Generates a variogram model, or adds to an existing model. print.variogramModel prints the essence of a variogram model.

vgm(psill = NA, model, range = NA, nugget, add.to, anis, kappa = 0.5, ..., covtable,
Err = 0)
# S3 method for variogramModel
print(x, ...)
# S3 method for variogramModel
plot(x, cutoff, ..., type = 'l')
as.vgm.variomodel(m)

## Arguments

psill (partial) sill of the variogram model component, or model: see Details model type, e.g. "Exp", "Sph", "Gau", "Mat". Calling vgm() without a model argument returns a data.frame with available models. range parameter of the variogram model component; in case of anisotropy: major range smoothness parameter for the Matern class of variogram models nugget component of the variogram (this basically adds a nugget compontent to the model); if missing, nugget component is omitted the variogram model to which we want to add a component (structure) anisotropy parameters: see notes below a variogram model to print or plot arguments that will be passed to print, e.g. digits (see examples), or to variogramLine for the plot method if model is Tab, instead of model parameters a one-dimensional covariance table can be passed here. See covtable.R in tests directory, and example below. numeric; if larger than zero, the measurement error variance component that will not be included to the kriging equations, i.e. kriging will now smooth the process Y instead of predict the measured Z, where Z=Y+e, and Err is the variance of e object of class variomodel, see geoR maximum distance up to which variogram values are computed plot type

## Value

If a single model is passed, an object of class variogramModel extending data.frame.

In case a vector ofmodels is passed, an object of class variogramModelList which is a list of variogramModel objects.

When called without a model argument, a data.frame with available models is returned, having two columns: short (abbreviated names, to be used as model argument: "Exp", "Sph" etc) and long (with some description).

as.vgm.variomodel tries to convert an object of class variomodel (geoR) to vgm.

Edzer Pebesma

## Details

If only the first argument (psill) is given a character value indicating a model, as in vgm("Sph"), then this taken as a shorthand form of vgm(NA,"Sph",NA,NA), i.e. a spherical variogram with nugget and unknown parameter values; see examples below. Read fit.variogram to find out how NA variogram parameters are given initial values for a fitting a model, based on the sample variogram. Package automap gives further options for automated variogram modelling.

## Note

Geometric anisotropy can be modelled for each individual simple model by giving two or five anisotropy parameters, two for two-dimensional and five for three-dimensional data. In any case, the range defined is the range in the direction of the strongest correlation, or the major range. Anisotropy parameters define which direction this is (the main axis), and how much shorter the range is in (the) direction(s) perpendicular to this main axis.

In two dimensions, two parameters define an anisotropy ellipse, say anis = c(30, 0.5). The first parameter, 30, refers to the main axis direction: it is the angle for the principal direction of continuity (measured in degrees, clockwise from positive Y, i.e. North). The second parameter, 0.5, is the anisotropy ratio, the ratio of the minor range to the major range (a value between 0 and 1). So, in our example, if the range in the major direction (North-East) is 100, the range in the minor direction (South-East) is 0.5 x 100 = 50.

In three dimensions, five values should be given in the form anis = c(p,q,r,s,t). Now, $p$ is the angle for the principal direction of continuity (measured in degrees, clockwise from Y, in direction of X), $q$ is the dip angle for the principal direction of continuity (measured in positive degrees up from horizontal), $r$ is the third rotation angle to rotate the two minor directions around the principal direction defined by $p$ and $q$. A positive angle acts counter-clockwise while looking in the principal direction. Anisotropy ratios $s$ and $t$ are the ratios between the major range and each of the two minor ranges. The anisotropy code was taken from GSLIB. Note that in http://www.gslib.com/sec_gb.html it is reported that this code has a bug. Quoting from this site: The third angle in all GSLIB programs operates in the opposite direction than specified in the GSLIB book. Explanation - The books says (pp27) the angle is measured clockwise when looking toward the origin (from the postive principal direction), but it should be counter-clockwise. This is a documentation error. Although rarely used, the correct specification of the third angle is critical if used.''

(Note that anis = c(p,s) is equivalent to anis = c(p,0,0,s,1).)

The implementation in gstat for 2D and 3D anisotropy was taken from the gslib (probably 1992) code. I have seen a paper where it is argued that the 3D anisotropy code implemented in gslib (and so in gstat) is in error, but I have not corrected anything afterwards.

## References

http://www.gstat.org/

Pebesma, E.J., 2004. Multivariable geostatistics in S: the gstat package. Computers \& Geosciences, 30: 683-691.

Deutsch, C.V. and Journel, A.G., 1998. GSLIB: Geostatistical software library and user's guide, second edition, Oxford University Press.

