ANALISIS SEMIVARIOGRAMAS

Exploration Chapter 13: Semi-Variogram Analysis

Chapter 13

Semi-Variogram Analysis

In this Chapter

Introduction The Semi-Variogram Preparing Linear Semi-Variograms Preparing 3-D Semi-Variograms

Introduction

Gemcom for Windows is a workspace system designed specificallyfor spatially related data. Therefore, it is important to providefacilities for the statistical analysis of data based on the spatialrelationship between the data values. Geostatistics is the namecommonly given to this type of statistical analysis, where anassumption is made that sample or data values are affected both bytheir location and their relationship with the surrounding data.Variables that follow this behaviour are known as regionalizedvariables and the study of them is called geostatistics.

The main application of geostatistics has been for estimating orereserves. It is now being used more and more in other fields, suchas in environmental assessments, where predictions andestimations need to be made from spatial data.

Geostatistics normally is performed in two stages:

Analyse the spatial relationships between values. The main toolfor this analysis is the semi-variogram. Semi-variograms are

Section II: Workspaces Gemcom for Windows

one of the statistical analysis tools provided in Gemcom forWindows.

Use the results of the analyses to predict values in areas withno data. This is called interpolation. Gemcom for Windowsprovides some general interpolation tools for estimating valuesusing grids. For more information on interpolation, seeChapter 19: Gridding and Contouring.

The Semi-Variogram

Geostatistics uses most of the standard tools of statisticians toanalyse the relationships between samples. Such tools includemeans, standard deviations, the variance, and presenting theseresults as a function of distance and direction.

The semi-variogram is a graph that shows the variability betweenpairs of samples against the distance between them in a specificdirection. The graph's horizontal axis shows the separation distancebetween pairs of samples, while the graph's vertical axis shows thevariance of the differences in values for specific separationdistances. Generally, sample pairs are grouped together into rangesof distance separation, as samples usually are never regulardistances apart. These ranges of distances are called the lagdistances. For convenience, the vertical axis usually shows half thevariance value, hence the term semi-variogram (see the formulabelow).

When the graph is derived from sample data, it is called anexperimental semi-variogram. When the semi-variogram is derivedsolely from theoretical data, it is called a model semi-variogram.

Calculating, displaying and modelling the semi-variograms is athree- stage process:

1. Calculation. The experimental semi-variogram is calculatedfrom the workspace. This can be done in two ways: alongtraverses or drillholes; or in any three-dimensional direction.


2. Display. The experimental semi-variogram is displayed as agraph using QuickGraf, Gemcom's graph display and plottingutility.

3. Modelling. The model semi-variogram is fitted to theexperimental semi-variogram using an interactive process. Thisis also done using QuickGraf.

The parameters that you specify during the modelling stage canthen be used during the kriging process to control the interpolationof values into areas with no data.

Figure 13-1: A typical semi-variogram


Calculation

The general formula for calculating a semi-variogram for a set of nsamples spaced h distance apart is as follows:

[ ]2 1 2Gammahn

g x g x h( ) ( ) ( )= - +

where

Gamma(h) is the semi-varianceg(x) - g(x+h) is the difference between the values of the sample

pairsh is the distance between the sample pairsn is the number of samples

The semi-variogram is computed for as many sets of h asappropriate to the data, and the results are plotted on a graph.

Model Semi-Variograms

The model semi-variogram is the ideal shape for the curveillustrating the theoretical relationship between sample pairs asthe distance between them increases. The curve begins at or nearthe origin, as samples that have coincidental locations should bethe same and thus have no variance. The semi-variance shouldincrease to the right, as the distance between the samplesincreases. The curve will gradually flatten and the semi-variancevalue will become constant. At this separation, there is no longerany relationship between sample pairs and they can be consideredindependent of each other. The distance at which this happens iscalled the range of influence, and the variance at this point is calledthe sill.

This ideal semi-variogram is called a spherical model. In practice,the curve may start with a small variance (as there are generallyvariances between two samples taken at the same location, oftencaused by sampling techniques). This is termed the nugget effect


and can be present in all semi-variograms. Also, more than one sillvalue can be present, in which case the model semi-variogram has anested structure.

