Tipos de Krigeage

8/12/2019 Tipos de Krigeage

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MlNlNG ENGlNEERlNG HANDBOOKux x asD (Xi,Xj) (5.6.19) ing with exogenic drift. A method of universal kriging thathe geologists interpretation of grade-zone trends as he ex

asD AV.(y(B,Xi)) (5.6.20) drift is zoned kriging. Cokriging is the method of kriginaccounts for the correlation of a primary variable with a seW ~ as AVE(y(B,B)) (5.6.21) ary variable, for exarnple, gold with silver or molybdenumcopper, etc. When wkriging is used with qualitative secowhere is the variance of samples in the deposit, r(Xi,Xj) is variables such asaiteration, rock type, or other geologic feathe vaiue of the variogram function between samples Xi ndXj, it s known as soft krig ng. Disjunctive kriging is a methodA VE(y(B,Xi)) is the average vaiue of the variogram betwtm the which attempts to estimate not only the local block gradblock and sample Xi and A VE(y(B3)) is the average vahe of also the shape of the tomage-grade distribution within the the variogram between ail points within the block.LognormalKriging Lognormal knging is a method ofnon-linear kriging that was developed to improve estimati011when 5.6.8.6 Volume-variance Effects and Recoverthe underlying data are distributed according to a lognonnai Functlonsprobability distribution. The basics of lognormal kriging include: The volume-variance effect refers to the inverse relatio( 1 the vario@am c o m ~ u t e d sing the natural logs ofth data, between the distnbution variante and h volume of block(1 the Lnging ystem t0 a waghted avage 0f volumevariance effst is by fige selatiothe natural logs of the data, and (3) the kriged log average is as followsthen transformed back to normal values usine lomormai trans-formation similar to that shown earlier in 4.-5.6Tl. The mathe-matics of lognormal kriging are complex and are discossed inRendu (1978) and Journel (1978, 1980).Complications n the practicai application of lognormai krig-ing are many, including a strict requirement for a lognormaidistribution and a variogram which is stationary over the fieldof estimation. Serious local and global biases may occur if eitherof these conditions are not met. In addition, there s a tendencyfor lognormai kriging to overestimate the high-grade end of thepopulation when the coefficient of variation is greater than 2.0.Lognormal kriging is recommended only for speclai purposeswhere the results can be monitored closely and adjusted topre-vent biases.Indicatori robability Kriging: Indicator knging and pmba-bility knging are related methods that are used to improve esti-mation when ore zones are erratic and grade distributiom arehighly variable and complex. Advantages of indicator kxiginginclude less smoothing of estimated grades than ordinary kngingand robustness in handling nonstandard grade distributions.The first step in indicator kriging is to set one or more cutoffswith which to define indicator variables. Given a cutoff g theindicator variable is set to 1 if the grade is above gc or O if thegrade is belowgc(the order of the 1,O) coding may be reversed);indicator variables are coded similarly for each desired cutoff.Variograms are modeled for each indicator variable and an ex-pected value for each indicator is estimated using ordinary krig-ing and the appropnate indicator variogram.The resulting indicator estimates, which may be interpreted

as either the probability that the block wili be above the cntoffor the percentage of the block that is above cutoff, are used toestimate the grade of the block as follows

where each j s the estimate for the indicator for cutoffj, gj isthe estimated grade for the intervalj oj+1, and n is the numberof indicator cutoffs. The interval grades gj are usually estimatedas the average of the cutoff grades for the interval, or, if theinterval is large, may be estimated from the kriged grade of thosedata in the intervalj to 1. The prior method is more precisewhen a large number of indicator cutoffs are defined; the latteris most often used for a single cutoff.Other Types of Kriging: Other types of kriging that arenot widely used include universal kriging, cokriging, disjunctivekriging, and soft kriging. Universal kriging is a method to incor-porate trends into the kriging equations. If the trends are definedaccording to a secondary variable, it is known as universal krig-

where oS s the variante of blocks in the deposit, SZsvariance of samples in the deposit, and P is the variasamples in the block. The variance of samples in the blocbe estimated from the variogram as follows

