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:
|
|
|
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 variogram cloud plot will appear on your screen to let you
know that the calculation is proceeding.
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 will be 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.
When
the calculation is finished you will be given the opportunity to see the final
graphs. New menus will appear at the top
of the screen:
When
you select 
from the 
menu, 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. Firstly as a symbol for each calculated point on the graph. The
symbol size is proportional to the number of pairs of samples which have been
averaged to obtain that point.
Choosing
the four directions, omitting the omni-directional, and clicking on the upper
of the plotting options – the scaled symbols plot – results in the graph on the
following page.
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.
If
we had chosen the shaded graph option, we would get the following graph.
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.
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 can fit the semi-variogram
model to the detailed graph above. However, a clearer picture would be obtained
by looking at the “all direction” graph.
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.