A Spatial and Geostatistical Environment for Variography
Isaaks & Co. 205 E. 3rd Ave. Suite 300 San Mateo, CA 94401
SAGE2001
General Purpose Software for Variography
Copyright 1999 Isaaks & Co. San Mateo, California 94401 ©
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Proprietary Notice Isaaks & Co. owns both this software program and its documentation. Both the program and the documentation are copyrighted with all the rights reserved by Isaaks & Co. See the License Agreement and Limited Warranty for complete information. Copyright © 1998 Isaaks & Co. 205 East Third Ave. Suite 300 San Mateo CA 94401 Phone (650) 655-9051 Fax (650) 655-9443 Email
[email protected] http://www.isaaks.com
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License Agreement and Limited Warranty Both the program and the documentation are protected under applicable copyright laws. Your right to use the program and the documentation is limited to the terms and conditions described herein. 1.0 GRANT OF LICENSE.Isaaks & Co. grants you the non-exclusive and nontransferable right during the term of this agreement to use this computer program only for the benefit of yourself and your organization. This includes the right to perform services for your customers. Under these terms you may use this computer program on one single central processing unit (CPU) used by a single user at a time. 2.0 COPYRIGHT. This computer program including the information contained in it is protected by United States and international copyright laws. You may not, without prior written consent of Isaaks and Co. reproduce, translate, or modify this computer program in any form, in whole or in part, or prepare any derivative work based on this computer program. 3.0 RESTRICTIONS.You may not in whole or in part, rent lease, license, disclose, or give this computer program to any person or organization for any purpose other than specified above unless such use is with the written permission of Isaaks & Co. You agree to take all responsible steps to prevent unauthorized use of this computer program. Except as specified above, this agreement does not grant you the right to any patents copyrights, rights, or licenses in respect of this computer program. 4.0 TERM AND TERMINATION. This license and this agreement are effective until terminated. You may terminate them at any time by destroying this computer program and its documentation. Isaaks & Co. has the right to terminate the license and this agreement immediately if you fail to comply with any term or condition of this agreement. Upon any termination, you must destroy this computer program and its documentation. 5.0 LIMITATION OF WARRANTIES AND LIABILITIES. Isaaks & Co. warrants to customer only that the Computer program, for a period of ninety (90) days after the initial delivery to customer hereunder (the warranty period), will perform substantially in accordance with the Documentation. Isaaks and Co. shall at its expense, provide all corrections or work-arounds for any errors which may be discovered in the computer program and reported by customer during Warranty Period 3
and will deliver a corrected version of the computer program to Customer no later than the delivery of the next code update, if required. For purposes hereof an “error” shall be a defect in the Computer program which causes the Computer Program to not operate substantially in accordance with the Documentation. This warranty shall not apply to the Computer Program it it has been modified, other than by Isaaks & Co. Isaaks & Co. does not and cannot warrant the performance or results that may be obtained by using the Licensed software. The Foregoing states Isaaks & Co’s sole and exclusive remedy for breachof warranty. Isaaks & Co. disclaims all other warranties, expressed or implied, including, but not limited to warranties of merchantability or fitness for any particular purpose. 6.0 CHOICE OF LAW.This agreement will be governed by the laws of the State of California as applied to transactions taking place entirely between California residents. 7.0 INTEGRATION. You have read and understand this agreement, and
acknowledge that it is the final, complete and exclusive agreement between you and Isaaks & Co. concerning its subject matter, superseding any prior related understanding or agreement. No waiver, variation, or different terms of this agreement will be enforceable against Isaaks & Co. unless Isaaks & Co. gives its prior consent signed by an officer of Isaaks & Co.
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Table of Contents 1.0 INTRODUCTION. ....................................................................... 8 2.0 GETTING STARTED ................................................................. 10 2.1 Installing the Demonstration Version of SAGE2001 ................. 10 2.2 Authorizing SAGE2001 for Unlimited Use............................... 11 2.3 Transferring the SAGE2001 Authorization License from One Computer to Another. ...................................................................... 12 3.0 CALCULATING SAMPLE VARIOGRAMS............................... 14 3.1 SAGE2001 Coordinate System and Definitions. ....................... 14 3.2 Importing Data Files................................................................. 15 3.2.1 The SAGE2001 File Format...............................................15 3.2.2 Importing the Data File...................................................... 16 3.2.3 Import File Delimiters. ...................................................... 16 3.2.4 The Import File Grid.......................................................... 16 3.2.5 Adding Column Headers....................................................17 3.2.6 Setting the Attribute Type.................................................. 17 3.3 Opening Existing SAGE2001 Files............................................ 18 3.4 Selecting Data Attributes and Constraints. ................................ 18 3.4.1 Selecting Attributes for the X,Y, and Z Coordinates........... 18 3.4.2 Selecting the Sample Variogram Attribute..........................19 3.4.3 Calculating a Down-The-Hole Sample Variogram. ............. 19 3.4.4 Specifying Additional Constraints:..................................... 20 3.5 Azimuths, Dips, Lags, and Tolerances. ..................................... 21 3.5.1 Directional Increments....................................................... 22 3.5.2 Lag Specifications. ............................................................ 22 3.5.3 Bandwidths. ...................................................................... 22 3.5.4 Tolerances....................... .................................................. 23 3.5.5 Customizing Azimuths, Dip Angles, and/or Lag Distances... 23 3.6 Sample Variogram Estimators, Transforms, and Output Files.... 24 3.6.1 Title. ................................................................................. 24 3.6.2 Estimators: ........................................................................ 25 3.6.3 Cutoff................................................................................ 26 3.6.4 Log Transform................................................................... 26 3.6.5 % of Samples..................................................................... 26 3.6.6 Output Files....................................................................... 26 4.0 MODELING SAMPLE VARIOGRAMS ...................................... 27 4.1 An Introduction to Variogram Modeling Using SAGE2001. ..... 27 4.1.1 Variogram Model Parameters by Least Squares.................. 27 4.1.2 Rotation Conventions. ....................................................... 28 4.2 Setup Parameters...................................................................... 30 4.2.1 Input File........................................................................... 30 4.2.2 Output Files....................................................................... 30 4.2.3 Project Title....................................................................... 31 5
4.2.4 Excluding Sample Variogram points.................................. 31 4.2.5 Weighting Scheme............................................................. 31 4.2.6 Structure Types. ................................................................ 31 4.3 Rotation Conventions............. .................................................. 32 4.3.1 Standard Conventions.......................................................... 32 4.3.2 Custom Conventions..........................................................34 4.4 Calculating the Parameters of the Variogram Model. ................ 34 4.4.1 Letting SAGE2001 Calculate the Variogram Model. .......... 34 4.4.2 Locking the Nested Structure Rotations..............................34 4.4.3 Report Formats.................................................................. 36 4.4.4 Overriding SAGE2001....................................................... 36 4.4.5 Interactive Variogram Modeling. ....................................... 36 5.0 VIEWING AND PRINTING REPORTS................. .................... 39 5.1 Opening the Print Report Window............................................ 39 5.2 The Print Report Window......................................................... 39 5.2.1 Viewing the Report on Screen............................................ 40 5.2.2 Printing the Report.............................................................40 6.0 VIEWING AND PRINTING VARIOGRAMS........ .................... 41 6.1 Opening the Plot Window. ....................................................... 41 6.2 The Plot Window................................................. .................... 41 6.2.1 Viewing Sample Variograms One at a Time....................... 42 6.2.2 Viewing Multiple Directions in One Display...................... 42 6.2.3 Display Options................................................................. 42 6.2.4 Re-scaling the Plot Axes.................................................... 43 6.2.5 Printing the Plot................................................................. 43 6.2.6 Inserting the Plot into a Report........................................... 43 7.0 A WORKING EXAMPLE. .......................................................... 44 7.1 Importing the Data File. ........................................................... 44 7.1.1 Adding Column Headers.................................................... 44 7.1.2 Saving the SAG file.......................................................... 46 7.2 Calculating the Sample Variograms...................... .................... 46 7.2.1 The Select Variogram Attributes & Constraints Window.... 47 7.2.2 The Calculate Sample Variograms – Directions Window.... 48 7.2.3 Calculate Sample Variograms – Finish Window................. 50 7.3 Viewing the Sample Variograms. ............................................. 52 7.3.1 The Plot Window............................................................... 53 7.4 Viewing and Printing the Sample Variogram Report................. 55 7.4.1 The Print Report Window.................................................. 55 7.5 Modeling the Sample Variograms............................................. 56 7.5.1 The Model Sample Variograms – Setup Window............... 57 7.5.2 The Model Sample Variograms – Rotation Convention Window....................................................................................... 58 7.5.3 The Model Sample Variogram – Parameters Window.........59 7.5.4 The Parameters of the Variogram Model............................ 60 6
7.5.4 7.5.5 7.5.6 7.5.7 7.5.8
The Parameters of the Variogram Model............................ 61 Interactive Variogram Modeling. ....................................... 62 The Plot Window............................................................... 63 Viewing and Printing The Variogram Model Report........... 64 Additional Postscript Output Files...................................... 64
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1.0
INTRODUCTION.
