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AI-GEOSTATS: Variograms

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  • Digby Millikan
    Hello, A summary of replies about my question concerning that there is uncertainty in our experimental variograms, I also raised the point that this
    Message 1 of 2 , Jan 15 5:36 PM
      Hello,
      A summary of replies about my question concerning that there is
      uncertainty in our experimental variograms, I also raised the point that
      this uncertainty is low for some depositional (mining) types
      (e.g. sedimentary) and high for other data types (e.g. high nugget effect
      trangressive gold deposit) and what data would look like for different
      variograms (this question also relating to stationarity and the intrinsic
      hypothesis often referred to in books about geostatistics).
      Answers thanks to Lorenz Dobler, Ruben Roa, Tom Mueller,
      Sean McKenna, Jeff Myers, Donald Myers and Julian Oritz.
      Jeff Myers and Julian Oritz also supplied papers on the subject.

      Thanks
      Regards Digby
      digbym@...
      http://www.users.on.net/digbym

      //===================================================

      Lorenz Dobler wrote:

      hello digby,

      maybe you should have a look at the ph-thesisi of william l. wingle:
      "evaluating subsurface uncertainty using modified geostatistical techniques"
      and the manual of hie softwarw UNCERT.

      regards Lenz

      //===================================================

      Ruben Roa wrote:

      This might be relevant to your question:
      Lahiri, Lee, and Cressie. 2002. On asymptotic distribution and asymptotic
      efficiency of least squares estimators of spatial variogram parameters.
      Journal of Statistical Planning and Inference 103:65-85.
      Rubén

      //===================================================

      Tom Mueller wrote:

      Hi Digby:

      In a paper (Mueller, T.G., F.J. Pierce, O. Schabenberger, and D.D. Warncke.
      2001. Map quality for site-specific management. Soil Sci. Soc. Am. J. 65:
      1547-1558), I suggest that the variance of the semivariogram at each lag may
      explain why map quality was poorer than expected in this study. I also
      suggest that this variance, particularly at earlier lags may be a good
      indicator of map quality. Bottom line is that semivariogram models are
      still just field average models of spatial autocorrelation. Since its not a
      Gaussian distribution (more like a Chi square distribution) it may not be
      appropriate to use CV's do describe the semivariogram cloud. Something I've
      been looking at recently is Hawkins and Cressie's robust estimator. It
      transforms the semivariogram distribution into a normal distribution to
      calculate the semivariogram and then rescales it back. I've found RMSE
      values are smaller with this approach more often than not with jack-knife
      analysis.

      Tom

      //====================================================

      Sean McKenna wrote:

      Digby, the following paper looked at variogram uncetainty as a function of
      the data by sequentially removing one data point at a time, recalculating
      the variogram and then examining the final distribution of gamma values at
      each lag. It may not be exactly what you are looking for, but you may find
      it interesting.

      Wingle, W.L., and E.P. Poeter, 1993, Uncertainty Associated with
      Semivariograms Used for Site Simulation, Ground Water, Vol. 31, No. 5, pp.
      725-734.

      //====================================================

      Jeff Myers wrote:

      Digby -

      This is a bit off the subject, but still related to the variogram questions
      you've been asking. I've attached a paper that describes a typecasting of
      error approach to select optimum variogram/kriging parameters. It can be
      used either to select the best variogram model or to select between
      different types of models (inverse distance, kriging, etc.) as long as you
      have a cutoff value. With this approach, it doesn't matter how good the
      model looks, just how well it performs. Apologies if I've sent this before.

      Jeff

      //====================================================

      Donald Myers wrote:

      Digby

      You have raised some interesting and provocative questions. Let me add a
      couple of observations

      1. The data always consitutes an incomplete picture. While we would like to
      believe (or at least we often act as though we would) that there is a unique
      choice of a variogram or covariance for any given problem, that is clearly
      not the case.

      2. Whether you use the ordinary sample variogram or the Cressie-Hawkins
      estimator or something else, it is at most an estimator.

