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AI-GEOSTATS: Review of answers on "Kriging versus Inv. Dist. Weighting"

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  • Tomislav Malvic RGNF
    Here is the review of answers on my mail Kriging versus Inv. Dist. Weighting from Nov 15: Dear all, This is my first try at geostat mailing list, and maybe
    Message 1 of 1 , Nov 22, 2002
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      Here is the review of answers on my mail "Kriging versus Inv. Dist.
      Weighting" from Nov 15:

      Dear all,
      This is my first try at geostat mailing list, and maybe my question will not
      be very "professional".
      I work with data set of porosity in one oil reservoir. Interpolations were
      done with three interpolation methods: Inverse distance weighting, Kriging
      (ordinary) and Cokriging (collocated). I done spatial analysis with
      semivariogram modelling for (co)Kriging.
      After all, I calculated true error for every included point as difference
      between real value and estimated value at the same place. I was confused
      when I saw that Kriging error was higher of Inverse Distance Weighting
      error! The lowest errors were gained by Cokriging (with the same
      semivariogram modell as used in Kriging).
      What could be reason for that? Maybe 14 points is too low set for proper
      modelling of directional semivariogram analysis (directions=0 and 90
      degrees). I tested several lag distances and distance with the highest range
      was chosen. If chosen distance is too low interpolation map contains mostly
      areas of "bull-eyes". Also, input points are moderately clustered.
      Thank you and best regards,
      Tomislav

      ***********************************************************
      As it is usual in Oil reservoir modelling, it seem you have a very dense
      geophysical information (may be acoustic impedance) and low dense
      information from wells.

      In these case is natural that the best result are obtained from collocated
      cokriging, similar result can be obtained from kriging with external drift
      and the homologous in the simulation scope.

      I think that 14 pint is to short for correct variogram modelling, and it is
      recommendable to use cross variogram in collocate cokriging. A solution for
      know the shape of the variogram could be using the background information,
      if it is close correlate to the wells information, bat I´m not sure that it
      is a good solution.

      Form more help please be more specific

      King regards

      Adrian Martínez Vargas
      Instituto Superior Minero Metalúrgico.
      Moa Holguín Cuba.
      CP 83329
      ***********************************************************
      A few thoughts

      1. As the "user" you have to make a number of choices when you use kriging
      ( I will assume that you were using Ordinary Kriging).
      a. You need to begin with some simple exploratory analysis (histogram of
      data, coded plot of the data (data value vs position coordinate(s), you
      might want to fit to a trend surface). This latter is to help decide whether
      you have to deal with a non-stationarity in the mean. In general this is to
      help understand and interpret what you see in the variogram fitting and
      cross-validation stages.

      2. You have to estimate the variogram/covariance
      i. While it is pretty common to begin with just the ordinary (second
      moment) sample variogram, there are a few others that are sometimes useful.
      Note that the software may have default values for the lag distance and
      number of lags plotted, these may or may not be the best choices for your
      data set. In general don't plot beyond about half the the maximum inter-data
      location distance (the number of pairs drops off significantly). You should
      at least look at directional sample variograms
      (if your data set is too small these may not be very good, too few pairs).

      3. You have to fit the variogram model
      i. This includes determining/choosing the type(s) , note that you
      may want to use a nested model, e.g., spherical, exponential, gaussian,
      power, etc
      ii. This includes determining the variogram model parameters, e.g.,
      range(s), sill(s), nugget, angle(s) of anisotropy and the major/minor
      range(s) (these latter if using an anisotropic model)
      iii. The simplest form of "fitting" is of course "visual", you may
      want to also consider weighted least squares if that is included in your
      software.

      4. You have to evaluate how well your variogram model fits the data,
      cross-validation is at least one way. If you use cross-validation then there
      are multiple statistics (some of these are especially sensitive to the
      choices for the search neighborhood) and in some cases the results may be
      contradictory. While the theory is based on the idea that there is a single
      "correct" variogram model, with only a finite data set there is no unique
      choice for the variogram. Use cross-validation to compare two choices rather
      than as a way to absolutely optimize the choice.

      b. You have to decide on the search neighborhood (this step is necessary
      both for the cross-validation and for the subsequent actual kriging)
      i. shape (usually this is limited in the software to circular or
      elliptical)
      ii. radius (for circular), lengths of axes (if elliptical)
      iii. angle of orientation (if elliptical)
      iv. is the search neighborhood sectioned? (usually taken 1, 2, 4 or
      8 sectors)
      v. minimum and maximum number of data locations to be used in the
      search neighborhood
      In the case of sectors, will empty sectors be allowed?

