nonparametric estimation of spatiai distributions

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 Mathematical Geology VoL 15 No. 3 1983 Nonparametric Estimation of Spatiai Distributions ~ A G Journel 2 The indicator approach whereby the data are used through their rank order allows a non- parametric approach to the data bivariate distribution. Such rich structural information allows a nonparametric risk-qualifled estimation of local and global spatial distributions. KEY WOR DS: indicator kdging, geostatistics, nonparam etric. INTRODUCTION Looking back at some 15 years of geostatistics präctice in the mineral industry and other fields such as hydrogeology, soff sciences, the environrnental sciences, and forestry survey, one can start appraising the real trnpact of the discipline and what work remains to be done. For reasonably well-behaved phenomena, with coefficients of variation of initial composite data not rauch higher than 1 e.g., porphyry copper grades, water table elevation) a record of successes can be established: experimental variograms can be obtained from available data and do depict the main features e.g., geology) of the phenomenon under study; linear kriging then provides im- proved, risk-qualified, local estimates of average characteristics such as block grades. Problems arise when dealing with highly variant phenomena where the data present long-tafled distributions with coefficient of variation in the range of 2-5, e.g., uranium, gold, diamonds, and trace poUutants such as plutonium. Raw variograms become extremely sensitive to high-val ued data, and are basically useless. Two avenues for solutions were proposed: 1) trim oft the high-valued data, caUed outliers, on some ground, whether geo- logical rock type, degree of alteration) or probabilistic which always refers to some normal-related hypothesis about the data distribution). 1Manuscri pt received 10 Decem ber 1981; revised 25 May 1982. 2Applied Earth Sciences Department, Stanford Urtiversity, Stanford, California 94305 U.S.A. 5 0020 5958/83/0600 0445S03.00/0 1983 Plenum Publishing Corporaüo n

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Non-parametric Estimation

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  • Mathematical Geology, VoL 15, No. 3, 1983

    Nonparametric Estimation of Spatiai Distributions ~

    A. G. Journel 2

    The indicator approach, whereby the data are used through their rank order, allows a non- parametric approach to the data bivariate distribution. Such rich structural information allows a nonparametric risk-qualifled, estimation of local and global spatial distributions.

    KEY WORDS: indicator kdging, geostatistics, nonparametric.

    INTRODUCTION

    Looking back at some 15 years of geostatistics prctice in the mineral industry and other fields such as hydrogeology, soff sciences, the environrnental sciences, and forestry survey, one can start appraising the real trnpact of the discipline and what work remains to be done.

    For reasonably well-behaved phenomena, with coefficients of variation of initial composite data not rauch higher than 1 (e.g., porphyry copper grades, water table elevation) a record of successes can be established: experimental variograms can be obtained from available data and do depict the main features (e.g., geology) of the phenomenon under study; linear kriging then provides im- proved, risk-qualified, local estimates of average characteristics such as block grades.

    Problems arise when dealing with highly variant phenomena where the data present long-tafled distributions with coefficient of variation in the range of 2-5, e.g., uranium, gold, diamonds, and trace poUutants such as plutonium. Raw variograms become extremely sensitive to high-valued data, and are basically useless. Two avenues for solutions were proposed:

    (1) trim oft the high-valued data, caUed outliers, on some ground, whether geo- logical (rock type, degree of alteration) or probabilistic (which always refers to some normal-related hypothesis about the data distribution).

    1Manuscript received 10 December 1981; revised 25 May 1982. 2Applied Earth Sciences Department, Stanford Urtiversity, Stanford, California 94305 U.S.A.

    445

    0020-5958/83/0600-0445S03.00/0 1983 Plenum Publishing Corporaon

  • 446 Joumel

    (2) smooth out the data by working on some smooth function of them, for ex- ample, their square roots, natural logarithms, or normal-score transforms.

    Except when done on solid geological ground, trimming oft bothersome data appears at least simplistic, and is plainly not acceptable when these data carry the most valuable structural information about the phenomena, not to mention their economical weight (e.g., the rare big stones in a diamond deposit).

    The characteristic of all smooth transforms, for example, In z(x) ifz(x) is the grade at point x, is that they are nonlinear. Unfortunately, nonlinear trans- forms call for nonlinear estimation techniques, and all these nonlinear estimation techniques require some hypotheses about the multivariate distribution of Z(x); compare, for example, the theories of disjunctive kriging (DK) and multivariate Gaussian kriging (MG) in Journel and Huijbregts (1981), p. 565-571, Matheron (1975), and Verly (1983). As usual, regardless of the underlying reality, these hypotheses are normal-related.

    If one reflects back on the history of the development of geostatistics, the reason for its practical success was that it always put the data before the model; linear geostatistics is entirely distribution-free, expected value and variance- free, and calls only for the shape of the variogram function. In this regard, the techniques of DK and MG represent a step aside from the original thrust of geostatistics.

