  The paper overviews and investigates several nonparametric methods of
estimating covariograms. It provides a unified approach and notation to compare
the main approaches used in applied research. The primary focus is on methods
that utilise the actual values of observations, rather than their ranks. We
concentrate on such desirable properties of covariograms as bias,
positive-definiteness and behaviour at large distances. The paper discusses
several theoretical properties and demonstrates some surprising drawbacks of
well-known estimators. Numerical studies provide a comparison of
representatives from different methods using various metrics. The results
provide important insight and guidance for practitioners who use estimated
covariograms in various applications, including kriging, monitoring network
optimisation, cross-validation, and other related tasks.
