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