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ISATIS 2011

Mining Case Studies


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Published, sold and distributed by GEOVARIANCES49 bis Av. Franklin Roosevelt, BP 91, 77212 Avon Cedex, France

Web: http://www.geovariances.com

Isatis Release 2011, March 2011

Contributing authors:

Catherine BleinsMatthieu Bourges

Jacques Deraisme

Franois Geffroy

Nicolas Jeanne

Ophlie Lemarchand

Sbastien Perseval

Jrme Poisson

Frdric Rambert

Didier Renard

Yves Touffait

Laurent Wagner

All Rights Reserved

1993-2011 GEOVARIANCESNo part of the material protected by this copyright notice may be reproduced or utilized in any form

or by any means including photocopying, recording or by any information storage and retrievalsystem, without written permission from the copyright owner.


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"... There is no probability in itself. There are only probabilistic models. The

only question that really matters, in each particular case, is whether this or

that probabilistic model, in relation to this or that real phenomenon, has or

has not an objective meaning..."

G. Matheron

Estimating and Choosing - An Essay on Probability in Practice

(Springer Berlin, 1989)


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1

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

1 About This Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

2 In Situ 3D Resource Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

2.1 Workflow Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

2.2 Presentation of the Dataset & Pre-processing . . . . . . . . . . . . . . . . . .16

2.3 Variographic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

2.4 Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68

2.5 Global Estimation With Change of Support . . . . . . . . . . . . . . . . . . .78

2.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88

2.7 Displaying the Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .129

3 Non Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145

3.1 Introduction and overview of the case study. . . . . . . . . . . . . . . . . . .146

3.2 Preparation of the case study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148

3.3 Global estimation of the recoverable resources . . . . . . . . . . . . . . . .165

3.4 Local estimation of the recoverable resources . . . . . . . . . . . . . . . . .1763.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .203

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222


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2


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5

Introduction


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6


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1 About This Manual

Note - The present document only contains case studies related to a specific field of application. The full CaseStudies Manual can be downloaded on Geovariances web site.

A set of case studies is developed in this manual. It is mainly designed:

for new users to get familiar with the software and gives some leading lines to carry a study through,

for all users to improve their geostatistical knowledge by following detailed geostatistical workflows.

Basically, each case study describes how to carry out some specific calculations in Isatis as precisely as possi-

ble. The data sets are located on your disk in a sub-directory, calledDatasets, of the Isatisinstallation directory.

You may follow the work flow proposed in the manual (all the main parameters are described) and then com-

pare the results and figures given in the manual with the ones you get from your test.

Most case studies are dedicated to a given field (Mining, Oil & Gas, Environment, Methodology) and therefore

grouped together in appropriate sections. However, new users are advised to run a maximum of case studies,

whatever their field of application. Indeed, each case study describes different functions of the package which

are not necessarily exclusive to one application field but could be useful for other ones.

Several case studies, namely In Situ 3D Resources Estimation(Mining), Property Mapping (Oil & Gas) and

Pollution(Environment) almost cover entire classic geostatistical workflows: exploratory data analysis, data

selections and variography, monovariate or multivariate estimation, simulations.

The other Case Studies are more specific and mainly deal with particular Isatis facilities, as described below:

Non Linear: anamorphosis (with and without information effect), indicator kriging, disjunctive kriging,

uniform conditioning, service variables and simulations.

Non Stationary & Volumetrics: non stationary modeling, external drift kriging and simulations, volume-

tric calculations, spill point calculation, variable editor.

Plurigaussian: an innovative facies simulation technique.

Oil Shale: fault editor.

Isatoil: multi-layer depth conversion with the Isatoil advanced module.


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8 Case Studies

Young Fish Survey, Acoustic Fish Survey: polygons editor, global estimation.

Image Filtering: image filtering, grid or line smoothing, grid operator.

Boolean: boolean conditional simulations.

Note - All case studies are not necessarily updated for each Isatis release. Therefore, the lastupdate and the correspondingIsatis version are systematically given in the introduction.


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In Situ 3D Resource Estimation 11

2 In Situ 3D Resource Esti-

mation

This case study is based on a real 3D data set kindly provided by Vale(Carajs mine, Brazil).

It demonstrates particular features related to the Mining industry:

domaining, processing of three dimensional data, variogram modeling

and kriging. A brief description of global estimation with change of

support and block simulations is also provided. A simple application of

use of local parameters in kriging and simulations is presented.

Reminder: while using Isatis, the on-line help is accessible anytime by

pressing F1 and provides full description of the active application.

Last update: Isatis version 2012


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12

2.1 Workflow Overview

This case study aims to give a detailed description of the kriging workflow and a brief introduction

to the grade simulation workflow of iron grades in an iron productive mine. This workflow over-

view lists the sequence of Isatis applications as they are ordered in the case study in order to runthrough it. The list is nearly complete but not exhaustive.

Next to each application, two links are provided:

m the first link opens the application description of the Users guide: this allows the user to

have a complete description of the application as it is implemented in the software;

m the second link sends the user to the corresponding practical application example in the case

study.

Applications in bold are the most important for achieving kriging and simulation:

l File/Import Users GuideCase Study

Import the raw drillhole data.

l File/Selection/Macro Users GuideCase Study

Creates a macro-selection variable for each assay of the raw data based on the lithological code.

It is used to define two domains rich ore and poor ore.

l File/Selection/Geographic Users GuideCase Study

Creates a geographic selection to mask 4 drillholes outside of the orebody.

l Tools/Copy Variable/Header to Line Users GuideCase Study

Copy the selection masking the drillholes header to all assays of the drillholes.

l Tools/Regularization Users GuideCase Study

Assays compositing tool. A comparison of regularization by length or by domains is made. This

step is compulsory to make data additive for kriging. The composites regularized by domains

are kept for the rest of the study.

l Statistics / Quick Statistics Users GuideCase Study

Different modes for making statistics are illustrated: numerical statistics by domain, graphic dis-

plays with boxplots or swathplots.

l Statistics/Exploratory Data AnalysisUsers Guide Case Study

Isatis fundamental tool for QA/QC, 2D data displays, statistical and variographic analysis.

l Statistics/Variogram Fitting Users guideCase Study

Isatis tool for variogram modeling. Different modes are illustrated:
http://../Online/tools/regularization.fmhttp://-/?-http://../Online/file/sel_macro.fmhttp://../Online/tools/regularization.fmhttp://-/?-http://../Online/Tools/migrate_point_point.fmhttp://../Online/tools/regularization.fmhttp://-/?-http://../Online/tools/regularization.fmhttp://-/?-http://../Online/statistics/eda_vmap.fmhttp://../Online/statistics/variogram_fitting.fmhttp://-/?-http://../Online/tools/regularization.fmhttp://../Online/tools/regularization.fmhttp://-/?-http://../Online/tools/regularization.fmhttp://../Online/tools/regularization.fmhttp://../Online/statistics/variogram_fitting.fmhttp://../Online/Statistics/eda.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/file/sel_macro.fmhttp://-/?-http://../Online/tools/regularization.fmhttp://../Online/tools/regularization.fmhttp://-/?-http://../Online/tools/regularization.fmhttp://../Online/tools/regularization.fmhttp://../Online/Tools/migrate_point_point.fm


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m manual: the user chooses by himself the basic structures (with their types, anisotropy, ranges

and sills) entering the parameters at the keyboard or for ranges/sills interactively in the Fit-

ting Window. This is used for modeling the variogramof the indicator of rich ore,

m automatic: the model is entireley defined (ranges, anisotropy and sills) from the definition of

the types and number of nested structures the user wants to fit. This is used for modeling the

Fe grade of rich ore.

l Statistics/Domaining/Border Effect Users GuideCase Study

Calculates statistical quantities based on domains indicator and grades to visualize the behav-

iour of grades when getting closer to the transition between domains.

l Statistics/Domaining/Contact Analysis Users GuideCase Study

Represents graphically the behaviour of the mean grade as a function of the distance of samples

to the contact between two domains.

l Interpolate/Estimation/(Co-)Kriging Users GuideCase Study

Isatis kriging application. It is applied here to krige (1) the indicator of rich ore and (2) the Fegrade of rich ore on blocks 75mx75mx15m. In order to take into account the geo-morphology of

the deposit, kriging with Local Parameters is achieved: the main axis of anisotropy and neigh-

borhood ellipsod are changed between the northern and southern part of the deposit.