SAGE2001 is a Windows 95/98/NT software toolkit designed for: 1. the calculation of directional sample variograms and, 2. the modeling of sample variograms. SAGE2001 is designed to work with many directional sample variograms at one time. Experience shows that at least 20 to 30 directional sample variograms are required to obtain a reasonably representative model of 3 dimensional spatial continuity. SAGE2001 enables the user to easily calculate 1 to 50 or more directional sample variograms in one run and then fit a model to all of the directional sample variograms simultaneously by regression. SAGE2001 makes the calculation of sample variograms easy, while providing the user with complete control over the calculations. • Either two or three-dimensional data may be processed. There is
• • • • • • • • • • •
no limit on theofsize of points the input data The selection data may be file. restricted to user specified limits on each coordinate (x,y,z) and/or to user specified limits on the variogram variable itself. The selection of data may also be restricted by specifying a correlated variable within user specified limits. The selection of data may also be restricted to specific rock types, alteration types, soil types, etc. The input data may be transformed using the log or indicator transform. Down-the-hole variograms can be calculated. Directional sample variograms can be fully defined by user specified azimuths, dip angles, tolerance angles, vertical bands, horizontal bands, lag distances, and lag tolerances Both omni-directional and omni-directional-horizontal (all directions are parallel to a plane, possibly dipping) may be calculated. Sample variogram estimators include the traditional average squared difference estimator; the correlogram; and the pair wise relative estimators. Smaller subsets of the input data may be selected randomly from large data files. Excellent report files and summary statistics are easily generated and viewed on screen or printed for future reference. Sample variograms can easily be viewed on screen, one at a time, or by displaying multiple directions on one screen. 8
Hardcopy of all graphic displays can be easily sent to the printer or clipboard. In addition to calculating directional sample variograms, SAGE2001 will also fit a variogram model to the calculated directional sample variograms by least squares or regression. For example, given 50 or more directional sample variograms, SAGE2001 can rapidly calculate the set of model parameters that minimize the average squared difference between the variogram model and all 50 directional sample variograms simultaneously. The regression or fitting of the variogram model to the directional sample variograms can be controlled by the user as follows: • The sample variogram(s) may srcinate from one, two, or three-dimensional data. • The variogram model may consist of one or two nested structures. • Each structure may be a spherical or exponential model. • A minimum number of pairs/sample variogram point may be specified. • The maximum lag distance may be specified. • Each sample variogram point may be weighted by the number of pairs. • Each directional sample variogram may be weighted by the inverse number of sample variogram points. • The user may select one of several popular conventions for the anisotropy rotation angles or define his own rotation conventions. Thus, the variogram models developed by SAGE2001 are compatible and may be used with any geostatistical software package. • Any or all of the variogram model parameters may be specified by the user. For example, the user may specify the nugget and the third rotation angle. SAGE2001 will leave the user specified parameters intact while it calculates values for the remaining unspecified parameters. The models of spatial continuity provided by SAGE2001 are generally far more accurate than those obtained using any other software package. This is because 30, 40, or 50 directional sample variograms provide a far better sample of spatial continuity than the 3 or 4 directions typically calculated and40, modeled by other software. Only simultaneously. SAGE2001 is capable of modeling 30, 50, or more sample variograms
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2.0 GETTING STARTED This section shows you how to set up the SAGE2001 software on your computer to begin calculating and modeling sample variograms.
2.1 Installing the Demonstration Versionof SAGE2001 The initial version of SAGE2001 that you install on your computer will be a demo version. The demo version is fully functional and can perform all of the functions that a fully licensed version can perform. You will be able to use the demo version of SAGE2001 for a period of 20 days from the date of installation with no obligations. SAGE2001 will write the demo expiration date to the main window so you will know the number of demo days remaining. To Install SAGE2001 on your computer use the program SETUP.EXE. This program will install the SAGE2001 executable and other files from the distribution diskettes to your hard disk. The distribution disk contains files in a compressed format. You cannot simply copy the files from the distribution disk to your hard disk and run the program. You must use the Setup program that decompresses and installs the files in the appropriate directories. The system requirements to run SAGE2001 on your computer are: • Any IBM compatible running Windows 95, 98, or NT. • A hard disk • A 3.5 inch floppy disk drive. To start Setup: • Place distribution disk No. 1 in the A: drive. • Click Start on the Windows taskbar, click Run, type A:SETUP in the textbox, then click OKAY. • Follow the Setup instructions on the screen. By default, the Setup program creates the C:\SAGE2001 directory. However, you can change the name of the directory during installation. The SAGE2001 icon will be installed in a new program group SAGE2001. To uninstall SAGE2001: • Click Start on the Windows taskbar, point to Programs, point to SAGE2001, and click Uninstall SAGE2001.
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2.2 Authorizing SAGE2001 for Unlimited Use. At the end of the 20 day demonstration period an authorization license will be required for further use of SAGE2001. The authorization license will provide unlimited use of SAGE2001 and can be validated any time during the 20 day demonstration period. To gain unlimited use of SAGE2001 follow these steps: 1.
Click on the Variographyà Unlock menu items as shown below.
2.
SAGE2001 will open the Unlock FormWindow as shown below.
3.
Use a pencil or pen to record the Code Entry Numberand Computer IDnumbers exactly as they are displayed. Store them in a safe place. Click Save Codes. This will save the codes so that they are
4.
compatible with the Authorization Code that will be returned to you. 11
5.
Then either email, telephone, or fax the Code Entry Numberand the Computer IDnumber to: • (email) ed&isaaks.com • (fax) (650) 558-9443 fax • (Telephone) (650) 558-9051 • or mail to: Isaaks & Co. rd 205 East 3 Ave., Suite 300 San Mateo, CA, 94401-4052 6. Isaaks & Co. will send you an Authorization Code upon receipt of payment. Re-open the Unlock Formas shown on the previous page and enter the Authorization Code in the text box. Click OKAY and the program will be authorized for unlimited use. For the most recent price list etc., contact Isaaks & Co. as described above or check the SAGE2001web pageat: http://www.isaaks.com Note that the complete version of SAGE2001 and this User’s Manual can also be downloaded from this web page.
2.3 Transferring the SAGE2001 Authorization License from One Computer to Another. The single user license authorizes SAGE2001 for use on one computer. However, the authorization license can be moved from the donor computer(this is the computer with the current license) to a recipient computer (this is the computer you are moving the license to) as described by the following steps: 1. Install a demo version of SAGE2001 on the recipient computer. 2. Start SAGE2001 on the recipient computerand click the License Transfer à Start Transfer menu items as shown below.
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SAGE2001 will ask you to insert a floppy diskette in the recipient computerdrive. SAGE2001 will then prepare this diskette for a license transfer. 3.
Start SAGE2001 on the donor computerand click the License Transferà Transfer This Licensemenu items as shown below.
4.
7.
Put the floppy diskette prepared in Step 3 into the donor computerdrive. SAGE2001 will then transfer the license to the prepared diskette. 5. Remove the diskette from thedonor computerand insert it back into the recipient computerdrive. Click the License Transferà Complete Transfermenu items on the recipient computeras shown below.
8. SAGE2001 will copy the license from the diskette to the recipient computer. That’s all there is to it!
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3.0 CALCULATING SAMPLE VARIOGRAMS 3.1 SAGE2001 Coordinate System and Definitions. SAGE2001 is based on the conventions of the CARTESIAN COORDINATE SYSTEM. For example, if the system of axes is oriented so that: • the X axis runs east/west with values increasing to the east, • the Y axis runs north/south with values increasing to the north, • then the Z axis will be vertical with values increasing upward. DIP ANGLES. A positive dip angle is measured upwards from the horizontal while a negative dip angle is measured downwards from the horizontal. AZIMUTH. The 0 or 360 degree azimuth lies parallel to the positive Y axis. (This is generally interpreted to be NORTH with EAST being parallel to the X axis. The 90 degree azimuth is parallel to the positive X axis). A useful conversion between the sine and cosine of the azimuth and the Cartesian coordinate system angle “α” or alpha (measured counter clockwise from the positive X axis towards the positive Y axis) is: sine(α) = cosine(azimuth) cosine(α) = sine(azimuth) This works for all angles α and azimuths. For those who would like a detailed explanation of spatial continuity and the mechanics of calculating sample variograms, the textbook, Isaaks and Srivastava , (1989) An Introduction to Applied Geostatistics, Oxford University Press, pages 44 to 66 is recommended. This reference will be particularly helpful in understanding the fundamental concepts of variography and terms such as separation vector, sample variogram estimator, correlogram, h-scatterplot, and lag statisticsetc.