where y is the average of the variogram for samples a block with the size and onentation of the mining blockThe volume-variance relationship is unimportant wheentire deposit is above the cutoff grade or where the ore snonselectively. Generally, however, the cutoff grade s hand only a portion of the mineralized grade diitribution istively mined as ore. The shape of the @e-tonnage distribas defined by the distribution variance, is then a criticalin detennining the grade and tomage above cutoKFor practical resource estimation purposes, the variamining blocks is generally larger than the variance of resource estimation blocks. The variante of mining blogenerally smailer than the variance of resource estimation for polygonal estimation.Polygonal estimation underestimates tons and overestgrade for low cutoffs. At higher cutoffs, tonnage and are both overestimated. Kriging tends to overestimate tounderestimate grade for low cutoffs. At higher cutoffs, toand grade are both underestimated.For polygonal estimation, the difference between estiand mined reserves is usualiy handled with dilution factorsa fixed tonnage is added with a grade that is less than the These dilution factors are adequate for correction of oreserves but are not accurate for smaller areas if localvary significantly from the average grade. Caution must aobserved since dilution factors wiii vary according to thegrade, the population variance, and the amount of varianduction between the polygonal and mine block distributishould be noted that polygonal reserve estimates may rdiiution factors for both volurne-variance effects and comining geometric effects.Kriging reserves are corrected for volume-variance according to the distribution of mining blocks within the rblock as 1) the variance and distribution of mining block9the reserve block is estimated, and 2) the tonnage andabove cutoff is estimated for the block. The mining blockbution parameters are most effectively determined by ~ m


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Indicator kriging I K )f . . .T h e v a lu e o f a W L ~ L ~ ;?t i : 5 1 r 1 :f semi-var iograrns for ihr : .L.: 1 T .i i ::

* *values in brackets are .::: e cr.zzr: 1 . i :III impac t o f chang ing corz s:z= :E ::::r ~ .-. :

. .F greater im pact wi l l be earr ,ec zv. : :z:r: k. .; core s am pl ing to s ampl ing eas r, i,;sr z; ;r ;:I the s tope-face.

4.16 I N D I C A T O R K R I G I NG IK )4 16 1 relwaste estimationInd ica to r k r ig ing i s an enhancement o f the s implekriging technique , the di fference being tha t i tdoes no t ca lcu la te a g rade o r m e ta l accumula t ionva lue bu t the p ropor t ion o f the b lock w hich canbe expected t o conta in va lues abo ve a given cut-off . The technique is par t icular ly appl icablewh ere s t r i ct o re lw as te boundar ies ex i st wi th ing iven b locks , e .g . l a rge coppe r porphyr ie s wh eregrade zon ing i s the ma jor con t ro l , and in lowgrade depos i t s where the cu t-o ff va lue is o f ma jo rconcern. Simple kr iging es t imates the expectedgrade o r m e ta l accumula tion o f a g iven b lockand , as such, i s a t rue ref lec t ion o f it s va lueproviding tha t the block is mined as a comple teun i t . I t does no t , how ever , a l low the use r tode te rmine w he the r the a ss igned va lue o f theblock is heavi ly biased by a s ingle high value o r i fi t conta ins areas of both or e and w as te m ateria l.W e c a n e x a m i n e t h e p ro b l e m f u r t h e r b y c on -s ide r ing a b lock o f mine ra li zed g ro und which hasbeen evaluated by dril1 holes at i ts four corners .T h e grade assigned to th e block wi l l , in th isinstance , be a s sumed to be the a r i thme t ic meano f the four holes i .e . the th ickness is cons tant) .In th e case o f b lock A in Figure 4.40, the averagegrad e is 5.75 g / t and , g iven a cu t-o ff g rade o f g /t , i t wo u ld thus be cons ide red a s o re . Al thoughblock B has an ident ica l grade and w ou ld a lso beclassified as ore, this decision is hig hly depe nden to n o ne value , a fac t which is no t reflec ted di rec tlyin th e resul t .Suppo se no w tha t the type o f ma te ri a l used tova lue the b lock i s t aken in to accoun t so tha t o remater ia l i s indica ted by the va lue and wa s te by


