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I am trying to implement a form of 2 stage least squares, in step 1 I ignored the spatial correlation between the observations, now in step 2 I look at the spatial correlation of the residuals of step 1. Observations are spread around in an area of 300 by 80 km, the minimum separation is 50 metres. The kriged map has to be on a 250 by 250 metre scale.

The variogram of the residuals is plotted below. Depending on the x-scale, I get two different pictures of the same variogram:

  • Plot one, max distance = 10 kilometres , nice variogram that can be fitted with an exponential model.
  • Plot two, same variogram but max distance = 50 kilometres, after the plateau from 2km to 10 km, the variogram starts rising again.

Is it still valid to use the fitted exponential variogram for ordinary kriging?

enter image description here

Extra info of the plot:

  • Observed variogram is in blue
  • Exponential model of the variogram in red
  • The grey lines are variograms of a monte carlo simulation where the observations are shuffled over space
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  • $\begingroup$ This question is difficult to understand because (1) you do not distinguish between experimental and model variograms and (2) it is impossible to tell what you mean by "unbounded." All exponential variogram models are bounded; all experimental variograms are bounded; everything in your graphs is clearly bounded. What exactly does "unbounded" mean then? When you clarify these points, please indicate more about your project. In particular, please tell us what the typical distances will be between prediction points and data points. $\endgroup$ – whuber Jan 4 '15 at 17:21
  • $\begingroup$ I guess I misunderstood the term unbounded. The title of the question is now more difficult, but I hope it is clear. $\endgroup$ – Kasper Jan 4 '15 at 17:42
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Ordinary kriging (OK) uses least squares methods to make predictions of values at unsampled locations. The model variogram provides information about the correlations among all locations, both sampled and unsampled.

It is well known that the results of OK depend most on the variogram values at the smallest lags (distances between locations), which also tend to be its smallest values (denoting greatest correlation).

The lack of local variation in the experimental variogram shown in the graphics suggests it is based on a very large number of data points. Over a 300 by 80 km region, that would translate to inter-point distances of a few kilometers or less. When kriging with local search neighborhoods, then, it would be unlikely to use points more than around 10 kilometers away.

Another consideration is that the experimental variogram appears to reach a "sill" quickly, by about 2000 meters (two kilometers). This indicates that almost all the improvement available by using OK will be accomplished in predicting values at points within 2000 meters or less of data points.


Recommendations

All these thoughts suggest there is little or nothing to be gained by attempting to fit the variogram model to data out to 50 km, and likely nothing is gained even by looking more than two to five km out. Instead, attention should be focused at obtaining the best possible fit of the model to the data within the first two km.

Given there are so much data, it would be worthwhile to construct directional model variograms rather than using a single (isotropic) variogram for this model. Another approach, when there are plenty of data, is to explore the possibility that the variogram changes across the study area--that is, to drop the assumption of stationarity.

Variography has great value as an exploratory tool. Exploration is carried out visually, in part by looking at plots of the experimental and model variograms. Unfortunately, the expanded vertical axes in these plots prevent close examination of either variogram. Limiting the upper values to about $0.015$ will triple the vertical resolution and make it much easier to see the data.

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