l Statistics/Gaussian Anamorphosis ModelingUsers GuideCase Study

Isatis tool for normal score transform and modeling of histogram on composites support. This

step is compulsory for any non linear application including simulations. It is applied here on Fe

in the rich ore domain.

l Statistics/Support Correction Users GuideCase Study

Isatis tool for modeling grade histograms on block support. Useful for global estimation and for

non linear techniques (see Non Linear case study).

l Tools/Grade Tonnage Curves Users GuideCase Study

Calculates and represent graphically the grade tonnage curves. From the different possible

modes we compare the kriged panels and the distribution of grades on blocks obtained after sup-

port correction.

l File/Create Grid File Users GuideCase Study

Creates a grid of blocks 25mx25mx15m, on which we will simulate the ore type (1 for rich ore,

2 for poor ore) and the grades of Fe-P-SiO2.

l Tools/Migrate Grid to Point Users GuideCase Study

Transfers the selection variable defining the orebody from the panels 75mx75mx15m to the

blocks 25mx25mx15m.
http://../Online/interpolate/kriging.fmhttp://../Online/interpolate/kriging.fmhttp://../Online/interpolate/kriging.fmhttp://../Online/statistics/anamorphosis.fmhttp://-/?-http://../Online/statistics/support_correction.fmhttp://-/?-http://../Online/interpolate/kriging.fmhttp://../Online/statistics/support_correction.fmhttp://-/?-http://../Online/statistics/support_correction.fmhttp://-/?-http://-/?-http://../Online/interpolate/kriging.fmhttp://-/?-http://-/?-http://../Online/statistics/support_correction.fmhttp://-/?-http://-/?-http://../Online/statistics/support_correction.fmhttp://-/?-http://-/?-http://../Online/statistics/anamorphosis.fmhttp://-/?-http://-/?-http://../Online/statistics/support_correction.fmhttp://../Online/interpolate/kriging.fmhttp://-/?-http://../Online/interpolate/kriging.fmhttp://../Online/interpolate/kriging.fm


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14

l Interpolate/Conditional Simulations/Sequential Indicator/Standard Neighborhood Users

GuideCase Study

Simulations of the indicator of rich ore by SIS method.

l Statistics/Gaussian Anamorphosis ModelingUsers GuideCase Study

That application is run again, for the purpose of a multivariate grade simulation, to transform

Fe-P-SiO2grades of composites. The P grade distribution is modelled differently from Fe and

SiO2, because of the presence of many values at the detection limit. The zero-effect distribution

type is then applied. It results that the gaussian value assigned to P has a truncated gaussian

distribution.

l Statistics/Exploratory Data Analysis Users Guide Case Study

The Exploratory Data Analysis is used for calculating the experimental variogram on the gauss-

ian transform of P.


The variogram fitting is used with the TruncationSpecial Option for modeling the gaussian

experimental variogram of the gaussian transform of P.

l Statistics/Statistics/Gibbs Sampler Users guideCase Study

The Gibbs Sampler algorithm is used to generate the final gaussian transforms of P with a true

Gaussian distribution instead of a truncated one.

l Statistics/Exploratory Data Analysis Users Guide Case Study

The Exploratory Data Analysis is used now for calculating the experimental variogram on the

gaussian transform of Fe-P-SiO2.


The variogram fitting is used for modeling the threevariate gaussian experimental variograms of

the gaussian transform of Fe-P-SiO2. The Automatic Sill Fittingmode is used: the sills of all

basic structures are automatically calculated using a least square minimization procedure.

l Statistics/Modeling/Variogram Regularization Users guideCase Study

The threevariate variogram model of the gaussian grades is regularized on the block support. A

new experimental variogram is then obtained.


The variogram fitting is used for modeling the threevariate gaussian experimental variograms of

the gaussian transform of Fe-P-SiO2on the block support (25mx25mx15m). TheAutomatic Sill

Fittingmode is used.
http://../Online/interpolate/sc_turning_bands.fmhttp://../Online/interpolate/sc_turning_bands.fmhttp://../Online/statistics/anamorphosis.fmhttp://-/?-http://../Online/statistics/variogram_fitting.fmhttp://../Online/statistics/variogram_fitting.fmhttp://../Online/statistics/variogram_fitting.fmhttp://../Online/statistics/variogram_fitting.fmhttp://../Online/statistics/variogram_fitting.fmhttp://-/?-http://../Online/statistics/variogram_fitting.fmhttp://-/?-http://../Online/statistics/variogram_fitting.fmhttp://-/?-http://../Online/statistics/variogram_fitting.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/statistics/variogram_fitting.fmhttp://-/?-http://-/?-http://../Online/statistics/anamorphosis.fmhttp://-/?-http://-/?-http://../Online/statistics/variogram_fitting.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/statistics/eda_vmap.fmhttp://../Online/interpolate/sc_turning_bands.fmhttp://../Online/interpolate/sc_turning_bands.fm


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l Statistics/Modeling/Gaussian Support Correction Users guideCase Study

Transforms the point anamorphosis and the variogram model referring to the gaussian variables

regularized on the block support. The result is a gaussian anamorphosis on a block support and a

variogram model referring to the block gaussian variables (0-mean, variance 1). These steps are

compulsory for carrying out Direct Block Simulations.

l Interpolate/Conditional Simulations/Direct Block Simulations Users GuideCase Study

Simulations using the Turning Bands technique in the discrete gaussian model framework

(DGM).

l Statistics/Variogram on Grid Users GuideCase Study

Calculates, for QC purpose, the experimental variograms on the simulated gaussian block val-

ues.

l Statistics/Data Transformation/RawGaussian Transformation Users guideCase Study

Transforms the block gaussian simulations into raw block values.

l Tools/Copy Statistics/ Grid-> Grid Users GuideCase Study

Calculates rich ore tonnage and metal quantities in the panels 75mx75mx15m from the simu-

lated blocks 25mx25mx15m.

l File/Calculator Users GuideCase Study

Transforms the previous results into real ore tonnages and metals.

l Tools/Simulation Post-Processing Users GuideCase Study

Presents examples of Post-Processing of simulations.

l

3D viewer Users GuideCase StudySome brief description of the 3D viewer module.
http://../Online/statistics/variogram_fitting.fmhttp://../Online/interpolate/sc_turning_bands.fmhttp://../Online/interpolate/sc_turning_bands.fmhttp://../Online/statistics/variogram_fitting.fmhttp://../Online/interpolate/sc_turning_bands.fmhttp://../Online/interpolate/sc_turning_bands.fmhttp://../Online/interpolate/sc_turning_bands.fmhttp://../Online/display/display3d.fmhttp://-/?-http://../Online/interpolate/sc_turning_bands.fmhttp://-/?-http://../Online/interpolate/sc_turning_bands.fmhttp://../Online/interpolate/sc_turning_bands.fmhttp://-/?-http://../Online/statistics/variogram_fitting.fmhttp://-/?-http://../Online/interpolate/sc_turning_bands.fmhttp://-/?-http://../Online/statistics/variogram_fitting.fmhttp://../Online/display/display3d.fmhttp://-/?-http://../Online/interpolate/sc_turning_bands.fm


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2.2 Presentation of the Dataset & Pre-processing

The data set is located in the Isatis installation directory (sub-directory Datasets/Mining) and con-

stituted of two different ASCII files:

l borehole measurements are stored in the ASCII file called boreholes.asc;

l a simple 3D geological model resulting from previous geological work (block size: 75 m hori-

zontally and 15 m vertically) is provided in a 3D grid file called block model_75x75x15m.asc.).

