Full Waveform Inversion of Viscoelastic Media Based on Gradient Preconditioning

In viscoelastic media, reverse time migration (RTM) requires accurate $Q$ value for energy compensation and phase correction. Full waveform inversion (FWI) has the potential to provide accurate $Q$ value. In the wide frequency band seismic data, the quality factor $Q$ almost does not change with frequency, which is called constant $Q$ model, but its computational complexity is large. In the narrow frequency band data, when the memory variable order $L$ is assumed to be 1 in the standard linear solid (SLS) model, the attenuation simulation effect of the subsurface $Q$ value is acceptable, and the calculation amount is small. In this article, we derive the back-propagation wavefield equation and gradient formula of FWI of viscoelastic medium (QEFWI) based on the SLS model with memory variable order $L =$ 1 and analyze the effectiveness and feasibility of this method. In viscoelastic medium, low velocity body will produce low-frequency noise in seismic record, which will lead to heterogeneous gradient energy. This low-frequency noise is similar to the low-frequency noise produced near the source of shallow strata. This will reduce the accuracy of the inversion results. The gradient preconditioning method using the pseudo-Hessian operator can make the gradient energy more balanced. Therefore, we introduce the pseudo-Hessian operator in gradient preconditioning and derives the gradient preconditioning formula for the FWI in viscoelastic media, which solves the problem of the nonuniform gradient energy. Examples show that the method can suppress the nonuniform gradient energy in the FWI of viscoelastic media.