The First International Conference on Statistical Methods in Engineering, Science, Economy, and Education 1st SESEE-2015Department of Statistics, Yogyakarta September 19-20 2015
Statistics Application On Terestrial Phenomena Of Metallic
Winarno. E., et al., “Statistics Application On Terestrial Phenomena Of Metallic Mining’s Activity”
ISBN:978-602-99849-2-7
Uniqueness and specific properties of a mineral and coal are not random, but there is a relation between genesisly. Therefore, a statistical approach that is widely used in terrestrial phenomena also known as geostatistics.
Geostatistics is statistic-science, which is applied for terrestrial phenomena; with the nearby data has a huge influence. The farther distance, the less data influenced.
2. Scale of Theory
2.1. Resources and Reserves
Classification of resources and the reserve has been published by the Australasian IMM / AMIC base on classification accuracy improvement and the results of geological investigations (Table 2).
Table 2 indicates that the increased resources into reserves to account for economic factors, mining, processing, market, environment, and government regulation.
Table 2. AIMM/AMIC Classification of identified minerals resources Identified mineral
resources (in situ)
Ore reserves (mineable)
Increasing level of geological knowledge and confidenec Inferred
Indicated Probable Consideration of economic, mining, metallurgical, marketing,
enviromental, social and govermental factors
Measured Proved Sources :Mineral Deposit Evaluation, A.E. Annel (1991)
Base on Table 2, Diehl and David in A.E. Annels (1991) develop a classification of ore deposits involves a degree of uncertainty (assurance) and degree of accuracy (error tolerance) for each of the different deposits. It was stated comprehensively in Table 3 below.
Table 3. Ore Reserve Classification
Identified Undiscovered
Demonstrated
Measured Indicated (Possible)
Proved Probable Inferred Hypothetical Speculative
+ 10%*) + 20% + 40% + 60%
> 80%$) 60-80% 40-60% 20-40% 10-20% < 10%
Economically significant resources Resources base Sources :Mineral Deposit Evaluation, A.E. Annel (1991)
*) : Error tolerance $) : Assurance
Table 3 state that for proven reserves has an error tolerance of + 10% and above 80% accuracy rate.
2.2. Error Tolerance and Level of Accuracy
In statistical theory, the probability of a confidence levels of the data distribution can be formulated by
. . ) 1
( X Z
/2X Z
/2P
with: :
X = Average of sample data = Standard deviation of population data μ = population mean n = a mount of samples
1-α = level of confidence α = fault tolerance (accuracy) Zα/2 = normal table value
1 1 8 Winarno. E., et al., “Statistics Application On Terestrial Phenomena Of Metallic Mining’s Activity”
ISBN:978-602-99849-2-7
Copyright © 2015by Advances on Science and Technology
Various values of the confidence level can be described by a normal distribution table and summarized in Table 4.
Table 4. Z value at Various Confidence Levels Level
Confidence (1- α)
Accuracy (α)
Value Z α/2
Deviation Reserve
65% 35% 0.935 0.935.
80% 20% 1.282 1.282.
90% 10% 1.645 1.645.
95% 5% 1.960 1.960.
99% 1% 2.575 2.575.
Reserve estimates, especially for the ore, it is generally formulated as follows:
ReservesVolume.x.Density..x.Grade.Deviation
Deviation is the backup has been taken of the level of confidence (accuracy) and the estimation error tolerance.
2.3. Geostatistics
Geostatistics is the statistics of spacially or temporally correlated data. Geostatistics is concerned with the study of phenomena that fluctuate in space. Geostatistics offers a collection of deterministic and statistical tools aimed at understanding and modelling spacial variability.
2.3.1. Variography
The semivariogram is the basic geostatistical tool for visualizing, modelling, and exploiting the spacial autocorrelation of a regionalized variable. As the name implies, a semivariogram is measure of variance, illustrated in figure 2 and the formula.
γ*(h) = ∑ [Z(x) – Z(x+h)]2/2n with :
γ*(h) = semivariogram at site h units distance h = distance, m Z(x+h) = Z variable between site x and a site h units distance 2n = number of variable pairs For modelling the semivariogram through iterative or least-square methods, practitioners recommend actual inspection of the observed semivariogram and the fitted model. If the surface represented by two points is continuous and h is small distance, one expects that the difference is small as well. If h were very large relative to the spacial degree of change in the variable, then the difference might be expected to increase.
