Aligned but Irregularly Spaced Data

To construct the experimental semi-variogram in the direction of alignment ${/boma /alpha}$, the data are grouped into distance classes. Every data pair which is separated by a distance $[r /pm /varepsilon(r)]$ is used to estimate the value $/gamma(r)$.

In practice, it is easier to consider a constant tolerance $/varepsilon(r)$ whatever the distance $r$, i.e., $/varepsilon(r) =$ constant, $/forall r$. Otherwise, $/varepsilon(r)$ should be smaller for smaller distances and larger for greater distances.

This grouping of data pairs into distance classes causes a smoothing of the experimental semi-variogram $/hat/gamma(r)$ relative to the underlying theoretical semi-variogram $/gamma(r)$, cf. Figure 4.12. If the $n$ available data pairs separated by a distance $r_i /in [r /pm /varepsilon(r)]$ are used instead of those separated by the strict distance $r$, then it is not $/gamma(r)$ that is being estimated but rather a linear combination of the $/gamma(r_i)$; more precisely, the theoretical mean value

Figure 4.12: Smoothing Effect when Estimating a Variogram.

The significance of the effect of this smoothing over the interval $[r /pm /varepsilon(r)]$ decreases as the tolerance $/varepsilon(r)$ becomes smaller with respect to the range of the theoretical model $/gamma(r)$ to be estimated.

In practice, the following points should be heeded.

(i)

Take the pseudo-periodicities of the data locations into account when defining the tolerances $/varepsilon(r)$ of the classes.

(ii)

Ensure that the section of interest of the variogram (e.g., the increase to the sill value) has been calculated using at least three or four classes; thus, the tolerance $/varepsilon(r)$ should not be too large.

(iii)

Ensure that each distance class $[r /pm /varepsilon(r)]$ contains enough pairs so that the corresponding estimator of the variogram is reliable; thus, the tolerance $/varepsilon(r)$ should not be too small.

(iv)

Detect any risk of bias due to preferential location of data.