In addition to the spherical model, there are several other types ofmodel semi-variograms that can occur:

Exponential model Linear model Logarithmic model Gaussian model Nugget effect model

For detailed information on these models, see Models inChapter 23: QuickGraf.

Figure 13-2: Ideal model semi-variogram


Experimental Semi-Variograms

Experimental semi-variograms can be made from any workspacethat contains information about the location of the sample values.This includes the point workspace, the traverse workspace, thedrillhole workspace and the polygon workspace.

Semi-variograms are calculated for specific geological or structuraldirections, such as along dip, down plunge, along strike,perpendicular to strike, along drillholes, etc. The direction that youwant to use will govern the type of semi-variogram you select.

You can produce two types of semi-variograms from theworkspaces. The type of semi-variogram you select depends on theway you want to determine the distance and directionalrelationship between the samples. They are:

Linear semi-variograms. This type of semi-variogram canonly be calculated directly from traverse or drillholeworkspaces. The only relationship between the sample pairsthat is considered is along the trace of the traverse or drillhole.

3-D semi-variograms. This type of semi-variogram iscalculated from data that has been extracted from a workspaceinto an extraction file. The semi-variogram is calculated along athree dimensional vector defined by an azimuth and a dipangle, within defined tolerances. You can calculate 3-D semi-variograms for up to twelve different directions at a time.


Modelling Experimental Semi-Variograms

You can use QuickGraf to fit model semi-variograms to theexperimental semi-variograms. The modelling process is aninteractive process that allows you to fit a number of different typesof semi-variogram models to the experimental semi-variogram.

You can fit semi-variogram models to experimental semi-variograms using any combination of the following three methods:

Defining the model type. You must select the type for each ofthe models that you want from the list of available model types.

Defining the number of semi-variogram models. You cannest up to three type of models together.

Defining the model parameters. You can do this using acombination of data entry screens and mouse positioning,depending on the type of semi-variogram models.

For complete information on the model types that are available aswell as on the procedures for modelling semi-variograms, seeModels in Chapter 23: QuickGraf.

Preparing Linear Semi-Variograms

Linear semi-variograms are calculated either along the line of atraverse or along the trace of a drillhole. The data for linear semi-variograms is taken directly from either a traverse or drillholeworkspace. The sample values and locations are obtained directly fromtables in the workspace according to selection criteria that you candefine. Sample locations are determined from values either in FROMand TO fields or in DISTANCE fields. The relationship between samplepairs is determined directly from their sequential position along eachtraverse or drillhole, regardless of the drillhole direction or orientation.


Figure 13-3: Calculation of linear semi-variograms

You can use the Linear Semi-Variogram command to definelinear semi-variogram profiles and produce linear semi-variogramsbased on the data in a traverse or drillhole workspace.

The semi-variogram calculation produces a log file that contains alist of the records used for the calculation identified by the contentsof the ID field that you defined. In addition, it also produces twofiles used by QuickGraf to plot the semi-variogram:

DDHVAR.GRF. This file contains the tabulation of the semi-variogram table.

DDHVAR.DAT. This file contains the statistics summary of thedata set used for the semi-variogram calculation.

Both of these files are text files and are located in theGCDBaa\GRAPHS subdirectory.


Linear Semi-Variogram Profiles

Before you can view the semi-variogram table or prepare semi-variogram plots you must have at least one linear semi-variogramprofile defined. Follow this procedure to create a new profile:

1. Select Workspace }} Analysis }} Linear Semi-Variogram. The LinearSemi-Variogram dialog box will appear.

2. Click Add. Enter a name for the linear semi-variogram profileand click OK.

3. The Linear Semi-Variograms dialog box appears. This dialogbox consists of four parts, represented by four tabs, which allowyou to specify which data is to be used to prepare the semi-variogram and to apply a variety of selection criteria,transformations and normalization options to that data:

Data Location Selections Parameters

4. Enter the required parameters for each of these tabs. Theseparameters are outlined in detail in the sections below.

5. Once all the desired parameters have been entered, click OK.The Select Records to Process dialog box will appear. Selectthe desired option as outlined in Chapter 4 of the Gemcom forWindows User Manual, Volume I (Core).