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3.2 Importing Data Files. 3.2.1 The SAGE2001 File Format.
SAGE2001 requires data files to be in a special ascii format with the file names ending with the *.sag suffix. The SAG format contains 4 header records as shown by the following example. SAGE2001-001 6 0, 1, 1, 1, 1, 0 ID, East, North, Elevation, Cu, Zone, NVW,92475.00,104872.00,6800.00,2.473,20.000 NVW,92475.00,104872.00,6758.00,.981,5.000 SVW,92402.00,104802.00,6840.00,1.015,2.000 The first record contains the version number. The second record contains the number of columns in the data set. The third record contains an indicator for each column where 0 (zero) indicates a categorical variable, while 1 (one) indicates a continuous variable. The fourth record contains the header labels for each column. All columns must be delimited by commas in a SAG file. The import facility in SAGE2001 makes it very easy to create a SAG file.
Figure1: Clicking items to open the import file dialogue box. 15
3.2.2 Importing the Data File.
If your file is not in the SAG format, you must IMPORT your data file into SAGE2001. To import a data file, click the Variography→ Sample Variograms→ Import Filemenu items as shown in Figure 1.
Figure 2: The standard open file dialogue box. SAGE2001 will then open a standard Open File Dialogue Box as shown in Figure 2. Click the file you wish to import and then click Open. 3.2.3 Import File Delimiters.
SAGE2001 will import ASCII files in FREE format with various types of delimiters. For example, the srcinal data file may be delimited as follows: • Space delimited • Comma delimited • Tab delimited • Strings may or may not be enclosed within double quotation marks 3.2.4 The Import File Grid.
SAGE2001 will respond to a click on the OPEN button of the Open File dialogue box (Figure 2) by reading the import file and writing the first 5 16
records to a File Import Gridshown in Figure 3. The number of records read will be shown on the upper right corner of the grid.
Figure 3: Adding column headers using the Import File Grid.
3.2.5 Adding Column Headers.
Column headers may be added to each column of the import file by selecting Use row 6 for column headers in the Column Headersframe. Then type the name or column header for each column in row 6 of the grid. Figure 3 shows a partially finished example where the column headers Sequence, BHID, East,and Northhave been typed by the user. Note that Row 1 may be used as column headers by default. 3.2.6 Setting the Attribute Type.
Each column of data in a SAGE2001 data file must designated as either a categorical or numerical type. SAGE2001 will initially classify each column. However, the user may change the type by clicking on the drop down list box in row 7. Columns containing variables such as Drill Hole ID, Rock Type, Alteration Zone, Mineralogical Zone etc., should be set to Categorical. This will enable subsequent variogram calculations to be restricted to specific categories.
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Finally, click the NEXT button, and SAGE2001 will create a new file. The new file will be named with the same prefix as the import file, but will have the SAG suffix appended. For example, the imported file robson.dat(Figure 2) would be written to disk as robson.sag.
3.3 Opening Existing SAGE2001 Files. To open an existing SAGE2001 data file (*.sag), click the Variography → Sample Variograms→ Open Filemenu items. SAGE2001 will
Figure 4: Opening an existing SAGE2001 data file (*.sag). respond by opening up a standard Open File dialogue box. However, only files with the suffix sag will be listed. Click the name of the sag file you wish to open and then click the Open button. SAGE2001 will respond by reading the selected file and report whether or not the file has been read successfully. The program then advises the user to click the Next button to continue.
3.4 Selecting Data Attributes and Constraints. 3.4.1 Selecting Attributes for the X,Y, and Z Coordinates.
•
X-Coordinate.Click the drop-down-list-box arrow as shown in Figure 5. Select the appropriate attribute from the drop down list by clicking on it. For example, the drop-down-list shown in Figure 5 contains the attributes East, North, Elevation, and Cu. The attribute 18
• •
East has been selected from the list. SAGE2001 will respond by displaying the maximum and minimum values of the attribute in the text boxes to the right. A new maximum and/or minimum may be specified by typing over the displayed values. Y-Coordinate.Similarly, select the appropriate attribute from the drop down list. Specify a new maximum or minimum if required. Z-Coordinate.Similarly, select the appropriate attribute from the drop down list. Specify a new maximum or minimum if required.
Figure 5: Selecting variables and constraints for sample variogram calculations. 3.4.2 Selecting the Sample Variogram Attribute.
•
Attribute#1.Select the appropriate attribute or variable from the drop down list. Specify a new maximum or minimum if required. This is the attribute for which sample variograms will be calculated.
3.4.3 Calculating aDown-The -Hol e Sample Variogram.
•
Bore Hole ID. Specifying a proper categorical attribute for Bore Hole ID will cause SAGE2001 to calculate adown-the-holesample
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variogram. Down-the-hole sample variograms are calculated in a special way: • All tolerance angles are ignored. • All bandwidths are ignored. • Only samples with the same Bore Hole IDare paired. • Variogram values from different Bore Holes but with the same lag distance are averaged together. The result is one sample variogram where the lag spacing is generally set equal to the drill hole composite length. The directions of the down-the-hole sample variogram will correspond to a mixture of the various bore hole directions. However, if all the Bore Hole samples are in the same direction, then the sample variogram can be considered as a directional sample variogram in the same direction as the bore holes. The BHID ID attribute must be a categorical variable whose values are identification labels unique to each bore hole. 3.4.4 Specifying Additional Constraints:
•
Attribute#2.Additional constraints may be added to the selection of Attribute#1 by specifying a second attribute from the Attribute#2 drop-down-list-box . Note that Attribute#2 is a continuous variable. Specify the maximum and/or minimum constraints for Attribute#2 in the text boxes to the right. For example, suppose the input data contains both a copper and arsenic value for each sample. The user wishes to calculate a sample variogram of copper (selected from the Attribute#1 drop-down-list-box) but wishes to include only those copper values with an associated arsenic value greater than 0.05. To do this the user would select arsenic from the Attribute#2 dropdown-list-box , and set the corresponding minimum to 0.05. • Attribute#3.This attribute enables the restriction of Attribute#1 values to specific categorical attributes such as rock type, soil type, alteration type, etc. For example, in Figure 5, Lithology is a categorical variable that has been selected from the Attribute#3 drop-down-list-box. SAGE2001 has automatically displayed all the Lithology categoriesin the scrollable list box to the right. The user has selected Lithologies 31 and 34 from the list of zones. This will restrict the selection of Attribute#1 values to Lithologies 31 and 34 for the purposes of calculating sample variograms. Clicking the Next button will open the next window for specifying the directional parameters.
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3.5 Azimuths, Dips, Lags, and Tolerances.
Figure 6: Specifying Azimuths, Dips, Lags, and other tolerances for the com utation of directional sam le vario rams. SAGE2001 is designed to calculate sample variograms in many directions simultaneously. To facilitate the entry of the many directional parameters, SAGE2001 requires the user to enter only the angular increments and one global unit lag distance. For example, a directional increment of 45 degrees specified for the azimuth will cause SAGE2001 to calculate sample variograms in the directions given by azimuths 0, 45, 90, 135, 180, 225,270, and 315 degrees. 21
However, if the user wishes to either specify or modify specific azimuths, dips, or lags, he can do so using the grid shown at the bottom of Figure 6. The following text provides more details on how to respond to the text boxes shown in Figure 6. 3.5.1 Directional Increments.
•
•
Azimuth.Select the azimuth increment from the drop-down-list box. For dip angles of 0 degrees, the sample variogram at an azimuth of X degrees is identical to it’s compliment at X+180 degrees. Although redundant, these sample variograms are retained since they are useful for the SAGE2001 modeling algorithm. Note that for dip angles different from zero, sample variograms with an azimuth of X are not symmetric to those with an azimuth of X+180 degrees and the same dip angle. Dip Angle.Select the dip angle increment from the drop-down-listbox. The total number of directional sample variograms is determined by the number of azimuth directions times the number of dip directions except for the –90 degree dip. Only one directional sample variogram is calculated for the –90 dip. For example, an azimuth increment of 30 degrees combined with a dip increment of 30 degrees will provide a total of 37 directional sample variograms
3.5.2 Lag Specifications.
• •
Unit Distance.Type in the unit lag distance appropriate for the data set at hand. This unit distance is initially applied to all directions. Max. No. of Lags.Type in the maximum number of lags to calculate each directional sample variogram. The magnitude of the largest separation vector between two sample variogram points is given by the Max. No. of Lags times the Unit Distance.