For the validity of variogram models on the sphere, see Huang, Chunfeng, Haimeng Zhang, and Scott M. Robeson. On the validity of commonly used covariance and variogram functions on the sphere. Mathematical Geosciences 43.6 (2011): 721-733.

show.vgms to view the available models, fit.variogram, variogramLine, variogram for the sample variogram.

## Examples

vgm()
#>    short                                      long
#> 1    Nug                              Nug (nugget)
#> 2    Exp                         Exp (exponential)
#> 3    Sph                           Sph (spherical)
#> 4    Gau                            Gau (gaussian)
#> 5    Exc        Exclass (Exponential class/stable)
#> 6    Mat                              Mat (Matern)
#> 7    Ste Mat (Matern, M. Stein's parameterization)
#> 8    Cir                            Cir (circular)
#> 9    Lin                              Lin (linear)
#> 10   Bes                              Bes (bessel)
#> 11   Pen                      Pen (pentaspherical)
#> 12   Per                            Per (periodic)
#> 13   Wav                                Wav (wave)
#> 14   Hol                                Hol (hole)
#> 15   Log                         Log (logarithmic)
#> 16   Pow                               Pow (power)
#> 17   Spl                              Spl (spline)
#> 18   Leg                            Leg (Legendre)
#> 19   Err                   Err (Measurement error)
#> 20   Int                           Int (Intercept)vgm("Sph")
#>   model psill range
#> 1   Nug    NA     0
#> 2   Sph    NA    NAvgm(NA, "Sph", NA, NA)
#>   model psill range
#> 1   Nug    NA     0
#> 2   Sph    NA    NAvgm(, "Sph") # "Sph" is second argument: NO nugget in this case
#>   model psill range
#> 1   Sph    NA    NAvgm(10, "Exp", 300)
#>   model psill range
#> 1   Exp    10   300x <- vgm(10, "Exp", 300)
vgm(10, "Nug", 0)
#>   model psill range
#> 1   Nug    10     0vgm(10, "Exp", 300, 4.5)
#>   model psill range
#> 1   Nug   4.5     0
#> 2   Exp  10.0   300vgm(10, "Mat", 300, 4.5, kappa = 0.7)
#>   model psill range kappa
#> 1   Nug   4.5     0   0.0
#> 2   Mat  10.0   300   0.7vgm( 5, "Exp", 300, add.to = vgm(5, "Exp", 60, nugget = 2.5))
#>   model psill range
#> 1   Nug   2.5     0
#> 2   Exp   5.0    60
#> 3   Exp   5.0   300vgm(10, "Exp", 300, anis = c(30, 0.5))
#>   model psill range ang1 anis1
#> 1   Exp    10   300   30   0.5vgm(10, "Exp", 300, anis = c(30, 10, 0, 0.5, 0.3))
#>   model psill range ang1 ang2 ang3 anis1 anis2
#> 1   Exp    10   300   30   10    0   0.5   0.3# Matern variogram model:
vgm(1, "Mat", 1, kappa=.3)
#>   model psill range kappa
#> 1   Mat     1     1   0.3x <- vgm(0.39527463, "Sph", 953.8942, nugget = 0.06105141)
x
#>   model      psill    range
#> 1   Nug 0.06105141   0.0000
#> 2   Sph 0.39527463 953.8942print(x, digits = 3);
#>   model  psill range
#> 1   Nug 0.0611     0
#> 2   Sph 0.3953   954# to see all components, do
print.data.frame(x)
#>   model      psill    range kappa ang1 ang2 ang3 anis1 anis2
#> 1   Nug 0.06105141   0.0000   0.0    0    0    0     1     1
#> 2   Sph 0.39527463 953.8942   0.5    0    0    0     1     1vv=vgm(model = "Tab",  covtable =
variogramLine(vgm(1, "Sph", 1), 1, n=1e4, min = 0, covariance = TRUE))
vgm(c("Mat", "Sph"))
#> [[1]]
#>   model psill range kappa
#> 1   Nug    NA     0   0.0
#> 2   Mat    NA    NA   0.5
#>
#> [[2]]
#>   model psill range
#> 1   Nug    NA     0
#> 2   Sph    NA    NA
#>
#> attr(,"class")
#> [1] "variogramModelList" "list"              vgm(, c("Mat", "Sph")) # no nugget
#> [[1]]
#>   model psill range kappa
#> 1   Mat    NA    NA   0.5
#>
#> [[2]]
#>   model psill range
#> 1   Sph    NA    NA
#>
#> attr(,"class")
#> [1] "variogramModelList" "list"