A relative spectral sequence for topological Hochschild homology of spectra

In the situation where we have a sequence of two spectrum extensions, we describe a spectral sequence converging to the topological Hochschild homology of the top spectrum over the base spectrum, in terms of invariants of the intermediate extensions. The main application is to the case of topological Hochschild homology of a ring extension, where the base spectrum is taken to be the sphere spectrum. We get a spectral sequence starting with appropriate algebraic Tor groups of the topological Hochschild homology groups of a ring, converging to the topological Hochschild homology of an algebra over it.