Nugget Effect versus Screen Effect

Nugget Effect versus Screen Effect
Yücel Tandogdu
Eastern Mediterranean University, Dep. Of Applied Mathematics
Magusa (Famgusta), North Cyprus
Mersin 10, TURKEY.
E-mail: [email protected]
1. Introduction
In the stochastic estimation of a variable of interest in earth sciences, each variable is
stationary with respect to time, but changing with respect to space, hence the term regionalized
variable (RgV). Spatial characteristics of the variable are identified by the semi-variogram function
h). These are the nugget variance (S), the range of influence (a), and the sill value (C). They are
used in the estimation process as input to the linear unbiased estimation technique known as
Kriging.
2. The Nugget Effect
The notation Q(x) is used for a RgV, where x∈G⊂R3. Then a RgV is a function Q:G×Ω→R1 .
Here Ω is the sample space. Q(x) is a random variable at a fixed point x in an ore deposit having a
volume G. But Q(x) is a rnadom function (RF) over the whole deposit G expressing the random and
structured aspects of the RgV. That is, for each pair of points xi and xi+h at h distance apart within
the deposit, the corresponding random variables Q(xi) and Q(xi+h) tend to be dependent to some
extent expressing the spatial structure of the RgV (Matheron). The semi-variogram function is
defined as
(1)
γ (h) =
1
1 n
E{[Q( x + h) − Q( x)] 2 } or γ (h) =
[q ( xi ) − q ( xi + h)] 2
∑
2
2n i =1
which can be computed using sample data. γ(h) measures the lack of dependence between
q(x) and q(x+) (Journel et al). Using one of the many theoretical γ(h) models, one can determine the
spatial parameters (S, a, C) for the variable under study. S is mainly attributed to sampling, essaying
errors and sampling interval. The nugget effect is defined as ε=S/C.
3. Kriging and the Nugget Effect
Kriging is the name given to the estimation technique used in geostatistical studies. It uses the
concept of best linear unbiased estimation known as BLUE. In doing so, it minimizes the variance
of the errors σk2 between the actual and estimated values at sample locations. When ε→0 estimates
obtained by kriging are better than those obtainable by conventional methods. For ε=0 the kriging
system is given by (Rendu)
( 2)
 n
 ∑ w j γ ( q i , q j ) + λ = γ ( q i , Q )
j =1
 n
∑ w j = 1
 j =1
for
i = 1, 2, L , n
Here
w
: Weights assigned to each sample
γ(qi ,qj) : Average semi-variogram values between the samples
γ(qi,Q) : Average semi-variogram values between the samples and the block being estimated.
However, obtaining ε=0 is not possible in application. S will always appear on the semivariogram and the first part of the kriging system given in equation (2) will become
n
(3)
∑ w γ (q , q
j =1
j
i
j
) − wi S i + λ = γ (qi , Q)
for
i = 1,2,L, n
The kriging variance is computed by
( 4)
σ k2 = −γ (Q, Q) + ∑ wi γ (qi , Q) + λ
When ε=0 kriging distributes weights to samples closer to the center of the block being
estimated (screen effect). As ε increase more weight is assigned to distant samples. While low ε is
desirable for a sound spatial study, the resulting screen effect hampers the distribution of weights to
distant samples which the earth scientist may think are worth taking into account. Trials have shown
that when S>0.1C, σ2k starts becoming greater than the estimation variance (Tandogdu) which
assumes equal weights for all data values.
REFERENCES
Matheron, G. (1971). The Theory of Regionalized Variables and its Application. Les Cahiers du
Centre de Morphologie Mathematique de Fontainebleau.
Journel, A. G. and Huijbregts, CH. J. (1978). Mining Geostatistics. Academic Press. London.
Rendu, J. M. (1981). An Introduction to Geostatistical Methods of Mineral Evaluation. South
African Institute of Mining and Metallurgy. Johannesburg.
Tandogdu, Y. (1996). Estimation Variance in Estimating a Regionalized Random Variable.
Proceedings of Environmental Statistics and Earth Science, 157-162. Brno, Czech Republic.
SOMMAIRE
En science du sol; l’estimation stochastique d’une variable presente deux aspects qui sont en
contradiction: l'effect nugget et l'effect ecran. L'outil utilisé pour l'analyse de l'espace est la fonction
de semi-variogramme (γ). Kriging utilise des parametres d'espace obtenues de γ pour estimer le
variable d'interet dans une region donnée ou dans un bloque de sol. La variable de nugget trouvée
dans γ entraine la concentration de poids aux echantillons du centre quand elle est trop basse, et la
dispersion du poids aux echantillons distants quand elle augmente (effect ecran). Une solution prati
que pour le problem du poids est proposée.