The variofit application is used to fit model semivariograms to experimental and jackknifed experimental semivariograms (Generated by vario). This can be done manually or automatically using least-squares regression or latin-hypercube sampling techniques. Ergodic variations of the model semivariogram from simulation series may also be evaluated.

The variofit application is composed of three sections; the main menu-bar, the status and log text area, and the drawing or graph area. The menu-bar is used to select all vario commands, the log/status area is used by the program to report important messages or results, and the drawing area is the display area for the semivariogram graph.

The Main Menu:

Figure 9.1

File:

Open:

View Data:

View Results:

Save:

The graph lines will be saved using one of the two file formats specified in the "File Output" section (single format, and multiple jackknifed format).

Save as:

Save Preferences:

WARNING: if "variofit.prf" already exists, you will be warned that it is about to be over-written. If you do not want the old version destroyed you must move it to a new file (e.g. the UNIX command mv variofit.prf variofit.old.prf would be sufficient). When you press OK the old version will be over-written! Moving the old file cannot be done currently from within the application. To rename the you will have to execute the UNIX mv command from a UNIX prompt in another window. If "variofit.prf" does not exist in the current directory, it is created. This is an ASCII file and can be edited by the user. See Appendix C for details.

Print Setup:

Print:

Quit:

Quit Without Saving:

[TOP] [SYNTAX]

Regression:

Calculation Method:

1) Open a experimental semivariogram data file (File:Open).
2) Select Regression:Calculation Method:Automatic
3) Display the Regression:Parameters dialog.
4) Select the desired Solution Method.
5) Select the number of model nests desired and the model type for each nest.
6) Set the Starting Range and the Ending Range for the iterative solutions to iterate upon.
7) Set the number of iterative solutions (Calculation Steps - Division) for the algorithm to iterate upon.
8) Select Regression:Calculate from the main menu-bar to start the regression.

NOTE: Any user input values for Range, Sill, or Nugget will be overwritten when an automatic solution is calculated.

WARNING: There are still bugs in this algorithm and the optimized solution may be an extremely poor fit!

WARNING: The automatic solver uses an iterative approach (there are more unknowns than equations) to optimize the model solution. Because of this it is not recommended that this solver be used to fit three or four nested structures. Using the default value for Division Steps (100) a two-nested model takes approximately 100 times as long to solve as a single-nested model (i.e. 30 sec. vs. 0.3 sec). For the same data set, a three-nested model would take 50 minutes, and a four-nested model several days!

1) Open a experimental semivariogram data file (File:Open).
2) Select Regression:Calculation Method:Manual.
3) Display the Regression:Parameters dialog.
4) Select the number of model nests desired and the model type for each nest.
5) Define the desired values for each appropriate Range, Sill, and Nugget value.
6) Press the Apply button to enter all values in the dialog.
7) Select Regression:Calculate from the main menu-bar to start the regression.

Figure 9.2

Figure 9.3

To fit a model manually using the slider fitting tool, select the Manual Calculation Method and select the Regression:Slider Fitting menu item. This will produce the dialog shown in Figure 9.4.

Figure 9.4

Parameters:

Figure 9.5

Because the graph can get very cluttered when there are several experimental semivariograms displayed, the Visible Models button was added. When pressed, the pop-up dialog shown in Figure 9.6 is created. With this dialog, different experimental semivariograms can be turned on and off.

Figure 9.6

When an automatic solution is desired several other parameters need to be defined. The automated method uses an iterative solver because there are more unknown's than equations. Because of this, some limits must be placed on the iteration. By default, the maximum possible range of the data set (Starting Range) is considered to be the maximum diagonal distance across the data set, and the minimum possible range is considered 0.0 (Ending Range). In the iterative solution, this possible range of Range values is divided into 100 (default value for Calculation Steps - Divisions) subdivisions. For the first iteration, a "best-fit" model will be calculated assuming the maximum range. For the next iteration, the range will be set to the maximum range - (maximum range / 100.0). This will be repeated until the minimum range (0.0) is used. For each iteration, the least-squares error for each model semivariogram will be saved. The model semivariogram with the smallest squared-error, is considered the best-fit. In many situations, it may be reasonable to modify these values. For example, by examining the experimental semivariogram in Figure 9.2, it is clear the range is less than 65 meters. If the data sets maximum possible range was 500 meters a great deal of computational effort would be wasted examining ranges greater than 70 meters. A similar argument can be made for the Ending Range. The number of Divisions defines the fineness of the steps used to determine the range. If the range is fairly narrow fewer divisions may be reasonable; if a very wide range is possible more divisions may be necessary.

