  This paper concerns the Time-Domain Full Waveform Inversion (FWI) for
dispersive and dissipative poroelastic materials. The forward problem is an
initial boundary value problem (IBVP) of the poroelastic equations with a
memory term; the FWI is formulated as a minimization problem of a least-square
misfit function with the (IBVP) as the constraint. In this paper, we derive the
adjoint problem of this minimization problem, whose solution can be applied to
computed the direction of steepest descent in the iterative process for
minimization. The adjoint problem has a similar numerical structure as the
forward problem and hence can be solved by the same numerical solver. Because
the tracking of the energy evolution plays an important role in the FWI for
dissipative and dispersive equations, the energy analysis of the forward system
is also carried out in this paper.
