Transient analytical point-source response of a layered acoustic medium; Part I

The exact transient responses (e.g., reflection or transmission responses) of a transient point source above a stack of parallel acoustic homogeneous layers between two half-spaces can be analytically obtained in the form of a finite integral strictly in the time domain. (The theory is presented in part II of this paper, this issue.) The transient acoustic potential of the point source is decomposed into transient plane waves, which are propagated through the layers at any angle of incidence as well in the time domain; finally, they are superposed to obtain the total point-source response. The theory dealing with transient analytic plane wave propagation is described here. It constitutes an essential part of computing the synthetic seismogram by the new transient method proposed in part II. The plane-wave propagation is achieved by an exact discrete recursion that automatically handles the conversion of homogeneous waves into inhomogeneous transient plane waves at layer boundaries. A particularly efficient algorithm is presented, that can be viewed as a natural extension of the popular normal-incidence Goupillaud (1961)-type algorithm to the nonnormal incidence case.