Yerbilimleri, 31 (1), 45–52
Hacettepe Üniversitesi Yerbilimleri Uygulama ve Araştırma Merkezi Dergisi
Journal of the Earth Sciences Application and Research Centre of Hacettepe University
Geostatistical grade modeling of Choghart North anomaly iron ore
deposit through disjunctive kriging
Choghart kuzey demir cevheri yatağı anomalisi için ayırıcı kriging
yöntemiyle jeoistatistiksel tenör modellemesi
Ali Akbar DAYA, Hossein HASSANI
Faculty of Mining and Metallurgy, Amirkabir University of Technology, Tehran, IRAN
Geliş (received) : 01 Aralık (December) 2009
Kabul (accepted) : 02 Şubat (February) 2010
ABSTRACT
This paper is devoted to the application of the disjunctive kriging method in the Choghart north anomaly iron ore
deposit in Yazd province, Iran. The local distributions of the values of a regionalized attribute at unsampled locations can be assessed by disjunctive kriging. The case study consists of borehole samples measuring the iron concentration. A Gaussian isofactorial model is fitted to these data and disjunctive kriging was used to assess the local probabilities that the actual concentrations exceed a threshold value and to divide the ore into an economic
and uneconomic parts. The tools and concepts are complemented by a set of computer programs and applied
to the case study. The study showed that disjunctive kriging can be applied successfully for modeling the grade
of an ore deposit.
Keywords: Choghart north anomaly, disjunctive kriging, geostatistics, nonlinear kriging.
ÖZ
Bu çalışma, İran’ın Yazd bölgesinde bulunan Choghant kuzeyi demir cevheri anomalisine ayırıcı kriging yönteminin
bir uygulamasını içermektedir. Bölgeselleştirilmiş bir değişkenin, örneklenmemiş noktalardaki değerlerinin yerel dağılımları ayrıca kriging ile kestirilebilmektedir. Sunulan çalışmada sondaj örneklerinde yapılan demir derişimleri kullanılmıştır. Bu verilere Gauss izofaktoryel model uydurulmuş ve gözlenen derişimlerin belirli bir eşik değerden yüksek olma olasılıklarının kestirilmesi ve cevheri ekonomik ve ekonomik olmayan kısımlara ayırabilmek amacıyla ayırıcı
kriging yöntemi kullanılmıştır. Bu çalışma kapsamında kullanılan araç ve kavramlar bir dizi bilgisayar programı yardımıyla uygulanmıştır. Çalışmanın sonuçları, ayırıcı kriging yönteminin cevher tenörlerinin modellenmesinde başarıyla uygulanabildiğini göstermiştir.
Anahtar Kelimeler: Choghant kuzey anomalisi, ayırıcı kriging, jeoistatistik, doğrusal olmayan kriging.
H. Hassani
E-mail: hhassani@aut.ac.ir
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INTRODUCTION
The aim of much drill hole sampling and mapping is to enable miners and their advisors to
predict values of ore deposit properties at sites
that have not been sampled. Kriging in its various forms is one way of doing this, providing estimates of minimum and known variance, and
there is now a large literature on its applications in earth science.
Miners must decide for each block from its
sampling estimate whether to extract it for processing (if its concentration exceeds the economic threshold) or send it to waste, and for
this they want to know the probability that the
true value exceeds the threshold. To solve this
problem Matheron (l976) developed the method known as ‘disjunctive kriging’. The main
advantage of disjunctive kriging over the simpler techniques, whether of kriging or prediction
from a classification, is in providing these probabilities. These probabilities should enable a
miner or his advisor to assess the risks associated with imprecise estimates. The probability that a critical value is exceeded depends on
the distribution function. For a normal distribution these probabilities can be read from published tables or calculated numerically using standard procedures.
Disjunctive kriging involves transforming data
to a normal distribution and then determining
for each point of interest the probability that
the true value exceeds the threshold. Yates et
al. (1986a, 1986b) have provided a full derivation of Matheron’s method and illustrated it with
examples in hydrology.
Other geostatistical methods used to calculate
the probability that the true value exceeds the
threshold are conditional simulations (Chile’s
and Delfiner, 1999), multigaussian kriging (Verly,
1983; Emery, 2005) or non-parametric estimators such as indicator kriging and its flavors
(Deutsch and Journel, 1998). Disjunctive kriging
has been applied mostly in the bivariate Gaussian model (Rendu, 1980; Yates et al., 1986a,
1986b; Yates and Yates, 1988; Webster and Oliver, 1989, 2001; Wood et al., 1990; Oliver et al.,
1996; Chica-Olmo and Luque-Espinar, 2002).