      Moreover, for a given data set there is not even a unique choice for the
      sample variogram. That is, one must make choices about the lag distances and
      the tolerances. These always represent a compromise, to best determine the
      shape of the variogram model it is desirable to have as many plotted points
      as possible but on the other hand one would like each plotted point to be as
      "reliable" as possible. These two always work against each other (for a
      given data set).

      Strictly speaking, all of the various variogram estimators (or
      covariance estimators) replace an "ensemble" average with a "spatial"
      average, the validity of this depends on the ergodicity assumption which is
      not testable for a given data set.

      It is obvious that there is some degree of interdependence between the
      plotted values for one lag versus the plotted values for another lag (e.g.,
      the same data location is used for multiple lags). There was a paper in Math
      Geology several years ago that computed a more "appropriate" number of
      pairs for each lag, however this was dependent on an assumption of
      multivariate normality.

      It is pretty common practice to not plot the sample variogram for a
      distance that exceeds half the maximum distance across the data location
      set. In fact in many cases we model the variogram against an even shorter
      distance (and for various reasons, some practical and some theoretical)

      3. The kriging equations (whichever form you are using) do not use the
      variogram (or covariance) estimator, i.e., its values but rather use a
      model. The kriging equations use directly (and indirectly) only two pieces
      of information, the spatial correlation function and the mean (the latter
      only indirectly). As we all know, the mean and the variance are not
      sufficient to characterize a probability distribution so it is not
      surprising that the variogram and the mean do not always do a good or
      adequate job for a given data set.

      4. Some of the points or questions you raise pertain more to how well the
      variogram estimator works for a given data set rather than as a property of
      the variogram model.

      5. Strictly speaking, the plotted values of the sample variogram (or other
      estimator) are averages and hence we should consider comparing them to
      averaged values of the model. Several years ago I had a student look at this
      and found that it didn't seem to make much difference. It is common practice
      to plot the sample variogram value against the center point of the tolerance
      interval but there is not theoretical reason why this has to always be done.

      6. Scale or support in the data always plays a role and it certainly can
      affect the C.V.

      I agree with you that it would be interesting to be able to associate in
      some way particular variogram model types or variogram model parameters with
      given types of data sets, i.e., data from particular phenomena. First of
      all, I am sure that it has never been done (there was a paper about this
      idea at a meeting in Paris in the spring of 1995, the meeting was held at
      the UNESCO, not sure it was published) but I really question whether it is
      possible. Part of the problem is that even if you collected examples from
      all the published papers there is no standardization as to how data was
      collected nor how it was treated.

      Donald E. Myers

      //====================================================

      Julian Ortiz wrote:

      Hi,

      You could be interested in looking at a paper we published with Dr.
      Clayton Deutsch on
      Math Geology, regarding the calculation of the uncertainty in the
      variogram.
      We proposed two different methods (that ended up giving the same
      result):
      A simulation approach to evaluate the uncertainty (somehow related to a
      spatial bootstrap)
      and a analytical approach, both based on the multi-Gaussian hypothesis.

      Hope this helps.

      Ortiz C., J. and C. V. Deutsch. "Calculation of uncertainty in the
      variogram", Mathematical Geology,
      V.34, N.2, Feb. 2002, pp.169-183.

      Regards,

      Julian.

      //====================================================


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    • godboutl@pac.dfo-mpo.gc.ca
      Hi All I am interested to test for correlation between time series that exhibits autocorrelation. The problem with this is that there is autocorrelation
      Message 2 of 2 , Jan 20 12:04 PM
        Hi All

        I am interested to test for correlation between time series that exhibits
        autocorrelation. The problem with this is that there is autocorrelation
        within a series (temporal) as well as well between series (spatial), plus
        error in the variables. Previous workers dealt with temporal
        autocorrelation using the effective sample size (Dutilleul et al.). But as
        far as know none have dealt with both type of autocorrelation at once.

        Is anyone know how to handle this?


        Thanks

        Lyse Godbout
        Fisheries and Oceans Canada
        Salmon Stock Assessment
        Pacific Biological Station
        Nanaimo BC V9R 5K6
        Tel (250) 756-7193
        Fax (250) 756-7053
        [mailto:Godboutl@...-mpo.gc.ca]



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