Is my RF stationary?

First of all, why did I choose such a topic, when I was planning to write on subjects related to the inverse problem? If we choose to solve the inverse problem deterministically, the concept of stationarity is irrelevant. However, the most recent advances in relation to the inverse problem have been stochastic and stationarity is a notion that must be tackled if we want to study stochastic processes.

Stationarity is a notion that I considered, until recently, exclusively as an hypothesis I had to assume as soon as I was considering applying a geostatistical method to my data. Some geostatistical methods explicitely treat non-stationarity such as kriging with a drift, but I will talk about those some other time.

I hadn’t really sat down to think about why this hypothesis was necessary. In geostatistics, it is a necessary hypothesis to be able to relate the covariance to the variogram, which comes down to say that if the variogram does not have a sill, you will not be able to calculate the covariance function from that variogram function. Stationarity, is not however, required to calculate the variogram itself. The only condition necessary to calculate the variogram is actually the “intrinsic hypothesis” which is defined as the second-order stationarity of the increments [Z(x)-Z(x+h)], where x is the location and h is the distance between the two data points, Z.

But the most important point that I have come upon regarding stationarity recently is this:

“Stationarity is a property of the RF model […] It is not a characteristic of the phenomenon under study. Stationarity is a decision made by the user, not a hypothesis that can be proven or refuted from data.” (1)

And if I understand this citation well, it stems from the fact, that to verify stationarity, we would need to be able to take multiple measurements at one same point. However, in the earth sciences, repeatability is generally not possible.

Many more things could be said about stationarity. If the notion interests you in any way, I suggest this small video about how ergodicity and wide and strict-sense stationarity are related.

(1) Goovaerts, Pierre (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press, New York.