[GIS] Keeping unchanged observed values after kriging interpolation

kriging

Is there a way to force kriging to keep unchanged measured values of observation points?

For example: if the observed value of a location is 1, after the interpolation it could be 0.9 or even 0.5, depending on the other locations' values. I'd like to have 1 for that observed location after the prediction too.

I know that, considering the kriging formula, this is a stretch but predicting values where it is actually known seems to me a stretch as well. I found that the only prediction method that preserves observation points' values is IDW but this is not the best method to predict values for the other prediction locations.

I'm using R and the package "intamap".

I'm using the "automatic" method provided by "intamap" package. I need to use it for two reasons: the first is for the very strong skewness of my data and the second reason is that kriging interpolation is the final step of method I'm developing that is devoted to be user friendly for inexpert persons.

"Intamap" package claims to meet these two needs. The reason why I need to keep original data unchanged is that observation points represent the probability values of a phenomenon to occur. Pay attention, I'm not talking about indicator kriging, because observed values are just probabilities. So I need to provide inexpert users the geographical distribution of different degree of probabilities. If I have two observation points with 1 and 0.5 probability values, I need these values to be unchanged and that kriging interpolates the values in the prediction locations between them.

I hope to have been clear in my explanation. How can I proceed now?

Best Answer

Kriging is an exact interpolator. You don't have to do anything. Check your settings if it is not.

from wikipedia

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When you use "default values" you aren't really kriging, you're just applying the kriging algorithm--which as you have found, is poor when used with these data.

(I will step up on a soapbox for a brief rant: in my opinion, the fastest way to get bad results with a computer program is to accept its default parameters. ArcGIS is one of the richest, most powerful environments for getting bad results this way. The moral is do not use software for important work until you understand how to control it. Down from the soapbox now...)

For kriging to work you have to conduct an intensive preliminary statistical analysis of the data known as "variography." How well this ultimately performs depends on the data as well as your geostatistical skills. (Entire books have been written about variography, including the seminal Mining Geostatistics by Journel & Huijbregts and Variowin by Yvan Pannatier.) Although people have succesfully kriged as few as seven data points (in a monograph by Robert Jernigan published by the US EPA in the late 1980's), and in principle you can krige using just two or three points (I have done this to demonstrate the algorithm), rules of thumb in the literature range from a minimum of 20 points to 100 points and the consensus appears to be around 30 points.

In your case--although you do not describe the data--you have some clear problems, including a highly skewed distribution and a distinct lack of evidence of stationarity. These require special statistical treatment or specialized forms of kriging (such as a spatial generalized linear model). You will not get good results when kriging such data until you have a very large amount of data.

The legend suggests you might be trying to create a density grid rather than actually interpolate data: although the outputs of the two procedures may look the same, they do distinctly different things and have distinctly different interpretations. You interpolate when the data are considered samples from some hypothetical continuous surface. Interpolation predicts the unsampled values. Standard examples include elevation measurements (which sample the earth's surface) and temperature measurements (which sample a "temperature field"). You compute a density when you have complete information about the amount of something and you wish to represent a smoothed version of that amount per unit area. (In contrast with interpolation, there do not exist any unsampled values to predict.) The standard example is a population density: the data are counts of all individuals within an area; the output is a map of population density.

R Kriging – Fix Empty Interpolation Points with gstat Package in R

I don't really recommend doing kriging in lat-long degree coordinates. Project your data to a cartesian coordinate system and try again. For example, if I convert to 3857 (web mercator) then it works.

> int_pnts=st_transform(int_pnts, 3857)
> input_data_sf = st_transform(input_data_sf, 3857)
> lzn.vgm <- variogram(log(OM)~1, input_data_sf)
> lzn.fit <- fit.variogram(lzn.vgm, vgm('Mat'), fit.kappa = TRUE)

Some warnings we'll ignore, and then kriging produces meaningful results:

> lzn.kriged <- krige(log(OM) ~ 1, input_data_sf, int_pnts, model = lzn.fit)
[using ordinary kriging]
> head(lzn.kriged)
Simple feature collection with 6 features and 2 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: -10757430 ymin: 4738785 xmax: -10757310 ymax: 4738918
Projected CRS: WGS 84 / Pseudo-Mercator
  var1.pred   var1.var                  geometry
1 0.6545492 0.04198732 POINT (-10757334 4738803)
2 0.3821123 0.04326622 POINT (-10757376 4738785)
3 0.8089032 0.04263483 POINT (-10757315 4738882)
4 0.6558402 0.04288256 POINT (-10757384 4738896)

CRS 3857 is not the best choice, its just something that is likely to "work" in terms of "producing a result" in most places on the globe. You should maybe find a local coordinate system for your data, or use a UTM zone appropriate for your longitude.

Best Answer

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