NOTE: The program supports only models with up to four structures. It is felt that only in an extremely unusual case would even four structures be justified, much less five or more.

NOTE: The range for each model structure must be between the Starting and Ending Range parameters. Do not over constrain the solution.

NOTE: For multi-nested structures, the number of Calculation Steps- Divisions is the main control on how long the automatic solution takes. Reducing the number of divisions can reduce the solution time significantly!

Slider Fitting:

Solution Method:

Figure 9.7

Least-Squares - This method fits a "best-fit" model curve to the data. No adjustments are made for the nugget, variance, or the relative number of pairs per lag.

Nugget LS - This method determines the nugget by determining the y-intercept of a line drawn through the first two (h) lag values. Once the nugget is determined the standard least-squares approach used above is applied.

Variance LS - This method defines the sill as equaling the variance of the data set. This forces the model curve to honor the data set variance; this is also commonly done by geostatisticians when manually fitting model semivariograms. Once the nugget is determined the standard least-squares approach used above is applied.

Nugget & Variance LS - This method combines the Nugget LS and Variance LS methods described above.

Weighted LS - This method is a normal least-squares regression, except that it weights the error for each lag, based on the number of sample pairs used to generate that lag (h) value (i.e., this method presumes that it is more important to honor a data point representing 500 data pairs than one representing 3 data pairs). The previous four methods do not consider the number of data pairs, therefore they weight all lag values equally.

Calculate:

NOTE: This Regression is not equivalent to the regression option described in Chapter 5 for plotgraph.

[TOP] [SYNTAX]

Jackknife:

Modify:

Figure 9.8

Individual Display:

Figure 9.9

Figure 9.10

[TOP] [SYNTAX]

Latin-Hypercube:

Parameters:

Figure 9.11

Unlike the Regression:Parameters dialog however, only two nested structures can be modeled. Because of the lack of data inherent when this process is used, modeling even two structures is probably suspect. For most cases there is probably little justification for using more than one structure. As with the Regression:Parameters dialog No Model, Linear, Linear with Sill, Spherical, Exponential, Gaussian, and Logarithmic models can be used for each structure. Instead of specifying single Range, Sill, and Nugget values though, the range of values must be specified, because this is a randomized model fitting procedure.

For each variable; Range, Sill and Nugget, a Minimum and Maximum value must be specified (The Minimum may equal the Maximum range, e.g. a constant is implied). This specifies the range of allowable or reasonable values. These entries are preset with default values:

Range: The Minimum Range is set by the longest lag that has its confidence interval fall completely below the confidence interval of the jackknifed experimental semivariogram variances (No lower lag values can fall above the variance either), or failing this, to the minimum confidence interval range of the shortest lag (Realistically, it might be best to set this value to 0.0 in this case since the possibility of pure nugget effect cannot be disregarded) (Figure 9.12). The Maximum Range is set by the shortest lag that has it confidence interval that lies completely above the variance confidence interval. In many models this never occurs, and the Maximum Range is set to the maximum extent of the maximum lag confidence interval (Figure 9.12, the Maximum Range is set to 436 feet).

Figure 9.12

Variance: The Minimum Variance is set to the lower bound of the jackknifed experimental semivariogram confidence interval, and the Maximum Variance is set to the upper bound (Figure 9.12, Minimum = 0.2613, Maximum = 0.2841).

Nugget: The Minimum Nugget is set to 0.0. The Maximum Nugget is set to the upper bound of the shortest lag confidence interval, or the upper bound of the jackknifed experimental semivariogram confidence interval; which ever is the least (Figure 9.12, Minimum = 0.0, Maximum = 0.2762).

Each interval (Range, Variance, and Nugget) can be divided into one to ten Divisions; by default these Divisions are of equal width. If non-equal Division intervals are desired, press the Define button for the appropriate parameter. When this is done, one of the pop-up dialogs shown in Figures 9.13a, 8.13b, and 8.13c will be created. In this dialog the desired absolute division intervals can be assigned. If the Range Increments have been adjusted and equal increments are desired, press the Set Divisions Equal button. By default the Range and Nugget are divided into four Divisions and the Variance one division. In this case, when the Latin-Hypercube sampled semivariograms are calculated (Select the Latin-Hypercube:Calculate main menu-bar option), 16 model semivariograms will be calculated (4 Range x 1 Variance x 4 Nugget divisions x Number of Solutions per Division). An example is shown in Figure 9.14a. For each model semivariogram, the model Range, Variance, and Nugget is randomly selected from within each Range Interval (Equally Spaced selections from within the Range Interval are Possible in addition to the Random Range Selection Option).

