The truncated Newton using 1st and 2nd order adjoint-state method: a new approach for traveltime tomography without rays
Abstract
Traveltime tomography algorithms generally use ray tracing. The use of rays in tomography may not be suitable for handling very large datasets and perform tomography in very complex media. Traveltime maps can be computed through finite-difference approach (FD) and avoid complex ray-tracing algorithm for the forward modeling (Vidale 1998, Zhao 2004). However, rays back-traced from receiver to source following the gradient of traveltime are still used to compute the Fréchet derivatives. As a consequence, the sensitivity information computed using back-traced rays is not numerically consistent with the FD modeling used (the derivatives are only a rough approximation of the true derivatives of the forward modeling). Leung & Quian (2006) proposed a new approach that avoid ray tracing where the gradient of the misfit function is computed using the adjoint-state method. An adjoint-state variable is thus computed simultaneously for all receivers using a numerical method consistent with the forward modeling, and for the computational cost of one forward modeling. However, in their formulation, the receivers have to be located at the boundary of the investigated model, and the optimization approach is limited to simple gradient-based method (i.e. steepest descent, conjugate gradient) as only the gradient is computed. However, the Hessian operator has an important role in gradient-based reconstruction methods, providing the necessary information to rescale the gradient, correct for illumination deficit and remove artifacts. Leung & Quian (2006) uses LBFGS, a quasi-Newton method that provides an improved estimation of the influence of the inverse Hessian. Lelievre et al. (2011) also proposed a tomography approach in which the Fréchet derivatives are computed directly during the forward modeling using explicit symbolic differentiation of the modeling equations, resulting in a consistent Gauss-Newton inversion. We are interested here in the use of a new optimization approach named as the truncated Newton (TCN) (Métivier et al. 2012) with a more accurate estimation of the impact of the Hessian. We propose an efficient implementation for first-arrival traveltime tomography. In TCN, the model update Δm is obtained through the iterative resolution of the Newton linear system H Δm = - g. Based on a matrix-free conjugate gradient resolution, the iterative solver requires only the computation of the gradient and of Hessian-vector products. We propose a generalization of the computation of the gradient using the adjoint-state method that allows to consider receivers located anywhere. Then the Hessian-vector products are computed using an original formulation based on a 2nd-order adjoint-state method, at the cost of an additional forward modeling. The TCN algorithm is composed of two nested loops: an internal loop to compute Δm, and an external loop where a line search is performed to update the subsurface parameters. TCN thus considers locally the inversion of the traveltime data using an estimation of the full Hessian (both 1st and 2nd order terms) at an acceptable cost. Tomography with TCN is an improvement over the simple gradient-based adjoint-state tomography due to its good convergence property, to the better consideration of illumination, and is a promising tool for multi-parameter inversion as rescaling is given by the Hessian.
- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2013
- Bibcode:
- 2013AGUFM.S33A2388B
- Keywords:
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- 7270 SEISMOLOGY Tomography;
- 0935 EXPLORATION GEOPHYSICS Seismic methods;
- 0902 EXPLORATION GEOPHYSICS Computational methods: seismic;
- 8180 TECTONOPHYSICS Tomography