Help understanding the kriging formula

EliseosMucaki

Hello,

I was reading the documentation on kriging on the ArcGIS website, and I am having trouble understanding the difference between the two formulas presented on these separate pages discussing Kriging.

How Kriging works—Help | ArcGIS for Desktop  

What are the different kriging models?—Help | ArcGIS Desktop 

From my understanding, it seems like the first page describes the general kriging formula, while the second has more to do with autocorrelation. But it's not clear. I was hoping someone could try and clarify what the two formulas represent, or provide a source that does.

Thank you.

Eric6346

That particular formula is the definition of the kriging model.  The variable s always represents an arbitrary spatial location.  Z(s) is the measured value at location s. Since the goal is to interpolate, you want to predict Z(s) at every s where you didn't take a measurement.

The equation is saying that the measured value Z(s) is equal to the sum of a mean function µ(s) and a spatially autocorrelated error function ε(s). For your paper, I would write something along those lines.

The differences between the various kriging models usually depend on how these mean and error terms are defined and estimated. 

There is also a common convention in statistics that is very important but also very easy to miss if you don't know to look for it.  Look at the following two nearly identical models:

• Z(s) = µ(s) + ε(s)  [Universal kriging]
• Z(s) = µ + ε(s)      [Ordinary kriging]

The only difference in the model definition between universal and ordinary kriging is  µ versus µ(s).  Since s refers to a location and µ refers to the mean, the notation µ(s) indicates that the mean depends on the location.  Similarly, the notation µ indicates that the mean does not depend on the location; in other words, the mean is constant at every location.  This is very significant because a mean value that changes from location to location is usually called a trend.  The real difference between those two models is that the first supports a trend and the second doesn't, and all of this gets indicated by a tiny difference in notation.

But even once the model definition is understood, there are still formulas and equations for estimating the mean value or trend, more formulas for estimating the autocorrelated error term (this is where the semivariogram comes into play), more formulas for neighborhoods, predictions, transformations, etc.  It's a lot to unpack, but that's why the Geostatistical Analyst help is very long and spread across many topics.  

-Eric