# CoordinateSystems: Coordinate systems In RandomFields: Simulation and Analysis of Random Fields

## Description

Implemented Coordinate Systems.

## Implemented coordinate systems

• Cartesian coordinate system

• Earth coordinate systems
The earth is considered as an ellipsoid; The first angle takes values in [0, 360), the second angle takes values in [-90, 90].

• Spherical coordinate systems
The earth is considered as an ellipsoid; The first angle takes values in [0, 2π), the second angle takes values in [-π/2, π/2].

## Transformations between the system

• Earth to cartesian
The 3-dimensional resulting coordinates are either given in ‘km’ or in ‘miles’.

• Gnomonic and orthographic projections
The 2-dimensional resulting coordinates are either given in ‘km’ or in ‘miles’. The projection direction is given by the `zenit`.

• Earth to spherical
In this case the Earth is considered as a ball.

Cartesian systems cannot be transformed to earth or spherical coordinate systems, nor a spherical system to earth coordinates.

## Options

`coord_system`

character. One of the values `"auto"`, `"cartesian"`, `"earth"`

If `"auto"`, then the coordinates are considered as `"cartesian"` except the names of the given coordinates indicate a different system. Currently, only `"longitude"` and `"latidute"` (or abbreviations of them) are excepted as names for given coordinates and indicate an earth coordinate systems. See the examples below.

Default: `"auto"`

`coordidx`

integer vector of column numbers of the variables in the data frame. `varidx` can be set alternatively to `coordnames`. This parameter gives the coordinate columns in a data frame by starting column and ending column or the sequence. An `NA` in the second component means ‘until the end’.

`coordnames`

vector of characters that can be set alternatively to coordidx. This parameter gives the coordinate columns in a data frame by names. If it is `"NA"`, then, depending on the context, either an error message is returned or it is assumed that the first columns give the coordinates.

`coordunits`

any string. If `coordinate_system = "earth"` and longitude and latitude are transformed to 3d cartesian coordinates, `coordunits` determines whether the radius is given in kilometers (`"km"`) or miles (`"miles"`). If empty, then `"km"` is chosen.

Default: `""`

`new_coord_system`

One of the values `"keep"`, `"cartesian"`, `"earth"`, `"plane"`.

1. `"keep"`
The `coord_system` is kept (except an explicit transformation is given, see `RMtrafo`.

Note that some classes of models, e.g. completely monotone functions and compactly supported covariance models with range less than π are valid models on a sphere. In this case the models are considered as models on the sphere. See spherical models for lists.

2. `"cartesian"`
If `coord_system` is `"earth"` the coordinates are transformed to cartesian coordinates before any model is considered.

3. `"orthographic"`, `"genomic"`
If `coord_system` is `"earth"` the locations are projected to a plane before any model is considered.

Default: `"keep"`

`new_coordunits`

internal and should not be set by the user.

Default: `""`

`polar_coord`

logical. If `FALSE` the spherical coordinates agree with the earth coordinate parametrization, except that radians are used for spherical coordinates instead of degrees for the earth coordinates.

If `TRUE` the spherical coordinates signify polar coordinates.

Default : `FALSE`

`varidx`

integer vector of length 2. `varidx` can be set alternatively to `varnames`. This parameter gives the data columns in a data frame, either by starting column and ending column. An `NA` in the second component means ‘until the end’.

`varnames`

vector of characters that can be set alternatively to `varidx`. This parameter gives the data columns in a data frame by names.

if `varnames` equals `"NA"` then for keywords ‘data’, ‘value’ and ‘variable’ will be searched for keywords. If none of them are found, depending on the context, either an error message is returned or it is assumed that the last columns give the data.

`varunits`

vector of characters. The default units of the variables.

Default: `""`

`xyz_notation`

logical or `NA`. Used by `RMuser` only.

`NA` : automatic choice (if possible)

`FALSE` : notation (x, y) should not be understood as kernel definition, not as xyz notation

`TRUE`: xyz notation used

`zenit`

two angles of the central projection direction for the gnomonic projection (https://en.wikipedia.org/wiki/Gnomonic_projection, https://de.wikipedia.org/wiki/Gnomonische_Projektion) and the orthographic projection, (https://en.wikipedia.org/wiki/Orthographic_projection_in_cartography, https://de.wikipedia.org/wiki/Orthografische_Azimutalprojektion).

If `any(is.na(zenit))` then either the value of either of the components may not be `NA`, whose value will be denoted by p.

If p=1 then the mean of the locations is calculated; if p=Inf then the mean of the range is calculated.

Default: `c(1, NA)`

## References

Covariance models in a cartesian system

• Schlather, M. (2011) Construction of covariance functions and unconditional simulation of random fields. In Porcu, E., Montero, J.M. and Schlather, M., Space-Time Processes and Challenges Related to Environmental Problems. New York: Springer.

Covariance models on a sphere

• Gneiting, T. (2013) Strictly and non-strictly positive definite functions on spheres. Bernoulli, 19, 1327-1349.

Tail correlation function

• Strokorb, K., Ballani, F., and Schlather, M. (2014) Tail correlation functions of max-stable processes: Construction principles, recovery and diversity of some mixing max-stable processes with identical TCF. Extremes, Submitted.

`RMtrafo`, `RFearth2cartesian`, `RPdirect`, `models valid on a sphere`, `RFoptions`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57``` ```RFoptions(seed=0) ## *ANY* simulation will have the random seed 0; set ## RFoptions(seed=NA) to make them all random again z <- 1:4 x <- cbind(z, 0) y <- cbind(0, z) model <- RMwhittle(nu=0.5) RFcov(model, x, y, grid=FALSE) ## standard is (cartesian) model ## same as above, but explicit: RFcov(model, x, y, grid=FALSE, coord_sys="cartesian") ## model is not valid on a sphere; x,y coordinates are ## transformed from earth coordinates to spherical coordinates RFcov(model, x, y, grid=FALSE, coord_sys="earth") ## now the scale is chosen such that the covariance ## values are comparable to those in the cartesian case RFcov(RMS(model, s= 1 / 180 * pi), x, y, grid=FALSE, coord_sys="earth") ## projection onto a plane first. Then the scale is interpreted ## in the usual, i.e. cartesian, sense, i.e. the model does not ## really make sense RFoptions(zenit = c(2.5, 2.5)) RFcov(model, x, y, grid=FALSE, coord_sys="earth", new_coord_sys="orthographic") ## again, here the scale is chosen to be comparable to the cartesian case ## here the (standard) units are [km] (z1 <- RFcov(RMS(model, s= 6350 / 180 * pi), x, y, grid=FALSE, coord_sys="earth", new_coord_sys="orthographic")) ## as above, but in miles (z2 <- RFcov(RMS(model, s= 6350 / 1.609344 / 180 * pi), x, y, grid=FALSE, coord_sys="earth", new_coord_sys="orthographic", new_coordunits="miles")) stopifnot(all.equal(z1, z2)) ## again, projection onto a plane first, but now using the ## gnomonic projection ## here the (standard) units are [km] (z1 <- RFcov(RMS(model, s= 6350 / 180 * pi), x, y, grid=FALSE, coord_sys="earth", new_coord_sys="gnomonic")) ## as above, but in miles (z2 <- RFcov(RMS(model, s= 6350 / 1.609344 / 180 * pi), x, y, grid=FALSE, coord_sys="earth", new_coord_sys="gnomonic", new_coordunits="miles")) stopifnot(all.equal(z1, z2, tol=1e-5)) ```