MVARIOGRAM procedure

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


Options

PRINT = strings
Controls printed output from the fit (model, summary, estimates, correlations, fittedvalues, monitoring); default mode, summ, esti

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

SMOOTHNESS = scalar
Value of power parameter for the stable model; default * i.e. estimate

WEIGHTING = string
Method to be used for weighting (counts, cbyvar, equal); default coun

CONSTANT = string
How to treat the constant (estimate, omit); default esti

WINDOW = scalar
Window in which to plot a graph; default 0 i.e. no graph

TITLE = text
Title for the graph

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


Parameters

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

COUNTS = variates or matrices
Counts for the points in each variogram (not required if WEIGHTING=equal)

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

DIRECTION = variates
Directions in which each variogram was computed

ESTIMATES = variates
Estimated parameter values

FITTEDVALUES = variates
Fitted values

EXIT = scalars
Exit status from the nonlinear fitting

SAVE = pointers
Saves the model name and estimates in a pointer that can be used in KRIGE


Description

Procedure MVARIOGRAM uses the directives FIT, FITCURVE and FITNONLINEAR to fit various models to the experimental variogram. Models must be authorized in the sense that they cannot give rise to negative variances when data are combined. Technically they are conditionally negative semi-definite (CNSD); see Webster & Oliver (1990, 2001), Journel & Huijbregts (1978) or Chiles & Delfiner (1999) for an explanation.

   The MODEL option specifies the model that is to be fitted. 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

and hole effect models

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

Finally, the affinepower function can be fitted to an experimental variogram that appears unbounded and geometrically anisotropic, but that might be made isotropic by a simple linear transformation of the spatial coordinates

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

In all these models, the intercept term (or nugget variance) c0 can be omitted by setting the CONSTANT option to omit; the default is estimate.

   For the stable model the SMOOTHNESS option controls the power parameter for the model. By default, the parameter is estimated, however, you can supply a value to fix the parameter for the model fitting.

   The data for the procedure can be taken directly from the FVARIOGRAM directive, with parameters DISTANCES, VARIOGRAMS and COUNTS corresponding 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 MODEL=affinepower 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 WEIGHTING option controls the weights that are used when fitting the model. The default setting counts uses the values supplied by the COUNTS parameter, cbyvar uses the COUNTS divided by the values in VARIOGRAM, and equal uses equal weights (of one).

   The procedure generates rough starting values for the parameters before calling FITNONLINEAR to convergence. If the solution does not converge there are two likely reasons. The model may be unsuited for the particular experimental variogram. For example, a bounded model is specified when the variogram is clearly unbounded, or vice versa. You should choose only models that have approximately the right shape. Alternatively, the starting values are too far from a sensible solution. Here you should estimate starting values by inspection and insert them into MVARIOGRAM.

   Printed output is controlled by the PRINT option, and includes all the usual settings as in FIT, FITCURVE or FITNONLINEAR. You can also produce a high-resolution graph of the data and the fitted model, by setting the WINDOW option to the number of a suitable window. By default WINDOW is zero, and no graph is produced. 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). Alternatively, you can use the ESTIMATES parameter to save the parameter estimates, and plot the variogram and model later with the DVARIOGRAM procedure.

   The SAVE parameter saves the parameter estimates and associated information required by the KRIGE directive. The FITTEDVALUES parameter saves the fitted values, and the EXIT parameter saves the exit "status code" from FIT, FITCURVE or FITNONLINEAR. A zero value indicates success (see the Guide to GenStat, Part 2, Section 3.7.4).

 

Options: PRINT, MODEL, SMOOTHNESS, WEIGHTING, CONSTANT, WINDOW, TITLE, XUPPER, PENDATA, PENMODEL.

Parameters: VARIOGRAM, COUNTS, DISTANCE, DIRECTION, ESTIMATES, FITTEDVALUES, EXIT, SAVE.


Method

The model is fitted using directives FIT, FITCURVE or FITNONLINEAR as appropriate.


Action with RESTRICT

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


References

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

Journel, A.G. & Huijbregts, C.J. (1978). Mining Geostatistics. Academic Press, London.

Webster, R. & Oliver, M.A. (1990). Statistical Methods in Soil and Land Resource Survey. Oxford University Press.

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.