3.5.3 Bandwidths.
The horizontal and vertical bandwidths control the horizontal and vertical tolerances allowed in the pairing of sample variogram points. Figure 7 shows the definition of the band thickness. The thick black arrow represents the separation vector. A sample variogram point at the tail of the separation vector can only be paired with sample variogram points located within the vertical and horizontal bands. Note that thehorizontal bands are not actually horizontal unless the separation vector is horizontal. More accurately, the horizontal bands always have the same dip angle and dip direction as the separation vector. Thus, they are always parallel to the separation vector. This includes dips between 90 and –90 degrees inclusive. 22
3.5.4 Tolerances.
•
Lag. Type in the required lag tolerance. The tolerance must be specified as a fraction of the unit lag distance and must be between
Figure 7: The vertical and horizontal bandthickness.
•
0.0 and 0.50. This lag will be initially applied to all directions. Angular. Type in the required angular tolerance in degrees. For example, Figure 6 shows an angular tolerance of ± 22.5 degrees. The tolerance angle is applied both horizontally and vertically forming a three dimensional cone whose axis is coincident with the separation vector.
3.5.5 Customizing Azimuths, Dip Angles, and/or Lag Distances.
SAGE2001 automatically prepares a grid or spreadsheet and populates it with the actual azimuths, dip angles and lag distances calculated from the information entered in the Directional Increment and Lag Specificationstext boxes. For example, the top row (shaded gray) of the spreadsheet at the bottom of Figure 6 shows the azimuths (0, 30, 60, 90, 120, etc.) of the directional sample variograms. Similarly, the first column (also shaded gray) shows the dip angles (0, -45, -90) for the directional sample variograms. The interior entries in the spreadsheet are the unit lag distances for each directional sample variogram. For example, Figure 6 shows a lag of 65 units (a user edit) for the directional sample 23
variogram with an azimuth of 60 degrees and a dip of –45 degrees. Note that the user may edit each or all of the azimuths, dips, and unit lag distances shown in the spreadsheet. This provides the user with total control over the specification of azimuth, dip, and lag distance for each directional sample variogram.
3.6 Sample Variogram Estimators, Transforms, and Output Files.
Figure 8: The final screen requiring parameters for the calculation o sample variograms.
3.6.1 Title.
Type in the name of the data or project name. This title will appear in the reports and plots.
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3.6.2 Estimators:
Click on the drop-down-list-box and select one of the following sample variogram estimators:
•
Standardized Traditional Estimator.The equation
of the
standardized traditional estimator is given by: 2
N (h)
1
∑ [Z 2N(h)
=
γ (h)
i
− Z i+ h ] / σ Z2
(1)
i =1
where h is the separation vector, Z i is the sample value at location
i , Z i + h is at location i + h which is separated from location i by 2
the vector h , and σ Z is the global variance of Z .
•
The Pairwise Relative Estimator. γ (h )
N (h)
1
=
[ Z i − Z i + h ]2 /[(Z i + Z i + h ) / 2]2
2 N ( h)
•
(2)
∑ i =1
The C orrelogram Est ima tor. γ (h ) = 1.0
-
N( h)
1 N (h)
∑ (Z ∗ Z i
i +h
+ m ( −h ) ∗ m ( + h ) ) / σ ( − h )σ ( + h )
i =1
(3) where the tail mean
m( − h ) is given by: N ( h)
m(− h ) = and the head mean
1 N (h)
∑Z
i
(4)
i =1
m( + h ) is given by: N (h)
m( + h ) =
1 N (h)
∑Z
i+ h
(5)
i =1
and σ ( −h ) is the lag standard deviation of Z i , and σ ( + h ) is the lag standard deviation of Z i + h , i = 1, N (h) .
•
The Indicator Estimator . This estimator has exactly the same form as the standardized traditional estimator given above except that the
variable
Z i has been transformed to an indicator variable as follows: 25
1 if Z i ≤ cutoff Ii = 0 otherwise
i = 1, N
(6)
3.6.3 Cutoff.
Specify the required cutoff to define the indicator transform as defined in the equation above. 3.6.4 Log Transform..
.Click this check box if you wish to compute a sample variogram of Y using the transform Yi = ln( Z i ) i = 1, N 3.6.5 % of Samples.
This box can be used to select a random subset of the input data. For example, to select all of the samples, set the % of samples to 100. A value of 50 will select approximately 50 percent of the available sample data. Note that the samples are selected randomly. This option is useful for very large data sets such as blast holes that may contain as many as 300,000 or more records. 3.6.6 Output Files.
SAGE2001 writes 3 files summarizing the calculated sample variograms. These are: • Report. The report file written by SAGE2001 is an ASCII file with the suffix rpt. This file is an excellent documentation file that summarizes both the input parameters and the output lag statistics. This file may be easily printed and filed if a permanent record of the sample variogram calculations is required • Autofit. This file is written by SAGE2001 for the purposes of modeling the sample variograms. It is an ASCII file ending with the suffix pli and contains sample variogram data. The file is read by SAGE2001 at the time of modeling. • Plot. This is also an ASCII file written by SAGE2001. The file is used to display the sample variograms on the computer screen. The file suffix is plt. The user must supply file names for all three files before SAGE2001 will calculate the sample variograms. Click the Finish button to begin the sample variogram calculations.
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4.0 MODELING SAMPLE VARIOGRAMS 4.1 An Introduction to Variogram Modeling Using SAGE2001. 4.1.1 Variogram Model Parameters by Least Squares.
In addition to calculating directional sample variograms, SAGE2001 can also fit a variogram model to the directional sample variograms by least squares or regression. For example, given 37 or more directional sample variograms, SAGE2001 can rapidly calculate the set of variogram model parameters that minimize the average squared difference between the variogram model and all 37 directional sample variograms simultaneously. The parameters determined by SAGE2001 are currently limited to the following set: 1. 2.
C0 -- the nugget C1 – the coefficient of the first nested structure.
3.
RX 1 -- the range of the first structure in the direction of the rotated X axis.
4.
RY 1 -- the range of the first structure in the direction of the rotated Y axis.
5.
RZ 1 -- the range of the first structure in the direction of the rotated Z axis.
6.
AZ 1 -- the rotation angle of the first structure around the Z axis.
7.
AX 1 -- the rotation angle of the first structure around the rotated X axis.
8.
AY 1 -- the rotation angle of the first structure around the rotated Y axis. C2 – the coefficient of the second nested structure.
9.
'
'
'
'
'
'
10. RX 2 -- the range of the second structure in the direction of the rotated X axis. '
11. RY 2 -- the range of the second structure in the direction of the rotated Y axis. '
12. RZ 2 -- the range of the second structure in the direction of the rotated Z axis.
27
13. AZ 2 -- the rotation angle of the second structure around the Z axis. '
14. AX 2 -- the rotation angle of the second structure around the rotated X axis. '
15. AY 2 -- the rotation angle of the second structure around the rotated Y axis. Note that the rotation angles for the second structure are generally very different from those of the first structure. Thus, the maximum number of unique variogram model parameters determined by SAGE2001 is 15. However, any subset of these parameters may be specified by the user in which case SAGE2001 will calculate values for the remaining parameters not specified by the user (if they exist). Thus, SAGE2001 provides the user with a broad range of control over the modeling process from full user control to letting SAGE2001 do all the work. 4.1.2 Rotation Conventions.
Unfortunately, coordinate transformations by rotation are not unique. For example, if one wishes to rotate a coordinate system of axes to a new predetermined position, the rotations required to get there can be defined many different ways. For example, if one requires a rotation around the Z axis so that the final rotated Y axis is at an azimuth of 20 degrees measured in the non-rotated system, then the required rotation can be specified as either: • a left1 hand rotation of 20 degrees, • or, a right hand rotation of 340 degrees. Either one of these rotation algorithms will provide the correct result.
1
The right and left hand rules are unambiguous, simple to understand, and easily applied. For example, to rotate around the Z axisaccording to the right hand rule, imagine grabbing the Z axis with the right hand so that the thumb is pointing in the direction of increasing Zvalues. Then the fingers will be pointing in the direction of a positive right hand rotation. A negative right hand rotation angle indicates a rotation in the opposite direction. Note that a positive right hand rotation angle is the same direction as a negative left hand rotation angle around the same axis.
