Manual Mine Sight Geoestadistica

MineSight for Modelers ®

Geostatistics Workbook E005 Rev. B

© 2002, 2001, 1994, and 1978 by MINTEC, inc. All rights reserved. No part of this document shall be reproduced, stored in a retrieval system, or transmitted by any means, electronic, mechanical, photocopying, recording or otherwise, without written permission from MINTEC, inc. All terms mentioned in this document that are known to be trademarks or registered trademarks of their respective companies have been appropriately identified. MineSight® is a registered trademark of MINTEC, inc. acQuire® is a registered trademark of Metech Pty Ltd

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Table of Contents

MineSight for Modelers – Geostatistics Table of Contents Using This MineSight Workbook......... ..............................................................Intro-1 MineSight Overview............................................................................................Over-1 Geostatistics Overheads.........................................................................................1 Classical Statistics.................................................................................................1-1 Variograms..............................................................................................................2-1 Point Validation/Cross Validation for Variogram Evaluation..............................3-1 Declustering............................................................................................................4-1 Model Interpolation (Inverse Distance Weighting and Ordinary Kriging)..........5-1 Debugging Interpolation Runs.............................................................................. 6-1 Point Validation/Cross Validation of Estimation Methods and/or Search Parameters...............................................................................................................7-1 Model Statistics/Geologic Reserves......................................................................8-1 Model Calculations..................................................................................................9-1 Quantifying Uncertainty........................................................................................10-1 Change of Support.................................................................................................11-1 Outlier Restricted Kriging......................................................................................12-1 Indicator Kriging to Define Geologic Boundary Above a Cutoff........................13-1 Muliple Indicator Kriging.......................................................................................14-1 Other Non-Kriging Interpolation Methods...........................................................15-1

Part #: E005 Rev. B

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Table of Contents

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Part #: E005 Rev. B

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Using this Mintec Workbook

Using this Mintec Workbook

Notes:

The objective of this workbook is to provide hands on training and experience with the MineSight Operations package. This workbook does not cover all the capabilites of MineSight, but concentrates on typical mine geologists duties using a given set of data.

Introduction to the Course To begin, we would like to thank you for taking the opportunity to enrich your understanding of MineSight through taking this training course offered by Mintec Technical Support. Please start out by reviewing this material on workbook conventions prior to proceeding with the training course documentation. This workbook is designed to present concepts clearly and then give the user practice through exercises to perform the stated tasks and achieve the required results. All sections of this workbook contain a basic step, or series of steps, for using MineSight with a project. Leading off each workbook section are the learning objectives covered by the subject matter within the topic section. Following this is an outline of the process using the menu system, and finally an example is presented of the results of the process. MineSight provides a large number of programs with wide ranges of options within each program. This may seem overwhelming at times, but once you feel comfortable with the system, the large number of programs becomes an asset because of the flexibility it affords. If you are unable to achieve these key tasks or understand the concepts, notify your instructor before moving on to the next section in the workbook.

What You Need to Know This section explains for the student the mouse actions, keyboard functions, and terms and conventions used in the Mintec workbooks. Please review this section carefully to benefit fully from the training material and this training course.

Using the Mouse The following terms are used to describe actions you perform with the mouse: Click -press and release the left mouse button Double-click - click the left mouse button twice in rapid succession Right-click - press and release the right mouse button Drag - move the mouse while holding down the left mouse button Highlight - drag the mouse pointer across data, causing the image to reverse in color Point - position the mouse pointer on the indicated item

Terms and Conventions

Part #: E005 Rev. B

Page Intro-1

Using this Mintec Workbook

Notes:

Proprietary Information of Mintec, inc. The following terms and conventions are used in the Mintec workbooks: Actions or keyboard input instructions - are printed in Times New Roman font, italics, embedded within “arrow brackets” and keys are separated with a + when used in combination, for example, to apply bold face to type is indicated by “”. Button/Icon - are printed in bold with the initial letter capitalized, in Times New Roman font, for example Print, on a button, indicates an item you click on to produce a hard copy of a file; or

, the Query icon, is clicked on to determine which

polyline you need to edit. Menu Commands - are printed in Arial font, bold, with a vertical bar, as an example “File I Open” means access the File menu and choose Open. Parameters - are printed in Arial font, lower case, in bullet format, as an example,

•

the project coordinate units (metric or imperial)

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the project type (3-D, GSM, BHS, SRV).

Select - highlight a menu list item, move the mouse over the menu item and click the mouse.

Questions or Comments? Note: if you have any questions or comments regarding this training documentation, please contact the Mintec Documentation Specialist at (520) 795-3891 or via e-mail at [email protected].

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Part #: E005 Rev. B

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MineSight Overview

MineSight Overview

Notes:

Learning Objectives When you have completed this section, you will know: A. The basic structure and organization of MineSight. B. The capabilities of each MineSight module. C. Ways to run MineSight programs. What Is MineSight? MineSight is a comprehensive software package for the mining industry containing tools used for resource evaluation and analysis, mine modeling, mine planning and design, and reserves estimation and reporting. MineSight has been designed to take raw data from a standard source (drillholes, underground samples, blastholes, etc.) and extend the information to the point where a production schedule is derived. The data and operations on the data can be broken down into the following logical groups. Digitized Data Operations Digitized data is utilized in the evaluation of a project in many ways. It can be used to define geologic information in section or plan, to define topography contours, to define structural information, mine designs and other information that is important to evaluate the ore body. Digitized data is used or derived in virtually every phase of a project from drillhole data through production scheduling. Any digitized data can be triangulated and viewed as a 3-D surface in MineSight. Drillhole Data Operations A variety of drillhole data can be stored in MineSight, including assays, lithology and geology codes, quality parameters for coal, collar information (coordinates and hole orientation), and down-the-hole survey data. Value and consistency checks can be performed on the data before it is loaded into MineSight. After the data has been stored in the system, it can be listed, updated, geostatistically and statistically analyzed, plotted in plan or section and viewed in 3-D. Assay data can then be passed on to the next logical section of MineSight which is compositing. Compositing Operations Composites are calculated by benches (for most base metal mines) or mineral seams (for coal mines) to show the commodity of interest on a mining basis. Composites can be either generated in MineSight or generated outside the system and imported. Composite data can be listed, updated, geostatistically and statistically analyzed, plotted in plan or section and viewed in 3-D. Composite data is passed on to the next phase of MineSight, ore body modeling. Modeling Operations Within MineSight, deposits can be represented by a computer model of one of two types. A 3-D block model (3DBM) is generally used to model base metal deposits, such as porphyry copper or other non-layered deposits. A gridded seam model (GSM) is used for layered deposits, such as coal or oil sands. In both models, the horizontal components of a deposit are divided into blocks that are usually related to a production unit. In a 3DBM, the deposit is also divided horizontally into benches, whereas in a Part#: E005 Rev B

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MineSight Overview

Notes:

Proprietary Information of Mintec, inc. GSM the vertical dimensions are a function of the seam and interburden thicknesses. For each block in the model, a variety of items may be stored. Typically, a block in a 3DBM will contain grade items, geological codes, and a topography percent. Many other items may also be present. For a GSM, the seam top elevation and seam thickness are required. Other items, such as quality parameters, seam bottom, partings, etc. can also be stored. A variety of methods can be used to enter data into the model. Geologic and topographic data can be digitized and converted into codes for the model, or they can be entered directly as block codes. Solids can also be created in the MineSight 3-D graphical interface for use in coding the model directly. Grade data is usually entered through interpolation techniques, such as Kriging or inverse distance weighting. Once the model is constructed, it can be updated, summarized statistically, plotted in plan or section, contoured in plan or section, and viewed in 3-D. The model is a necessary prerequisite in any pit design or pit evaluation process. Economic Pit Limits & Pit Optimization This set of routines works on whole blocks from the 3-D block model, and uses either the floating cone or Lerchs-Grossmann technique to find economic pit limits for different sets of economic assumptions. Usually one grade or equivalent grade item is used as the economic material. The user enters costs, net value of the product, cutoff grades, and pit wall slope. Original topography is used as the starting surface for the design, and new surfaces are generated which reflect the economic designs. The designs can be plotted in plan or section, viewed in 3-D, and reserves can be calculated for the grade item that was used for the design. Simple production scheduling can also be run on these reserves. Pit Design The Pit Design routines are used to geometrically design pits that include ramps, pushbacks, and variable wall slopes to more accurately portray a realistic open pit geometry. Manually designed pits can also be entered into the system and evaluated. Pit designs can be displayed in plan or section, can be clipped against topography if desired, and can be viewed in 3-D. Reserves for these pit designs are evaluated on a partial block basis, and are used in the calculation of production schedules. Production Scheduling This group of programs is used to compute schedules for long-range planning based upon pushback designs (or phases), and reserves computed by the mine planning programs. The basic input parameters for each production period include mill capacity, mine capacity, and cutoff grades. Functions provided by the scheduling programs include:

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Calculation and reporting of production for each period, including mill production by ore type, mill head grades and waste

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Preparation of end-of-production period maps

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Calculation and storage of yearly mining schedules for economic analysis

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Evaluation of alternate production rates and required mining capacity

Part#: E005 Rev. B

Proprietary Information of Mintec, inc. Ways to Run MineSight Programs MineSight consists of a large group of procedures and programs designed to handle the tasks of mineral deposit evaluation and mine planning. Each procedure allows you to have a great amount of control over your data and the modeling process. You decide on the values for all the options available in each procedure. When you enter these values into a procedure to create a run file, you have a record of exactly how each program was run. You can easily modify your choices to rerun the program. To allow for easier use, the MineSight Compass menu system has been developed. Just select the procedure you need from the menu. Input screens will guide you through the entire operation. The menu system builds run files behind the scenes and runs the programs for you. If you need more flexibility in certain parts of the operations, the menus can be modified according to your needs, or you can use the run files directly. The MineSight 3-D graphical interface provides a Windows-style environment with a large number of easy-to-use, intuitive functions for CAD design, data presentation, area and volume calculations and modeling.

MineSight Overview

Notes:

Basic Flow of MineSight The following diagram shows the flow of tasks for a standard mine evaluation project. These tasks load the drillhole assays, calculate composites, develop a mine model, design a pit, and prepare long-range schedules for financial analysis. There are many other MineSight programs which can be used for geology, statistics, geostatistics, displays, and reserves.

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MINESIGHT OVERVIEW

Flow of Tasks for a Standard Mine Evaluation Project

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PCF

Drillhole Assays

Composites

Digitized Data

Mine Model

Digitize Load Edit List Dump Plot 3-D Viewing

Pit Designs

Planning & Scheduling

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Initialize Update List

Enter Scan Load Edit List Dump Rotate Add Geology Statistics Variograms Plot Collars Plot Sections Special Calculations 3-D Viewing and Interpretation Load Edit List Dump Add Geology Add Topography Statistics Variograms Variogram Validation Plot Sections Plot Plans Special Calculations Sort 3-D Viewing and Interpretation Initialize Interpolate Add Geology Add topography List Edit Statistics Reserves Special Calculations Plot Sections Plot Plans Contour Plots Sort 3-D Viewing & Solids Construction Creat Pit Optimization Model Run Pit Optimization Pit Optimization Reserves Pit Optimization Plots Run Pit Design Pit Design Reserves Pit Design Plots Reserves 3-D Views

Long Range Short Range

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MineSight Capacities Drillholes • No limit to the number of drillholes; only limited by the total number of assays in the system • 99 survey intervals per drillhole • 524,285 assay intervals per file • 8,189 assay intervals per drillhole • 99 items per interval • Multiple drillhole files allowed (usually one is all that is required)

MineSight Overview

Notes:

Composites • 524,285 assay intervals per file • 8,189 composites per drillhole • 99 items per composite interval • Multiple composite files allowed (usually one is all that is required) Geologic Model • 3-D block model limit of 1000 columns, 1000 rows and 400 benches • Gridded seam model limit of 1000 columns, 1000 rows and 200 seams • 99 items per block • Multiple model files allowed (usually one is all that is required) Digitized Point Data • 4,000 planes per file - either plan or section • 20,000 features (digitized line segments) per plane • 100,000 points per plane • 99 features with the same code per plane and a unique sequence number • Multiple files allowed • Pit Optimization (Floating cone/Lerchs-Grossman programs) • 600 row by 600 column equivalent (rows * columns < 360000) • Multiple files are allowed Reserves • 20 material classes • 20 cutoff grades for each material class • 10 metal grades • Multiple reserves files allowed Slice Files for Interactive Planning and Scheduling • 2,000,000 blocks containing one item (the number of blocks allowed drops as the number of items per block rises) • Unlimited benches and sections • 30 items per block

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MineSight Overview

Notes:

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Proprietary Information of Mintec, inc. Blastholes • 524,285 blastholes per file with standard File 12 • 8,189 blastholes per shot with standard File 12 • 4,194,301 blastholes per file with expanded limit File 12 • 1,021 blastholes per shot with expanded limit File 12 • 99 items per blasthole • Multiple blasthole files allowed (usually one is all that is required)

Part#: E005 Rev. B

Introduction to Geostatistics Objective: To make you familiar with the basic concepts of statistics, and the geostatistical tools available to solve problems in geology and mining of an ore deposit

Geostatistics • Sample values are realizations of random functions • Samples are considered spatially correlated • Value of a sample is a function of its position in the mineralization of the deposit • Relative position of the samples is taken under consideration.

