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RECONCILIATION [Modo de
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Clayton v. Deutsch
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Quantitative Kriging Neighbourhood Analysis for the Mini Geologist — A Description of the Method With Worked Ca Examples J Vann1, S Jac Jackso kson n2 and O Bertoli3 ABSTRACT Ordinary kriging and non-linear geostatistical estimators are now well accepted methods in mining grade control and mine resource estimation. Kriging is also a necessary step in the most commonly used methods of conditional simulation used in the mining industry. In both kriging and conditional simulation, the search volume or ‘kriging neighbourhood’ is define def ined d by the user. user. The definit definition ion of thi thiss sea search rch can ha have ve a ve very ry significant impact on the outcome of the kriging estimate or the quality of the conditioning of a simulation. In particular, a neighbourhood that is too restrictive can result in serious conditional biases. The methodology for quantitatively assessing the suitability of a kriging neighbourhood invo in volv lves es som somee sim simple ple tes tests ts (wh (which ich we cal calll ‘Qu ‘Quant antifi ified ed Kri Krigin ging g Neighbourhood Neighbourhoo d Analysis’ or QKNA) that are well established in the geostatistical literature. The authors argue that QKNA is a mandatory step ste p in set settin ting g up an any y kri krigin ging g est estima imate, te, inc includ luding ing one use used d for condit con dition ioning ing a sim simula ulatio tion. n. Kri Krigin ging g is com commo monly nly des descri cribed bed as a ‘min ‘m inim imum um va vari rian ance ce es esti tima mato tor’ r’ bu butt th this is is on only ly tr true ue wh when en th thee neighbourhood is properly defined. Arbitrary decisions about searches are highly risky, because the kriging weights are directly related to the variogram model, data geometry and block/sample support involved in the kriging. The criteria to look at when evaluating a particular kriging neighbourhood neighbourho od are the following: 1.
the slope of the regression of the ‘true’ block grade grade on the ‘estimated’ block grade;
2.
the weight weight of the mean mean for a simple simple kriging; kriging;
3.
the distribution distribution of of kriging weights themselves themselves (including the proportion of negative negative weights); and and
4. the krig kriging ing vari variance. ance. Outside of the technical geostatistical literature, there is little in the published domain to describe the nature of QKNA and no practical presentation of case examples. In this paper we attempt to redress this by setting setti ng out the calcu calculatio lations ns requi required red for QKN QKNA A and defining defining some approaches to interpreting the results. Several practical worked mining case examples are also given. Finally some comments are made on using the results results of QKN QKNA A to ass assist ist with blo block ck siz sizee sel select ection ion,, cho choice ice of discretisation and mineral resource classification decisions.
INTRODUCTION This paper presents the methodology for quantitati quantitatively vely assessing the suitability of a kriging neighbourhood: ie the combination of the search strategy and block definition used in a kriging. In this paper ‘kriging’ refers refers to ordinary kriging (OK), unless otherwise indicated and the process of assessing a kriging neighbourhood (for any kind of kriging) is referred to as ‘Quantified Kriging
The cri criter teria ia for ass assess essing ing the qua qualit lity y of specified kriging neighbourhood (or ‘neighbour establish esta blished. ed. How Howev ever er,, outs outside ide of the spec literatur liter aturee (Arm (Armstro strong, ng, 1998 1998;; Dav David, id, 197 1977; 7; Chiles and Delfiner, Delfiner, 1999), there is little in the to describe QKNA or to guide geologists in imp this paper an attempt is made to redress this by calculations required for QKNA and defining so to interpreting the results. Several practical work examples are also given. Finally, some commen using the results of QKNA to assist with block choice choi ce of disc discretis retisation ation and mine mineral ral reso decisions. This paper assumes the reader has a basic u linear line ar geo geostati statistics stics.. Arms Armstron trong g (199 (1998), 8), Chile (1999), Isaaks and Srivastava (1989) or Journel (1978 (19 78)) can be ref referr erred ed to for the req requir uired ed variograms and kriging.
MOTIVATION The motivation for QKNA
Ordinary kriging (OK) and non-linear geostatis including uniform conditioning and multiple in are no now w wid widesp esprea read d an and d rou routin tinee me metho thods ds estimation and grade control. control. In this paper ‘krigin unless unle ss othe otherwis rwisee indi indicate cated. d. Krig Kriging ing (Math (Matheron eron 1963b; Journel and Huijbregts, 1978) is also a n the main methods of conditional simulation use industry, eg sequential Gaussian simulation (SGS (TB) and seq sequent uential ial indi indicato catorr simu simulatio lation n simulatio simu lation n (Jou (Journel rnel,, 197 1974; 4; Lant Lantuejo uejoul, ul, 2002 utilis uti lised ed by min mining ing ge geolo ologis gists ts in gra grade de estimation and risk analysis applications. In both kriging and conditional simulation, the is defined by the user (or at least it should be: Sign to vote on this title accepting defa ‘black boxupapproach’ may involve Arbitr Arb itrary ary sp speci ecifi ficat cation the nei neigh ghbo bourh urhoo oo Useful Notofuseful ion because the kriging weights are directly related t model, data geometry and block/sample support kriging. Whilst kriging is commonly and correctl
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