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From the results of the Spatial correlation operation, you can make a semi-variogram. In the semi-variogram, the discrete experimental semi-variogram values that are the outcome of Spatial correlation will be modeled by a continuous function so that a semi-variogram value g will be available for any desired distance h for the Kriging operation later on.
A semi-variogram model describes the relation between squared differences of pairs of point values and distance. For general information on semi-variograms, see the Additional Info below. The points in the pairs are at certain distances towards each other and optionally in a certain direction towards each other. You can choose a type of semi-variogram model (spherical, exponential, etc.), and you can fill out the model's parameters: nugget, sill, and range. According to the specified semi-variogram model and parameters, a line will be drawn in the graph window that displays experimental semi-variogram values.
The aim is to find the model and its parameters that fit your experimental semi-variogram values 'best'. You may need to experiment a little with different models, different values for sill, range and nugget, etc. Once, you have decided which model, and which values for nugget, sill, and range fit your data best, you can continue with the Kriging operation.
Besides a visual comparison of different models in a graph as described below, you can also use the Column Semi-variogram operation to calculate semi-variogram values g for specific distances and store the results in an output column.
Preparation:
Display the input table of the Spatial correlation operation in a table window.
Display the output table of the Spatial correlation operation in a table window.
Create point graphs, i.e. the experimental semi-variogram(s), from the columns in the output table of the Spatial correlation operation:
When you used the bi-directional method in Spatial Correlation, you can draw two graphs (e.g. in two graph windows):
In literature, the shown graph is called a discrete experimental semi-variogram.
Then, fit a semi-variogram model through the experimental semi-variogram values.
To add a semi-variogram model:
This Add Graph Semi-variogram Model dialog box will appear.
Dialog box options:
Semi-variogram: |
Choose a model which fits your experimental semi-variogram values and which later on can be used in the Kriging operation to calculate the values for the semi-variogram function g(h). You can choose between the Spherical model, Exponential model, Gaussian model, Wave model, Rational Quadratic model, Circular model, or the Power model. For more information on the mathematical formulae of the models, see Formulae of semi-variogram models. |
Nugget: |
When you found a nugget effect, specify the value. A nugget effect is the vertical jump from value 0 at the origin, to the semi-variogram value at extremely small separation distances. You are specifying parameter 'C0' for the selected model. |
Sill: |
Type a value for the sill; the sill is the plateau that the semi-variogram values reach at the range. You are specifying parameter 'C0 + C' for the selected model. |
Range: |
Type a value for the range; the range is the distance at which the semi-variogram values do not increase anymore and reach a plateau. You are specifying parameter 'a' for the selected model (real value > 0). |
Slope: |
For the Power model only: type a value for the 'slope', i.e. specify parameter 'k' for this model. When you use the Power model with a power exponent of 1, the model becomes linear (a straight sloping line); then, this 'slope' parameter equals the direction coefficient (Dg/Dh) of the line. |
Power: |
For the Power model only: type a real value for the power exponent, i.e. specify parameter ‘m’ for this model. The power function is meaningful if 0 < power exponent < 2. When value 1 is used, the Power model becomes linear and the slope will be constant. If the power exponent is 2 the assumed stochastic model (‘randomness’) is not always justifiable and the interpolation can become pathological. |
A (continuous) line, i.e. the semi-variogram model, will be directly drawn in the graph window.
You are advised to visually experiment a little with models and sill, range, and nugget values to find the best line through your experimental semi-variogram values. You can edit a semi-variogram model by double-clicking the semi-variogram layer in the Graph Management pane: the Graph Options - Semi-variogram Model dialog box will appear.
To adapt the title, the X-axis, or the Y-axis (left or right) of the graph, double-click the appriopriate item in the Graph Management pane or in the graph window itself: the Title, X-axis, Y-axis (left), or the Y-axis (right) dialog box will appear. The grid lines of the graph can be modified in the X-axis and Y-axis dialog boxes.
To store semi-variogram values in a column and to evaluate which model and parameters give the best fit through your experimental semi-variogram values, choose the Semi-variogram command on the Columns menu in the table window. The Column Semivariogram dialog box will appear.
When finished, you can save the contents of the graph window by choosing the Save or the Save As commands from the File menu of the graph window.
Once you have decided which model and which values for sill, range and nugget fit your data, you can continue with the Kriging operation.
Figure 1 below shows a semi-variogram depicting a spherical model:
Fig. 1: A semi-variogram depicting a spherical model.
Remarks on semi-variograms:
Examples of different semi-variogram models:
Figure 2 below depicts a semi-variogram through which multiple models are fitted. In this example, only the Gaussian and the Power model (for the shorter distances) describe the experimental semi-variogram values reasonably; the Spherical model and the Exponential model show rather different values for the experimental and the modeled semi-variogram values.
The values that were used in this example for nugget, sill and range are:
The mathematical formula of each model are described in Formulae of semi-variogram models.
Fig. 2: Finding the best semi-variogram model through an experimental semi-variogram. The black dots are the outcomes of the Spatial correlation operation, i.e. the experimental semi-variogram values; the colored lines represent different semi-variogram models calculated for all possible distances.
See also:
Graph window : Graph Management
Graph Options - Semi-variogram Model