The difference both the expected variance (model, γ*) and the real inspection (γ) will be small, that can be expressed with :
n
i
Deviation
1
* ]
[ is small 2.3.2. Kriging
In the geostatistics theory, the deviation with small value, can be expressed by Kriging standard deviation (illustrated in figure 2).
118
Winarno. E., et al., “Statistics Application On Terestrial Phenomena Of Metallic Mining’s Activity”
ISBN:978-602-99849-2-7
Z6
Z2 Z5
ω
2ω
5ω
6Z1
ω
1a0 Z*
ω
3ω
7Z3
ω
4Z4 Z7
Figure 2. Point Kriging Orientation Grade of A block can be estimated with Z*
1n
i.
Zi ;with : ω : weigted factor ; Zi : real block with grade-i ; Z* : estimate block Figure 3 show that :
a. Var[Z,Z*] = σ2+
n
i n
i
j i j
i Cov z z
1 1
) , ( . .
-2.
n
i
i
iCov z z
1
*) , (
.
b. To reach the maximum or minimum value of Var[Z,Z*] = 0, need to support with the Lagrange Function : 2.μ.( 1)
1
n i
i with
n
i i 1
1. c. Var[Z,Z*] = σ2+
n
i n
i
j i j
i Cov z z
1 1
) , ( . .
-2.
n
i
i
i Cov z z
1
*) , (
. +2.μ.( 1)
1
n i
i
d. The differencial of Var [Z,Z*] : 2 0 d k
d
and k20 d
d
reachminimum value, and the result is :
k2
σ2-
n
i
i
iCov z z
1
*) , (
. + μ e. This function can be writen in matrix formula :
γ(Z1,Z1) γ(Z1,Z2) ... γ(Z1,Zn) ω1 γ(Z1,Z*)
γ(Z2,Z1) γ(Z2,Z2) ... γ(Z2,Z1) ω2 γ(Z2,Z*) ... ... ... ... .... = ...
γ(Zn,Z1) γ(Zn,Z1) ... γ(Zn,Zn) ω3 γ(Z3,Zn) 1 1 1 μ 1 Note : γ(Z1,Zn) = variogram point Z1 to point Zn
f. Solution of matrix get the value of ω (weigted factor) and μ (Langrange factor).
3. Reserves Estimate
3.1. Characteristics of Ore
Reserve estimation carried out simulations on the “ X nickel deposits”. Exploration activity that has been done is taking samples with a regular distance of 25 m by using a rotary drilling tool.
Topography of the hills with a slope of 300-500 and 50-230 meters above sea level. Based on how the formation, geology of ore deposits is a nickel laterite ore, mineral deposit is the result of the weathering of ultra basic rock peridotite, in general, contain elements of iron, cobalt and klorium.
This ultramafic rock outcrops generally have undergone weathering, yellow-brown mottled gray, black or white with a greenish tint on the outer edge or rim. In this area there are also small cracks, fractures are commonly filled by secondary minerals (silica and magnesite).
In general profiles ore deposits in the study area are as follows:
a. Top Soil, ground cover is reddish brown, there are the rest of the herbs.
1 2 0 Winarno. E., et al., “Statistics Application On Terestrial Phenomena Of Metallic Mining’s Activity”
ISBN:978-602-99849-2-7
Copyright © 2015by Advances on Science and Technology
b. Limonite, is the result of weathering of the soil soft yellowish brown color containing nickel and iron in the ratio is not necessarily.
c. Saprolite, is highly weathered soils have yellowish brown to greenish with many veins garnierit and onyx, has a relatively high nickel content.
d. Bed Rock, a peridotite host rock that has not weathered serpentinite.
3.2. Estimate Simulation
Estimated reserves of nickel ore deposits base on data illustrated in the figure 2 and Table 5.
