geostatistics

Introduction Geostatistics is used in fields such as mining, forestry, hydrology, and meteorology in order to understand how data values change over distance. Probably the most common use of geostatistics is to make estimations, such as the specific gravity of rock for an area where there are only a few known sample values. This is often done in three-dimensional space. A set of estimated points in space is known as a “model”. As George Box, a professor of Statistics at the University of Wisconsin in the United States, once said, “All models are wrong. Some are useful.

Geostatistics in Surpac 6.0 August 2007 www.gemcomsoftware.com Copyright © 2007 Gemcom Software International Inc (Gemcom) This software and documentation is proprietary to Gemcom and, except where expressly provided otherwise, does not form part of any contract Changes may be made in products or services at any time without notice Gemcom publishes this documentation for the sole use of Gemcom licensees Without written permission you may not sell, reproduce, store in a retrieval system, or transmit any part of the documentation For such permission, or to obtain extra copies please contact your local Gemcom office or visit www.gemcomsoftware.com While every precaution has been taken in the preparation of this manual, we assume no responsibility for errors or omissions Neither is any liability assumed for damage resulting from the use of the information contained herein Gemcom Software International Inc Gemcom, the Gemcom logo, combinations thereof, and Whittle, Surpac, GEMS, Minex, Gemcom InSite and PCBC are trademarks of Gemcom Software International Inc or its wholly-owned subsidiaries Contributors Rowdy Bristol Peter Esdale Phil Jackson Kiran Kumar Product Gemcom Surpac 6.0 Table of Contents Introduction Requirements Objectives Workflow Required Files Tutorial profile Important Concepts Understand the Domains Check the Input Data Understand the Estimation Method and Parameters 10 Check the Output Model 10 Domains 11 A Simple Example 12 Viewing Domains in Surpac 14 Extracting Data with a Domain in Surpac 16 Basic Statistics 19 The Histogram 20 Bimodal Distributions 22 Outliers 23 Displaying Histograms in Surpac 24 Removing Outliers in Surpac 27 Anisotropy 31 Isotropy vs Anisotropy 32 Geostatistical Estimation Using Isotropy 34 Geostatistical Estimation Using Anisotropy 38 Ellipsoid Visualiser 43 Variograms 53 Introduction to the Variogram 54 Calculating a Variogram 56 Modifying the Lag Distance 60 Omnidirectional Variograms 63 Directional Variograms 64 Calculating an Omnidirectional Variogram in Surpac 66 Modelling Variograms in Surpac 73 Variogram Maps 85 Primary Variogram Map 86 Secondary Variogram Map 94 Anisotropy Ellipsoid Parameters 96 Steps for Using Variogram Maps to Create Anisotropy Ellipsoid Parameters 103 Inverse Distance Estimation 106 Isotropic vs Anisotropic Inverse Distance Estimation 107 Steps to Performing Inverse Distance Estimation 108 The Impact of Inverse Distance Power 113 Ordinary Kriging 115 Page of 137 Table of Contents Impact of the Nugget Effect 116 Impact of the Range 117 Block Size Analysis 123 Debug Output from Ordinary Kriging 124 Using Kriging Efficiency and Conditional Bias Slope 125 Block Site Selection 127 Model Validation 128 Comparing Cross-sectional data with Model 129 Grade-Tonnage Curves 131 Basic Statistics of Model Values 133 Trend Analysis 134 Page of 137 Introduction Geostatistics is used in fields such as mining, forestry, hydrology, and meteorology in order to understand how data values change over distance Probably the most common use of geostatistics is to make estimations, such as the specific gravity of rock for an area where there are only a few known sample values This is often done in three-dimensional space A set of estimated points in space is known as a “model” As George Box, a professor of Statistics at the University of Wisconsin in the United States, once said, “All models are wrong Some are useful.” Requirements Prior to proceeding with this tutorial, you will need to have installed Surpac 6.0 or later from a CD Additionally, you should have a good understanding of the following concepts in Surpac: Geological Database Solid modelling Block modelling (how to create and constrain a model) Tcl scripts If you not have a good background in these subjects, many parts of this tutorial may be difficult to follow Objectives The primary objective of this tutorial is to help you become familiar with the methods for performing geostatistical operations with Surpac Also, this tutorial will introduce you to some general geostatistical concepts, and provide some guidance on making geostatistical decisions Ultimately, the models you create are your responsibility There are often more methods than those described here to obtain a model Workflow The process described in this tutorial is outlined below: 10 11 12 Introduction Required Files Important geostatistical concepts Domains Basic statistics Anisotropy Variograms Variogram maps Inverse distance estimation Ordinary kriging Block size analysis Model validation Page of 137 Required Files Workflow Required Files Overview This chapter will identify where you will find the files required for this tutorial Requirements Prior to performing the exercises in this chapter, you should have installed Surpac 6.0 or later from a CD The files and directory structure for each tutorial will only be present if you have installed the software from the CD If you have not installed the software from a CD: Create the following directory: c:/surpacminex/surpac_60/demo_data/tutorials/geostatistics Download the geostatistics tutorial/data (contained in a single zip file) from: http://www.surpac.com/Tutorials.asp Unzip the file geostatistics.zip into the directory you created Page of 137 Required Files Tutorial profile Tutorial profile Profiles are a collection of menubars and toolbars The tutorial profile contains a set of menus that assist you with learning various aspects of the software To display the tutorials profile: Right click in the blank area to the right of the menus From this popup menu, choose Profiles > tutorials A new menubar will be displayed, listing all available tutorials Choose Geostatistics > CD to geostatistics folder Page of 137 Required Files Tutorial profile This should set your working directory to: C:/surpacminex/surpac_60/demo_data/tutorials/geostatistics This directory contains all of the files required to perform the steps in this tutorial Summary The files you will need for the remainder of the tutorial should now be present in your work directory Refer to the Introduction to Surpac