Nonparametric Geostatistics

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1 Non-Parametric - An Introduction

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Nonparametric Geostatistics

Advanced Book Search Browse by Subject. Make an Offer. Find Rare Books Book Value. This roughness is characterized later-in probabilis- tic terms-by the indicator variograms The first step of such an approach is to characterize the spatial variability of the indicator random functions I x;z.

npsp: Nonparametric spatial (geo)statistics

Note that there is one such indicator random function for each argument cutoff z. Therefore a complete indicator structural analysis brings more information than a mere classical Z structural analysis. In practice the indicator semivariograms 7z h;z should be estimated first, from the indicator data i xa;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. For this median value, the standard binormal indicator correlogram is writ- ten Abramovitz and Stegun, , p.

Other functions

S2 z increases in [0,. S2 z decreases in [. Determine the value z u for which the expefimental sill S 2 z appears to be maximum and dose to. They would be best defined for quantiles z close to the median z M. Beware that S 2 z is usually not symmetrical around the median z M.

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. Minimization of this form yields the classical "simple" kriging system without any Lagrange parameter Journel and Huij- bregts, , p.

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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. In practice, it will be calculated by a discrete sum of the type In practice, in the presence of highly variant data distribution and "outlier" data values, the more robust M-type estimator may be preferred.

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. The nonparametric indicator approach may represent the long awaited comeback to a no-nonsense problem-solving approach.

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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. Declustering a data set. In Fig. Choosing an optimal cell size.

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. 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, 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. 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, , p.

Changcofsupporton theglobalhistogram. An additional hypothesis is needed to determine the shape of this v-support distribution. Various such hypotheses are proposed in Journel and Huijbregts , p. 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. 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.

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. 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. 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.

Nonparametric Methods for Point Processes and Geostatistical Data

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, Journel, A.

Matheron, G. Switzer, P. XLVII no.