Basic Statistics Definitions • • • • • • •

Statistics Geostatistics Universe Sampling Unit Support Population Random Variable

Classical Statistics • Sample values are realizations of a random variable • Samples are considered independent • Relative positions of the samples are ignored • Does not make use of the spatial correlation of samples

Topics • Basic Statistics • Data Analysis and Display • Analysis of Spatial Continuity (variogram)

Statistics • The body of principles and methods for dealing with numerical data • Encompasses all operations from collection and analysis of the data to the interpretation of the results

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Geostatistics Throughout this workbook, geostatistics will refer only to the statistical methods and tools used in ore reserve analysis

Sampling Unit Part of the universe on which a measurement is made (can be a core sample, channel sample, a grab sample etc.; one must specify the sampling unit when making statements about a universe)

Population • Like universe, population refers to the total category under consideration • It is possible to have different populations within the same universe (for example, population of drillhole grades versus population of blasthole grades; sampling unit and support must be specified)

Universe The source of all possible data (for example, an ore deposit can be defined as the universe; sometimes a universe may not have well defined boundaries)

Support • Characteristics of the sampling unit • Refers to the size, shape and orientation of the sample (for example, drillhole core samples will not have the same support as blasthole samples)

Random Variable A variable whose values are randomly generated according to a probabilistic mechanism (for example, the outcome of a coin toss, or the grade of a core sample in a diamond drill hole)

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F requency Distribution

F requency Distribution

Probability Density Function (pdf) • Discrete: 1. f(xi) ≥ 0 for xi∈R (R is the domain) 2. Σf(xi) = 1 • Continuous: 1.f(x) ≥ 0 2.∫f(x)dx = 1

Cumulative Density Function (cdf) Proportion of the population below a certain value: F(x) = P(X≤x) 1. 0≤F(x) ≤ 1 for all x 2. F(x) is non decreasing 3. F(-∞)=0 and F(∞)=1

Example

PDF

Assume the following population of measurements: 1, 7, 1, 3, 2, 3, 11, 1, 7, 5

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

CDF

2

3

4

5

6

7

8

9

10

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Descriptive Measures

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

1

2

3

4

5

6

7

8

9

10

11

• • • • • •

Measures of location: Mean Median Mode Min, Max Quartiles Percentiles

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Mean m = 1/n Σxi i=1,...,n Arithmetic average of the data values

Mean m= (1+ 7+ 1+ 3+ 2+ 3+ 11+ 1+ 7+ 5)/10= = 41/10= = 4.1

Mean m= (1+ 7+ 1+ 3+ 2+ 3+ 1+ 7+ 5)/9= = 30/9= = 3.33

Mean What is the mean of the example population: 1, 7, 1, 3, 2, 3, 11, 1, 7, 5 m =?

Mean What is the mean if we remove highest value?

Median M = x(n+1)/2 if n is odd M = [x n/2+x(n/2)+1]/2 if n is even Midpoint of the data values if they are sorted in increasing order

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Median What is the median of example population?

Median Sort data in increasing order: 1, 1, 1, 2, 3, 3, 5, 7, 7 ,11

M=? M=3

Other • • • • • • •

Mode Minimum Maximum Quartiles Deciles Percentiles Quantiles

Mode

Mode The value that occurs most frequently In our example: Mode=?

Quartiles

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11

Split data in quarters

Mode = 1

Q1 = 1st quartile Q3 = 3rd quartile In example: Q1=? Q3=?

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Quartiles

Deciles, Percentiles,Quantiles

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11 Q1= 1 Q3= 6

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11 D1= 1 D3= 1 D9= 7

Mode on the PDF

Mean on the PDF

Mode (also min) Mean(=4.1) Max

Median on the CDF

Descriptive Measures Measures of spread: • Variance • Standard Deviation • Interquartile Range

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Variance S2 = 1/(n-1) Σ(xi-m)2

Variance i=1,...,n

• Sensitive to outlier high values • Never negative

Variance Remove high value: 1, 1, 1, 2, 3, 3, 5, 7, 7 M=3.33 S2= 1/8 {(1-3.33)2+ (1-3.33)2+ (1-3.33)2+ (2-3.33)2+ (3-3.33)2+ (3-3.33)2+ (5-3.33)2+ (7-3.33)2+ (7-3.33)2 = = 1/8 (5.43+ 5.43+ 5.43+1.769+ 0.109+ 0.109+ 2.789+ 13.469+ 13.469) = = 48/8 = =6

Standard Deviation Example: S2= 11.21 S = 3.348

Example: 1, 1, 1, 2, 3, 3, 5, 7, 7 ,11 M=4.1 S2= 1/9 {(1-4.1)2+ (1-4.1)2+ (1-4.1)2+ (2-4.1)2+ (3-4.1)2+ (3-4.1)2+ (5-4.1)2+ (7-4.1)2+ (7-4.1)2+ (11-4.1)2 } = = 1/9 (9.61+ 9.61+ 9.61+ 4.41+ 1.21+ 1.21+ 0.81+ 8.41+ 8.41+ 47.61) = = 100.9/9 = = 11.21

Standard Deviation s = √s2 • Has the same units as the variable • Never negative

Interquartile Range IQR = Q3 - Q1 Not used in mining very often

S2 = 6 S =2.445

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Descriptive Measures Measures of shape: • Skewness • Peakedness (kurtosis) • Coefficient of Variation

Skewness

Skewness Skewness = [1/n Σ(xi-m)3] / s3 • Third moment about the mean divided by the cube of the std. dev. • Positive - tail to the right • Negative - tail to the left

Skewness

Example:

Remove high value:

1, 1, 1, 2, 3, 3, 5, 7, 7 ,11 M=4.1 Sk= 1/10 {(1-4.1)3+ (1-4.1)3+ (1-4.1)3+ (2-4.1)3+ (3-4.1)3+ (3-4.1)3+ (5-4.1)3+ (7-4.1)3+ (7-4.1)3+ (11-4.1)3 } = = 1/10 (-29.79-29.79-29.79-8.82-1.33 1.33+ 0.73+ 24.39+ 24.39+328.51) = = 277.2/10 = = 27.72

1, 1, 1, 2, 3, 3, 5, 7, 7 M=3.3 Sk= 1/10 {(1-3.3)3+ (1-3.3)3+ (1-3.3)3+ (2-3.3)3+ (3-3.3)3+ (3-3.3)3+ (5-3.3)3+ (7-3.3)3+ (7-3.3)3 } = = 1/10 (-12.17- 12.17- 12.17- 2.2- 0.03- 0.03+ 4.91+ 50.65+ 50.65) = = 67.44/9 = = 7.49

Positive Skewness

Peakedness • • • •

Peakedness = [1/n Σ(xi-m)4] / s4 Fourth moment about the mean divided by the fourth power of the std. dev. Describes the degree to which the curve tends to be pointed or peaked Higher values when the curve is peaked Usefulness is limited

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Coefficient of Variation CV = s/m • No units • Can be used to compare relative dispersion of values among different distributions • CV > 1 indicates high variability

Normal Distribution

Coefficient of Variation In our example: CV = 3.348/4.1 =0.817 Remove high value: CV = 2.445/3.33=0.743

Normal Distribution curve

f(x) = 1 / (s √2π) exp [-1/2 ((x-m)/s)2] • symmetric, bell-shaped • 68% of the values are within one std. dev. • 95% of the values are within two std. dev.

Std. normal distribution • mean = 0 and s = 1 • standardize any variable using: z = (x-m) / s

Normal Distribution Tables • The cumulative distribution function F(x) is not easily computed for the normal distribution. • Extensive tables have been prepared to simplify calculation • Most statistics books include tables for the std. normal distribution

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Example of cdf (normal) Find the proportion of sample values above 0.5 cutoff in a normal population that has m =0.3, and s = 0.2 Solution: • First, transform the cutoff, x0 , to unit normal. z = (x0 - m) / s = (0.5 - 0.3) / 0.2 = 1 • Next, find the value of F(z) for z = 1. The value of F(1) = 0.8413 from Table • Calculate the proportion of sample values above 0.5 cutoff, P(x > 0.5), as follows: P(x > 0.5) = 1 - P(x

≤ 0.5) = 1 - F(1) = 1 -0.8413 = 0.16

Lognormal Distribution Logarithm of a random variable has a normal distribution f(x) = 1 / (x β√2 π) e -u for x > 0, β> 0 where u= (ln x -α ) 2 / 2β 2 α= mean of logarithms β= variance of logarithms

• Therefore, 16% of the samples in the population are > 0.5

Conversion Formulas

Lognormal Distribution Curve

Conversion formulas between the normal and lognormal distributions: Lognormal to normal: • µ = exp (α+β 2 /2) • σ2 = µ2 [exp(β 2) - 1] Normal to lognormal: • α = logµ - β 2 /2 • β 2 = log [1 + (σ2 /µ 2)]

Three-Parameter LN Distribution Logarithm of a random variable plus a constant, ln (x+c) is normally distributed Constant c can be estimated by: c = (M2 - q1 q2 ) / (q1 + q2 + 2M)

Bivariate Distribution • Joint distribution of outcomes from two random variables X and Y: F(x,y) = Prob {X≤x, and Y≤y} • In practice, it is estimated by the proportion of pairs of data values jointly below the respective threshold values x, y.

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Statistical analysis • To organize, understand, and/or describe data • To check for errors • To condense information • To uniformly exchange information

Data Analysis and Display Tools • Frequency Distributions

• • • • •

Histograms Cumulative Frequency Tables Probability plots Scatter Plots Q-Q plots

Histograms

E rror Checking • • • •

Avoid zero for defining missing values Check for typographical errors Sort data; examine extreme values Plot sections and plan maps for coordinate errors • Locate extreme values on map; Isolated? Trend?

Data Analysis and Display Tools • • • • • • • •

Correlation Correlation Coefficient Linear Regression Data Location Maps Contour Maps Symbol Maps Moving Window Statistics Proportional Effect

Histogram in text file #

• Visual picture of data and how they are distributed • Bimodal distributions show up easily • Outlier high grades • Variability

CUM. UPPER

FREQ. FREQ LIMIT 0 20 40 60 80 100 ------------+......... +......... +. ........ +. ........ + ......... + 86 .093 .100 +*****. + 34 .130 .200 +** . + 48 .182 .300 +*** . + 73 .261 .400 +**** . + 86 .354 .500 +***** . + 80 .440 .600 +**** . + 84 .531 .700 +***** . + 74 .611 .800 +**** . + 70 .686 .900 +**** . + 60 .751 1.000 +*** . + 43 .798 1.100 +** . + 28 .828 1.200 +** . + 29 .859 1.300 +** . + 31 .893 1.400 +** .+ 25 .920 1.500 +* .+ 19 .941 1.600 +* . 16 .958 1.700 +* . 8 .966 1.800 + . 9 .976 1.900 + . 3 .979 2.000 + . 6 .986 2.100 + . 4 .990 2.200 + . 1 .991 2.300 + . 3 .995 2.400 + . 3 .998 2.500 + . 1 .999 2.600 + . 0 .999 2.700 + . 0 .999 3.500 + . 0 .999 3.600 + . 0 .999 3.700 + . 1 1.000 3.800 + . -----------+ .........+ .........+ ......... + .........+ . ........ + 925 1.000 0 20

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Histogram Plot

Histograms with skewed data • Data values may not give a single informative histogram • One histogram may show the entire spread of data, but another one may be required to show details of small values.

