A comparison of different scaling methods for least-squares inversion/migration

Wenyong Pan, Kristopher A. Innanen, Gary F. Margrave

The least-squares inverse problem, such as full waveform inversion and least-squares migration, can be performed iteratively using a gradient based method. While the convergence rate of this method is very slow for that it assumes the Hessian matrix as an identity matrix. The poorly scaled gradient can be enhanced considerably by multiplying the inverse Hessian matrix. Hessian matrix works as a nonstationary deconvolution operator to compensate the geometrical spreading effects and recover the deep reflectors amplitudes. It can also sharpen or focus the gradient by suppressing the multiple scattering effects and improve the resolution of the gradient. While direct calculation of the Hessian matrix is considered to be unfeasible in practical application for its expensively computational burden. Many Hessian approximations have been proposed to scale the gradient and improve the convergence rate of the gradient method. In this research, we compared different scaling methods in the least-squares inverse problem based on different Hessian approximations. The pseudo-Hessian, constructed by two virtual sources, can compensate the geometrical spreading effects obviously. While it is still not enough to balance the amplitude for that it ignores the receiver-side Green's functions. The Hessian approximation based on double illumination method, the linear and chirp phase encoded Hessian can balance the amplitude better for taking the receiver-side Green's functions into consideration. The chirp phase encoding method introduced in this research can approach the exact approximate Hessian better with the same number of simulations compared to the linear phase encoding method.