This paper is intended to investigate and to evaluate the potential and applicability of disjunctive kriging as a tool for modeling the grade of
the Choghart North Anomaly Iron Ore Deposit. All of the statistical and geostatistical calculations and graphical output generated for
this case study were made using the software
system implemented by the authors.
DISJUNCTIVE KRIGING
Principles of the disjunctive kriging of blocks
and estimating reserves were given in detail in
earlier works (Chile’s and Delfiner, 1999). Basically, in disjunctive kriging, the variable z0 to be
estimated is decomposed into a sum of disjoint (uncorrelated) components of sample values. When kriging of the separate components
is possible, the procedure is feasible, i.e., when
the joint probability density function of z0 (or the
transformed Yo) and each sample za (or Ya) is of
isofactorial type.
In practice, a continuous variable like the iron
grade of an ore deposit can always be transformed by anamorphosis into a Guassian equivalent Y, and then only a joint Guassian hypothesis for the probability density function (PDF) of
samples and blocks is required.
We start by transforming the measured variable, Z(x), to one Y(x) that has a standard normal
distribution such that
Z ( x ) = Φ[Y ( x )]
(1)
This is done using Hermite polynomials, which
are related to the normal distribution by
Rodriguez’s formula.
H k ( y) =
1
d k g( y )
k
k !g( y) dy
(2)
Where; g(y) is the normal probability density
function, k is the degree of the polynomial taking values 1, 2,… and (1/(k!)0.5 is a standardizing factor. The first two Hermite polynomials
are as follows.
H o ( y) = 1 ,
H 1 ( y ) = −y ;
(3)
Daya and Hassani
Thereafter the higher order polynomials obey
the recurrence relation.
H k ( y) = −
k −1
1
yH k −1 ( y ) −
H k −2 ( y )
k
k
The Hermite polynomials are orthogonal with
respect to the weighting function (-y2/2) on the
interval from −∞ to +∞ ; they are independent components of the normal distribution of
ever increasing detail. Many functions of Y(x)
can be represented as the sum of Hermite polynomials.
f {Y ( x )} = fo H o {Y ( x )} +
1 H 1 {Y ( x )} + f2 H 2 {Y ( x )} + ,
(5)
Because the polynomials are orthogonal, the
coefficients required for Eq. (1) can be calculated as given below.
Z ( x ) = Φ[Y ( x )] = Φ0 H 0 {Y ( x )} +
Φ1 H1 {Y ( x )}Φ2 H 2 {Y ( x )} + =
∞
∑ Φ H {Y (x)}
k
k=0
k
2
2
∑λ
ik
cov[ H k {Y ( x j )}, H K {Y ( xi )}] =
cov[ H k {Y ( x j )}, H k {Y ( xo )}] ; for all j
(6)
(7)
So from n points in the neighborhood of x0 estimations of the Hermite polynomials are:
(9)
Or alternatively the following expression can be
written.
n
∑λ ρ
i =1
ik
k
( xi − x j ) = ρ k ( x j − x0 ); for all j
(10)
In particular, the procedure enables us to estimate Z(x0) by
Zˆ ( x0 ) = Φ{Yˆ ( x0 )} = Φ0 +
Φ [ Hˆ k { y( x )}] +
1
1
0
Φ [ Hˆ { y( x )}] +
k
0
2
2
(11)
The kriging variance of Hˆ k {Y ( x )} is given by
the following expression.
n
The transform is invertible, and so the results
can be expressed in the original units of measurement. To krige the variable of interest, Z(x),
we simply krige the Hermite polynomials separately and sum their estimates to give the disjunctive kriging estimator.
Zˆ DK ( x ) = Φ0 + Φ1 Hˆ 1k {Y ( x )} +
Φ Hˆ k {Y ( x )} +
n
i =1
(4)
47
σk2 = 1 − ∑ λik ρ k ( xi − z0 )
(12)
i =1
The disjunctive kriging variance of fˆ[Y ( x0 )] is
given below.
∞
σ ( x0 ) = ∑ fk2σk2 ( x0 )
2
DK
(13)
k =1
Once the Hermite polynomials have been estimated at x0, the conditional probability, that
the true value there exceeds the critical value,
Zc can be estimated. The transformation Z(x)
=F[Y(x)] means that zc has an equivalent yc on
the standard normal scale. Since the two scales are monotonically related their indicators
are the same.
n
Hˆ kk {Y ( xo )} = ∑ λik H k {Y ( xi )}
i =1
(8)
The next step is inserting them into Equation 7.