Gemcom for Windows will perform the semi-variogram analysis,and the tabulated results will appear in the Linear Semi-Variogramtable (see Viewing the Linear Semi-Variogram Table onpage 2306). If you are not satisfied with the results, you can clickthe Parameters button at the bottom of the table to redisplay theprofile creation dialog box and alter your parameters as desired.


Data

This tab brings up a dialog box containing the following parameterswhich determine the table and fields to be used in performing thesemi-variogram calculation.

Description. Enter a brief description for the profile, ifdesired.

Table to be Used. Select the name of the table which containsthe data you wish to use to create the linear semi-variogram.

Field to be Used. Select the name of the field which containsthe data you wish to use to create the linear semi-variogram.

Figure 13-4: Linear Semi-Variograms dialog box (Data tab)


You can calculate linear semi-variograms on fields withnumeric data types (real, double or angle).

Cross-Reference Table. Select the table which contains thefield(s) you wish to use as a cross-reference for selecting data.This is an optional parameter. For more information on cross-referencing, see Chapter 7: Extracting Data.

Reference Position. Select one of the following options todetermine the reference position for the data:

Use FROM. Select this option to use the location parameter inthe FROM field as your reference position.

Use MIDDLE. Select this option to use the mid-point betweenthe FROM and TO fields as your reference position.

Use TO. Select this option to use the location parameter inthe TO field as your reference position.

Location

You can use the parameters in this tab to define the physical areafrom which data for the calculation is to be taken. Enter the lowerand upper bounds for the northing, easting and elevationcoordinates to create a bounding box in space.

The lower and upper default values for all coordinates of-99999999.000 and 99999999.000 respectively have the effect ofcreating a bounding box so large that all records in the workspaceare selected.

Selections

The parameters you enter in the Selections tab will determinewhich records from the physical bounding box you specified in theLocation tab will be used for the calculations. You can specify lowerand upper bounds or matching strings for fields from up to threetables: the Header table, the table to be used (if different from the


Header table), and the cross-reference table (if selected, and ifdifferent from the Header table).

Enter the following parameters as necessary for each field you wishto use to limit the selection criteria:

Field. Select the name of the field you wish to use to limit recordselection.

Axis. If the field you selected is a coordinate field, select the axis(X, Y or Z) for which to enter lower and upper bounds.

Lower Bound and Upper Bound. If the field you selected is anumeric field, enter a lower and upper bound for the data to beselected.

Figure 13-5: Linear Semi-Variograms dialog box (Selections tab)


Matching String. If the field you selected is a character field,enter a string to define which data is selected. You can use thewildcard characters * and ? in your string.

Parameters

This tab brings up a dialog box containing additional parametersnecessary for the creation of the semi-variogram analysis.

Enter the following parameters:

ID Field Name from Header Table. Select the name of thefield from the Header table which is used as the primary key inthe workspace. For a discussion of primary keys, seeWorkspace Structure in Chapter 3: The Gemcom for WindowsWorkspace.

Directional Filtering: Enter upper and lower bounds for theaverage dip and azimuth angles if desired.

Semi-Variogram Parameters. These are the parameters usedto determine the way all of the semi-variograms are calculated.

Threshold Pairs. This is the minimum number of samplepairs in a single lag distance that will produce a reliablepoint on the semi-variogram. Numbers of sample pairs thatare less than this threshold will be indicated on the semi-variogram with a different symbol.

Lag Distance. This is the size of each class (range ofdistance) used for the semi-variogram calculation. Forexample, if the lag distance is 10 feet, then each point on thesemi-variogram will be calculated for sample pairs fallingbetween 0 and 10 feet apart, 10 and 20 feet apart, 20 and 30feet apart, etc. A semi-variogram has 30 equally spaced lagdistances.


Starting Offset. This is the starting point of the first lagdistance. For example, if this is set to 50 feet and the lagdistance is set for 10 feet, then the semi-variogram will becalculated for distance ranges of 50 to 60 feet, 70 to 80 feet,80 to 90 feet, etc.

