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It is a straightforward approach having a set of coordinates (e.g., in 2D as {x,y}) and at least an associated variable (e.g., v) to calculate a variogram as a descriptor of the spatial dependency of the variable v through the field being studied.

enter image description here enter image description here

The question appeared to me is:
how to generate a realization of a dataset having a variogram? (inverse move!)
That is, there is at least one variogram available but neither dataset nor other description is available and the goal is to generate a realization of original (unknown) dataset that could have such a variogram.
What is the probability of having such a realization?

Updates / Comments: From variogram in the above context I mean empirical variogram. I suppose that fitting a variogram model is not an issue at least for this question. Also variogram is available as pairs (h, gamma).

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  • $\begingroup$ By variogram, you mean variogram model? $\endgroup$ – Paul Hiemstra Nov 30 '11 at 16:31
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You can use sequential simulation to generate realizations of a random field that has the covariance structure given in the variogram model. In R this can be done using gstat. See demo(ugsim) and demo(uisim) from R code examples from gstat.

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    $\begingroup$ +1 First, of course, you have to fit a variogram model to the empirical variogram shown. Note that geoR easily performs simulations too. $\endgroup$ – whuber Nov 30 '11 at 15:50
  • $\begingroup$ But I assume that the OP has a variogram already. Or else this would be a chicken and egg problem :). $\endgroup$ – Paul Hiemstra Nov 30 '11 at 16:07
  • $\begingroup$ The graphic in the question, Paul, shows an empirical variogram, not a variogram model. You cannot use an empirical variogram directly to drive a stochastic simulation (it wouldn't be positive definite, so the simulation would likely run into numerical problems). But there's no chicken/egg problem: one fits the variogram model to this empirical variogram and then proceeds. Some information is lost--a good variographic analysis considers much more than a single empirical variogram in vacuo --but that can't be helped here. $\endgroup$ – whuber Nov 30 '11 at 16:19
  • $\begingroup$ By chicken and egg I mean that without data to fit the variogram model, there is no variogram model to generate new data. And by variogram, I mean variogram model. $\endgroup$ – Paul Hiemstra Nov 30 '11 at 16:22
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    $\begingroup$ Part of my PhD was spent on automatic variogram fitting, so I think it is possible ;) $\endgroup$ – Paul Hiemstra Nov 30 '11 at 16:38

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