Computational Geophysics and Data Analysis 1Linear Inverse Problems

Linear(-ized) Inverse Problems

Linear inverse problems - Formulation- Some Linear Algebra- Matrix calculation – Revision- Illustration under(over)determined, unique case- Examples

Linearized inverse problems- Formulation- Examples

Partial derivatives

Scope: Formulate linear inverse problems as a system of equations in matrix form. Find the conditions under which solutions exist. Understand how to linearize a non-linear system to be able to find solutions.

Computational Geophysics and Data Analysis 2Linear Inverse Problems

Literature

Stein and Wysession: Introduction to seismology, Chapter 7

Aki and Richards: Theoretical Seismology (1s edition) Chapter 12.3

Shearer: Introduction to seismology, Chapter 5

Menke, Discrete Inverse Problemshttp://www.ldeo.columbia.edu/users/menke/gdadit/index.htmFull ppt files and matlab routines

Computational Geophysics and Data Analysis 3Linear Inverse Problems

Formulation

Linear(-ized) inverse problems can be formulated in the following way:

jiji mGd

(summation convention applies)

i=1,2,...,N number of dataj=1,2,...,M number of model parametersGij known (mxn)

We observe:- The inverse problem has a unique solution if N=M and det(G)≠0, i.e. the data are linearly independent- the problem is overdetermined if N>M- the problem is underdetermined if M>N

Computational Geophysics and Data Analysis 4Linear Inverse Problems

Illustration – Unique Case

In this case N=M, and det(G) ≠0. Let us consider an example

212

211

42

231

mmd

mmd

Let us check the determinant of this system: det(G)=10

Gmd

2

1

2

1

41

23

m

m

d

d

dGmGmGdG -1-1-1

5.0

0

3.01.0

2.04.0

2

1

2

1

2

1

m

m

d

d

m

m

Computational Geophysics and Data Analysis 5Linear Inverse Problems

Illustration – Overdetermined Case

In this case N>M, there are more data than model parameters.Let us consider examples with M=2, an overdetermined system would exist if N=3.

213

22

11

2

2

1

mmd

md

md

A physical experiment which could result in these data:Individual Weight measurement of two masses m1 and m2

leading to the data d1 and d2 and weighing both together leads to d3. In matrix form:

2

1

3

2

1

11

10

01

m

m

d

d

d

Gmd

Computational Geophysics and Data Analysis 6Linear Inverse Problems

Illustration – Overdetermined Case

Let us consider this problem graphically

A common way to solve this problem is to minimize the difference between data vector d and the predicted data for some model m such that

is minimal.

21

2

1

2

2

1

mm

m

m

2Gmd S

Computational Geophysics and Data Analysis 7Linear Inverse Problems

Illustration – Overdetermined Case

Using the L2-norm leads us to theleast-squares formulation of the problem. The solution to the minimization (and thus the inverse problem) is given as:

In our example the resulting (best) model estimation is:

dGG)(Gm~ T1T

3/5

3/2~m

and is the model with the minimal distance to all three lines in the plot.

best model

Computational Geophysics and Data Analysis 8Linear Inverse Problems

Illustration – Underdetermined Case

Let us assume we made one measurement of the combined weight of two masses:

Clearly there are infinitely many solutions to this problem. A modelestimate can be defined by choosing a model that fits the data exactly Am=d and has the smallest l2 norm ||m||. Using Lagrange multipliers one can show that the minimum norm solution is given by

221 dmm

1

1~

)(~ 1

m

dGGGm TT

Computational Geophysics and Data Analysis 9Linear Inverse Problems

Examples – Inversion of Gravity Data

Let’s go back to the problem of gravity, in 2-D the Bouguer anomaly at point x0 with arbitrary topography is given by (e.g. Telford et al., 1990)

To bring this into the form d=Gm we discretize the space

dxdzzxx

zzxxd

220

0

),(2)(

d(x0)

r(x,z)

x,x0

z

j=1 2 3 4 5

6 7 8 9 ...

