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[ai-geostats] kriging without a nugget

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  • Törneman Niklas
    Hi All, 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
    Message 1 of 6 , Mar 6, 2006
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      Hi All,
       
      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.
       
       
      cheers
       
       
      Nicholas

       
       
       
    • Collier, Perry (TS)
      Hi Niklas You ll get better/more detailed explanations than this, but kriging is what they call an exact interpolator in that the data values are honoured at
      Message 2 of 6 , Mar 6, 2006
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        Hi Niklas
         
        You'll get better/more detailed explanations than this, but kriging is what they call an 'exact interpolator' in that the data values are honoured at their locations (ie: 0 error variance).  As the nugget increases, more distant samples receive a higher weight and taken to the extreme (pure nugget) the samples receive the same weight and the estimation approaches a simple averaging.  See pp 305-307 of Isaaks & Srivastava's book "An Introduction to Applied Geostatistics" for the full story...
         

        Cheers
         

        Perry Collier

        Senior Geologist

         

        Rio Tinto Technical Services

        Phone: +61 7 3327 7676 

        Mobile: 0408 015 837

         


        From: Törneman Niklas [mailto:Niklas.Torneman@...]
        Sent: Tuesday, 7 March 2006 6:15 AM
        To: ai-geostats@...
        Subject: [ai-geostats] kriging without a nugget

        Hi All,
         
        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.
         
         
        cheers
         
         
        Nicholas

         
         
         
      • Pierre Goovaerts
        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
        Message 3 of 6 , Mar 6, 2006
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          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;
          see paper
          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.
          http://ode.engin.umich.edu/publications/papers/2002/DETC2002_DAC34091.pdf

          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,

          Pierre

          Pierre Goovaerts
          Chief Scientist at BioMedware
          516 North State Street
          Ann Arbor, MI 48104
          Voice: (734) 913-1098 (ext. 8)
          Fax: (734) 913-2201
          http://home.comcast.net/~goovaerts/

          ________________________________

          From: Törneman Niklas [mailto:Niklas.Torneman@...]
          Sent: Mon 3/6/2006 3:14 PM
          To: ai-geostats@...
          Subject: [ai-geostats] kriging without a nugget


          Hi All,

          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.


          cheers


          Nicholas
          ________________________________


          <http://www.sweco.se/>
        • M. Nur Heriawan
          Dear Pierre, You mentioned that the nugget effect represents measurement errors or small-scale variability in the field. How to differentiate between both of
          Message 4 of 6 , Mar 6, 2006
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            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).

            Thank you.


            Nur Heriawan
            ------------
            Earth Resources Exploration Research Group
            Institut Teknologi Bandung
            Indonesia


            --- Pierre Goovaerts <Goovaerts@...> wrote:

            > 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.

            M. Nur Heriawan
            http://www.mining.itb.ac.id/heriawan

            __________________________________________________
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          • Pierre Goovaerts
            Hello, The nugget effect is usually interpreted as a combination of small-scale variability and measurement errors. The only way to distinguish between both is
            Message 5 of 6 , Mar 6, 2006
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              Hello,

              The nugget effect is usually interpreted as a combination of small-scale
              variability and measurement errors. The only way to distinguish between
              both is to assess the magnitude of measurement errors in the lab,
              e.g. through replication of the measurement on subsamples.

              Wherever the interpolated node does not coincide with a sampled location,
              the nugget variance is filtered. So, I don't see how you could compare
              the common kriging with factorial kriging in a cross-validation mode.

              Regards,

              Pierre


              Pierre Goovaerts
              Chief Scientist at BioMedware
              516 North State Street
              Ann Arbor, MI 48104
              Voice: (734) 913-1098 (ext. 8)
              Fax: (734) 913-2201
              http://home.comcast.net/~goovaerts/

              ________________________________

              From: M. Nur Heriawan [mailto:mn_heriawan@...]
              Sent: Mon 3/6/2006 8:39 PM
              To: ai-geostats@...
              Subject: RE: [ai-geostats] kriging without a nugget



              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).

              Thank you.


              Nur Heriawan
              ------------
              Earth Resources Exploration Research Group
              Institut Teknologi Bandung
              Indonesia


              --- Pierre Goovaerts <Goovaerts@...> wrote:

              > 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.

              M. Nur Heriawan
              http://www.mining.itb.ac.id/heriawan

              __________________________________________________
              Do You Yahoo!?
              Tired of spam? Yahoo! Mail has the best spam protection around
              http://mail.yahoo.com
            • Isobel Clark
              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
              Message 6 of 6 , 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).
                 
                Isobel
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