Number of Class. This is the total number of the ranges ofdistance used for the semi-variogram calculations. Thenumber of class multiplied by the lag distance equals therange of influence.

Data Transformation. Enter the following parameters tospecify how any data transformation will be performed:

Transformation. Select one of the three following optionsfrom the list provided:

Figure 13-6: Semi-Variograms dialog box (Parameters tab)


None. This option provides no data transformation,resulting in a semi-variogram with normallydistributed data.

Log. This option is the three-parameter log-normaltransformation, which can be applied to log normallydistributed data. The transformation will cause thenatural log of the values to be normally distributed.

The three parameter log normal transformation isexpressed by the following formula:

Vn = Log (Vo * F + C)

where

Vn = new valueVo = old valueF = multiplication factorC = constant

Indicator. Selecting this option allows you to applyan indicator transformation to the data to create anindicator semi-variogram. The indicatortransformation allows you to replace data valueswith an indicator value of 1 (if the data value isgreater than or equal to the indicator cut-off value)or an indicator value of 0 (if the data value is lessthan the indicator cut-off value). Indicator semi-variograms are then calculated using the indicatorvalue instead of the data value.

Additive Constant. This constant is used to perform athree-parameter log-normal transformation. You cantransform your data values by entering a constant that willbe added to every data value. The default is 0.

Multiplication Factor. This factor is used to perform athree-parameter log-normal transformation. You cantransform your data selection by entering a multiplicationfactor. This is a factor by which every data value is


multiplied. If you have a range of data that is extremelyflat, you might want to accentuate any differences bychoosing a multiplication factor of 2, for example, to doubleall the data values (and therefore the differences betweenthem). The default is set to 1.0.

Indicator Cut-Off. This only applies to the sample valueswhen the Indicator Transformation option is selected. Eachsemi-variogram will have its own indicator cut-off value.

Viewing the Linear Semi-Variogram Table

To view the results of the linear semi-variogram calculation in atable format, follow this procedure:

1. Select Workspace }} Analysis }} Linear Semi-Variogram. The LinearSemi-Variogram profiles dialog box will appear (see Figure13-7).

2. Add a linear semi-variogram profile using the above procedure.

or

Select an existing profile and click View.

3. The Linear Semi-Variogram data table will appear. Thisdialog box is divided into three main areas delimited by boxes:

Variogram Parameters Statistics Display Options


Variogram Parameters

The variogram parameter area, the untitled area at the top of thedialog box, contains the following information pertaining to thedata set (population) used to create the semi-variogram.

Lower Azimuth and Upper Azimuth. Lower Dip and Upper Dip. Total pairs used.

Figure 13-7: The Linear Semi-Variogram table dialog box


Statistics

The semi-variogram Statistics table contains the followinginformation about each lag distance (interval):

From. The starting distance of the lag interval.

To. The ending distance of the lag interval.

Pairs. The number of sample pairs in the lag interval.

Drift. The general increase or decrease of sample values in thedirection of the semi-variogram as the lag distance increases.The general formula for the drift is:

Drift=Cumlative DifferenceNumber of Samples

Gamma (h). This is the semi-variance value. This value can becalculated and displayed using one of four options. The optionused is determined by the setting in the Display options area inthe lower left-hand corner of the dialog box.

Local mean. The mean of all of the sample values in the laginterval.

Distance. The average distance between all of the sample pairsin the lag interval. This value will lie between the lag distancefrom and the lag distance to.


Display (Normalization)

The effect of regional variations of values (for example, some areashaving many high values and other areas having many low values) cancause distortions to the experimental semi-variograms. Normalizationof the semi-variogram will help to minimize these effects.

The Display area, in the bottom left-hand corner of the dialog box,contains a list of normalization options which will determine howthe Gamma (h) value in the Statistics area will be displayed. Youcan select from among the following four options:

Local Mean Square. If you select this option, each sample valuewill be divided by the mean square of all values in the lag intervalbefore the variance between the samples pairs is calculated.

No Normalization. Selecting this option will display the semi-variance value with no normalization.

Population Mean-Square. If you select this option, eachsample value will be divided by the mean square of all thevalues in the data set before the variance between the samplespairs is calculated.

