Tutorial Session Three (Part 1) Semi-variograms
The example session with EcoSSe which is described below is
intended as an example run to familiarise the user with the package. This
documented example illustrates one possible set of analyses which may be
carried out. One of the most neglected aspects of statistical analysis ---
especially of spatial data --- is the purely visual assessment of the sample
data. It takes you through the following sequence of analyses:
Ø
Calculating
and interpreting a semi-variogram
There are many other facilities within the
package, which are given as alternative options on the menus. To start the
tutorial, choose EcoSSe from your Start menu. See Tutorial One for starting up
and specifying your ghost file output.
As you can see from the above I have elected to
read in a set of sample data by clicking on the 
option and selecting 
from the menu which appears. EcoSSe
will remember the last five data files accessed and include these in your
options. Three input file types can be read in. I will read in a standard
Geostokos data file.
The layout of such files is described in detail
in the main EcoSSe
documentation. The routine which reads in the data shows the first 10 lines of
your data file so that you can check it is going in OK. The routine also checks
whether we actually had the correct number of samples on the file and informs
you if there is any discrepancy.
Even if you select a file from the list of
previously analysed data files, EcoSSe will ask you to confirm your choice. This
is actually a quick way of getting back to your working directory, since you
can change your choice at this point. Be warned, though, that if you change
which file you want to read it must be the same type of file – that is, if you
are reading a standard Geostokos data file, you cannot change your mind at this
point and read in a CSV type file.
For this illustration, I have selected coalmine.dat for my input data file.
This is a set of 116 borehole samples drilled into an unspecified coal seam in
southern
As your data is read in, it is stored on a
working binary file. A progress bar will indicate how far the process has gone.
When data input is complete, your Window should look like this:
Semi-variogram
calculation
When the data has been read in, you will see
that the “greyed out” options on the main menu bar will be activated. We use
the menu bar to select an option, say:
The screen will prompt you to choose the three
variables for the analysis. You will see two dialog boxes: the one in the top
left hand corner lists the variables available for analysis in your data file:
the bottom right box shows the variables
already chosen (at this point, none!):
The routine, needs to have information on the
position of the samples and on the value at each sample location. This
particular data file only contains three variables. However, EcoSSe
does not know (as yet) which of these variables is which.
There is a lot of information on the screen. At
the bottom of the Window, you see the “status bar” which shows the name of the
current data file and the title read from that file. The “already chosen”
dialog box shows you that you are expected to select variables to be the “X
(east/west) co-ordinate”, “Y (north/south) co-ordinate” and “Measurement to be
analysed” for your semi-variogram.
The upper left dialog box lists the variable
names as they appeared in the data file and is prompting you to choose the
variable which will be the “X co-ordinate” on the graph. For this example, let
us choose Easting for the X co-ordinate:
We may then choose “Northing” for the Y
co-ordinate:
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Finally, we must choose the variable to be
analysed and state any relevant transformations to be made. For this data we
require no transformation of the variable “Coal Value (KJ)”, so click on 
.
The 
dialog now shows the complete set of chosen
variables and has moved to the upper left corner. You have the option to change
your mind here by clicking on 
.
This choice of variables is acceptable, so click
on 
to proceed. This may seem tedious to you at
the moment, but (later) try running the program with another set of data with
more variables. Or try a data set where the columns are in a different order.
The EcoSSe
input routine has been written to allow you this flexibility in building
your data files.
Now, we may finally proceed to calculating a
semi-variogram. For the complete data set, samples are paired up. The
difference between the values of the two samples is calculated and squared.
Plotting each of these points on a graph – squared difference versus distance – results in a “variogram cloud”.
For the semi-variogram interpretation and
modelling routines the “differences” are grouped together into “distance”
intervals. That is, all pairs of samples which are more or less the same
distance apart are grouped together and the differences averaged. To do this,
you must choose a distance interval and a number of groups. The maximum distance
considered will be the product of these two values.
You have the opportunity to specify your own
directions, in which case you will need to define direction as azimuth clockwise from North.
Alternatively, you may accept the four default directions: 0", 45", 90" and 135" - North, Northeast, East and Southeast. Finally, you can simply make a
graph which ignores direction entirely and groups all possible pairs of samples
into one semi-variogram. If you choose the “main points of the compass” you
will also get the “omni-directional” semi-variogram. For user defined
directions, you must also specify a tolerance angle to be allowed on either
side of your specified azimuth. The default directions allow 22.5" either side, so that the four directions cover all possibilities.