λik are the kriging weights, which are found by
solving the simple kriging equations.
[ Z ( x ) ≤ zc ] = [Y ( x ) ≤ yc ]
(14)
[Y ( x ) yc ] , which is the complement of
[Y ( x ) ≤ yc ] , the kth Hermite coefficient is as
below.
For
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fk = ∫
+∞
−∞
yc
H k ( y )g( y ) dy
∫
CASE STUDY
[ y ≤ yc ]H k ( y )g( y ) dy
−∞
(15)
The coefficient for k=0 is the cumulative distribution to yc.
f0 = G ( yc )
(16)
And for larger k
fk =
1
H k −1 ( yc )g( yc )
k
(17)
The indicator can be expressed in terms of the
cumulative distribution and the Hermite polynomials:
Ω [Y ( x ) ≤ yc ] = G ( yc )
∞
1
H k −1 ( yc )g( yc ) H k {Y ( x )}
∑
k
k =1
(18)
Its disjunctive kriging estimate is obtained by
as follow.
ˆ DK
Ω [ y( x0 ) ≤ yc ] = G ( yc )
L
1
H k −1 ( yc )g( yc ) Hˆ kk { y( x0 )}
∑
k
k =1
(19)
Where, L is some small numbers. The kriged
k
estimates Hˆ k { y( x0 } approach to zero rapidly
with increasing k, and so summation need extend over only few terms. This is the same as
ˆ DK
Ω [ z( x0 ) ≤ zc ] . In this instance, we are
interested in the probability of excess, and so
we compute.
Ω̂
DK
[ z( x0 ) zc ] Ω̂DK [ y( x0 ) yc ]
1
Hk −1( yc )g( yc ) Hˆ kk { y( x 0)}. (20)
1 − G ( yc ) −∑
k
k 1
L
The deposit under scrutiny in the case study
is an iron deposit, which is in the northwest of
the famous Choghart mine in the Central Iran
Zone (Figure 1). The deposit is composed of
magnetite-hematite with varying amounts of alkali amphiboles. It occurred in large low-grade
orebodies. There is no general consensus regarding the origin of this iron oxide deposit.
Some authors (Moor and Modabberi, 2003) believe that it has been formed directly from magmas filling volcanic diatrems or flowing as lavas, while others (Samani, 1988) suggest metasomatic replacement of preexisting rocks by
hydrothermal (deuteric) solutions charged with
iron that leached from cooling felsic plutons.
The orebody and the metamorphosed country
rock are cut by several diabasic dikes.
Hematite is the second most ubiquitous mineral after magnetite. Although some primary hematite is also found in the drill cores, most of
the hematite is secondary in origin. Some goethite and hydrous iron oxide occur on the surface, but disappear rapidly with increasing depth.
Calcite, dolomite, secondary hematite and
talc occur throughout the orebody as veinlets
and cementing material of oxidized ore. Rutile and goethite are probably the results of a total transformation of the earlier formed martite.
According to field observations, the host rocks
are metamorphic rocks, hornblendite–pyroxenite, volcanic rocks (rhyolite, andesite) and the
pyroclastic fill of the vent.
Principally, this deposit was explored by 26 boreholes (Figure 2). The data file gives the name
of each drill, the coordinate of drills, the grade
of each element, the measure depth and azimuth of each drill, the inclination of boreholes
and level, and lithology coding etc. In general,
the drilling grid is irregular (see Figure 2). As the
bench height for mining is fixed at 12 m, borehole samples were regularized at 12.5 m intervals.
Daya and Hassani
Figure1.
49
Figure 1. The location of the North Anomaly of Choghart deposit in central Iran.
Şekil 1. Choghart cevheri kuzey anomalisinin Orta İran’daki konumu.
Figure2.
APPLICATION
Data Transformation
Figure 2. Borehole sample location map of North
Anomaly of Choghart deposit.
Şekil 2. Choghart cevheri kuzey anomalisinden alınan örnek yerleri.
Usually, the random field that represents the
regionalized attributes under study does not
have any univariate distributions, and therefore must be transformed into a field with a distribution suited to a known isofactorial model.
In the Gaussian case, this procedure is known
as normal scores transformation. The problem
Figure3.
