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Re: AI-GEOSTATS: Cracks in the foundations?

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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 1 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 2 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 3 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 4 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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