... 20

h

h

zj

xj

mj

j

M

j

G

jji

ji m

zxx

zhd

ij

1

22

2

)(

2

Computational Geophysics and Data Analysis 10Linear Inverse Problems

Master-Event-Method

Let s assume we have have previously located an earthquake (x0,y0,z0) at time t0 and we recorded a new event at stations 1, ..., N

1

2

3Dti

x

z

y

Event 2

Event 1

Li

ui

gi

zuyuxu

uLLt

iziyix

iii

1

cos

This is a system of linear equations for 4 unknowns:

Computational Geophysics and Data Analysis 11Linear Inverse Problems

Master-Event-Method

x

z

y

Event 2

Event 1

Li

ui

gi

izi

iyi

ixii

ii

uG

uG

uGG

zmymxmm

td

4321

4321

1

Let us put this system into the common form d=Gm

z

y

x

uuu

uuu

t

t

NzNyNx

zyx

d

N

i

1

1 111

1

Computational Geophysics and Data Analysis 12Linear Inverse Problems

Vertical Seismic Profile

Let us consider a string of receivers in a borehole

v1

v2

vM

- We assume straigt rays- The ground is discretized with M layers of equal thickness dz with velocities vi

-The seismometers (N) are located at depths zj

Formulate the forward problem in matrix form d=Gm! Is the problem linear? What would happen if the rays are not modelled as straight lines?

seis

mom

ete

rs

Computational Geophysics and Data Analysis 13Linear Inverse Problems

Linearized Inversion

Let us formalize the situation where we are able to linearize a otherwise nonlinear problem around some model m0. In this case the forward problem is given by

NimmmFd Mi ,1),...,,( 21

this m-dimensional function is developed around some model m0=(m01, m02, ..., m0M) where we neglect higher-order terms:

))(,...,,(),...,,( 0002011

00201 mmmmmm

FmmmFd jM

M

j j

iMii

d0 GijDmj

d0 synthetic data of starting model (known)di=di-d0 data difference vector (residuals, misfit, cost ...)mj=mj-m0 model difference vector (gradient)

Computational Geophysics and Data Analysis 14Linear Inverse Problems

Linearized Inversion: Hypocenter location

Above a homogeneous half space we measure P wave travel times from an earthquake that happens at time t at (x,y,z) at i receiverlocations (xi, yi ,zi). So our model vector is m=(t,x,y,z)T. The arrival times are given by

this is a nonlinear problem! Now let us assume we have a rough idea about the time of the earthquake and its location. This is our startingmodel m0= (t0, x0, y0, z0)T.

2/1222 )()(1

)( zyyxxtmFt iiii

To linearize the problem we now have to find the partial derivatives of F with respect to all model parameters at m0.

Computational Geophysics and Data Analysis 15Linear Inverse Problems

Hypocenter location – partial derivatives

... we obtain : 2/1222 )()()(1

)( zzyyxxtmFt iiiii

0

004

0

003

0

002

01

2/120

20

20000

)(

)(

)(

1)(

)()(1

)(

ii

i

ii

i

ii

i

iiii

R

z

z

mFG

R

yy

y

mFG

R

xx

x

mFG

t

mFG

zyyxxtmFt

Ri0

Computational Geophysics and Data Analysis 16Linear Inverse Problems

Hypocenter location – partial derivatives

... let us now define a vector

ui=1/Ri0(xi-x0, yi-y0,-z0)

which is a vector pointing from the initial source location to receiver i. We obtain:

)()()(1

00000 zzuyyuxxutttt iziyixii

di m1 m2 m3 m4

which is exactly the form we obtained for the Master-Event Method, what is the difference, however?

