A Partition Modelling Approach to

Tomographic Problems

Thomas Bodin & Malcolm Sambridge

Research School of Earth Sciences,

Australian National University

Outline

Parameterization in Seismic tomography

Non-linear inversion, Bayesian Inference and Partition Modelling

An original way to solve the tomographic problem

• Method

• Synthetic experiments

• Real data

2D Seismic Tomography

We want

A map of surface wave velocity

2D Seismic Tomography

source

receiver

time

distv

We want

A map of surface wave velocity

We have

Average velocity along seismic rays

We want

A map of surface wave velocity

We have

Average velocity along seismic rays

2D Seismic Tomography

2D Seismic Tomography

We want

A map of surface wave velocity

We have

Average velocity along seismic rays

Regular ParameterizationCoarse grid Fine grid

Bad GoodResolution

Constrain on the model

Good Bad

Regular ParameterizationCoarse grid Fine grid

Bad GoodResolution

Constraint on the model

Good Bad

Define arbitrarily more constraints on the model

9

Irregular parameterizations

Chou & Booker (1979); Tarantola & Nercessian (1984); Abers & Rocker (1991); Fukao et al. (1992); Zelt & Smith (1992); Michelini

(1995); Vesnaver (1996); Curtis & Snieder (1997); Widiyantoro & van der Hilst (1998); Bijwaard et al. (1998); Bohm et al. (2000);

Sambridge & Faletic (2003).

Nolet & Montelli (2005)

Sambridge & Rawlinson (2005)Gudmundsson & Sambridge (1998)

Voronoi cells

Cells are only defined by their

centres

11

QuickTime™ and a decompressor

are needed to see this picture.

Voronoi cells are everywhere

12

QuickTime™ and a decompressor

are needed to see this picture.

Voronoi cells are everywhere

13

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

Voronoi cells are everywhere

Voronoi cells

Problem becomes highly nonlinear

Model is defined by:

* Velocity in each cell* Position of each cell

Non Linear Inversion

X2

X1

Sampling a multi-dimensional function

X1X2

Non Linear Inversion

Optimisation Bayesian Inference

Solution : Maximum Solution : statistical distribution

X2

X1

X2

X1

X2

X1

X2

X1

(e.g. Genetic Algorithms, Simulated Annealing)

(e.g. Markov chains)

Partition Modelling(C.C. Holmes. D.G.T. Denison, 2002)

• Cos ? • Polynomial function?

Regression Problem

A Bayesian technique used for classification and Regression problems in Statistics

n=3

The number of parameters is

variable

Dynamic irregular parameterisation

Partition Modelling

n=6 n=11

n=8 n=3

Partition Modelling

Mean. Takes in account all the

models

Partition Modelling

Bayesian Inference

Mean solution

Adaptive parameterisation

Automatic smoothing

Able to pick up discontinuities

Partition Modelling

Can we apply these concepts to tomography ?

Mean solution

True solution

Synthetic experiment

True velocity model Ray geometry

Data Noise σ = 28 sKm/s

Iterative linearised tomography

Inversion step Subspace method (Matrix inversion)

Fixed Parameterisation

Regularisation procedure

Interpolation

Inversion step Subspace method (Matrix inversion)

Fixed Parameterisation

Regularisation procedure

Interpolation

Ray geometry

Ray geometry

Observed travel timesObserved

travel times

Forward calculationFast Marching Method

Forward calculationFast Marching Method

Solution Model

Solution Model

Reference Model

Reference Model

Regular grid Tomographyfixed grid (20*20 nodes)

Damping

Smoothing

Km/s

20 x 20 B-splines nodes

Iterative linearised tomography

Inversion step Subspace method (Matrix inversion)

Fixed Parameterisation

Regularisation procedure

Interpolation

Inversion step Subspace method (Matrix inversion)

Fixed Parameterisation

Regularisation procedure

Interpolation

Ray geometry

Ray geometry

Observed travel timesObserved

travel times

Forward calculationFast Marching Method

Forward calculationFast Marching Method

Solution Model

Solution Model

Reference Model

Reference Model

Iterative linearised tomography

Inversion step

Partition Modelling

Adaptive Parameterisation No regularisation procedure No interpolation

Inversion step

Partition Modelling

Adaptive Parameterisation No regularisation procedure No interpolation

Ray geometry

Ray geometry

Observed travel timesObserved

travel times

Forward calculationFast Marching Method

Forward calculationFast Marching Method

Ensemble of ModelsEnsemble of Models

Reference Model

Reference Model

Point wise spatial average

Description of the method

I. Pick randomly one cell

II. Change either its value or its position

III. Compute the estimated travel time

IV. Compare this proposed model to the current one

)(

)(,1min)(

current

proposed

mP

mPacceptP

Each stepKm/s

Description of the method

Step 150 Step 300 Step 1000

Solution

Maxima Mean

Best model sampled Average of all the models sampled

Km/s

Regular Grid vs Partition Modelling

200 fixed cells 45 mobile cells

Km/sKm/s

Model Uncertainty

Standard deviation

1

0

Average Cross Section

True modelAvg. model

Computational Cost Issues

Monte Carlo Method cannot deal with high dimensional problems, but …

Resolution is good with small number of cells.

Possibility to parallelise.

No need to solve the whole forward problem at each iteration.

Computational Cost Issues

When we change the value of one cell …

Computational Cost Issues

When we change the position of one cell …

Computational Cost Issues

When we change the position of one cell …

When we change the position of one cell …

Computational Cost Issues

When we change the position of one cell …

Computational Cost Issues

Real Data

(Erdinc Saygin ,2007)

Cross correlation of seismic

ambient noise

Real Data

Maps of Rayleigh

waves group velocity at

5s.

Damping

Smoothing

Km/s

40

Changing the number of Voronoi cells

The birth step

Generate randomly the location of a new cell nucleus

Real Data

Variable number of Voronoi cells

Average model (Km/s) Error estimation (Km/s)

Real Data

Variable number of Voronoi cells

Average model (Km/s)

Conclusion

Adaptive Parameterization

Automatic smoothing and regularization

Good estimation of model uncertainty