The VARIOGRAM Procedure


Semivariance is defined on the basis of the spatial increment vector $\bm {h}$. If the variance characteristics of $Z(\bm {s})$ are independent of the spatial direction, then $Z(\bm {s})$ is called isotropic; if not, then $Z(\bm {s})$ is called anisotropic. In the case of isotropy, the semivariogram depends only on the length h of $\bm {h}$ and $\gamma _ z(\bm {h}) = \gamma _ z(h)$. Anisotropy is characterized as geometric, when the range $a_0$ of the semivariogram varies in different directions, and zonal, when the semivariogram sill $c_0$ depends on the spatial direction. Either type or both types of anisotropy can be present.

In the more general case, an SRF can be anisotropic. For an accurate characterization of the spatial structure it is necessary to perform individual analyses in multiple directions. Goovaerts (1997, p. 98) suggests an initial investigation in at least one direction more than the working spatial dimensions—for example, at least three different directions in $\mathcal{R}^2$. Olea (2006) supports exploring as many directions as possible when the data set allows.

You might not know in advance whether you have anisotropy or not. If the semivariogram characteristics remain unchanged in different directions, then you assume the SRF is isotropic. If your directional analysis reveals anisotropic behavior in particular directions, then you proceed to focus your analysis on these directions. For example, in an anisotropic SRF in $\mathcal{R}^2$ you should expect to find two distinct directions where you observe the major axis and the minor axis of anisotropy. Typically, these two directions are perpendicular, although they might be at other than right angles when zonal anisotropy is present.

If you can distinguish a maximum and a minimum sill in different directions, then you have a case of zonal anisotropy. The SRF exhibits strongest continuity in the direction of the lowest sill, which is the direction of the major anisotropy axis. If the sill does not change across directions, then the major axis direction of strongest continuity is the one in which the semivariogram has maximum range. See An Anisotropic Case Study with Surface Trend in the Data for a detailed demonstration of a case with anisotropy when you use PROC VARIOGRAM.

You can find additional information about anisotropy analysis in the section Anisotropic Models in Chapter 55: The KRIGE2D Procedure.