    The question then should be: Is it P0ssible to design nonparametric, dis- tribution-free techniques that can do the job of either DK or MG? This job is essentiaUy twofold

    (1) handle highly variant phenomena without having to trim off important high-valued data;

    (2) provide risk-qualified estimates of unknown values of z(x) and more gen- eraUy of spatial distributions of these values z(x) within delimited areas.

    Position of the Problem

    Consider a phenomenon characterized by the spatio-temporal regionalization of an attribute z, for example, the grade z(x) at location x, with x E deposit D.

    Classically, that regionalization is interpreted as a particular spatial out- come of a random function .[Z(x), x E 1)). The only data available are {z(x) = z, t~ = 1 to N). In addition, the mineralization within D being reasonably homo- geneous, the random function Z(x) will be considered as stationary within D.

    The experimental variability of the N data z is so high (coefficient of vari- ation greater than 2), that it precludes any interpretation of the extremely noisy raw experimental variogram, written in its theoretical form

    27'z(h) = E{[Z(x + h) - Z(x)] 2} (1)

  • Nonparametric Estimation of Spaal Distributions 447

    Typical goals for the study of the regionalization {z(x), x E D} are

    (1) risk-qualified estimation of unknown values z(x), or zv(x ) averaged over blocks of size v, to allow mapping

    (2) estimation of spatial distributions of z(x), or zu(x), within areas A C D, to allow estimation of the proportion of such values, z(x) or zv(x), which are above any given cutoff grade (i.e., the problem of local recoverable re- serves estimation).

    The lndicator Function

    At each point x C D, consider the following step function of z, defined as

    i(x;z) = I l , if z(x)

  • 448 Journel

    Tonnage point recovery factor in A

    1 - $(A ; z) = Proportion (z (x) > A ; x E A) (4)

    Quantity of metal recovery factor in A

    q(A;z ) -- u d~(A;u) (5)

    Mean ore grade at cutoffz

    m(A;z ) = q(A;z) / [1 - ~b(A;z)l (6)

    Remarks

    The in-situ mean grade within A is written

    m (A ;o) = q(A ;o) = u dc~(A ;u) (7)

    If the area A is reduced to a point x, it foltows

    dp(x;z) = i (x ;z )

    m(x ;o ) =z(x) = u d i (x ;u ) (8)

    Expressions (5) and (7) call for a Stieltjes integral with the density d(~(A ; z). Although the function ~(A;z) may not be derivable for all z, this Stieltjes in- tegral is always defined in practice, since

    z(x) < ~, for all x E D

    The previous expressions do not provide the block recovery functions cor- responding to selection performed not on point grades z(x) but on block v- average grades

    problem of estimation of block recovery functions from point- The difficult support information is addressed in Appendix B.

    It thus appears that the critical function to be estimated is ~b(A;z), for knowledge of such an estimate would provide in its turn

    estimates of z (x), or zo(x), for mapping purposes,

    estimates of point local recoverable reserves, or in nonmining terms, esti- mates of local spatial distributions.

  • Nonparamet f i c Estimation of Spatial Distributions 449

    LINEAR ESTIMATORS OF HISTOGRAMS

    An estimate of ~(A;z) has to be, one way or another, a function (linear or not) of the N available data: actual values z(x) or indicator values i(xo~;z ). The linear family of estimators are considered first for, as is usual in statistics, it can be approached without having to make any hypothesis about the dis- tributions of the random functions Z(x) or l(x;z).

    Disjunctive Coding of a Histogram

    Out of the available N data over the deposit D, eonsider the subset of size n available within area A

    (z(x), x ~ A, = 1 to n}

    A possible estimate of the proportion q~(A; z) is the naive, equal weighted, cumulative histogram of these n data (Fig. 2). This histogram appears as a rank- ordered series of n step-functions i(xc,; z), a = 1 to n, each with equal amplitude 1In and located at the actual data value z(x) = z

    cB(A;z)=--1 ~ i(x;z) n =1

    (9)

    The previous expression is none other than the "disjunctive coding" of the ex- perirnental cumulative histogram, an expression coined by the French statisti- cian, J. P. Benzecri. The previous histogram (9) is naive in the sense that

    (1) It equally weights each datum without regard to the spatial continuity of z(x) within A and to the degree of clustering of the n data used.

    (2) It does not make any use of surrounding data z(x~), x~ q~ A but still close to A.

    (A;z)

    i/n ~ I "--I , r - - - -r - , __...1 '

    0 z (~) z (~) " z (d) n rank-ordered data ---+

    Fig. 2. Disjunctive eoding of an experimental histogram.