Firstly, a new study has to be created using the File / Data File Managerfacility; then, it is advised

to verify the consistency of the units defined in the Preferences / Study Environment / Unitswin-

dow. In particular, it is suggested to use:

l Input Output Length Options:

Default Unit...= Length (m) Default Format...= Decimal (10,2)

l Graphical Axis Units:

X Coordinate = Length (km)

Y Coordinate = Length (km)

Z Coordinate = Length (m)

2.2.1 Borehole data

2.2.1.1 Data import

The boreholes.asc file begins with a header (commented by #) which describes its contents:#

# structure=line , x_unit=m , y_unit=m , z_unit=m

#

# header_field=1 , type=alpha , name="drillhole ID"

# header_field=2 , type=xb , f_type=Decimal , f_length=8 , f_digits=2 , unit="m"

# header_field=3 , type=yb , f_type=Decimal , f_length=8 , f_digits=2 , unit="m"# header_field=4 , type=zb , f_type=Decimal , f_length=8 , f_digits=2 , unit="m"

# header_field=5 , type=numeric , name="depth" , ffff=" " , bitlength=32 ;

# f_type=Decimal , f_length=8 , f_digits=2 , unit="m"

# header_field=6 , type=numeric , name="inclination" , ffff=" " ,bitlength=32 ;

# f_type=Decimal , f_length=8 , f_digits=2 , unit="deg"

# header_field=7 , type=numeric , name="azimuth" , ffff=" " , bitlength=32

;

# f_type=Decimal , f_length=8 , f_digits=2 , unit="deg"

#

# field=1 , type=xe , f_type=Decimal , f_length=8 , f_digits=2 , unit="m"

# field=2 , type=ye , f_type=Decimal , f_length=8 , f_digits=2 , unit="m"

# field=3 , type=ze , f_type=Decimal , f_length=8 , f_digits=2 , unit="m"

#

# field=4 , type=numeric , name="Sample length" , ffff=" " , bitlength=32;

# f_type=Decimal , f_length=6 , f_digits=2 , unit="m"

# field=5 , type=numeric , name="Fe" , ffff=" " , bitlength=32 ;

# f_type=Decimal , f_length=6 , f_digits=2 , unit="%"

# field=6 , type=numeric , name="P" , ffff=" " , bitlength=32 ;


# field=7 , type=numeric , name="SiO2" , ffff=" " , bitlength=32 ;



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# field=8 , type=numeric , name="Al2O3" , ffff=" " , bitlength=32 ;


# field=9 , type=numeric , name="Mn" , ffff=" " , bitlength=32 ;


# field=10 , type=alpha , name="Lithological code ALPHA" , ffff=" "# field=11 , type=numeric , name="Lithological code INTEGER" , ffff=" ", bitlength= 8 ;

# f_type=Integer , f_length= 4 , unit=" "

#

# ++++ --------- +++++++++ --------- +++++++++ --------- +++++++++# ++++++++++ --------- +++++++++ --------- +++++++++ --------- +++++++++ --------- +++++++++ --------- ---------

*---- 1 026 1400.00 -195.00 804.21 144.46 90.00 0.00

1 1400.00 -195.00 799.71 4.50 65.90 0.13 0.200.90 0.07 6 6

2 1400.00 -195.00 795.32 4.39 66.70 0.12 0.100.90 0.08 6 6

3 1400.00 -195.00 791.22 4.10 67.70 0.11 0.200.50 0.08 3 3

The samples are organized along lines and the file contains two types of records:

l The header record (for collars), which starts with an asterisk in the first column and introduces a

new line (i.e borehole).

l The regular record which describes one core of a borehole.

The file contains two delimiter lines which define the offsets for both records.

The dataset is read using the File / Import / ASCIIprocedure and stored in two new files of a new

directory called Mining Case Study:


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18

l The file Drillholes Header, which contains the header of each borehole, stored as isolated

points.

l The file Drillholes, which contains the cores measured along the boreholes.

(snap. 2.2-1)

You can check in File / Data File Manager(by pressing s for statistics on the Drillholesfile) that

the data set contains 188 boreholes, representing a total of 5954 samples. There are five numericvariables (heterotopic dataset), whose statistics are given in the next table (using Statistics/Quick

Statistics...):

We will focus mainly on Fe variable. Also note the presence of an alphanumeric variable called

Lithological code Alpha.

Number Minimum Maximum Mean St. Dev.

Al2O3 3591 0.07 44.70 1.77 4.14

Fe 5069 4.80 69.40 60.51 14.19

Mn 5008 0. 30.70 0.58 1.75

P 5069 0. 1. 0.06 0.08

Si O2 3594 0.05 75.50 1.54 4.32


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2.2.1.2 Borehole data visualization without the 3D viewer

Note - To visualize boreholes with the Isatis 3D viewer module, see the dedicated paragraph at theend of this case study.

All the 2D Display facilities are explained in detail in the Displaying & Editing Graphics chapterof the Beginner's Guide.

To visualize the lines without the 3D viewer, perform the following steps:

l Click onDisplay / New Page,

l In the Contents, for the Representation Type, choose Perspective,

l Double-click onLines. AnItem Contents for: Lines window appears:

m In the Data area, select the file Mining Case Study/Drillholes, without selecting any vari-

able as we are looking for a display of the boreholes geometry.

m Click onDisplay, andOK. The Lines appear in the graphic window.

l To change the View Point, click on theCameratab and choose for instance:

m Longitude = -46

m Latitude = 20.

l Using theDisplay Boxtab, deselect the toggleAutomatic Scalesand stretch the vertical dimen-

sion Z by a factor of 3.

l Click onDisplay.

l You should obtain the following display. You can save this template to automatically reproduce

it later: just click onApplication / Store Page asin the graphic window.

(fig. 2.2-1)


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20

The data set is contained in the following portion of the space:

Most of the boreholes are vertical and horizontally spaced approximately every 150m. The vertical

dimension is oriented upwards.

2.2.1.3 Creation of domains

In order to demonstrate Isatis capabilities linked to domaining, a simplified approach is presented

here. It consists in splitting the assays into two categories:

m the first one called rich ore corresponds to the lithological codes 1, 3and 6,

m the second one called poor ore corresponds to the lithological codes 10and above

A macro-selection final lithology[xxxxx]is created using File / Selection/Macro ...

After asking to create aNew Macro Selection Variableand defining its name final lithologyin the

Data File, you have to click onNew.

(snap. 2.2-2)

Minimum Maximum

X 0.009 km 3.97 km

Y -0.35 km 3.77 km

Z -54.9 m +811.8 m


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For creating Rich ore, Poor ore and Undefinedindices, you should give the name you want

(this has to be repeated three times). Then in the bottom part of the window you will define the

rules to apply. For each rule, you will have then to choose which variable it depends to, here Litho-

logical Code Integer, and the criterion to apply among the list you get by clicking on the buttonproposingEqualsas default:

m in the case of Rich oreyou choose Is Lower or Equals to 9

m in the case of Poor oreyou choose to match 2 rules (see snap shot on the previous page).

m in the case of Undefinedyou choose to match any of two rules (see next snap shot).

(snap. 2.2-3)

2.2.1.4 Drillholes selection

From the display of the drillholes, we can see that 4 are outside of the area covered by the other

drillholes. We will mask these drillholes for the rest of the study by using the File / Selection / Geo-

graphicmenu.

The procedure "File / Selection / Geographic" is used to visualize and to perform a masking opera-tion based on complete boreholes or more selectively on composites within a borehole.

We create the selection mask drillholes outsidein the Drillholes header file.


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(snap. 2.2-5)

By picking with the mouse left button the 4 boreholes, their symbols are blinking, they can then be

masked by using the menu button of the mouse and clicking on Mask, the 4 masked boreholes are

then represented with the red square (according to the menu Preferences / Miscellaneous).

In the Geographic Selectionwindow the number of selected samples (i.e.boreholes) is appearing

(184 from 188). To store the selection you must click onRun.

0 1000 2000 3000 4000

X (m)

0

1000

2000

3000

4000

Y

(m

)


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24

(snap. 2.2-6)

This selection is defined on the drillhole collars. In order to apply this selection to all samples of the

drillholes, a possible solution is to use the menu Tools / Copy Variable / Header Point -> Line.

(snap. 2.2-7)

2.2.1.5 Borehole data compositing

The compositing (or regularization) is an essential phase of a study using 3D data, especially in the

mining industry, although the principle is much more general. The idea is that geostatistics will

consider each datum with the same importance (prior to assigning a weight in the kriging process

0 1000 2000 3000 4000

X (m)

0

1000

2000

3000

4000

Y

(m

)


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for example) as it does not make sense to combine data that does not represent the same amount of

material.

