Column semi-variogram

Table window	Columns menu
Semi-variogram column

Functionality

For distance values (h) in a Distance column, the Column Semi-variogram operation calculates semi-variogram values (g) according to a semi-variogram model and its parameters as specified by the user. The calculated semi-variogram values are stored in an output column. You can calculate multiple columns with semi-variogram values, e.g. using different semi-variogram models or using different parameters for one semi-variogram model.

The aim of storing modelled semi-variogram values in a column is for instance to be able to assess the 'Goodness of Fit (R²)' between modelled semi-variogram values and experimental semi-variogram values.

This operation is generally used on the output table of Spatial Correlation (see Spatial Correlation : functionality), and presumably after you have fit some semi-variogram models through the experimental semi-variogram values in a graph window (see Graph window : Add Semi-variogram Model).

Input column requirements:

The input column should be a column with positive values representing distance values.

You can generally use the Distance column, the AvgLag column, or the AvgLag1 / AvgLag2 columns stored in the output table of the Spatial Correlation operation:

the Distance column contains the center value of each distance interval; the width of the distance interval is equal to the user-specified lag spacing.
the AvgLag column contains for each distance interval, the average distance of all point pairs within that distance interval.
the AvgLag1 column is available when you used the bidirectional method in Spatial Correlation; it contains the average distance value of point pairs in that distance interval and in the specified direction sector.
the AvgLag2 column is available when you used the bidirectional method in Spatial Correlation; it contains the average distance value of point pairs in that interval and in the perpendicular direction sector.

For more information, refer to Spatial Correlation : functionality.

Furthermore, you need to define a complete semi-variogram model.

Domain of output column:

The output column will use system domain Value.

Calculating R²:

To test which semi-variogram model fits the experimental semi-variogram values best, you can calculate goodness of fit (R²), for instance as:

where:

	are the experimental semi-variogram values calculated by the Spatial Correlation operation; these values are present in columns SemiVar, SemiVar1, SemiVar2 in the output table of the Spatial Correlation operation;
g	are the semi-variogram values calculated by the Column Semi-variogram operation, i.e. the output column name of the Column Semi-variogram operation.
N	total number of distance classes/intervals.

The numerator of the fraction gives the sum of the squared differences between the experimental semi-variogram values and the semi-variogram values calculated by a user-specified semi-variogram model.

The denominator of the fraction gives the sum of the squared differences between the experimental semi-variogram values and the average experimental semi-variogram value of all distance classes/intervals.

The maximum value of R² is 1, meaning an exact match of semi-variogram values calculated using a certain semi-variogram model and parameters, and experimental semi-variogram values.

Dialog box

Dialog box options:

Distance column:	Select the column that you want to use as the distance column. You can generally select one of the Distance interval columns from the output table of the Spatial Correlation operation: Distance , AvgLag , AvgLag1 , or AvgLag2 .
Semi-variogram:	Choose a model which fits your experimental semi-variogram values; the model and specified parameters define the calculation of semi-variogram values for the values in the selected distance column. 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 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.
Output column:	Type a name for the output column that will contain the calculated semi-variogram values. A dependent column is created.

The Column Properties dialog box appears. The default domain for the output column is system domain Value.

Command line

The Column Semi-variogram operation can be directly executed by typing the following expression on the command line of the table window:

OUTCOL

ColumnSemiVariogram(DistanceColumn, SemiVarModel)

where:

OUTCOL	is the name of the output column containing the calculated semi-variogram values.
ColumnSemiVariogram
	is the command to start the Column Semi-variogram operation.
DistanceColumn	is the column containing distance values. Any column with a value domain is allowed. You can generally use one of the Distance interval columns from the output table of Spatial Correlation. For more information on the contents of these columns, refer to Spatial correlation : functionality.
SemiVarModel	Model(nugget, sill, range) \| Power(nugget, slope, pow) This expression defines the semi-variogram model that should be used and its parameters.
Model	Spherical \| Exponential \| Gaussian \| Wave \| RatQuad \| Circular
nugget	value for the nugget, according to semi-variogram.
sill	value for the sill, according to semi-variogram.
range	value for the range, according to semi-variogram; real value > 0.
slope	when using the Power model: value for the 'slope'. When pow is specified as 1 (i.e. thus using a linear model), then slope is the direction coefficient, i.e. Dg/Dh.
pow	when using the Power model: an exponent 0 < real value < 2. When you use value 1, the Power model will become linear. If the power exponent is 2 the assumed stochastic model (�randomness�) is not always justifiable and the interpolation can become pathological.

When the definition symbol = is used, a dependent output column is created; when the assignment symbol := is used, the dependency link is immediately broken after the column is calculated.

Examples:

Examples of complete expressions are for instance:

OUTCOL1	=	ColumnSemiVariogram(Distance, Spherical(0, 7200, 4500))
OUTCOL2	=	ColumnSemiVariogram(Distance, Gaussian(200, 7550, 2400))
OUTCOL3	=	ColumnSemiVariogram(Distance, Power(200, 0.4, 1.2))

Algorithm

For the value encountered in a specified Distance column, a semi-variogram value is calculated according to a specified semi-variogram model and specified model parameters.

Mind:

With the Column Semi-variogram operation, only some discrete semi-variogram values are calculated, namely one semi-variogram value for each distance interval. When you would make a line graph of a Distance column against a column that is the result of the Column Semi-variogram operation, you will see a straight line between any two consecutive calculated semi-variogram values.

To obtain a line graph with continuous semi-variogram values, first make a point graph of a distance column against a semi-variogram column that is the result of the Spatial Correlation operation, then add a semi-variogram model through these experimental semi-variogram values. For more information, see Graph window : Add semi-variogram model.

Table window

Semi-variogram column

Functionality

Dialog box

Command line

Algorithm