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in this case is that the variable is not normally
(Gaussian) distributed, and therefore, it has to
be transformed from the original distribution
into a standard Gaussian distribution, the process being known as a Gaussian anamorphosis
transformation. The histogram of the iron concentrations (Figure 3) is not normal. Under these conditions, as suggested by Chile`s and Delfiner (1999), a Guassian model is preferred. In
a first attempt, the data are transformed to values with a standard normal distribution. As, to
give an example from the data of the authors,
the raw distribution of Fe is shown in Figure 3,
and Figure 4 shows the transformed Gaussian
distributions.
Variography
Figure2.
Taking Figure2.
spatial variety and randomicity into account, the variogram function can reflect the
spatial variable structure of a regional variable.
Regarding the variogram analysis of the normal
scores data, Figures 5 and 6 of the variogram
in different directions do not allow one to detect
an anisotropy, so only an omnidirectional model is considered. The model consists of a pure
nugget effect with 0.32 plus a spherical scheme with sill 0.90 and range 150 m. This model
is required since disjunctive kriging estimation
will be based on it as well as on the block anamorphosis calculated before.
Figure3.
Figure3.
Figure5.
Figure 3. Raw histogram of borehole sample iron
concentrations.
Şekil 3. Kuyu örneklerine ait demir derişim yatay sütun diyagramı.
Figure 5. Variogram in different azimuths forFigure
Chaghart
6.
North anomaly.
Şekil 5. Choghart cevheri kuzey anomalisinde farklı
azimutlar için varyogramlar.
Figure4.
Figure4.
Figure 7
Figure 4. Transformed histogram of borehole sample
iron concentrations.
Şekil 4. Kuyu örneklerinde demir derişimlerine ait
dönüştürülmüş histogram.
Figure 6. Variogram in different dips for Chaghart
North anomaly.
Şekil 6. Choghart cevheri kuzey anomalisinde farklı
eğimler için varyogramlar.
Figure 8
Daya and Hassani
Estimation by Disjunctive Kriging
To verify the practical usefulness of disjunctive
kriging to estimate block values, the theory has
been applied to the Choghart north anomaly
iron ore deposit. The estimate of the grade of
the survey samples only is not enough. The fact
that the grade of a block is estimated at 10%,
e.g., does not mean that this grade, as it will be
known at the time of extraction, cannot exceed
a threshold of 20%. So forecasting the probability of the Y(x) past the threshold of 20% is significant. Consider that in this case study 20% is
cut off. It means that more than 20% is ore and
lower is waste. Figures 7 and 8 show the grade
and estimated error variance maps computed
by disjunctive kriging. According to these two
figures, a decision can be made. Figures 9 and
10 give the probability and probability error variance maps for a threshold of 20 %.
CONCLUSIONS
51
distributions in the values of a spatial attribute (Fe in this case), using disjunctive kriging instead of simple or ordinary kriging. Implementation of disjunctive kriging is fairly simple when
done within the scope of a bivariate isofactorial
model. In practice, given the wide range of possibilities in the modeling, trial-and-error procedures are often necessary to choose a type of
bivariate distribution and fit its parameters. This
choice may be based on the experience of the
practitioners and their understanding of the
data. From the case study, we concluded that
disjunctive kriging can be used to model the
uncertainty of mapping iron ore concentrations
in an ore deposit. It is hoped that this example,
taken from very different application fields will
encourage practitioners to apply disjunctive kriging with models that extend the capabilities of
the bigaussian model. We end by saying that if
the data are relatively marginally skewed and
the goal is to predict nonlinear functional, then
disjunctive kriging would be suitable.
This work addressed the problem of assessing
the uncertainty and estimating the conditional
Figure 7
Figure 7
Figure 7. Estimated block averages of iron by DK.
Şekil 7. Ayırıcı kriging yöntemiyle tahmin edilen demir ortalamaları.
Figure 8. Estimated block averages error variance by
DK.
Şekil 8. Ayırıcı kriging yöntemiyle tahmin edilen ortalamaların hata varyansları.
Figure 9. Conditional probabilities that block averages of iron are more than 20%.
Figure
8
Şekil 9. Figure
Demir ortalamalarının
%20’den büyük oldu8
ğu koşullu olasılıklar.
Figure 10. Conditional probabilities error variance that
block averages of iron are more than 20%.
Şekil 10. Demir ortalamalarının %20’den büyük olduğu hata varyanslarının koşullu olasılıkları.
Figure 9
Figure 9
F
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