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I'd like to know if I understood correctly the following. In the fit.variogram method of the gstat library, there is a fit.method argument. In the documentation, it says that:

The default method uses weights $N_h/h^2$ with $N_h$ the number of point pairs and $h$ the distance. This criterion is not supported by theory, but by practice. For other values of fit.method, see table 4.2 in the gstat manual.

Let's say we have a variogram that includes sets of points separated by some distances $h: 100, 300, 600$ and $900$ and each $N_{h}$ contains 100, 1000, 1200, 900 points, respectively. Does the weighting method described above mean that the weights are calculated as $100/100, 1000/300, 1200/600, 900/900$. That is, we are placing less weight on points that are separated by larger distances. Is this correct? I know it says that this method is not supported by theory, but it seems a bit counterintuitive.

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The intuition behind it is that estimates with more point pairs (= more data) get more weight, and that estimates at smaller lags get more weight (focus on behavior near origin).

In case you'd have a linear variogram model without sill, this criterion would correspond to the Nh/(gamma(h)^2) of Cressie's classic "Fitting variogram models by weighted least squares" Math Geol paper. The problem with using this criterion in general is that during the iteration, the weights would change because they depend on the parameter vector, leading to pretty weird effects.

In your description, you forgot to square the h values, so weights are 100/(100^2), 1000/(300^2), etc.

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