Table 5. Drillhole Data Sample Coordinate Grade,
ppm
Distance
From Z (63E, 137N) to no sample
No. Code E N
1 225 61 139 477 4,5
2 437 63 140 696 3,6
3 367 64 129 227 8,1
4 52 68 128 646 9,5
5 259 71 140 606 6,7
6 436 73 141 791 8,9
7 366 75 128 783 13,5
a. Determined semivariogram value from one point to another, γ(Z1,Z1) ; γ(Z1,Z2) ; γ(Z1,Z3) ; γ(Z1,Z4) ; γ(Z1,Z5) ; γ(Z1,Z6) ; γ(Z1,Z7) ; γ(Z2,Z2) ; γ(Z2,Z3) ; γ(Z2,Z4) ; γ(Z2,Z5) ; γ(Z2,Z6) ; γ(Z2,Z7) ; γ(Z3,Z3) ; γ(Z3,Z4) ; γ(Z3,Z5) ; γ(Z3,Z6) ; γ(Z3,Z7) ; γ(Z4,Z5) ; γ(Z4,Z6) ; γ(Z4,Z7) ; γ(Z5,Z6) ; γ(Z5,Z7) ; γ(Z6,Z7) ; γ(Z7,Z7) ; γ(Z0,Z1) ; γ(Z0,Z2) ; γ(Z0,Z3) ; γ(Z0,Z4) ; γ(Z0,Z5) ; γ(Z0,Z6) ; and γ(Z0,Z7)
b. From output of Geostatistical program (GS+7), Solution the kriging matrices :
c. Estimated value Z
1n i Zi*
.
Z* = (0,173)(477) + (0,318)(696) + (0,129)(227) +(0,086)(646) + (0,151)(606) + (0,57)(791) +(0,086)(783) =592 ppm
d. Kriging variance
k2
σ2-
n
i
i
i Cov z z
1
*) , (
. + μ
2
Zk = 10 – (0,173)(2,61) – (0,318)(3,39) – (0,129)(0,89) – (0,086)(0,58) – (0,151)(1,34) – (0,057)(0,68) – (0,086)(0,18) + 0,907 2 Zk = 8,96 ppm2 and Zk = 2,99 ppm
e. The reserves of nickel ore on Z* block for one ton resources is : (592 + 2,99) ppm
4. Discussion
Reserves estimate of nickel ore on Z* block (see figure 2), can be discribed with 3 methods are : a. Nearest Neightbor Point (Poligon), for :
- Grade of nickel ore, equal to nearest point with minimum distance. Here, grade of nickel ore on 0 block equal with Z2 = 696 ppm (distance point 0 to point 2 is nearest distance, Table 5)
- Deviation standard are computed to all point, point 1 until point 7, σ = 198,11ppm b. Inverse Distance Weight (IDS), for :
- Grade of nickel ore are estimated base on distance weighted factor. The estimated value is
ni i
n i
i
Z d
Z d
1 1
*
1
/ 1 .
Here, grade of nickel ore on 0 block equal with Z2 = 488,87 ppm
- Deviation standard are computed to all point, point 1 until point 7, σ = 198,11ppm
120
Winarno. E., et al., “Statistics Application On Terestrial Phenomena Of Metallic Mining’s Activity”
ISBN:978-602-99849-2-7
c. Ordinary Kriging
- Grade of nickel ore on 0 block , Z2 = 592 ppm (sub chapter 3.2).
- Deviation standard, σ = 2,99 ppm
From the result estimate of grade nickel ore above, Ordinary Kriging has moderat value, not to big and not to small. And deviation standard value has smallest value, that can be concluded has best accurate value.
That concluded can be reached if :
1. The base data must be valid and reliable, both in prospecting stage (sampling tecnics), exploration (procedure of data processing), exploitation (prediction and validation on target area), metalurgical processing (feed, process and product quality control), and marketing (product quality control).
2. In the future advantage, resources or reserves can be improved to increase added value with optimally exploitation.
5. Conclusion
To reached the higher accuracy prediction of resources or reserves metallic deposits can be expreseed with standard deviation value. Too smallest standard deviation value, too accurate prediction result.
References
[1] Winarno, E., Aplikasi dan Pengaruh Cut-Off Grade dalam estimasi cadangan bahan galian , Jurnal Ilmu Kebumian Teknologi Mineral, Vol.23 Nomer 2 Mei-Agustus 2010, UPN “Veteran”
Yogyakarta, 73-80 (, 2010).
[2] Winarno, E., K. Gunawan, T. Wahyuningsih, R.Z. Mirahati, Accuracy Statement Of Ore Deposits Reserves Estimation, International Symposium on Earth Science and Technology, Bandung, (2012).
[3] Annel, A.E, Mineral Deposit Evaluation : A practical approach, Chapman & Hall, London, p.99- 212 (1991).
[4] Clark., I., Practical Geostatistics, Chapman & Hall, London, (1979)
[5] Carras, S., Sampling Evaluation And Basic Principles of Ore Reserve Estimation, Carras Mining
& Associates, Australia, p.40-83 (1986).
The First International Conference on Statistical Methods in Engineering, Science, Economy, and Education 1st SESEE-2015Department of Statistics, Yogyakarta September 19-20 2015