Ore Body Modelling Assignment 2

8/9/2019 Ore Body Modelling Assignment 2

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ORE BODY MODELLING (ERTH3301)

ASSIGNMENT 2 23/09/2013

BRUNA KHARYN DE ASSUNCAO BARBOSA

STUDENT NUMBER 43282812

ANSWERS

QUESTION 1

Omnidirectional Experimental Variograms:

GC (2D)

Figure 1 Omnidirectional Variogram Lag 5 Figure 2: Omnidirectional Variogram Lag 10

Figure 3: Omnidirectional Variogram Lag 15 Figure 4: Omnidirectional Variogram Lag 20


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Figure 5: Omnidirecional Variogram Lag 25



Figure 8: Omnidirectional Variogram Lag 40

Optimal Variogram and Model:

Figure 9 Isotropic Model (Lag= 25)


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GC

Model type Spherical

Nugget 6.3600

Sill Structure 1: 14.0000

Range Structure 1: 93.0000Major/ semi-major 2.5993

Major/ minor 1.2360

Table 1: 2D GC and TK Isotropic Model Parameters

Variogram Maps:

Figure 10: Variogram Map - Normal View Figure 11: Variogram Map (GC) - Plan View

Figure 12 Variogram Map (Thickness) Plan View


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Discussion I

Guibal (Guibal, 2001) states that: "variography is the calculation of experimental variogramsand the subsequent fitting of appropriate variogram models. (...) We wish to fit the

experimental variogram with a model that captures the main features of spatial variability, i.e.

the attributes that are important for the estimation of grades in space. (...) Because variographyreflects the actual spatial distribution of grades, it can be a powerful exploratory tool for the

geologist."

In order to guarantee the selection and modelling of a proper variogram, we should firstly

analyse and select certain model/kriging parameters such as lag value (h), number of lags, lag

tolerance, isotropy or anisotropy, direction and angular tolerance etc.The first parameter to be chosen for this exercise is the lag value. The general rule for this

selection is "the lag distance should be at least equal to the sample spacing" (Coombes, 1996).

That is because h values that are too low or too high do not truly represent the variability of the

samples. Low lag values will only provide appropriate information at every number of lags,

since "variograms at intermediate lags are based on relatively low numbers of

sample pairs" (Coombes, 1996). The result of effect of low lag values can be seen in the firstvariograms, especially of figures 1 and 2, where they are too erratic or "noisy".

On the other hand, variograms calculated with too high h values end up neglecting important

points and masking real variability information. That makes variograms too "smooth", such as

can be seen in figures 7 and 8.In the studied case we have a 20m X 20m grid. So the variogram of h= 25 seems appropriate

enough, showing a nice curve that is smooth, has low nugget and long range, which will be

good for modelling.

Another point to be analysed in this exercise was the matter of directional behaviour. When

choosing the lag value, the experimental variograms were calculated assuming the body was

isotropic, that is, had the same variation in all directions. However, by plotting and looking at

variogram maps for gold and thickness (figures 11 and 12), it is clear that there is a change in

variability according to direction. Isotropic variogram maps should not show any important

changes in colours. In this case, though, there is obvious variation, indicating anisotropy.

Thus, the model presented in figure 9 is not valid. New, directional variograms needed to becalculated and modelled to express the true variability of the deposit.

Directional Variograms and Models:

Figure 13: Directional Variogram (GC) - Normal View Figure 14: Directional Variogram (TK) - Normal View


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Figure 15: Directional Variogram (GC) Plane View Figure 16: Directional Variogram (TK) Plane View

GC Thickness

Model type Spherical Spherical

Nugget 7.0194 0.5433Sill Structure 1: 2.8762

Structure 2: 10.1268

0.4869

Range Structure 1: 53.3500

Structure 2: 100.2200

73.2770

Major/ semi-major 4.2194 2.7784

Major/ minor 6.4061 1.0000Table 2 Variogram Model Parameters (GC and TK)

Discussion II

Figures 13 to 16 show the final directional variograms that were selected for grade and

thickness as the most appropriate from a number of variograms that were tested. Table 2

summarises the parameters used in the modelling process.The resulting variograms both have low nugget effect, that is, their variability at short distances

is not too high. The range, which is the distance at which the variance stabilises and the samplesare no longer correlated, is long. That means that samples separated by a large distance are

likely to be somewhat similar to each other.

These variograms indicate a reasonably continuous distribution of the variables, with a gradual

variation up to a maximum.

As to the modelling process, we can see that for some cases the model has a higher sill than the

expected (green line). That is because real variability does not necessarily follow mathematicalexpectations. The variogram calculations indicate that the sill should always be achieved at the

variance value. Nevertheless, reality is messy and the modelling of the true sill depends muchon the modellers sensibility.

