Geostatistics: Interpolation or extrapolation

Interpolate: to estimate values of (a function) between two known values. Extrapolate: to infer (values of a variable in an unobserved interval) from values within an observed interval.

To krige or not to krige?

That's not a question in Journelian geostatistics! Stanford's prominent geostatistician in 1992 proclaimed, "The very reason for geostatistics or spatial statistics in general is the acceptance (a decision rather) that spatially distributed data should be considered a priori as dependent one to another, unless proven otherwise (my emphases in bold!)". Did all geostatistical scholars agree with Journel's decision? After all, the a priori acceptance of continued mineralization between (independently) measured grades makes no scientific sense whatsoever. Assume spatial dependence, insert kriged estimates between measured grades, and compute kriging variances of kriged estimates. Surely, the geostatistical scholars must be joking!

And what's the difference between interpolation and extrapolation when Webster's values of a function and values within an observed interval are missing? Indeed, Darrell Huff (How to Lie with Statistics) and W J Reichmann (Use and Abuse of Statistics) would have taken the same dim view of interpolation by kriging between measured values when the ordered set does not display a significant degree of spatial dependence as they do of extrapolation beyond observed intervals. Nassim Nicholas Taleb's "Fooled by Randomness" makes mandatory reading for geostatistical scholars who thrashed randomness by kriging and rigging the rules of classical statistics.

Ironically, kriging creates an illusion of spatial dependence where it does not exist simply because kriged estimates are functionally dependent variables. Ignoring the requirement of functional independence demands a bit of irrational thinking to explain why kriging variances shrink. Here's how Armstrong's prickly pear problem is solved!

When Armstrong and Champigny noticed that kriging variances converge on zero, they issued the legendary caution against oversmoothing.. Did the authors really believe that the fundamental requirement of functional independence in classical statistics could be violated a little bit but not a lot? Despite all my complaints about the mind-boggling vagaries of geostatistics, CIM Bulletin did review and publish a paper on perfect smoothing!