  • 450 Joumel

    In the presence of spatial continuity between the data (which is the very reason for the whole geostatistical approach), all data z(x), x E D should be considered, although with different weights. Intuitively, a datum within A should receive a greater weight, eren more so if it is not redundant with a close-by other datum. The degree of redundancy as weil as the notion of close- ness should be linked to the spatial continuity (i.e., degree of roughness)of the regionalization {z(x), x E D}. In other words, the foUowing estimates should be considered

    n *(A;z)= ~ Xai(x;z), with x EA (10)

    ~=1

    or even better

    N *(A;z)= ~ ~(z ) " i(x;~), with x eD (11)

    Ot=l

    and usually N n. Expression (10) appears exactly as a discrete approximation of the integral

    (3) defining (A ;z). Consequently, numerical integration teehniques can be ap- plied to the determination of the n weights ?~a. One such technique, the "declus- terizing-cell technique," is proposed in Appendix A. As opposed to kriging, these techniques are nonprobabilistic: they have the advantage of not requiring any probabilisitic model or any probabilistic hypothesis such as stationarity. They have the disadvantage however of not providing easy quantification of the error involved [(A ;z)- *(A ;z)]. Such error is linked to the roughness of the sur- faces i(x; z), x E D, for all z. This roughness is characterized later-in probabilis- tic terms-by the indicator variograms (17). The last and not least disadvantage, particularly if both area A and sample size n are small, is that the deterministic integration techniques do not make use of nearby data x ~ A.

    The lndicator Approach

    Using expression (11) of the estimate O*(A;z), a simple linear kriging ap- proach is considered, with the indicator data i(x;z), a = 1 to N. The first step of such an approach is to characterize the spatial variability of the indicator random functions I(x;z). Note that there is one such indicator random function for each argument (cutoff) z.

    Structural Analysis

    For a given z, the random function I(x; z) appears as binomial distributed with expected value

  • Nonpatametrie Esmaon of Spaal Distributions 451

    E{I(x ;z)) = 1 Prob {Z(x) ~< z) + 0 - Prob {Z(x) > z)

    = Prob (Z(x)

  • 452 Journel

    Z(x+h)

    z (x+h)l- "-i

    o

    \ ,

    7 ~. . . _ . , J ~

    ~~45o

    d = [z(x + h) - z(x)] cos 45

    d2 = 1 [z(x + h) - z(x)] 2

    E(d 2) : E{[Z(x + h) - Z(x)] 2} : 7z(h)

    = inertia moment/first bisector

    Fig. 3. h scattergram of [z(x + h), z(x), x~D].

    robust estimation of 7z (h). In practice the indicator semivariograms 7z(h;z) should be estimated

    first, from the indicator data i(xa;z ). Indeed, the expressions of these semi- variograms do not call for knowledge of the univafiate distribution function F(z) = E{I(x;z)}.

    These indicator semivariograms are extremely robust with regard to tail data, for their estimation does not call for the data values z themselves but rather for their rank order (indicator values) with regard to a given cutoff z. For example, if z = 1%, all data values above 1% have the same impact on 7/(h; 1%) whether they are equal to 1.5% or 15%, or 30%! There is thus no need to trirn oft the bothersome, so-called outtier value, 30%.

    The best defined experimental indicator variograms correspond to cutoffs z close to the median zM, for then roughly 50% of the indicator data are equal to 0 and the rest are equal to 1.

    Programming-wise, since the configuration of indicator data i(x;z) at all cutoffs, and of actual data values z(x), are all identical, a single run of a classi- cal variogram program can provide both 7z(h) and all 71(h;z). An indicator variography, although richer and more robust than a mere Z variography, does not require additional software and is not appreciably more expensive.

    Stability of the Indicator Correlogram

    Consider a stationary random function Z(x) standard binormal, that is, such that any pair of random variables Z(x), Z(x + h) is bivariate normally dis- tributed, with zero means, unit variances, and correlogram

    oz (h) = F,{Z(x + h) . Z(x)} (19)

    Switzer (1977) has shown that the corresponding indicator correlogram oi(h; z)

  • Nonparametrie Estimation of Spatial Distfibutions 453

    is insensitive to the cutoff value for z close to the median value z M = 0

    p I (h ;z )~p i (h ;o ) , forz close toz M (20)

    more precisely for z C [-+.67] : interquartile range. For this median value, the standard binormal indicator correlogram is writ-

    ten (Abramovitz and Stegun, 1965, p. 937)

    pI(h;o) = (2/fr) arc sin Pz (h) (21)

    It can be shown that the approximation (20) applies also to any random function Z(x) such that its univariate normal transform Y(x) is also bivariate normally distfibuted.