Figure 9.13a, Figure 9.13b, and Figure 9.13c

Figure 9.14a

NOTE: As long as no parameters are changed in this dialog, the randomly selected model semivariograms will always be exactly the same; this is because the same initial Randomize Seed is used. This allows the user to repeat previous work. To generate a new series of random model semivariograms, select a new Randomize Seed (any integer value).

NOTE: Apparently invalid model semivariograms are drawn, because in some cases the models are reasonable, but the decision algorithm in the software is overly sensitive. Carefully examine each model before discarding it.

Figure 9.14b

NOTE: Draw Valid Range Area is not currently installed.

Individual Display:

Figure 9.15

Figure 9.16

NOTE: Only Valid models are saved.

Save:

Save as:

Calculate:

[TOP] [SYNTAX]

Series:

Display:

Figure 9.17

Figure 9.18a; Figure 9.18b; Figure 9.18c; and Figure 9.18d.

Parameters:

prefix.#.suffix

Figure 9.19

An example series might be:

sample.1.gam
sample.2.gam
sample.3.gam
sample.4.gam

[TOP] [SYNTAX]

Graph:

Border:

Error-Bar Style:

Fonts:

Labels:

Legends:

Figure 9.20

Mesh:

Style:

[TOP] [SYNTAX]

Log:

[TOP] [SYNTAX]

Plot:

[TOP] [SYNTAX]

Help:

[TOP] [SYNTAX]

Example of Using Variofit:

Loading a Pre-Existing Experimental Semivariogram:

The difficulty in using this package comes from generating the model semivariogram correctly. Once this has been done, the appearance of the graph can be modified using many of the same tools described for plotgraph in Chapter 5 .

There are three methods to load a file in variofit. The first is to execute variofit from the UNIX prompt and open the file from the menu, the second is to pass the file as a command line argument, and the third is to define the file name in the program preference file. To open a file from the main menu, execute variofit from the UNIX prompt:

> variofit

> variofit [optional arguments] filename

> variofit -ymin 0.0 clark.gam

NOTE: This show Figure 9.3 although other variables then those passed on the command line here were defined. These variables could have been set using the menus, or using a preference file (Appendix C). A preference file is used to define user preferred variable default values. Every time variofit runs, it searches the current working directory for the file variofit.prf. If it exists, variofit reads the file and sets the variables as specified. This is the third way to open a file, because one of the arguments in the preference file is the name of the graph file.

Calculating a Single Model Semivariogram - Automatic Mode:

To fit a model semivariogram to the experimental semivariogram in Figure 9.3 (clark.dat), an initial guess might be to fit a single-nested spherical model (Select a Spherical model for nest 1) to the data set. The sill appears to approximately equal the data set variance; for this reasons, using the Variance LS Solution Method may be appropriate. When Regression:Calculate is pressed the model semivariogram in Figure 9.21a is generated.

This model is a reasonable approximation based on the constraints used, but it is not a good fit. The calculated nugget of 1.07 is much larger then one would expect. Extending the slope of the first few lags suggests a value closer to 0.4! The model, also, in zones tends to consistently over- or under-estimate the experimental semivariogram lag values. A better model selection may be a two-nested spherical structure (Select a Spherical model for nest 2). For nests one and two, select the Spherical model type. When the model is recalculated (Press Regression:Parameters), the model shown in Figure 9.21b is a better fit.

This, model honors the early portion of the experimental semivariogram more accurately then the single nested model and is therefore a better fit. It does still tend to overestimate the variance over the 15 to 45 meter lag distance. Using the automatic fitting algorithm, however, this is about the best that will be done.

Figure 9.21a, and Figure 9.21b

Note that this automatic model fitting process is not completely automatic. The modeler is responsible for selecting the appropriate model structure, the algorithm only fits a curve based upon set constraints. If the constraints are poorly defined, the model fit will also be poor.

Calculating a Single Model Semivariogram - Manual Mode:

Although this model could be further refined, there is probably little reason to do so; with the kriging process, there would probably be little difference between a map generated with this model and a more refined model. Practically , there would probably be little difference between the model shown in Figure 9.21b and Figure 9.21c.

Figure 9.21c

Calculating Multiple Latin-Hypercube Sampled Model Semivariograms:

As an example, load (File:Open) the experimental jackknifed semivariogram file well.jack.gam. This series of experimental semivariograms describes the uncertainty in the spatial distribution of materials based on the wells shown in Figures 1.3 and 1.4. The data set (well.jack.gam) was calculated by vario ( Chapter 8 example). Once the file is loaded, the jackknifed experimental semivariograms and their uncertainty will be displayed (Figure 9.22a).

