**
Software Tutorial --- Ordinary Kriging in 3 dimensions
**

The example session with **PG2000 **which is described in this
and Part 1 is intended as an example run to familiarise the user with the
geostatistical facilities within the package. This documented example illustrates
one possible set of analyses which may be carried out. It takes you through the
following sequence of analyses:

{#} Cross-validation of the semi-variogram model for 3D data

{#} Kriging a 3D grid of point values for mapping

There are many other facilities within the package, which
are given as alternative options on the menus. This part of the documentation
assumes that you have worked through the Software Tutorial
on semi-variograms in 3D where we constructed and modelled semi-variograms
on the copper values listed in the **copper** data set.

**Cross validation of the semi-variogram model**

For this Tutorial, we have decided to continue with some geostatistical estimation using the model which we have fitted.

As you can see, I have chosen the option to .
There is a bit of confusion in the literature in the naming of the process.
Some authors call this *jack-knifing*.
This nomenclature is misleading, since the procedure bears little relationship
to what statisticians would expect by jack-knifing. Other authors use the two
words hyphenated or as a single word. We have chosen the above form to emphasis
the meaning of the procedure. We attempt to *validate*
our semi-variogram by dropping out each sample value and (cross) estimating the
value at that location from the neighbouring samples. We then compare the
estimated value with the actual value, and the difference between them with the
supposed geostatistical error.

There are different ways of comparing the actual error with
the Kriging error. We have chosen a simple method by calculating the ratio
between the two – i.e. actual error divided by Kriging standard deviation. If
certain basic assumptions are satisfied, *and
we have chosen the correct semi-variogram model*, these (error) statistics
should average zero and have a standard deviation of one. We use the mnemonic
**XVAL**
for cross validation. Running this option will also give you an idea of how
long Kriging is likely to take on your computer.

**PG2000** remembers which variables you were
studying for the duration of a single run. If you model your semi-variogram and
do not continue immediately to kriging, you should store your semi-variogram
model on file for later access. If you start a new run of **PG2000** you can come to this
routine directly. There is no need to go through the whole procedure to get
this far! In that case you will have to select the variables in the same way
that you did for the semi-variogram analysis.

To carry out cross validation – which includes kriging the
estimates – you need a semi-variogram model. In the previous Tutorial we have
fitted a model. If you continue straight on to this tutorial without closing
down **PG2000**,
the software will remember this and offer it to you as the base model. If you
have started a new run for this tutorial or have not fitted a model already,
you will need to define the semi-variogram model.

You have various options as to how you define the model:

If you have previously stored a semi-variogram model on file, you can click on and read it back in. If this model was stored during semi-variogram model fitting, you will need to add information about anisotropies or trend.

If you have not fitted a model yet and have no idea how to proceed, please click on and go read Tutorials on semivariograms in 2 or 3 dimensions!!

If you did not store the semi-variogram previously, you will need to manually re-enter the model now.

Click on . A new option bar will appear:

From this select your model type. For copper, I have previously fitted a Spherical model so I click on . The first thing the software needs to know is how many component Sphericals you might need. You can have up to 3 Spherical components mixed together. For copper, we only fitted one component, so we enter this in the relevant box:

Once we type 1 in this box, the dialog will refresh with the parameters we have to enter:

Note that you still have the option at any stage to cancel the analysis, import a model from file or change the type of model. We enter the model fitted to the vertical (down-hole) direction:

Then we click on to introduce the fact that the horizontal directions will be different from this. If you need help in understanding anisotropy, you may click on the button at this time or any time up to accepting the model.

To define anisotropy in three dimensions, we need the following parameters:

Strike direction (azimuth)

Dip direction (azimuth); should be strike direction + or – 90 degrees

Down dip direction, degrees down from horizontal

Plunge direction

In addition, we will need anisotropy factors for the strike, dip and 'across strike' directions. For a Spherical semi-variogram, the range(s) of influence will be multiplied by these factors.

For our example, our semi-variogram model has a range of influence of 20 metres in the vertical direction. This is our minor axis on the anisotropy. In all the horizontal directions, we have a range of influence of 280 metres, i.e. 14 times that of the vertical direction.

In the anisotropy dialog, I leave the default of 0 degrees for the strike direction and change the anisotropy factor from 1 to 14:

Once I click on , the currently greyed buttons for dip azimuth will activate:

and I enter 14 for the anisotropy factor. I click on . Both azimuth buttons become grey again, but only this one retains the printing. Notice that the button is now active. You can still edit any parameters either than strike and dip direction until you press the button. If you need to change strike and dip, you will need to accept here and go back and redefine the model from scratch.

When you close this dialog, you will notice that the button on the semi-variogram model dialog has changed to . Our model now looks like:

You can edit the anisotropy parameters in this dialog, but I wouldn't advise it.

If you have a significant trend that should be defined so that the kriging can allow for it. In the presence of trend, we would use Universal Kriging. The level of trend is specified in a dropdown menu list:

Always bear in mind that when there is a trend in the original sample data, all parameters refers to the semi-variogram of the residuals.