Histograms with skewed data

Cumulative Frequency Tables CUTOFF CU

SAMPLES ABOVE

PERCENT ABOVE

.000 .200 .400 .600 .800 1.000 1.200 1.400 1.600 1.800 2.000 2.200 2.400 2.600 2.800 3.000 3.200 3.400 3.600

2399.00 1717.00 1240.00 840.00 522.00 310.00 205.00 133.00 72.00 35.00 21.00 11.00 6.00 2.00 2.00 2.00 2.00 2.00 2.00

100.00 71.57 51.69 35.01 21.76 12.92 8.55 5.54 3.00 1.46 .88 .46 .25 .08 .08 .08 .08 .08 .08

MEAN ABOVE .5129 .6858 .8365 1.0025 1.1917 1.4012 1.5682 1.7165 1.9206 2.1697 2.3614 2.6118 2.8667 3.6550 3.6550 3.6550 3.6550 3.6550 3.6550

C.V.

.8782 .6133 .4809 .3889 .3229 .2663 .2266 .2106 .2002 .1966 .1947 .2006 .2134 .0174 .0174 .0174 .0174 .0174 .0174

Min. data value = .0000 Max. data value = 3.7000 Std. Deviation = .450 C.V. = Coeff. of variation = Standard deviation / mean 2399 Intervals used out of 2412

Probability Plots

Probability Plot

• Shows if distribution is normal or lognormal • Presence of multiple populations • Proportion of outlier high grades

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Scatter Plots

Scatter Plot

• Simply an x-y graph of the data • It shows how well two variables are related • Unusual data pairs show up • For skewed distributions, two scatter plots may be required to show both details near origin and overall relationship.

L inear Regression

L inear Regression

• y = ax + b a = slope, b = constant of the line

Different ranges of data may be described adequately by different regressions Culocal mean sample If p tends to ∞ => nearest neighbor method (polygonal) Traditionally, p = 2

Ordinary kriging Definition: Ordinary kriging is an estimator designed primarily for the estimation of block grades as a linear combination of available data in or near the block, such that estimate is unbiased and has minimum variance.

Kriging Estimator • z* = Σwi z(xi ) i = 1,...,n where z* is the estimate of the grade of a block or a point, z(xi) refers to sample grade, wi is the corresponding weight assigned to z(xi), and n is the number of samples.

Inverse Distance Advantages: • Easy to understand • Easy to implement • Flexible in adapting weights to different estimation problems • Can be customized Disadvantages: • Susceptible to data clustering • p? • No anisotropy • No error control

Ordinary kriging B.L.U.E. for best linear unbiased estimator. Linear because its estimates are weighted linear combinations of available data Unbiased since the sum of the weights adds up to 1 Best because it aims at minimizing the variance of errors.

Kriging Estimator Desirable Properties: • Minimize σ2 = F (w1, w2, w3,…,wn) • r = average error = 0 (unbiased) • Σ wi = 1

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E rror variance Using R. F. model, you can express the error variance as a function of R.F. parameters: σ2R= σ2z + Σ Σ(λi λj Ci,j ) - 2 Σ λi C i,o where σ2z is the sample variance Ci,j is the covariance between samples Ci,o is the covariance between samples and location of estimation. See Isaaks and Srivastava pg 281-284

Ordinary Kriging

E rror variance σ2R= σ2z + Σ Σ(λi λj Ci,j ) - 2 Σ λi C i,o • Error increases as variance of data increases • Error variance increases as data become more redundant • Error variance decreases as data are closer to the location of estimation

Kriging System (point) Previous equation in matrix form:

• Minimize error σ2R= σ2z + Σ Σ(λi λj Ci,j ) - 2 Σ λi C i,o • Σ λi = 1 • Use Lagrange method (Isaaks and Srivastava, pg 284-285). Result: Ci,o = Σ(λi Ci,j) + µ Σ λi = 1

Point Kriging (cont.)

Kriging System (block)

• Matrix C consists of the covariance values Cij between the random variables Vi and Vj at the sample locations. • Vector D consists of the covariance values Ci0 between the random variables Vi at the sample locations and the random variable V0 at the location where an estimate is needed. • Vector λ consists of the kriging weights and the Lagrange multiplier.

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Block Kriging (cont.)

Kriging Variance

• In point kriging, the covariance matrix D consists of point-to-point covariances. In block kriging, it consists of block-to-point covariances. • Covariance values CiA no longer a point-to-point covariance like Ci0 , but the average covariance between a particular sample and all of the points within A: CiA = 1/A Σ Cij In practice, the A is discretized using a number of points in x, y and z directions to approximate CiA .

Block Discretization To be considered: • Range of influence of the variogram used in kriging. • Size of the blocks with respect to this range. • Horizontal and vertical anisotropy ratios.

Disadvantages of kriging • • • •

computer required prior variography required more time consuming smoothing effect

σ2ok = CAA - [Σ(λi CiA) + µ] Data independent

Advantages of kriging • Takes into account spatial continuity characteristics • Built-in declustering capability • Exact estimator • Calculates the kriging variance for each block • Robust

Assumptions • No drift is present in the data (Stationarity hypothesis) • Both variance and covariance exist and are finite. • The mean grade of the deposit is unknown.

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E ffect of scale

E ffect of shape

Nugget Effect

E ffect of range

E ffect of Anisotropy

Search Strategy • Define a search neighborhood within which a specified number of samples is used

• If anisotropy, use an ellipsoidal search • Orientation of this ellipse is important • If no anisotropy, search ellipse becomes a circle and the question of orientation is no longer relevant

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Search Strategy

Search strategy (cont.)

• Include at least a ring of drill holes with enough samples around the blocks to be estimated • Don’t extend the grades of the peripheral holes to the undrilled areas too far • Increasing vertical search distance has more impact on number of samples available for a given block, than increasing horizontal search distance (in vertically oriented drillholes) • Limit the number of samples used from each individual drillhole

Octant or Quadrant Search

Importance of kriging plan An easily overlooked assumption in every estimate is the fact the sample values used in the weighted linear combination are somehow relevant, and that they belong to the same group or population, as the point being estimated. Deciding which samples are relevant for the estimation of a particular point or a block may be more important than the choice of an estimation method.

Declustering

Declustering

Clustering in high grade area:

Clustering in mean grade area:

Naïve mean= (0+1+3+1+7+6+5+6+2+4+0+1)/ 12 = 3

Naïve mean= (7+1+3+1+0+6+5+1+2+4+0+6)/ 12 = 3

Declustered mean= [(0+1+3+1+2+4+0+1) + (7+6+5+6)/4] /9 = =2

Declustered mean= [(7+1+3+1+2+4+0+6) + (0+6+5+1)/4] /9 = =3

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Declustering

Declustering

Clustering in low grade area: Naïve mean= (7+1+6+1+0+3+4+1+2+5+0+6)/ 12 = 3 Declustered mean= [(7+1+6+1+2+5+0+6) + (0+3+4+1)/4] /9 = =3.33

Declustering • Cell declustering

• Data with no correlation, do no need declustering (pure nugget effect model) • If variogram model has a long range and low nugget, you may need to decluster.

Cell Declustering Each datum is weighted by the inverse of the number of data in the cell

• Polygonal

Polygonal

Declustered Global Mean • DGM = Σ(wi . vi ) / Σwi i=1,...,n where n is the number of samples, wi are the declustering weights assigned to each sample, and vi are the sample values. The denominator acts as a factor to standardize the weights so that they add up to 1.

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Cross Validation • To check how well the estimation procedure can be expected to perform. • Temporarily discard the sample value at a particular location and then estimate the value at that location using the remaining values.

Cross validation Check: • Histogram of errors • Scatter plots of actual versus estimate

Cross validation • It may suggest improvements • It compares, does not determine parameters • Reveals weaknesses/shortcomings

Cross validation Remember: • All conclusions are based on observations of errors at locations were we do not need estimates. • We remove values that, after all, we are going to use.

Quantifying Uncertainty One approach: • Assume that the distribution of errors is Normal • Assume that the ordinary kriging estimate provides the mean of the normal distribution • Build 95 percent confidence intervals by taking ±2 standard deviations either of the OK estimate

Quantifying Uncertainty Kriging Variance σ2ok = CAA - [Σ(λi CiA) + µ] Advantages Does not depend on data It can be calculated before sample data are available (from previous/know variography) Disadvantages Does not depend on data If proportional effect exists, previous assumptions are not true

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Quantifying Uncertainty

Same Kriging Variance!!!

Quantifying Uncertainty Combined Variance = sqrt (local variance * kriging variance)

Quantifying Uncertainty Other approach Incorporate the grade in the error variance calculation: Relative Variance = Kriging Variance /Square of Kriged Grade

Quantifying Uncertainty Relative Variability Index(RVI) = SQRT(Combined Variance) / Kriged Grade

where local variance of the weighted average (σ2w ) is: σ2w = Σw2i * (Z0- zi )2 i = 1, n (n>1) where n is the number of data used, wi are the weights corresponding to each datum, Z0 is the block estimate, and zi are the data values.

Change of Support

Change of Support N=4 M = 8.825

N = 16 M = 8.825

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Change of Support

Change of Support >10 N = 2 = 50% M = 11.15

Change of Support • The mean above 0.0 cutoff does not change with a change in support • The variance of block distribution decreases with larger support • The shape of the distribution tends to become symmetrical as the support increases • Recovered quantities depend on block size

Krige’s Relation

>10 N = 5 = 31% M =18.6

Affine Correction Assumptions: • The distribution of block or SMU grades has same shape as the distribution of point or composite samples. • The ratio of the variances, i.e., variance of block grades (or the SMU grades) over that of point grades is non-conditional to surrounding data used for estimation.

Krige’s Relation (cont’d)

σ2p = σ2b + σ2 p∈b

Total σ2 = between block σ2 + within block σ2

σ2p = Dispersion variance of composites in the deposit (sill) σ2b = Dispersion variance of blocks in the deposit σ2 p∈b = Dispersion variance of points in blocks

σ2p = calculated directly from the composite or blasthole data

This is the spatial complement to the partitioning of variances which simply says that the variance of point values is equal to the variance of block values plus the variance of points within blocks.

σ2 p∈b = calculated by integrating the variogram over the block b σ2b = calculated using the Krige’s relation: σ2b = σ2p - σ2 p∈b

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Krige’s Relation (cont’d)

Calculation of A.C.

How to calculate σ2 p∈b ?

K2 = σ2b / σ2p

Integrating the variogram over a block provides variance of points within the block

(from the variogram averaging):

σ2 p∈b = γblock = 1/n2 Σ Σ γ(hi,j)

≤ 1

K2 = [ γ(D,D) - γ(smu,smu) ] / γ(D,D) = 1 - [ γ(smu,smu) / γ(D,D) ] ≤ 1 Affine correction factor, K = √K2 ≤ 1

Affine Correction (cont.)

Affine correction of Variance

Use affine correction if: (σ2p -σ2b) /σ2 p ≤ 30%

Indirect Lognormal method Assumption: all distributions are lognormal; the shape of distribution changes with changes in variance. Transform: znew = azbold

Indirect Lognormal method Disadvantage: If the original distribution departs from log normality, the new mean may require rescaling: znew = (mold/mnew) zold

a = Function of (m, σnew ,σold ,CV) b = Function of (σnew,σold,CV), see the notes CV: coefficient of variation = σold / mold

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Change of Support (other) Hermite Polynomials: • Declustered composites are transformed into a Gaussian distribution • Volume-variance correction is done on the Gaussian distribution • Then this distribution is back transformed using inverse Hermite Polynomials

Change of Support (applications) Design a search strategy: • Decluster composites/variogram • Define SMU units • Apply change of support from composites to SMU • Calculate the SMU GT curves. • “Guess at a search scenario • Krige blocks => create GT curves • Compare GT curves of block estimates to GT curves of SMUs • Adjust search scenario etc.. GT: grade tonnage curves

C. of S. for Ore Grade/Tonnage Estimation

Change of Support (other) Conditional Simulation: • Simulate a realization of the composite (or blasthole) grades on a very closely spaced grid (for example, 1x1) • Average simulated grades to obtain simulated block grades

Change of Support (applications) Reconciliation between BH model and Exploration model: • Calculate GT curves of exploration model • Apply change of support from BH model to Exploration model • Calculate the adjusted BH model GT curves. • Compare GT curves of block estimates to GT of adjusted BH model estimates.

Equivalent Cutoff Calculation (zp - m) / σp = (zsmu - m) / σsmu

zp = the equivalent cutoff grade to be applied to the point (or composite) distribution m = mean of composite and SMU distribution σp = square root of composite dispersion variance zsmu = the cutoff grade applied to the SMU m = mean of composite and SMU distribution σsmu = square root of SMU dispersion variance

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Equivalent Cutoff Calculation

Numeric Example

zp = ( σp / σsmu ) zsmu + m [1 - ( σp / σsmu )]

Let the mean of composites = 0.0445, and the specified cutoff grade zsmu = 0.055

The ratio σp / σsmu is basically the inverse of the affine correction factor K.

