Methodology - Response of Flood Events to Land Use and Climate Change - page 58

In Kriging, to predict the values at unknown points, the following equation
would be applied:

b Zs ðÞ¼ X

N

k i : z ð s i Þ

ð 3 : 11 Þ

i ¼ 1

where:
b Zs ðÞ is the predicted value at the unsampled location S 0 ; N is the number of
measured sample data points within the neighborhood defined for S 0 ; k i are the
weights associated with each sample point; and zs ðÞ is defined as the observed
value at location S i .
In Kriging, the average difference between the predicted and the measured
value should be zero. To achieve this, the value of the following equation has to be
minimized:

! 2

b Zs ðÞ¼ X

N

k i : z ð s i Þ ð 3 : 12 Þ

i ¼ 1

The minimization of the above formula generates the Kriging equations:

C k ¼ g ð 3 : 13 Þ

where:
the variance-covariance matrix is defined as C which is calculated from the
data points, multiplied by the column vector containing the weights being calcu-
lated, which is defined as k equal g which is a column containing the fitted
semivariance for the predicted location.

Fitting a Model to the Empirical Semivariogram

The next step in Kriging procedures is to fit a model to the data points by fitting the
empirical semivariogram. In the previous section, it was explained how the
empirical semivariogram provides information on the spatial autocorrelation of
datasets. However, it does not provide information for all possible directions and
distances. For this reason, and to ensure that Kriging predictions have positive
Kriging variances, it is necessary to fit a model to the empirical semivariogram.
Fitting a model or variography with the structural analysis is computed as:

h i

Semivariogram ð distanceh Þ¼ 0 : 5 average ð valueatlocationi valueatlocationj Þ 2

The equation represents the difference squared between the values of the paired
locations. Figure 3.2 shows the pairing of one point (the dark blue point) with all
other measured locations. This process continues for each measured point. Each
pair of locations has a unique distance, and there are often many pairs of points.
To plot all pairs quickly becomes unmanageable. Instead of plotting each pair,

the pairs are grouped into lag bins; for example, computing the average