    Direct Estimation of F(z)

    From expression 16, it appears that the stil of the indicator semivafiogram, if it exists, is equal to the corresponding indicator variance which is itself linked to the univariate distribution function F(z)

    Sill S2(z) = 3"/(oo; z) = Cz(o;z) = F(z) - F2(z)

    The sflls S 2 (z) can be estimated from the experimental indicator semivariograms, so the corresponding distribution funcfion values F(z) can be retrieved by solv- ing the second-degree equation

    F2(z) - F(z) + S2(z) = 0 (22)

    Remarks

    The following order relations hold true for S 2 (z)

    * S2(z) is maximum and equal to .25 for z equal to the median z M of the distfibution F (z ), that is, F (z M ) = .5

    . S2(z) increases in [0, .25] when z increases to z M

    . S2(z) decreases in [.25, 0] when z decreases from ZM

    In practice, the following procedure can be implemented to produce an estimate F*(z) of the distfibution function.

    Determine the value z u for which the expefimental sill S 2 (z) appears to be maximum and dose to .25, then

    F*(ZM) = .5

    F*(z) = {1 + e(z) [1 - 4S2"(z)] '/2}/2 (23)

    with e(z) = Sign (z - ZM).

  • 454 Journel

    The sills S2*(z) are "bull's eyed" around the plateau of the experimental semivariograms 7~(h; z) and should respect the previous order relations,

    Experimental indicator semivariograms are bound to be iU defined for extreme cutoffs z-+ +oo. They would be best defined for quantiles z close to the median z M.

    On idea would be to plot S2*(z) versus z, for values z close to ZM, then in- vestigate extrapolation procedures of this curve toward the two tails, where S:(z) ~ O. For example, if z(x) is a low concentration grade, extrapolation can be guided by the theoretical tail behavior S2(z)= F(z)- F2(z) of a lognormal distribution F(z ).

    Beware that S 2 (z) is usually not symmetrical around the median z M. If, in addition to being stationary, the indicator random functions I(x;z) are also as- sumed to be ergodic, then

    q~(A;z) = i (x;z)dxE(I(x;z)} =F(z) when A-++oo (24)

    Consequently, an estimate of F(z) can also be obtained by estimating the in- tegral (A ;z) defined over a large stationary area A (cf. the following section).

    KRIGING FOR SPATIAL DISTRIBUTIONS

    In practice, the variables of interest are the realizations z(x) or i(x; z)and not their random function counterparts, Z(x), I(x;z). Similarly, it is not the model distribution F(z) which is looked for, but the proportion (A ;z) defined over any given area A. In statistical jargon, the problem is to predict the out- come ~b(A ;z) of the random function ~(A; z), and not to estimate the param- eters of the probabilistic model Z(x).

    Note that

    (~(A;z)} = EU(x;z)} ax = F(z) (25)

    for all areas A, under the hypothesis of stationarity. Also note that, even if area A was exhaustively sampled and hence qS(A ;z) is exactly known, there would still be a nonzero variance E([cb(A ;z) - F(z)] 2) characterizing the dispersion of the possible outcomes around the model F(z). Of course such variance is irrele- vant to the problem of predicting the actual unique outcome ~b(A ;z).

    The most general linear estimate of ~(A;z) is of type (11) calling for a dif- ferent weight ~~a(z) for all N data available, whether inside or outside A. Such an estimate would call for a so-caUed "complete kriging" system and would be im- practical as soon as N 50.

    The idea is to differentiate the weights of only those N' data within a

  • Nonparametfic Estimafion of Spatial Distributions 455

    neighborhood of A, and weight equally all other data. More precisely a "simple kriging" system is considered, whereby it is not 4~(A;z) which is estimated, but rather its residual [~(A ;z)- F*(z)] using the residual indicator data [i(x;z)- F*(z)l

    N' 4)*(A ;z) - F*(z) = ~ Xt(z) [i(x ;z) - F*(z)] (26)

    ~=1

    with F*(z) being an unbiased estimate of F(z) = E{(b (A ; z)}. Such an estirnate F*(z) can be either

    (a) the naive, equal weighted, cumulative histogram of all N data present in the deposit or stationary zone D, provided these N data are uniformly located within D (no clusters)

    (b) the previous estimate F*(z) deduced from the indicator semivariogram sills (formula 23).

    (c) any other, possibly nonprobablistic, declustefized estimate of F(z)(Appen- dix A).