Population Variance. Selecting this option displays the semi-variance of the sample pairs divided by the variance of all thesamples in the data set.


Printing the Linear Semi-Variogram Table

You can now use the Report button located in the lower right-handcorner of the dialog box to print out the semi-variogram table thatyou have created and are currently viewing on-screen.

To print the table, follow this procedure:

1. Click Report in the Linear Semi-Variogram dialog box.

2. Select the report destination (file, printer or screen) from theSelect Report Destination dialog box (see Chapter 4: Dialogboxes, Volume I: Core).

3. The report will be generated. Regardless of which display optionyou were using to display the Gamma (h) value on-screen whenyou clicked on Report, all four variations of that value willappear in the report.

Viewing the Linear Semi-Variogram in GraphicalFormat

When you perform a linear semi-variogram calculation, you canalso view the graphical representation of your data on-screen in theQuickGraf utility by clicking the Graph button located in the lowerright-hand corner of the Linear Semi-Variogram dialog box.

You can also use QuickGraf to create various semi-variogrammodels to which you can fit your data. For more details aboutworking with this utility, see Chapter 23: QuickGraf.

Preparing 3D Semi-Variograms

Three-dimensional semi-variograms are calculated from point dataalong lines with given azimuths and dip angles. The sample valuesand locations are obtained directly from data that has been


extracted from workspaces and is in extraction files. (Forinformation on preparing extraction files, see Chapter 10: TheExtraction File.) Sample locations are obtained from the northing,easting and elevation coordinates of each point in the extractionfile. Values are obtained from either of the elevation, the real value,or the integer value in the extraction file.

Figure 13-8: Calculation of 3-D semi-variograms

Up to twelve directional semi-variograms can be calculatedsimultaneously for specified directions. You can impose furtherfiltering on the values in the extraction file when you compute thesemi-variogram, and you can impose elevation limits for each of thedirectional semi-variograms independently.

The 3D semi-variogram calculation produces two files that are usedby QuickGraf to plot the semi-variogram:

3DVAR.GRF. This file contains the semi-variogram table.

3DVAR DAT. This file contains the statistics summary of thedata set used for the semi-variogram calculation.

Both files are text files and are located in the GCDBaa\GRAPHSsubdirectory.


3D Semi-Variogram Profiles

In order to calculate a 3D semi-variogram analysis, you must firstdefine a 3D semi-variogram profile. Note that in order to create aprofile, you must have created at least one extraction file. For moreinformation on extraction files, see Chapter 10: Extracting Data.

To create a new profile, follow these steps:

1. Select Workspace }} Analysis }} 3D Semi-Variogram from ExtractionFile. This will bring up the 3D Semi-Variogram Profiles list.

2. Click Add. Type in a name for your profile and click OK.

3. In the file name dialog box that appears, select the extractionfile which contains the data you wish to use to create your 3Dsemi-variograms. Click Open.

4. Gemcom for Windows will read the extraction file, displaying itsprogress in a status window. Click OK when the process iscompleted to close the status window.

5. The 3D Semi-Variogram Parameters dialog box will comeup. This dialog box will display the name and description of theextraction file, as well as the following information about thevalues within the extraction file:

# Values. This is the total number of values (records) in theextraction file.

# Values


Variable to be used. Select the numeric variable in theextraction file that will be used for the sample values. Youcan use the real value, the integer value, or any of thecoordinate values.

Semi-Variogram parameters. Enter the followingparameters:

Threshold pairs. This is the minimum number ofsample pairs within a single lag distance that willproduce a reliable point on the semi-variogram.Intervals that contain fewer sample pairs than thisthreshold will be indicated on the semi-variogram with adifferent symbol.

Lag distance. Enter the size of each interval (range ofdistance) used for the semi-variogram calculation. Forexample, if the lag distance is 10 feet, then each point onthe semi-variogram will be calculated for sample pairsfalling between 0 and 10 feet apart, 10 and 20 feet apart,

Figure 13-9: 3D Semi-Variogram Parameters dialog box


20 and 30 feet apart, etc. A semi-variogram has 30equally spaced lag distances.

