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
We want
A map of surface
wave velocity
source
dist
v
time
We have
Average velocity
along seismic rays
receiver
2D Seismic Tomography
We want
A map of surface
wave velocity
We have
Average velocity
along seismic rays
2D Seismic Tomography
We want
A map of surface
wave velocity
We have
Average velocity
along seismic rays
Regular Parameterization
Coarse grid
Fine grid
Resolution
Bad
Good
Constrain on
the model
Good
Bad
Regular Parameterization
Coarse grid
Fine grid
Resolution
Bad
Good
Constraint on
the model
Good
Bad
Define arbitrarily more constraints on the model
Irregular parameterizations
Gudmundsson & Sambridge (1998)
Sambridge & Rawlinson (2005)
Nolet & Montelli (2005)
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).
9
Voronoi cells
Cells are only
defined by their
centres
Voronoi cells are everywhere
Q u ic k T im e ™ a n d a
decom pr e ssor
a r e n e e d e d t o s e e t h is p ic t u r e .
11
Voronoi cells are everywhere
Q u ic k T im e ™ a n d a
decom pr e ssor
a r e n e e d e d t o s e e t h is p ic t u r e .
12
Voronoi cells are everywhere
Q u ic k T im e ™ a n d a
decom pr essor
a r e n e e d e d t o s e e t h is p ic t u r e .
13
Voronoi cells
Model is defined by:
* Velocity in each cell
* Position of each cell
Problem becomes
highly nonlinear
Non Linear Inversion
Sampling a multi-dimensional function
X1
X2
X2
X1
Non Linear Inversion
X1
X2
Solution : Maximum
Solution : statistical distribution
X2
X2
X1
Optimisation
(e.g. Genetic Algorithms,
Simulated Annealing)
X1
Bayesian Inference
(e.g. Markov chains)
Partition Modelling
(C.C. Holmes. D.G.T. Denison, 2002)
A Bayesian technique used for classification and
Regression problems in Statistics
Regression Problem
• Cos ?
• Polynomial function?
Partition Modelling
Dynamic irregular parameterisation
The number of
parameters is variable
n=3
Partition Modelling
n=6
n=11
n=8
n=3
Partition Modelling
Bayesian Inference
Mean. Takes in
account all the models
Mean solution
Partition Modelling
Adaptive
parameterisation
Automatic
smoothing
Mean solution
True solution
Able to pick up
discontinuities
Can we apply these concepts to tomography ?
Synthetic experiment
Km/s
True velocity model
Data Noise
σ = 28 s
Ray geometry
Iterative linearised tomography
Reference
Model
Forward calculation
Fast Marching Method
Inversion step
Ray
geometry
Subspace method (Matrix inversion)
Solution
Model
 Fixed Parameterisation
 Regularisation procedure
 Interpolation
Observed
travel times
Regular grid Tomography
fixed grid (20*20 nodes)
Damping
20 x 20
B-splines nodes
Smoothing
Km/s
Iterative linearised tomography
Reference
Model
Forward calculation
Fast Marching Method
Inversion step
Ray
geometry
Subspace method (Matrix inversion)
Solution
Model
 Fixed Parameterisation
 Regularisation procedure
 Interpolation
Observed
travel times
Iterative linearised tomography
Reference
Model
Point wise
spatial
average
Forward calculation
Fast Marching Method
Inversion step
Ray
geometry
Partition Modelling
Ensemble
of Models
 Adaptive Parameterisation
 No regularisation procedure
 No interpolation
Observed
travel times
Description of the method
Each step
Km/s
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
 P(m proposed ) 
P (accept)  min1,

P
(
m
)
current


Description of the method
Step 150
Step 300
Step 1000
Solution
Maxima
Best model sampled
Km/s
Mean
Average of all the models sampled
Regular Grid vs Partition Modelling
200 fixed cells
Km/s
45 mobile cells
Model Uncertainty
True model
Avg. model
Average
Cross Section
1
0
Standard deviation
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 …
Computational Cost Issues
When we change the position of one cell …
Computational Cost Issues
When we change the position of one cell …
Real Data
(Erdinc Saygin ,2007)
Cross correlation
of seismic ambient
noise
Real Data
Damping
Maps of
Rayleigh
waves group
velocity at 5s.
Smoothing
Km/s
Changing the number of Voronoi cells
The birth step
Generate randomly the location
of a new cell nucleus
40
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