    N' is the number of indicator data i(x ; z) being differentiated in the neigh- borhood ofA. Usually n < N'

  • 456 Journel

    Order Relations

    *(A ; z), being an estimate of a cumulative proportion (A ;z), must verify the corresponding order relations, that is

    ~b*(A; z) C [0, 1 ], for all areas A and for all z

    ~*(A;z)

  • Nonparametric Estimation of Spaal Distributions 457

    p- I (A,A;z)=~ fA dU fA Pi(U- U';z)du'

    Stil S2(z) = Var {I(x;z)}

    = F(z) - F2(z) "~ F*(z) - F *~ (z),

    Mean indicator-correlogram values

    (relation 16)

    R cm arks

    (a) The system (30) is written in terms of correlogram instead of semivariogram for the usual reasons of convenience: the diagonal terms, O(0;z) = 1, are also the pivot terms.

    (b) If K cutoff values zk, k = 1 to K, are considered to discretize the range of outcomes z(x), K kriging systems need to be solved, each with its particular correlogram pi(h; zx). However, as was noted already for the indicator variography, the data configuration remains the same for all cutoffs z, a feature that considerably reduces the computer time requirement.

    (c) If K different correlograms ox(h;z), k = 1 to K, are used, the weighting schemes X(zk) depend on the cutoff value zk, and consequently the order relations (28) may not be ensured.

    One quick solution is to use a single weighting scheme, that corresponding to the median cutoffZM:

    Xt(z) ~- ~(zM) (32) for all z and all a = 1 to N'

    Another approximation consists in correcting sequentially for the order relation problems, for example, by considering the following corrected estimates

    **(A;zl) = Max {~b*(A ; Z l ) , 0}

    **(A ;zk) = Max {**(A ;zk_l), *(A ;zk))

    ~**(A;ZK) = Min **(A;ZK) , 1} (33)

    with {Zl

  • 458 Journel

    Estimation of Point Recovery Functions

    Availability of an estimate O*(A ; z) allows estimation of the point recovery functions within area A, according to formulas (4), (5), (6)

    tonnage recovery factor

    1 - *(A, z)

    quantity of metal recovery factor

    q*(A;z)= u" d*(A;u)

    This Stieltjes integral is, in practice, approximated by the following discrete sum

    K(z) q*(A;z)~ ~ zk[ ~ (A;Zg+l)- 4)*(A;zk)] (34)

    k=l

    with z, k = 1 to K(z) being values discretizing the interval of integration [z, +oo], and z~ a central value of the interval [zg, zk+ 1 ].

    mean ore grade

    m*(A ;z) = q*(A;z)/[1 - *(A ;z)]

    Estimation of In Situ Mean Grade Within A (cf. relation 7)

    m*(A;O)=q*(A;O) = u d*(A;u)

    approximated by a discrete sum as in (34).

    (35)

    Using the median *(A; z) is written

    Disjunctive Coding Interpretation of*(A; z)

    indicator kriging approximation (32), the estimate

    *(A;z)= ~--" ~(ZM)' i (xa;z)+ 1- ~' X(ZM) F*(z) (36) Ot=l ~=1

    As a function of z, the estimate *(A ; z) is seen as a rank-ordered series of the N step functions i(xa;z ), each of unequal amplitude. The amplitudes attached to the N' data inside the kriging neighborhood of A will generally be larger than those given to the (N- N') outside data (Fig. 5).

    For "cosmetic" appearance, each of the N step functions i(x;z) may be replaced by any continuous distribution, for example, uniform distributions over the intervals [z (- 1), z()] which amounts to linear interpolations between steps.

  • Nonparametdc Estimation of Spaal Distribuons 459

    ~*(A;z)

    B ~'(2) t

    (1)

    z~l) z(h) N rank-ordered data --+ z~N)

    (i) Large step, if x ~ Neighborhood of A (2) Small step, otherwlse

    Fig. 5. Disjunctive coding interpretations of 4~*(a;z) (under the median ap- proximation).

    Estimation of I(x; z)

    By reducing the area A to a single point x, the unknown indicator value i(x;z) = ~b (x;z) can be estimated through a simple kriging procedure in all points similar to that used for estimating ~(A ;z). More precisely, the estimate is written

    i*(x;z) = ~ ~(z) i(xa;z)+ 1- ~_, Xt(z) F*(z) 07) ~=I t~=l

    The corresponding simple kriging system being

    N t

    Z /3=1

    and

    XB(z)Pi(X -x/3;Z)=Pi(X -x ;z) for all a= 1 toN' (38)

    O}K(Z) = E{[I(x;z) - I*(x;z)] 2}

    =S2(z) 1- Z X~(z)pi(x-x;z OL=I

    (39)

    Conditional Probability Interpretation

    The elementary indicator datum i(x;z) (see Fig. la) can be interpreted as the following conditional probability

    Prob {Z(x) ~< z/Z(x) = z} = i(x ;z) (40)

    Consequently, the estimate i*(x; z) appears as an estimate of the unknown con- ditional probability

    Prob (Z(x)

  • 460 Joumel

    Similarly, the estimate ~*(A;z) appears as an estimate of the composite condi- tional distribution

    f 1 --- ~ Prob (Z(x)

  • Nonpatametfie Esfimafion of Spaal Distributions 461

    pected absolute conditional error

    E*{[ Z(x) - z [/surrounding data}

    The E-type estimate z*(x) is the outcome value z which minimizes the ex- pected squared conditional error

    E* {[Z(x) - z] 2/surrounding data}

    (a) For estimation and mapping of point value z(x), there is no a priori reason to prefer the estirnation variance criterion to the absolute deviation criterion. In practice, in the presence of highly variant data distribution and "outlier" data values, the more robust M-type estimator may be preferred.