Figure 9.22a

The Latin-Hypercube technique allows the modeler to select a limited number of model semivariograms which cover the range of solutions which honor the confidence intervals of jackknifed experimental semivariograms and the modelers opinion of reasonable values. After loading the file, the modeler must specify how to sample the jackknifed experimental semivariograms to generate model semivariograms. For this data set there, is definitely not enough data to suggest a two-nested structure. Model this data set with only one nest. Select a Spherical model for nest one. A spherical model is selected because it has given reasonable results for similar conditions. Based on the data available, it would be difficult to argue one model was better than another. Because the selection criteria for range needs work, reset the Minimum Range to 10.0 (data implies pure-nugget effect possible, a 0.0 range though is felt highly unlikely (expert opinion)), and the Maximum Range to 230.0 (Confidence interval of third lag suggests range could be greater then 230 feet but it is unlikely to be much longer). Setting these limits sets reasonable ranges for the Range variable. Once these parameters are set, select Latin-Hypercube:Calculate. Sixteen model semivariograms will be generated and displayed (4 Range Divisions * 1 Variance Division * 4 Nugget Divisions * 1 Solution per Division). Every time Latin-Hypercube:Calculate is pressed, using these parameters these exact 16 model semivariograms will be calculated. To generate a different set of random sampled model semivariograms, change the Randomize Seed value to another integer value (Set to 10, it was 1 (default value)). If this is done, a new series of model semivariograms will be calculated (Figure 9.22b). Note that in this series, two models are drawn with dashed red lines. This indicates that these models do not honor the uncertainty limits of the experimental semivariograms and should not be used for future calculations. To save these valid models, Select the Latin-Hypercube:Save menu option and supply an appropriate file name. This file can be used for input for the sisim module (Chapter 14 ).

Figure 9.22b

NOTE: For different levels of uncertainty, the confidence intervals can be reduced or expanded in the Jackknife:Modify dialog.

Running from the Command Line:

Syntax:

variofit [-am #] [-cs #] [-er #.#] [-esp #] [-fnt1 " "] [-fnt2 " "] [-fnt3 " "] [-fnt4 " "] [-fnt5 " "] [-fnt6 " "] [-fnts1 #.#] [-fnts2 #.#] [-fnts3 #.#] [-fnts4 #.#] [-fnts5 #.#] [-fnts6 #.#] [-help] [-jcl #.#] [-jebd #] [-jebr #] [-jev #] [-jir #] [-jjr #] [-lc {#}] [-lgf " "] [-lgmp #] [-lgms #] [lgpa #] [-lgpp #] [-lgps #] [-lhck #] [-lhdr #] [-lhmax1 #.#] [-lhmax2 #.#] [-lhmax3 #.#] [-lhmin1 #.#] [-lhmin2 #.#] [-lhmin3 #.#] [-lhmt1 #] [-lhmt2 #] [-lhout " "] [-lhrj #] [-lhseed #] [-lhseg1 #] [-lhseg2 #] [-lhseg3 #] [-lhsel #] [-lhspd #] [-lpbm #.#] [-lpc #] [-lpd #] [-lpf " "] [-lph #] [-lplm #.#] [-lpo #] [-lppsext " "] [-lpq " "] [-lpr] [-lps #] [-lprm #.#] [-lptm #.#] [-lty {#}] [-ltk {#.#}] [-m1 #] [-m2 #] [-m3 #] [-m4 #] [-md #] [-ms #] [-mox #.#] [-moy #.#] [-mx #.#] [-my #.#] [-ng #.#] [-out " "] [-prf " "] [-r1 #.#] [-r2 #.#] [-r3 #.#] [-r4 #.#] [-rfh #] [-run] [-s1 #.#] [-s2 #.#] [-s3 #.#] [-s4 #.#] [-scl #.#] [-sdc #] [-sdm #] [-sdx #] [-sex " "] [-sfl {#}] [-sp " "] [-sr #.#] [-sse #] [-sss #] [-ssz {#.#}] [-st #] [-sttl " "] [-sty {#}] [-sv #] [-svm #] [-ttl " "] [-xfmt " "] [-xlabel " "] [-xmax #.#] [-xmin #.#] [-xMt #.#] [-xmt #] [-xto #.#] [-xy #.#] [-yfmt " "] [-ylabel " "] [-ymax #.#] [-ymin #.#] [-yMt #.#] [-ymt #] [-ys #.#] [-yto #.#] [filename]

Meaning of flag symbols:

# = integer
#.# = float
" " = character string.
{} = variable is an array. Values must be seperated by a ',' and no spaces are allowed. Do not use the "{ }" symbols on the command line.