This approach is an iterative algorithm

Computational Geophysics and Data Analysis 17Linear Inverse Problems

Linearized Travel-Time Inversion

We learned in seismology that for a given ray parameter p the delay time t(p) is given by the difference of the travel time T and the distance X(p) the ray ermerges times p

)(

0

2/122 )(2)()()(pzs

dzpzcppXpTp

Graphically this can be interpreted as:

p=dT/dX

X

pX T

t(p)

Computational Geophysics and Data Analysis 18Linear Inverse Problems

Linearized Travel-Time Inversion

… the important property of t(p) is the fact that it decreases monotonically with increasing p so it is a function easier to handle than the travel-times (which may contain triplications).

t(p) is nonlinearly related to the velocity model c(z). So in order to invert for it we would have to linearize. We obtain

)(

0

2/122 )(2)(pzs

dzpzcp

Now the perturbation in t(p) (the data residual) is linearly relatedto the perturbation in the velocity model c(z). This integral can easily be brought into the form d=Gm by subdividing the Earth into layers (e.g. of equal thickness).

p=dT/dX

X

pX T

t(p)

)(

01/222

)( )(c

c(z)2)(

pzs

dzzcpz

p

Computational Geophysics and Data Analysis 19Linear Inverse Problems

Partial Derivatives

Let us take a closer look at the matrix Gij for linearized problems. What useful information is contained in this matrix (operator)? When d=g(m), then the linearization leads to

The actual (relative) values of Gij determine how the model parameters influence the data (or data difference).

Example: Gik are small for all i. This implies that the model Parameter mk has almost no influence on the data. It can be varied Without changing them. Therefore, its resolution is poor.

mGd

And the matrix Gij contains the partial derivatives

j

iij m

gG

Computational Geophysics and Data Analysis 20Linear Inverse Problems

Resolution – Hypocenter Location

Example: Earthquake hypocenter location

)()()(1

00000 zzuyyuxxutttt iziyixii

di m1 m2 m3 m4

Remember the elements of Gij where the components of the unit vector which points from the original (known) hypocenter to the receiver. Small uiz with respect to the other ones means bad resolution in depth:The depth resolution of shallow earthquakes Far away is poor.

Computational Geophysics and Data Analysis 21Linear Inverse Problems

Linear Dependence

ilik cGG

When two columns of Gij are linearly dependent then for all i

What are the consequences for a model perturbation in parameters k and l?Linear dependence implies

kl mc

m 1

lilkikllilkkik mGmGmmGmmG !

)()(

In words: Parameters mk and ml cannot be independently determinedas they compensate each other. This is called a trade-off.

Computational Geophysics and Data Analysis 22Linear Inverse Problems

Trade-Off Fault Zone Waves

Computational Geophysics and Data Analysis 23Linear Inverse Problems

Calculating Partial Derivatives (1)

),...,,( 02010 Mj

iij mmm

m

FG

Generally we need to calculate the partial derivatives

… depending on the formulation of the forward problem …

1. For explicit functions Fi, for example:

j

M

j

G

jji

ji m

zxx

zhd

ij

1

22

2

)(

2

zuyuxut iziyixi

1

Gravity problem Master-Event Method

… we can directly calculate the partial derivatives.

Computational Geophysics and Data Analysis 24Linear Inverse Problems

Calculating Partial Derivatives (2)

2. The data di are given implicitly through

0),...,,,( 21 Mii mmmdf

The arguments being the model and data parameter of the starting model m0. Often the data d0 are obtained by finding the roots of the topmost equation.

we differentiate with respect to mj

0

j

i

i

i

j

i

m

d

d

f

m

f

i

i

j

i

j

iij d

f

m

f

m

dG

/

Computational Geophysics and Data Analysis 25Linear Inverse Problems

Calculating Partial Derivatives (3)

3. In more complicated cases the partial derivatives have to be obtained by numerical differentiation.

Note that for the evaluation of each element of Gij a solution of the forward problem is necessary! In cases where the number of model parameters is large or where the forward problem is very involved this is impractical. But at least this method always works (approximately).

j

ijjiij m

dmmdG

00 ,...)(...,

Computational Geophysics and Data Analysis 26Linear Inverse Problems

Summary

Most inverse problems can be formulated as discrete linear problems either as

… or – if the problem is linearized - …

jiji mGd

jiji mGd

In which case the Gij contains the partial derivatives of the problem. The elements of Gij contain useful information on the resolution of the model parameters and linear dependence may indicate trade-offs between model parameters.