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Used for analysing relationships between measurements at a number of locations.
VARIOGRAM ANALYSIS
<P>
Measurements are often taken at a number of locations and the relationship
between observations at the various locations can be explored by
variogram analysis.
<P>
A variogram summarises the relationship between differences in pairs
of measurements and the distance of the corresponding points from each other.
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Here we introduce some of the concepts and
approaches used in variogram analysis. Explaining terms such as
kriging, range, sill and nugget.
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Examples of fields in which the technique may be used include.
<UL>
<LI>meteorology;
<LI>geology;
<LI>ecology;
</UL>
Usually, but not always, measurements at points close together are
more similar than those further apart. It can be useful to quantify this
spatial variability in order to understand its form or to help
in planning a sample survey of an area.
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A variogram describes the spatial correlation between observations.
It tells us how different, on average, are measurements at
various distances apart.
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For the example used (microwave backscatter in a field),
there are 400 data items and these provide 79800 pairs of
observations (400 x 399 / 2). To estimate average differences
in Z as a function of the distance separating points, we group
pairs of points by their separation distances (h).<P>
<p>
We calculate the mean squared difference between pairs of measurements
within a group.
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Note : It is advisable to have at least 150
observations to provide reliable variogram estimates.
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Generally variance usually increases with distance apart of measurements.
<BR>
<B>Definitions</B>
<BR>
Range & Sill : variance levels off (the sill)
beyond a certain distance (the range)
beyond which observations appear independent.
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Nugget Effect : When a variogram is extrapolated back to zero distance,
it may approach a non-zero variance. The amount by which the
variance differs from zero is known as the nugget effect.
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It can be helpful to fit a smooth curve descibing the change in variance (v)
with distance (h). Some of the models used include :
<BR>
Linear : has no sill and is a very simple model</B>
<BR>
<B>Spherical : is widely used</B>
<BR>
<B>Exponential</B>
and <B>Gaussian.</B>
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We can use the variogram to estimate values at points other than
where measurements were taken. This process is termed
Kriging.
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Here we can use the variogram to estimate the underlying
backscatter w, if we regard the observed values Z as being composed
as follows.
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<CENTER>Z = w + nugget variance</CENTER>
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To summarize :
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Variogram analysis :
<UL>
<LI>provides a useful tool for summarising spatial data;
<LI>provides a measure of spatial dependence between samples;
<LI>can be used with kriging to estimate the value of the surface
at an unsampled location.
</UL>
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