Full Waveform Inversion (FWI) employs full waveform information to inverse the sub-surface properties through a iterative process. This method has been widely studied in recent years but still cannot be practiced effectively in industry. FWI with a steepest descent method assumes the Hessian matrix as an identity matrix and suffers from slow convergence rate. The Hessian matrix can compensate the geometrical spreading effects and then improve the convergence rate of FWI. While calculating the inverse Hessian matrix directly is thought to be unfeasible because of its extensive computational cost. Even though the researchers have developed various methods to approximate the full Hessian matrix, this problem remains to be addressed. It is known that FWI and Reverse Time Migration (RTM) share the same algorithmic structure and the gradient calculation in FWI is formally identical to a RTM image with a cross-correlation imaging condition. In this research, we found that auto-correlation of the forward modeling wavefields, namely, the source illumination is actually equivalent to the diagonal part of the pseudo-Hessian. And the gradient scaled by the auto-correlation of the forward modeling wavefields is equivalent to a RTM image based on the deconvolution imaging condition. Furthermore, deconvolution imaging condition based gradient is much more close to the reflectivity. Hence, it is possible for us to estimate the model perturbation through the traditional impedance inversion method. Combining FWI and traditional impedance inversion forms the Iterative Modelling Migration and Inversion (IMMI) method by Margrave et al.(2012). Finally, we practiced this strategy on a portion of Marmousi model and the phase encoding method was introduced to construct the gradient and diagonal pseudo-Hessian. And the iteration-dependent ray parameter setting strategy in the iterative process has also been involved to reduce the computational burden and balance the update.
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