In this paper the time domain scattering problem of a stratified dissipative half-space excited by a point source is considered. Both the direct and inverse scattering problem are treated. The dissipative wave equation is reduced to an equation in one spatial dimension by spatial Fourier series and a Hankel transform. The transformed wave equation is then solved by a combination of a time domain wave-splitting technique and a Green function technique. The Green functions represent the internal fields inside the stratified medium. The splitting is expressed in terms of a Neumann operator K which has an explicit expression as a convolution integral with a Bessel function kernel. Based upon this wave splitting, Green functions can be defined and a set of first order partial differential equations for Green functions are derived. The equations are well suited for a numerical treatment and a number of numerical examples for both the direct and inverse problems are presented. A brief discussion of the invariant imbedding method applied to the direct and inverse scattering problem is also given. It is shown that one-sided reflection data are sufficient to reconstruct the phase velocity and the dissipation coefficient simultaneously by using two different values of the Hankel transform parameter.