Spatial interpolation - Spatial Data Analysis: An Introduction for GIS Users - page 142

observations are separated. For example, two observations may be 5 km apart and one
may be directly north of the other. In simple terms, the variogram is estimated by
calculating the squared dif erences between all the available paired observations and
obtaining half the average for all observations separated by that lag (or within a lag
tolerance (e.g. 5 ± 2.5 km) where the observations are not on a regular grid). Semi-
variance refers to half the squared dif erence between data values. An example of
variogram estimation is given below. Figure 9.9 gives a simple example of a transect
along which observations have been made at regular intervals. Lags ( h ) of 1 and 2 are
indicated. In this case, therefore, half the average squared dif erence between observa-
tions separated by a lag of 1 is calculated and the process is repeated for a lag of 2 and
so on. In many cases the distance between observations will not be regular, so ranges
of distances are grouped. h e selection of the bin size (e.g. 0-5 km, > 5-10 km,
> 10-15 km, … or 0-10 km, > 10-20 km, …) is important. Smaller bin sizes will result
in more noisy variograms, while a bin size that is too large will smooth out too much
spatial structure and it will not be possible to capture spatial variation of interest. In
other words, the plotted values in a variogram with too small a bin size will appear to
be widely scattered, while the values in a variogram with a larger bin size will tend to
be more similar to neighbouring values on the plot. Finding an appropriate bin size is
important in characterizing spatial structure and in guiding the selection and i tting of
a model, as detailed below.
h e variogram can be estimated for dif erent directions to enable the identii cation
of directional variation (anisotropy). In other words, rather than consider all observa-
tions 5 km from a given observation, we may consider only observations that are
directly north or south (for example) of the observation of interest within a particular
angular tolerance (e.g. north or south ± 45 degrees). An example of an anisotropic
phenomenon is temperature—Hudson and Wackernagel (1994) showed that average
January temperature in Scotland decreased systematically from west to east (a function
of the warming ef ect of the Gulf Stream), but there was no systematic trend in a north
to south direction.
In summary, the variogram characterizes the degree of dif erence in values as a
function of the distance by which they are separated. h e experimental variogram,
ˆ (

g h , relates semivariances to distances (and directions)—it has distance and direction
(the lag) on the x axis and semivariance on the y axis. If a property is spatially auto-
correlated, we would expect the semivariance to increase as the distance between
observations increases.

)

Lag h = 1

x x x x x x

Lag h = 2

Figure 9.9 Transect with paired points selected for lags of 1 and 2 units.