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2459[ai-geostats] RE: kriging without a nugget

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  • Isobel Clark
    Mar 7, 2006
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      Hello all
      The 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 effect
      Option (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).
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