Unbiased reconstruction of the large-scale structure

We present a new unbiased minimal variance (UMV) estimator for the purpose of reconstructing the large-scale structure of the Universe from noisy, sparse and incomplete data. Similar to the Wiener filter (WF), the UMV estimator is derived by requiring the linear minimal variance solution given the data and an assumed a priori model specifying the underlying field covariance matrix. However, unlike the WF, the minimization is carried out with the added constraint of an unbiased reconstructed mean field. The new estimator does not necessitate a noise model to estimate the underlying field; however, such a model is required for evaluating the errors at each point in space. The general application of the UMV estimator is to predict the values of the reconstructed field in unsampled regions of space (e.g. interpolation in the unobserved Zone of Avoidance), and to dynamically transform from one measured field to another (e.g. inversion of radial peculiar velocities to over-densities). Here, we provide two very simple applications of the method. The first is to recover a 1D signal from noisy, convolved data with gaps, for example CMB time-ordered data. The second application is a reconstruction of the density and 3D peculiar velocity fields from mock SEcat galaxy peculiar velocity catalogues.