NOTES:

1) All parameters in [] brackets are optional.
2) Quotes must be used around character strings.
3) Filename, if given, must be listed last.
4) If no default is given, the feature is not currently supported on command line.

If no entry is required for flag, flag command executed.

-am	=	active experimental model (for calculation)	default = 1
-cs	=	calculation steps for automatic solution	default = 100
-er	=	ending range	default = 0.0
-esp	=	exageration scale priority	default = 0
		0 1 = = favor y-exageration scale (-ys) favor x/y ratio
-fnt1	=	main title font	default = Helvetica-Bold
-fnt2	=	secondary title font	default = Helvetica-Bold
-fnt3	=	axes label font	default = Helvetica
-fnt4	=	division font	default = Helvetica
-fnt5	=	annotation font	default = Helvetica
-fnt6	=	mouse position font	default = Helvetica
-fnts1	=	main title font size	default = 24.0
-fnts2	=	main title font size	default = 15.0
-fnts3	=	main title font size	default = 15.0
-fnts4	=	main title font size	default = 12.0
-fnts5	=	main title font size	default = 10.0
-fnts6	=	main title font size	default = 12.0
-help	=	give this help menu
-jcl	=	jackknife error-bar/band confidence level	default = 90.0%
-jebd	=	plot jackknife error-bands (lag variance)	default = 1
		0 1 = = False True
-jebr	=	plot jackknife error-barss (g(h) variance)	default = 1
		0 1 = = False True
-jev	=	plot jackknife error variance bars	default = 1
		0 1 = = False True
-jir	=	plot intermediate jackknife results	default = 1
		0 1 = = False True
-jjr	=	plot full unjackknifed semivariogram	default = 1
		0 1 = = False True
-lc {}	=	line color	default = variable
		0 1 2 3 4 5 6 7 = = = = = = = = Black White Red Green Blue Magenta Yellow Cyan
-lgf	=	log file name	defalut = "log.dat"
-lgmp	=	legend parameter position	default = 1
		0 1 2 3 = = = = Top left Top right Bottom left Bottom right
-lgms	=	legend model displayed	default = 1
		0 1 = = False True
-lgpp	=	legend parameter position	default = 1
		0 1 2 3 = = = = Top left Top right Bottom left Bottom right
-lhck	=	check latin-hypercube samples in range	default = 1
		0 1 = = False True
-lhmax1	=	latin-hypercube maximum range	default = max. lag
-lhmax2	=	latin-hypercube maximum range	default = max. variance
-lhmax3	=	latin-hypercube maximum range	default = max. variance
-lhmax1	=	latin-hypercube maximum range	default = min. lag
-lhmin2	=	latin-hypercube maximum range	default = min. variance
-lhmin3	=	latin-hypercube maximum range	default = min. variance
-lhmt1	=	latin-hypercube nest #1 model structure	default = 3
		0 1 2 3 4 5 6 7 = = = = = = = = No Model (Not available for nest #1) Linear Linear With Sill Spherical Exponential Logarithmic Power (NOT INSTALLED) Gaussian
-lhmt2	=	latin-hypercube nest #2 model structure (See -lhmt1)	default = 3
-lhout	=	latin-hypercube output file	default = "junk.lhc"
-lhrj	=	draw only latin-hypercube samples in range	default = 0
		0 1 = = False True
-lhseed	=	latin-hypercube random seed	default = 1.0
-lhseg1	=	latin-hypercube range divisions	default = 4
-lhseg2	=	latin-hypercube C divisions	default = 1
-lhseg3	=	latin-hypercube nugget divisions	default = 4
-lhsel	=	latin-hypercube range selection	default = 1
		0 1 = = equally spaced ramdom
-lhspd2	=	latin-hypercube solutions per division	default = 1
-lpbm	=	page bottom margin	default = 1.5
-lpc	=	number of copies to print	default = 1
-lpd	=	print destination	default = 0
		0 1 = = Printer File
-lpf	=	print filename	default = "junk.ps"
-lph	=	print header page	default = 0
		0 1 = = False True
-lplm	=	page left margin	default = 1.5
