [ms]
Distance [m]
Raypaths q2NW
20
25
30
35
Dep
th [
m]
40
45
500 50
Distance [m]150 200100
l1 l2 l3SE
Laurent Marescot, 2010
Source: ETHZ
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Combined Seismic Tomographic and Ultrashallow Seismic Reflection Study
of an Early Dynastic Mastaba, Saqqara, Egypt
Metwaly et al., 2005,
Archeological Prospection, 12, 245-256
Laurent Marescot, 2010
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Investigation of a Monumental
Macedonian Tumulus by Three dimensional Seismic Tomography
Polymenakos et al., 2004, Archeological Prospection, 11, 145-158
Laurent Marescot, 2010
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with:
log( )
log( ) log( )
( ) is the sensitivity matrix
meas calc
a a
iij
j
g mG
m
1T T
m G G W G d
m
d
Electric
tomography
inversion
Laurent Marescot, 2010
• Iterative solution!
• Large Computing may be required
• G is a partial derivative matrix (cannot be pre-evaluated!)
• Mixed-determined inverse problem:
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Non-linear Inverse ProblemKey Concepts
dGIGGmT1T ][
Laurent Marescot, 2010
Final notes:
- ε is called the damping factor and is sometimes written λ
- The value of the damping factor is usually decreased at each iteration
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Further Reading
The following documents were used to prepare this presentation:
More information on non-linear inversion with examples:
MARESCOT L., 2003. A weighted least-squares inversion algorithm: application to geophysical
frequency-domain electromagnetic data. Bull. Soc. vaud. Sc. nat. 88.3: 277-300.
More information on geoelectrical tomography:
MARESCOT L., 2006. Introduction à l’imagerie électrique du sous-sol. Bull. Soc. vaud. Sc. nat.
90.1: 23-40.
These documents can be downloaded at: http://www.tomoquest.com
Laurent Marescot, 2010