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AI-GEOSTATS: Cracks in the foundations?

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  • Betty Gibbs
    Hi you all, Recently I published an article by Dr. Stephen Henley in my newsletter (Earth Science Computer Applications) called: Geostatistics - Cracks in the
    Message 1 of 5 , Apr 17, 2001
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      Hi you all,
      Recently I published an article by Dr. Stephen Henley in my newsletter (Earth
      Science Computer Applications) called: "Geostatistics - Cracks in the
      foundations?".  I am looking for comments on the statements he makes about
      limitations of geostatistical methods for building ore body models - and
      whether these same limitations apply for other types of data. I would like to
      publish comments in a future issue. Feel free to make comments - but let me
      know if you do *not* want your comments published in the newsletter.

      A copy of the article can be found on the InfoMine/Softwaremine site at:
      <http://www.infomine.com/softwaremine/articles/>www.infomine.com/softwaremi
      ne/articles/

      I will be happy to send sample issues of the newsletter if you send me your
      mailing address. 

      Thanks for any help you can give.
      Betty


      **************************************************
      Betty Gibbs Earth Science Software Information
      Gibbs Associates
      P.O. Box 706, Boulder, CO 80306
      Ph & Fax: 303-444-6032
      e-mail: gibbsb@... http://www.earthsciswinfo.com


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    • Chuck Ehlschlaeger
      Dear Betty, et. al., ... I am ignorant of geostatistical methods on ore bodies. However, I have been using variations of geostatistical techniques on
      Message 2 of 5 , Apr 17, 2001
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        Dear Betty, et. al.,



        Betty Gibbs wrote:
        >
        > Recently I published an article by Dr. Stephen Henley in my newsletter (Earth
        > Science Computer Applications) called: "Geostatistics - Cracks in the
        > foundations?". I am looking for comments on the statements he makes about
        > limitations of geostatistical methods for building ore body models - and
        > whether these same limitations apply for other types of data. I would like to

        I am ignorant of geostatistical methods on ore bodies. However, I have
        been using variations of geostatistical techniques on geographic data,
        especially digital elevation models. Recently, a paper of mine
        "Representing multiple spatial statistics in generalized elevation
        uncertainty models: moving beyond the variogram" was recently accepted
        to be published in the International Journal for Geographic Information
        Science.

        The paper discusses at length the limitations of traditional
        geostatistical techniques for data with measurable surfaces. Of course,
        I propose a solution that deals with those limitations. (Otherwise, it
        wouldn't be "moving beyond the variogram".)

        I don't know how these techniques might be useful to ore bodies, but you
        did ask about "limitations for other types of data".

        The following contains the abstract.

        sincerely, chuck

        ABSTRACT: Spatial data uncertainty models (SDUM) are necessary tools
        that quantify the reliability of results from geographic information
        system (GIS) applications. One technique used by SDUM is Monte Carlo
        simulation, a technique that quantifies spatial data and application
        uncertainty by determining the possible range of application results. A
        complete Monte Carlo SDUM for generalized continuous surfaces typically
        has three components: an error magnitude model, a spatial statistical
        model defining error shapes, and a heuristic that creates multiple
        realizations of error fields added to the generalized elevation map.
        This paper introduces a spatial statistical model that represents
        multiple statistics simultaneously and weighted against each other. This
        paper’s case study builds a SDUM for a digital elevation model (DEM).
        The case study accounts for relevant shape patterns in elevation errors
        by reintroducing specific topological shapes, such as ridges and
        valleys, in appropriate localized positions. The spatial statistical
        model also minimizes topological artifacts, such as cells without
        outward drainage and inappropriate gradient dis tributions, created by
        the realization heuristic. Multiple weighted spatial statistics enable
        two conflicting SDUM philosophies to co-exist. The two philosophies are
        `errors are only measured from higher quality data’ and `SDUM need to
        model reality.’ This article uses an automatic parameter fitting random
        field model to initialize Monte Carlo input realizations followed by an
        inter-map cell swapping heuristic to adjust the realizations to fit
        multiple spatial statistics. The inter-map cell swapping heuristic
        allows spatial data uncertainty modelers to choose the appropriate
        probability model and weighted multiple spatial statistics which best
        represent errors caused by map generalization. This article also
        presents a lag based measure to better represent gradient within a SDUM.
        This article will cover the inter-map cell swapping heuristic as well as
        both probability and spatial statistical models in detail.
        --
        Chuck Ehlschlaeger 212-772-5321
        Dep. of Geography fax: 212-772-5268
        Hunter College chuckre@...
        695 Park Ave. secure: chuckre@...
        New York, NY 10021 http://www.geo.hunter.cuny.edu/~chuck/

