Description Usage Arguments Details Value Note Author(s) References See Also Examples

Generates a variogram model, or adds to an existing model.
`print.variogramModel`

prints the essence of a variogram model.

1 2 3 4 5 |

`psill` |
(partial) sill of the variogram model component, or model: see Details |

`model` |
model type, e.g. "Exp", "Sph", "Gau", "Mat". Calling vgm() without a model argument returns a data.frame with available models. |

`range` |
range parameter of the variogram model component; in case of anisotropy: major range |

`kappa` |
smoothness parameter for the Matern class of variogram models |

`nugget` |
nugget component of the variogram (this basically adds a nugget compontent to the model); if missing, nugget component is omitted |

`add.to` |
the variogram model to which we want to add a component (structure) |

`anis` |
anisotropy parameters: see notes below |

`x` |
a variogram model to print |

`...` |
arguments that will be passed to |

`covtable` |
if model is |

`Err` |
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 |

`m` |
object of class |

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.

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.

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.

Edzer Pebesma

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | ```
vgm()
vgm("Sph")
vgm(NA, "Sph", NA, NA)
vgm(, "Sph") # "Sph" is second argument: NO nugget in this case
vgm(10, "Exp", 300)
x <- vgm(10, "Exp", 300)
vgm(10, "Nug", 0)
vgm(10, "Exp", 300, 4.5)
vgm(10, "Mat", 300, 4.5, kappa = 0.7)
vgm( 5, "Exp", 300, add.to = vgm(5, "Exp", 60, nugget = 2.5))
vgm(10, "Exp", 300, anis = c(30, 0.5))
vgm(10, "Exp", 300, anis = c(30, 10, 0, 0.5, 0.3))
# Matern variogram model:
vgm(1, "Mat", 1, kappa=.3)
x <- vgm(0.39527463, "Sph", 953.8942, nugget = 0.06105141)
x
print(x, digits = 3);
# to see all components, do
print.data.frame(x)
vv=vgm(model = "Tab", covtable =
variogramLine(vgm(1, "Sph", 1), 1, n=1e4, min = 0, covariance = TRUE))
vgm(c("Mat", "Sph"))
vgm(, c("Mat", "Sph")) # no nugget
``` |

```
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)
model psill range
1 Nug NA 0
2 Sph NA NA
model psill range
1 Nug NA 0
2 Sph NA NA
model psill range
1 Sph NA NA
model psill range
1 Exp 10 300
model psill range
1 Nug 10 0
model psill range
1 Nug 4.5 0
2 Exp 10.0 300
model psill range kappa
1 Nug 4.5 0 0.0
2 Mat 10.0 300 0.7
model psill range
1 Nug 2.5 0
2 Exp 5.0 60
3 Exp 5.0 300
model psill range ang1 anis1
1 Exp 10 300 30 0.5
model psill range ang1 ang2 ang3 anis1 anis2
1 Exp 10 300 30 10 0 0.5 0.3
model psill range kappa
1 Mat 1 1 0.3
model psill range
1 Nug 0.06105141 0.0000
2 Sph 0.39527463 953.8942
model psill range
1 Nug 0.0611 0
2 Sph 0.3953 954
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 1
[[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"
[[1]]
model psill range kappa
1 Mat NA NA 0.5
[[2]]
model psill range
1 Sph NA NA
attr(,"class")
[1] "variogramModelList" "list"
```

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