Methodology - Response of Flood Events to Land Use and Climate Change - page 57

each other. Therefore, in this case, the Kriging method is an interpolation tech-
nique which is similar to the IDW method. The only difference between Kriging
and the deterministic method such as IDW is that the procedure to construct the
new values is not only based on the distance between the samples and the pre-
diction location, but also on the estimate according to the spatial arrangement of
the sampled points. This means that the measurement is based on spatial auto-
correlation between these points [ 6 ]. Kriging is represented by the Spherical,
Circular, Exponential, Gaussian, and Stable methods. With these options, Kriging
uses the mathematical function specified with the argument to fit a line or curve to
the semivariance data in the semivariogram. Ordinary Kriging assumes that the
variation in z-values is free of any structural component. These five models are
provided to ensure the necessary conditions of the variogram model are satisfied,
which is not always possible with interactive, manual variogram fitting. These
methods and conditions are discussed in McBratney and Webster [ 22 ]. They
described the methodology process as:

1. First, the variance is calculated based on the average variance of all point pairs

within each interval of the cell size.

2. Then, the variogram is fitted to the variance points using the Levenberg-

Marquardt Method [ 23 ] of a nonlinear least squares approximation. A mini-
mum of three points of data are required for the fit. In mathematics and
computing, the Levenberg-Marquardt algorithm (LMA) provides a numerical
solution to the problem of minimizing a function, generally nonlinear, over a
space of parameters of the function. These minimization problems arise
especially in least squares curve fitting and nonlinear programming [ 24 ].

3. Next, by increasing the cell size, the number of sample points per cell size

interval will increase, thereby providing enough data points to estimate the
semivariogram.

4. Once the semivariogram is estimated, a smaller cell size can be used in

creating the actual output grid.

5. After the construction of the empirical semivariogram, a model is fitted to the

plotted values using a defined function.

In general, Kriging goes through a two-step process:

• Variograms and covariance functions are created to estimate the statistical
dependence (spatial autocorrelation) values, which depend on the model of
autocorrelation (fitting a model).
• Prediction of unknown values. Therefore, The Kriging model is defined as:

Z ðÞ¼ l þ e ð s Þ ð 3 : 10 Þ

where:
Z(s) is the value at an unknown location of s; l is a constant mean for the data;
and e ðÞ is defined as dependent random errors.