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Quantifying uncertainties in seismic tomography

Quantifying uncertainties in seismic tomography. Michel Cuer, Carole Duffet, Benjamin Ivorra and Bijan Mohammadi Université de Montpellier II, France. SIAM Conference on Mathematical & Computational Issues in the Geosciences, Avignon, France. June 7-10, 2005. Outline. Introduction

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Quantifying uncertainties in seismic tomography

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  1. Quantifying uncertainties in seismic tomography Michel Cuer, Carole Duffet, Benjamin Ivorra and Bijan Mohammadi Université de Montpellier II, France. SIAM Conference on Mathematical & Computational Issues in the Geosciences, Avignon, France. June 7-10, 2005

  2. Outline • Introduction • The reflection tomography problem • Uncertainty analysis on the solution model • Global inversion • Conclusions SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  3. Outline • Introduction • The reflection tomography problem • Uncertainty analysis on the solution model • Global inversion • Conclusions SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  4. x(km) Source 1 times(s) times(s) receivers sources x(km) v(km/s) z(km) The reflection tomography problem • Forward problem:ray tracing • Inverse problem: minimizing the misfits between observed and calculated traveltimes inverse problem forward SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  5. v1(km/s) Int1 (km) v2(km/s) Int2(km) v3(km/s) sources receivers Modelisation • Modelisation: • Model: 2D model parameterization based on b-spline functions • interfaces: z(x), x(z) • velocities: v(x), v(x)+k.z • Acquisition survey: • sources: S=(xs,0) • receivers: R=(xr,0) • Data: traveltimes modeled by the forward problem based on a ray tracing SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  6. Ray tracing algorithm • Ray tracing for specified ray signature : • source (S) and receiver (R) fixed • ray signature known (signature = reflectors where the waves reflect) • Fermat’s principle: • analytic traveltime formula within layer (P = impact point of the ray) • Fermat’s principle: ( C = trajectory between S and R) (in particular) SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  7. R=(xr,0) S=(xs,0) V1 P1 t = minimize t(S,R) P3 P1,P2,P3 Int1 V2 P2 Int2 t=t(P1,P2,P3) Ray tracing: an optimization problem A ray is a trajectorythat satisfies Fermat ’s principle for a given signature z(km) x(km) SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  8. The reflection tomography problem: an inverse problem • Search a model which • fits traveltime data for given uncertainties on the data • and fits a priori information • The least square method with the a priori covariance operator in the data space the a priori covariance operator in the model space This classical approach give the estimate SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  9. Outline • Introduction • The reflection tomography problem • Uncertainty analysis on the solution model • Global inversion • Conclusions SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  10. Uncertainty analysis on the solution model • Linearized approach: • Jacobian matrix: • Acceptable models = mest + m with: Jm small in the error bar, m in the model space. • Motivations: • find error bars on the model parameters SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  11. Uncertainty analysis: two approaches • We propose two methods to access the uncertainties • Linear programming method • Classical stochastic approach SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  12. Under the constraints Linear programming method (Dantzig, 1963) • Solve the linear programming problem where: • t=0.003(s), •  m= ( mv,  mz) avec mv = min(|vm-0.8|,|3-vm|) et mz =  z/2 SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  13. Stochastic approach (Franklin, 1970) • Solve the stochastic inverse problem where: with t=0.003(s),  m= ( mv,  mz) avec mv = min(|vm-0.8|,|3-vm|) et mz =  z/2 SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  14. Stochastic approach • Linearized framework: analysis of the a posteriori covariance matrix • uncertainties on the inverted parameters • correlation between the uncertainties SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  15. sources Application on a 2D synthetic model x(km) • Acquisition survey: • 20 sources (xs=200m) • 24x20 receivers (offset=50m) • data: • 980 traveltimes data • uncertainty 3ms • model: • 1 layer • 10 interface parameters • 1 constant velocity times(s) x(km) V=2.89(km/s) z(km) SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  16. Uncertainty analysis on the solution model Linear programming VELOCITY Boundaries of the model INTERFACE Velocity: V=2.89(km/s) SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  17. x(km) V=2.89(km/s) z(km) Uncertainty analysis on the solution model Linear programming SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  18. Uncertainty analysis on the solution model Stochastic inverse VELOCITY Boundaries of the model INTERFACE Velocity: V=2.89(km/s) SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  19. x(km) V=2.89(km/s) z(km) Uncertainty analysis on the solution model Stochastic inverse SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  20. Linear programming V=2.89(km/s) Stochastic inverse z(km) x(km) Comparison of the two approaches • Results obtained by the two methods are similar SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  21. Comparison of the two approaches • Results obtained by the two methods are similar • Linear programming method is more expensive but as informative as classical stochastic approach • However, these two approaches may furnish uncertainties on the model parameters : error bars SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  22. Outline • Introduction • The reflection tomography problem • Uncertainty analysis on the solution model • Global inversion • Conclusions SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  23. Global inversion • Global Optimization Method: • We choose a classical optimization method (e.g.: Levenberg-Marquardt, Genetic, …) • Our algorithm improve the initial condition to this method using Recursive Linear Search. • Reference: “Simulation Numérique” Mohammadi B. & Saiac J.H. , Dunod, 2001 SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  24. Global Inversion Results V=2.89(km/s) Initial model Global inversion V=2.75(km/s) z(km) x(km) SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  25. Linear programming Stochastic inverse Global inversion vPL = +/- 26.40 m/s vIS = +/- 3.3 m/s Global Inversion Results V=2.89(km/s) V=2.75(km/s) z(km) x(km) SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  26. Cost Function Iterations Global Inversion Convergence SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  27. Outline • Introduction • The reflection tomography problem • Uncertainty analysis on the solution model • Global inversion • Conclusions SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  28. Conclusions • We propose two methods to access a posteriori uncertainties: • Linear programming method • Classical stochastic approach • The two approaches of uncertainty analysis furnish similar results in the linearized framework . • These two approaches to quantify uncertainties may be applied to others inverse problems SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

  29. Conclusions • Linearized approach explores only the vicinity of the solution model • Future work: global inversion can allow to overcome the difficulties to quantify uncertainties in the nonlinear case. • Estimations given by the stochastic inverse approach could (to do) be used as initial iterate in linear programming problems SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005

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