Choosing the main points of the compass:
Note that choosing this option reveals the two
input boxes for “interval between points” and the number of intervals. There has been much discussion of the choice
of lag interval in recent times. Sometimes, you just have to experiment. If you
have absolutely no idea how far apart your samples are, do a nearest neighbour
analysis before you try constructing a semi-variogram. This will give you some
idea of the ‘natural’ spacing amongst your samples.
For the coalmine data, the average inter-sample
spacing is around 440 metres, with a mode at around 225 metres.
A progress bar and two variogram cloud plots
will appear on your screen to let you know that the calculation is proceeding. The
software goes through the data set and make all
possible pairs of one sample with another. For each, the distance between the
sample locations is calculated. In addition, the difference in value between
the samples is calculated and squared.
The above is the classic variogram cloud, where
the horizontal axis is the distance between the sample locations and the
vertical axis is the square of the difference between the sample values. Each
point on the graph represents one pair. The graph below is a more modern
presentation, where relative location of each pair is shown – i.e. relative
distance eat/west and relative distance north/south.
When all pairs have been found and the
differences, distances and relative orientations found, the screen is modified
in three ways:
The top left plot is overwritten in the
following way: each square on the graph represents a 225 metre interval
east/west and a 225 metre interval north/south. All pairs found in a particular
square are averaged – to be strict, the “differences in value squared” for all
those pairs is averaged. That average is then halved. This is called a ‘semi-variogram
map’. The top right hand graph shows the number of sample pairs which were
found in each of those 225 by 225 metre squares. The more traditional method of
presenting the semi-variograms is shown in the bottom right hand plot.
With our choices, all pairs of samples between
112.5 and 337.5 [225 ± 112.5] metres apart will be grouped together. For each
of these pairs, the difference in value was calculated and squared. All of
these values will be added together and divided by twice the number of pairs.
This calculation will result in one point to be plotted on our final
semi-variogram graph. This process will be repeated for all pairs of samples
between 337.5 and 552.5 miles apart, and so on. Different directions are
illustrated by different symbols in this graph.
New menus will appear at the top of the screen:
You can continue to look at your
semi-variograms in the four part screen or choose 
to magnify the bottom left hand plot into the whole
screen.
You will probably find this more useful for
model fitting. The symbols in this graph are scaled to illustrate the number of
pairs of samples which were found in that interval. The largest symbol in this
graph has 102 pairs grouped together into one interval. The smallest point has
only 2 pairs in its calculation and is, one would think, somewhat less
reliable.
You can plot any, some or all of your
calculated semi-variograms at once. Select 
from the 
menu and a new dialog will be displayed.
This dialog lists all of the semi-variograms
which have been calculated (and for which the routine found pairs of
samples). You may plot any combination
of these calculated semi-variograms on the screen at once. Next to each
calculated semi-variogram is a check box. At the top of the dialog, you will
see the message “You may select one or more of these at one time”. Check the
boxes for the ones you want to plot.
At the right hand side of the dialog, you will
see two small graphs. These indicate the type of graph you can plot. There are
two ways in which the graph can be plotted.
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Firstly
as a symbol for each calculated point on the graph, as we have seen above. |
Secondly
we can provide a rougher, but perhaps more visually informative, graph by
shading areas. |
Plotting the four directional semi-variograms
using the shaded option, results in:
This display is produced as follows:
q
For each calculated
semi-variogram, we join the first point to the third, the third to the fifth
and so on.
q
The second point is joined
to the fourth, the fourth to the sixth and so on.
q
The area between these two
lines is shaded.
This display is easier to interpret, especially
for beginners. It does not, however, give any information about the number of
pairs of samples in each interval. One single pair of samples giving an erratic
When you have looked at the graphs to your
heart’s content, you will be offered the choice to fit a model to the
calculated (or experimental)
semi-variogram.
Note in particular the last option which
enables you to store the experimental semi-variograms - not the model - on
a text file for input to (say) a report quality graphics package. An EcoSSe option (read in experimental
semi-variograms) exists, which allows you to read this file back in and
continue with the modelling stage.
Apart from an odd couple of erratic points, we
see that the four semi-variograms look very similar. We can, therefore, assume
that we have no apparent anisotropy.
In this case, we can elect to combine the four directions and plot only the “omni-directional”
semi-variogram. Select 
from the 
menu and select the 
option.
Fitting a semi-variogram model
It would now be appropriate to fit a model to
the experimental graph, so that we can proceed to estimating unsampled locations.
