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A Partition Modelling Approach to Tomographic Problems

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

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A Partition Modelling Approach to Tomographic Problems

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  1. A Partition Modelling Approach to Tomographic Problems Thomas Bodin &Malcolm Sambridge Research School of Earth Sciences, Australian National University

  2. 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

  3. 2D Seismic Tomography We want A map of surface wave velocity

  4. 2D Seismic Tomography source We want A map of surface wave velocity We have Average velocity along seismic rays receiver

  5. 2D Seismic Tomography We want A map of surface wave velocity We have Average velocity along seismic rays

  6. 2D Seismic Tomography We want A map of surface wave velocity We have Average velocity along seismic rays

  7. Regular Parameterization Coarse grid Fine grid BadGood Resolution Constrain on the model GoodBad

  8. Regular Parameterization Coarse grid Fine grid BadGood Resolution Constraint on the model GoodBad Define arbitrarily more constraints on the model

  9. 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).

  10. Voronoi cells Cells are only defined by their centres

  11. Voronoi cells are everywhere

  12. Voronoi cells are everywhere

  13. Voronoi cells are everywhere

  14. Voronoi cells Model is defined by: * Velocity in each cell * Position of each cell Problem becomes highly nonlinear

  15. Non Linear Inversion X1 X2 X2 X1 Sampling a multi-dimensional function

  16. Non Linear Inversion X2 X2 X1 X1 Solution : Maximum Solution : statistical distribution Optimisation Bayesian Inference (e.g. Genetic Algorithms, Simulated Annealing) (e.g. Markov chains)

  17. 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?

  18. Partition Modelling Dynamic irregular parameterisation The number of parameters is variable n=3

  19. Partition Modelling n=6 n=11 n=8 n=3

  20. Mean. Takes in account all the models Partition Modelling Bayesian Inference Mean solution

  21. Partition Modelling • Adaptive parameterisation • Automatic smoothing • Able to pick up discontinuities Mean solution True solution Can we apply these concepts to tomography ?

  22. Synthetic experiment True velocity model Ray geometry Data Noise σ = 28 s Km/s

  23. Iterative linearised tomography Forward calculation Fast Marching Method Reference Model Ray geometry • Inversion step • Subspace method (Matrix inversion) • Fixed Parameterisation • Regularisation procedure • Interpolation Solution Model Observed travel times

  24. Regular grid Tomographyfixed grid (20*20 nodes) Damping 20 x 20 B-splines nodes Smoothing Km/s

  25. Iterative linearised tomography Forward calculation Fast Marching Method Reference Model Ray geometry • Inversion step • Subspace method (Matrix inversion) • Fixed Parameterisation • Regularisation procedure • Interpolation Solution Model Observed travel times

  26. Iterative linearised tomography Forward calculation Fast Marching Method Reference Model Ray geometry • Inversion step • Partition Modelling • Adaptive Parameterisation • No regularisation procedure • No interpolation Point wise spatial average Ensemble of Models Observed travel times

  27. Description of the method Pick randomly one cell Change either its value or its position Compute the estimated travel time Compare this proposed model to the current one Km/s Each step

  28. Description of the method Step 150 Step 300 Step 1000

  29. Solution Km/s Maxima Mean Best model sampled Average of all the models sampled

  30. Km/s Regular Grid vs Partition Modelling 200 fixed cells 45 mobile cells Km/s

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

  32. 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.

  33. Computational Cost Issues When we change the value of one cell …

  34. Computational Cost Issues When we change the position of one cell …

  35. Computational Cost Issues When we change the position of one cell …

  36. Computational Cost Issues When we change the position of one cell …

  37. Computational Cost Issues When we change the position of one cell …

  38. Real Data (Erdinc Saygin ,2007) Cross correlation of seismic ambient noise

  39. Real Data Damping Maps of Rayleigh waves group velocity at 5s. Smoothing Km/s

  40. Changing the number of Voronoi cells The birth step Generate randomly the location of a new cell nucleus

  41. Real Data Variable number of Voronoi cells Average model (Km/s) Error estimation (Km/s)

  42. Real Data Variable number of Voronoi cells Average model (Km/s)

  43. Conclusion • Adaptive Parameterization • Automatic smoothing and regularization • Good estimation of model uncertainty

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