Course given at the University of Fribourg (19 April 2010 ...tomoquest.com/attachments/File/Marescot_Intro_to_Inversion_UNIFR... · Introduction to Inversion in Geophysics ... •

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


Agenda





3


4

Definition: Model

A model is a simple and ideal view of a physical reality


5

Definition: Contrast

To characterize different material using geophysics, a

contrast must exist (i.e. a difference in the physical

properties)


6

Definition: Anomaly


7

Noise in Geophysics


8

2D

3D

2D and 3D Models


Agenda





9


Forward and Inverse Problems

10


RX Tomography (CT-Scan)






Forward and Inverse Problems


Agenda





15


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


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


or


19

Are two data points

enough?



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


2

1

2

22 :i

ienormL e

pre

i

obs

ii dde

)()( GmdGmdeeTTE

dGGGmT1Test ][Solution of inverse problem:

Min

0m

E

dGm gestor




eeTE 0

m

E


b

a

Z

Z

Z

T

T

T

NN 1

1

1

2

1

2

1

NN bZaT

bZaT

bZaT

22

11

mGd

dGGGmT1Test ][


or


dGGGmT1Test ][

2

1

21

1

2

1

21

111

1

1

1

111

T

T

ZZZ

Z

Z

Z

ZZZb

a

N

N

N



Over-determinedEven-determined Under-determined

dGGGmT1Test ][dGm

1

Linear Inverse ProblemEffect of Number of Data


26

Enough data?

Linear Inverse ProblemA Mixed Problem


27

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


28

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


29

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 ][


Linear Inverse ProblemEffect of Sampling on Solution Variance

Because of measurement errors, inversion solution is NOT UNIQUE!


dWGWεGWGm e

T1

me

Test ][

Mixed (Marquardt-Levenberg)

W are weighting matrices

Linear Inverse ProblemIncluding a priori Information


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

33

Linear Inverse ProblemKey Concepts

dGIGGmT1Test ][


Agenda





34


35

Non-linear Inverse ProblemSeismic Measurement Principle


Non-linear Inverse ProblemDiscrete Problem


A Mixed Problem…


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


Non-linear Inverse ProblemOptimal Solution

Linear Problem Non-linear Problem


Non-linear Inverse ProblemDescent Methods


E(m)=c1

E(m)=c2 < c1 E(m)=c3 < c2

m0

m4

m3

m2

m1

Non-linear Inverse ProblemDescent Methods


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

)(



43

mGddΔdcalcobs

dGIGGmT1T ][

Compare with the linear inverse solution:

dGIGGmT1Test ][

mGd

j

i

ijm

mgG

)(



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 ][


45

Seismic

tomography

inversion


46

Initial Traveltimes

50

40

30

20

10

00 50 100 150 200 250

T-D

ist/

20

00 [

ms]

Distance [m]


Source: ETHZ

47

Final Traveltimes

50

40

30

20

10

00 50 100 150 200 250

T-D

ist/

20

00 [

ms]

Distance [m]


Source: ETHZ

48

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


Source: ETHZ

49

Combined Seismic Tomographic and Ultrashallow Seismic Reflection Study

of an Early Dynastic Mastaba, Saqqara, Egypt

Metwaly et al., 2005,

Archeological Prospection, 12, 245-256


50

Refraction

Metwaly et al., 2005, Archeological Prospection, 12, 245-256


51

Investigation of a Monumental

Macedonian Tumulus by Three dimensional Seismic Tomography

Polymenakos et al., 2004, Archeological Prospection, 11, 145-158


Non-linear Inverse ProblemError vs Resolution


53

Non-linear Inverse ProblemGeoelectrical Measurement Principle


54

Non-linear Inverse Problem

Current distribution


55

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


Complex Model Geometries

Finite Element Method


57

Slope Instabilities

Source: Marescot et al., 2008,

Engineering Geology 98


58

Slope Instabilities


• Iterative solution!

• Large Computing may be required

• G is a partial derivative matrix (cannot be pre-evaluated!)

• Mixed-determined inverse problem:

59

Non-linear Inverse ProblemKey Concepts

dGIGGmT1T ][


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

60

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


Documents

Course given at the University of Fribourg (19 April 2010 ...tomoquest.com/attachments/File/Marescot_Intro_to_Inversion_UNIFR... · Introduction to Inversion in Geophysics ... •