    (b) Since kriging is a linear operation, and provided that the same kriging neigh- borhood is used for all points x E A, it follows

    1 fA ~*(A,z) = i*(x;z)dx (46)

    This entails, in turn, a linearity property for the E-type estimators

    , 1 fAz,(x)d x m (A;0) = (47)

    The same does not hold true for the M-type estimators. (c) The unbiasedness, E{Z2(x)- Z*(x)} = 0, of the E-type estimator is war-

    ranted by the unbiasedness of the conditional probability estimators

    E{I(x;z) - I*(x;z)} = 0, for all z

    (d) All three estimates, z*(x), z*(x), and the classical kriged value z~:(x) using the Z variogram, appear distribution free and provide a qualffication of the error estimation. The M- and E-type estimates present the advantage of be- ing robust with regard to outlier data z(x). Hence, although it has not been designed for this purpose, the IK approach seems more appropriate than ordinary kriging when dealing with highly variant phenomena.

    CONCLUSIONS

    In recent years, research in geostatistics has strayed from the two funda- mental rules that made the initial success of the discipline

    Data always supersede unnecessary models Robustness and simplicity prevail over baroque mathematics.

  • 462 Journel

    Although addressing real critical problems and introduced in the mid-1970's nonlinear Geostatistics still lacks general acceptance, for it is hea~_ly distribution dependent and is plainly heavy, particularly under its DK form.

    The nonparametric indicator approach may represent the long awaited comeback to a no-nonsense problem-solving approach. Data are used through their rank order with regard to any given cutoff, allowing for a more comprehen- sive structural analysis, and are yet more robust with regard to outlier values. Indicator kriging allows risk-qualified estimation of spatial distributions, from which local recoverable reserves can be assessed.

    The IK approach does not require any new software, nor any new mathe- matical insight: it can be implemented overnight by anyone who is already

    trained in linear geostatistics. Under its rigorous form, multiple indicator kriging system (30), the IK technique still faces severe order-relations problems not yet fully solved.

    APPENDIX A: THE CELL-DECLUSTERIZING TECHNIQUE

    The proportion q~(A ;z) appears as a spatial integral

    ~b(A;z) = i (x;z)dx (see relation 3) (A1)

    As such its estimation can be done by classical numerical integration techniques. If the variability of the attribute z(x) is homogeneous and isotropic over

    the area A, and if the n data available inside A were uniformly located within A, there would be no reason to over- or under-weight any particular data; hence the "optimal" estimate of ~b(A;z) will be the naive equal-weighted mean intro- duced in (9)

    $(A;z)--L ~ i(~;z) n ~=1

    Consider now a more common case where the n data are not any more uniformly located within A, but rather rend to be regrouped in L clusters. The idea is to split the area A into L subareas corresponding to the L elusters and apply an equal-weighting scheine to each subarea

    ~(A;z)=l=l Alt 1 L i (x;z)dx = t~_l Al~b(At;z), L L

    A = y~~ At, n = ~ nr, and 1=1 /=1

    with

    =-- n l 1 Z i(x;z) (m) ~*(A;z) nl =1

  • Nonparametfic Estimation of Spaal Distribuons 463

    Fig. AI. Declustering a data set.

    Outl ine of area A /

    This cell has 5 data I receiving a weight

    each

    In most practical cases, at least in the mining industry, underlying the clusters there exists a uniform grid; that is, apart from the clusters, a basic grid cell of size a can be defined which most of the time contains at least 1 datum (Fig. A1).

    The previous declusterization procedure then reduces to the following steps

    (a) overlay over area A a regular grid with cell size a (b) count the number of data n t falling within each elementary cell a I (c) weight each data falling in cell al by 1/n l

    1 z nl 1 ~*(A;z) =~- Z Z -- i(x~;z) (A3)

    1=10tl=l nl

    Remarks

    The same set of weights applied to the actual data values z(x), x E A, pro- vides a "declusterized" estimate of the mean grade, m(A, O) = 1lA fA z(X)dx, over area A.

    If it is known that all chisters are located in preferentially rich areas (high z valued), various ceU sizes a may be tried, and the cell-size ao providing the low- est declusterized estimate m*(A ; 0) should be retained.

