FVARIOGRAM directive

Forms auto variograms for individual variates or cross variograms for pairs of variates.

Options

PRINT = string

Controls printed output (statistics); default stat

Y = variate

Y positions (needed only for 2-dimensional irregular data)

X = variate

X positions or interval (not needed for 2-dimensional regular data i.e. when DATA is a matrix)

YMAX = scalar

Maximum lag in the y direction (2-dimensional regular data only)

XMAX = scalar

Maximum lag in the x direction

STEPLENGTH = scalar

Length of each step in which lag is incremented

DIRECTIONS = scalar or variate

Directions (degrees) along which to form the variogram (relevant only for 2-dimensional irregular data)

SEGMENTS = scalar or variate

Angles subtended by the segments (degrees) over which averaging is to be done (relevant only for 2-dimensional irregular data)

Parameters

DATA = variates or matrices

Measurements as a variate or, for data on a regular grid, as a matrix

VARIOGRAMS = variates or matrices

Structure to store the sample variogram or cross-variogram

COUNTS = variates or matrices

Numbers of comparisons involved in the calculation of each (cross-) variogram

DISTANCES = variates or matrices

Mean lag distances at each step

CROSSCOMPARISON = variates or matrices

Accompanying measurements from which to form a cross-variogram

Description

The FVARIOGRAM directive forms an experimental variogram from a set of values of a variable, Z, distributed in one or two dimensions using the usual computing formula:

where z(x_i) and z(x_i + h) are the values at positions x_i + h, and m(h) is the number of paired comparisons contributing to the estimate. For data on a regular grid or transect h is an integer multiple of the sampling interval. For irregularly scattered data h is discretized so that for each nominal lag there is a range of distance equal to the increment and an angular range set by the user. The nominal lag is at the centre of both ranges.

The data are specified using the DATA parameter. If they are on a regular grid, they should be supplied in a matrix defined with a variate of column labels to provide the x-values and a variate of row labels to provide the y-values. Alternatively, if they are irregularly scattered, then they should be supplied in a variate, and the X and Y options should be set to variates to supply their spatial coordinates.

The experimental variogram is controlled by five options. For irregular data the maximum distance to which the variogram is calculated is set by the XMAX option for all directions. For regular data XMAX defines the maximum lag distance in the X direction, and YMAX must also be given to limit the distance in the Y direction. The increment in distance is set by the STEP option. The variogram may be computed in one or more directions. These are given by the DIRECTIONS option in degrees counterclockwise from east in the usual convention. Each direction is at the centre of an angular range, which is defined by the SEGMENTS option. DIRECTIONS and SEGMENTS should be set to scalars if the variogram is to be calculated for only one direction, or to variates if there are to be several.

A variogram can be computed without regard to direction by setting DIRECTIONS to 0 and SEGMENTS to 180. This is advisable if variation seems to be isotropic, i.e. the same in all directions, or if there are too few data to compute (h) for two or more directions separately. The lag then becomes a scalar |h| = h in distance only. Experience suggests that some 300 data are needed to distinguish anisotropy.

By default some statistics are printed concerning the variogram, but these can be supressed by setting option PRINT=*. Other information can be saved using the various parameters, in variates if there is a single direction, or in matrices with one column for each direction if there are several: VARIOGRAMS stores the ordered set of semivariances; DISTANCES stores the mean lag distances at which the semivariances have been computed; and COUNTS stores the numbers of paired comparisons from which the semivariances have been computed.

Options: PRINT, Y, X, YMAX, XMAX, STEPLENGTH, DIRECTIONS, SEGMENTS.

Parameters: DATA, VARIOGRAMS, COUNTS, DISTANCES, CROSSCOMPARISON.

Action with RESTRICT

You can restrict a DATA variate to form the variogram from only a subset of its units.