Properties of Kriging
We started this chapter by examining the desirable properties of an estimator. We note that kriging takes into account the following elements:
1. The relative positions of the estimated blockV and sample locations xi,
through the termsγ ( xi,V )orγ ( xi
−
x0) in the case of estimation of a point.2. The distances between samples, through the termsγ ( x i
−
xj).3. The structure of the spatial variability peculiar to the mineralisation under consideration, i.e.γ
( )
h . The kriging weightsλ i aretailored to this modelledspatial continuity. Exact Interpolation Exact Interpolation
Unlike some estimators, for example trend surfaces, or simple linear regression, kriging is anexact interpolator . This means that when we estimate point values, kriging restores at the data points, the measured value. This is a property of the kriging, not anad hoc adjustment.
This property can be easily checked in the kriging system. If x 0
=
xi then thesolution is: λ λ i j j i
=
=
≠
1 0 and forIn fact, this corresponds with our intuitive notion of how a good estimator should behave. We can go back to first principles, and consider the squared error that we are attempting to minimise:
E Z x [ *() Z x() ]
0 0
2
−
This error is obviously minimised when the pointestimate at a sampled location is identical to the (known) sample value:
Z x *( ) Z x( )
0
=
0and this minimum is clearly equal to 0. This is one attribute of kriging that makes it especially suited for contouring applications, because the data points are honoured
exactly .
Note: This property is a property of point kriging . Point kriging simply replaces the point-block variogram values in matrix B, above, with a vector of (sample)point- (unknown)point variogram values. We do not expect block kriging estimates to ‘honour the data’ because we know (from the information effect) that some degree of smoothing isrequired to minimiseconditional bias.
Exactitude Exactitude
We do not expect block kriging estimates to ‘honour the data’ because we know (from the information effect) that some degree of smoothing is required to minimise conditional bias.
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Unique Solution Unique Solution
The kriging system always has aunique solution , providing that the variogram model used is positive definite. This is another reason why only admissible functions can be used for variogram models.
Kriging Systems Do Not Depend On the Data Values Kriging Systems Do Not Depend On the Data Values The kriging weights depend upon the data in the sense that the variogram model we choose is intimately linked to the histogram and spatial continuity of the
samples themselves; however, the kriging equations contain no reference to the data values themselves. This means that the set of weights we will obtain for a given sampling geometry and a specified variogram model are the same, regardless of the particular grades of the samples32.
For example, if we consider a zone with one defined variogram model and a regular rectangular sampling grid, there will be one set of kriging weights to derive (since each block will be informed by identically located samples, in a relative sense). Historically, this property could be used to vastly increase the efficiency and speed of grade control kriging. These advantages are not so striking with modern computers!
Because of this property, it is important that we take care to define the variogram properly and we must have enough data to be confident of theγ
( )
h model we select.Combining Kriging Estimates Combining Kriging Estimates
Theoretically, if we discretise a blockV into a very large number of points (say 100 or more) and perform a point kriging for each, the average of these point estimates equates to the block estimate. However, this process is very inefficient and is never used in practice. Using the point-block covariances (via the variogram model) allows us to much more efficiently obtain block kriged estimates.
Influence of the Nugget Effect on Kriging Weights Influence of the Nugget Effect on Kriging Weights
In the chapter on variography, we emphasised that the short scale behaviour of the variogram was critical in kriging. In particular, the nugget effect has a strong
influence on the kriging weights:
Screen Effect Screen Effect
If we have a small nugget effect, i.e. very continuousγ
( )
h , then the weighting will be heavily biased towards the block being estimated and its immediate neighbours. This is called thescreen effect , because the nearby samples are considered to ‘screen’ the outer samples from receiving significant weights when there is a small nugget effect. Again, this makes sense intuitively, because we would desire our estimator to give the closest samples the majority of weight in the case of pronounced spatial continuity. A strong screen effect means that the kriging is not very smoothed.32 This is true for IDW also. The difference is that the weights in IDW are arbritrary, whereas those for
OK are derived from data correlations. Discretising
Discretising
In IDW, discretisation involves estimation of an array of point values and recombination. In kriging the discretisation is used to calculate point-block variogram values. Block
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The Case of Low Nugget Effect, High Continuity The Case of Low Nugget Effect, High Continuity
If we have a more continuous model forγ
( )
h , for example in for a topographic variable, the closest samples provide all the information required to obtain a goodestimate.
In the extreme case of absolute (‘table-top’) continuity, the closest sample is sufficient, and polygons/nearest neighbour methods work well! In this case there is a very strong screen effect and the smoothing is complete.
The Case of High Nugget Effect, Low Continuity The Case of High Nugget Effect, Low Continuity
If we have a more discontinuous model forγ
( )
h, for example in some precious metal deposits, more and more distal samples will receive non-zero weights. In the extreme case of pure nugget effect, all samples receive equal weight.In fact, the case of pure nugget effect implies that no local estimation can be made: the best estimate of a blockV is the mean grade of the deposit. In this case there is a total absence of screen effect and the smoothing is complete.