Sorry, I hadn't updated my new definitions page. attached is the current
version, and it has also been updated at
http://acd.ucar.edu/~caron/definitions.html.
Coordinate Variables in Netcdf : Definitions
Draft 7/21/97 by John Caron, with help from Brian Eaton and Russ Rew
The following tries to make formal definitions using the language of
abstract algebra. A standard reference is Algebra, Saunders MacLane and
Garrett Birkhoff, The Macmillan Company, 1967.
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A dimension, d, is a named range of integers: d = {0,1,..size-1} (or d
{1,2,..size} if you prefer). A dimension is completely specified by the pair
(name, size).
An index domain, D, is a set constructed from the cartesian product of one
or more dimensions: D = d1 x d2 x .. x dn, where di are dimensions. The
points of D are thus tuples of integers. A projection Dp of D is a cartesian
product of a subset of the dimensions {di} that D is constructed from. (So a
point in Dp is just a point in D with 0, 1, or more indices missing). We
will also call the function p that maps D to Dp a projection of D.
A variable is a function v(D) -> C, where D is an index domain, v denotes
the function, and C is the range or codomain. The image of a function is the
set of points in C that are the values of the function. Since we consider
here only index domains, which are a finite set of points, the image of a
function is also always a finite set of points. In the context of netcdf
files, the values of a function have any of the possible data types of a
netcdf variable: double, int, string, etc. The number of dimensions in the
domain of a function is its dimensionality.
A vector function is an ordered list of scalar functions with the same
domain, called component functions. A vector function thus maps points in D
to a tuple of values of its component scalar functions. In practice the
component functions may have domains that are projections of D. Formally
this is done by composing the component function with a projection function:
cf_formal = cf_actual * p, where * is functional composition and p is the
projection function which maps D to the domain of cf_actual.
An embedding E is an invertible map from a finite set C to Rn, the cartesian
n-product of the real numbers R. Each set of real numbers in Rn is called an
axis, so that the embedding E(C) -> Rn is a map from S onto n axes.
A coordinate function is a scalar or vector function whose codomain C is
embedded into R. An ordered list of coordinate functions can be considered a
vector function by replacing any vector coordinate function by its list of
component (scalar) functions. A coordinate system is an ordered list of
coordinate functions with the same domain, which is one-to-one as a vector
function. A coordinate system thus assigns unique physical values to the
points in its domain: it maps an n-tuple of integers in "index space" to a
unique m-tuple of reals, strings, etc. in "physical space", called a
location. The number of coordinate functions is the rank of the coordinate
system, and each is associated with a different axis of Rn . Formally, we
can write a coordinate system as a function Cs(D) -> C, or equivalently
Cs(D) = (F1(D), F2(D), ...,Fn(D)) -> (C1, C2, ..., Cn), where Fi(D) -> Ci is
the ith coordinate function. The values of the coordinate functions are the
coordinates of the coordinate system.
For a variable v(Dv) -> Cv, and coordinate system Cs(Ds) -> Cs, Cs may be a
coordinate system for variable v when Ds is a projection of Dv. When Ds
Dv, Cs is a complete coordinate system for v, since then Cs assigns a unique
location to every value of v.
A spatial coordinate system is a coordinate system whose locations are in
3-dimensional space. A georeferencing coordinate system is a spatial
coordinate system which provides enough information to place its locations
in reference to the earth. A temporal coordinate system is one which
provides enough information to place its locations in real, physical time.
