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

SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005 Outline nIntroduction nThe reflection tomography problem nUncertainty analysis on the solution model nGlobal inversion nConclusions

SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005 Outline nIntroduction nThe reflection tomography problem nUncertainty analysis on the solution model nGlobal inversion nConclusions

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

SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005 nModelisation: –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=(x s,0) receivers: R=(x r,0) nData: traveltimes modeled by the forward problem based on a ray tracing Modelisation sources receivers v2(km/s) v3(km/s) v1(km/s) Int1 (km) Int2(km)

SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005 nRay 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 ) Ray tracing algorithm (in particular)

SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005 Ray tracing: an optimization problem A ray is a trajectory that satisfies Fermat ’s principle for a given signature S=(x s,0) R=(x r,0) P1P1 P2P2 P3P3 V1V1 V2V2 Int 1 Int 2 t=t(P 1,P 2,P 3 ) t = minimize t(S,R ) P 1,P 2,P 3 x(km) z(km)

SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005 The reflection tomography problem: an inverse problem nSearch a model which –fits traveltime data for given uncertainties on the data –and fits a priori information nThe 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 Outline nIntroduction nThe reflection tomography problem nUncertainty analysis on the solution model nGlobal inversion nConclusions

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

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

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

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

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

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

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

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

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

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

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

SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005 Outline nIntroduction nThe reflection tomography problem nUncertainty analysis on the solution model nGlobal inversion nConclusions

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

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

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

SIAM Conference on Mathematical & Computational Issues in the Geosciences, 2005 Outline nIntroduction nThe reflection tomography problem nUncertainty analysis on the solution model nGlobal inversion nConclusions

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

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