        "I'm not satisfied with any of the explanations yet, including my own."
        -- Dr. Robert Wayne, evolutionary biologist studying the red wolf


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      • Chuck Ehlschlaeger
        Dear AI-GEOSTATS members, Back in April, Betty Gibbs introduced us to an article by Dr. Stephen Henley entitled Geostatistics - Cracks in the foundations? . I
        Message 3 of 5 , Oct 27, 2001
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          Dear AI-GEOSTATS members,

          Back in April, Betty Gibbs introduced us to an article by Dr. Stephen
          Henley entitled "Geostatistics - Cracks in the foundations?". I
          responded to AI-GEOSTATS with a message describing an article I wrote
          for the International Journal of Geographic Information Science (IJGIS).
          http://www.geo.hunter.cuny.edu/~chuck/RGSIBG/beyondVariogramSmallMaps.pdf
          is a digital copy of the final draft of that paper.

          Other people, including Steve Krajewski, Isobel Clark, Brian Gray, Dan
          W. McCarn, and Digby Millikan responded to the Infomine newsletter at
          http://www.infomine.com/softwaremine/articles/ with some well thought
          out responses to Dr. Henley's paper. Dr. Henley then provided good
          feedback to these comments. I learned a lot from "Geostatistics - Cracks
          in the foundations?" and about different perspectives from these views.

          I noticed that there were two main goals not explicitly stated within
          these discussions. Many geostatistical analyses have the goal of
          estimating the sum of the magnitude of a variable over a specific region
          where the slope of the variable isn't very important. Other analyses,
          for example, Dan McCarn's description of uranium deposits, need a good
          estimate of a variable's slope in order to provide useful application
          results. The following is a snippet from Dan's comment:

          "For roll-front uranium deposits, a geostatistical approach is avoided
          because of the deposit characteristics. The leading edge of the roll has
          generally the highest concentration of uranium and is bounded by a very
          distinct discontinuity on the reduced side. The trailing edge displays a
          more gradual tapering of grade on the oxidized side."

          Many geostatisticians are trying to design a better "probability model"
          than the BLUE model that traditional geostatistics is based on. They
          correctly realize that non-linear magnitude models are necessary to
          better represent spatial structure.

          However, many applications are more sensitive to the slopes and shapes
          of the goal surface than the magnitude of goal surface values. Digital
          elevation models, for example, are often used for viewshed analysis,
          watershed delineation, and countless other applications where the value
          of the surface is irrelevant except for determining slope and shape. I
          described the process of estimating the appropriate slopes and shapes to
          be "fitting the spatial statistical model". Roll-front uranium deposits
          seem to need a combination of a well-fitting non-linear probability
          model along with a well-fitting spatial statistical model. Comments
          about the variogram and madogram statistics in InfoMine suggest that
          geostatisticians are focusing their efforts on trying to create a
          probability model that will generate appropriate spatial structures for
          a specific application.

          Variograms, and its many variants, are really only useful for
          representing the spatial structure of the magnitude of a variable, but
          not the slopes and shapes of that variable. My IJGIS paper provides a
          specific example of a semivariogram's inability to represent specific
          shapes. Other spatial statistics, with no specific relationship to the
          probability model, are necessary to recreate these shapes and slopes
          during stochastic simulation.

          In order to model these surfaces, I designed a modular system where the
          probability model can be designed independently from the various spatial
          statistical models. I use an "inter-map cell swapping" algorithm to fit
          the spatial statistical model while preserving the probability model.
          This modularization provides several benefits: The most important is
          that it is relatively easy to design probability models without worrying
          whether they generate the appropriate shapes. The same is true for
          spatial statistical models.

