Seismic modeling using a first-order acoustic wave equation with vector reflectivity in isotropic and anisotropic media

We have developed a new first-order acoustic wave equation parameterized by velocity and vector reflectivity for isotropic and anisotropic media. This equation has been contrasted with the first-order wave equation using the variables velocity and density and we have demonstrated the equivalence between the two equations. The full acoustic seismic wavefield can be generated without explicit knowledge of density if an estimate of reflectivity is known. To numerically solve the proposed equation we have employed a scheme derived from the Lie product formula where the time evolution operator of the analytic solution is written as a product of exponential matrices, and each exponential matrix term is approximated by the Taylor series expansion. Moreover, we have presented numerical results demonstrating the equivalence between the two equations for a known earth model. In addition, we have also shown how the numerical solution of the proposed equation has allowed a straightforward implementation of perfectly matched layer (PML) absorbing boundary condition. To demonstrate the efficiency and applicability of the PML scheme we also have compared the results of numerical modeling using PML with the results obtained by the second-order wave equation parameterized by velocity and vector reflectivity with traditional attenuation absorbing boundary condition.