Finite-frequency kernels for wave propagation in porous media based upon adjoint methods



We determine finite-frequency kernels for wave propagation in porous media based upon adjoint methods. These sensitivity kernels may be obtained based upon two numerical simulations for each source: one calculation for the current model and a second, ‘adjoint’, calculation that uses time-reversed signals at the receivers as simultaneous, fictitious sources. The adjoint equations for a porous medium are identical to the usual Biot equations, with the exception of the adjoint source term, which involves time-reversed measurements of the differences between simulations and data. Isotropic poroelastic wave propagation is governed by eight primary model parameters which appear in the original Biot equations. These parameters include four moduli: the shear modulus of the frame plus three bulk moduli. We consider two alternative parameterizations: one involving density-normalized moduli corresponding to squared wave speeds, and a second involving the poroelastic shear and compressional wave speeds. The alternative parameterizations lead to density-sensitivity kernels that are small, reflecting the fact that the traveltime of a poroelastic wave is governed by wave speed, not density. We systematically investigate and illustrate the sensitivity of the fast and slow compressional waves and the shear wave to the poroelastic model parameters using a 2-D spectral-element method. The poroelastic sensitivity kernels presented herein form the basis of tomographic imaging and inversion in porous media.