Wiener Reconstruction of the Large-Scale Structure

The formalism of Wiener filtering is developed here for the purpose of reconstructing the large-scale structure of the universe from noisy, sparse, and incomplete data. The method is based on a linear minimum variance solution, given data and an assumed prior model which specifies the covariance matrix of the field to be reconstructed. While earlier applications of the Wiener filer have focused on estimation, namely suppressing the noise in the measured quantities, we extend the method here to perform both prediction and dynamical reconstruction. The Wiener filter is used to predict the values of unmeasured quantities, such as the density field in unsampled regions of space, or to deconvolve blurred data. The method is developed, within the context of linear gravitational instability theory, to perform dynamical reconstruction of one field which is dynamically related to some other observed field. This is the case, for example, in the reconstruction of the real space galaxy distribution from its redshift distribution or the prediction of the radial velocity field from the observed density field. When the field to be reconstructed is a Gaussian random field, such as the primordial perturbation field predicted by the canonical model of cosmology, the Wiener filter can be pushed to its fullest potential. In such a case the Wiener estimator coincides with the Bayesian estimator designed to maximize the posterior probability. The Wiener filter can be also derived by assuming a quadratic regularization function, in analogy with the "maximum entropy" method. The mean field obtained by the minimal variance solution can be supplemented with constrained realizations of the Gaussian field to create random realizations of the residual from the mean.