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Geostatistical ore reserve estimationDDH 1 DDH 2 DDH 1 DDH 2

v g 5 75rs vg 5 75ElDH 4 DDH 3 DDH 4 DDH 3

Fig. 4.40 Tw o blocks showing gold grades ocornerboreholes. Block A has a mean indicator value o 1. 0and Block B a value of 0.25, for a cut-off grade o 5 g/

the value of O. Th en , in the first exa mp le, thesample values would be categorized, accordingto a cut-off grade of 5 glt , as fol lows:Dril1 hole Grade Category Indicator value

g/t ore2 5 g/t ore3 6 g/t ore4 6 g/t ore

T h e average grade of the block su ggests that i tis or e and the indicator values suggest th at i t canbe expected to contain 100 ore-grade material .I f we now consider the second example , wehave:Drillhole Grade Category Indicator value

1 g/t waste O2 g/t waste O

3 g/t waste O4 19 g/ t ore

Althou gh the average grade of the block sug-gests that the block is ore, the mean indicatorvalue is 0.25 suggesting that only 25 is o f ore-grade material . This fact must be consideredduring the const ruct ion of the f ina l gradeltonnage curve to avoid an overest imat ion o f theore tonnage.

The appl ica t ion of the above technique ingeostatistical ore-reserve estimation procedures

was first prop osed by Journe l (1983) and furthrefined by Lemmer (1984). The mathematicexpression for an indicator variable is i(x;which i s based on the grade z(x) of a samppoint x and on the cut-off grade z Hence:

O nc e al1 the grade (or accum ulation) data habeen transformed in this way, experimentsemi-variograms are generated and mathematicmo dels fitted as described in sections 4.5 and 4.Where a spherical scheme model is deemed aplicable, the model equation is:yi h;t) = 1, 1[1.5 h/a) 0. 5 hl a) ~]or h < a

= l o f o r h 2 a where 1 and are equivalent to C o and C ingrade semi-variogram. Th ese indicator semvariogram parameters are then used to produblock indicator values using simp le kriging tecniques (section 4.1 O), the m et ho do lo gy beinreferred to as indicator kriging (IK). Th is krigeindicator value for a block thus represents threcovery function for that block at a specifiecut-off. T h e above procedure can be repeated fa rang e o f cut-off values.

T he final prod uct of the exercise wil l thus beblock plan of kriged grades and indicator valuwh ich is of considerable benefi t to an ope ratiowhere selective mining is possible within thconfines of ore-evaluation blocks. R evised bloctonnages can thus be computed and used produ ce a m or e real istic grade-tonnage curv(the tonnages being determined after repeateuse of IK for a range o f cut-off grades).

4 16 2 Semivariogram modelling withIKLemmer (1986) states that indicator semvariograms are much more robust with respeto anomalous out l ie rs than grade or accumul


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tion semi-variograms. Indicator semi- REFERENCESvariograms can thus be modelled with moreconhdence. Also, when the nugget effect is notto o small, as in the case of Witwa tersrand goldaccumulations cm glt), then an approximatelinear relationship exists be tween yI h;z) andy h). As a result, t he mod ellin g o indicatorsemi-variograms can be used t o provide a morereliable model for the associated grade o r accu-mulation semi-variogram. Th e metho d involvesth e plotting o f yI h;z) against y h) fo r succes-sive lags for a specific cut-off). A least-squaresbest-fit line is then plotted thr oug h t he points.Th e equation o f this line is thus:

where A z) is the intercept on the yI h;z) axisand B z) is the gradient of the line.Co and C can no w be predicted by using I andI fr om the indicator semi-variogram and b y set-ting h a and h in equations 3) and 1)respectively, as below:When

h a yI h;x) I I and y h) C o Ctherefore

Hence

When

andI A r) B x) Co)

hence

The range is assumed to be the same in bothsemi-va riograms. ~ h i sredicted rnodel can nowbe superimposed on the experimental gradelmetal accumulation semi-variogram to testwhether a good fit has been achieved.

Boakye, E. B. 1989) Statistical and geostatisevaluation of an alluvial gold deposit in the ceregion of Ghana. Unpublished PhD thesis, Unisity of Wales Cardim, 334 pp.

