Full Waveform Inversion (FWI) has been widely studied in recent years but still cannot be practiced in industry effectively. Generally, its failure can be attributed to expensively computational cost, slow convergence rate, cycle skipping problem and so on. For traditional FWI, the gradient is calculated shot by shot based on the adjoint state method. The computational burden rises significantly when considering large 2D velocity model or 3D experiments. A linear phase encoding strategy is employed to construct the gradient in the time-ray parameter domain. The phase encoding approach forms supergathers by summing densely distributed individual shots and can reduce the computational burden considerably. Furthermore, we propose the gradient be calculated using one single ray parameter per FWI iteration, with the ray parameter value varied for different iterations. The computational cost is reduced further within this strategy. The gradient is a poorly scaled image which can be considerably enhanced by multiplying the inverse Hessian. The Hessian matrix serves as a nonstationary deconvolution operator to compensate the geometrical spreading effects and suppress the multiple scattering effects. While explicit calculation of the gradient is also considered to be unfeasible. Under the assumption of high frequency limitation, the diagonal Hessian can work as a good approximation and it can also be constructed by the phase encoding method. In this research, preconditioning the gradient using the diagonal part of the phase encoded Hessian forms one pseudo Gauss-Newton step. Several numerical examples are presented to analyze the gradient contributions and compare different Hessian approximations. Finally, a modified Marmousi model is illustrated for full waveform inversion. we compared the effects for fixed ray parameter and varied ray parameter and analyzed the sensitivity to the ray parameter range, sensitivity to the Gaussian noise and sensitivity to the number of encoded sources. And the inversion results with different scaling methods are also provided for comparison.
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