Methods for approximating hessian times vector operation in full wavefield inversion

A method for determining a discrete physical properties model of a subsurface region by iteratively inverting measured geophysical data acquired from the subsurface region, comprising using a Hessian matrix operating on an objective function, then times a vector, called Hessian times vector, to determine an update for an initial model, wherein the Hessian times vector is approximated, using a computer, with a single forward wave simulated propagation and a single gradient of the objective function computation in a modified subsurface model thereby requiring only three forward-wave or reverse-wave propagations, comprising:

• forming a scaled vector by multiplying the vector by a number ε, where 0<ε≦1, and adding the scaled vector to the initial model;
• simulating a first synthetic data set using the model with the scaled vector added to it;
• forming the objective function to measure misfit between a second synthetic data set, simulated using the initial model, and said first synthetic data set;
• computing a gradient of the formed objective function with respect to the physical property parameters, dividing the gradient by ε, and using that as an estimate of the Hessian times vector;
• computing a direction in model parameter space for an update to the initial 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 estimate of the 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 initial model to form an updated model, and using the updated model for geophysical prospecting.

A computer program product, comprising a non-transitory computer usable medium having a computer readable program code embodied therein, said computer readable program code adapted to be executed to implement 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, said method comprising:

• using a Hessian matrix of an objective function, then times a vector, called Hessian times vector, to determine an update for the model, wherein the Hessian times vector is approximated 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.