  Future high resolution, high sensitivity Sunyaev-Zeldovich (SZ) observations
of individual clusters will provide an exciting opportunity to answer specific
questions about the dynamical state of the intra-cluster medium (ICM). In this
paper we develop a new method that clearly shows the connection of the SZ
signal with the underlying cluster model. We include relativistic temperature
and kinematic corrections in the single-scattering approximation, allowing
studies of hot clusters. In our approach, particular moments of the temperature
and velocity field along the line-of-sight determine the precise spectral shape
and morphology of the SZ signal. We illustrate how to apply our method to
different cluster models, highlighting parameter degeneracies and instrumental
effects that are important for interpreting future high-resolution SZ data. Our
analysis shows that line-of-sight temperature variations can introduce
significant biases in the derived SZ temperature and peculiar velocity. We
furthermore discuss how the position of the SZ null is affected by the
cluster's temperature and velocity structure. Our computations indicate that
the SZ signal around the null alone is rather insensitive to different cluster
models and that high frequency channels add a large leverage in this respect.
We also apply our method to recent high sensitivity SZ data of the Bullet
cluster, showing how the results can be linked to line-of-sight variations in
the electron temperature. The tools developed here as part of SZpack should be
useful for analyzing high-resolution SZ data and computing SZ maps from
simulated clusters.