-lpo	=	print orientation	default = 0
		0 1 = = Portrait Landscape
-lppsext	=	search extention for postscript files	default = "*.ps"
-lpq	=	print queue	default = "ps"
-lpr	=	print file at specified orientations
-lprm	=	page right margin	default = 1.0
-lps	=	print output	default = 0
		0 1 = = Black & white Color
-lptm	=	page top margin	default = 1.5
-lsfl {}	=	fill line symbol	default = 0
		0 1 = = False True
-lsc {}	=	line symbol color	default = variable
		1 2 3 4 5 6 7 = = = = = = = Black White Red Green Blue Magenta Yellow Cyan
-lssz {}	=	line synbol size	default = 9.0
-lsty {}	=	line symbol type	default = 0
		-1 0 1 2 3 4 = = = = = = No Symbol Circle Cross Diamond Square X
-ltk {}	=	line thickness	default = 1.0
-lty {}	=	line type	default = 0
		-1 0 1 2 = = = = No Line Solid Dashed Double Dashed
-m1	=	model types nest #1	default = 3
		0 1 2 3 4 5 6 7 = = = = = = = = No Model (Not available for nest #1) Linear Linear With Sill Spherical Exponential Logarithmic Power (NOT INSTALLED) Gaussian
-m2	=	model types nest #2 (See -m1)	default = 0
-m3	=	model types nest #3 (See -m1)	default = 0
-m4	=	model types nest #4 (See -m1)	default = 0
-md	=	dash mesh	default = 0
		0 1 = = False True
-mox	=	X mesh origin	default = 0.0
-moy	=	Y mesh origin	default = 0.0
-ms	=	use mesh	default = 0
		0 1 = = False True
-mx	=	X mesh frequency	default = 1/10 DX
-my	=	Y mesh frequency	default = 1/10 DY
-ng	=	nugget	default = 0.0
-out	=	output *.out filename	defalut = "junk.out"
-prf	=	preference file name	defalut = "variofit.prf"
-r1	=	range for nest #1	default = 0.0
-r2	=	range for nest #1	default = 0.0
-r3	=	range for nest #1	default = 0.0
-r4	=	range for nest #1	default = 0.0
-rfh	=	screen refresh	default = 0
		0 1 = = On exposure On update
-run	=	run and calcualte variofit without X-interface
-s1	=	sill for nest #1	default = 0.0
-s2	=	sill for nest #1	default = 0.0
-s3	=	sill for nest #1	default = 0.0
-s4	=	sill for nest #1	default = 0.0
-scl	=	series confidence level	default = 95.0
-sdc	=	draw series lag confidence bars	default = 1
		0 1 = = False True
-sdm	=	draw series lag mean	default = 1
		0 1 = = False True
-sdx	=	draw series lag extents	default = 1
		0 1 = = False True
-sex	=	series extention	default = ".gam"
-sp	=	series prefix	default = " "
-sr	=	search range	default = Max. diagonal
-sse	=	ending series	default = 1
-sss	=	starting series	default = 1
-st	=	automatic solution type fitting algorithm	default = 4
		0 1 2 3 4 = = = = = Least-Squares Nugget Least-Squares Variance Least-Squares Nugget & Variance Least-Squares Weighted Least-Squares
-sttl	=	Secondary title	default = " "
-sv	=	solver method	default = 0
		0 1 = = Automatic Manual
-svm	=	experimental semivariogram read from file	default = 1
		0 1 = = False True
-ttl	=	Main title	default = Filename
-xfmt	=	Number of decimal places for X-axis	default = ".2f"
-xlabel	=	X-axis label	default = "X"
-xmax	=	Graph X-maximum	default = Data Maximum
-xmin	=	Graph X-minimum	default = Data Minimum
-xMt	=	X main tic frequency	default = 1/10 DX
-xmt	=	Number of minor X tics	default = 5
-xto	=	X axis label origin	default = 0.0
-xy	=	xy ratio	default = 1.5
-yfmt	=	Number of decimal places for X-axis	default = ".2f"
-ylabel	=	X-axis label	default = "Y"
-ymax	=	Graph Y-maximum	default = Data Maximum
-ymin	=	Graph Y-minimum	default = Data Minimum
-yMt	=	X main tic frequency	default = 1/10 DY
-ymt	=	Number of minor Y tics	default = 5
-ys	=	Y-axis exageration relative to X-axis	default = Calculated
-yto	=	X axis label origin	default = 0.0