Therefore, if data is measured on different support sizes, a first, essential task is to convert the

information into composites of the same dimension. This dimension is usually a multiple of the size

of the smallest sample, and is related to the height of the benches, which is in this case 15m.

l This operation can be achieved in different ways:

m the boreholes are cut into intervals of same length from the borehole collar, or in intervals

intersecting the boreholes and a regular system of horizontal benches. It is performed with

the Tools / Regularization by Benches or by Lengthfacility, consists in creating a replica of

the initial data set where all the variables of interest in the input file are converted into com-

posites.

m the boreholes are cut into intervals of same length, determined on the basis of domain defini-

tion. Each time the domain assigned to the assay is changed a new composite is created. The

advantage of that method is to get more homogeneous composites. It is performed with the

Tools / Regularization by Domainsfacility.

m We will work on the 5 numerical variables Al203, Fe, Mn, P and SiO2.

m The regularization by lengthis performed on 5 numerical variables Al203, Fe, Mn, P and

SiO2and on the lithological code, in order to keep for each composite the information on the

most abundant lithology and the corresponding proportion. The new files are called:

- Composites 15m by length header for the header information (collars).

- Composites 15m by lengthfor the composite information.

m Regularization mode: By Lengthmeasured along the borehole: this is the selected option as

some boreholes are inclined, with a constant length of 15m.

m Minimum Length: 7.5m. It may happen that the first composite, or the last composite (or

both) do not have the requested dimension. Keeping too many of those incomplete samples

will lead us back to the initial problem of having samples of different dimensions being con-sidered with the same importance: this is why the minimum length is set to 7.5 m (i.e. half of

the composite size).


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26

(snap. 2.2-8)

m Three boreholes are not reproduced in the composite file as their total length is too small

(less than 7.5m): boreholes 93, 163 and 171. There are 1282 composites in the new output

file.

l The regularization by domainwill calculate composites for two domains rich oreand poor

ore. The macro selection defining the domains in the input file is created with the same indicesin the output composites file. The selection mask drillholes outsideis activated to regularize

only the boreholes within the orebody envelope. Only Fe, P, SiO2 are regularized. The new files

are called:

m Composites 15m header for the header information (collars).

m Composites 15m for the composite information.

m The Undefined Domainis assigned to the Undefinedindex. It means that when a sample is

in the Undefined Domain the composition procedure keeps on going (see on-line Help for

more information).

m The Analysed Length is kept for each grade element.

m The option Merge Residualis chosen, which means that the last composite is merged with

the previous one if its length is less than 50% of the composite length.


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(snap. 2.2-9)

There are 1556 composites on the 184 boreholes in the new output file. From now on all geostatis-tical processes will be applied on that regularized by domains composites file.

Using Statistics / Quick Statisticswe can obtain different types of statistics, as for example:

The statistics on the Fe grades by domains. You note that after compositing there are no more

Undefined composites.


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28

(snap. 2.2-10)

(snap. 2.2-11)

l Graphic representations withBoxplotsby slicing according the main axes of the space.


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(snap. 2.2-12)


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2.2.2 Block model

2.2.2.6 Grid import

The block model_75x75x15m.asc file begins with a header (Isatis format, commented by #) which

describes its contents:#

# structure=grid, x_unit="m", y_unit="m", z_unit="m";

# sorting=+Z +Y +X ;

# x0= 150.00 , y0= -450.00 , z0= 310.00 ;

# dx= 75.00 , dy= 75.00 , dz= 15.00 ;

# nx= 28 , ny= 47 , nz= 31 ;

# theta= 0 , phi= 0 , psi= 0

# field=1, type=numeric, name="geographic domain", bitlength=32;

# ffff="N/A", unit="";

# f_type=Integer, f_length=9, f_digits=0;

# description="Creation Date: Mar 21 2006 15:13:15"

##+++++++++

0

0

0

The file contains only one numeric variable named geographic domain which equals 0, 1 or 2:

l 0 means the grid node lies outside the orebody,

l 1 means the grid node lies in the southern part of the orebody,

l 2 means the grid node lies in the northern part of the orebody.

Launch File/Import/ASCII... to import the grid in the Mining Case Studydirectory and call it 3D

Grid 75x75x15 m.

You have now to create a selection variable, called orebody, for all blocks where the geographic

code is either 1 or 2, by using the menuFile / Selection / Intervals.


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34

(snap. 2.2-15)

2.2.2.7 Visualization without the 3D viewer

Note - To visualize with the Isatis 3D viewer module, see the dedicated paragraph at the end of thiscase study.

Click onDisplay / New Pagein the Isatis main window. In the Contents window:

l In the Contents list, double click on the Raster item. A newItem contents for: Raster windowappears, in order to let you specify which variable you want to display and with which color

scale:

m Grid File...: select orebodyvariable from the 3D Grid 75x75x15 m file,

m In the Grid Contents area, enter 16 for the rank of the section XOY to display.

m In the Graphic Parameters areabelow, the default color scale is Rainbow.

m In theItem contents for: Rasterwindow, click on Display.

m Click on OK.


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l Your final graphic window should be similar to the one displayed hereafter.

(fig. 2.2-2)

The orebody lies approximately north-South, with a curve towards the southwestern part. Thenorthern part thins out along the northern direction and has a dipping plane striking North with a

western dip of 15 approximately. This particular geometry will be taken into account during vario-

graphic analysis.

500 1000 1500 2000

X (m)

0

1000

2000

3000

Y

(

m

)


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36

2.3 Variographic Analysis

This step describes the structural analysis performed on 3D data set. In a first stage we consider the

Fegrade only of the rich ore (univariate analysis) on the 15 m composites. The estimation requires

to estimate for each block the proportion of rich ore and its grade. The analysis has then to be made:

l on the indicator of rich ore variable, which is defined on all composites

l and on the rich ore Fe grade, which is defined on rich ore composites.

The Exploratory Data Analysis (EDA) will be used in order to perform Quality Control, check sta-

tistical characteristics and establish the experimental variograms. Then variogram models will be

fitted.

2.3.1 Variographic analysis of rich ore indicator

The workflow that has been applied illustrates some important capabilities of Exploratory Data

Analysis, the decisions that are taken would probably require more detailed analysis in a real study.

The main steps of the workflow, that will be detailed in the next pages are:

l Calculation of the rich ore indicator.

l Variogram map in horizontal slices to confirm the existence of anisotropy.

l Calculations of directional variograms in horizontal plane. For simplification we keep 2 orthog-

onal directions East-West (N90) and North-South (N0).

l Check that the main directions of anisotropy are swapped when looking to northern or southern

boreholes.

l Save the Indicator variogram in the northern part (where are most of the data), with the idea

that the variogram in the Southern part is the same as in the North by inverting N0 and N90directions of the anisotropy. In practice this will be realized at the kriging/simulation stage by

the use of Local Parameters for the variogram structures.

l Variogram Fitting using a combination of Automatic and Manual mode.

2.3.1.1 Calculation of the indicator

Use File / Calculator to assign the macro-selection index corresponding to rich ore to a float vari-

able Indicator rich ore.


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38

(snap. 2.3-2)

Highlight the Indicator rich ore variable in the main EDA window and open the Base Map and His-

togram:


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(fig. 2.3-1)

The mean value gives the proportion of rich ore samples.

The variogram map allows to check potential anisotropy. After clicking on the variogram map, the

Define Parameters Before Initial Calculations being on, you should choose the parameters as

shown in the next figure. You define parameters for horizontal slices, i.e. Ref.Plane UVwith No

rotation.

Switch off the buttonDefine the Calculations in the UW Planeand in the VW Plane, using the cor-

responding tabs.

With 18 directions each direction makes an angle of 10 with the previoius one. By asking a Toler-

ance on Directionsof 2sectors, the variograms are calculated from pairs in a given direction +/-25.

0 1000 2000

X (m)

0

1000

2000

3000

4000

Y

(m)

0.0 0.5 1.0

Indicator rich ore

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Frequencies

Nb Samples: 1556

Minimum: 0.000

Maximum: 1.000

Mean: 0.627

Std. Dev.: 0.484


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40

(snap. 2.3-3)


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(snap. 2.3-4)

After pressing OKyou get the representation of the Variogram Map. In the ApplicationMenu ask

Invert View Orderto have variogram map and extracted experimental variograms in a landscape

view.

In theApplicationMenu ask Graphic Specific Parametersand change the Color Scaleto Rain-

bow Reversed.

In the variogram map representation drag with the mouse a zone containing all directions. With themenu button ask Activate Direction. You will then visualize the experimental variograms in the 18

directions of the horizontal plane. It exhibits clearly anisotropic behaviour.


