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We develop a numerical solver for three-dimensional wave propagation incoupled poroelastic-elastic media, based on a high-order discontinuous Galerkin(DG) method, with the Biot poroelastic wave equation formulated as a firstorder conservative velocity/strain hyperbolic system. To derive an upwindnumerical flux, we find an exact solution to the Riemann problem, including theporoelastic-elastic interface; we also consider attenuation mechanisms both inBiot's low- and high-frequency regimes. Using either a low-storage explicit orimplicit-explicit (IMEX) Runge-Kutta scheme, according to the stiffness of theproblem, we study the convergence properties of the proposed DG scheme andverify its numerical accuracy. In the Biot low frequency case, the wave can behighly dissipative for small permeabilities; here, numerical errors associatedwith the dissipation terms appear to dominate those arising from discretisationof the main hyperbolic system.  We then implement the adjoint method for this formulation of Biot's equation.In contrast with the usual second order formulation of the Biot equation, weare not dealing with a self-adjoint system but, with an appropriate innerproduct, the adjoint may be identified with a non-conservative velocity/stressformulation of the Biot equation. We derive dual fluxes for the adjoint andpresent a simple but illuminating example of the application of the adjointmethod.

A Discontinuous Galerkin method for three-dimensional elastic and poroelastic wave propagation: forward and adjoint problems