    In Fig. A2, the variabllity of the estimate m*(A; 0) as function of the cell-

    m*(A;O)

    Fig..6,2. Choosing an optimal cell size.

    &(A;O)

    optimal ! ! ! ! |

    a 0 a

  • 464 Journel

    size a (with fixed geometry) is sketched; for smaU and large size a, the estimate m*(A; 0) is nothing else than the naive equal-weighted mean r~ (A; 0) of all data within A.

    This declusterization technique does not take into account the influence of neighboring data: z(x), x ~ A, and does not account for the particular type of continuity prevailing over A. Therefore it should not be used for estima- tion of spatial distributions ~b(A ;z) over small areas A, where the number n of internal data is small (in practice less than 50). Nor should it be used when the spatial variability of z(x) over A present such features as anisotropies or hole effects (pseudo-periodicities).

    Consequently, although the cell-declusterizing technique allows estimation of spatial distribution over large areas, it does not replace geostatistics for point mapping (estimation of point values z(x)), nor for estimation of spatial distribu- tions over small areas A. In mining practice, a panel A, within which local re- coverable reserves must be assessed, contains usuaUy no more than 10 or 20 data spread over two or three drillholes. Last but not least, the error linked to the estimate (A - 3) cannot be directly assessed, for the derivative of the spatial function i(x;z) is not defined.

    APPENDIX B: ESTIMATION OF BLOCK RECOVERY FUNCTIONS

    The distribution O(A;z) is related to grades z(x) of point support (more exactly of support equal to that of the composite data). In many applications, it is the distribution of block grades, zv(x ) = 1/o fr(x) z(u) du, which is required, that is, the following block o-support spatial distribution

    ~v(A ;z) -- iv(x;z ) dx, with (B1)

    = ~1, if zv(x)

  • Nonpaxametric Estimation of Spaal Distributions 465

    As in the point-point case, it appears that knowledge of all covariances and cross-variances of the type (B3) is dual to knowledge of all point-block v bi- variate probability distributions.

    Unfortunately, and as opposed to the point-point case, the covariance Ki, io(h;z ) or its variogram counterpart cannot be inferred, since there are no data iv (x;z ) on block v support. One could design a model that would link the point-point and point-block v covariances, allowing thus the determination of the tatter: one such model, the discrete Gaussian model, is currently used in the DK approach (G. Matheron, 1975).

    The drawback of such point-block v covariance modeling is that it occrs prior to the kriging process and thus renders this kriging optimization process model dependent.

    Support Correction for Global Recovery

    It is common practice in linear geostatistics to correct the data histogram for support effect. In Figure B1, a histogram of data is drawn together with an estimate of the corresponding block v-support histogram. The data histogram is supposed to be representative of the whole stationary field D. Both histograms have the same mean, estimated by m*(D; 0), see notation (7), but have different variances. The block v-support variance is estimated to be the dispersion variance of v-support within D; this dispersion variance can be easily derived from the Z- variogram (Journel and Huijbregts, 1981, p. 61-67)

    D 2. (v/D) = D 2. (O/D) - 7z(v , v), with

    fO m*(D;O)= u dq~*(D; O)

    D2*(O/D) = [u - m*(D; 0)] 2 db*(D; u) (B4)

    frequenay

    block-v support variance: D2(v/D)

    su ~ p~ort variance

    0 m*(D;0) z

    Fig. BI. Changc of support on the globalhistogram.

  • 466 Joumd

    being, respectively, the mean and variance of point-support grades within the deposit D, and esfimated from the data distribution q)*(D; z).

    ~z(v, v) = - j du "rz(U - u') du'

    being the mean semi-variogram value within the support volume v

    ~z(u - u') = e{[z (u ' ) - z(u)] ~}

    being the semivariogram of Z(u). Thus the block variance estimate D2*(v]D) is fully determined from relation (B4). But, knowledge of a mean m*(D; 0)and a variance D2*(v[D) are not enough to determine the block v distribution of Fig. B1. An additional hypothesis is needed to determine the shape of this v-support distribution. Various such hypotheses are proposed in Journel and Huijbregts (1981, p. 467-475). The hypothesis of permanence of shape undeflying the af- fine correction of variance consists in assuming that, once corrected for their difference of variances, the two standardized distributions are identical. More precisely, the block v-support distribution ~*(D; z) is wfitten

    q~*(D;z) = ~* [D; m* + (a l /a2) (z - m*)], for all z, with

    fo ~ jo ~ m* = m*(D; O) = u d~*(D;u) = u d~*(D;u): mean grade overD

    (B5)

    estimated from the distribution O*(D; z)

    o~ =D2*(O/D) = (u - m*) z dP*(D;u)

    variance attached to the distribution p*(D;z)