28
Rotations are a frequently the source of trouble. Each kriging and simulation program contains a specific rotation algorithm, probably defined by the author of the computer code. For example, one kriging program may be written so that a positive rotation angle around the Z axis is defined as left hand A second kriging program may define a positive rotation angle around the Z axis as right hand. Now, in order for a kriging program to calculate variogram values correctly, the variogram model must be defined with the identical rotation algorithm hardwired in the kriging program. For example, consider a kriging program written so that the rotation around the Z axis isright hand. Next, consider a variogram model where the rotation around the Z axis is defined as +20 degrees left. If one simply specifies the rotation around the Z axis in the kriging plan as +20 degrees, the kriged estimates will be wrong. In this example, it is easy to make the correction mentally and specify the correct rotation angle as 340 or -20 degrees to get the correct estimates. However, if the variogram rotation algorithm is more complicated, it may be next to impossible to calculate the correction. Remember the rotation algorithm will likely consist of rotations around all three axes and the order in which the rotations are applied to the axes is not interchangeable. For example, an initial rotation around the Z axis followed by a rotation around the rotated X axis followed by a rotation around the rotated Y axis is not the same rotation algorithm as an initial rotation around the Z axis followed by a rotation around the rotated Y axis followed by a rotation around the rotated X axis. Thus, you must know the rotation algorithm of your software in order to use SAGE2001 correctly. This is because SAGE2001 provides many rotation algorithms in an attempt to provide variogram models compatible with all kriging and simulation packages. Thus, the onus is on you to determine the correct rotation algorithm for your geostatistical software.
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4.2 Setup Parameters
Figure 9: Setting up SAGE2001 for modeling sample variograms.
4.2.1 Input File.
Click the Sample Variogrambutton to open a standard File/Open dialogue box. Only files with the extension .pli will be shown in the box. Select the file name and then click the Open button to open the file. 4.2.2 Output Files.
• •
Plot. Click the Plot button and type in a name for the plot file. This file is used by SAGE2001 to plot the sample variograms and the model on the computer screen. Report.Click the Report button and type in a file name for the report file. This file contains a detailed summary of the input parameters and the output variogram model. The azimuths and dip angles are
also provided for the anisotropy axes of each structure. This report is an excellent summary that can be either viewed on screen or printed 30
and kept as a permanent record of the variogram model and the input parameters to SAGE2001. 4.2.3 Project Title.
This text box will contain the project or report title specified previously in the sample variogram screen. The user has the option to change the title by typing in a new one. 4.2.4 Excluding Sample Variogram points.
•
•
Minimum # pairs.Type in the minimum number of pairs each sample variogram point must have to be considered in the modeling process. In other words, any sample variogram point with fewer pairs than the value specified here will not be considered in determining the variogram model parameters. Maximum Drift.The drift is defined as the difference between the head mean and the tail mean (see Equations 4 and 5). Any sample variogram point with an absolute drift value exceeding the value
specified here will not be considered in determining the variogram model parameters.. 4.2.5 Weighting Scheme.
Select the weighting scheme from the drop-down-list box. The options are: • No weighting. • Weighting by number of pairs. • Weighting by the inverse number of sample variogram points per direction. SAGE2001 models all of the directional sample variograms simultaneously. Thus, this option may be useful if some of the directional sample variograms contain many more sample points than others. 4.2.6 Structure Types.
SAGE2001 will model the sample variograms using either one or two nested structures. Click the appropriate option boxto select either one or two structures. Select each Structure Typefrom the drop-down-list-box. If two structures are required, then two structure types must be selected. The structure types are: • The spherical model. • The exponential modelwith practical range P : -3h/P
•
γ (h) 1.0 = − e The exponential modelwith the traditional range T :
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(7)
γ ( h) = 1.0 −
e -h/T
(8)
Note that 3T = P Also, note that the first edition of GSLIB (Deutsch and Journel) defined the exponential model given by (8), but the second edition, defines the exponential model given by (7).
4.3 Rotation Conventions.
Figure 10: Setting up the rotation conventions for the variogram model. This screen enables the user to specify a rotation algorithm for the variogram model so that the model can be used with external kriging and simulation software such as GSLIB etc. 4.3.1 Standard Conventions.
The user may select one of four different standard rotation conventions or rotation algorithms2, or alternatively the user may select the Custom
•
2
An abbreviated version of the rotation algorithm is provided by the two groups of 3 letters each within the brackets shown in Figure 10, i.e. (ZYX, RRR). The order of the rotations is denoted by the first 32
option button and define his own custom rotation algorithm.. The standard rotation conventions are as f ollows: • SAGE2001 (ZYX, RRR). This rotation algorithm provides a variogram model that is compatible with the SAGE2001 kriging and simulation software. • MEDSYSTEM. (ZXY, LRL)This rotation algorithm will provide a variogram model compatible with the Medsystem geostatistical software. However, the Medsystem also provides several rotation algorithms from which the user must select one. One of these options is called the GSLIB option. Unfortunately, this option has been mislabeled or it has been implemented incorrectly. The Medsystem implementation is (ZXY, LRL) while GSLIB is actually (ZXY, LRR). Thus, to correctly use SAGE2001 variogram models in the MEDSYSTEM geostatistical software, select the G SLIB option in the Medsystem software and theMEDSYSTEM (ZXY, LRL) option in the SAGE2001 software. • Vulcan by Maptek. (ZXY, LRL). This rotation algorithm will provide a variogram model compatible with the Vulcan geostatistical software. Maptek has implemented GSLIB in their software package. However, they made a slight change to theGSLIB rotation algorithm by changing the direction of the rotation around the third rotation axis. • GSLIB (ZXY, LRR) This rotation algorithm will provide a variogram model compatible with the GSLIBgeostatistical software. Many who read this may think that the rotation algorithm forGSLIB is (ZXY, LRL). We agree that the written GSLIB documentation certainly appears to suggest the rotation algorithm is (ZXY, LRL). However, the documentation is not clear and it turns out that all GSLIB software actually executes the (ZXY, LRR) rotation algorithm. This appears to be an oversight in GSLIB.
three letters in the brackets, for example ZYX. The first letter indicates the first rotation axis, the second letter indicates the second rotation axis and the third letter, the third rotation axis. The second group of 3 letters, for example LRL indicates the rotation directions. The first letter indicates either a left L or right R hand rotation around the first rotation axis. The second letter indicates either a left L or right R hand rotation around the second rotation axis. The third letter indicates either a left L or right R hand rotation around the third rotation axis. 33
4.3.2 Custom Conventions.
Clicking this option button will enable the user to design his own rotation algorithm with one limitation --- the first rotation must be around the Z axis. However, the direction may be either left or right as the option buttons show. The option buttons within the User defined convention frame are programmed in such a way that the user cannot select an invalid combination of axes. For example, it is impossible to select a (ZXX) combination. Note however, that the rotation algorithms(ZXZ) or (ZYZ) are valid combinations. In fact, if you think about it you will realize that these rotations are more geologically intuitive than the older more familiar rotation algorithms. Finally, any combination of left and right rotation directions may be selected for the three rotations.
4.4 Calculating the Parameters of the Variogram Model. Figure 11 shows the last screen controlling the variogram modeling
process. Initially, The parameter text boxes will be blank on the screen shown in Figure 11 except for the Sill, which will be set to 1.0 by default. Besides Back, Help, and Exit, the user must choose from the following options: 4.4.1 Letting SAGE2001 Calculate the Variogram Model.
Click on Fit, which will cause SAGE2001 to go to work and calculate the variogram model parameters. This may take a few minutes, however, the program will open up and display a progress bar. Note that sometimes the progress bar will appear to be stalled. However, it will likely continue to move at some point, so do not click on CANCEL unless you really want to exit the program. When SAGE2001 has determined the variogram model parameters, they will be written to the screen in the appropriate text boxes. The azimuth and dip (measured in the srcinal data coordinate system) of the rotated anisotropy axes for each structure are also written to the screen in the Output Anisotropy Axes frame. The total sum of squared differences or error SSE is also written to the screen. Note that all this information is also written to the report files (*.r pt) and plot (*.plt) files named earlier. 4.4.2 Locking the Nested Structure Rotations.
A click on the Lock Rotationscheckbox will provide identical rotation angles for each nested structure. In other words, the anisotropy axes of each nested structure will be oriented identically. For example, the 34
Figure 11: The Variogram Model Parameters. mining software package DATAMINE requires identical rotation angles for all nested structures. By default, SAGE2001 models each structure with a unique orientation for the anisotropy axes. This generally enables a better fit to the sample variogram points since the extra freedom gained by allowing individual orientations provides more degrees of freedom in fitting the model to the sample variogram points.
35
4.4.3 Report Formats.
The format controlling how the variogram model will appear in the various plots and reports can be specified in the Report Formatstext boxes: • C0, C1 Decimals type the number of decimal places required for the nugget and coefficients C1 and C2. • Range Decimals type the number of decimal places for the variogram ranges. • Max Lag Distance type the maximum distance for the separation vector h. This will be the maximum length of the X axis or h axis for all directional variogram plots. Note that SAGE2001 will not use or consider any data with a lag distance larger than Max Lag Distance. 4.4.4 Overriding SAGE2001.