manual for more information on profiles Page of 137 Important Concepts Understand the Domains Important Concepts Overview Although geostatistics is not an exact science, there are some important concepts which can reduce estimation errors These concepts can be divided into four regions: Domains Validation of input data Understanding estimation methods and parameters Validating the output model Requirements There are no requirements for reading this chapter, but you may find some of the principles easier to understand if you: • have some understanding of basic statistics • know what a geostatistical model is, or • have previously performed a geostatistical estimation Understand the Domains It is important to recognise separate “regions” or “domains” within a model Once you have identified the domains, it is important to group all sample data contained within each domain into distinct subsets After that, you can analyse each subset individually, and use data from each separate domain to make estimations within that domain Check the Input Data The saying “Garbage in = Garbage out” is certainly true in geostatistics Although sampling theory and laboratory quality control practices are important concepts which impact the quality of any estimation made using a set of data values, these subjects are outside the scope of this tutorial Assuming that the quality of the data is as good as you’re going to get, there are a couple of potentially hazardous characteristics of the data which you should look for: “bimodalism” and “outliers” You can look for both of these features with a histogram A data set is said to be “unimodal” if the histogram shows a single peak If there are two peaks, the data is said to be “bimodal” If you use some of the more common estimation techniques to create a model based on a bimodal distribution, it is likely to contain more estimation errors than a model created from a unimodal data set Additionally, “outliers”, or values which are significantly distant from the majority of the data, can cause estimation errors Page of 137 Important Concepts Understand the Estimation Method and Parameters Understand the Estimation Method and Parameters There are a large number of estimation methods, and a large number of parameters within each method Before using a particular estimation method, you should have a good background in basic statistics, as well as basic geostatistical principles Using geostatistics can be likened to flying a jet plane Although there are “autopilot” modes, where you just press a few buttons and something happens, it is important that the pilot understand the theory of aerodynamics to understand what impact a particular control has upon the end result Check the Output Model A final method you should use to check the quality of estimation is to take time to examine the output Histograms of estimated values, contours of plans, cross sections of block models, colour coded and rotated in three-dimensional space are all methods which can be used to verify the output values Summary Geostatistics is the study of how data varies in space It is an inexact science which is used to make estimations at locations where no data exists It is important to recognise that validation of input and output data are as important as understanding geostatistical theory and the estimation method being used Page 10 of 137 Block Size Analysis Impact of the Range Block Size Analysis Overview An important end product of a geostatistical evaluation is a “model”, or a set of points in space which contain estimated values The points are representative of the centroid of a block of material The determination of the spacing between these points, or the block size is often a critical factor in a geostatistical estimation By evaluating two parameters derived from ordinary kriging, you can determine the optimal block size The ultimate selection of the block size, however, may be based on other factors, such as minimum mining width The analysis of the block size is described in this chapter through the following: Debug output from ordinary kriging Using kriging efficiency and conditional bias slope Block site selection Requirements Prior to proceeding with this chapter, you should understand the following concepts: • Surpac menubars • Surpac string files • Surpac block models • isotropy and anisotropy • anisotropy ellipsoid • ordinary kriging Page 123 of 137 Block Size Analysis Debug Output from Ordinary Kriging Debug Output from Ordinary Kriging When using ordinary kriging to estimate values in a block model, the debug output will contain two parameters which will be used in the analysis of block size Run the macro kriging_debug.tcl This macro: a b c creates a block model performs ordinary kriging within a constraint to estimate a value for a single block uses debug output in the report file This output file, block_size.not will be displayed Near the bottom of the file, you will see the following parameters: Estimated grade: 4.452 Kriging variance: 0.250 Twice std dev.: 1.000 Block variance: 0.907 Kriging efficiency: 0.724 Slope of regression: 0.112 Lagrange multiplier: Conditional bias slope: 0.053 0.930 The values of kriging efficiency and conditional bias slope are used in analysing block size block variance – kriging variance Kriging Efficiency = block variance Block variance – kriging variance + | lagrange multiplier | Conditional Bias Slope = Block variance – kriging variance + 2x | lagrange multiplier | Page 124 of 137 Block Size Analysis Using Kriging Efficiency and Conditional Bias Slope Using Kriging Efficiency and Conditional Bias Slope Ideally, both the kriging efficiency and the conditional bias slope should have a value of 1.00 In practice, this is impossible, but what can be done is to compare values of these two parameters for a variety of block sizes To this in Surpac, you would perform the following functions for a set of block sizes: select the X, Y, and Z coordinates of a location where you wish to perform the analysis select the X, Y, and Z block dimensions create a block model using an origin such that the coordinates of the centroid of the first block in the model are the same as the coordinates in step performs ordinary kriging within a constraint to estimate a value for the first block ensure “debug output” option is used in the report file note the values for kriging efficiency and conditional bias slope in the debug file Here is the data from the kriging_debug.tcl example: • coordinates to perform the analysis: Y=7340 X=1660 Z=110 • block dimensions: Y=10 X=10 Z=10 • block model origin: Y=7335 X=1655 Z=105 • ordinary kriging within constraint: Y< 7345, X