By applying the transmissionpropagationdiffraction operator theory (TPDOT; in earlier publications abbreviated as TPOT) we have derived an analytical representation of the wavefield in two 3D inhomogeneous acoustic half-spaces separated by a piecewise-smooth interface. We scrutinize the conventional approach and reformulate the mechanical particle motion forward problem in terms of a novel wave amplitude form of counter-propagating wave events. The new wave motion statement is composed of two systems of integral equations describing propagation and transmission of the seismic wavefields. The first system represents the propagationdiffraction operator equation. We express the propagationdiffraction operator as the feasible Kirchhoff-type surface integral with a feasible fundamental solution in the kernel. The feasible fundamental solution generalizes the conventional free space Green's function for an arbitrary domain case. The second system represents the transmission (refraction/reflection) operator equation. This convolutional interface equation generalizes the SommerfeldWeyl spatial-frequency plane wave decomposition in an infinitesimally thin interface layer. We transform the wave amplitude formulation by Neumann iteration into infinite series of counter-propagating multiply transmitted (reflected/refracted) wave modes, representing an analytical solution to the problem. Since the wave motion statement of the forward problem enables analysis of separate wave modes of the total wavefield, it has an advantage over the particle motion statement. The suggested wavefield representation therefore solves both forward and inverse wave motion problems. Calculation of separate wave modes of TPDOT can be done by the software package of the tip-wave superposition method (TWSM) introduced in our earlier papers (in co-authorship). The proposed TPDOT provides the theoretical foundations for TWSM.