The user may override the calculation of variogram model parameters by SAGE2001 by typing the model parameter in the appropriatetext box(s) and clicking on the fix checkbox(s)immediately to the right of the text box(s). If a fix checkbox is checked, SAGE2001 will leave that parameter intact while it calculates values for any unchecked text boxes. For example, if the user types the value 0.30 in the Nugget text box, clicks the adjacent fix checkboxfollowed by a click onFit, SAGE2001 will calculate values for the unchecked parameters, but will not change the Nugget. If the user specifies all of the parameters and checks all the fix checkboxes and then clicks Fit, SAGE2001 will write the model parameters to the output files and return the identical values to the screen. 4.4.5 Interactive Variogram Modeling.
After the user has clicked Fit and the variogram model calculations are complete, SAGE2001 will automatically open a plot window and begin plotting the directional sample variograms and the model as shown in Figure 12. Note that the plot window opened by SAGE2001 may occupy the full computer screen. If this happens, the window showing the variogram model parameters (the left window in Figure 12) will be hidden by the plot window. However, the hidden window can be exposed by resizing the plot window as shown in Figure 12. Simply click and drag the sides or corners of the plot window to the required size. SAGE2001 will remember the size of the plot window the next time it is opened. For an explanation of the Plot Windowsee Section 6.1and Figures 14 and 15.
36
Figure 12: The Setup for Interactive Variogram Modeling.
SAGE2001 makes it easy for the user to interactively test various variogram model parameters and model by trial and error. For example, The right window in Figure 11 enables the user to view the current model and judge how well it fits the sample variogram points. The user can then type in new parameters in the appropriate text boxes and click the corresponding fix check boxes. SAGE2001 will then update all unchecked parameters and provide a new variogram model with a simple click on the fit button. Note that the plot window shown on the right side of Figure 11 must be closed before SAGE2001 will accept user input in the left window shown in Figure 11. Viewing the directional variograms corresponding to the anisotropy axes. The azimuth and dips of the anisotropy axes for each structure of the model are written to the Variogram Parameter Screen (Figure 11). For example, the azimuth and dips of the anisotropy axes of the second structure shown in Figure 11 are: • Rotated X, Azimuth = 60, Dip = 24 • Rotated Y, Azimuth = 328, Dip = 4
37
• Rotated Z, Azimuth = 228, Dip = 66 The ranges along each of these axes is also provided by SAGE2001 (Figure 11): • Range along rotated X = 329.153 • Range along rotated Y = 997.152 • Range along rotated Z = 118.810 The directional variogram models corresponding to these directions (or nearly so) can be viewed simultaneously in the plot window by clicking on the Select multiple setsoption button on the plot window (seeSection 6.2.2). Make sure the correct directional variograms are selected. For example, the correct directional variograms corresponding to the azimuths and dips given above are: • The directional variogram corresponding to an azimuth of 60 and dip 24 is Azimuth=240, Dip=-30.This is because a positive dip is upward and a negative dip is downward. Thus, the direction AZ=240, Dip=-30 is identical to the direction Az=60, Dip=+30. Since this example does not provide a variogram plot in the direction Az=60, Dip=+24, the closest direction is Az=240, Dip=-30. • The closest directional variogram corresponding to the direction Az=328, Dip=+4 is Az=328, Dip=0. • The closest directional variogram corresponding to the direction Az=228, Dip=+66 is Az=60, Dip=-60.
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5.0
VIEWING AND PRINTING REPORTS.
5.1 Opening the Print Report Window. Click on the Variography→Print Report menu items. SAGE2001 will open the File Open dialogue box. Only files with the suffix rpt will be shown in the File Open dialogue box. Click on the report file you wish to open. Then click on the Open button. SAGE2001 will respond by formatting the report file and opening the Print Reportwindow.
5.2 The Print Report Window.
Figure 13: The Print Report window.
39
The Print Reportwindow enables the user to view the report on screen or to print the report. Reports may be printed one page at a time or by all pages at once. 5.2.1 Viewing the Report on Screen.
To review the report on screen, Click the Windows maximizebutton in the extreme upper right corner of the window. This will maximize the window within your computer screen. To increase the size of the report page shown in the window, select the magnification factorfrom the Zoom drop-down-list-box, or type in the required magnification factor directly. A new page can be selected by clicking the tiny arrow heads at each end of the horizontal slider bar. Clicking the left arrow displays the previous page in the report while a click on the right arrow head displays the next page. 5.2.2 Printing the Report. Current To the current click the Print the print complete report, page, click the Print all Pages button.Pagebutton. To print
40
6.0
VIEWING AND PRINTING VARIOGRAMS
6.1 Opening the Plot Window. Click on the Variography→ Display Variogramsmenu items. SAGE2001 will open the File Open dialogue box. Only files with the suffix plt will be shown in the File Open dialogue box. Click on the plot file you wish to open. Then click the Open button. SAGE2001 will respond by opening the Plot window and displaying the sample variograms.
Figure 14: The variogram plot window.
6.2 The Plot Window. 41
6.2.1 Viewing Sample Variograms One at a Time.
When SAGE2001 opens the plot window, the program will immediately begin to display all of the sample variograms automatically. It does this by rotating through all of the directional sample variograms automatically. This feature may be turned on and off by clicking the Auto On and Auto Off buttons. When the Auto feature is Off, the previous or next directional sample variogram may be manually displayed in the plot window by clicking the Next or Previousbutton. 6.2.2 Viewing Multiple Directions in One Display.
To view multiple directional sample variograms simultaneously in one plot , click the Select multiple setsoption button. The scrollable list box immediately below the option button lists the directional sample variograms that may be viewed. Amaximum of 5 directions may be viewed at one time. Select the directions by clicking on them. Then display the directional variograms in the plot window by clicking the Graph button. Figure 15: A blow up of the Plot controls shown in Figure 14.
6.2.3 Display Options.
Additional information may be plotted simultaneously with the directional sample variograms. For example: • Show Pairs.Click this checkbox to show the number of pairs associated with each sample variogram point. • Connecting lines. Click this checkbox to connect the sample
•
variogram points with straight lines. Variogram Model.Click this checkbox to display the variogram model in the corresponding direction. Note that if a variogram model 42
has not been fit to the directional sample variograms, this option will not be available. 6.2.4 Re-scaling the Plot Axes.
The axes of the sample variogram plot may be re-scaled by typing in new maximums and minimums in the X (lag distance) and Y (variogram value) text boxes . Note that values for the X and Y tic-intervalsmust also be typed in the appropriate text boxes. Click the Apply button to re draw the plot with the new maximums and minimums. 6.2.5 Printing the Plot.
Click the Print button to print the current display in the Plot window. If the display contains multiple plots, the various directions will be displayed using different colors. The printer output will also be colored if the printer is color capable. 6.2.6 Inserting the Plot into a Report.
The graphics display shown in the Plot window can easily be inserted into a report document using the Windows clipboard. • Click the Copy button located between thePrint and Exit buttons. This will put a copy of the sample variogram display on the Windows clipboard. • Open your report document and position the cursor at the location in the document where you wish to insert the variogram figure. • Press the Ctrl and V keys simultaneously. This will copy the graphics display from the Windows clipboard to your report. It’s that easy! NOTE, If the image pasted into your document from the clipboard appears truncated and some of the text such as axes labels, title etc., are missing, re-install the printer driver for your default Windows printer. This may seem totally unrelated, but it will probably fix the problem. Once you have re-installed the default printer driver, the clipboard image will copy exactly as it appears on the SAGE2001 screen. This is because the control used by SAGE2001 to copy the image to the clipboard use the Windows printer driver. The re-installation of the printer driver with SAGE2001 loaded on your machine generally resolves any problems between the driver and SAGE2001.
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7.0 A WORKING EXAMPLE. The following sections contain an example using SAGE2001 to calculate and model sample variograms. The data set is called TEST.TXTand is included with the program software. You will find this file in the same directory where SAGE2001 is installed. The data are 3 dimensional and srcinate from a mineral deposit. This example is meant to provide an illustration of how SAGE2001 can be used to calculate and model sample variograms.