QUESTION 2

The block model is a geological representation of a deposit that evaluates grades and other

variables from the drill hole data by interpolating the drilling data, limited within a certain

wire frame model, using geostatistical techniques. The purpose of the block model is to

associate grades with the volume model. Two of the main features of the block model are the

size of the blocks and the model rotation. The former is defined according to the spacing/

drilling patterns, being as small as possible; the latter is used to adjust the block axis to be


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parallel to the strike of the ore body, to ensure a good intersection between the model and

the ore body and exclusion of excessive waste.

The blocks extents and attributes are summarised in Table 3 and 4.

Type Y X Z

M inimum Coordinat es 7320 1570 200

Maximum Coordinates 7620 1970 201

User Block Size 10 10 1

Min. Block Size 10 10 1

Rotation 0.000 0.000 0.000

Total Blocks 1200Table 3: 2D block model extents

Att ribute Name Type Decimals Background Descript ion

gold_id Real 2 0 Gold value estimated by inverse distance

gold_nn Real 2 0 Gold value estimated by nearest neighbour

gold_ok Real 2 0 Gold value estimated by kriging

gold_thk Real 2 0 Estimated mineralisation thickness

zok_ads Real 2 0 Average distance to samples

zok_cbs Real 2 0 Conditional bias slope

zok_dns Real 2 0 Distance to nearest sample

zok_ke Real 2 0 Kriging efficiency

zok_kv Real 2 0 Kriging variance

zok_ns Integer - 0 Number of informing samplesTable 4: 2D block model defined attributes

QUESTION 3

For any technique applied in an estimation, it is important to choose certain parameters thatdefine a search strategy, which controls the samples that really contribute for the estimation

(informing samples).

The first parameter is the search type that could be either ellipsoid or octant. The octant, or

the quadrant for 2D, defines a maximum of N points in each of the eight octants, or four

quadrants, to be used in the interpolation calculations showing high performance with

clustered data. For data that shows anisotropy, such as in this case, the ellipse method is

better because it controls the shape of the search space that surrounds the interpolation

point and uses only a subset of the scatter points in the vicinity of the target for the estimation

calculations. That is why we use the ellipse for this data.


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The minimum and maximum number of samples control how many of the points found in the

search radius are actually used in the kriging calculations. If fewer than the minimum value

are found, a default value is used. If greater than the maximum, the closest points are used.

The maximum search ellipse is used in conjunction with the minimum and maximum number

of samples to select the appropriate samples for kriging, and it should generally be a value

slightly above the range of the variogram of the major axis.

The search anisotropy ratio is associated with the sizes of major and minor ellipse axis, whose

lengths are based on the spatial continuity defined in the variography analysis.

The search parameters used are presented in Table 5.

Parameter Feature

Search Type Ellipse

Min. Number of Samples 3

Max. Number of

Samples

12

Max. Search Ell ipse 100

Search Anisotropy

Ratios

2.83

Table 5: Search parameters


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QUESTION 4

Discussion

Nearest Neighbour:

Assigns values to blocks in the model by assigning the values of the nearest sample point to

the block of interest. It is a very simple technique and it is effective if training data is large.

It is, however, of computational complexity, has problems with memory limitations and is

easily compromised by irrelevant data.

Inverse Distance Estimation

It is one of the earliest estimation methods, using interpolation based on an empirical

observation that the weight of each sample is proportional to an inverse power of the

distance.The degree of smoothing, or variance reduction may be controlled by changing the weighting

power and search parameters. A lower power results in the estimation being more

continuous. Higher weighting powers on the other hand result in estimations that seem more

erratic. The appropriate parameters may only be found through trial and error.

Indicator Kriging:

Indicator kriging and probability kriging are related methods that are used to improve

estimation when ore zones are erratic and grade distributions are highly variable and

complex. Advantages of indicator kriging include less smoothing of estimated grades thanordinary kriging and robustness in handling nonstandard grade distributions.

The first step in indicator kriging is to set one or more cut-offs with which to define indicator

variables. Given a cut-off gc, the indicator variable is set to 1 if the grade is above gcor 0 if the

grade is below gc; indicator variables are coded similarly for each desired cut-offs.

The resulting indicator estimates may be interpreted as either the probability that the block

will be above cut-off or the percentage of the block that is above cut-off.

Variograms are modelled for each indicator variable and an expected value for each indicator

is estimated using ordinary kriging and the appropriate indicator variogram.

