DVARIOGRAM procedure

Plots fitted models to an experimental variogram (S.A. Harding, D.A. Murray & R. Webster).


Options

MODEL = string
Defines which model to plot (power, boundedlinear, circular, spherical, doublespherical, pentaspherical, exponential, besselk1, gaussian, affinepower, linear, cubic, stable, cardinalsine; default powe

WINDOW = scalar
Window in which to plot a graph; default 1

TITLE = text
Title for the graph


Parameters

VARIOGRAM = variates
Experimental variogram to which the model or matrices has been fitted, as a variate if in only one direction or as a matrix if there are several

DISTANCE = variates
Mean lag distances for the points in each or matrices variogram

DIRECTION = variates
Directions in which each variogram was computed

ESTIMATES = variates
Estimated parameter values

XUPPER = scalar
Upper limit for the x-axis in the graph

PENDATA = scalar
Pen to be used to plot the data; default 1

PENMODEL = scalar
Pen to be used to plot the model; default 2


Description

DVARIOGRAM plots fitted models to an experimental variogram using estimates produced by MVARIOGRAM.

   The data for the procedure can be taken directly from the FVARIOGRAM directive and MVARIOGRAM procedure. The parameters DISTANCES and VARIOGRAMS correspond to those with the same names in FVARIOGRAM. The data will be in variates if the variogram was calculated in only one direction. If it is in several, they can either be in matrices (as generated by FVARIOGRAM) or in variates. For the affinepower model, directions must be supplied using the DIRECTIONS parameter. These should be in a variate with one value for each column if the other data are in matrices; alternatively, they should be in a variate of the same length as the other variates. The MODEL option specifies the fitted model that is to be plotted, and the estimates for the model are supplied in a variate using the ESTIMATES parameter. The model estimates can be taken directly from MVARIOGRAM using the ESTIMATES parameter. The number of values within the variate for the estimates will depend on the model that has been fitted. For the MODEL option the following models are available. There are bounded isotropic models with finite ranges. These all take the value c + c0 for h ≥ a, and the following values for h < a

boundedlinear
c0 + ch/a

circular
c0 + c {1 - (2/π)arccos(h/a)

   
+ (2h/(πa))√(1-h2/a2)}

spherical
c0 + c {1.5h/a - 0.5(h/a)3 }

doublespherical
c0 + c1 {1.5h/a1 - 0.5(h/a1)3 }

   
+ c2 {1.5h/a2 - 0.5(h/a2)3 } for h ≤ a1

   
c0 + c1 + c2 {1.5h/a2 - 0.5(h/a2)3} for a1 < h < a2

   
where c = c1 + c2

pentaspherical
c0 + c {1.875h/a - 1.25(h/a)3 + 0.375(h/a)5}

cubic
c0 + c {7(h/a)2 - 8.75(h/a)3 + 3.5(h/a)5

   
- 0.75(h/a)7}

There are also bounded asymptotic models

besselk1
c0 + c {1 - h/a K1(h/a) }

(Whittle's elementary correlation, Whittle 1954)

exponential
c0 + c {1 - exp(-h/a)}

gaussian
c0 + c {1 - exp(-h2/a2)}

stable
c0 + c {1 - exp(-(h/a)b))}

unbounded models

power
c0 + g hα

(power function with exponent α strictly between 0 and 2)

linear
c0 + c h

which is a special case of the power function with exponent 1

hole effect models

cardinalsine
c0 + c × (1 - a/h × sin(h/a)) .

and an unbounded and geometrically anisotropic model, i.e. one that might be made isotropic by a simple linear transformation of the spatial coordinates

affinepower
c0 + ( √{ a2cos2(θ-φ) + b2sin2(θ-φ) } h )power

where θ represents the direction (specified by the DIRECTION parameter) converted from degrees to radians.

   The placement of the graph within the graphical frame can be controlled using the WINDOW option. The TITLE option can supply a title for the plot. Option XUPPER can define an upper value for the x-axis (i.e. distance), and PENDATA and PENMODEL can supply the numbers of the pens to be used to plot the experimental variogram and the fitted model respectively (by default 1 and 2).

 

Options: MODEL, WINDOW, TITLE.

Parameters: VARIOGRAM, DISTANCE, DIRECTION, ESTIMATES, XUPPER, PENDATA, PENMODEL.


Action with RESTRICT

If the data variates are restricted, only the units not excluded by the restriction will be plotted.


References

Chiles, J-P. & Delfiner, P. (1999). Geostatistics: Modeling Spatial Uncertainty. Wiley, Chichester.

Webster, R. & Oliver, M.A. (2001). Geostatistics for Environmental Scientists. Wiley, Chichester.

Whittle, P. (1954). On stationary processes in the plane. Biometrika, 41, 434-449.