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 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

We want A map of surface wave velocity We have Average velocity along seismic rays

Regular Parameterization Coarse grid Fine grid Bad Good Resolution Constrain on the model Good Bad

Regular Parameterization Coarse grid Fine grid Bad Good Resolution 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 Voronoi cells are everywhere

12 Voronoi cells are everywhere

13 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 X1 X2

Non Linear Inversion Optimisation Bayesian Inference Solution : Maximum Solution : statistical distribution 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 s Km/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 Observed travel times Forward calculation Fast Marching Method Forward calculation Fast Marching Method Solution Model Reference Model

Regular grid Tomography fixed 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 Observed travel times Forward calculation Fast Marching Method Forward calculation Fast Marching Method Solution 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 Observed travel times Forward calculation Fast Marching Method Forward calculation Fast Marching Method Ensemble of Models 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 Each step Km/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/s

Model Uncertainty Standard deviation 1 0 Average Cross Section True model Avg. 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 …

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