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42

(snap. 2.3-5)

We will now calculate the experimental variograms directly from the main EDA window by click-

ing on the Variogrambitmap at the bottom of the window. In the next figure we can see the param-

eters used for the calculation of 4 directional variograms in the horizontal plane and the verticalvariogram.

(snap. 2.3-6)


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(snap. 2.3-7)

(snap. 2.3-8)

For sake of simplicity we decide to keep only 2 directions N0, showing more continuity and the

perpendicular direction N90.

The procedure to follow is:


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44

l In theList of Options, change from OmnidirectionaltoDirectional.

l InRegular Direction chooseNumber of Regular Directions2and switch onActivate Direction

Normal to the Reference Plane. Click Ok and go back to the Variogram Calculation Parameters

window.

(snap. 2.3-9)

You have then to define the parameters for each direction. Click the parameter table to edit:

l You have then to define the parameters for each direction. Click the parameter table to edit. Forapplying the same parameters on the 2 horizontal directions, you must highlight these directions

in theDirections listof the Directions Definition window.

l The two regular directions choose the following parameters:

m Label for direction 1: N90 (default name)

m Label for direction 2: N0

m Tolerance on direction: 45 (in order to consider all samples without overlapping)

m Lag value: 90 m (i.e. approximately the distance between boreholes)

m Number of lags: 15(so that the variogram will be calculated over 1350 m distance)

m Tolerance on Distance(proportion of the lag): 0.5

m Slicing Height:7.55 m (adapted to the height of composites)

m Number of Lags Refined: 1

m Lag Subdivision: 45m (so that we can have the variogram at short distance from the drill-

holes closely spaced).

l The normal direction with the following parameters:

m Label for direction 1: Vertical

m Tolerance on angle: 22.5

m Lag value: 15 m

m Number of lags: 10

m Tolerance on lags (proportion of the lag): 0.5


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l In the Application Menu ask for Graphic Specific Parametersand click on the toggle button

for the display of the Histogram of Pairs.

(snap. 2.3-10)

Because the general shape of the orebody is anisotropic, we will calculate the variogram restricted

to the northern part and to the southern part of the orebody.

To do so you will use capabilities of the linked windows of EDA, by masking samples in the Base

Map. Automatically the variograms will be recalculated with only the selected samples.

For instance in the Base Map you drag a box around data in the Southern part (as shown on the fig-

ure) and with the menu button of the mouse you ask Mask. You will then get the variogram calcu-

lated from the northern data.


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46

(snap. 2.3-11)

In the next figure we compare the variograms calculated from the northern and the southern data.

The main directions of anisotropy are swapped between North and South.


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48

(snap. 2.3-13)

We decide now to fit a variogram model on the northern variogram, which is calculated with the

most abundant data. Then we will apply the same variogram to the southern data by making the

main axes of anisotropy swapped. This will be realized by means of local parameters attached to the

variogram model and to the neighborhood.

In the graphic window containing the experimental variogram in the northern zone, click onAppli-

cation / Save in Parameter Fileand save the variogram under the name Indicator rich ore North.

2.3.1.3 Variogram Modeling of the Indicator rich ore

You must now define a Model which fits the experimental variogram calculated previously. In the

Statistics / Variogram Fittingapplication, define:

l the Parameter File containing the set of experimental variograms: Indicator rich ore North.

l the Parameter File in which you wish to save the resulting model: Indicator rich ore

Click on Show Advanced Parameters.


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(snap. 2.3-14)


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50

l Set the toggles Fitting Window and Global WindowON; the program displays automatically

one default spherical model. The Fitting window displays one direction at a time (you may

choose the direction to display through Application/Variable & Direction Selection...), and the

Global windowdisplays every variable (if several) and direction in one graphic.

l

To display each direction in separate views, click in the Global Window on Application /Graphic Specific Parameters and choose the Manualmode. Choose for Nb of Columns3,

then Add, in turn for each Current Column, in the Selectionby picking in the View Contents

area the First Variable, the Second Variableand theDirection.

(snap. 2.3-15)

l when pressing theEditbutton next to the variogram model, the Model Definition sub-window

opens and the user can choose the basic structures. The model must reflect:

m the variability at short distances, with a consistent nugget effect,

m the main directions of anisotropy,

m the general increase of the variogram.

The model is automatically defined with the same rotation definition as the experimental vario-

gram. Three different structures have been defined (in the Model Definition window, use the Add

button to add a structure, and define its characteristics below, for each structure):

l Nuggeteffect,

l Anisotropic Exponentialmodel with the following respective ranges along U, V and W: 700 m,

550 m and 70 m,

l Anisotropic Exponentialmodel with the following respective ranges along U, V and W: 500

m, 5000 m and nothing (which means that it is a zonal component with no contribution in the

vertical direction).

Do not specify the sill for each structure at this stage, instead:


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l click Nugget effect in the main Variogram Fitting window, set the toggle buttonLock the Nug-

get Effect Components During Automatic Sill FittingONand enter the value .065.

l set the toggleAutomatic Sill FittingON. The program automatically computes the sills and dis-

plays the results in the graphic windows.

l A final adjustement is necessary, particularly to get a total sill of 0.25, which is the maximum

admissible for a stationary indicator variogram. Set the toggleAutomatic Sill FittingOFFfrom

the main Variogram Fitting window, then in the Model Definition window set the sill for the

first exponential to 0.14and the sill for the second exponential to 0.045.

The final model is saved in the parameter file by clickingRunin the Variogram Fitting window.

(snap. 2.3-16)

2.3.2 Variographic Analysis of Fe rich ore2.3.2.4 Experimental Variogram of Fe rich ore

Launch Statistics/Exploratory Data Analysis... to start the analysis on the variable Fe using the

selection for the rich ore composites.


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52

(snap. 2.3-17)

You will calculate the variograms in 2 directions of dipping plane striking North with a western dip

of 15. In the Calculation Parameters you will ask inList of OptionsaDirectional.Click thenReg-

ular Directionsa new window Directionspops up where you will define the Reference Direction

and switch onActivate Direction Normal to the Reference Plane.

(snap. 2.3-18)

ClickReference Direction, in3D Direction Definition window set the convention to User Defined

and define the rotation parameters as shown in the next figure.


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(snap. 2.3-19)

The reference direction U (in red) correspond to the N121 main direction of anisotropy.

The calculation parameters are then chosen as shown in the next figure.


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(snap. 2.3-21)

For using the linked windows the following actions have to be made:


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56

l ask to display the histogram (accept the default parameters),

l in the Graphic Specific Parametersof the graphic page containing the experimental variogram,

set the toggle button Variogram Cloud (if calculated) OFF, and click on the radio button Pick

from Experimental Variogram.

l in the Calculation Parametersof the graphic page containing the experimental variogram, set

the toggle button Calculate the Variogram CloudON.

l In the graphic page click on the experimental point with 43 pairs and ask in the menu of the

mouse Highlight. The variogram is then represented as a blue square, and all data making the

pairs represented the part painted in blue in the histogram.

(snap. 2.3-22)

The high variability due to pairs made of the samples with low values is responsible of the peak in

the variogram. It can be proved by clicking in the histogram on the bar of the minimum values and

clicking with the menu of the mouse on Mask, the variograms are automatically calculated and

dont show anymore the anomalous point as shown on the next figure.

(snap. 2.3-23)


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l We now re-calculate the variograms with 2 directions, omni-directional in the horizontal plane

and vertical, with the parameters shown hereafter you enter by clickingRegular Directions....

(snap. 2.3-24)


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(snap. 2.3-26)

In the Global window, you represent the variograms in two columns, the automatic variogram

looks satisfactory, so you click Runin the Variogram Fitting window to save it.

(fig. 2.3-2)

2.3.3 Analysis of border effects

This chapter may be skipped in a first reading as it does not change anything in the Isatis study. It

helps to decide whether kriging/ simulation will be made using hard or soft boundary.

In order to understand the behaviour of Fe grades when the samples are close to the border between

rich and poor ore, we can use two applications:


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60

l Statistics / Domaining / Border effectcalculates bi-point statistics from pairs of samples belong-

ing to different domains. The pairs are chosen in the same way as for experimental variogram

calculations.

l Statistics / Domaining / Contact Analysis calculates the mean values of samples of 2 domains

as a function of the distance to the contact between these domains along the drillholes.

