# Uncertainty Analysis of the Solution Model of 3D Seismic Reflection Tomography

Thursday, April 25, 2002 - 11:00am - 12:00pm

Keller 3-180

Delphine Sinoquet (Institut Français du Pétrole)

Joint work with Carole Duffet.

Reflection tomography aims to determine the velocity model that best fits the travel time data associated with reflections of seismic waves in the subsurface. This solution model is only one model among many possible models. Indeed, the uncertainties on the observed travel times (resulting from an interpretative event picking on seismic sections) and more generally the underdetermination of the inverse problem lead to uncertainties on the solution model. An a posteriori uncertainty analysis is then crucial to delimit the range of possible solution models that fit, with the expected accuracy, the data and the a priori information. The large size non linear least-square problem is solved classicaly by a Gauss-Newton method based on successive linearizations of the forward operator. A linearized a posteriori analysis is then possible by an analysis of the a posteriori covariance matrix (inverse of the Hessian matrix). The computation of this matrix is generally expensive (the matrix is huge for 3D problems) and the physical interpretation of the results is difficult. A formalism based on macro-parameters (linear combinations of model parameters) allows to compute uncertainties on relevant geological quantities for a reduced computational time (the matrices to be manipulated are reduced to the macro-parameter space). A first application on a synthetic example with basic macro-parameters shows their potentialities. The generality of the formalism allows a wide flexibility for the construction of the macro-parameters. Nevertheless, this approach is only valid in the vicinity of the solution model (linearized framework) and complex cases may require a non linear approach.

Reflection tomography aims to determine the velocity model that best fits the travel time data associated with reflections of seismic waves in the subsurface. This solution model is only one model among many possible models. Indeed, the uncertainties on the observed travel times (resulting from an interpretative event picking on seismic sections) and more generally the underdetermination of the inverse problem lead to uncertainties on the solution model. An a posteriori uncertainty analysis is then crucial to delimit the range of possible solution models that fit, with the expected accuracy, the data and the a priori information. The large size non linear least-square problem is solved classicaly by a Gauss-Newton method based on successive linearizations of the forward operator. A linearized a posteriori analysis is then possible by an analysis of the a posteriori covariance matrix (inverse of the Hessian matrix). The computation of this matrix is generally expensive (the matrix is huge for 3D problems) and the physical interpretation of the results is difficult. A formalism based on macro-parameters (linear combinations of model parameters) allows to compute uncertainties on relevant geological quantities for a reduced computational time (the matrices to be manipulated are reduced to the macro-parameter space). A first application on a synthetic example with basic macro-parameters shows their potentialities. The generality of the formalism allows a wide flexibility for the construction of the macro-parameters. Nevertheless, this approach is only valid in the vicinity of the solution model (linearized framework) and complex cases may require a non linear approach.