If the ratio σp / σsmu = 1.23, what is the equivalent cutoff grade?

This ratio is ≥1.

zp=1.23 (0.055) + 0.0445 (1 - 1.23) =0.0574 Therefore, the equivalent cutoff grade to be applied to the composite distribution is 0.0574.

Equivalent Cutoff • if the specified cutoff grade is less than the

mean, the equivalent cutoff grade becomes less than the cutoff

Change of Support (applications) Other: • Almost required in MIK

• if the specified cutoff grade is greater than the mean, the equivalent cutoff grade becomes greater than the cutoff.

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Simple Kriging

Cokriging

Z*sk = Σλi [Z(xi ) - m] + m

i = 1,...,n

•Z*sk - estimate of the grade of a block or a point •Z(xi ) - refers to sample grade •λi - corresponding simple kriging weights assigned to Z(xi ) •n - number of samples •m = E{Z(x)} - location dependent expected value of Z(x).

Cokriging ....................................... [Cov{dibj}] [1] [0] [Cov{didi}] ....................................... [Cov{bjbj}] [0] [1] [Cov{dibj}] ....................................... [ 1 ] [ 0 ] 0 0 ....................................... [ 0 ] [ 1 ] 0 0 .......................................

Cokriging-steps for Drill and Blasthole data ..... ........... [8i] [Cov{x0di}] ..... ........... [*j] [Cov{x0bj}] x ..... = ........... 1 µd ..... ........... 0 µb ..... ...........

[Cov{didi}] = drillhole data (dhs) covariance matrix, i=1,n [Cov{bjbj}] = blasthole data (bhs) covariance matrix, j=1,m [Cov{dibj}] = cross-covariance matrix for dhs and bhs [Cov{x0di}] = drillhole data to block covariances [Cov{x0bj}] = blasthole data to block covariances [8i] = Weights for drillhole data [*j] = Weights for blasthole data

•Suitable when the primary variable has not been sampled sufficiently. •Precision of the estimation may be improved by considering the spatial correlations between the primary variable and a better-sampled variable. •Example: extensive data from blastholes as the secondary variable - Widely spaced exploration data as the primary variable.

• Regularize blasthole data into a specified block size. Block size could be the same as the size of the model blocks to be valued, or a discreet sub-division of such blocks. A new data base of average blasthole block values is thus established. • Variogram analysis of drillhole data. • Variogram analysis of blasthole data. • Cross-variogram analysis between drill and blasthole data. Pair each drillhole value with all blasthole values. • Selection of search and interpolation parameters. • Cokriging.

µd and µb = Lagrange multipliers

Universal Kriging

Outlier Restricted Kriging •Determine the outlier cutoff grade •Assign indicators to the composites based on the cutoff grade 0 if the grade is below the cutoff 1 otherwise •Use OK with indicator variogram, or simply use IDS , or any other method to assign the probability of a block to have grade above the outlier cutoff. •Modify Kriging matrix.

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ORK matrix

Nearest Neighbor Kriging Utilize nearest samples (assign more weight)

Non-Linear kriging methods • Indicator

kriging •Probability kriging •Lognormal kriging •Multi-Gaussian kriging •Lognormal short-cut •Disjunctive kriging Parametric (assumptions about distributions) or non-parametric (distribution-free)

Indicator Kriging Suppose that equal weighting of N given samples is used to estimate the probability that the grade of ore at a specified location is below a cutoff grade. The proportion of N samples that are below this cutoff grade can be taken as the probability that grade estimated is below this cutoff grade. Indicator kriging obtains a cumulative probability distribution at a given location in a similar manner, except that it assigns different weights to surrounding samples using the ordinary Kriging technique to minimize the estimation variance.

Why Non-Linear • To overcome problems encountered with outliers • To provide “better” estimates than those provided by linear methods • To take advantage of the properties on nonnormal distributions of data and thereby provide more optimal estimates • To provide answers to non-linear problems • To provide estimates of distributions on a scale different from that of the data (the “change of support” problem)

Indicator Kriging The basis of indicator kriging is the indicator function: At each point x in the deposit, consider the following indicator function of zc defined as: 1, if z(x) < zc i(x;zc ) = 0, otherwise where: x is location, zc is a specified cutoff value, z(x) is the value at location x.

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Indicator Kriging Examples: Separate continuous variables into categories: I(x) = 1 if k(x) ≤ 30, 0 if k(x) >30 Characterize categorical variables and differentiate types: I(x) = 1 for heterozygote, 0 for homozygote

Indicator Kriging (applications) Some data may represent a spatial mixture of two or more statistical populations (for example, clay and sand. • Separate populations: I(x) = 1 for clay, 0 for sand. • Then calculate the probability of an unsampled location to be clay or sand. • Krige (local estimates) unsampled locations using only data belonging to that population • Final estimate can be a weighted (by probabilities) average of the local estimates.

Multiple Indicator Kriging Same as indicator kriging but instead of one cutoff, we use a series of cutoffs.

Indicator Kriging (applications) Some drill holes have encountered a particular horizon, some were not drilled deep enough, some penetrated the horizon but the core or the log is missing: Use I(x) = 1 for drill hole assays above the horizon and I(x) = 0 for assays below the horizon. Use indicator kriging and calculate the probability of the missing assays to be 1 or 0.

Indicator Kriging (applications) Extreme values: Separate population to 1 and 0 based on outlier cutoff. Proceed then as though you are dealing with two spatially mixed populations.

Multiple Indicator Kriging THE INDICATOR FUNCTION: At each point x in the deposit, consider the following indicator function of zc defined as: 1, if z(x) < zc i(x;zc ) = 0, otherwise where: x is location, zc is a specified cutoff value, z(x) is the value at location x.

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Indicator Function at point x

T he ϕ(A;zc) function ϕ(A;zc ) = 1/A∫A i(x;zc ) dx ∈ [0,1]

Proportion of Values z(x)≤ zc within area A

Local Recovery Functions Tonnage point recovery factor in A: t*(A;zc) = 1 - ϕ(A;zc) Quantity of metal recovery factor in A: q*(A;zc) = ∫zc u d ϕ(A;u) A discrete approximation of this integral is given by q*(A;zc) = Σ1/2 (zj + zj-1) [ϕ*(A;zj) -ϕ*(A;zj-1) ] j=2,...,n

Local Recovery Functions

E stimation of φ(A;zc)

This approximation sums the product of median cutoff grade and median (A;zc) proportion for each cutoff grade increment. The mean ore grade at cutoff zc gives the mean block grade above the specified cutoff value.

ϕ(A;zc) proportion of grades z(x) below cutoff zc within panel A. (unknown since i(x;zc) known at only a finite number of points). ϕ(A;zc) = 1/n Σ i(xj ;zc) j=1,...,n or ϕ(A;zc) =Σ λj i(xj ;zc) xj ∈ D j=1,...,N where n is the number of samples in the panel A, N is the number of samples in search volume D, j are the weights assigned to the samples, Σλj = 1, and usually N >> n.

Mean ore grade at cutoff zc : m*(A;zc) = q*(A;zc) / t*(A;zc)

Ordinary kriging is used to estimate ϕ(A;zc) from the indicator data i(xj ;zc). We use a random function model for i(xj ;zc), which will be designated by I(xj ;zc).

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Indicator Variography γI(h;zc ) = 1/2 E [ I(x+h);zc ) - I(x;zc ) ]2

Median Indicator Variogram Indicator variogram where cutoff corresponds to median of data γm(h;zm ) = 1/2n Σ [ I(xj+m+h);zm ) - I(xj;zm ) ]2 j=1,…,n

Order Relations

Advantages of MIK • It estimates the local recoverable reserves within each panel or block. • It provides an unbiased estimate of the recovered tonnage at any cutoff of interest. • It is non-parametric, i.e., no assumption is required concerning distribution of grades. • It can handle highly variable data. • It takes into account influence of neighboring data and continuity of mineralization.

Disadvantages of MIK

Change of Support

• It may be necessary to compute and fit a variogram for each cutoff. • Estimators for various cutoff values may not show the expected order relations. • Mine planning and pit design using MIK results can be more complicated than conventional methods. • Correlation between indicator functions of various cutoff values are not utilized. More information becomes available through the indicator cross variograms and subsequent cokriging. These form the basis of the Probability Kriging technique.

Function ϕ*(A;zc) and grade-tonnage relationship for each block is based on distribution point samples (composites). Selective mining unit (SMU) volume is much larger than sample volume, therefore, one must perform a volume-variance correction to the initial grade-tonnage curve of each block.

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Affine Correction Equation for affine correction of any panel or block is given by ϕ*v (A;z) = ϕ* (A;zadj) where zadj=adjusted cutoff grade = K(z - ma)+ma Use affine correction if:

Grade Zoning • Grade zoning is usually applied to control the extrapolation of grades into statistically different populations • Often grade zones or mineralization envelopes correspond to different geologic units

(σ2p -σ2b) /σ2 p ≤ 30%

Grade Zoning (cont’d) Determine how the grade populations are separated spatially • Is there a reasonably sharp discontinuity between the grades of the different populations? • Or is there a larger transition zone between the grades of the different populations?

Grade Zoning (cont’d) Transition zone between grade populations:

Grade Zoning (cont’d) Discontinuity between grade populations:

Grade Zoning (cont’d) • Discontinuity between the grade populations is best modeled using a deterministic model, i.e., digitized the outlines • Transition zone between the grade populations is best modeled using a probabilistic model, i.e., indicator kriging

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Grade Zoning (cont’d) Characterizing the contact between different spatial populations: • Calculate the difference between the average grades within each population as a function of distance from the contact: Dzi = zi - z(-i)

Grade Zoning (cont’d) • If the average difference in grade Dzi vs distance from the contact is small for small distances but increases with increasing distance, then there is likely a transition zone between the different populations:

Grade Zoning (cont’d) • If the average difference in grade Dzi vs distance from the contact is more or less constant, then there is probably a discontinuity between the different populations :

Grade Zone Bias Check • Often mineralization envelopes lead to biased ore reserve models. To check: • Interpolate using the nearest neighbor (polygonal) method) • Use the search parameters corresponding to the model of spatial continuity • Disregard all grade zoning • Compare at 0.0 cutoff grade, the tons and grade of the polygonal model to those of the mineralization envelope model.

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Geostatistics Overheads

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Classical Statistics

Classical Statistics

Notes:

Prior to this section you loaded the drillhole data to MineSight. If you are calculating statistics on the composites, you calculated the composites. In this section you can compute classical statistics on the assays and composites. This is not required for later work.

Learning Outcome In this section you will learn:

•

How to calculate general statistics

•

How to produce a histogram

•

How to produce probability plots

•

How to produce scatter plots

•

How to calculate insitu statistics

•

How to calculate proportional effect

Classical Statistics The most commom statistical operations available within MineSight are:

•

Mean and standard deviation

•

Histograms

•

Cumulative frequency plots

•

Correlations

•

Cumulative probability plots

Use classical statistics to:

•

Analyze data to determine descriptive parameters

•

Make inferences about an entire population based on samples

Some difficulties involved with the application of classical statistics to mineral projects are:

•

Mineral deposit data is generally not independent. It is for this reason that geostatistics was developed.

•

Different geologic zones may have different statistical populations. Mixing zones may produce incorrect statistics.

•

Different types of samples have different volumes and should be kept separate for analysis, e.g., drillhole assays and bulk samples.

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•

Although samples may be equal in size, they may not have an equal volume of influence. Drilling tends to be closer spaced in higher grade areas so the statistics may be indicating a higher proportion of ore than actually exists.

General Statistics On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - Assay Data Statistical Analysis This panel provides input for the items to be analyzed, an optional weighting item and associated multiplier, and an optional selection item for limiting the data to be analyzed. . Panel 2 - Assay Data Statistical Analysis This panel provides input for the basic statistical analysis parameters. A frequency interval of .1 will be used and all values below 0 will be ignored. We’ll report 40 frequency intervals in this example, and use ‘cu’ as the run and report file extensions. Panel 3 - Optional Data Selection for Assay Statistics This panel allows you to define portions of the data to include or exclude from the analysis based on item values. There are also titling options for the resulting histogram plot; enter a title such as “Assay statistics weighted by length”. Panel 4 - 3D Coordinate Limits for Data Selection This panel provides the option of limiting the area of data selection, either through project coordinates or through the use of a boundary file. Panel 5 - Histogram Plot Attributes This panel provides options for setting up your histogram display and plot. Results and Histogram Plot The results of the statistical analysis are presented in the report file (rpt401.cu) in tabular form and as a symbolic histogram.