An example command might be (typed on one line):

variofit -xmin 0.0 -ymin 0.0 -ymax 12.0 -md 1 -ms 1 -mx 1.0 -my 1.0 -xMt 1.0 -yMt 1.0 -ttl "Semivariogram of Elevation Data" -sttl "UNCERT variofit module" -xlabel "distance (feet)" -ylabel "gamma h" -xfmt ".1f" -yfmt ".1f" -esp 1 -xy 1.0 water.out

Setting up the Input Data File:

Single Models:

SEMIVARIOGRAM #[Model Number][/][Number of Models in File]

for (i=1;i<=# of Models)

[Lags in Model]
for (j=1;j<=# of Lags)

[Minimum Lag][Maximum Lag][Average Lag][(h)][Pairs]

end for
[Data Set Variance]
[Data Set Mean]
[Calculation Type: 0 = Point, 1 = Gridded]
if (calculation type == Point)

[Lag]
[Search Direction]
[Plunge]
[Horizontal Bandwidth]
[Vertical bandwidth]

else

[X Step increment]
[Y Step increment]
[Z Step increment]

endif
[Spatial Equation: 0 = Semivariogram, 1 = Cross-Semivariogram, ... (See -set flag in vario (Chapter 8))]
if (spatial equation == Indicator or Soft-Indicator Covariance)

[Low Cutoff]
[High Cutoff]

endif
[Data Set File Name]
if (calculation type == Point)

[X Column]
[Y Column]
[Z Column]
[Head Column]
if (spatial equation == Covariance, Cross-Semivariogram, Correlogram, or Soft Indicator Covariance)

[Tail Column]

endif

endif

end for

Jackknifed Models:

JACKKNIFED SEMIVARIOGRAM
[Data Set Mean] [Data Set Variance]
[# of Lags]
for (i=1;i<=# of Lags)

[Lag #] [Mean of Lag] [Lag Standard Deviation] [.7;(h) Standard Deviation]

end for
[# of Experimental Semivariograms]
for (i=1;i<=# of Semivariogram)

[# of Lags for this Semivariogram]
for (j=1;j<=# of Lags)

[Minimum Lag][Maximum Lag][Average Lag][(h)][Pairs]

end for

end for

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Output Data File Formats:

Single Models:

	!-------------------------------------
	!Model Segment 1:        Spherical
	!    Range  = 9.600000
	!    Sill   = 1.109714
	!
	!Model Segment 2:        Spherical
	!    Range  = 49.600000
	!    Sill   = 1.133128
	!
	!Nugget = 0.307158
	!
	!Variance = 2.550000
	!Mean     = 3.041230
	!
	!Sum squared error = 0.169223
	!-------------------------------------
	MULTIPLE LINES
	3
	2 1 5  1.0 -1 5  5.0 1 Variance
	    0.00000     2.55000
	   80.00000     2.55000
	0 -1 2  1.0 0 2  5.0 1 Variance
	200 0 4  3.0 -1 4  5.0 1 Variance
	    0.00000     0.30716
	    0.40000     0.39018
	
	   .
	   .
	   .

basically this file can be used as a reference to lookup model parameters, and it a valid graph file for plotgraph (Chapter 8).

Latin-Hypercube Sampled Models:

[# of Thresholds]
for (i=1;i<=# thresholds)

[Threshold] [Prior CDF]
[# of models for threshold]
for (j=1;j<=# of models)

[# of nests] [nugget]
for (k=1;k<=# of nests)

[Model Type] [Range] [Sill] [X-Anis] [Y-Anis]
end for

[Major X-Y Search Direction] [Plunge Angle]

end for

end for

For an example file, see data file well.aniso.2.lhc.

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Variofit Mathematics:

Experimental Semivariogram Calculation:

Automated Semivariogram Fit:

(9-1)

This process is not perfect, but it does yield reasonable results and it can give the experienced user a good starting equation when fitting complicated nested models.

In the case of a Spherical Model with a nugget and one sill, the model semivariogram is defined as (Journel and Huijbregts, 1978):

; for h < h <= a. (9-2)

; for h > a. (9-3)

; for h = 0. (9-4)

C = point variance.
Co = nugget or random variance.
a = range.
h = sample distance.

