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Modeling of mineral deposits using geostatistics and experimental design

Luis P. Braga(UFRJ)Francisco J. da Silva(UFRRJ)Claudio G. Porto(UFRJ)Cassio Freitas(IBGE)

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August [email protected]://www.slideshare.net/bragaprof/stanford-2009-1872654

Eight International Geostatistics Congress. Santiago, Chile

Goal: Improve the mineral resources evaluation of a deposit in the initial stages of exploration.International Association for Mathematical Geosciences, Stanford, USA, 23-28 August [email protected]://www.slideshare.net/bragaprof/stanford-2009-1872654


How: Through experimental design techniques applied to variogram based estimation methods.International Association for Mathematical Geosciences, Stanford, USA, 23-28 August [email protected]://www.slideshare.net/bragaprof/stanford-2009-1872654


Outline of the presentation

a)Creating a synthetic study case: Simulate the grade on a regular 3D mesh based on data of a lateritic Ni deposit and calculate the amount of resources.

b)Applying designed experiments: Varying the values of the four main parameters of the semivariogram, according to an experimental design. c)Testing the method with kriging: Using a sample, estimate the total resources with kriging by changing the semivariogram parameters values, according to an experimental design. Calculate the different resource totals and compare with a).

d)Testing the method with simulation: Repeat c) with simulation as an interpolator.

e)Discussion. International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009a) Simulate the grade on a regular 3D mesh based on a sample of a lateritic Ni deposit, and calculate the amount of resources.Figure 1


a)The data consists of 76 drillholes, located in a grid of 100mx100m having in total 2021 drillholes samples which were collected downhole at 1m interval. The experimental and the adjusted semivariogram in the principal directions were obtained by a geologist.International Association for Mathematical Geosciences, Stanford, USA, 23-28 August 2009Figure 2 aFigure 2 cFigure 2 b


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009a) The central values(0) of a spherical semivariogram model parameters, as well as, the minimum(-) and maximum(+) acceptable values to the geologist are presented in Table 1. Table 1


Minimum(-)

Central(0)

Maximum(+)

Range(NS)

180

200

240

Range(EW)

280

300

340

Range(vertical)

7,5

9

10,5

Sill

0,119

0,139

0,149

Nugget effect

0,000

0,01

0,030

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009


a) A three dimensional model of the selected region of the deposit, shown in Figure 1, was built using the Gauss Simulation algorithm as implemented in the package GSTAT of the R environment. It consists of 5400 blocks with support 10m x 20m x 5m. The semivariogram used in the simulation was the one with the central values, as in Table 1.



a) The simulated variable is the grade in percentage. Density is considered constant and equal to 1Kg of metal per cubic meter ore. After simulating the grades one can easily calculate the amount of nickel through the basic equation R= VDT (where V stands for volume, D for density and T for grade). This formula is applied to each block and then summing it up. The obtained value was 31,365 Kg.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Figure 3


a) In the Figure 3 it is depicted the vertical section of the simulated deposit. In figures 4 to 6 it is shown horizontal sections at different depths.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Horizontal SectionFigure 4




b) Experimental design principles can be used in uncertainty assessment for more realistic sensitivity analysis. There are several schemes that can be adopted. We opted for the Box-Behnken design with four quantitative factors and three levels (the nugget effect will be kept constant and equal to 0.01) which focuses in the central values, for instance: N-S range, E-W range, Vertical range and Sill. As response variable we took the total resources of the deposit, coded as R.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009b)For each interpolator the sequence presented in Table 2 was applied, obtaining different resources values. For each range and level an average of the resources obtained are calculated, as shown in the next slide, leading to the average effect table.


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009N-S range(-)


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Table 3 Average Effect Table


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009b)From Table 3 we build the average effect plots thatindicate the best combination regarding the estimationof the resources.We are not proceeding the full designed experimentsphases, that is, the regression between the variablewith its factors, but only keeping the factors levels assignment strategy.




c) The first experiment was done with ordinary kriging to generate a three dimensional model of the deposit with the same dimensions, that is 5400 blocks with support 10m x 20m x 5m. In the sequence we can see horizontal sections for some levels obtained from run 1 for variable Ni. We have obtained all levels for every run, but for sake of conciseness, just a few are shown.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Horizontal Section


Figure 7 Horizontal Section at level -5m , Kriged according to run 1

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009bbbbbbbbbbbbbbbHorizontal Section



International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Horizontal Section





c) Next, we show the same level for different runs. Although, the difference between the same levels, from the different runs, are less visible, there may be large discrepancies for the total volumes, with some up to 2,500 Kg. ! In the Table 4 we show the total amount of Ni as calculated for each run.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Figure 10 (a) Run 2Figure 10 (b) Run 9Figure 10 (c) Run 19


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Resources 9, 18 and 27 are equal because they correspond to the same choice of parameters - central values.Table 4


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009c) As mentioned before we calculate the average of the Resources for each set of runs, as arranged by type of parameter and its value. It will allow us to build the so called average effect plot. For each row we calculate the average of the corresponding Resources indicated by its run number. That is, for N-S RANGE(-) we take the average between R[1], R[2], R[10], R[11], R[19] and R[20], that happens to be 30,539Kg. The same procedure is repeated for each row, generating a designed average.


