\name{Coordinate systems}
\alias{coordinates}
\alias{coordinate system}
\alias{coordinate systems}
\alias{zenit}
\title{Coordinate systems}
\description{
Implemened Coordinate Systems
}
\section{Implemented coordinate systems}{
\itemize{
\item Cartesian coordinate system
\item Earth coordinate systems\cr
The earth is considered as an ellipsoid;
The first angle takes values in \eqn{[0, 360)},
the second angle takes values in \eqn{[-90, 90]}.
\item Spherical coordinate systems\cr
The earth is considered as an ellipsoid;
The first angle takes values in \eqn{[0, 2\pi)},
the second angle takes values in \eqn{[-\pi/2, \pi/2]}.
}
}
\section{Transformations between the system}{
\itemize{
\item Earth to cartesian\cr
The 3-dimensional resulting coordinates
are either given in \sQuote{km} or in \sQuote{miles}.
\item Gnomonic an orthographic projections\cr
The 2-dimensional resulting coordinates
are either given in \sQuote{km} or in \sQuote{miles}.
The projection direction is given by the \code{zenit}.
\item Earth to spherical\cr
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.
}
\section{Options}{
\describe{
\item{\code{coord_system}}{character.
One of the values \code{"auto"}, \code{"cartesian"}, \code{"earth"}
% \code{"polar"}, \code{"cylindric"}, \code{"spherical"},
If \code{"auto"}, then the coordiates are considered as
\code{"cartesian"} except the names of the given coordinates
indicate a
different system. Currently, only \code{"longitude"} and
\code{"latidute"} (or abbreviations of them) are excepted
as names for given coordinates
and indicate an earth coordinate systems. See the examples below.
Default: \code{"auto"}
}
\item{\code{coordnames}}{integer vector of length 2 or an increasing
sequence of integers or character.
This parameter gives the coordinate columns in a data frame, either
by starting column and ending column or the sequence or by names.
In the first case, single code{NA}s might be included, meaning
\sQuote{from the beginning} or \sQuote{until the end}. If both
values are \code{NA}, then, depending on the context, either
an error message is returned or it is assumed that the first
columns give the coordinates.
}
\item{\code{coordunits}}{any string.
If \code{coordinate_system = "earth"} and longitude and latitude
are transformed to 3d cartesian coordinates, \code{coordunits}
determines
whether the radius is given in kilometers (\code{"km"}) or miles
(\code{"miles"}).
If empty, then \code{"km"} is chosen.
Default: \code{""}
}
\item{\code{new_coord_system}}{
One of the values \code{"keep"}, \code{"cartesian"}, \code{"earth"},
\code{"plane"}.
% \code{"polar"}, \code{"cylindric"}, \code{"spherical"},
\enumerate{
\item \code{"keep"}\cr
The \code{coord_system} is kept (except an explicite transformation
is given, see \command{\link{RMtrafo}}.
Note that some classes of models, e.g. completely monotone
functions and compactly supported covariance models with
range less than \eqn{\pi} are valid models on a sphere.
In this case the models are considered as models on the sphere.
See \link{spherical models} for lists.
\item \code{"cartesian"}\cr
If \code{coord_system} is \code{"earth"} the coordinates are transformed
to cartesian coordinates before any model is considered.
\item \code{"orthographic"}, \code{"genomic"} \cr
If \code{coord_system} is \code{"earth"} the locations are projected
to a plane before any
model is considered.
}
Default: \code{"keep"}
}
\item{\code{new_coordunits}}{internal and should not be set by the user.
Default: \code{""}
}
\item{\code{polar_coord}}{logical.
If \code{FALSE} the spherical coordinates agree with
the earth coordinate parametrisation, except that
we radians are used for spherical coordinates instead
of degrees for the earth coordinates.
If \code{TRUE} the spherical coordinates signify polar coordinates.
Default : \code{FALSE}
}
\item{\code{varnames}}{integer vector of length 2 or an increasing
sequence of integers or character.
This parameter gives the data columns in a data frame, either
by starting column and ending column or the sequence or by names.
In the first case, single code{NA}s might be included, meaning
\sQuote{from the beginning} or \sQuote{until the end}. If both
values are \code{NA}, then for keywords \sQuote{data},
\sQuote{value} and \sQuote{variable} will be searched for.
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.
}
\item{\code{varunits}}{vector of characters.
The default units of the variables.
Default: \code{""}
}
\item{\code{xyz_notation}}{logical or \code{NA}. Used by
\code{\link{RMuser}} only.
\code{NA} : automatic choice (if possible)
\code{false} : notation (x, y) should not be understood as
as kernel definition, not as xyz notation
\code{true}: xyz notation used
% \code{2}:this value is only used by calls of RFcov and should not
% be used by a user
}
\item{\code{zenit}}{two angles of the central
projection direction for the gnomonic projection
(\url{http://en.wikipedia.org/wiki/Gnomonic_projection},
\url{http://de.wikipedia.org/wiki/Gnomonische_Projektion})
and the orthographic projection,
(\url{http://en.wikipedia.org/wiki/Orthographic_projection_in_cartography},
\url{http://de.wikipedia.org/wiki/Orthografische_Azimutalprojektion}).
If \code{any(is.na(zenit))}
then either the value of either of the components may not be \code{NA},
whose value will be denoted by \eqn{p}.
If \eqn{p=1} then the mean of the locations is calculated;
if \eqn{p=Inf} then the mean of the range is calculated.
Default: \code{c(1, NA)}
}
}
}
\references{
Covariance models in a cartesian system
\itemize{
\item Schlather, M. (2011) Construction of covariance functions and
unconditional simulation of random fields. In Porcu, E., Montero, J.M.
and Schlather, M., \emph{Space-Time Processes and Challenges Related
to Environmental Problems.} New York: Springer.
}
Covariance models on a sphere
\itemize{
\item Gneiting, T. (2013) Strictly and non-strictly positive definite
functions on spheres.
\emph{Bernoulli}, \bold{19}, 1327-1349.
}
Tail correlation function
\itemize{
\item 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.
\emph{Extremes}, \bold{} Submitted.
}
}
\seealso{
\command{\link{RMtrafo}},
\command{\link{RFearth2cartesian}},
\command{\link{RPdirect}},
\command{\link[=spherical models]{models valid on a sphere}},
\command{\link{RFoptions}}
}
\examples{
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) models
## same as above, but explicite:
RFcov(model, x, y, grid=FALSE, coord_sys="cartesian")
## model is valid not on a sphere; x,y coordinates are
## transformed from earth coordinates to sphereical coordinates
RFcov(model, x, y, grid=FALSE, coord_sys="earth")
## now comparable the scale chosen sucht that the covariance
## values are comparable to those int 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:
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 comparable to cartesian case
## here the (standard) units are [km]
RFcov(RMS(model, s= 6350 / 180 * pi), x, y, grid=FALSE,
coord_sys="earth", new_coord_sys="orthographic")
## as above, but in miles
RFcov(RMS(model, s= 3750 / 180 * pi), x, y, grid=FALSE,
coord_sys="earth", new_coord_sys="orthographic",
new_coordunits="miles")
\dontshow{FinalizeExample()}
}
\author{Martin Schlather, \email{schlather@math.uni-mannheim.de}
\url{http://ms.math.uni-mannheim.de/de/publications/software}
}
\keyword{spatial}