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Classical Statistics

Notes:

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Classical Statistics

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Proprietary Information of Mintec, inc. There is also information provided regarding the number of intervals excluded based on the selection criteria input through the procedure panels:

Finally, a plottable histogram is also generated; this can be plotted using the M122MF program, which is controlled with the MPLOT panel. The MPLOT panel can also be used to create deferred plot files from the histogram.

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Proprietary Information of Mintec, inc. Exercise 1 Use procedure p40201.dat to generate composite statistics. Use the same parameters as in the assays. Use item length as the weighting item. Compare histograms and statistics.

Classical Statistics

Notes:

Exercise 2 Generate composite statistics for those composites that have ALTR = 1 and 2 only. Hint: this can be done with a change to Panel 3. Exercise 3 Repeat the exercise for those composites that have ROCK = 1 and 2 only. Exercise 4 1.

Generate lognormal composite data statistics for those composites that have ROCK =

Statistics within Geology Types From the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - 3-D Composite Data Statistical Analysis This panel provides input for the file type, items for analysis, and optional weighting and selection items. Using File 9, enter ROCK as the first item. This item will be used to determine the cutoffs. Enter CU as the second item, which will be used for statistical analysis. Keep item length as a weighting item. Panel 2 - 3-D Composite Data Statistical Analysis This panel provides input for the basic statistical analysis parameters. A frequency interval of 1 will be used because the CU statistics will be reported at cutoffs of ROCK item. Also . Panel 3 - Optional Data Selection for Assay Statistics This panel allows you to define portions of the data to include or exclude from the analysis based on item values. There are also titling options for the resulting histogram plot; enter a title such as “15-m bench composites by rock type”. Panel 4 - 3D Coordinate Limits for Data Selection This panel provides the option of limiting the area of data selection, either through project coordinates or through the use of a boundary file. Panel 5 - Histogram Plot Attributes This panel provides options for setting up your histogram display and plot. Results and Histogram Plot The results of the statistical analysis are presented in the report file (rpt402.rck) in tabular form and as a symbolic histogram. Since the first item for analysis was ROCK, the histogram is by rock type; thus, there are only two frequency intervals in the histogram. The report file tabulates the bench by bench mean and standard deviation for CU.

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Notes:

There is also information provided regarding the number of intervals excluded based on the selection criteria input through the procedure panels:

The MPLOT panel will come up, allowing you the option of plotting the resulting histogram. . Exercise Generate composite Cu statistics within ALTR codes.

Probability Plots On the MineSight Compass Menu tab, . Fill out the panels as described.

Classical Statistics

Notes:

Panel 1 Select File or Drillholes to Use . Panel 2 Parameters for Probability Plot . There are also titling options for the resulting probability plot; enter a title such as “CU probability plot - rock types 1 and 2”. Panel 3 Optional Data Selection This panel allows you to define portions of the data to include or exclude from the analysis based on item values. There are also options for a selection item and/or boundary file for further data limiting. . Panel 4 Optional Plot Files This panel provides inputs for the overlay of existing plot files. Leave this panel blank. Panel 5 Optional Plot Parameters This panel provides options for setting up your probability plot features. You can try out different parameters until you get a display you like. Results and Probability Plot The MPLOT panel will appear, giving you the option to preview, plot directly to the plotter, or generate a deferred plot file for later use. .

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Exercise 1 Generate probability plots for those assays that have ALTR = 1 and 2 only. Hint: this can be done with a change to Panel 3. Exercise 2 Repeat the exercises using the composites. Exercise 3 Overlay the composite probability plot on the assay probability plot and compare. Hint: Output only the cumulative probability curve for composites and overlay it on the full assay probability plot.

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Classical Statistics

Notes:

Exercise 4 Generate probability plots without using log transformation.

In situ Data Statistics On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - In-Situ Data Statistical Analysis . Panel 2 - In-Situ Data Statistical Analysis This panel provides input for the basic statistical analysis parameters. . Panel 3 - Optional Data Selection for Statistical Analysis This panel allows you to define portions of the data to include or exclude from the Part #: E005 Rev. B

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Proprietary Information of Mintec, inc. analysis based on item values. You can also specify an optional boundary file in this panel. . Panel 4 - Parameters for in-Situ Statistical Analysis .

Results and Contour Data Plot The results of the in-situ statistical analysis are presented in the report file (rpt403.cu) in tabular form.

When you close the report file, the MPLOT panel will come up, giving you the opportunity to preview the contour data plot, send it to the plotter, or generate a deferred plot file for later use. Page 1-10

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Classical Statistics

Notes:

Exercise Generate In-situ statistics for assays in elevation range 2500 to 3000.

Correlation and Scatter Plots . Fill out the panels as described. Panel 1 - Bivariate Analysis - Scatter Graph and Q-Q Plot This panel provides inputs for the file type, interval selection and other options. We will use File 11 assays and all the drillhole assay intervals; specify ‘cu’ as the filename

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Classical Statistics

Notes:

Proprietary Information of Mintec, inc. extension for both the run and report files. Leave the titling options blank for this exercise. Panel 2 - Bivariate Analysis - Scatter Graph and Q-Q Plot This panel provides input for the items to be analyzed; . Panel 3 - Bivariate Analysis - Optional Plot Parameters This panel provides input for the basic plot parameters; . Panel 4 - Q-Q (Quantile-Quantile) Plot Options This scatter plot exercise does not use the Q-Q options; leave this panel blank. Panel 5 - Optional Data Selection This panel allows you to define portions of the data to include or exclude from the analysis based on item values. You can also specify an optional boundary file and/or selection item in this panel. Use the RANGE command to specify ROCK Types 1 and 2 only. Results and Scatter Plot The report file shows a summary of correlation statistics for CU and MO grades.

There are also symbol plots for the scatter plot and histograms for each of the specified analysis items.

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Classical Statistics

Notes:

When you close Part #: E005 Rev.the B report file, the MPLOT panel will come up, giving you the

Page 1-13

Classical Statistics

Notes:

Proprietary Information of Mintec, inc. opportunity to preview the scatter plot, send it to the plotter, or generate a deferred plot file for later use.

Plot Proportional Effect On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - In-Situ Data Statistical Analysis . Panel 2 - In-Situ Data Statistical Analysis This panel provides input for the basic statistical analysis parameters. . Panel 3 - Optional Data Selection for Statistical Analysis Page 1-14

Part #: E005 Rev. B

Proprietary Information of Mintec, inc. This panel allows you to define portions of the data to include or exclude from the analysis based on item values. You can also specify an optional boundary file in this panel. Use the RANGE command to specify ROCK Types 1 and 2 only.

Classical Statistics

Notes:

Panel 4 - Parameters for in-Situ Statistical Analysis Specify the parameters of the 3-D grid for the statistical analysis of Cu. Also specify plot parameters for the selected slice to be plotted.

Panel 5 - Generate Contours for Selected Level In this panel, . Exit MPLOT Panel Plot the correlation between the std and mean. On the MineSight Compass Menu tab, Panel 1 Panel 2 Panel 3 Panel 4 Leave this panel blank. Results and Scatter Plot Preview plot:

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Classical Statistics

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Notes:

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Variograms

Variograms

Notes:

Prior to this section you calculated the composites. In this section you can develop variograms of the composites. Following this, you can initialize the mine model and do Kriging.

Learning Outcome In this section you will learn:

•

How to make an h-scatterplot

•

Types of variograms within MineSight

•

Procedure for creating variograms

The Variogram In geostatistics, geologic samples such as assays or thickness values are not independent samples. Samples in proximity to one another are usually correlated to some degree. As the distance between samples increases this degree of correlation declines until the samples are far enough apart where they can be considered to be independent of one another. The variogram is a graph that quantifies the spatial correlation between geologic samples. It is a plot with the average squared assay difference between all pairs of samples “h” distance apart plotted along the y-axis (h), and the distance “h” plotted along the x-axis. Logically you would expect this squared difference (h) to increase as the distance “h” between the sample pairs increases. Once you reach a distance where the sample pairs are independent, the average squared difference is not related to the distance “h” anymore and the curve levels off. This distance where the samples are no longer correlated is called the range of the variogram and the value of (h) where it levels off is called the sill. Theoretically the sill is equal to the variance of samples. The distance over which the samples are correlated can be and usually is different in different directions. This is called Anisotropy and simply states that mineralization may be more continuous in one direction than another. Therefore, variograms are computed in different directions. At DISTANCE h=0 (i.e., 2 samples at the same location) the sample values should be identical. In reality they usually are not. This is described in geostatistics as the Nugget effect. Its value should be small if correct sampling and assaying procedures are used. Variogram Models The variogram model is the equation of a curve that best fits the variogram generated with your data. Variogram models (singel or nested) available in MineSight are:

•

Spherical

•

Exponential

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Variograms

Notes:

Proprietary Information of Mintec, inc.

•

Linear

H - Scatter Plots On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - H-Scatter Plot This panel provides input for the file type and item to be analyzed, as well as some data selection parameters. . Panel 2 - H-Scatter Plot Parameters This panel provides the input for a number of plot parameters such as the plot title, symbol type and size. . Leave the rest of this panel blank. Panel 3 - Optional Data Selection This panel allows you to define portions of the data to include or exclude from the analysis based on item values. There is also an optional selection item available for further data limiting. Select only ROCK item value 1 for plotting by using the RANGE command. Panel 4 - Coordinate Limits This panel allows you to limit the data further, either by specifying coordinates or a boundary file. Leave this panel blank. Panel 5 - Parameters to Determine Data Pairs This panel provides input for the data pair selection parameters. . Results and H-Scatter Plot The results of the data pair statistical analysis are presented in the report file (rpt311.cu) in tabular form. There are also symbolic plots for H-Scatter and frequency distribution histograms.

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Variograms

Notes:

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Variograms

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Notes:

When you close the report file, the MPLOT panel will come up, giving you the opportunity to preview the H-Scatter data plot, send it to the plotter, or generate a deferred plot file for later use.

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Variograms

Notes:

Exercise Generate h-scatter plots for pairs 0-50m apart in different horizontal directions.

Calculating Variograms and Modeling On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - Experimental Variograms for 3-D Composites This panel provides input for file type selection, analysis item and variogram type. Use File 9 and normal variogram type to compute the initial variograms for the item CU. Appropriate values for the minimum and maximum are zero and five, respectively. Specify a filename extension of ‘001’ for the report and output files Panel 2 - Optional Variogram Parameters This panel provides input for optional parameters such as title and selection item. Specify a title such as ‘Horizontal Variograms Rock type 1’. Leave the rest of the panel blank. Part #: E005 Rev. B

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Variograms

Notes:

Proprietary Information of Mintec, inc. Panel 3 - Optional Data Selection This panel allows you to define portions of the data to include or exclude from the analysis based on item values. Select only ROCK item value 1 by using the RANGE command. Panel 4 - 3D Coordinate Limits for Data Selection This panel allows you to limit the data further, either by specifying coordinates or a boundary file. Leave this panel blank. Panel 5 - Parameters for Multi-Directional Variograms This panel is used to enter the specific variogram parameters. Compute 4 normal variograms (4x1), starting at horizontal angle 0.0 with 45 degree increments and at a vertical angle 0.0. Use 10 intervals with 50m lag distance. Use 22.5 degree horizontal windowing angle and 10 degree vertical angle. Results and Variogram Plot In the report file (rpt303.001), a summary appears for each variogram calculated. The variogram value is under the column V(H). It is plotted along the y-axis on the graph. The distance “h” is under the column DISTANCE. It is plotted along the x-axis on the graph.

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Modeling Variograms On the MineSight Compass Menu tab, . Fill out the panels as described.

Variograms

Notes:

Panel 1 - Variogram File Input This panel provides input for the MineSight variogram data file for modeling. Specify the file we just created, dat303.001. Results and Variogram Plot Program M300V1 will display on the screen a list of the 4 directional variograms plus the 2-D Global variogram and the 3-D Global variogram. .

Exercise 1 Interactively fit a spherical variogram model to the experimental variogram by following these steps: 1. . Exercise 2 Interactively modify the spherical variogram model you just created by following these steps: 1. . Exercise 3 Try an exponential model and compare the fit (visually) with the spherical. Exercise 4 Try to model the directional variogram at 0, 45, 90, and 135 degrees. Exercise 5 If the directional variograms are difficult to model, add an absolute lag tolerance of 25. Don’t use composite values above 3. Use extension 002. Exercise 6 Try a different horizontal angle increment (30°). Use extension 003. Exercise 7 Try to compute variograms using different vertical angle orientations. Compare horizontal variograms to the vertical ones. Use extension 004. Exercise 8 Compute variograms using different variogram type options, such as correlogram. Use extension 005.

