A time-domain numerical modeling of transversely isotropic Biot ă poroelastic waves is proposed in two dimensions. The viscous dissipation ă occurring in the pores is described using the dynamic permeability model ă developed by Johnson-Koplik-Dashen (JKD). Some of the coefficients in ă the Biot-JKD model are proportional to the square root of the frequency. ă In the time-domain, these coefficients introduce shifted fractional ă derivatives of order 1/2, involving a convolution product. Based on a ă diffusive representation, the convolution kernel is replaced by a finite ă number of memory variables that satisfy local-in-time ordinary ă differential equations, resulting in the Biot-DA (diffusive ă approximation) model. The properties of both the Biot-JKD and the ă Biot-DA models are analyzed: hyperbolicity, decrease of energy, ă dispersion. To determine the coefficients of the diffusive ă approximation, two approaches are analyzed: Gaussian quadratures and ă optimization methods in the frequency range of interest. The nonlinear ă optimization is shown to be the better way of determination. A splitting ă strategy is then applied to approximate numerically the Biot-DA ă equations. The propagative part is discretized using a fourth-order ADER ă scheme on a Cartesian grid, whereas the diffusive part is solved ă exactly. An immersed interface method is implemented to take into ă account heterogeneous media on a Cartesian grid and to discretize the ă jump conditions at interfaces. Numerical experiments are presented. ă Comparisons with analytical solutions show the efficiency and the ă accuracy of the approach, and some numerical experiments are performed ă to investigate wave phenomena in complex media, such as multiple ă scattering across a set of random scatterers. (C) 2014 Elsevier Inc. All ă rights reserved.