David, M. 1988) Handbook ofA ppli ed Advanced Gtatistical Ore Reserve Estimation, Elsevier Devements in Geomathematics Q lsevier, Amsterd216 pp.

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Journel, A. G. 1983) Non-parametrics estimatiospatial distributions. International AssociationMathematical Geology Journal, 15, 445-468.

Lemmer, 1. C. 1984) Estimating local recoverreserves via IK, in Geostatisticsfor Natural ResouCharacterization eds G. Verly et al.), Reidal, Ddrecht, pp. 349-364.

Lemmer, 1 C. 1986) Grade-indicator Plots for GReef Data: Their Use in the DeterminationVariogram Parameters, in Extended Abstracts, Congress 86, Geol. Soc. South Afr., Johannesbpp. 237-240.

Matern, B. 1960) Spatial Variation, Almaenna Flaget, Stockholm, 144 pp.Matheron, G. 1971) T h e Theory of Regionalized Vables and its Applications, Centre de MorpholoMathematique, Fontainbleau, 21 1 pp.

Royle, A. G. 1977a) Global estimates of oreserves. Trans . I M M , 86 A9-A17.

Royle, A. G. 1977b) How to use geostatistics forreserve classification. World Mining, February, 56.

Smith, 1. H. 1987) Geology, exploration and evation of the gold deposits of Suriname. UnpublisPhD thesis, University of Wales Cardiff), 639

BIBLIOGRAPHYArmstrong, M. 1984) Improving the estimation

modelling of the Variogram. in Geostatistics or Nral Resources Characterization, Vol. 1 eds G. Veral.) , Reidal, Dordrecht, pp. 1-20.


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Geo s ta t i s t i ca l o re reserve es t ima t io nBlais, R. A. and Carlier, P. A. (1968) Applications ofgeostatist ics in ore evaluation. C I M M , 9 (Spec.Vol . O re Reserve Est imat ion and G rade Control ) ,

48-61.B rooke r , P. 1 (1975) Optimal Block Estimation byKriging, in Aust . I M M , Proceedings, No. 253.B rooke r , P. 1 (1976) Block Estimation at Various

Stages of Deposi t De velopm ent , in Proceedings of the14th International A P C O M Symp osiu m, Perrn. StateUniversi ty , pp. 995-1003.Clarke, 1 (1979) Practica1 Geostatistics, Applied Sci-ence , New York , 129 pp .Clarke, 1. and White, B. (1976) Geostatistical Mod-ell ing o f an Or ebo dy as an Aid to M ine Planning, inProceedings of t l ~ e 4th lrlternarional Sym po siu m, Perin.State Universi ty . pp. 1004-1012.Crozel, D. and David, D. (1985) Global est imationvariance formulae and calculation. Matl i . Geol . ,17( 8), 785-796.Dagbert , M. and David, M. (1983) New develop-me nts in the categorization of ore reserves by geo-statistical methods, in A I M E Fa11 Meet ing, Sal t LakeC i t y .Da vid , M . (1970) Geostatistical or e est imation astep by step case study, C I M M E 12, (Spec. Vol.Decision Ma king in the Mineral Industry) 185-191.

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Parker, H . M . (1979) Volum e variance relationshiuseful tool for mine planning. Eng. Mining J . 1106-23.R aym ond , G . F. (1979) O re est imat ion problemsan erratically mineralized ore bod y. CIMAVI Bull..(806), 90-8.Rendu, J . M. (1970) Geostat i s t ical approach to reserve calculation. Eng. Mining J. 171, 112-18.Royle, A. G. (1975) A Practical Introdu ction Geostatist ics, unpublished course notes, Depment of Mining and Mineral Sciences, Leeds Uversity.Roy le, A. G. (1979) Estimating sm all blocks of oho w to d o it wi th confidence. World Minirlg, 32, 57.Royle , A . G . , New ton , M . J . and Sar in, H . (1972) G eostatist ical factors in design of m ine sapl ing programmes. T ra ns . I M M , 81, A81-A88.

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Tipos de Krigeage