linear model of coregionalization

Isaaks & Co Specialists in Spatial Statistics

1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 [email protected] 1

Linear Model of Coregionalization

Variography

User’s Manual

S37x.exe

Autofit.exe



The LMOC Toolbox

The variography toolbox for the linear model of coregionalization (LMOC) contains two

executable programs:

1. S37x.exe for calculating directional sample variograms.

2. Autofit.exe for modeling directional sample variograms.

S37x.exe

S37x.exe is a command line C program for calculating directional sample variograms for

the LMOC for two variables. The directional sample variograms consist of:

1. Directional sample auto-variograms for the primary variable

2. Directional sample auto-variograms for the primary variable

3. Directional sample cross-variograms between the primary and secondary

variables.

Sample Variogram Estimators

S37x.exe provides three different estimators for calculating directional sample

variograms.

1. The Traditional variogram is given by:

( ) 0.5* {( )( )}H T H TE A A B B h (1)

where HA is the primary variable at the head of the separation vector h ; TA is the

primary variable at the tail of the separation vector h ; HB is the secondary

variable at the head of the separation vector h ; TB is the secondary variable at the

tail of the separation vector h . Note that if variable B = A, then we have the

traditional auto-variogram for variable A. Similarly, if variable A = B we have

the same for B. Note that the variables A and B must be co-located for this

estimator.

2. A version of the Correlogram given by:

2( ) 1.0 { } { } { } *H T H T ABE A B E A E B h (2)

where , , , and H T H TA A B B are values of the variables A and B at the head and tail

of the separation vector h . Note that 2

AB is the global variance/covariance. Also

note that if variable B = A, then we have (1.0 - auto-correlogram)* 2

AA estimate

for variable A. Similarly, if variable A = B we have the same for B. Note that the

co-location of A and B is not necessary for this estimator. However, if A and B

are not co-located, then one will have to estimate 2

AB using some other method to

pair the variables A and B. For example, one could pair A with the nearest

neighbor B using a limited search radius.



3. The Psuedo variogram is given by:

2( ) 0.5* {( ) }*H TE A B W h (3)

where W is a “re-standardization factor” which re-standardizes the sill of the

psuedo cross variogram to 2

AB . The re-standardization factor W is given by:

2

2 2{ } 2 { } { } { }

2 ABWE A E A E B E B

(4)

Note that W is a global value. It is calculated using global values of A and B. Note that

when A = B or B = A, W = 1.0;

The re-standardization of the psuedo cross-variogram is necessary to obtain the correct

relationship between the primary, secondary, and cross-variogram sills of the LMOC.

Finally, note that the co-location of A and B is not necessary for the psuedo cross

variogram estimator. However, if A and B are not co-located, then

one will have to estimate 2

AB using some other method to pair the variables A and B. For

example, one could pair A with the nearest neighbor B using a limited search radius.

S37x.exe Input and Output files

The following diagram illustrates the various input and output files associated with

S37x.exe

S37x.par

S37x.exe

Data File

Printer File (*.prt) Plot File (*.plt)

Input files

Output Files



The Data File

The data file must be an ASCII text file. The file may be either free or fixed format

delimited with spaces or commas. The first row should be data, not column names or

headers. The file should contain the x, y, z coordinates, and values for the variables A and

B. Each row in the file should have exactly the same number of variables. Thus, missing

values must be coded using values such as –1, -2, -9, -999, or some other negative

number.

The S37x.par File

The file S37x.par is a parameter file for controlling program execution. A template of this

file will be generated by S37x.exe and put into your current working directory if it does

not already exist. An example of a template file that has been edited with some responses

is shown in Figure 1:

Figure 1: An example of the S37x.par parameter file. Note that S37x.exe automatically

calculates 37 directional sample variograms for each variable. For example, the azimuth

increment is 30 degrees while the dip increment is also 30 degrees. This provides 36

directional sample variograms, plus the vertical sample variogran for a total of 37. The

user must specify a lag distance for each of the 37 directions. Note that if the data file

contains a geology or rock type code, one can restrict the variogram calculations to any

combination of codes. Also, note the pairing of sample values can also be controlled by



using or specifying horizontal (the horizontal band is actually not horizontal. It’s strike is

perpendicular to the azimuth, and it dips at the same angle as the separation vector) and

vertical bands

The Printer File (*.prt)

The printer file (*.prt) is an ascii text file that provides all the statistics and

documentation pertaining to the variogram calculations. This file is generated by

S37x.exe each time the program is run. It is a useful file for recording and archiving the

details of the calculations.

