1. Isaaks & Co
Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 1
Linear Model of Coregionalization
Variography
User’s Manual
S37x.exe
Autofit.exe

2. Isaaks & Co
Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 2
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 , , , andH 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.

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Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 3
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 AB
W
E 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

4. Isaaks & Co
Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 4
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

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Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 5
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.

6. Isaaks & Co
Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 6
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

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Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 7
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.

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Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 8
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.

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Specialists in Spatial Statistics
1042 Wilmington Way Redwood City CA 94062 Phone 650-369-7069 ed@isaaks.com 9
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.