Hi Isobel
I didn't know the existence of the two schools of thought! So thanks
for the clarification.
The point is the I interpret smoothing (filtering) properties of
kriging by means of the dual representation of kriging
interpolator given for example in Goovaerts's book "Geostatistics for
natural resources evaluation".
According to this representation an estimate is equal to the mean
plus a linear combination of covariance terms (nugget
effect included). Clearly according to this representation if you
want to filter out nugget effect,
making your interpolator no more exact, you need to filter out the
nugget explicitly.
Then, I'm wondering if in presence of really short range components
of spatial continuity (compared to
the overall range of spatial continuity) It could be better to use a block
kriging (with a block size equal to range of the short range
components) in order to get more spatially homogeneous
maps from the point of view of smoothing (above all with high
clustered and low density data sampling geometry).
Sebastiano
At 14.35 20/02/2008, you wrote:
Just to clarify the point about the nugget effect.
The nugget effect reflects the uncertainty (or randomness) which
cannot be removed from the phenomenon. No matter how closely you
sample, there will generally be some difference between values. I
have had a couple of chances to be involved with 'contiguous'
sampling exercises, where samples are taken side-by-side.
Part of the nugget effect is the inherent variability of the
phenomenon being measured. The nugget effect will also include
'random' errors introduced by the sampling and/or analysis of the samples.
The value allocated to the nugget effect is the intercept of your
semi-variogram model on the axis -- if you let it hit the axis (zero
distance).
There seem to be two schools of thought on what value the
semi-variogram actually takes at zero distance:
(1) the nugget effect exists at all distances, except zero. This is
the basis on which Matheron originally developed his theory of
regionalised variables. At zero distance (exactly) the
semi-variogram is zero. At all other distances, the nugget effect is
included in the model.
(2) the nugget effect exists at zero distance. The implication of
this is that the nugget effect is all "sampling error".
Software packages vary according to which of the above they
implement. My personal preference is for the former mainly because
it gives (strangely enough) more conservative confidence levels. If
your software package does (2) but you want to do (1), you can
achieve this by adding (say) a spherical component with a very short
range of influence and a sill equal to the nugget effect. The ONLY
effect this will have will be to change the value taken at zero
distance. For any other distance, the nugget effect will exist
exactly as before.
For those of you who prefer algebra:
(1) gamma(0) = 0
gamma(h) = C0+spatial component for all h not equal to 0
Guarantees exact interpolation and honours the data.
(2) gamma(0) = C0
gamma(h) = C0+spatial component for all h not equal to 0
Smooths estimates close to sampled locations but does not
(necessarily) honour the data.
To be absolutely clear, I do not advocate ignoring the nugget
effect. That is why I use approach (1). Approach (2) effectively
removes the nugget effect from the kriging system and reduces the
kriging variance significantly.
My point is that you should know what your software is doing at zero
and the effect that can have on your estimation.
Peter's other point is exactly correct. If you map with a grid of
points and your sampled locations are not on that grid, the kriging
will have no possibility to honour the data -- since the data points
are never 'estimated'.
Isobel
<http://www.kriging.com>http://www.kriging.com
