| Summary: | I present a new quasi-linear method for reconstructing cosmological density and velocity fields from all-sky redshift surveys. The method is used to reconstruct the velocity field, dipole, bulk flows and distortion parameter <i>b</i> = Ω<sup>0.6</sup>/<i>b </i>from the PSCz survey. Analytic expressions for the cosmic variance and shot noise uncertainties on the reconstructed velocity field are presented. It is found that the uncertainties are reduced if reconstruction is carried out in the Local Group frame. The uncertainty on the dipole is also found. A generalised version of the Path Interchange Zel'dovich Approximation (PIZA) is presented. PIZA is a simple Lagrangian reconstruction method based on the Zel'dovich Approximation and the Least Action Principle, which reconstructs cosmological fields given the present day real space positions of galaxies. The generalizations take account of redshift space distortions, incomplete sky coverage, and the selection function. The method can be used to estimate <i>b</i> from radial velocities, bulk flows and the dipole. Generalised PIZA has been tested using a set of PSCz-like simulations. The reconstructed radial peculiar velocity field is compared with that of the simulation and that reconstructed by linear theory. The generalized PIZA is applied to the <i>IRAS </i>PSCz Survey. The dipole, bulk velocities and peculiar velocity field, and the derived value of <i>b</i> are presented. The Local Group is found to have an average displacement of 1225<i>kms</i><sup>-1</sup> in the direction of (1.b)=(264°, 42°). From this it is found that <i>b</i> = 0.512 ± 0.141.
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