7.1 Importing the Data File. The srcinal data file TEST.TXT is an ASCII text file that is space delimited and contains 10 fields of data as shown by the following 5 records: 1 DDH-01
6324.4
5435.4
5995.1
317.0
7.5
0.744
20
-1.00
1 DDH-01
6324.4
5435.4
5987.6
324.5
7.5
0.731
20
-1.00
1 DDH-01
6324.4
5435.4
5980.1
332.0
7.5
0.726
20
-1.00
1 DDH-01
6324.4
5435.4
5972.6
339.5
7.5
0.830
20
-1.00
1 DDH-01
6324.4
5435.4
5965.1
347.0
7.5
0.734
20
-1.00
7.1.1 Adding Column Headers
The first step is to import the file TEST.TXTand add column headers. To do this click on the Variography Sample VariogramsImport File drop down menu items as shown in the following Figure
SAGE2001 will respond by opening the standard Windows Open File Dialogue box. Select the fileTEST.TXT and click Open. SAGE2001 will respond by reading the file TEST.TXT and automatically open the Import File Grid shown in the next Figure.
44
At the top of the grid, SAGE2001 tells you it is displaying the first 5 rows of 5,072 data. Note that the grid has been scrolled horizontally so that columns 5 to 10 are displayed in the figure. Type and/or click the following buttons: 1. Click on the Use row 6 for column headers option button in the Column Headersframe. 2. Type the column header “Sequence” in row 6 column 1. 3. Type the column header “BHID” for Bore Hole ID in row 6, column 2. Note that SAGE2001 has automatically classified this variable as categoricalbecause it contains letters. 4. Type the column header “Easting” in row 6, column 3. 5. Type the column header “Northing” in row 6, column 4. 6. Type the column header “Elevation” in row 6, column 5. 7. Type the column header “Depth” in row 6, column 6. 8. Type the column header “Length” in row 6, column 7. 9. Type the column header “Copper” in row 6, column 8. 10. Type the column header “Lithology” in row 6, column 9. Note that all of the lithology codes are numeric. Thus, SAGE2001 was not able to classify this variable as categorical. Since lithology should be classified as categorical, click on the drop down list box in row 7, column 9 and click categorical. This will classify the variable as a categorical 11. Type thelithology column header “Density”variable. in row 6, column 10.
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7.1.2
Saving the SAG file.
Finally, click the Next button. Windows will respond by opening the standard Windows Save File Dialogue Box. Type in the file name TEST and then click on the SAVE button. This will create and save a file called TEST.SAG in the required SAGE2001 format. That’s all there is to importing data files.
7.2 Calculating the Sample Variograms. When the Next button is clicked on the previous Import File Grid Window, SAGE2001 responded by automatically opening the Select Variogram Attributes & Constraints Window. However, if you are at the opening window of SAGE2001 and a sag file already exists on your hard disk drive, you can open it by clicking on the Variography Sample Variograms Open File drop down menu as shown in the following figure.
SAGE2001 will respond with the standard Open File Dialogue box. Select the required sag file and click on Open. SAGE2001 will respond by opening the Select Variogram Attributes & Constraints window.
46
7.2.1 The Select Variogram Attributes & Constraints Window.
This window enables the user to select the variables and constraints to use in calculating the sample variograms.
Type or click the follow to continue with the example exercise: 1. SAGE2001 automatically selected the Easting, Northing, and Elevationvariables for the X, Y, and Z Coordinate text boxes, so there are no coordinate variables to select. 2. Click on the drop down menu list for Attribute#1 and select Copper. This is the variable for which sample variograms will be calculated. Type0.0 in the Minimumtext box. This will eliminate all copper values less than 0.0. 3. Click on the drop down menu list for Attribute#2 and select Length. Then type 2.0 in the Minimum text box. Length is a variable representing the length of the diamond drill core for each copper assay. Thus, by specifying a minimum of 2.0 and a maximum of 7.5, we eliminate all samples that are shorter than 2.0 and longer than 7.5 meters. 4.
Click on the drop down menu list for Attribute#3 and select all lithologies except -1 as shown in the figure above. This
47
will eliminate all samples with a lithologic code of -1 from the sample variogram calculations. Note that one could constrain the selection of samples by their X, Y, and/or Z coordinates. If you wish to calculate adown-the-holesample variogram, click on the Bore Hole IDtext box and select the variableB HID. At this time we do not wish to calculate a down the hole variogram, so for this example, do not click the Bore Hole ID text box. Finally, click the Next button to go to the next window. SAGE2001 will respond by closing the Select Variogram Attributes & Constraints window and open the Calculate sample variograms – directions window. 7.2.2 The Calculate Sample Variograms – Directions Window.
This window is used to specify the azimuths, dips, and tolerance parameters for calculating the sample variograms.
48
Type or click the following to continue with the example variogram calculations. Use the arrow keys to move around in the grid: 1. Leave the Azimuth and Dip text boxes in the Directional Increments frame with the values 30 and 30. This will provide directional sample variograms at 30 degree increments on azimuth and on dip. Thus, the total number of sample variograms will be 37. Remember there is only one directional sample variogram with a dip of –90. 2. Type in a unit distanceof 25 in the lag Specsframe. 3. Type in 47 for the Maximum No. of Lags. 4. Set the Horizontal Bandwidth to 16. 5. Set the Vertical Bandwidthto 25 6. Set the Lag tolerance to 0.45 or 45% of the unit lag distance. 7. Leave the Angular Toleranceat 22.5 degrees. 8. Click on the Customize Azimuth, Dip Angle, and Unit Distance Values.We do this because we want to change some of the lag distances for some of the directions. 9.
Change all lag distances for the –30 Dip from 25 to 30 meters. 10. Change all lag distances for the –60 Dip from 25 to 20 meters. 11. Change the lag distance for the –90 Dip from 25 to 7.5 meters. Note that we can also change any of the azimuth or dip angles in the grid as well as the unit lag distances Finally, click the Next button to close this window and open the Calculate sample variogram – Finish window.
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7.2.3 Calculate Sample Variograms – Finish Window
This window requires the user to specify a few more parameters for the calculation of the sample variograms and to provide names for the output files
Click or type the following to continue with the example variogram calculations. 1. Type in the title Example Using Test.Sag.This title will appear on all the report and plot files. 2. Click on the Estimatordrop down list box and select (click) on the correlogram estimator. If the Indicatorestimator is selected, then you must also supply the cutoff or threshold value defining the indicator in the Cutofftext box. 3. We wish to use 100% of the available samples,so we leave the % of Samples box at 100. This box can be used to randomly select a subset of the available samples, e.g., 50%. 4.
Next click on the Report button and provide a suitable name for the report file. These will always have the suffix rpt, e.g., Test.rpt. 50
5.
Click on the Autofitbutton and type in the file name Test. This will create the file Test.pliwhich is an ascii file that is used by SAGE2001 to calculate the variogram model. SAGE2001 only knows how to modelp li files. It cannot model any other type of file. However, note that the file is ASCII and the format is quite simple. Thus, it is very easy to calculate sample variograms using other software and format the output identical to a pli file. Then SAGE2001 can be used to model the sample variograms. 6. Click on the Plot button and type in a name f or the plot files, e.g., Test.plt. Although these files are similar to the pli files they are slightly different. SAGE2001 uses these files to plot the sample variograms to the computer screen. Finally, click on the Finish button to begin the sample variogram calculations. SAGE2001 will open a progress barand let the user know the percentage of the job completed. For this example, the sample variogram calculations may take something like 2 or 3 minutes assuming your CPU is at least 200 Megahertz or more. When SAGE2001 has completed the sample variogram calculations, a message box will appear with the message Analysis Complete. Note a small message that also appears on the bottom of the Calculate Sample Variograms – Finish Window. A return code of 0 indicates that no errors were detected during the calculations. The return message also indicates the actual number of samples that were used to calculate the sample variograms. For example , the f ile TEST.Sag contains 5,072 data values, but because of the imposed constraints, only 5,047 of these samples were actually used to calculate the sample variograms. Click the message boxOKAY button and SAGE2001 will respond by closing the Calculate Sample Variograms – Finishwindow and opening the Plot window.
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7.3 Viewing the Sample Variograms. SAGE2001 automatically opens the Plot window when the OKAY button on the Analysis Completemessage box is clicked. You can also open the plot window by clicking on the Display Variogramsmenu item as shown in the following figure.
Windows will respond by opening a standard Open File Dialogue box from whichyou can select a plt file and display the sample variograms using the Plot Window.
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7.3.1 The Plot Window.
This is the plot window that SAGE2001 uses to display the sample variograms and the model if it exists. However, for this exercise we have not calculated a model yet, so for now, we will use the plot window to simply view the calculated sample variograms. Automatic Plot Update. Probably the first thing you will notice is that SAGE2001 automatically rotates through the sample variograms by changing the plot every few seconds. To turn off the automatic rotation, click the Auto Offbutton. To turn it back on, click the Auto Onbutton. You can also manually rotate through the sample variogram plots by clicking on the Previousand Next buttons when Auto rotate is off. Resizing the Window.The plot window can be resized using the standard Windows resizing methods. 1. Use the Maximize/Restorebutton which is the center button of the three buttons located in the extreme upper
right corner the window. This window is probably bestyou wayhave to increase andofdecrease the plot sizethe once adjusted the smaller window size to your preference. 53
2.
