Isotropy

Isotropy is a term used in spatial statistical analysis to describe the property of a spatial process that produces outcomes that are independent of direction in space. In science and technology, an isotropic phenomenon is one that appears the same in all directions. For example, a material such as a fabric or sheet of paper that has the same strength in all directions would be said to be isotropic, as would a universe that from the point of view of a given observer appeared the same in any direction.

In GIS, the word is used to describe a spatial process that exhibits similar characteristics in all directions. In spatial statistical analysis, the second-order variation, the variation that results from interactions between the spatially distributed objects or variables, is usually modeled as a spatially stationary process. Informally, such a process is said to be stationary if its statistical properties do not depend upon its absolute location in space, usually denoted by the vector of coordinates. This also implies that the covariance between any two values of a variable y measured at any two locations, s1 and s2, depends only on the distance and direction of their separation and not on their absolute locations. For the process to be isotropic, we carry the argument further by assuming that this covariance depends solely on the distance separating s1 and s2. It follows that an isotropic process is one that not only exhibits stationarity, but in which the covariance depends only upon the distance between the objects, or data values, and does not at all depend upon direction.

[Page 245]An alternative way of illustrating the same idea is to say that the process is invariant under any rotations in geographic space around some point of origin. If there is directional dependence, the process is said to be anisotropic. In geostatistics, and especially in spatial interpolation by kriging, isotropy is often assumed when the experimental semivariogram is computed and then modeled. However, provided there are sufficient data, it is also possible to compute, model, and use different semivariograms for different directions of variation, thereby taking anisotropy into account.