; for h <= a (9-5)

= experimental semivariogram value at h,

(9-6)

(9-7)

(9-8)

(9-9)

Equation (9-9) is minimized by setting its first two derivatives with respect to C and Co equal to zero:

therefore

(9-10)

and

therefore

(9-11)

Using a matrix form, the two unknowns, C and Co, can be calculated by:

(9-12)

This equation is in the form:

Where

is the equation to be solved. Once the combination of C and Co which minimize the solution are determined, the sum of squared errors is calculated, and a new range (a) is selected. This is done iteratively until the range (a) which yields the minimum error is found. Generally, it is sufficient to evaluate "a" to plus or minus 1/100th of the maximum distance between data points (Instead of using the maximum distance across the data set, another distance may be more appropriate and yield more stable estimates). The form of the solution matrices in equation (9-12) are the same regardless of the type model that is fit to the data. The only difference is that the term fi, is evaluated using the appropriate model. These are defined as follows (Journel and Huijbregts, 1978):

Exponential Model:

(9-13a)

Gaussian Model:

(9-13b)

Logarithmic Model:

(9-13c)

Linear Model:

(9-13d)

Linear With Sill Model:

(9-13e)

Spherical Model:

(9-13f)

where for the linear models, m is a slope term. NOTE: In the future Power and Hole Effect models will be included in the software.

Power Model:

(9-13g)

When describing the experimental semivariogram, often a nugget and a single point variance (i.e. one sill) term is inadequate. Models that fit such data have what is called a nested structure, and may be described by several point variance terms (i.e., C0, C1, C2,...). So, for a two-nested semivariogram:

(9-14)

where:

C0 = nugget.
C1 = point variance of the first nest with range a1.
C2 = point variance of the second nest with range a2, where a2 >= a1.
di, eii = an experimental model, as f used in equations 9-13a-g. Note that these models may be of different types.

Under this arrangement, three terms, C0, C1, and C2, must be determined. The solution matrix is:

(9-15)

For a model with three nests, four terms, C0, C1, C2, and C3 must be determined. The format is similar:

(9-16)

where

C0 = nugget.
C1 = point variance of the first nest with range a1.
C2 = point variance of the second nest with range a2, where a2 >= a1.
C3 = point variance of the second nest with range a3, where a3 >= a2 >= a1.
di, ei, fi = n experimental model, as f used in equations 9.13a-g. Note that these models may be of different types.

and the solution matrix is:

(9-17)

To demonstrate how these matrices were developed, the final case developed where four nests are used will be derived. In this case there are nine unknowns, the five point variance terms C0, C1, C2, C3, and C4, and four ranges. In this case, as in the previous cases the point variance terms are determined explicitly, but the four range terms are determined through an iterative process. For the system where:

(9-18)

where

C0 = nugget.
C1 = point variance of the first nest with range a1.
C2 = point variance of the second nest with range a2, where a2 >= a1.
C3 = point variance of the third nest with range a3, where a3 >= a2 >= a1.
C4 = point variance of the fourth nest with range a4, where a4 >= a3 >= a2 >= a1.
di, ei, fi, gi = an experimental model, as f is used in equations 9-13a-g. These models may be of different types.

(9-19)

where:

(9-20)

(9-21)

(9-22)

(9-23)

(9-24)

(9-25)

Modified Least-Squares Regressions:

(9-26)

(9-27)

Latin-Hypercube Sampling:

An alternative approach to random selection of a large number of possible semivariogram models is to use latin-hypercube sampling. This reduces the number of simulations required to insure that the "flavor" of all alternatives is addressed (McKay et al, 1979).

This approach can yield a daunting number of simulations, many of which will bear little resemblance to one another if the data set is small. Such a situation results in the obvious conclusion that some data sets provide so little information about a site that more data should be collected before further assessment is undertaken. If the data are more abundant, the range of possible models will be constrained, and the simulated models may represent a modest range of possible subsurface interpretations. If the jackknifed semivariogram has small error-bars, the entire process of using a variety of semivariograms for simulation of one site can be omitted because the process is not likely to indicate a larger uncertainty associated with the interpretation of such well characterized sites.

Recall that the objective of this approach is not to make a single best estimate of the subsurface interpretation, but to evaluate the possible range of subsurface character based on available data. From a purely mathematical approach this may be computationally intractable, however incorporation of expert opinion into the process makes it possible to limit the reasonable alternatives.

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Bibliography (variofit):

Clark, I., 1979, Practical Geostatistics, Elsevier Applied Science, London and New York.

Journel, A.G. and Ch. J. Huijbregts, 1978, Mining Geostatistics, Academic Press, London.

McKay M.D., R.J. Beckman and W.J. Conover. 1979. "A Comparison of Three Methods for Searching of Input Variables in the Analysis of Output From a Computer Code", Technometrics. Vol. 21, No. 2, pp 239-245.

Wingle, W.L., and E.P. Poeter, 1993, Evaluating Uncertainty Associated with Semivariograms Applied to Site Characterization. Ground Water, Vol. 31, No. 5, pp 725-734.

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