Digby
Some simplistic definitions which might help:
(o) "stationary" implies that every location in the study area has a potential statistical distribution with the same mean and standard deviation. That is, all your samples come from the same distribution.
(o) "intrinsic hypothesis" implies that the difference between two values taken at two different locations come from a distribution which depends only on the distance (and possibly relative direction) of the two locations. In short, differences are stationary.
Under the first scenario, you would have to have a sill at the overall population variance. However your sampled area may be too small for you to actually reach that theoretical sill. The variance of samples within a limited area will be seriously underestimated by statistical techniques because of the predominance of correlation
between the samples. So, you might get a semi-variogram which appears (italics) not to have a sill and a calculated variance which appears to come some way down that slope.
Under the second scenario, where data follow the intrinsic hypothesis are are not necessarily stationary: you will get a sill if the differences do tend towards a maximum over some specific distance. You may not get a sill at all -- either because of the above-mentioned or because there is no sill.
A simple example of an intrinsic situation without a sill. Think of a gas disseminating freely in 3D. Maybe too difficult, so use a 1D analogy.
Stand up in a fairly large empty space, with a coin and a pen&paper (or get a friend to help!). Take a coin and toss it. Heads take 1 step left. Tails take one step right. Note your location. You have two possibilities: 1 left, 1 right, equal probability.
Toss your coin again: step left or right according to coin. You now have 3 possibilities: 2 steps left, back to zero, 2 steps right. probabilities 1/4, 1/2, 1/4.
Repeat: 3 left, 1 left, 1 right, 3 right, probabilities 1/8,3/8,3/8,1/8
Repeat until bored. After (say) k steps you could be anywhere from k left (-k?) to k right (+k) with a binomial distribution of probabilities starting at 1/2^k. Each step has a completely different distribution of possibilities with the same mean (0) and a different standard deviation.
OK, you are probably getting bored by now. Let's look at the differences at one lag: for any one step the difference between where you are now and where you were a step ago is simply 1 left, 1 right, equal probabilities. 2 lags: 2 left, zero, 2 right. And so on.
If you calculated a semi-variogram on such data
you would get a straight line with no sill. Sample variance is meaningless in this context, because every sample has a different potantial variance. However, it is fiercely "intrinsic".
This "random walk" approach can be used to characterise other sill-less semi-variograms by varying the probability (say use a dice and only go left if it is 5 or 6). The coin-toss version is exactly analagous to Brownian motion of gasses or some disseminated mineralisations.
Sorry to be so long-winded. Hope this helps.
Isobel
Do stationary hypothesis variograms have a sill,
and intrisic hypothesis variograms have no sill.
Does this mean intrisic hypothesis variograms
indicate a trend.
+
+ To post a message to the list, send it to [email protected]
+ To unsubscribe, send email to majordomo@ jrc.it with no subject and "unsubscribe ai-geostats" in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list
+ As a general service to list users, please remember to post a summary of any useful responses to your questions.
+ Support to the forum can be found at http://www.ai-geostats.org/