We choose to plot only one calculated
semi-variogram – the one which includes all pairs regardless of direction. In
‘symbol’ mode, we can show the number of
pairs which went into each calculated point by the size of the symbol
plotted.
Note that in this graph we can clearly see the
number of pairs in each ‘point’ falling off as the distances get larger than
half the extent of the study area (around 5,000 metres).
Now that we have a reasonable experimental
semi-variogram, we can venture to fit a model to it. This is a mathematical
function which can be used in the kriging routines to produce “optimal” linear
estimates.
A dialog for semi-variogram model definition is
provided:
At present the software offers a limited number
of semi-variogram models which have proved useful in past applications. The
names of the models available will appear in the dialog at the top of the
screen. EcoSSe will prompt you for the
values of the necessary parameters for the chosen model. Full description of
all available models in given in the full documentation and in Practical
Geostatistics 2000.
In this example we chose a Spherical model by
clicking on 
.
The dialog activates the table in which the parameters will appear. You can
manually change the values of the parameters at any stage except when
“adjusting the model”.
All models have the possibility for a “nugget
effect” or discontinuity at zero distance. The first prompt asks you to specify
this value by placing the cursor in the appropriate position and clicking the
right button on your mouse.
As the cursor tracks across the screen, the
value of the nugget effect will change in the left hand dialog:
Once you click the right mouse button, a red
blob will appear at your chosen nugget effect.
You will then be prompted to move the cursor to
choose where the graph levels off, i.e. the range of influence and final sill.
The cursor will change to a hand grabbing 
Horizontal
and vertical bars will appear around your cursor to help you visualise the
correct place to stop and right click your mouse.
When you finally click the right button on your
mouse, a line will be drawn on the graph showing the specified model. You will
notice that the model semi-variogram shows as a wide fuzzy line. This is
intended to reflect the intuitive, subjective nature of model fitting.
If the model is totally the wrong shape, you
might want to start again by clicking on 
which will take you back to the list of
possible models and clear the parameter list.
If you are happy with the model, click on 
.
This will remove the ‘blobs’ from the graph and return you to the main
semi-variogram menu. At any stage, you may save the experimental semi-variogram
and/or the model for the semi-variogram. If you have already saved a model on a
previous run, you can import it and replot the graph
using the model fitting option.
As a measure of the goodness of fit of the
model to the calculated points, the Cressie goodness
of fit statistic is quoted at the bottom of the dialog. We use a modified
version of this statistic which is standardised by the total number of pairs
included in the graph. This gives a figure which is not influenced by the
number of pairs and can, perhaps, be more objectively interpreted. The
statistic is calculated as follows:
It is a good idea to get this as low as
possible, but not at the expense of a good visual fit.
Given that the above model is not an attractive
fit to the points, we can improve the fit by changing the various parameters of
the semi-variogram. You can manually enter different values for the parameters
or you can choose the 
button. The latter option gives you the
possibility to right click on any ‘blob’ and move it around to improve the
model:
Once you select the point to be moved (remember
to click the right button), the
prompt will change to:
As you move the mouse, the model line will
follow your hand. The parameters in the dialog will change as you move,
including the Cressie goodness of fit statistic.
Right click to fix the blob in place. You can select another blob, right click,
move and right click to fix. You can do this as many times as you like until
you have the best model.
When you want to stop moving blobs around,
click on 
.
This will return you to the model fitting layout. To get a picture with no
blobs, click on 
which will return you to the main
semi-variogram menus.
Before quitting this routine, we might want to
store the calculated semi-variograms in case we want to look at them again
later. You can store any combination of
the calculated semi-variograms on a file.
EcoSSe
will prompt you for the name of the file. The default name is the original data
file name with an extension of .SXP. You can change the default
extension simply by typing in a new one. Alternatively you can change the whole
name to something entirely different.
Choosing which semi-variograms to store is
identical to choosing which ones to plot:
The stored semi-variograms can be read back in
at any time using the option on the main menu:
Storing
a semi-variogram model
You may also want to store your model on a file
for future use in modelling or in the kriging routines:
You will be prompted for the name of the output
file on which the model will be stored. This has a default name the same as
your original data file and an extension of .par. Since we have several variables on this data file, we will choose to
call the semi-variogram model file CV.par:
The model file is a flat text file listing all
the possible semi-variogram parameters and can be accessed by Wordpad, Notepad or some such for reporting or editing.
Having done all we need to do with the
semi-variogram calculation and modelling:
which returns you to the main menu bar.
This Tutorial is continued in Tutorial
Session 3 (Part 2) Kriging.