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While studying how to develop a simple kriging model with a linear semivariogram, the various tutorials point towards creating a covariogram using $\sigma(h) = \sigma(0) - \gamma(h)$, but the value of $\sigma(0) \to \infty$ (i.e. the sill of the semivarigram) for a linear semivariogram.

So, how does one go about proceeding with a simple kriging model using a linear semivariogram (the variogram has been constructed by me using regression)? (i.e. How does one go about evaluating the covarigram using the above equations or does one use a different set of equations for Kriging with linear semivariograms?) Thanks in advance!

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  • $\begingroup$ The nature of this question is not apparent. Is it about estimating the slope and nugget of the variogram, cross-validating the variogram, choosing appropriate search procedures, setting up the kriging equations, solving the kriging equations, or something else? Please edit this post to explain what you mean by "developing." $\endgroup$
    – whuber ♦
    Commented Jun 1, 2015 at 21:16
  • $\begingroup$ @whuber Edited the question. I have evaluated the linear variogram (slope, nugget = $\gamma (0)$). But with the evaluated variogram, I am not able to find the sill to be used for evaluating the covariogram (mentioned in the question); as a result of which I can not find the value of $\sigma(0)$, which is the sill (it $\to \infty$ for linear semivariograms). $\endgroup$
    – Vishal Anand
    Commented Jun 1, 2015 at 21:33
  • $\begingroup$ There is no sill: that is the entire point of using variograms instead of covariance functions. In the absence of any nugget effect or measurement error, necessarily $\gamma(0)=0$ and the variogram has the form $\gamma(h)=\rho h$ for some positive $\rho$. $\endgroup$
    – whuber ♦
    Commented Jun 1, 2015 at 21:40
  • $\begingroup$ @whuber That is very true. And hence I put up this question, since the Kriging tutorials ask to create a semivariogram and then construct a covariogram Page6, after which the covariance matrix is constructed. When a new point is to be interpolated(kriged), this matrix is used for finding the value at the given point. Thus, how does one calculate the $\sigma (0)$ for the covariogram computation. I could not search for any online references to this; I will be really thankful for your answer to this. $\endgroup$
    – Vishal Anand
    Commented Jun 1, 2015 at 21:53
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    $\begingroup$ It's just like the Ordinary Kriging system but without the Lagrange multipliers. Much of the online material on kriging is awful--you have to search hard to find anything good--so it's usually better to consult one of the more reputable textbooks. The material from NKU at ceadserv1.nku.edu/longa//modules/geostats/lec/latex2html/… seems good, though. $\endgroup$
    – whuber ♦
    Commented Jun 1, 2015 at 21:58

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The linear variogram does not have a covariance equivalence, because it continues increasing to infinity. If you need to work with the linear variograms (or with variograms that have no sill in general), use ordinary kriging.

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