          For the applications I am trying to represent uncertainty for, this
          technique seems to be meeting my objectives. Several people have
          requested digital copies of the IJGIS paper. I have finally gotten
          around to putting it on the net.

          sincerely, chuck

          Chuck Ehlschlaeger wrote:
          >
          > Dear Betty, et. al.,
          >
          > Betty Gibbs wrote:
          > >
          > > Recently I published an article by Dr. Stephen Henley in my newsletter (Earth
          > > Science Computer Applications) called: "Geostatistics - Cracks in the
          > > foundations?". I am looking for comments on the statements he makes about
          > > limitations of geostatistical methods for building ore body models - and
          > > whether these same limitations apply for other types of data. I would like to
          >
          > I am ignorant of geostatistical methods on ore bodies. However, I have
          > been using variations of geostatistical techniques on geographic data,
          > especially digital elevation models. Recently, a paper of mine
          > "Representing multiple spatial statistics in generalized elevation
          > uncertainty models: moving beyond the variogram" was recently accepted
          > to be published in the International Journal for Geographic Information
          > Science.
          >
          > The paper discusses at length the limitations of traditional
          > geostatistical techniques for data with measurable surfaces. Of course,
          > I propose a solution that deals with those limitations. (Otherwise, it
          > wouldn't be "moving beyond the variogram".)
          >
          > I don't know how these techniques might be useful to ore bodies, but you
          > did ask about "limitations for other types of data".
          >
          > The following contains the abstract.
          >
          > sincerely, chuck
          >
          > ABSTRACT: Spatial data uncertainty models (SDUM) are necessary tools
          > that quantify the reliability of results from geographic information
          > system (GIS) applications. One technique used by SDUM is Monte Carlo
          > simulation, a technique that quantifies spatial data and application
          > uncertainty by determining the possible range of application results. A
          > complete Monte Carlo SDUM for generalized continuous surfaces typically
          > has three components: an error magnitude model, a spatial statistical
          > model defining error shapes, and a heuristic that creates multiple
          > realizations of error fields added to the generalized elevation map.
          > This paper introduces a spatial statistical model that represents
          > multiple statistics simultaneously and weighted against each other. This
          > paper’s case study builds a SDUM for a digital elevation model (DEM).
          > The case study accounts for relevant shape patterns in elevation errors
          > by reintroducing specific topological shapes, such as ridges and
          > valleys, in appropriate localized positions. The spatial statistical
          > model also minimizes topological artifacts, such as cells without
          > outward drainage and inappropriate gradient distributions, created by
          > the realization heuristic. Multiple weighted spatial statistics enable
          > two conflicting SDUM philosophies to co-exist. The two philosophies are
          > `errors are only measured from higher quality data’ and `SDUM need to
          > model reality.’ This article uses an automatic parameter fitting random
          > field model to initialize Monte Carlo input realizations followed by an
          > inter-map cell swapping heuristic to adjust the realizations to fit
          > multiple spatial statistics. The inter-map cell swapping heuristic
          > allows spatial data uncertainty modelers to choose the appropriate
          > probability model and weighted multiple spatial statistics which best
          > represent errors caused by map generalization. This article also
          > presents a lag based measure to better represent gradient within a SDUM.
          > This article will cover the inter-map cell swapping heuristic as well as
          > both probability and spatial statistical models in detail.

          --
          Chuck Ehlschlaeger N 40 46' 07.7", W 73 57' 54.4"
          Dep. of Geography 212-772-5321, fax: 212-772-5268
          Hunter College chuckre@...
          695 Park Ave. secure: chuckre@...
          New York, NY 10021 http://www.geo.hunter.cuny.edu/~chuck/
          "We don't have time to have meetings about how to fix
          problems, just fix them." -- George Tenet, CIA Director

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        • dwmccarn@aol.com
          Dear Chuck: On reflection, I guess that I should have been less absolute in my statement (You never know how your own words may come back to haunt you!). A
          Message 4 of 5 , Oct 27, 2001
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            Dear Chuck:

            On reflection, I guess that I should have been less "absolute" in my
            statement (You never know how your own words may come back to haunt you!). A
            number of companies (including COGEMA) have very successfully applied
            geostatistical techniques to the evaluation of ore zones in roll-front
            environments. One of the approaches is to use a curvilinear block which
            matches the sinuosity of the ore body geometry (a significant handicap if you
            can't model the shape). Although it doesn't directly overcome the problem of
            leading-edge discontinuity of the ore body, with practice, well justified
            estimates can be obtained. I am intrigued, however, by your suggestion of
            using a well-fitting non-linear probability model. But for two companies
            that I've worked with, other means to obtain ore reserve estimates have been
            applied. In other sandstone U environments besides roll-fronts, the ore
            tends to be more or less continuous, and spatial statistical models are quite
            appropriate. One of these ore bodies, however, had a disappointingly very
            "flat" variogram which had very little spatial structure over a large range
            and direction. This more or less justified their use of an "average" grade
            adjusted by windsorising the high outliers when specific criteria are met.