Click on or to get the routine to accept the semi-variogram model for kriging.

Until you do this, you still have the full option list:

Now we have variables to study and a semi-variogram model which tells the software how the values are related to one another. For cross validation, we will take each sample in turn and remove it from the data set. The neighbouring samples will be used to produce an estimate at this location. We can then compare estimated value with the actual value found in the sample at that location.

Before we can go any further, we need to define the "neighbourhood". That is, how far do we want the software to search for samples to be included in the estimation process. Since we have an "isotropic" semi-variogram model, it seems sensible to select an isotropic search radius. Since we have a Spherical model, PG2000 will suggest that we use the range of influence of the model as a default search radius. Since our model is anisotropic, the software offers an anisotropic search ellipsoid:

The default search radius is always the range of influence
of the first component fitted – providing the model is Spherical. For
Exponential and Gaussian models the search radius is adjusted to a realistic
distance. For models without a sill, **PG2000** cannot guess what an appropriate search
radius would be and uses a similar default to that in the inverse distance
interpolation routines.

The default search radius, given our semi-variogram model,
is 280 metres along strike and down dip and 20 metres in the vertical (across
strike) direction. You can change these but the software will still
*weight* the samples according to the
semi-variogram model anisotropy.

The software will then enquire as whether you wish the results stored on file for later analysis:

If you answer "yes" to this question, you will be prompted
for an output file name. The default name is that of the original data file
with the extension **.XVL**. You can change the data file name, the
extension or both if you so wish. This file will be written in the correct
format to be read back in as a data file. The cross validation outputs a table
of values on the **ghost.lis** file as the estimation proceeds. The final
column "error statistic" is the ratio of the actual error to the Kriging
standard error.

As the cross validation is carried out, a post plot will be
drawn of the "error statistics". The contour levels for this graph have been
chosen so that a value in the highest (+2.5) and the lowest (-2.5)
contour bands should occur one time in one hundred. This plot is an excellent
device for visually spotting outliers in the sample data. These need not be
outliers in the usual statistical sense. That is, they may be quite acceptable
*values* as such. What the cross
validation will show is whether they are acceptable values *
in the context of the neighbouring samples*.

The left hand box on the screen summarises the various calculated values. A direct comparison can be made between the average actual value (23.85 %Cu) and the average Kriged value (23.83 %Cu). The standard deviation of actual copper values is 1.80 and that of the estimated values is 1.25 %Cu. This is illustrative of the smoothing which takes place when ordinary kriging is used as an estimation technique. A weighted average of a set of samples will have a smaller standard deviation than values when taken singly.

The average (typical?) Kriging standard error is 1.25 %Cu, although the individual standard errors vary around this value. Finally, looking at the all-important "error statistics", we find an average of 0.0031 and a standard deviation of 1.1954. Ideally, we are looking for zero and one. It is your decision as to whether 1.19 is close enough to 1.0 to be accepted (sic). Please refer to full documentation for further discussion of this point.

Note that the post plot is only two dimensional, so it is harder to pick up outliers visually, especially with vertical boreholes.

The contours are chosen so that black and purple points should appear about 1% of the time.

With error statistics avergaing 0.0047 with a standard deviation of 0.9883, we will accept the above model. If you do not wish to accept, you will need to change the semi-variogram model.

If you choose to store your cross validation results on a file, this file can be read back into PG2000. You can produce scattergrams of, for example, estimated values versus actual values to see how well the kriging is performing. You can also do a probability plot of the 'error statistics' to see if Normal confidence intervals would be appropriate.

**Interpolating a map with kriging**

Interpolating a grid of points with kriging will produce an estimated (or "predicted") map of the values over the study area. This map reflects the actual values measured at the actual sample locations and uses a weighted average estimator for grid points which have not been sampled. Weights are determined by a set of equations which combine:

{#} the spatial continuity as modelled by the semi-variogram

{#} any anisotropy identified and modelled in the semi-variogram

{#} any trend component identified and defined by the user

{#} the spatial layout of the samples relative to the points being estimated

{#} the spatial layout amongst the samples themselves (clusters, irregularities etc)

The chosen weights will minimise the "estimation variance", which may be interpreted as a measure of the estimation error.

**PG2000** will remember everything which has been
defined during this run. We have already defined which variables we have been
analysing:

Click on to proceed. The routine also needs contour levels:

and to know whether you want the results stored on a "grid" file:

**PG2000** will suggest contour levels based on the
variability of the sample values. You can change these if you so desire.
Alternatively you can run with the default contours and draw prettier maps by
reading the grid files back in. The
default name for a grid file is the original data file name with the extension
**.GEA**.
Please note that "grid" files are not in the same format as
"data" files. If you want to read them back
in, you must use the option:

You need to confirm semi-variogram model, search parameters and the area which is to be studied.

The semi-variogram we defined previously in the cross validation section. If you come directly to the kriging routines without passing through any others, you will need to respecify (or import) your semi-variogram model including the local trend component.