The Plot File (*.plt)

This file is generated by S37x.exe each time the program is run and is used by the

program Autofit.exe to fit variogram models to the directional sample variograms.



Autofit.exe

Autofit.exe is a command line C program for the modeling of the LMOC. Autofit.exe

will automatically fit a linear model of co-regionalization to the set of directional sample

variograms calculated by S37x.exe. The LMOC may be a three dimensional model with 1

or 2 nested structures and a unique three dimensional anisotropy for each structure. This

model may be used directly for the co-kriging or co-simulation of two co-regionalized

variables.

Autofit Input and Output Files

Plot File (*.plt)

This file is generated by S37x.exe each time S37x.exe is run. It contains all of the

directional sample variogram information such as azimuth, dip, lag, variogram value,

number of pairs, lag distance etc. This information is used by Autofit.exe to fit the

variogram models.

Plot File (*.plt)

((*.plt)

Autofit.exe

Autofit.par

Gam2.ps

Input files

Output Files

Model.ps Gam1.ps Gam3.ps



File Autofit.par

A template of this file will be automatically generated and placed in the current working

directory if it does not already exist.

Figure 2: An example of the initialization file for Autofit.par. This file can be used to fix

one or more of the variogram model parameters. Autofit.exe reads this file at the

beginning of program execution. If any parameter is negative (with the exception of the

rotation angles) Autofit.exe will automatically fit a value for that parameter. However, if

Autofit reads a positive value for a parameter, then that value is retained for the

variogram model. Thus, if one changes the primary nugget parameter from –1.00 to

0.50, the final variogram model will have a primary nugget value of 0.05.

Note that the default parameters for the rotation angles are –999. This is because negative

rotation angles from –360 to 0 are legitimate rotation angles. Thus, any rotation angle

specified larger than –360 will be interpreted as a legitimate rotation angle and will be

held fixed at that value.

Also, note that one can specify the number of decimal places that will be displayed in the

postscript output for each variable and for the cross variables. This is useful when the

resolution of one variable may be quite different from the other.



File Model.ps

This file is a postscript file written by Autofit.par. It contains the statistics and variogram

model parameters for each direct and the cross variogram model. This file also contains

postscript graphics illustrating the orientation of the anisotropy ellipsoids.

Files Gam1.ps, Gam2.ps. and Gam3.ps

These are also postscript files showing plots of all 37 directional sample variograms and

the trace of the model in each plot direction. The number of pairs is posted alongside

each sample variogram point. The equation of the model with the ranges in the plot

direction is also provided for each direction. These files can be conveniently viewed and

printed using Ghostscript and Ghostview.

Running Autofit.exe

The following figures provide an example of user responses to the program during

program execution:

Figure 3: The user responded to the Input Sample Variogram? prompt with the file name

6800.plt. Then the user specified 25 as the minimum number of pairs; and 9999 as the

maximum allowable drift. Then the program read in 1,476 sample variogram points. The

maximum lag distance read was 1,383. Thus, the user specified 1,000.0 as the maximum

distance to show on the postscript plots. Note the plot distances will be annotated using

one decimal place because the distance 1000.0 was specified using one decimal place.



Figure 4: Continuing with the example in Figure 3, the user next responded by

specifying 2 structures; the exponential model for each structure; a weighted fit by the

number of pairs; and the GsLib Rotation Scheme.

Figure 5: Example of user responses to program prompts for plot titles. These titles will

appear on each of the postscript plots.

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