A dedicated constrained optimization method for 3D reflexion tomography

Abstract : Seismic reflection tomography is a method for the determination of a subsurface velocity model from the traveltimes of seismic waves reflecting on geological interfaces. From an optimization viewpoint, the problem consists in minimizing a nonlinear least-squares function measuring the mismatch between observed traveltimes and those calculated by raytracing in this model. The introduction of a priori information on the model is crucial to reduce the under-determination. The contribution of this paper is to introduce a technique able to take into account geological a priori information in the reflection tomography problem expressed as constraints in the optimization problem. This constrained optimization is based on a Gauss-Newton Sequential Quadratic Programming approach. At each Gauss-Newton step, a solution to a strictly convex quadratic optimization problem subject to linear constraints is computed thanks to an augmented Lagrangian relaxation method. Our choice for this optimization method is motivated and its original aspects are described. The efficiency of the method is shown on applications on a 2D OBC real data set and on a 3D real data set: the introduction of constraints coming both from well logs and from geological knowledge allows to reduce the under-determination of the 2 inverse problems. Introduction Reflection tomography allows to determine a velocity model from the traveltimes of seismic waves reflecting on geological interfaces. This inverse problem is formulated as a nonlinear least-squares function which measures the mismatch between observed traveltimes and traveltimes computed by ray tracing method. This method has been successfully applied to real data sets (Ehinger et al, 2001, Broto et al, 2003). Nevertheless, the under-determination of the inverse problem generally requires the introduction of additional information on the model to reduce the number of admissible models. Penalization terms modelling this information can be added to the seismic terms in the objective functions but the tuning of the penalization weights may be tricky. In this paper, we propose to handle the a priori information by the introduction of equality and inequality constraints in the optimization process. This approach allows to introduce lot of constraints of different types, provided we have at our disposal an adequate constrained optimization method. We developed an original method designed for the tomographic inverse problem which presents the following characteristics: it is a large scale problem (10000-50000 unknowns), the forward operator is nonlinear and its computation may be expensive (large number of source-receiver couples, up to 500000), the problem is ill-conditioned. In the first part of this paper, the chosen method is motivated and its original aspects are shortly described (for further details, refer to Delbos et al, 2004). Applications on a 2D PP/PS real data set and on a 3D PP real data set are presented in a second part.
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Delphine Sinoquet, Frédéric Delbos, Jean Charles Gilbert. A dedicated constrained optimization method for 3D reflexion tomography. EAGE 66th Conference & Exhibition, Jun 2004, Paris, France. ⟨hal-02284254⟩

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