2.3.3.6 Statistics on Border effect

Launch Statistics / Domaining / Border effect and choose in the file Composites 15m, theMacro

Selection Variablefinal lithology[xxxxx], that contains the definition of all domains, and the vari-

able of interest Fe.

In the list ofDomainsyou may pick only some of these, in this case Rich oreand Poor ore, while

you ask to Mask Samples from Domain choosing Undefined.

In the Calculation Parameterssub-window we define the parameters for 3 directions by pressing

the corresponding tabs in turn and switching on the toggleActivate Direction. For the 3 directions

the parameters are:

Switch on the three toggle buttons for the Graphic Parameters and click on Run.


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(snap. 2.3-27)

Three graphic pages corresponding to the three statistics are then displayed:


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62

l Transition Probability, that, in the case of only 2 domains, is not very informative.

(snap. 2.3-28)


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64

l Mean Diff[Z(x+h)-Z(x)], that shows that when going from Rich ore to Poor ore as well as

going from Poor ore to Rich ore the grade difference is influenced by the proximity of both

domains.

(snap. 2.3-30)

2.3.3.7 Contact AnalysisLaunch Statistics / Domaining / Contact Analysis and choose in the file Composites 15m, the

Macro Selection Variablefinal lithology[xxxxx], that contains the definition of all domains, and

the variable of interest Fe. You set the variables Direct Distance Variableand Indirect Distance

Variable to None, which means that the contact point is determined when the domain changes

down the boreholes.

In the list ofDomainsyou pick Rich orefor Domain 1 and Poor ore for Domain 2, while you let

Use Undefined Domain Variableto Off.

The statistics are calculated as a function of the distance to the contact along the drillhole, you have

the possibility to select only some of the drillholes according to a specific direction with an angular

tolerance. In this case, as most of the drillholes are vertical, we select all drillholes by choosing a

tolerance of 90 on the vertical direction defined by thre rotation anglesAz=0,Ay=90,Ax=0(Math-

ematician Convention). The samples are regrouped byDistance Classesof 15m.

Dir

Dir

Dir

0 500 1000 1500

Distance (m)

-40

-30

-20

-10

0

10

20

30

40

Diff

Fe,

x+

h

in

Ric

h

ore,

x

NOT

Dir

Dir

Dir

0 500 1000 1500

Distance (m)

-40

-30

-20

-10

0

10

20

30

40

Diff

F

e,

x+

h

in

Ric

h

o

re

|

x

in

Poor

or

Dir

Dir

Dir

0 500 1000 1500

Distance (m)

-40

-30

-20

-10

0

10

20

30

40

Diff

F

e,

x+

h

in

Poor

ore,

x

NOT

Dir

Dir

Dir

0 500 1000 1500

Distance (m)

-40

-30

-20

-10

0

10

20

30

40

Diff

Fe

x+

h

in

Poor

ore

|

x

in

Ric

h

ore


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(snap. 2.3-31)

Two graphic pages are then displayed:

l

Contact Analysis (Oriented)contains two views:m Directfor statistics calculated in the Reference Direction

m Indirect for statistics calculated in the opposite of the Reference Direction

In the Application menu of the graphic pages we ask the Graphical Parameters, as shown

below,to display the Number of Pointsand the Mean per Domain.

(snap. 2.3-32)


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l Contact Analysis (Non-Oriented) displays the average of the two previous ones.

(snap. 2.3-34)

From these graphs it appears that the poor grades are influenced by the proximity to rich grades.

In conclusion we decide for the kriging and simulations steps to apply hard boundary when dealing

with rich ore.


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2.4 Kriging

We are now going to estimate on blocks 75mx75mx15m the tonnage and Fe grades of Rich ore.

Therefore, we will perform two steps:

l Kriging of the Indicator of Rich ore to get the estimated proportion of rich ore, from which the

tonnage can be deduced.

l Kriging of the Fe grade of rich ore using only the rich ore samples. Each block is then estimated

as if it would be entirely in rich ore, by applying the estimated tonnage, we can then obtain an

estimate of the Fe metal content.

2.4.1 Kriging of indicator of rich ore with local parameters

After the variographic analysis it was found that the variogram model has an horizontal anisotropy

that has a different orientation in the northern and southern part of the orebody. We will then use

that orientation as local parameter recovered from the grid file in a variable called RotZ. As a first

attempt, that should be sufficient in this case because of the orebody shape, we will use two values90 for blocks in the southern area and 0 for the northern area, both areas being defined by means

of the geographic codevariable (respectively 1and 2). These values are stored in the grid file by

using File / Calculator.

(snap. 2.4-1)


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Then you launchInterpolate / Estimation / (Co)Kriging.

(snap. 2.4-2)

You need to specify the type of calculation to Block and the number of variables to 1, then:

l Input File: Indicator rich ore(Composites on 15mwith the selection None).

l The names of the variables in the output file (3D Grid 75 x 75 x 15 m), with the orebodyselec-

tion active:

m Kriging indicator rich orefor the estimation of Indicator rich ore

m Kriging indicator rich ore std devfor the kriging standard deviation


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l The variogram model contained in the Parameter File calledIndicator rich ore.

l The neighborhood: open theNeighborhood...definition window and specify the name (Indica-

tor rich ore for instance) of the new parameter file which will contain the following parameters,

to be defined from theEdit...button nearby. The neighborhood type is set by default to moving:

(snap. 2.4-3)

m The moving neighborhood is an ellipsoid with No rotation, which means that U,V,W axes

are the original X,Y,Z axes;

m Set the dimensions of the ellipsoid to 800m, 600m and 60m along the vertical direction;

m Switch ONthe Use Anisotropic Distancesbutton.

m Minimum number of samples: 4;

m Number of angular sectors: 12

m Optimum Number of Samples per Sector: 5

m Block discretization: as we chose to performBlockkriging, the block discretization has to bedefined. The default settings for discretization are 5 x 5 x 1, meaning each block is sub-

divided by 5 in each X and Y direction, but is not divided in Z direction. TheBlock Discret-


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ization sub-window may be used to change these settings, and check how different discreti-

zations influence the block covariance Cvv. In this case study, the default parameters 5x5x1

will be kept.

m Press OKfor theNeighborhood Definition.

l The Local Parameters: open theLocal Parameters Loading... window and specify the name oftheLocal Parameters File(3D Grid 75x75x15m). Fore theModel All StructuresandNeighbor-

hoodtabs switch ONUse Local Rotation (Mathematician convention) then 2D and define as

Rotation/Zthe variable Rot Z.

(snap. 2.4-4)


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It is possible to check both the model and the neighborhood performances when processing on a

grid node, and to display the results graphically: this is the purpose of the Testoption at the bottom

of the (Co-)Kriging main window. When pressing it, a graphic page opens where:

l The Indicator rich ore variable is represented with proportional symbols,

l The neighborhood ellipsoid is drawn on a 2D section.

By pressing once on the left button of the mouse, the target grid is shown (in fact a XOY section of

it, you may select different sections through Application/Selection For Display...). The user can

then move the cursor to a target grid node: click once more to initiate kriging. The samples selected

in the neighborhood are highlighted and the weights are displayed. We can see here that the nearest

samples get the higher weights. It is also important to check that the negative weights due to screen

effect are not too important. The neighborhood can be changed sometimes to avoid this kind of

problem (more sectors and less points by sector...).

You can also select the target grid node by giving the indices along X, Y and Z with theApplication

menu Target Selection(for instance 6, 11, 16). You can figure out how the local parameters used

for the neighborhood are applied.

(snap. 2.4-5)


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(snap. 2.4-6)

Note - From Application/Link to 3D viewer, you may ask for a 3D representation of the searchellipsoid if the 3D viewer application is already running (see the end of this case study).

Close the Test Window and press RUN.

7814 grid nodes have been estimated. Basic statistics of the variables are displayed below.

(fig. 2.4-1)

The kriging standard deviation is an indicator of the estimation error, and depends only on the geo-

metrical configuration of the data around the target grid node and on the variogram model. Basi-

cally, the standard deviation decrease as an estimated grid node is closer to data.

Some blocks have the kriged indicator above 1. These values will be changed into 1 by means of

File / Calculator.


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74

(snap. 2.4-7)

Note - In the main Kriging window, the optional toggle Full set of Output Variables allows tostore in the Output File other kriging parameters: slope of regression, weight of the mean,

estimated dispersion variance of estimates etc...