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Proprietary Information of Mintec, inc. Exercise 9 Run variograms (as in Exercise 5) for rock type 2. Use extension 006.

Variograms

Notes:

Exercise 10 Run variograms for different windowing angles and/or band widths. Check reports and see how the number of pairs used changes.

Calculating Down-hole Variograms On the MS Compass Menu tab, select the Group Statistics, and the Operation Calculation; from the procedure list, select procedure p30101.dat - Down-hole Variograms. Fill out the panels as described. Panel 1 - File and Variogram Type Selection This panel provides input for the source file; you have the option to use the assay or composite files. You can also specify the variogram type in this panel. Use the assay file 11 for analysis with variogram type 1 for a normal variogram. Panel 2 - Input Parameters In this panel you can specify the item for analysis and optional minimum and maximum values. Use Cu for variogram analysis, with the default values for min and max. Specify the filename extension ‘cu’ for the report and output files. Panel 3 - Optional Variogram Parameters This panel provides titling and selection item options. Specify a title such as ‘Downhole variogram - Rock type 1’ Panel 4 - Optional Data Selection for Down-hole Variograms This panel allows you to define portions of the data to include or exclude from the analysis based on item values. Select only ROCK item value 1 by using the RANGE command. Panel 5 - 3D Coordinate Limits for Data Selection This panel allows you to limit the data further, either by specifying coordinates or a boundary file. Leave this panel blank. Panel 6 - Parameters for Down-hole Assay Variograms This panel provides input for the variogram parameters. Compute variograms for each hole with horizontal direction 0, windowing angle 90, and vertical direction -90, windowing angle 15°. Use 20 intervals with 5m lag distance. In the report file, a summary appears for each variogram calculated, as well as a summary for a combined variogram. The variogram value is under the column V(H). It is plotted along the y-axis on the graph. The distance “h” is under the column DISTANCE. It is plotted along the x-axis on the graph. On the following page is the plot of the variogram points of the combined variograms.

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Variograms

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Notes:

Exercise 1 Model the combined down-hole variogram. Pick up a nugget value. Exercise 2 Calculate and model a down-hole variogram for rock type 2. Pick up a nugget value. Exercise 3 Generate down-hole variograms using composite data.

Variogram Data Contouring On the MineSight Compass Menu tab, . Fill out the panels as described.

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Variograms

Panel 1 - GNV2CN Parameters for Variogram Data Contouring This panel provides inputs for the source and destination files. Use the variogram data file generated for ROCK1 (dat303.002), and specify the output file pts303.002. .

Notes:

Panel 2 - Setup Plot Parameters This panel provides input for the plot parameters; . Leave the remainder of the panel blank.

2250

Minimum Easting

2750

Maximum Easting

5250

Minimum Northing

5750

Maximum Northing

1000

Pl ot Scal e (RF Scal e, eg, 1000=1:1000)

0.7

Label Height for Grid Coordinates (CMs)

250

Grid Spacing (Metres)

Panel 3 - Generate Contours from Variogram Data This panel provides input for the contouring parameters. . Panel 4 - Pen Specification and Optional User Plot Files This panel provides inputs for the overlay of existing plot files. Leave this panel blank. Panel 5 Standard Title Box Set up This panel provides inputs for titling your plot. Fill in appropriate title block information. Results and Contour Plot The MPLOT panel will appear, giving you the option to preview, plot directly to the plotter, or generate a deferred plot file for later use. .

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Notes:

Exercise 1 Overlay an ellipse to the variogram contours using a 200m major axis and a 125m minor axis. What is the major axis orientation? Adjust lengths until the ellipse fits the contours. If you find an orientation for which you don’t have variograms, rerun the variograms programs for the new orientation. Exercise 2 Repeat for rock type 2.

Variogram Parameter File On the MineSight Compass Menu tab, . Fill out the panels as described. Having studied the individual variograms, down-the-hole variograms, and the contour maps for each rock type, decide on one set of variograms.

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Variograms

For example:

Notes:

Rock typ e

Nugget Sil l

Range (major)

Range (minor)

1

0.014

0.240

70 (10 )

40 (100 ) Exponential

2

0.007

0.085

80 (45 )

60 (135 ) Exponential

o

o

Model type o

o

Assume for the vertical axis the same ranges as the minor axis. If you use an exponential model, use three times the range as search distances in the interpolation programs. Panel 1 - Variogram Parameter File This panel provides input for the name of the variogram parameter file to set up and an optional description line. . Panel 2 Variogram Parameters This panel provides entry for the variogram parameters: model type, nugget, sill, range, and direction of major axis. , recalling that the sill in the table includes the nugget effect. Results The resulting variogram file will be brought up in the screen.

Exercise 1 Set up variogram parameter file for Rock Type 2. Exercise 2 Set up variogram parameters for both Rock Types 1 and 2 in the same file. Hint: specify the geology label as ROCK in the first panel.

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Variograms

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Notes:

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Point Validation/Cross Validation for Variogram Evaluation

Point Validation/Cross Validation for Variogram Evaluation

Notes:

Prior to this section, you calculated the variograms and modeled them. In this section you can use the Kriging method to determine the error between the estimated and the actual known value of composite data at selected locations, using different variograms. The theoretical variogram that produces the smallest error can be assumed as the better fit.

Learning Outcome In this section you will learn how to use point validation for variogram evaluation

Interpolation Controls There is a large range of parameters for controlling the point interpolation.

•

Search distance N-S, E-W, and by elevation

•

3-D ellipsoidal search

•

Minimum and maximum number of composites to use

•

Maximum distance to the nearest composite

•

Use or omit geologic control

•

Inverse distance powers and variogram parameters

Point interpolation program M524V1 outputs the results for each composite used to an ASCII file. These results are evaluated using program M525TS and the statistical summaries are output to the report file. In this case we will assume search parameters as indicated by the variograms. If a specific search scenario has been determined for the model interpolation, it should be used instead.

Point Validation From the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - File and Area Selection This panel provides input for the composite file type, area specifications, and optional filename extensions. . Panel 2 - Point Interpolation This panel provides input for the item and search parameters to be used in interpolation. . Panel 3 - Optional Ellipsoidal Search Parameters Ellipsoidal Search and use of anisotropic distances are optional. For this run, Panel 4 - Optional Data Selection This panel will only come up if you choose the anisotropic distances option on the previous panel. The angles for this example are ROT = 10, DIPN = 0 and DIPE = 0. Panel 5 - Optional Parameters . If the variogram parameter file is not entered in this panel, you will be prompted to enter the variogram parameters on subsequent panels. Panel 6 - Optional Data Selection for Point Interpolation This panel allows you to define portions of the data to include or exclude from the analysis based on item values. There is also an optional selection item and geologic matching item available for further data limiting. . Panel 7 - Optional IDW powers and Other Parameters . Results This section of the report shows summary statistics for actual composite grades versus the results from different interpolations.

This section of the report (on the next page) shows the statistics of the differences between actual and kriging values The histogram is the histogram of the errors.

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Point Validation/Cross Validation for Variogram Evaluation

Notes:

This section of the report file (on the next page) shows correlation statistics between the actual and kriging values.

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Point Validation/Cross Validation for Variogram Evaluation

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Notes:

Exercise Modify the variogram parameter file. Use nugget of 0.02. Rerun the point validation procedure. Compare results (you should get a lower correlation and a higher standard error).

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Declustering

Declustering

Notes:

Prior to this section you calculated the composites and sorted statistics. In this section you will use cell declustering technique to decluster the composite data. This is not required for later work.

Learning Outcome In this section you will learn:

•

How to decluster composite values

•

How to produce a histogram of declustered composite values

Declustering There are two declustering methods that are generally applicable to any sample data set. These methods are the polygonal method and the cell declustering method. In both methods, a weighted linear combination of all available sample values is used to estimate the global mean. By assigning different weights to the available samples, one can effectively decluster the data set. In this section you will be using the cell declustering method which divides the entire area into rectangular regions called cells. Each sample received a weight inversely proportional to the number of samples that fall within the same cell. Clustered samples will tend to receive lower weights with this method because the cells in which they are located will also contain several other samples. The estimate one gets from the cell declustering method will depend on the size of the cells specified, If the cells are very small, then most samples will fall into a cell of its own and will therefore receive equal weights of 1. If the cells are too large, many samples will fall into the same cell, thereby causing artificial declustering of samples.

Declustering Composite Data On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - Declustering Composite Data This panel provides input for the item to decluster and an optional selection item. . Panel 2 - Optional Data Selection for Composite Declustering This panel allows you to define portions of the data to include or exclude from the analysis based on item values. Leave this panel blank to use all the values of Cu. Panel 3 - Declustering Composite Data This panel allows you to adjust the grid size for declustering. .

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Declustering

Notes:

Proprietary Information of Mintec, inc. Results The report file shows summary statistics for the original and the declustered samples.

Exercise 1 Obtain declustered data using cell sizes 45 x 45 and 40 x 40. Exercise 2 Create a graph of the cell sizes vs mean values. The cell size that gives the lowest value should be the best choice. Exercise 3 Try declustering using Rock Type 1 only.

Histogram of Declustered Data On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - ASCII Data Statistical Analysis . Panel 2 - ASCII Data Statistical Analysis . Panel 3 - ASCII Data Statistical Analysis This panel provides input for the analysis parameters. . Page 4-2

Part #:E005 Rev. B

Proprietary Information of Mintec, inc. Panel 4 - Histogram Plot Attributes Set up the Histogram Plot Attributes as desired.

Declustering

Notes:

Results and Histogram Plot The report file presents the declustered data analysis in tabular and symbolic histogram form.

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Declustering

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Notes:

When you exit the report file, the MPLOT panel will appear, giving you the option to preview, plot directly to the plotter, or generate a deferred plot file for later use. .

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Model Interpolation

Model Interpolation

Notes:

Prior to this section you calculated and sorted the composites. You initialized the mine model and added any necessary geology. In this section you can use inverse distance weighting to add grades to the mine model. This is required before displaying the model, calculating reserves or creating pit designs.

Learning Objective In this section you will learn:

•

The types of interpolations available

•

The use of controls on the interpolation

•

How to interpolate grades with MineSight

Types of Interpolations There are several methods of interpolation provided to you.

•

Polygonal assignment

•

Inverse distance weighting

•

Relative elevations

•

Trend plane

•

Gradients

•

Kriging

Interpolation Controls There is a large range of methods for controlling the interpolation available.

•

Search distance N-S, E-W, and by elevation

•

Minimum and maximum number of composites to use for a block

•

Maximum distance to the nearest composite

•

Use or omit geologic control

IDW Interpolation On the MineSight CompassMenu tab and Fill out the panels as described.

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Model Interpolation

Notes:

Proprietary Information of Mintec, inc. Panel 1 - M620V1/V2 IDW Search Parameters This panel provides input for the composite files to use, the area to interpolate, and optional filename extensions. For this example, use model 15, composite file 9, and specify “idl” as the filename extension for both the run and report files. Panel 2 - M620V1/V2 IDW Search Parameters . Panel 3 - Interpolation Control Items This panel lets you to specify the items and methods for interpolation. . Panel 4 - Store Local Error This panel provides options for the stroage of local interpolation error values. Leave this panel blank for this example. Panel 5 - Optional Search Parameters Ellipsoidal Search and use of anistropic distances are optional. . Panel 6 - Optional Data Selection This panel will only come up if you choose the anistropic distatncces option on the previous panel. The angles for this example are ROT = 10, DIPN = 0 and DIPE = 0. Panel 7 - Optional Geologic Codes This panel provides options for up to three block limiting items and two code matching items. Use only Rock Type 1 by specifying Rock as a block limiting item and entering the value 1 as the corresponding integer code. Also use ROCK as a code matching item. Panel 8 - Optional Data Selection Since we have already limited the interpolation through block and code matching, we can leave this panel blank. Results The results of the interpolation are saved to the model file 15; to view the results, create a model view in MineSight 3-D.

Exercise Rerun for Rock Type 2. Change search distances and use option omit. Change the following panels: Panel 1 - M620V1/V2 IDW Search Parameters . Page 5-2

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Proprietary Information of Mintec, inc. Panel 2 - M620V1/V2 IDW Search Parameters .

Model Interpolation

Notes:

Panel 3 - Interpolation Control Items Use the OMIT option for this second interpolation pass. Panel 5 - Optional Search Parameters s. Panel 6 - Optional Data Selection The angles for this example are ROT = 45, DIPN = 0 and DIPE = 0. Panel 7 - Optional Geologic Codes Use only Rock Type 2 by specifying ROCK as a block limiting item and entering the value 2 as the corresponding integer code. Again, the results for the interpolation of Rock type 2 can be checked by creating a Model View in MineSight 3-D.

