set the variogram of the Kriging object
a Kriging object
a variogram, specified by name (Spherical, Exponential, Gaussian, Wave, RationalQuadratic or Circular) or as a function call with three positive numerical arguments
The SetVariogram command sets the variogram associated with the given Kriging object.
This variogram is used in the Kriging interpolation process to determine the weights for various points. For example, it is used when you interpolate at a given point. The variogram gives the variance of the difference between field values at two locations at a given distance.
If the variogram is specified as a name only, then the parameters will be estimated by the FitVariogramParameters command.
A Kriging object by default has its variogram set to the Spherical model with parameters determined by FitVariogramParameters, and it will keep this default until SetVariogram is called.
The variogram has three parameters: the nugget, sill, and range.
The nugget is the limit of the variogram as the distance approaches zero, and corresponds to a base level of uncertainty and variation. The nugget must be nonnegative.
The sill is the maximum variance achieved for any distance. The sill must be greater than or equal to the nugget; so in particular, it must be nonnegative.
The (effective) range is the distance at which the variogram reaches 95% of the sill, or for some variograms, the distance where the sill is reached. More precisely, it is a distance a so that if a≤h, then γ⁡h deviates from the sill by less than 5%. Points at a distance greater than the range are not used to compute interpolated values. The range must be positive.
The available variogram models are as follows. Each is a family of functions of h, the distance, with parameters c__n, c__s, and a corresponding to the nugget, sill, and range, respectively. After defining each variogram model, you see a graph of the variogram with parameters c__n=0.5, c__s=5, and a=4, displayed for h from 0 to 6. Dashed lines indicate the nugget, sill, and range, and a gray rectangle indicates the bounds between which the variogram is guaranteed to lie at distances greater than the range. After this graph, you see a heat map of a sample generated with GenerateSpatialData according to the same variogram over a square with sides of length 6.
The Spherical⁡c__n,c__s,a model has the following equation.
The Exponential⁡c__n,c__s,a model has the following equation.
The Gaussian⁡c__n,c__s,a model has the following equation.
The Wave⁡c__n,c__s,a model has the following equation.
The RationalQuadratic⁡c__n,c__s,a model has the following equation.
The Circular⁡c__n,c__s,a model has the following equation.
We generate some values that have the specified variogram.
points,data ≔ Kriging:-GenerateSpatialData⁡Spherical⁡1,10,1
k ≔ Kriging⁡points,data
k≔Krⅈgⅈng ⅈntⅇrpolatⅈon obȷⅇct wⅈth 30 samplⅇ poⅈntsVarⅈogram: Sphⅇrⅈcal(1.94850490681965,21.53887094302,.52789833)
Setting the variogram explicitly is not required, as a variogram with estimated parameters is set by default. However, it leads to better results if the correct variogram is set.
Krⅈgⅈng ⅈntⅇrpolatⅈon obȷⅇct wⅈth 30 samplⅇ poⅈntsVarⅈogram: Sphⅇrⅈcal(1,10,1)
The Interpolation[Kriging][SetVariogram] command was introduced in Maple 2018.
For more information on Maple 2018 changes, see Updates in Maple 2018.
Interpolating at a point
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