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Table 3 Average Effect Table


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009true value = 31365




The results stress the relationship between the quality of the semivariogram and that of the resource estimation. The N-S and E-W semivariograms are not good and their impact on estimation is straightforward. Central choices of the range imply in worse estimates.



The vertical semivariogram is the best one and there is no relevant difference between conservative and central choices, but for optimist choices the impact is negative. The sill estimates are dependent on the semivariogram analysis.



It is important to remind that the estimation is done by averaging a selection of runs, instead of just one kriging interpolation. The experimental design gives us a new range of interpolators based on the quality of the semivariograms. In this case, the average of fifteen different kriging interpolations, see row V range (0) in Table 4, which gives an amount of 30,625 Kg of Ni.



The combined estimation was superior to the ordinary kriging alone. For example, a natural choice would be the central values for every semivariogram parameter, that is, run 9 with 30,558 Kg which is poorer than the former average (30,625 Kg), as the true value is 31,365 Kg.



Besides improving resources estimation, the designed averaged kriging (DAK) does better than ordinary kriging to estimate blocks, which enable us to produce more realistic maps of the deposit. Consider the histogram of the errors of blocks, obtained by comparing the true amount of Ni in each block with the one estimated by DAK (Vrange(0)).

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Figure 15 Histogram of Errors (DAK))Figure 16 Histogram of errors - run 9


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009The histogram of the errors of the geologist choice of parameters (central values) , which corresponds either to run 9, 18 or 27, represents a good match, but DAK performs better.Table 5 Errors statistics for DAK(Vrange(0)): True - EstimatedTable 6 Errors statistics for single Kriging (Run 9): True - Estimated


The summary of the errors is given in Table 5, where density is equal to 1, therefore the maximum amount of Ni in a block being 1000kg.



The methodology may be applied to improve the performance of any semivariogram based interpolator. The aleatorization of the runs is quite natural in the case of simulation as interpolator, but it may also be done with kriging like interpolators by using any technique of resampling.

In the next section we present an analogous experiment with simulation as an interpolator.



d)Following the same guidelines used in the previous section, 27 simulations of the deposit were generated according to Tables 1 and 2. As the seed of the random generator was initialized only once, we did not obtain the same values for runs 9, 18 and 27. For each run just one simulation was obtained.



d) In the sequence one can observe horizontal sections for some levels obtained from the run 1 for variable Ni. We have obtained all levels for every run, but for the sake of conciseness, just a few are shown.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009 Figure 17 (a)Horizontal Sectionlevel -10mRun 1Figure 17 (c)Horizontal Sectionlevel -10mRun 3Figure 17 (b)Horizontal Sectionlevel -10mRun 2


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Table 7


The equivalent runs 9, 18 e 27 now have different values because of the aleatorization of the simulation process.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Figure 18 (a)Figure 18 (b)Figure 18 (d)Figure 18 (c)313653136531365


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009The extension of the method to simulation will be called designed averaged simulations (DAS). The results preserved the relation between the quality of the semivariogram and that of the resource estimation.


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009The vertical semivariogram is the best one and there is no relevant difference between conservative and central choices, but for optimist choices the impact is negative. The resources estimation is better achieved with DAS(Vrange(0)) than simulation.




The DAS(Vrange(0)) for the vertical range shows a total of 30,726 Kg, while the average of the equivalent simulations runs 9, 18 and 27 shows 30,463 Kg, with the true value equal to 31,365 Kg.

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009We also evaluated the simulated values for each block, as we did before with kriging, comparing each designed averaged simulated block (DAS) with the true one.Table 8 Errors Statistics for DAS(Vrange(0)): True - EstimatedTable 9 Error Statistics for simulation (Vrange(0)): True - Estimated


Both present good results, but DAS (Vrange(0)) is less biased and has lower variance than Vrange(0).

International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009d) conclusions:1)The gain in the evaluation process was almost 10%2)The method orientates which simulations or interpolations must be kept.3)The method allows a better selection of directional semivariograms and its parameters levels.4) Future work includes tests with other samples andsimulation methods.


International Association for Mathematical Geosciences, Stanford, USA, 23-28 August, 2009Send comments to:[email protected]://www.slideshare.net/bragaprof/stanford-2009-1872654THANK YOU !


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