Ordinary Kriging

It is a distance weighting technique where weights are selected via the variogram according

to the samples distance and direction from the point of estimation. The weights are not only

derived from the distance between samples and the block to be estimated, but also the

distance between the samples themselves. This tends to give much lower weights to

individual samples in an area where the samples are clustered.


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Lognormal Kriging:

It is a method of non-linear kriging that was developed to improve estimation when the

underlying data are distributed according to a lognormal probability distribution.

The variogram is computed using the natural logs of the data; the kriging system is solved to

provide a weighted average of the natural logs of the data; the kriged log average is then

transformed back to normal values using lognormal transformation.

It has complex mathematics and presents many complications in its application, such as the

strict requirement for a lognormal distribution and a variogram which is stationary over the

field of estimation. Serious global and local biases may occur if either these conditions are not

met. In addition, there is a tendency for lognormal kriging to overestimate the high-grade end

of the population when the coefficient of variation is greater than 2.0.

Lognormal kriging is recommended only for special purposes where the results can be

monitored closely and adjusted to prevent biases.

Method Advantages Disadvantages

Nearest Neighbor Simplicity

Effective if training data is large

Computation Complexity

Memory limitation

Computationally Slow

Easily fooled by irrelevant

attributes

Inverse Distance Simplicity

Fast calculation

Reasonable Results

Choice of weighting function may

introduce ambiguity

Not sensible to cluster regions

Does not have a measure of error

Ordinary Kriging Sensitive to clustering

Unbiased estimation

Error estimation and mapping

Difficulty to define the variogram used.

Not a suitable method for data sets

which present boundaries

Log-normal kr iging Suitable for distributions that are

positively skewed

Improves estimation when the

underlying data are distributed

according to a lognormal probability

distribution

Complex mathematics

Many complications in its application

Strict requirement for a lognormal

distribution and a variogram stationarity

Possible biases and overestimations

Indicator kriging Improves estimation when ore zones

are erratic and grade distributions

are highly variable and complex

Less smoothing of estimated grades

than ordinary kriging

Robustness in handling nonstandard

grade distributions.

Time and effort to do full indicator

variography

Order relation problems needs to be

controlled

Table 6: Summary of Estimation Methods


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QUESTION 5

Input data:Origin File: Comps Classified1.str

Gold Thickness

Number of samples 194 194

Minimum value 0.19 1

Maximum value 34.51 8

Mean 2.996837 3.197165

Median 1.82 3

Variance 19.944073 0.861087

Standard Deviation 4.465879 0.927948

Table 7: Input data statistics

Figure 17: Histogram for gold (input) Figure 18: Histogram for thickness (input)

Output data:

Figure 19: Histogram for thickness (output) Figure 20: Histogram for gold (output) - Inverse

distance


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Figure 21: Histogram for gold (output) - Nearest

neighbourFigure 22: Histogram for gold (output) - Ordinary

kriging

Origin File: Woody Gc 2d 10.mdl

Variable Gold Id Gold Nn Gold Ok Gold Thk

Number of samples 756 756 756 756

Minimum value 0.31275 0.19 0.355292 1

Maximum value 21.9529 34.51 16.315715 8

Mean 2.69648 2.572256 2.701896 3.146296

Variance 8.87698 17.30514 7.706975 0.613591

Standard Deviation 2.97943 4.159945 2.776144 0.78332

gold_id 1 0.8796 0.9754 -0.1316

gold_nn 0.8796 1 0.7841 -0.0672

gold_ok 0.9754 0.7841 1 -0.1661

gold_thk -0.1316 -0.0672 -0.1661 1Table 8: Output data statistics and estimation results

Q-Q Plot 1Thickness Q-Q Plot 2GC ID


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Q-Q Plot 3GC NN Q-Q Plot 4 GC OK

Discussion

The true value of gold and thickness (input data) and the estimate value (output data) should

be the same or near the same. And the statistic parameters such as mean, variance and

standard deviation, should decrease, since the data values were treated with the estimation

methods. It is clear that all estimated data have better statistics parameters after comparing

Table 7 to Table 8. And it is also noticeable the efficiency of the methods, just comparing

those parameters to each other. From that we can notice that the best estimation method in

this case was the ordinary kriging, and that the nearest neighbour algorithm did not present

satisfactory results. It is also noticeable that the thickness value were already well distributed,

so the estimation methods did not change its statistics parameters much.Q-Q plots are used to compare two data sets where samples are not necessarily paired. They

use the values of the percentiles of the analysed samples, plotted against each other in order

to check how close the data sets distributions are. In the Q-Q plot charts above, it is possible

to compare the distributions of input and output thickness and gold grade. The estimated and

input values for thickness follow the 1:1 line very closely for most of the values. That indicates

they are very close, and there are not many deviations.

