The problem considered in this paper is that of approximating a sample covariance matrix by one with a Toeplitz structure. The importance stems from the apparent sensitivity of spectral analysis on the linear structure of covariance statistics in conjunction with the fact that estimation error destroys the Toepliz pattern. The approximation is based on appropriate distance measures. To this end, we overview some of the common metrics and divergence measures which have been used for this purpose as well as introduce certain alternatives. In particular, the metric induced by Monge-Kantorovich transportation of the respective probability measures leads to an efficient linear matrix inequality (LMI) formulation of the approximation problem and relates to approximation in the Hellinger metric. We compare these with the maximum likelihood and the Burg method on a representative case study from the literature.