Ordinary Kriging Ordinary kriging is an estimator designed primarily for the local estimation of block grades as a linear combination of the available data in or near the block, such that the estimate is unbiased and has minimum variance. It is a method that is often associated with the acronym B.L.U.E. for best linear unbiased estimator. Ordinary kriging is linear because its estimates are weighted linear combinations of the available data, unbiased since the sum of the weights is 1, and best because it aims at minimizing the variance of errors. The conventional estimation methods, such as inverse distance weighting method, are also linear and theoretically unbiased. Therefore, the distinguishing feature of ordinary kriging from the conventional linear estimation methods is its aim of minimizing the error variance.

Kriging with MineSight Before producing an interpolation using kriging, you developed a variogram. Three types of variograms are allowed:

•

Spherical

•

Linear

•

Exponential

On the MineSight CompassMenu tab, . Fill out the panels as described. Panel 1 - M624V1: Kriging Search Parameters This panel provides input for the model and composite files to use, the area to interpolate, and optional filename extensions. For this example, use model file 15, composite file 9, and specify ‘kr1’ as the filename extension for both the run and report Part #: E005 Rev. B

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Model Interpolation

Notes:

Proprietary Information of Mintec, inc. files. Panel 2 - M624V1: Kriging Search Parameters . Panel 3 - Interpolation Control Items This panel allows you to specify the items and method for interpolation. Interpolate the CU composites using kriging to the model item CUKRG. Use the RESET option for this first interpolation pass; for multiple passes, use the OMIT option. Panel 4 - Optional Input Parameters/Composite Type Use this panel to specify the variogram parameter file (if used), and optional block discretization parameters. Specify the storage of the kriging variance in item CUKVR. Panel 5 - Variogram Parameters This panel provides entry for the variogram parameters: model type, nugget, sill, range, and direction of major axis. recalling that the sill in the table includes the nugget effect. Panel 6 - Optional Search Parameters Ellipsoidal Search and use of anisotropic distances are optional. For this run, Panel 7 - Optional Data Selection This panel will only come up if you choose the anisotropic distances option on the previous panel. The angles for this example are ROT = 10, DIPN = 0 and DIPE = 0. Panel 8 - Optional Block Limiting and Geologic Matching This panel provides options for up to three block limiting items and two code matching items. Use only Rock Type 1 by specifying ROCK as a block limiting item and entering the value 1 as the corresponding integer code. Also use ROCK as a code matching item. Panel 9 - Optional Data Selection Since we have already limited the interpolation through block and code matching, we can leave this panel blank. The results of the interpolation are saved to the file 15 - you can check the results visually by creating a Model View in MineSight 3-D.

Exercise Repeat calculations for Rock Type 2. Use variograms calculated in Section 2. Change search distances as you did for IDW.

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Debugging Interpolation Runs

Debugging Interpolation Runs

Notes:

Learning Outcome In this section you will learn how to debug your interpolation runs. You will learn how to:

•

make a list of the composites used for interpolating a block,

•

make a visual representation of your search parameters, and

•

find out how small changes in search parameters affect your interpolation.

Kriging Debug Procedure Exercise 1 Fill panels as described in the following: Panel 1 – M624V1: Kriging Search Parameters This panel allows you to select the desired model file (file 15, msop15.dat), the desired composite file (file 9, msop09.dat), the area to interpolate (for this example, bench 35, row 75, column 85) and filename extensions (we’ll use ‘dbg’ for this example). Panel 2 – M624V1: Kriging Search Parameters This panel accepts input for the search distances and parameters – for this and subsequent examples, we’ll use the search parameters and variogram parameters used for rock type 2. Leave the rest of the panel entries blank. Panel 3 – Interpolation Control Items This panel provides input for the items to be interpolated. For this example, we’ll interpolate the composite CU values into the model item CUKRG, using the kriging option (calc type 0). Accept the defaults for the rest of the panel. Panel 4 – Optional Input Parameters/Composite Type This panel provides input for the variogram parameter file, if you have one – if not, the variogram parameters are entered on a subsequent panel. This panel also provides other optional interpolation parameters; leave this panel blank for this example. Panel 5 – Variogram Parameters This panel accepts the variogram parameters if no variogram parameter file has been previously entered. Use the following parameters:

model

nugget

Sill (without nugget)

Ranges

Directions (MEDS)

EXP

0.007

0.078

80/60/60

45/0/0

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Debugging Interpolation Runs

Notes:

Proprietary Information of Mintec, inc. Panel 6 – Optional Limiting and Search Parameters This panel accepts a number of optional search parameters relating to block limiting, anisotropic distances and search ellipses. Panel 7 – Optional Data Selection Panel 8 – Optional Composite Data Selection This panel allows you to define portions of the data to include or exclude from the analysis based on item values, and outlier definition options. Select all available data by leaving this panel blank. Panel 9 – Input Parameters for ellipsoid generation. You can type in the names of the output files and objects. Leave blank for this exercise. Examine Results This is a list of the composites used:

DIST 84.24 86.58 93.25 93.25 103.42 103.42 103.69 132.10 133.60 133.60 138.02 138.02 145.09 145.09 158.24 159.50

EAST 2628.40 2628.40 2628.40 2628.40 2628.40 2628.40 2728.30 2618.80 2618.80 2618.80 2618.80 2618.80 2618.80 2618.80 2829.60 2829.60

NORTH 5432.60 5432.60 5432.60 5432.60 5432.60 5432.60 5439.10 5558.90 5558.90 5558.90 5558.90 5558.90 5558.90 5558.90 5537.40 5537.40

ELEV 2435.0 2450.0 2465.0 2405.0 2390.0 2480.0 2480.0 2435.0 2420.0 2450.0 2465.0 2405.0 2390.0 2480.0 2435.0 2420.0

VALUE 0.6300 0.3700 0.3100 0.4700 0.6100 0.4100 0.7300 0.0100 0.0000 0.0100 0.0000 0.0100 0.0000 0.0000 0.3300 0.2300

DH 53 53 53 53 53 53 54 61 61 61 61 61 61 61 62 62

Block (75, 75) Calculated = 0.3906 Note all of the composites are well inside the 240m maximum search distance. Distances reported are adjusted by anisotropy. If you want to see the real distances, rerun procedure without the anisotropic option on. You should now notice that the order of the composites has changed. Exercise 2 What do you notice?

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Debugging Interpolation Runs

Exercise 3 What do you notice?

Notes:

Exercise 4 Check report and study results:

DIST 77.99 84.20 85.52 89.38 89.38 95.47 95.47

EAST 2728.30 2628.40 2628.40 2628.40 2628.40 2628.40 2628.40

NORTH 5439.10 5432.60 5432.60 5432.60 5432.60 5432.60 5432.60

ELEV 2480.0 2435.0 2450.0 2465.0 2405.0 2390.0 2480.0

VALUE 0.7300 0.6300 0.3700 0.3100 0.4700 0.6100 0.4100

DH 54 53 53 53 53 53 53

Exercise 5 Open report: DIST 84.20 85.52 89.38 89.38 95.47 95.47

EAST 2628.40 2628.40 2628.40 2628.40 2628.40 2628.40

NORTH 5432.60 5432.60 5432.60 5432.60 5432.60 5432.60

ELEV 2435.0 2450.0 2465.0 2405.0 2390.0 2480.0

VALUE 0.6300 0.3700 0.3100 0.4700 0.6100 0.4100

DH 53 53 53 53 53 53

It seems like there are no more composites available.

Part #: E005 Rev. B

Page 6-3

Debugging Interpolation Runs

Notes:

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Ellipsoidal Search Exercise Do not use octant/quadrant options. In MineSight 3-D, you can import ellipse as a MineSight object (import file ellips.msr). Adjust properties of object if needed.

Page 6-4

Part #: E005 Rev. B

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Pt Validation/Cross Validation of Est Methods - Search Parameters

Point Validation/Cross Validation of Estimation Methods and/or Search Parameters

Notes:

In this section you will use inverse distance weighting and Kriging methods to determine the error between the estimated and the actual known value of composite data at selected locations. Then, you will decide which method is more appropriate. You will also validate search parameters.

Learning Outcome In this section you will learn:

•

The types of interpolations available in point validation

•

The use of controls on the interpolation

•

How to interpolate point grades with MineSight

Types of Point Interpolations Each composite is interpolated using different powers of inverse distance weighting method and Kriging. The results are then summarized showing the differences between the estimated and actual known data values. The following interpolations are done by default by the program.

•

Inverse distance weighting (IDW) of power 1.0

•

IDW of power 1.5

•

IDW of power 2.0

•

IDW of power 2.5

•

IDW of power 3.0

•

Kriging

Interpolation Controls There is a large range of parameters for controlling the point interpolation.

•

Search distance N-S, E-W, and by elevation

•

3-D ellipsoidal search

•

Minimum and maximum number of composites to use

•

Maximum distance to the nearest composite

•

Use or omit geologic control

•

Inverse distance powers and variogram parameters

Part #: E005 Rev. B

Page 7-1

Pt Validation/Cross Validation of Est Methods - Search Parameters

Notes:

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Point interpolation program M524V1 outputs the results for each composite used to an ASCII file. These results are evaluated using program M525TS and the statistical summaries are output to the report file.

Point Validation On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - File and Area Selection This panel provides input for the composite file to use, and for the area of the model to validate. . Panel 2 - Point Interpolation This panel provides input for the validation item and search parameters. Panel 3 - Optional Ellipsoidal Search Parameters Ellipsoidal Search and use of anisotropic distances are optional. For this run, Panel 4 - Optional Data Selection This panel will only come up if you choose the anisotropic distances option on the previous panel. The angles for this example are ROT = 10, DIPN = 0 and DIPE = 0. Panel 5 - Optional Parameters . If the variogram parameter file is not entered in this panel, you will be prompted to enter the variogram parameters on subsequent panels. Panel 6 - Optional Data Selection for Point Interpolation This panel allows you to define portions of the data to include or exclude from the analysis based on item values. There is also an optional selection item and geologic matching item available for further data limiting. . Panel 7 - Optional Parameters Use the default IDW powers for each case; generate a detailed report for case 3, using a 0.1 frequency interval and 40 intervals. Results This report (on the next page) shows summary statistics for actual composite grades versus the results from different interpolations.

Page 7-2

Part #: E005 Rev. B

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Pt Validation/Cross Validation of Est Methods - Search Parameters

Notes:

This section of the report shows the statistics of the differences between actual and kriging values The histogram is the histogram of the errors.

Part #: E005 Rev. B

Page 7-3

Pt Validation/Cross Validation of Est Methods - Search Parameters

Notes:

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This section of the report file shows correlation statistics between the actual and Kriging values.

This section of the report file shows correlation statistics between the actual and inverse distance values.

Exercise Change some of the search parameters and rerun the above procedure. What do you observe?

Page 7-4

Part #: E005 Rev. B

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Model Statistics/Geologic Reserves

Model Statistics/Geologic Reserve

Notes:

Prior to this section you added the grades, topography, and necessary geology into the mine model. In this section you will summarize the mine model data with frequency distributions and calculated the geologic resources.

Learning Outcome In this section you will learn:

•

How to calculate grade and tonnages above different cutoffs

•

How to calculate grade and tonnages between cutoffs

•

How to produce a histogram plot of model values

•

How to generate reserves by bench or geological resources

•

How to generate probability plots from the model

Model Statistics From the MS Compass Menu tab, . Fill out the panels as described. Panel 1 - 3D Model Data Statistical Analysis This panel is used to specify the model file for analysis - file 15 in this example. Panel 2 - 3D Model Data Statistical Analysis This panel provides input for the item(s) to be analyzed, along with optional weighting and selection items. For this example, . Panel 3 - 3D Model Data Statistical Analysis This panel is used to enter the frequency analysis parameters; use a minimum of zero, and 40 intervals with an interval size of 0.1. . Panel 4 - Optional Data Selection This panel allows you to define portions of the data to include or exclude from the analysis based on item values.< Select ROCK item values 1 and 2 by using the RANGE command. Enter 16.2 as multiplier for resource calculation>. This is the Ktonnage/ block for our project. Panel 5 - 3D Model Data Statistical Analysis This panel provides the opportunity to limit the data selection based on model parameters, as well as titling options. Let the area selection default to the entire model, but enter a title such as ‘Inverse Distance Weighting’. Panel 6 - Histogram Plot Attributes Set up the Histogram Plot Attributes as desired.