Starting offset. Enter the starting point for the firstlag distance. For example, if this is set to 50 feet and thelag distance is set to 10 feet, then the semi-variogramwill be calculated for distance ranges of 50 to 60 feet, 70to 80 feet, 80 to 90 feet, etc.

Number of classes. Enter the total number of intervals(ranges of distance) to be used for the semi-variogramcalculations. The number of classes multiplied by the lagdistance equals the range of influence.

Data transformation. Enter the following parameters todetermine the type of data transformation to be performed.

Transformation. Select one of the three followingoptions from the list provided:

None. This option provides no datatransformation, resulting in a semi-variogram with normally distributed data.

Log. This option is the three-parameter log-normal transformation, which can be appliedto log normally distributed data. Thetransformation will cause the natural log ofthe values to be normally distributed.

The three parameter log normaltransformation is expressed by this formula:

Vn = Log (Vo * F + C)

where

Vn = new valueVo = old valueF = multiplication factorC = constant


Indicator. Selecting this option allows youto apply an indicator transformation to thedata to create an indicator semi-variogram.The indicator transformation allows you toreplace data values with an indicator value of1 (if the data value is greater than or equal tothe indicator cut-off value) or an indicatorvalue of 0 (if the data value is less than theindicator cut-off value). Indicator semi-variograms are then calculated using theindicator value instead of the data value.

If you selected the Log Normal data transformation, youwill also have to enter the following parameters:

Additive Constant. You can transform yourdata values by entering a constant that will beadded to every data value. The default is 0.

Multiplication Factor. You can transformyour data selection by entering a factor bywhich every data value is multiplied. If youhave a range of data that is extremely flat,you might want to accentuate any differencesby choosing a multiplication factor of 2, forexample, to double all the data values (andtherefore the differences between them). Thedefault is set to 1.0.

6. Once you have entered the required semi-variogramparameters, click OK. The 3D Semi-Variogram Definitionsdialog box will appear. In this dialog box, you will define a set ofparameters for each of up to twelve individual directional semi-variograms.

7. In order to be able to enter the parameters for a particularvariogram, you must select the variogram number and ensurethat the variogram is enabled by verifying that the Enable boxhas a checkmark in it. If you do not see a checkmark, click thecheckbox to enable the variogram.


8. You are now able to enter the following parameters for thechosen variogram:

Azimuth. Enter the direction, in degrees clockwise fromnorth, along which the semi-variogram will be calculated.

Dip. Enter the direction, in degrees from the horizontal,that defines the dip of the semi-variogram. Negative anglesindicate a dip downwards from the horizontal, and positiveangles indicate a dip upwards from the horizontal.

Spread Angle. As it is often unlikely that the directionalvectors between each sample pair will exactly coincide withthe directional vector of the semi-variogram, you mustdefine a spread angle or tolerance that will allow for thesedeviations. The tolerance is applied equally to the azimuthand dip angles, and defines a conical search. A spread angleof 45 degrees would provide a total tolerance of 90 degrees.

Figure 13-10: 3D Semi-Variogram Definitions dialog box


Dip ofsemi-variogram

Tolerance angle

Conical search

Azimuth ofsemi-variogram

Elevation (Z)

N

E

Figure 13-11: Semi-Variogram directional tolerance

Lower elevation. This defines the lowest elevation valuefor a point in the extraction file that will be used tocalculate the semi-variogram. Any points with elevationslower than this value will not be used.

Upper elevation. This defines the highest elevation valuefor a point in the extraction file that will be used tocalculate the semi-variogram. Any points with elevationsgreater than this value will not be used.

Lower cut-off. This defines the smallest sample value inthe extraction file that will be used. Values less than thiswill not be used.

Upper cut-off. This defines the largest sample value in theextraction file that will be used. Values greater than thiswill not be used.

Indicator cut-off. This only applies to the sample valueswhen the Indicator Transformation option is selected. Eachsemi-variogram will have its own indicator cut-off value.

Half Width. This defines the width of the local corridorwindow within which the sample pair must fall.


Half Height. The defines the height of the local corridorwindow within which the sample pair must fall.