            R = Thickness * Grade * Area * density * probability of encountering ore
            zones in the area

            I think that the above is "out of the book" for Russian & Kazakh
            practitioners.

            Regards,

            Dan ii

            In a message dated 10/27/2001 5:18:11 PM Mountain Daylight Time,
            chuckre@... writes:

            << "For roll-front uranium deposits, a geostatistical approach is avoided
            because of the deposit characteristics. The leading edge of the roll has
            generally the highest concentration of uranium and is bounded by a very
            distinct discontinuity on the reduced side. The trailing edge displays a
            more gradual tapering of grade on the oxidized side." >>


            Dan W. McCarn, AIPG CPG #10245, Wyoming PG#3031
            10228A Admiral Halsey NE
            Albuquerque, NM 87111
            +1 (505) 822-1323
            +1 (505) 710-3600
            http://resumes.yahoo.com/dwmccarn/geologist
            11 Sept 2001

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          • Donald E. Myers
            Re: the article by S. Henley ( Geostatistics-cracks in the foundations? ) There was a debate of sorts in Mathematical Geology in 1987 but I don t think it
            Message 5 of 5 , Oct 28, 2001
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              Re: the article by S. Henley ("Geostatistics-cracks in the foundations?")

              There was a debate of sorts in Mathematical Geology in
              1987 but I don't think it contributed much to understanding geostatistics
              including its strengths and weaknesses. Here are a few more thoughts.

              GENERAL COMMENTS

              All of the concerns raised in the article are well-known in the geostatistical
              literature, in particular in Mathematical Geology and Water Resources
              Research as well as in the publications emanating from the various
              APCOMS. Unfortunately the problems, strengths and weaknesses of
              geostatistics are not well described in the article, in some cases there are
              clear errors or mis-representations.

              I. Although geostatistics is clearly and deservedly associated with G. Matheron
              (and some of his early students such as A. Journel, J.-P. Chiles,
              P. Delfiner as well as others from the Fontainebleau center), his
              work is similar to or duplicates the work of Matern and Ghandin. Matheron
              acknowledges this in some of his writings. In addition as almost always
              happens, his work builds on work done earlier by many people in a
              number of fields. Finally, geostatistics is not a stagnant field, it
              has developed and evolved due to the work of many people, not always
              in directions directly related to Matheron's original work (this is
              not to belittle or denigrate Matheron's contribution but to recognize
              that geostatistics as it exists today is not the same as 30 years ago).
              Henley's article seems to imply that little has changed in the interim.

              II. Although there are aspects of the applications in hydrology that
              incorporate state equations derived from basic principles, geostatistics
              has most often been used for the analysis of spatial data when no
              state equations are available. This means that the problem is really
              ill-posed, i.e., the solution is not unique and to obtain a unique solution
              one must impose some form of model. The stochastic model implicit in
              Matheron's work serves this purpose. However, one should not be mis-led
              in thinking that this is totally artificial. There are clear connections
              with Bayesian statistics (see the work of Wahba as early as 1970),
              Thin Plate Splines and the more general interpolation methodology
              known in the numerical analysis literature as "Radial Basis Functions"
              which is a deterministic approach to the problem. Again Henley seems to
              ignore this background.

              MORE SPECIFIC COMMENTS

              III. Henley makes a great deal out of the point that kriging is a "linear
              method". This is true and not true. The kriging estimator (Simple,
              Ordinary, Universal) for the value at a non-data location or the
              spatial average over an area or volume (e.g., average grade in a mining
              block) is a linear combination of the data. Written as an interpolating
              function it is NOT linear, i.e., not a linear function of the position
              coordinates. Moreover in the case of multivariate normality, the
              Simple Kriging estimator is the conditional expectation (which is THE minimum
              variance estimator, linear or otherwise). Of course multivariate
              normality is a strong condition.