2.4.2 Kriging of Fe rich oreIn the Standard (Co)Kriging menu specify the type of calculation to Block and the number of

variables to 1, then enter the following parameters:

l Input File: Fe(Composites on 15mwith the selection final lithology{rich ore}).

l The names of the variables in the output file (3D Grid 75 x 75 x 15 m), with the orebodyselec-

tion active:

m Kriging Fe rich orefor the estimation of Fe;

m Kriging Fe rich ore std devfor the kriging standard deviation.


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l The variogram model contained in the Parameter File calledFe rich ore.

l The neighborhood: open theNeighborhood...definition window and specify the name (Fe rich

ore for instance) of the new parameter file which will contain the following parameters, to be

defined from theEdit...button nearby. The neighborhood type is set by default to moving:

m The moving neighborhood is an ellipsoid with No rotation, which means that U,V,W axesare the original X,Y,Z axes;

m Set the dimensions of the ellipsoid to 800m, 300m and 50m along the vertical direction;

m Switch ONthe Use Anisotropic Distancesbutton.

m Minimum number of samples: 4;

m Number of angular sectors: 12

m Optimum Number of Samples per Sector: 3

m Block discretization: as we chose to performBlockkriging, the block discretization is kept to

the default 5 x 5 x 1.

l Apply Local Parameters but only for the Neighborhood, where you use Rot Zvariable for 2D

Rotation /Z.

(snap. 2.4-8)


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76

After Runyou can calculate the statistics of the kriged estimate by asking in Statistics / Quick Sta-

tisticsto apply as Weightthe weight variableKriging indicator rich ore. 7561 blocks from 7814

have been kriged. By using a weight variable you will obtain the statistics weighted by the propor-

tion of the block in rich ore.

(snap. 2.4-9)

(fig. 2.4-2)


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The mean grade is close to the average of the composites grade (65.84). Therefore in the next steps,

carrying out non linear methods which require the modeling of the distribution, we will not apply

any declustering weights.


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2.5 Global Estimation With Change of Support

The support is the geometrical volume on which the grade is defined.

Assuming the data sampling is representative of the deposit, it is possible to fit a histogram model

on the experimental histogram of the composites. But at the mining stage, the cut-off will be

applied on blocks, not on composites. Therefore, it is necessary to apply a support correction to the

composite histogram model in order to estimate an histogram model on the block support.

Note - When kriging too small blocks with a high error level, applying a cut-off to the krigedgrades will induce biased tonnage estimates due to the high smoothing effect. It is then

recommended to use non-linear estimation techniques, or simulations (see the Non Linear case

study). For global estimation, an other alternative is to use the Gaussian anamorphosis modeling,

as described here below.

2.5.1 Gaussian anamorphosis modeling

Gaussian anamorphosis is a mathematical technique which allows to model histograms, taking thechange of support from composites to blocks into account.

Note - From a support size point of view, composites will be considered as points compared toblocks.

The technique will not be mathematically detailed here: the reader is referred to the Isatis on-line

help and technical references. Basically, the anamorphosis transforms an experimental dataset to a

gaussian dataset (i.e. having a gaussian histogram). The anamorphosis is bijective, so it is possible

to back transform gaussian values to raw values. A gaussian histogram is often a pre-requisite for

using non linear and simulation techniques. The anamorphosis function may be modelled in two

ways:

l by a discretization with n points between a negative gaussian value of -5 and a positive gaussian

value of +5.

l by using a decomposition into Hermite polynomials up to a degree N. This was the only possi-

bility until the Isatis release V10.0. It is still compulsory for some applications, as will be

explained later on.

Open the Statistics/Gaussian Anamorphosis Modelingwindow.


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(snap. 2.5-1)

l In Input... choose the Composites 15 m file with the selection final lithology{Rich ore};

choose Fefor the raw variable.

l Do NOTask for a Gaussian Transform.

l Name the anamorphosis function Fe rich ore.

l InInteractive Fitting...choose the TypeStandardand switch ONthe toggle buttonDispersion

with theDispersion Lawset to Log-Normal Distribution. In this mode the histogram will be

modelled by assigning to each datum a dispersion, that accounts for some uncertainty that is


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globally reflected by an error on the mean value. The variability of the dispersion is controlled

by the Variance Increaseparameter, related to the estimation variance of the mean. By default

that variance is set to the statistical variance of the data divided by the number of data.

(snap. 2.5-2)


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l Click on the Anamorphosis and Histogram bitmaps. You will visualize the anamorphosis func-

tion and how the experimental histogram is modelled (black bars are for the experimental histo-

gram and the blue bars for the modelled histogram).

(snap. 2.5-3)

Close the Fitting Parameters window.

l Press RUN in the Gaussian Anamorphosis window: because you have not asked for Hermite

Polynomials, the following error message window is displayed to advise you on the applications

requiring these polynomials.

(snap. 2.5-4)


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2.5.2 Block anamorphosis on SMU support

Using the composite histogram and variogram models, we are now going to take the change of sup-

port into account using Statistics/Support Correction...:

(snap. 2.5-5)

The Selective Mining Unit (SMU) size has been fixed to 25 x 25 x 15 m. Therefore, the correction

will be calculated for a block support of 25 x 25 x 15 m. Each block is discretized by default in 3x3

for the X and Y direction (NX = 3 and NY = 3); no discretization is needed for the vertical direction(NZ = 1) as the composites are regularized accordingly to the bench height (15 m). Changing the

discretization along X and Y may allow to study the sensitivity on change of support coefficients.

Switch ONthe toggle button Normalize Variogram Sill. As the variogram sill is higher than the


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variance, the consequence is to reduce a little bit the support correction (r coefficient a bit higher

than without normalization).

Press Calculateat the bottom of the window. The block support correction calculations are dis-

played in the message window:

(snap. 2.5-6)

The block variogram value Gamma (v,v) is calculated and is the base for calculating the real

block variance and the real block support correction coefficient r. We can see that the supportcorrection is not very important (r not very far from 1), it is because of the variogram model whose

ranges are rather large compared to the smu size. The calculation is made at random, so different

calculations will give similar results, but different. If the differences in the real block variance are

too large, the block discretization should be refined by increasing NX and NY. By pressing Calcu-

late...several times, we statistically check if the discretization is fine enough to represent the vari-

ability inside the blocks. Press OK.

Save theBlock Anamorphosisunder the name Fe rich ore block 25x25x15and press RUN.

2.5.3 Grade Tonnage Curves

Launch Tools / Grade Tonnage Curves. You will ask to display two types of curves, calculated

from:


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(snap. 2.5-8)

For the second curve, on blocks histogram:


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86

(snap. 2.5-9)

After clicking the bitmaps at the bottom of the Grade Tonnage Curveswindow (M vs. z,T vs z,Q

vs. z,Q vs.T,B vs z) you get the graphics like for instance T(z), M(z):


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(snap. 2.5-10)

(snap. 2.5-11)

These curves show as expected that the selectivity is better from true blocks 25x25x15 than from

kriged panels 75x75x15, that have a lower dispersion variance.

The legend is displayed in a Separate Windowas was asked in the Grade Tonange Curves win-

dow. By clickingDefine Axesyou switch OFFAutomatic Boundsto change theAxis Minimumand

Axis MaximumforMean Gradeto 60and 70respectively.

50 55 60 65

Cutoff

0

10

20

30

40

50

60

70

80

90

100

Tota

l

Tonnage

50 55 60 65

Cutoff

60

61

62

63

64

65

66

67

68

69

70

M

ean

Gra

de


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2.6 Simulations

This chapter aims at giving a quick example of conditional block simulations in a multivariate case.

Simulations allow to reproduce the real variability of the variable.

We will focus on the Fe-P-SiO2grades of rich ore of blocks 25mx25mx15m. Two steps will then be

achieved:

l simulation of the rich ore indicator. Sequential Indicator method will be applied to generate sim-

ulated model where each block has a simulated code 1for rich ore blocks and 2for poor ore

blocks. A finer grid would be required to be more realistic, for sake of simplicity we will make

the indicator simulation on the same blocks 25mx25mx15m.

l simulation of rich ore Fe grade, as if each block would be entirely in rich ore. By intersecting

with the indicator simulation, we will get the final picture.