9. Repeat Steps 7 and 8 for any of the remaining twelve semi-variograms you wish to use. When you have defined all desiredsemi-variograms, click OK. Gemcom for Windows will performthe semi-variogram calculations and bring up the 3D Semi-Variogram table.

The 3D Semi-Variogram Table

The semi-variogram calculation produces a tabulation of the semi-variogram (see Figure 13-12). This table contains information abouteach lag distance for each of the directions calculated. To view thecalculations for a particular semi-variogram, select the desired

Figure 13-12: 3D Semi-Variogram dialog box


semi-variogram number from the Semi-Variogram list at the topof the dialog box.

The rest of this dialog box is divided into three main areasdelimited by boxes:

Variogram Parameters Variogram Statistics Display Options

Variogram Parameters

The variogram parameter area, at the top of the dialog box,displays the following parameters for the particular semi-variogram calculation currently selected:

Azimuth Dip Spread Angle Lower Elevation and Upper Elevation Total pairs used

Statistics

The Semi-variogram Statistics table contains the followinginformation about each lag distance (interval) for the currentlyselected directional semi-variogram:

From. This is the starting distance of the lag interval, and iscommon for all directional semi-variograms defined using thecurrent data set.

To. This is the ending distance of the lag interval, and iscommon for all directional semi-variograms defined using thecurrent data set.

Pairs. The number of sample pairs in the lag interval.


Drift. The general increase or decrease of sample values in thedirection of the semi-variogram as the lag distance increases.The general formula for the drift is:

Drift=Cumlative DifferenceNumber of Samples

Gamma (h). This is the semi-variance value. This value can becalculated and displayed using one of four options. The optionused is determined by the setting in the Display options area inthe lower left-hand corner of the dialog box.

Local mean. The mean of all of the sample values in the laginterval.

Distance. The average distance between all of the sample pairsin the lag interval. This value will lie between the lag distancefrom and the lag distance to.

Display Options (Normalization)

The effect of regional variations of values (for example, some areashaving many high values and other areas having many low values)can cause distortions to the experimental semi-variograms.Normalization of the semi-variogram will help to minimize theseeffects.

The Display Options area, in the bottom left-hand corner of thedialog box, contains a list of normalization options which willdetermine how the Gamma (h) value in the Statistics area will bedisplayed. You can select from among the following four options:

Local Mean Square. If you select this option, each samplevalue will be divided by the mean square of all values in the laginterval before the variance between the samples pairs iscalculated.

No Normalization. Selecting this option will display the semi-variance value with no normalization.


Population Mean-Square. If you select this option, eachsample value will be divided by the mean square of all thevalues in the data set before the variance between the samplespairs is calculated.

Population Variance. Selecting this option displays the semi-variance of the sample pairs divided by the variance of all thesamples in the data set.

Printing the 3D Semi-Variogram Table

You can now use the Report button located in the lower right-handcorner of the dialog box to print out the semi-variogram table that youhave created and are currently viewing on-screen.

To print the table, follow this procedure:

1. Click Report in the 3D Semi-Variogram dialog box.

2. Select the report destination (file, printer or screen) from theSelect Report Destination dialog box (see Chapter 4: Dialogboxes, Volume I: Core).

3. The report will be generated. The data for all enabled directionalsemi-variograms will be included in the report, as will all fourvariations of the Gamma (h) value.

Viewing the 3D Semi-Variogram in Graphical Format

When you perform a linear semi-variogram calculation, you canalso view the graphical representation of your data on-screen in theQuickGraf utility by clicking the Graph button located in the lowerright-hand corner of the 3D Semi-Variogram dialog box.


Clicking Graph will bring up the following dialog box:

Figure 13-13: 3D Semi-Variogram Display Parameters dialog box

Select one of the two available options:

Output all Semi-Variogram Columns. This option willcreate a graph plotting all defined semi-variograms.

Output current Semi-Variogram Columns. This option willcreate a graph plotting only the semi-variogram currentlydisplayed in the 3D semi-variogram table (as selected from theSemi-Variogram pull-down list at the top of the 3D Semi-Variogram dialog boxsee Figure 13-12).

For more details about working with this utility, see Chapter 13:QuickGraf.

ANALISIS SEMIVARIOGRAMAS

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