Modeling of acoustic pulse propagation in nonideal fluids requires the inclusion of attenuation and its causal companion, dispersion. For the case of propagation in a linear, unbounded medium Szabo developed a convolutional propagation operator which, when introduced into the linear wave equation, accounts for attenuation and causal dispersion for any medium whose attenuation possesses a generalized Fourier transform. Utilizing a one dimensional Finite Difference Time Domain (FDTD) model Norton and Novarini showed that for an unbounded isotropic medium, the inclusion of this unique form of the convolutional propagation operator into the wave equation correctly carries the information of attenuation and dispersion into the time domain. This paper addresses the question whether or not the operator can be used as a basic building block for pulse propagation in a spatially dependent dispersive environment. The operator is therefore used to model 2-D pulse propagation in the presence of an interface separating two dispersive media. This represents the simplest description of a spatially dependent dispersive media. It was found that the transmitted and backscattered fields are in excellent agreement with theoretical expectations demonstrating the effectiveness of the local operator to model the field in spatially dependent dispersive media. [Work supported by ONR/NRL.]