The neighbouring samples will be used to produce an estimate at each unsampled grid point. Before we can go any further, we need to define the "neighbourhood". That is, how far do we want the software to search for samples to be included in the estimation process.

**PG2000** cannot guess what an appropriate search
radius would be. As a simple default, with models such as the Spherical, a
default based on the semi-variogram parameters is offered.
When the value at a specified grid point is
being estimated, all samples within this circle of the point will be used in
the Kriging process. If there are too many samples within this circle, those
closest to the "unsampled" location will
be selected. The default search radius,
given our semi-variogram model, is 280 metres in the Along Strike and Down dip
directions (in our case horizontal East and North), and 20 metres in the
"across strike" direction (in our case vertical).

We need to define the area which is to be mapped. Firstly we define the limits of the area and, if appropriate, a polygonal boundary within which values are to be mapped. The default area offered will be the extent of the sample data:

Note, of course, that you now have three sets of co-ordinates.

You can manually change the rectangular limits of the area
or you can read in a file containing the polygonal boundary. The default name
for a boundary is always the original data file name with the extension **.bln**.
This boundary will operate in the X and Y directions and remain constant for
all Z values. If your boundary changes
with Z, you will need to do each volume separately.

Once you have chosen the volume to be studied, you must define the grid spacing to be used. Points will be calculated at each grid node and represented on the screen as a shaded rectangle of the appropriate size. The volume is represented by slices at each Z value (third co-ordinate).

The program offers a default which will give you about 25 grid points in the 'X' direction. We can alter the grid spacing or the limits of the map by changing the number in the relevant box.

Note you also have a check box at the bottom of the dialog to specify whether you want plotting to pause between each level, so that you can see the slices.

If you make a change and want to check how many grid points you have before proceeding, click on and the rest of the parameters will be updated. You may also change minimum and maximum X and Y values at this stage. Once you click on the map parameters will be defined.

Interpolating a grid of points produces sketch maps on the screen. The shading information for the contour levels will appear in the left hand box and the map itself in the right. A shaded square will be displayed on the map to show you which point is being estimated in addition to the information in the prompt box. You may copy the screen to your printer at any stage during the estimation process.

As specified above, we will see 3 maps starting at Z = 77 and mapping every 25 metres up. That is, one map at elevation 77, one at 102 and one at 127. Each map will be shown as shaded rectangles at 5 metre spacing in both X and Y directions.

Since I have asked the software to pause at each level, I can capture and display each level.

If you pause between levels, you have the options to plot the sample data used to estimated this level, terminate the kriging run or continue to the next level.

When the Kriging has been completed, you have the following options:

To display the data locations, click on .

If you click the "error" maps will be displayed showing the standard errors associated with the estimated grid points, on a level by level basis.

You can copy the plots with +
and paste them into another application. Some systems (notably Windows NT)
require pressing +.
This will place a copy of the Window in the
clipboard. You can *import* the picture
into a Word processing application such as Microsoft Word, a spreadsheet
application like Lotus or Excel, or paste +
into many applications, such as MSPaint.

See overleaf for both of the above mentioned plots.

If you have elected to write a grid file, these values are
also stored on the **.GD3** file so that you can redraw the maps with
different contours by reading back the grid file. You can see from the default
error map overleaf that this option can be rather useful.

**Finishing up**

Clicking on this menu item or on will end your run with the software. You will see the closing down dialog box:

The above Tutorial session should serve only to illustrate a
possible use of the various routines from **PG2000**.
Try running the program again, choosing
your own responses. try looking at reef width instead of grade. This variable
has a standard two parameter lognormal distribution. Try reading in one of the
other data files which are provided, say, **samples.dat**.

**General Notes**

There are a few points which you may have noted in following
the Tutorial session above. Most of the routines communicate between themselves,
without you having to worry about getting the right information from one to the
other. For example, after you read in the complete contents of the data file,
the routines ask which of the variables you actually want to analysis. This
information is then stored internally and may be accessed by any of the other
routines. This is a feature of most of **PG2000**, in that it will recall what you chose
previously and ask whether this is to change or not. You should bear this in
mind if you are analysing more than one data file in a single run. In
particular, the boundary used in mapping will be remembered. If you change data
file or even which variables you analyse this will not automatically update.

A copy of this run should have been made on a file called
**GHOST.LIS**
unless you changed the name at the beginning of the run. Send this file to your
printer if you want a record of the analysis or look at it with Wordpad or
Notepad.

**PG2000** — like any computer software — is not
completely error-free. Neither is it fool-proof. You can always get out of the
software by right clicking on the Taskbar. This will invoke the 'End Task'
facility to close the Window without damaging the rest of your system. If you cannot
figure out what went wrong, note down as much information as you can about the
program you were running, the data you were using and exactly where it broke
down. Contact your supplier locally or Geostokos direct for assistance,
software@kriging.com. Send us the **ghost.lis**
file and (if you can) the data you were analysing at the time.