  We develop a new method to estimate the redshift of galaxy clusters through
resolved images of the Sunyaev-Zel'dovich effect (SZE). Our method is based on
morphological observables which can be measured by actual and future SZE
experiments. We test the method with a set of high resolution hydrodynamical
simulations of galaxy clusters at different redshifts. Our method combines the
observables in a principal component analysis. After calibrating the method
with an independent redshift estimation for some of the clusters, we show -
using a Bayesian approach - how the method can give an estimate of the redshift
of the galaxy clusters. Although the error bars given by the morphological
redshift estimation are large, it should be useful for future SZE surveys where
thousands of clusters are expected to be detected; a first preselection of the
high redshift candidates could be done using our proposed morphological
redshift estimator. Although not considered in this work, our method should
also be useful to give an estimate of the redshift of clusters in X-ray and
optical surveys.