      Now to your questions and concerns
      It appears that you are describing a simple form of cross-validation and you
      are looking only at the "mean error". As you know the "mean" sort of
      balances positive and negative values so that the mean could be zero (or
      close to zero) even though the individual errors are large in magnitude..
      Among other things I would look at plot of the errors (i.e., plot data
      locations with each point coded by the error from IWD or from kriging). I
      would look at a histogram of the errors and also of the "normalized" errors
      (in the case of kriging, divide each error by the kriging standard
      deviation).

      While it is not a theoretical contradication for the IWD results to be
      better than the kriging results I suggest that this may indicate one of
      several things, the variogram model is not well fitted and/or the search
      neighborhood parameters need to be changed.

      Note also that IWD can be sensitive to the choice of the exponent but this
      will depend on the particular data set.

      Finally if one compares in a theoretical sense, IWD vs kriging, the form of
      the estimator is the same for both. The weights reflect to some degree the
      interdependence between the value at the data location and the value at the
      place where an estimate is desired. However kriging does one more thing, it
      incorporates the independence between the values at the data locations in
      the search neighborhood. While it would be possible to use a modified notion
      of "distance", most software implementing IWD treat it as "isotropic" and
      hence it does not "decluster" the data locations. Think of the estimation
      location as the center of a circle (i.e., the search neighborhood),
      contrast the case of two data locations close together but both the same
      distance from the center vs the case where they are spread apart at an
      angle of perhaps 90 degrees. In IWD the two data locations will have the
      same weights in both cases but in kriging (even with an isotropic variogram)
      the weights are different. When they are close together the weight is
      essentially "split" between them, i.e., because they are very close they are
      "highly correlated" and hence instead of two separate pieces of information
      there is really only one.

      A couple of possibly helpful references


      1991, Myers,D.E., On Variogram Estimation. in Proceedings of the First
      Inter. Conf. Stat. Comp., Cesme, Turkey, 30 Mar.-2 April 1987, Vol II,
      American Sciences Press, 261-281

      1991, Myers,D.E., Interpolation and Estimation with Spatially Located Data,
      Chemometrics and Intelligent Laboratory Systems 11, 209-228
      (The previous) one is was written as a "tutorial" and it uses the free
      "GEOEAS" package)

      1982, V. Kane, C. Begovich, T. Butz and D.E. Myers,Interpretation of
      Regional Geochemistry. Computers and Geosciences, 8, no. 2, 117-136

      (The previous one discusses optimization of the exponent in IWD)

      Warrick, A.W., Zhang, R., El-Haris, M.K. and Myers, D.E., Direct comparisons
      between kriging and other interpolators. in Proceedings of the Validation of
      Flow and Transport Models in the Unsaturated Zone, Ruidoso, NM, 23-26 May,
      1988, 505-510

      Donald E. Myers
      http://www.u.arizona.edu/~donaldm
      ***********************************************************
      Hi Tomislav,

      Don't be surprised. It is my experience that cross-validation
      might sometimes indicate that best interpolation results are obtained
      using the simplest techniques. If your observations are not
      too clustered and display no anisotropy, inverse square
      distance could yield good results.
      Now, you didn't explain which secondary information was used
      for cokriging and how many neighboring values were used
      in the different interpolators.

      Regards,

      Pierre Goovaerts

      Dr. Pierre Goovaerts
      Consultant in (Geo)statistics
      and Senior Chief Scientist with Biomedware Inc.
      710 Ridgemont Lane
      Ann Arbor, Michigan, 48103-1535, U.S.A.

      E-mail: goovaert@...
      Phone: (734) 668-9900
      Fax: (734) 668-7788
      http://alumni.engin.umich.edu/~goovaert/
      ***********************************************************
      Tomislav,

      Try getting a semivariogram range from the (presumably)
      dense secondary data. This could be seismic derived porosities.
      Construct a semivariogram for porosity using that range and
      a sill more in line with the variance that you see in
      the sampled points. Then re-perform the kriging.

      14 points is a bit on the low side, but unfortunately more
      the norm than the exception for most oil and gas datasets,
      especially for offshore locations. Would there be analogous
      data that you can use? At every lag distance try to get at
      least 50 pairs for calculation of the variogram values.

      Regards,

      Syed
      ***********************************************************
      This paper may be of interest to you.
      http://www.uky.edu/~mueller/Internal/map%20quality%20for%20SSFM.pdf
      If you send me your address, i will send you another paper that will be
      published in January.

      tom
      ***********************************************************
      I wish to thank everybody for very useful answer and help.
      With my best regards,

      Tomislav Malvic, M.Sc.
      Reservoir Geologist
      INA-Oil Industry plc. (INA-Naftaplin)
      Subiceva 29
      HR-10000 Zagreb
      CROATIA


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