              IV. Henley claims that it is necessary that the error distribution be normal,
              this is absolutely wrong.

              V. While there are individuals who would identify themselves as "geostatisticians"
              it is more likely that individuals using geostatistics as well as
              contributing to new developments in the field would call themselves:
              mining engineers, petroleum engineers, geologists, soil scientists,
              hydrologists, statisticians, mathematicians, ecologists, geographers,
              etc. Hence the constant reference to (and laying blame on) "geostatisticians"
              is mis-leading.

              VI. Henley fails to distinguish between "estimation variance" and
              "kriging variance" (the latter being the minimized value of the former,
              i.e., the weights in the kriging estimator are obtained by this
              minimization).

              VII. Henley fails to distinguish between the sample/experimental
              variogram and the (theoretical) model. The sample variogram is AN
              estimator of the model (and certainly a number of authors would
              say that it is not the only choice). Ultimately however it is the
              variogram (theoretical) that is of interest and which is use in
              the kriging equations (to obtain the coefficients in the kriging
              estimator).

              VIII. It is true that there are similarities between the Inverse
              Distance Weighting (IWD) estimator and the kriging estimator. (1) As
              noted by Henley, both are weighted averages of the data (but in the
              case of IWD the coefficients are always non-negative, this is not
              true for the kriging estimator), (2) in the usual formulation, IWD
              is isotropic (only distance is used, not direction), (3) IWD is
              not a "perfect/exact" interpolator as is the kriging estimator since
              one can estimate at a data location using the data value at that location
              in IWD (this would involve a zero distance), (4) while IWD in some
              sense incorporates the spatial correlation between the value at the location
              to be estimated and a data location (all pairings), IWD does NOT
              incorporate the spatial correlation between the values at pairs of
              data locations hence some information is ignored in IWD. Finally
              one might attempt to optimize the choice of the exponent in IWD, one
              size does NOT fit all (see a paper by Kane et al, Computers & Geosciences
              1982).

              IX. It is certainly true that kriging will "smooth" the data, howeve
              this is true of all interpolation methods/algorithms. This is one
              reason why some advocate the use of conditional simulation as an
              alternative.

              X. All statistical techniques are subject to problems resulting from
              a lack of data, many of the problems Henley identifies or asserts are
              related to insufficient data. Unfortunately, data costs money and one
              will almost never have enough (of either). The related problem is how the
              data is collected. In more typical applications of statistics, one
              "designs" the experiment to ensure that the underlying statistical
              assumptions are satisifed. This simply will not work in geostatistics.

              XI. Henley correctly identifies "stationarity" or rather the lack of it
              as a critical problem. However he does not quite describe it correctly. The
              constant mean condition (which is only part of the definition of
              second order stationarity) pertains to the implied random function, not
              to the data. Since one has data from at most one realization of the
              random function one can not statistically "test" this assumption. This
              is one of the places where the lack of data is a serious problem,e.g,
              to decide whether to partition the region of interest into separate
              regions to obtain "stationarity" on each separately. Note that
              the condition on the variogram/covariance (being only a function of
              the separation vector) is critical and not implied by the constant
              mean condition.

              XII. The statement "The problem with this method was that the semivariogram
              itself was sensitive to the form of the deterministic surface.
              Therefore, it required a number of iterations of kriging, variogram
              computation, and model fitting, to converge towards a consistent
              solution." contains a germ of truth but it is also confused and mis-leading.
              What Henley is presumably referring to is that if one fits a trend surface to
              the data and then computes a sample variogram using the residuals, the
              resulting sample variogram is biased (see a paper by N. Cressie in
              Mathematical Geology for example and a much earlier paper by another
              author in the proceedings of the NATO conference of 1975). Then there
              was a dissertation in the Dept Hydrology (J. Samper) about 1987) on
              a maximum likelihood method (an iterative application) for fitting the
              sample variogram to residuals. This problem is not completely resolved
              because as is frequently the case in geostatistics or rather in the
              application of geostatistics there is a discrepancy between theory and
              application. It is well known (see Matheron's 1971 Fountainebleau Summer
              School Lecture Notes) that the optimal estimator of the drift, i.e., the
              non-constant mean of the random function, is obtained by kriging. However
              to apply this means that one must first have the variogram model, yet
              one can not model the variogram using the original data. The problem is
              circular. In practice the problem in fact often addressed by fitting a
              trend surface to the data and fitting the variogram to the residuals.

