The empirical variogram is a standard tool in the investigation and modelling of spatial
covariance. However, its properties can be difficult to identify and exploit in the
context of exploring the characteristics of individual datasets. This is particularly true
when seeking to move beyond description towards inferential statements about the
structure of the spatial covariance which may be present. A robust form of empirical
variogram based on a fourth-root transformation is used. This takes advantage of the
normal approximation which gives an excellent description of the variation exhibited
on this scale. Calculations of mean, variance and covariance of the binned empirical
variogram then allow useful computations such as confidence intervals to be added to
the underlying estimator. The comparison of variograms for different datasets provides
an illustration of this. The suitability of simplifying assumptions such as isotropy and
stationarity can then also be investigated through the construction of appropriate test
statistics and the distributional calculations required in the associated p-values can be
performed through quadratic form methods. Examples of the use of these methods in
assessing the form of spatial covariance present in datasets are shown, both through
hypothesis tests and in graphical form. A simulation study explores the properties of
the tests while pollution data on mosses in Galicia (North-West Spain) are used to
provide a real data illustration
