RE: [ai-geostats] kriging without a nugget
- Hi Niklas,
This is a very good question; in fact one of the participants to my last short course
asked the same question since he was using ARCview with the option
"nugget effect excluded" and was surprised to see that his observations were
not honored by the kriging predictions. This is related to the issue
of how to define the nugget effect. On almost all figures in the literature the semivariogram
model seems to start on the vertical axis at a value equal to the nugget effect, while
in fact the value of the model is set to zero for h=0 in the kriging system. This ensure
that kriging is an exact interpolator, which is usually a desirable property.
When interpolated nodes correspond to sampled locations, this exactitude
property can create spikes in the kriged map; in other words these locations
contrast with the general smoothness of the interpolated map produced by kriging
and it is one reason why the option to filter the noise, even at the sampled locations
was introduced (A general presentation of the filtering properties of kriging
can be found in my book p. 172-174). Another application of the filtering
method is the use of kriging for finding minimum or maximum in numerical models;
Sasena, M.J., Parkinson, M., Goovaerts, P., Papalambros, P.Y. and M. Reed. 2002. <http://ode.engin.umich.edu/publications/papers/2002/DETC2002_DAC34091.pdf>
Adaptive experimental design applied to an ergonomics testing procedure. Proceedings of
DETC'02 ASME 2002 Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Montreal, Canada, September 29- October 2, 2002.
More generally, the question is whether the nugget effect represents measurement errors
(variability at the sampled locations) which you might want to filter, or whether it
represents small-scale variability in the field.
Note that the discontinuities in the map will disappear if you use a simulation method.
Hope it helps,
Chief Scientist at BioMedware
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Ann Arbor, MI 48104
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From: Törneman Niklas [mailto:Niklas.Torneman@...]
Sent: Mon 3/6/2006 3:14 PM
Subject: [ai-geostats] kriging without a nugget
I am sort of a beginner within this field and my question might seem a bit simple. Any help would be appreciated however.
In commercial all purpose software's such as SURFER there is an option to exclude the nugget effect from the kriging interpolation.
The purpose is to ensure that measured values are honoured at their locations. This seems understandable to me since the absence of a nugget ensures that the variance is zero at a distance of zero from the measured point, i.e. the measured value=interpolated value.
However, in most projects that I work with (soil pollution problems) there is a significant nugget effect. My question is simply how the interpolation is affected if the nugget effect is excluded when in reality there is a clear nugget present in the data.
One reason for this question (apart from a personal interest) is that I am trying to motivate the use of other methods (i.e. SGS) and more specialized software such as SGeMS and GS+. One good easily explainable motivation for this would be if the above mentioned methodology of excluding the nugget is inappropriate, which is suspect that it is.
- Dear Pierre,
You mentioned that the nugget effect represents
measurement errors or small-scale variability in the
field. How to differentiate between both of them?
Last year I was studying about factorial kriging.
Actually on that time I wanted to use factorial
kriging to filter the nugget variance. I assumed that
my nugget variance only represented small-scale
variability, because I was sure (I already checked)
that my data set did not have any measurement errors.
I also was surprised when I got the estimation result.
Beforehand I guessed that the estimation result will
be more precise as I excluded the nugget variance, but
actually the result showed the contrary. When I
plotted the estimated versus real values, the
correlation coefficient was much less (compared to
include the nugget variance in estimation).
Earth Resources Exploration Research Group
Institut Teknologi Bandung
--- Pierre Goovaerts <Goovaerts@...> wrote:
> Hi Niklas,M. Nur Heriawan
> This is a very good question; in fact one of the
> participants to my last short course
> asked the same question since he was using ARCview
> with the option
> "nugget effect excluded" and was surprised to see
> that his observations were
> not honored by the kriging predictions.
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- Hello allThe real issue here is not what your philosophy is but what your software does with the semi-variogram model at zero distance.There are (to my knowledge) two possibilities in current software packages:(a) force the model to go through zero at zero distance, that is gamma(0)=0(b) allow the model to hit the vertical axis, that is gamma(0)=nugget effectOption (a) makes kriging an exact interpolator. If you krige exactly at a sample location, you will get the sample value and a kriging variance of zero. This is what Matheron orignally specified and will be found in all of the early geostatistics text books.Option (b) means that kriging will not exactly 'honour' your data, but will put the most weight on the sample and some weights on the other samples.If you have software that runs on option (b) the only way to honour your sample values is to have a zero nugget effect. You do not have to remove the nugget effect from your model, just add another (say spherical) component to your model whose sill equals the real nugget effect and whose range of influence is below your closest sample spacing. If you do not know which option is implemented in your software, run a kriging with nugget effect is and with this alternative. If there is no difference in the results, your software does option (a) gamma(0)=0.As discussed in the other emails, nugget effect includes all 'random' variation at scales shorter than your inter-sample distances -- measurement errors, reproducibility issues and short scale variations. Measurement and reproducibility/replication errors can be quantified by standard statistical analysis of variance methods such as described in any experimental design textbooks. Remember, in this case, that it is the 'errors' that need to be independent of one another -- not the actual sampled values. Small scale variation can only be addressed by closer sampling, for example the famous geostatistical crosses.If you can quantify "sampling errors" and have (b)-type software, you can use a combination where a short-range spherical (say) replaces the smaller scale variability and the nugget effect reflects the 'true' replication error. It is then your choice as to whether you filter out the replication error by removing that nugget effect from your model.An important point to bear in mind is that if you use (b)-type software and/or remove the nugget effect when kriging, your calculated kriging variances will be too low by a factor of 2*nugget effect. If you divide the nugget effect as suggested, your kriging variance will be too low by a factor of 2*replication error.One more comment: some packages analyse and model the semi-variogram but use a covariance (sill minus semi-variogram) when kriging. It is odds-on that these packages will be type (b).Isobel