Kriging values greater than the most extreme value of the input layer?

Question

I'll admit I don't understand kriging to the fullest. Yet, I've read that the weights should add up to 1 as in IDW and Natural Neighbors.

However, surrounding the biggest elevation points of my vector layer, I get some raster cells with greater values.

Answer 1

There are different types of kriging, which make different assumptions.

Simple kriging assumes a known mean for the total extent and ordinary kriging assume a local mean defined within a radius. With those you should not predict larger values than your largest observed value.

Universal kriging assumes a general polynomial trend model. The polynomial might fit with out of range value (kriging works then on the residues)

Answer 2

Simple kriging may stay inside the data range, depending on the simple kriging mean, but ordinary kriging typically will not:

> library(gstat)
> library(sp)
> p = SpatialPoints(cbind(c(0,0,0,0), 1:4))
> p$z = c(0,1,1,0)
> p
  coordinates z
1      (0, 1) 0
2      (0, 2) 1
3      (0, 3) 1
4      (0, 4) 0
> krige(z~1, p, SpatialPoints(cbind(0,2.5)), vgm(1, "Sph", 2))
[using ordinary kriging]
  coordinates var1.pred  var1.var
1    (0, 2.5)  1.055556 0.3880208
> krige(z~1, p, SpatialPoints(cbind(0,2.5)), vgm(1, "Sph", 2), beta = 0)
[using simple kriging]
  coordinates var1.pred  var1.var
1    (0, 2.5) 0.9975884 0.3807338
> krige(z~1, p, SpatialPoints(cbind(0,2.5)), vgm(1, "Sph", 2), beta = .5)
[using simple kriging]
  coordinates var1.pred  var1.var
1    (0, 2.5)   1.06873 0.3807338

the reason for ordinary kriging to go beyond the data range is that it has to estimate the mean effect from the data, and hence (some) kriging weights become negative.