Variogram Model
A variogram model characterizes the variability between data points as a function of seperation distance. In kriging the variogram model is used to provide estimates of values at any given point a weighted sum of input values at surronding points, with weights assigned based on spatial trends correlations. The weights are derived from fitting a model variogram (theoretical model) to the empirical variogram (experimental variogram), which are graphical depictions of seperation distance, or lag, and semivariance (or dissimilarity) in values between paris of input data.Experimental Variogram
The first step in the variogram building process is specifying an experimental framework from which to fit a theoretical model. This is done through the following steps:1) An Influence Radius is specified which provides the proximity cutoff limit for considering a pair of scatter points in the analysis.
2) Number of Lags is specified. Each lag is centered about a lag line (represented by the blue lines in the figure). The spacing of the lag lines is determined from the influence radius and number of lags by:
Note that the innermost lag line is set at a radius of around the central point. Each lag extends in and out (radially) a certain distance from the lag line. This distance is known as the lag tolerance. In this case, the lag tolerance for each lag is equal to xlag.
3) The value for each lag is computed based on the 'Semi-variogram' defined by:
where N is the number of pairs of scatter points represented by the lag, i is the index of the scatter point pairs, and the f variables represent the values at the two points (where the central indicates the central point and radial indicates the radial point).
The user may choose to alter the radius of influence and the number of lags and may also choose between an isotropic or anisotropic variant of the experimental model.
Anisotropic Option
Theoretical Variogram
The second step in the variogram building process is fitting a theoretical model to the experimental framework. The model function is selected and the parameters of the function are adjusted until the best bit is achieved. This process may be automated or the model and parameters may be adjusted by the user.The panel along the righthand-side (RHS) of the interface is used to create the theoretical variogram.
Model Parameters
There are a number of parameters that are common to a number of the models. They are 1) the nugget, 2) the sill (denoted as ‘variance’ in the ‘Variogram’ window), and 3) the range.The nugget, a, is the minimum variance. The sill, b, is the average variance of points at a distance from the point in question that there is no correlation between the points. The range, c, is the distance from a point at which there is no correlation between the point and any others.
Theoretical Model
There are four models available:Spherical model defined by:
Exponential model defined by:
Gaussian model defined by:
Power model defined by:
where 0 < c < 2.