Predicting the dynamic responses of structures and their interactions through the soil, whether embedded in or resting on unbounded media, requires advanced and efficient methodologies. This contribution presents an enhanced Integral Transform Method (ITM) that provides a semi-analytical solution for an elastic half-space with a spherical cavity in the wavenumber-frequency domain. A fully curvilinear description establishes a common azimuthal coordinate, enabling the direct coupling of angular wavenumbers. This permits parallel computation, significantly reducing computational effort. Additionally, it eliminates numerical artifacts caused by the discrete Fourier transform used in the original formulation. The resulting wavenumber stiffnesses at each boundary are coupled with the Finite Element Method (FEM) to model arbitrarily complex and spatially limited structures. The method is tested by comparison with the semi-analytical solution of a half-space in Cartesian coordinates. Furthermore, computational efficiency and numerical disturbances are discussed and compared with the original ITM.