  Quasiparticle Interference (QPI) imaging is a powerful tool for the study of
the low energy electronic structure of quantum materials. However, the
measurement of QPI by scanning tunneling microscopy (STM) is restricted to
surfaces and is thus inherently constrained to two dimensions. This has proved
immensely successful for the study of materials that exhibit a
quasi-two-dimensional electronic structure, yet it raises questions about how
to interpret QPI in materials that have a highly three dimensional electronic
structure. In this paper we address this question and establish the methodology
required to simulate and understand QPI arising from three dimensional systems
as measured by STM. We calculate the continuum surface Green's function in the
presence of a defect, which captures the role of the surface and the vacuum
decay of the wave functions. We find that defects at different depths from the
surface will produce unique sets of scattering vectors for three dimensional
systems, which nevertheless can be related to the three-dimensional electronic
structure of the bulk material. We illustrate the consequences that the
three-dimensionality of the electronic structure has on the measured QPI for a
simple cubic nearest-neighbour tight-binding model, and then demonstrate
application to a real material using a realistic model for PbS. Our method
unlocks the use of QPI imaging for the study of quantum materials with three
dimensional electronic structures and introduces a framework to generically
account for $k_z$-dispersions within QPI simulations.