              The real problem is the fact that the sample variogram only estimates
              the variogram if the mean of the random function is constant. When the
              mean is not constant and in particular when the fitted trend surface is
              not just a constant, the sample variogram will exhibit a very rapid growth
              rate (quadratic or greater). There are no valid variogram models with
              this property. Hence if the sample variogram exhibits this growth condition
              this property is taken as evidence of a non-stationarity. Note that this
              property or characteristic is not absolute, it may only appear for large lag
              distances and hence one may be able to fit a variogram model to the
              sample variogram using only the information for short lags.


              XIII. The statements "These methods have found little practical application
              because of their complexity, and the inherent instability of their
              solutions. Furthermore, the resulting kriging system is no longer
              linear and thus loses its ideal "BLUE" properties." are partially correct and
              partially incorrect.

              It is well known that the form of the kriging estimator and the kriging
              equations when using generalized covariances is exactly the same as for
              Universal Kriging. It is also well known that the kriging estimator
              (written in the dual form), when using the right choice of a generalized
              covariance, is the same as the thin plate spline. One must use the so-called
              "spline covariance", this was implemented in the BLUEPACK software back
              in the 80's and is also in the current ISATIS. It is NOT correct to
              say that the resulting kriging system is not linear, the form of the kriging
              estimator is unchanged and the kriging equations ARE still linear. The
              kriging estimator is still a BLUE contrary to Henley's claim.

              There are several reasons why generalized covariances are not widely used
              (for a particularly good presentation see the recent book by Delfiner
              and Chiles). One is that to write software is much more complicated, i.e.,
              to determine the order of the non-stationarity. A second is that as
              contrasted with the family of known variograms there are only a few
              known generalized covariances (although every variogram corresponds to
              a generalized covariance) and these are all isotropic. Hence in practice
              one is likely to revert to using a variogram. Third, it is not so difficult
              to use geostatistics/kriging because of the availability of both free
              and moderately inexpensive software, the theoretical questions are largely
              taken care of in writing the software. There is essentially no free
              software that incorporates the use of generalized covariances (ISATIS is
              rather expensive because it is intended primarily by petroleum and mining
              companies)

              XIV. The first half of the statement "Although regionalised variable theory
              does not require normal (gaussian) distribution of the data, it does
              assume normal distribution of error terms." is almost true but the second half
              is completely false. Henley consistently fails to distinguish between
              properties of a particular data set and the properties of the random function.
              The kriging equations are derived without making any distributional
              assumptions. It is true that kriging (and any other interpolation method)
              will tend to smooth the data. This characteristic is exagerated when the
              distribution of the data is not approximately symmetric (but not necessarily
              normal) and if the distribution of the data has "fat tails". Both lognormal
              kriging and indicator kriging are ways to deal with this problem.

              In claiming that the kriging estimator is no longer "BLUE" when using the
              a log transformation, Henley is presumably referring to the fact that
              if one simply exponentiates the result then there is a bias. Under an
              assumption of multivariate lognormality one can compute the bias and hence
              compensate for it. This has received considerable attention in the geostatistics
              literature. The problem of course is that is not possible to absolutely
              know whether the underlying random function (for a particular data set) is
              multivariate lognormal (univariate lognormal is a much weaker condition).
              All one can do is to determine whether that assumption is reasonable.

              Finally, it is quite plausible to ask whether there might be a better way
              to approach the problem. There is nothing really wrong with Matheron's
              work, the problem is knowing whether the assumptions implicit in its
              use are valid. Since those assumptions pertain to the random function
              , RATHER THAN THE DATA, it is not possible to completely answer the
              question. The real question, particularly for the practitioner is whether
              geostatistics produces useful results. "useful results" does not have
              a very precise meaning in general but it will have a very real meaning
              to the practitioner. Geostatistics is not a cure-all nor is it useful
              for all problems. It is not a black box scheme and must be used with
              some care. In some cases, i.e., for some data sets and some objectives,
              a user need only have access to a software package such as GEOEAS,
              G-STAT or even the add-ons available in ARCVIEW, S-PLUS, etc. In other
              cases it may be necessary to seek the advice and assistance of someone
              more experienced and with a stronger understanding of the mathematics/
              statistics in order to adequately apply the geostatistics to the data
              set.

              For a slightly different take on the same general question see a recent book by M.L. Stein

              Donald E. Myers
              http://www.u.arizona.edu/~donaldm


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