  We propose and test a new method based on Richardson-Lucy deconvolution to
reconstruct three-dimensional gas density and temperature distributions in
galaxy clusters from combined X-ray and thermal Sunyaev-Zel'dovich
observations. Clusters are assumed to be axially symmetric and arbitrarily
inclined with respect to the line-of-sight. No equilibrium assumption other
than local thermal equilibrium is needed. We test the algorithm with synthetic
observations of analytically modeled and numerically simulated galaxy clusters
and discuss the quality of the density and temperature reconstructions in
idealised situations and in presence of observational noise, deviations from
axial symmetry and cluster substructure. We find that analytic and numerical
gas density and temperature distributions can be accurately reconstructed in
three dimensions, even if observational noise is present. We also discuss
methods for determining the inclination angle from data and show that it can be
constrained using X-ray temperature maps. For a realistic cluster and including
observational noise the three-dimensional reconstructions reach a level of
accuracy of about 15%.