The plot window size can also be adjusted by dragging the edges or corners of the window using the mouse.
Show Pairs.Click on the Show Pairscheckbox to show the number of pairs for each sample variogram point. Connecting Lines. Click on the Connecting lines checkbox and SAGE2001 will join all the sample variogram points with straight lines. Re-scaling the Plot Axes. The axes of the sample variogram plot can be re-scaled by setting the text boxes in the lower right hand corner of the plot window: 1. Leave Minimum h= 0 and G(h) = 0. 2. Set Maximum h= 400. 3. Set Maximum G(h)= 1.5 4. Set tic-interval h= 80 5. Set tic interval G(h) =0.25 This will re-scale the sample variogram plot as shown in the figure above.
Now, as we rotate through the sample variograms, we should be noting the following: 1. Is the scale of the X-axis (max h=400) of the sample variogram plot appropriate? SAGE2001 will not use any sample variogram points at lag distances larger than a specified maximum lag distance. Thus, it is important to choose this distance appropriately. For example, if the specified length is too long, then useless sample variogram points may be included in the modeling process. This could have a negative impact on the resultant model parameters. 2. What is an appropriate minimum for the required number of pairs? View the various directional sample variogram plots and try to determine a threshold value that will eliminate spurious sample variogram points near the srcin, but will also retain those points which appear to be critical to the shape of the variogram model at shorter lag distances near the srcin.
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7.4 Viewing and Printing the Sample Variogram Report. The report on the sample variogram calculations is opened by clicking on the Print Reportmenu item as shown by the following figure.
Windows will respond by opening the standard Open File Dialoguebox. Select the file test.rpt and click on the Open button. SAGE2001 will open the Print Reportwindow. 7.4.1 The Print Report Window.
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View and print the report file Test.rptas follows: 1. Maximize the window by clicking on the Maximizebutton in the extreme upper right hand corner of the window. 2. Select 100% from the Zoom(%)drop down list box. This will increase the size of the report on the screen so it can be read. 3. Select or scroll through the pages of the report by either moving the slider bar or by clicking on the tiny arrow heads at the ends of the slider bar. 4. Print the current page by clicking on the Print Current Page button. 5. Print all pages by clicking on the Print all Pages Button. Finally, close the window by clicking on the Close button.
7.5 Modeling the Sample Variograms. The sample variogram file required for modeling purposes is opened by clicking on the Model Variograms menu item as shown in the following figure.
Windows will respond by opening the Model Sample Variograms – Setup window.
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7.5.1 The Model Sample Variograms – Setup Window.
This window collects the file names and other parameters that will control the modeling of the sample variograms.
Type or click the following to continue with the example exercise: 1. Click on the Input file Sample Variogram button. Windows will respond by opening the standard Open File Dialoguebox. Select the file Test.pliand click Open. 2. Similarly, select or name the output files by clicking on the Plot and Reportoutput file buttons. 3. Type in the title Example Using Test.Sagif it does not appear in the Analysis Titlecheckbox. 4. Set the Minimum # pairsto 200 5. Leave the Maximum driftentry at 999999. This is such a large number that it will have no effect on the selection of sample variogram points. 6. Select the by pairs Weighting scheme. 7. Select 2 structures. 8. Select the Exp(practical range) Structure Type for each structure Now we are ready to go to the next window. Click theNext button. 57
7.5.2 The Model Sample Variograms – Rotation Convention Window.
This window is used to tell SAGE2001 what rotation conventions it should use to model the sample variograms.
For this exercise, we will use the custom rotation convention ZYZ-RRR: 1. Click on the Customoption button. 2. Click the Y option button for the second rotation axis. 3. Click the Z option button for the Third rotation axis. 4. Leave all the direction buttons Right. Click the Next button to open the Model Sample Variogram Parameterswindow.
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7.5.3 The Model Sample Variogram – Parameters Window.
This is the last window requiring input to model the sample variograms.
To continue with the exercise, type in the following: 1. Change the C0, C1 Decimalstextbox entry to 3. This will print the nugget and the C1 and C2 coefficients with 3 decimal places in all reports and plots. 2. Change the Max Lag Distancetextbox entry to 400. SAGE2001 will not use sample variogram points at lag distances larger than the value specified here. Thus, it is important to choose this distance appropriately. 59
Click Fit to begin the modeling calculations. SAGE2001 will open a Progress Barand report the percentage completed. Note that sometimes it will appear that the program is stuck in an endless loop because the progress bar is not moving. Be patient. SAGE2001 is probably not stuck in an endless loop, but is modeling the sample variograms. Eventually (within a minute or two), the progress bar will show some sign of activity. The movement of the progress bar from 0 to 100% can be quite sporadic and punctuated with relatively long pauses at times When SAGE2001 has finished calculating the variogram model two things will happen: 1. The variogram model parameters will be written to the Model sample variogram – Parameters window. 2. SAGE2001 will open a message box with the message Analysis Complete. Click on the OKAY button to continue. SAGE2001 will respond by opening the plot window and rotating through the directional sample variograms and their models. You should Plot window and the resize the plot window so you can view both the Model sample Variogram – Parameters window on your computer screen as shown in the following figure.
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7.5.4 The Parameters of the Variogram Model.
The following figure shows the Modeling sample variogram – Parameterswindow after SAGE2001 has finished calculating the model parameters and written them to the window.
For example, the nugget is 0.1297, and the coefficients C1 and C2 are 0.517, and 0.353 respectively. Note that the sill is 1.00 because we have used the correlogram which has a theoretical sill of 1.0. Note that SAGE2001 has calculated the Azimuth and Dip of the anisotropy axis for each structure. For example, consider the second structure. The orientation of the anisotropy axes is: 61
1.
2.
3.
The azimuth and dip of the Rotated X axis is 43 and +29 degrees respectively. The closest directional sample variogram we have to this orientation is Az=240, Dip=-30. The azimuth and dip of the Rotated Y axis is 314 and -2 degrees respectively. The closest directional sample variogram we have to this orientation is Az=300, Dip= 0. The azimuth and dip of the Rotated Z axis is 229 and +62 degrees respectively. The closest directional sample variogram we have to this orientation is Az=60, Dip=-60.
7.5.5 Interactive Variogram Modeling.
Suppose that after examining the plot window we decide that we do not like the variogram model provided by SAGE2001. For example, we feel the nugget is too small. We would prefer a nugget equal to 0.25 rather than 0.129. To re-model the sample variograms and force a nugget of 0.25, do the following: 1. IMPORTANT – First, close the plot window! 2. Type the value 0.25 in the Nugget text box. 3. Click the Fix check box next to the nugget textbox. A checked fix check box will cause SAGE2001 to leave that parameter intact while it calculates values for any remaining parameters with uncheckedfix check boxes. 4. Click the Fit button. SAGE2001 will calculate a new variogram model with a nugget of 0.25. 5. You may check as many of thefix check boxes as you like. Thus, you have the option of specifying the complete variogram model if you wish. This provides you with the option of assuming complete control over the modeling of the sample variograms. Alternatively, you may let SAGE2001 have complete control over modeling the sample variograms or some combination whereby you share control over the modeling of the sample variograms with SAGE2001.
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7.5.6 The Plot Window.
The plot window can also be used to view several directional sample variograms simultaneously. For example, the following figure shows the directional sample variograms and their models corresponding to the anisotropy axes of the second structure.
To view these directional sample variograms and their models click the following: 1. Click the Select Multiple Setsoption button. 2. Click the directional variogram with Az = 240, Dip = -30 from the list box. 3. Click the directional variogram with Az = 300, Dip = 0 from the list box. 4. Click the directional variogram with Az = 60, Dip = -60 from the list box. 5. Click the Graph button. SAGE2001 will show the three directional sample variograms and the model using different colors. The black curve is obviously the major axis in the above figure, while the blue (triangles) curve is the minor axis. 63
7.5.7 Viewing and Printing The Variogram Model Report
The variogram model report is viewed using the same procedures described in Section 7.4. The variogram model report should be printed and retained as a permanent record of the variogram model. The report contains a complete description of the variogram model parameters and rotation conventions.
7.5.8 Additional Postscript Output Files.
SAGE2001 also writes two additional postscript files to the working directory. These files are always calledROSE.PS and GAM.PS.Thus, they are overwritten each time SAGE2001 is run. Although these files duplicate the report files discussed earlier, they do provide additional graphics that may be helpful in visualizing the orientation of the anisotropy ellipsoids in 3 dimensional space. For example, the following three figures are examples from the file ROSE.PS.
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