For the GC Q-Q plots it is noticeable that kriging did not achieve such a good estimation of the

data. Despite being clearly better than nearest neighbour and inverse distance estimations,

kriging also resulted in a few deviations. That might be explained by the skewness of the gold

grade data as proved by the histograms (Figures 17to 22). Kriging does not deal very well thistype of distribution.


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QUESTION 6

Plan views

Figure 23: Plan view - Inverse Distance

Figure 24: Plan view - Nearest Neighbour


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Figure 25: Plan view - Ordinary Kriging

Nearest Neighbour Inverse Distance Squared Ordinary Kriging

Cut-off grade Mineralised

Tonnes

Gold (g/t) Mineralised

Tonnes

Gold (g/t) Mineralised

Tonnes

Gold (g/t)

>0 33288 0.22 0 0 0 0

>0.25 93423 0.32 64866 0.42 47766 0.44

>0.5 27892 0.61 46645 0.61 56031 0.6

>0.75 28861 0.87 30552 0.88 33953 0.87

>1.0 23788 1.12 38570 1.13 44080 1.13

>1.25 9538 1.34 33193 1.35 33649 1.39

>1.5 36917 1.6 31645 1.63 27208 1.61

>1.75 23294 1.83 20045 1.87 21518 1.86>2 12084 2.14 19884 2.12 16302 2.12

>2.25 25802 2.33 17157 2.37 21480 2.36

>2.5 15314 2.55 11818 2.65 10165 2.62

>2.75 14288 2.84 10564 2.9 12825 2.87

>3.0 107445 6.93 126996 6.06 126958 5.95

Grand Total 451934 2.5 451934 2.6 451934 2.59Table 9: Grade and tonnages


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Figure 25: Grade tonnage chart - NN

Figure 26: Grade tonnage chart - ID

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Cut-off grade

Grade-Tonnage Chart - Nearest Neighbour

Cumulative tonnes

Cumulative grade

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Grade-Tonnage Chart - Inverse Distance

Cumulative tonnes

Cumulative grade


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Figure 27: Grade tonnage chart - OK

Figure 27: Grade tonnage chart for all estimates

Discussion

Plan views:

Plan views are a visual form of analysis employed in model validation, in which we should

check for large differences between drillhole composites and the estimated grade. If those

exist, they must be analysed and explained.

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Grade-Tonnage Chart - Ordinary Kriging

Cumulative tonnes

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Title

Cummulat

ivetonnes

Cut-offs

Combined Grade-tonnage chart

Cumulative tonnes (OK)

Cumulative tonnes (ID)

Cumulative tonnes (NN)

Cumulative grade (OK)

Cumulative grade (ID)

Cumulative grade (NN)


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Grade tonnage curve:

One of the final steps of modelling the grades of a deposit is checking the grade tonnage

curves. To generate these curves, we first define a series of cut-off grades. The blocks are

classified in relation to those cut-offs and those which have grades equal or higher to the cut-

offs are identified and their tonnes are accumulated. The grades are then weighted by density

and averaged to find an overall grade for the block model.

The estimated grades and the tonnage of the cut-offs are selected and plotted to create the

grade-tonnage curve.

This curve is important to compare estimates from different models or different phases, such

as exploration and grade control. We can see that each of the charts present certain variations

for each estimation method that was used (inverse distance, nearest neighbour, and ordinary

kriging).

Another example of application of the grade tonnage curve is expressed by (SILVA & SOARES,

2001): In the mining industry, from the geology and mine planning to management, the

grade tonnage curve has been applied in economic and financial analysis, probably being oneof the most important tools used in determining the volume and grade of material in face of

variations of the cut-off grade. That is, the curve provides information to assess the optimal

utilization of the concept of cut-off grade. One of the most important methods to do that,

Lanes model of cut-off grade policy requires values from this type of curve to be applied in

the estimation of the optimal cut-off.


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References

Coombes, J. (1996). Handy Hints For Variography. Latest Developments in VisualisingSpatial Continuity from Variogram Analysis, AusIMM Conference Diversity - the Key to

Prosperity", (pp. 295-300). Perth.

Guibal, D. (2001). Variography, a Tool for the Resource Geologist.Mineral Resource and Ore

Reserve The AusIMM Guide to Good Practice, pp. 85-90.

SILVA, F., & SOARES, A. (2001). Grade Tonnage Curve: How Far Can It Be Relied Upon?

Technical Report - Somincor - Sociedade Mineira de Neves-Corvo.

Ore Body Modelling Assignment 2

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