Part #: E005 Rev. B

Page 8-1

Model Statistics/Geologic Reserves

Notes:

Proprietary Information of Mintec, inc. Results and Histogram Plot This section of the report file (rpt608.cu1) shows the grade and tonnage of CUID, CUKRG, CUPLY and MOID values at specified cutoffs.

This section of the report file shows the tonnage and grade of CUID values at each bench. The grade is reported at whatever the minimum value specified on Panel 3.

Page 8-2

Part #: E005 Rev. B

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Model Statistics/Geologic Reserve

Notes:

When you close the report file, the MPLOT panel will come up, giving you the opportunity to preview the contour data plot, send it to the plotter, or generate a deferred plot file for later use.

Part #: E005 Rev. B

Page 8-3

Model Statistics/Geologic Reserves

Proprietary Information of Mintec, inc.

Notes:

Exercise Generate model statistics for items from IDW, polygonal and Kriging methods separately. Use the same cutoff intervals. We will use the data output files from each run to make grade tonnage plots. Use filename extensions cui, cup, and cuk respectively.

Grade/Tonnage Plots From the MineSight Compass Menu tab, . Fill out the panel as described. Panel 1 Select Files or Parameters We will plot grade/tonnage curves from polygonal, IDW and kriging methods on the same graph. Specify the data files produced in the previous section for each of these options: dat608.cup, dat608.cui,and dat608.cuk. Differentiate between the curves with contrasting symbol and linetypes, and enter an appropriate title such as ‘Polygonal vs IDW (+) vs Kriging (#)’. Results and Grade/Tonnage Plot The MPLOT panel will appear, giving you the option to preview, plot directly to the plotter, or generate a deferred plot file for later use. .

Page 8-4

Part #: E005 Rev. B

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Model Statistics/Geologic Reserve

Notes:

Plot IDW and Kriging Histograms Together On the MineSight Compass Menu tab, . Fill out the panel as described. Panel 1 Plotting panel . Plotting files are USERF. Use appropriate shift commands, as shown in the figure on the next page.

Part #: E005 Rev. B

Page 8-5

Model Statistics/Geologic Reserves

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Notes:

Results and Histogram Plot The plot is opened in program M122MF. .

Page 8-6

Part #: E005 Rev. B

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Model Statistics At and Between Cutoffs

Model Statistics/Geologic Reserve

Notes:

On the MineSight Compass Menu tab, . Fill out the panels as described. Panel 1 - 3D Model Data Statistical Analysis This panel provides input for file selection - use file 15 for this example. Panel 2 - 3D Model Data Statistical Analysis This panel provides input for the item(s) to be analyzed, along with optional weighting and selection items. For this example, use CUID as the first (base) item; weight the results by the TOPO item with a multiplier of 0.01. Panel 3 - 3D Model Data Statistical Analysis This panel provides input for the item parameters and cutoff grades.

Calculate probabilities Assign the probability of occurrence of outlier grades to the blocks. This step requires that you have an additional item in file 15 to store the probabilities. The item should be initialized with min =0, max =1, precision = 0.01 or 0.001. Use Inverse Distance Weighting to assign the probabilities.

Perform ORK

Compare results with regular kriging Compare results from ordinary kriging and outlier restricted kriging. Run model statistics and create grade-tonnage curves.

Page 12-2

Part #: E005 Rev. B

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Indicator Kriging to Define Geologic Boundary above a Cutoff

Indicator Kriging to Define Geologic Boundary above a Cutoff

Notes:

Learning outcome In this section you will learn: - How to calculate the indicator function (0 or 1) based on a grade cutoff - How to calculate the probability of a block having a grade value above the Cutoff - How to view the probabilities (from block model) in MineSight

Indicator Kriging The basis of the technique is transforming the composite grades to a (0 or 1) function. All composite grades above cutoff can be assigned a code of 1 whereas all the composites below can be assigned a code of 0. Then a variogram can be formed from the indicators which can be used for Kriging the indicators. The resulting Kriging estimate represents the probability of each block having a grade value above the cutoff.

Assign Indicators On the MineSight Compass Menu tab, Panel 1 Labels of Composite Items to use Panel 2 Optional Data Selection Panel 3 Limits for Data Selection Leave this panel blank. Panel 4 Special Project Calculations

Variogram of Indicators On the MineSight Compass Menu tab, Panel 1 Experimental Variograms for 3-D Composites Panel 2 Optional Variogram Parameters Leave this panel blank

Part #: E005 Rev. B

Page 13-1

Indicator Kriging to Define Geologic Boundary above a Cutoff

Notes:

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Panel 3 Optional Data Selection Panel 4 3-D Coordinate Limits for Data Selection Leave this panel blank Panel 5 Parameters for Multi-Directional Variograms

Model the Indicator Variogram On the MineSight Menu tab,

Page 13-2

Part #: E005 Rev. B

Proprietary Information of Mintec, inc.

Indicator Kriging to Define Geologic Boundary above a Cutoff

Krige Indicators On the MineSight Menu tab,

Notes:

Panel 1 Files and Model Specification Area Panel 2 Krige Search Parameters Panel 3 Interpolation Control Items Panel 4 Optional Input Parameters Leave this panel blank. Panel 5 Variogram Parameters Panels 6, 7, 8 Leave those panels blank.

View Results in MineSight

Part #: E005 Rev. B

Page 13-3

Indicator Kriging to Define Geologic Boundary above a Cutoff

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Notes:

Page 13-4

Part #: E005 Rev. B

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Multiple Indicator Kriging (M.I.K.)

Multiple Indicator Kriging (M.I.K.)

Notes:

Prior to this section, you must have initialized and loaded the composite and 3-D model files. You must also have calculated the classical statistics and the grade variograms of the composites.

Learning Outcome In this section you will learn: How to determine indicator cutoffs How to calculate indicator variograms How to model indicator variograms How to determine indicator class means How to assign indicators to composite data How to setup indicator variogram parameter files How to calculate affine correction How to do multiple indicator Kriging run How to calculate indicator Kriging reserves

Overview Multiple Indicator Kriging (M.I.K.) is a technique developed to overcome the problems with estimating local recoverable reserves. The basis of the technique is the indicator function which transforms the grades at each sampled location into a [0,1] random variable. The indicator variograms of these variables are estimated at various cutoff grades. The technique consists of estimating the distribution of composite data. The distribution is then corrected to account for the actual selective mining unit (SMU) size. This yields the distribution of SMU grades within each block. From that distribution, recoverable reserves within the block can be retrieved. Accumulation of recoverable reserves for these blocks over a volume gives the global recoverable reserves for that volume.

Uses of Indicators Indicators can be used to:

•

deal with outliers

•

model multiple populations

•

estimate categories (descriptive or qualitative variables)

•

estimate distributions

•

estimate confidence intervals

Part #: E005 Rev. B

Page 14-1

Multiple Indicator Kriging (M.I.K.)

Notes:

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Incremental Statistics Results:

Page 14-2

Part #: E005 Rev. B

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Multiple Indicator Kriging (M.I.K.)

Notes:

Determine M.I.K Cutoffs This procedure is used to analyze where the M.I.K. cutoffs should be for a given grade distribution and the number of M.I.K. cutoffs specified. The basis of analysis is the metal contained in each indicator class. It is possible that the user may try this procedure several times until he or she is satisfied with the results. Panel 1 M.I.K. Cutoff Grade Determination This file must have been generated using statistics within cutoff option.

Part #: E005 Rev. B

Page 14-3

Multiple Indicator Kriging (M.I.K.)

Notes:

Proprietary Information of Mintec, inc. Results:

Calculating Indicator Variograms Panel 1 Compute Indicator Variograms Panel 2 Optional Variogram Parameters Specify the cutoffs at which indicator variograms are to be computed:

Cutoffs 0.42

0.56

0.68

0.78

0.88

0.98

1.12

1.30

1.46 1.66

Panel 3 Optional Data Selection Limit the data input to Rock Type 1 only. Panel 4 Optional Coordinate Limits You have the option of limiting the area of data selection. Leave this panel blank. Panel 5 Parameters for Multi-Directional Composite Variograms There will be a set of indicator variograms for each of the 10 cutoffs. There will also be a set of variograms for the grade item. This is the last set. In the report file, a summary Page 14-4

Part #: E005 Rev. B

Proprietary Information of Mintec, inc. appears for each variogram calculated. This is a 3-D omni-directional variogram for the first indicator cutoff (IND1).

Multiple Indicator Kriging (M.I.K.)

Notes:

Modeling Indicator Variograms

Part #: E005 Rev. B

Page 14-5

Multiple Indicator Kriging (M.I.K.)

Proprietary Information of Mintec, inc.

Notes:

Variogram Parameter File Panel 1 Output and Description File Variogram parameters will be written to the output file specified (vario.mik). Use 10 cutoffs. The mean beyond the last cutoff is 2.0 whereas the max value is 3.7. Affine correction factor must be equal to or greater than 1 (use 1.15). Panel 2 Variogram Parameters You can use the figures from the table.

Page 14-6

Part #: E005 Rev. B

Proprietary Information of Mintec, inc.

Multiple Indicator Kriging (M.I.K.)

Notes: Cutoff

Mean

0.42 0.56 0.68 0.78 0.88 0.98 1.12 1.30 1.46 1.66

0.1907 0.4802 0.6132 0.7241 0.8232 0.9199 1.0279 1.2050 1.3727 1.5539

Variogram type 3 3 3 3 3 1 1 1 1 1

Nugget

Sill-nugget

Range

0.05 0.05 0.05 0.06 0.06 0.05 0.03 0.03 0.03 0.02

0.20 0.20 0.20 0.18 0.17 0.15 0.13 0.10 0.06 0.03

75 75 63 63 57 51 41 41 41 32

Multiple Indicator Kriging Panel 1 Select Files / Area Panel 2 M24IK Search Parameters Panel 3 MIK Interpolation Control Items The program computes the grade and the percent of ore above the specified cutoff for each block and stores them into the 3-D block model. Panel 4 MIK Input Parameters MIK variogram parameters file must be specified (vario.mik). Panel 5 Optional Data Selection Include Rock Type 1 data only. Panel 6 Optional Search Parameters Leave this panel blank. Panel 7 Optional Geologic Codes Kriging will be done in Rock Type 1 blocks only. Use geologic matching (item ROCK).

Part #: E005 Rev. B

Page 14-7

Multiple Indicator Kriging (M.I.K.)

Notes:

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MIK Reserves Panel 1 3-D Model Data Statistical Analysis Panel 2 Model Data Statistical Analysis Since MIK computes ore percent in each block, a second weighting item is necessary for MIK cutoffs greater than 0. Panel 3 Model Data Statistical Analysis Panel 4 Optional Data Selection This is the Ktonnage/block for our project. Panel 5 Mine Model Statistical Analysis Panel 6 Histogram Plot Attributes Report:

Page 14-8

Part #: E005 Rev. B

Proprietary Information of Mintec, inc.

Other Non-Kriging Interpolation Methods

Other Non-Kriging Interpolation Methods

Notes:

Learning Outcome In this section you will learn: How to use the Trend Planes method of interpolation How to use the Gradient interpolation technique

Trend Plane Search Interpolation M621V1 uses a similar interpolation scheme to M620V1. However, the search is along the dip and strike of a trend plane specified by a mine model code associated with a certain plane strike azimuth and plane dip angle as well as with certain distances along the strike, dip and off the plane. The program can be run from procedure p62101.dat. The procedure can be found in MineSight Compass Menu, under Group 3d modeling I Operations Calculations. Exercise Try to use the same interpolation parameters as you did with Inverse distance and Kriging methods.

Gradient Interpolation Technique M625V1 uses gradients to neighboring points for weighting the sorted composites during interpolation of mine model values. M625V1 uses the tangent plane or gradient method to interpolate block values in a mine model. Tangent planes are calculated between all composites and a specified number of neighboring composites. Each plane must satisfy the following conditions:

•

the plane must pass through the function value at the point in question (i.e., through the Z (grade) value),

•

and the angles the plane makes with vectors or lines to all of the various points in the neighborhood must be minimized.

The angles are weighted by a function of how far or near the various neighboring points are from the point of interest. After the tangent planes are generated, block values are calculated from neighboring composites (now with gradients). User needs to specify how many neighbors are used in this calculation. The gradient information for each composite is evaluated at the block location and the calculation of the block value is weighted by the distance from the block to each composite. Exercise Try to use the same search parameters as in previous methods.

Part #: E005 Rev. B

Page 15-1

Other Non-Kriging Interpolation Methods

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Notes:

Page 15-2

Part #: E005 Rev. B