    02 = D2*(v/D) = o21 - 7z(V, v)

  • Nonparametric Estimation of Spatial Distribuons 467

    To transform this estimate into a block v-spatial distribution estimate, ~5 v (A;z), a relative variance-correction factor, similar to that defined in (B6), may be applied

    K2 . . . . _ ~z(V , v ) (v/A ) - D ~ E [0, 1],

    m~4 = u db*(A ; u),

    with

    ~0 ~ D2*(O/A) = (u - m]) 2 dp*(A;u)

    (B7)

    being the mean and variance attached to the estimated distribution qS*(A;z). The corresponding affine correction of variance thus provides the following esti- mate for the block v-spatial distribution within area A

    qb*(A;z) = ~*[A;m] +(1/[1 - K2(v/A)] 1/2} (z - m I)] (BS)

    for allz, all v A , allA CD.

    Remarks

    (a) Formula (B7) assumes that the average variance of point grades within a block v c A, does not depend on the particular data values conditioning the environment of A.

    (b) The affine correction of variance (B8) is done posterior to the kriging pro- cess which provides b*(A;z), thus leaving this process "block distribution model-free."

    (c) The correction (B8) is extremely simple to perform.

    APPENDIX C: INDICATOR CO-KRIGING (ICK)

    At each data point xa, the whole series of indicator data i(x,~;z), for all z, is available. Consequently, when estimating the unknown value i(x;z) one may want to use, not only the data i(x,x;z ) for the same cutoffz; but also the data i(x&;zk) for all other cutoffs Zk 4 = z, that is, one may want to replace the kriged estimate (26) for a cokriged estimate

    K N' *(A;z)- F*(z) = ~ ~ ~k(z) [i(xk;Zk)- F*(z~)] (C1)

    k=l O~k= 1

    The range of variability of z(x) being discretized into K cutoff values ze. Determination of the optimal set of weights (kak(k), k = 1 to K, o~tc = 1

    to N'} calls for solving a cokriging system, which would make use of the whole indicator covariance matrix [Kx(h;zk, zk,), k, k' = 1 to K] see, for example, the cokriging theory in Journel and Huijbregts (1981, p. 324).

  • 468 Joumel

    Although, theoretically possible, such cokriging would be practically ex- tremely tedious, as soon as K>3 (and K of the order of 10 or 20 may be re- quired to discretize the range of variability).

    Important Remark

    Practice in polymetallic deposits has shown that cokriging brings very little improvement over kriging, if all metals are consistently assayed at each data loca- tion. This is precisely the case with indicator kriging, since at each data location xt , the whole series i(x, zk), k = 1 to K, is always present. Therefore it is ex- pected that indicator cokriging will bring also little improvement over simple indicator ktiging.

    There is one case, though, where indicator cokriging is feasible. If the initial randorn function Z(x) is standard bivariate normally distributed, all bivariate dis- tributions of any pair [Z(x + h), Z(x)] are normal and fully determined by the sole Z-covariance function, pz(h) = E~Z(x + h)Z(x)}. As a consequence, the indicator covariance matrix [K~(h; zk, zk')] is solely a function of that z covari- ance, whatever the number K and the cutoffs z, zk' considered. Hence the co- kriging system can be written solely in function of pz(h). This is nothing else than disjunctive kriging! The price, though, to pay for such a dubious improve- ment over simple kriging is rather high: DK requires a bivariate normal distribu- tion hypothesis for either Z(x) or its normal score transform Y(x).

    As to leave the field of nonparametric geostatistics, one may be better off going all the way toward a multivariate normal hypothesis for either Z(x) or Y(x); that is, a multivariate Gaussian (MG) approach should be considered in- stead of DK. Besides its simplicity, MG is the ordy approach which guarantees all order relations of the type (28) (see Verly, 1983).

    REFERENCES

    Abramovitz, M. and Stegun, I., 1965, Handbook of mathematical functions: Dover Pub- lishers, New York, 1046 p.

    Journel, A. and Huijbregts, Ch., 1978, Mining geostatistics: Academic Press, London, 600 p. Matheron, G., 1975, A simple substitute for conditional expectation: Disjunctive kriging

    (p. 221-236), in Guarascio et al. (Eds.) Geostat 1975, NATO ASI: Reidel Publ. Co., Dordrecht, Netherlands.

    Switzer, P., 1977, Estimation of distribution functions from correlated data: Bull. Inter. Stat. Inst., v. XLVII no. 2, p. 123-137.

    Verly, G., 1983, The multi-Gaussian approach and its applications to the estimation of local recoveries, in Proceedings of the MGUS Conference, Denver, January 1982: Jour. Math. Geology, v. 15, no. 2, p. 263-290.