  We present and test a new method for the reconstruction of cosmological
initial conditions from a full-sky galaxy catalogue. This method, called
ZTRACE, is based on a self-consistent solution of the growing mode of
gravitational instabilities according to the Zel'dovich approximation and
higher order in Lagrangian perturbation theory. Given the evolved
redshift-space density field, smoothed on some scale, ZTRACE finds via an
iterative procedure, an approximation to the initial density field for any
given set of cosmological parameters; real-space densities and peculiar
velocities are also reconstructed. The method is tested by applying it to
N-body simulations of an Einstein-de Sitter and an open cold dark matter
universe. It is shown that errors in the estimate of the density contrast
dominate the noise of the reconstruction. As a consequence, the reconstruction
of real space density and peculiar velocity fields using non-linear algorithms
is little improved over those based on linear theory. The use of a
mass-preserving adaptive smoothing, equivalent to a smoothing in Lagrangian
space, allows an unbiased (although noisy) reconstruction of initial
conditions, as long as the (linearly extrapolated) density contrast does not
exceed unity. The probability distribution function of the initial conditions
is recovered to high precision, even for Gaussian smoothing scales of ~ 5
Mpc/h, except for the tail at delta >~ 1. This result is insensitive to the
assumptions of the background cosmology.
