How dimensions are specified

We consider the dimensions of a data object to be those quantities which are used to select and categorize the data values. To understand the nature of these dimensions, it may help to identify them with the indices of a multi-dimensional array. (This paradigm has its limits, though: data which cannot be described in terms of a regular array can nevertheless be described in terms of these dimensions.) We have proposed four levels of these dimensions:

Level 0

dimensions describe the components of the data (these dimensions are unrelated to the position of a datum in a coordinate space). Simple scalar data such as temperature would have a single Level 0 dimension containing a single grid point (i.e, A Level 0 of order 1 with a rank of 1). A two-component horizontal wind vector would also have a single Level 0 dimension, but with two grid points (order of 1, rank of 2). A wind stress tensor (a $3 \times 3$ matrix) would have two Level 0 dimensions, each of which would have three grid points (order of 2, ranks of 3 and 3). In terms of array indices, Level 0 dimensions select components at a fixed position. That is, a wind stress tensor is considered to be a datum, and two Level 0 indices are needed to select a single component of the tensor. Likewise, a wind vector as a whole is a single item, but a Level 0 index would select which component of the wind (North-South or East-West) is desired. For temperature, there is only one component to select; this would correspond to a single array index which can take only a single value. (Because a $N \times M$ array is indistinguishable from a $N \times M \times 1$ array, such an index is superfluous and can be omitted.)

Level 1

dimensions are those that occur within a single data record.

Level 2

dimensions are those that occur across data records. The Level 1 dimensions, together with the Level 2 dimensions, locate a datum in a coordinate space. The difference between the two levels is that Level 1 dimensions vary within a data record in the dataset, while Level 2 dimensions vary between data records. If each data record were read into a separate array variable, then every array would have to have an index for each Level 0 and Level 1 dimension. Suppose, for example, that a set of temperatures is written out in a series of two-dimensional longitude-latitude grids, one for each day. The resulting dataset would have two Level 1 dimensions--longitude and latitude--plus one Level 2 dimension: time.

This distinction between the two levels may seem artificial, but it has two advantages: first, it enables a program to read the data either as a single large array variable (whose indices would correspond to each of the Level 0, Level 1, and Level 2 dimensions), or as a series of separate variables (whose array indices correspond to each of the Level 0 and Level 1 dimensions, with the number of arrays calculated from the Level 2 dimensions). Second, if the data are read in as separate arrays, then those arrays may be of different sizes. That is, the Level 1 dimensions may have different structures over different ranges of Level 2 dimensions. For example, the first five days of the longitude-latitude wind fields might consist of five $2 \times 72 \times 37$ arrays, while the next ten days might be a series of ten $2 \times 144 \times 46$ arrays.

Level 3

dimensions specify how the data in the dataset have been averaged, integrated, or summed over which subsets of Level 0, Level 1, and Level 2 dimensions. These are ``virtual dimensions'', in that they do not correspond to any indices in a data array, but otherwise their structure is very similar to that of the other, ``real'' dimensions. A set of monthly averaged data, for example, would have a Level 3 dimension corresponding to time and detailing the days of the month over which the average was taken, as well as what averaging method was used. (Note that instantaneous or point data, consisting of observations or calculations that are considered to occur at fixed points in coordinate space, have no averages and hence have no Level 3 dimensions.)

Dimensions may have multiple, parallel definitions. For example, pressure levels at which data are recorded can also be specified as altitudes above sea level. Which description is best? This is a decision best left to the dataset originator. The originator must define one quantity as the authoritative descriptor for a given dimension; any other descriptors for that same dimension will be duplicate, additional descriptors for the dimension. A user of the data will use the authoritative descriptor to obtain the definitive word on the grid point values of that dimension; users may use the additional information if they desire it, but that information is not required for understanding the quantity that dimension represents.