Methods for approximating hessian times vector operation in full wavefield inversion

摘要

Method for estimating the Hessian of the objective function, times a vector, in order to compute an update in an iterative optimization solution to a partial differential equation such as the wave equation, used for example in full wave field inversion of seismic data. The Hessian times vector operation is approximated as one forward wave propagation (24) and one gradient computation (25) in a modified subsurface model (23). The modified subsurface model may be a linear combination of the current subsurface model (20) and the vector (21) to be multiplied by the Hessian matrix. The forward-modeled data from the modified model are treated as a field measurement in the data residual of the objective function for the gradient computation in the modified model. In model parameter estimation by iterative inversion of geophysical data, the vector in the first iteration may be the gradient of the objective function.

主权项

1. A method for determining a discrete physical properties model of a subsurface region, referred to herein as the model or the subsurface model, by iteratively inverting measured geophysical data acquired from the subsurface region, comprising:
approximating a Hessian matrix of an objective function, then times a vector, called Hessian times vector, using a computer, with a single forward-wave simulated propagation and a single computation of gradient of the objective function, in a modified subsurface model, thereby requiring only three forward-wave or reverse-wave propagations; wherein said approximation is based on an approximate equation for a Born scattered pressure field, {tilde over (p)}d,F, where the Hessian times vector is approximated by a gradient computation using {tilde over (p)}d,F as an artificial residual; then computing a direction in model parameter space for an update to a current model by multiplying inverse of the Hessian matrix times a gradient of the objective function, wherein the inverse of the Hessian matrix is computed iteratively using a conjugate gradient method in which said approximate Hessian times vector is used to evaluate the Hessian matrix times a model perturbation vector; performing a line search to determine magnitude of the model update using said computed direction; and adding the model update to the current model to form an updated model, and using the updated model for geophysical prospecting.