2.6.1 Simulation of the indicator rich ore

You must first create the grid of blocks 25x25x15 with File / Create Grid File.

(snap. 2.6-1)


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To create in the grid file the orebody selection we use the migration capability (Tools/Migrate/Grid

to Point...) from the 3D Grid 75x75x15 mfile to 3D Grid 25x25x15with maximum migration dis-

tance of 55m.

(snap. 2.6-2)

Open the menuInterpolate / Conditional Simulations / Sequential Indicator / Standard Neighbor-

hood.


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(snap. 2.6-3)

For defining the two facies 1 for rich ore and 2 for the complementary you have to click on

Facies Definitionand enter the parameters as shown below.


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m model these gaussian variograms with a linear model of coregionalisation;

m regularize these variograms on the block support;

m perform a support correction on the gaussian transforms;

m perform the simulations using the discrete gaussian model framework, that allows to condi-

tion block simulated values to gaussian point data.

2.6.2.1 Gaussian Anamorphosis

We will perform the gaussian anamorphosis on the three grades of the rich ore domain in one go.

and independently. Note that the three anamorphosis functions must be stored together in the same

Parameter file called Fe-SiO2-P rich ore. Note in this case that we also ask to store the Gaussian

transforms in the composites file with the names Gaussian Fe/P/SiO2rich ore, ...


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(snap. 2.6-6)

AfterRunthe transformed values of Fe and SiO2 have a gaussian distribution, while for P the gaus-

sian transform has a truncated gaussian distribution. The gaussian values assigned to the samples

concerned by the zero effect are all equal to the same value (gaussian value corresponding to the

frequency of the zero effect).

2.6.2.2 Gaussian transform of P rich ore

The next steps consist of making the gaussian transform of P a true gaussian distribution. This is

achieved by using a Gibbs Sampler algorithm that will generate for all samples of the zero effect a

gaussian value consistent with the structure of spatial correlation with all gaussian values. Practi-

cally 3 steps must be carried out:

l calculation of the experimental variogram of the truncated gaussian values;

l variogram modelling of the gaussian transform using the truncation option;

l Gibbs Sampler to generate the gaussian transform with a true distribution and honouring the

spatial correlation.

Using EDA we calculate the histogram and the experimental variogram on the variable GaussianP rich ore(activating the selection final lithology{Rich ore}). In theApplicationmenu of the his-

togram you ask the Calculation Parametersand switch offtheAutomaticmode to the values shown

below:

(snap. 2.6-7)


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For the variogram you choose the same parameters as used for Fe (omnidirectional in the horizontal

plane and vertical), by asking in theApplication Menu / Calculation Parameters, in the Variogram

Calculation Parameters window click Load Parameters from Standard Parameter Fileand select

the experimental variogram Fe rich ore.

On the graphic display you see the truncated distribution with about 35% of samples concerned bythe zero effect, the gaussian truncated value is -0.393. The variance displayed as the dotted line on

the variograms is about 0.5. In the Application / Save in Parameter Filemenu of the graphic con-

taining the variogram you save it under the name Gaussian P rich ore zero effect.

(snap. 2.6-8)


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(snap. 2.6-9)

In the Variogram Fittingwindow you choose the Experimental VariogramsGaussian P rich ore

zero effect and you create a New Variogram Model, called Gaussian P rich ore. Note that the var-

iogram model refers to the gaussian transform (with the true gaussian distribution), it is transformed

by means of the truncation to match the experimental variogram of the truncated gaussian variable.

(snap. 2.6-10)

Click Edit, in theModel Definitionwindow you must first click Truncation.


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You will now generate gaussian values for the zero effect on P rich ore by using Statistics / Statis-

tics / Gibbs Sampler. Note that the gaussian values not concerned by the zero effect are kept

unchanged.

l TheInput Dataare the variogram model you just fitted Gaussian P rich oreand the Gaussian

P rich orevariable stored after the GaussainAnamorphosis Modelling.

l The Output Data are a new variogram model Gaussian P rich ore no truncation (which is in

fact the same as the input one without the truncation option) and a new variable in the Compos-

ites 15m file Gaussian P rich ore (Gibbs).

l You ask to perform 1000iterations.

(snap. 2.6-15)

You can check how the Gibbs Sampler has reproduced the gaussian distribution and the input vari-

ogram. You just have to recalculate the histogram and the variograms on the variable Gaussian P

rich ore (Gibbs). After saving in the Parameter File that experimental variogram, you can superim-

pose to it the variogram model with no truncation using Variogram Fitting menu. For the first dis-

tance the fit is acceptable.


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(snap. 2.6-16)

(snap. 2.6-17)

N0

1

157

472688

11201373

11951196900

1108 12221155

D-9

6

78

92

325

266

223

183

148

117

0 500 1000 1500

Distance (m)

0.0

0.5

1.0

1.5

Variogram:

GaussianPrichore

(Gibbs)


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2.6.2.3 Multivariate Gaussian variogram modeling

In Statistics / Exploratory Data Analysisyou calculate the variograms with the same parameters as

before (one monidirectional horizontal direction and one vertical direction) on the 3 gaussian trans-

forms.

In the graphic window you useApplication / Save in Parameter Fileto save these variograms under

the name Gaussian Fe-SiO2-P rich ore.

(snap. 2.6-18)

In Statistics/Variogram Fitting..., choose the experimental variogram you just saved. Create the

new variogram model with the same name Gaussian Fe-SiO2-P rich ore. Set the toggles Global

Window and ask to display the number of pairs in the graphic window (Application/Graphic

Parameters...).


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(snap. 2.6-19)

The model is made using the following method:


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l enter the name of the new variogram model Gaussian Fe-SiO2-P rich oreand Editit.

l in theModel Definitionwindow click onLoad Modeland choose the model made for Gaussian

P rich ore no truncation. The following window pops up:*

(snap. 2.6-20)

Clck on Clearbutton, then move the mouse to the second line Gaussian P rich ore, click on Link

and on OKin the Selectorwindow to put the variogram made on Gaussian P alone for the same

variable in the three variate variogram. Then you click onOKin theModel Loadingwindow.

l in the Variogram Fittingwindow click on Automatic Sill Fitting. The Global Window shows

the model that has been fitted. Press Runto save it in the parameter file.


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l You first have to launch Statistics / Modeling / Variogram Regularization. You will store in a

new experimental variogram Gaussian Fe-SiO2-P rich ore block 25x25x153 directional vari-

ograms using a discretization of 5x5x1. You will also ask to Normalize the Input Point Vario-

gram.

(snap. 2.6-22)

l Then you model the regularized variogram using Variogram Fittingand the Automatic Sill Fit-

ting mode, after having loaded the model made on the point samples Gaussian Fe-SiO2-P rich

ore. You note that the Nugget effect is put to zero. When you save the variogram model the

Nugget effect is not stored in the Parameter file


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(snap. 2.6-23)


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(snap. 2.6-25)

2.6.2.6 Direct Block Simulation

It is achieved by running the menuInterpolate / Conditional Simulations / Direct Block Simulation.

It takes some time to get 100 simulations. Depending on the computer it may be more than an hour.


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l The simulated variables are created with the following names Simu block Gaussian Fe rich

ore...in the 3D Grid 25x25x15. We store the gaussian values before transform to allow a check

of the experimental variograms on gaussian simulated values with the input variogram model,

that is defined on the gaussian variables.

l

TheBlock Anamorphosisand theBlock Gaussian Modelare those obtained from the GaussianSupport Correction.

l The Neighborhood used for kriging Fe rich ore is modified into a new one called Fe rich ore

simulation changing the radius along V to 800m. The reason is just because the Local Parame-

ters for the neighborhood are not implemented in the application Direct Block Simulation.

l Number of simulations: 100for instance .

l We ask to notPerform a Gaussian Back Transformation, for the reason explained above. The

back transform will be achieved afterwards.

l The turning bands algorithm is used with 1000Turning Bands.


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(snap. 2.6-26)

You can compare the experimental variograms calculated from the 100 simulations in up to 3 direc-

tions with the input variogram model. The directions are entered by giving the increments (number

of grid mesh) of the unit directional lag along X, Y, Z. For instance for the direction 1, the incre-

ments are respectively 1, 0, 0, which makes the unit lag 25m East-West.


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(snap. 2.6-27)

Three graphic pages (one per direction) are then displayed. The average experimental variograms

are displayed with a single line, th

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