Viscous attenuation can have a strong impact on seismic wave propagation, but is rarely taken into account in full waveform inversion (FWI). In the time domain, when viscoelasticity is considered, the displacement formulation of the wave equation is usually used. However, the adjoint state equations are quite different for the velocity-stress formulation than for the displacement formulation. In this paper, we derive the adjoint state equations for the viscoelastic wave equation based on the velocity-stress formulation. Using a modified definition of the memory variables commonly found in the literature, we define a Lagrangian from which the adjoint state equations and the misfit gradient are derived. The resulting expressions are similar to the displacement formulation, but differ by the source term and by the wavefield cross-correlations giving the misfit gradient. To validate our results, the misfit gradient obtained by the adjoint state method is compared to the misfit gradient calculated by finite-difference for a simplified problem, giving an excellent agreement. In short, this work gives the right adjoint state equations for the velocity-stress formulation, which is commonly used for time-domain viscoelastic modeling. Further studies are required to evaluate the performance of this approach in real FWI viscoelastic experiments.


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