C-squares

Spatial data are inherently (at least) 2-dimensional; without additional indexing, a numeric range query in 2 dimensions (e.g. x and y, or latitude and longitude) is required to retrieve data items within a particular area. Such queries are computationally expensive so it can be beneficial to pre-process (index) the data in some manner that reduces the inherent dimensionality from two to one dimension, for example as labelled cells of a grid; the grid labels can then be indexed by standard, one dimensional methods for rapid search and retrieval,[12] and/or searched by simple alphanumeric text searches. C-squares is an example of such a grid where the cell identifiers are designed to be human- as well as machine-readable, and to be concordant with recognizable and commonly intervals of latitude and longitude.

Spatial data, that is, data associated with particular geographic locations on the earth, have spatial "footprints" that ideally are recorded in metadata systems or data catalogues, to support spatial searching of the resources in question.[13] A "basic" generalization of any data footprint is the minimum bounding rectangle or MBR, that is, the smallest set of boundaries of latitude and longitude that completely contain the data. Such stored rectangles are relatively simple to query as a mathematical operation (look for overlap with an input "search" rectangle) but, with real world data, may not be a good surrogate for the true data "footprint" if the latter contains disjoint or sparsely populated data items, items on a diagonal line, or items with significant "holes", such as a vessel track around a continent.[3] Representation as a set of smaller "tiles", such those denoted by c-square codes at an appropriate resolution, can more accurately reflect the shape of such non-rectangular and/or non-contiguous datasets.[3][13]

"Binning" describes the process of converting continuously variable data (in the present case, spatial locations in degrees of latitude and longitude) into a set of discrete "bins" in order to apply the indexing and subsequent search/retrieval processes, other processing and reporting, etc. The optimal size of the spatial data "bins" can depend on the user's requirements (e.g. geographic coverage), desire for handling large vs. small datasets (which can affect the time required for processing or production of on-demand maps) and density of available data (with sparse input data, many "bins" may end up being empty). Large "bins" will have smaller data handling requirements and smooth out data deficiencies, but will result in a loss of resolution of fine scale data; small "bins" can result in large quantities of data to be handled (for example, a global coverage at 0.1 degree resolution requires over 6 million grid cells) which may be too much for easy handling. To date, 0.5 degree cells (50×50 km nominal size) have been found to be a reasonable compromise between resolution and data storage requirements for global coverages (259,200 cells), e.g. as used for Aquamaps,[6] while for more local applications, either 0.5 or 0.1 degree cells may be useful.

One advantage of binning spatial data as described above is that it offers the potential for data reduction in some use cases: for example rather than storing (say) hundreds of raw data points within the same bin, the data can be represented as an average value and/or number of data points held, or just as presence/absence (for example as a range map). By this means, the quantity of information required to be stored in the spatial index and associated information can be substantially reduced in many cases, with a concomitant improvement in performance (speed of spatial queries and result mapping as desired).

As a property of a discrete global grid, hierarchical notation ensures that the geocodes for finer resolutions of the mesh incorporate those of all their parents, permitting rapid search and/or data aggregation at any desired equal or higher level of the hierarchy. In the c-squares case, the code is extended by additional alphanumeric characters as the spatial resolution increases, with the corollary that resolution can be decreased if desired, merely by truncating the code by the relevant amount.

An equal angle spherical (global) grid, represented in a "real world" (orthographic) perspective.

Equal angle grids (the class that includes c-squares) have the advantage that transformation of spatial data (for example as simple coordinates of latitude and longitude) in and out of the grid notation can be simple, since the latitude–longitude grid is itself equal angle.[14] On the actual surface of the globe, the cells are approximately "square" only adjacent to the equator, and become progressively narrower and tapered (also with curved northern and southern boundaries) as they approach the poles, and cells adjoining the poles are unique in possessing three faces rather than four. By contrast, equal area grids attempt to preserve a constant area for all cells at the same hierarchical level (resolution), at the expense of losing concordance with familiar lines of latitude and/or longitude.

Local and/or national grids have been developed for use within a number of countries, for example the UK National Grid has been in use since 1946,[15] while a separate system is in use for Ireland. Discontinuities occur where such grids meet or overlap, and some areas (for example the more offshore portions of surrounding seas and the Channel Islands) are not covered at all. Global grids offer a solution to this problem (providing standard treatment for all areas of the globe) and also offer a potential format for collation of cross-national data into a single repository for analysis and reporting, for example see Vanhee et al., 2018.[8]

10-degree c-squares are specified as being identical to equivalent World Meteteorological Organization (WMO) square codes, refer illustration at right. These squares are aligned with 10-degree subdivisions of the global latitude–longitude grid, which for c-squares use is specified as employing the WGS84 datum. WMO (10 degree) squares are encoded with four digits, in the series 1xxx, 3xxx, 5xxx and 7xxx.[11] The leading digit indicates the "global quadrant" with 1 for north-east (latitude and longitude are both positive), 3 for south-east (latitude is negative and longitude positive), 5 for south-west (latitude and longitude are both negative) and 7 for north-west (latitude is positive and longitude negative). The next digit, 0 through 8, corresponds to the tens of latitude degrees either north or south; while the remaining 2 digits, 00 through 17, correspond to the tens of longitude degrees either east or west (by specification, 0 is treated as positive). Thus the 10 degree cell with its lower left corner at 0,0 (latitude,longitude) is encoded 1000, and acts as a bin to contain all spatial data between 0 and 10 degrees north (actually, 0 and 9.999...) and 0 and 9.999... degrees east; the 10 degree cell with its lower left corner at 80 N, 170 E is encoded 1817, and acts as a bin to contain all spatial data between 80 and 90 degrees north and 170 and 179.999... degrees east.

C-squares extends the initial WMO 10×10 square notation via a recursive series of "cycles", each 3 digits long (the final one may be 1 digit), separated by the colon character, the number of characters (and cycles) indicating the resolution encoded, as per these examples:

• 1000 ... 10×10 degree square (up to 1000×1000 km nominal)
• 1000:1 ... 5×5 degree square (up to 500×500 km nominal)
• 1000:100 ... 1×1 degree square (up to 100×100 km nominal)
• 1000:100:1 ... 0.5×0.5 degree square (up to 50×50km nominal)
• 1000:100:100 ... 0.1×0.1 degree square (up to 10×10 km nominal)
• 1000:100:100:1 ... 0.05×0.05 degree square (up to 5×5km nominal)

(etc.)

The nominal sizes given reflect the fact that at the equator, 1 degree of both latitude and longitude correspond to around 110 km, with the actual value for longitude declining between there and the poles, where it becomes zero (latitude actual: 110.567 km at the equator, 111.699 km at the poles; longitude actual: 111.320 km at the equator, 78.847 km at latitude ±45 degrees, 0 km at the poles); at a sample northern hemisphere latitude e.g. that of London (51.5 degrees north), a 1×1 degree square measures approximately 111×69 km.[16]

C-squares recursive subdivision principle - intermediate quadrant example (south-east global quadrant)

To produce the 1 or 3 digits in any cycle following the initial 4-digit, 10-degree square identifier, first an "intermediate quadrant", 1 through 4 is designated (refer diagram at right), where 1 indicates low absolute values of both latitude and longitude (regardless of sign), 2 indicates low longitude and high latitude, 3 indicates high latitude and low longitude, and 4 indicates high values for both; "low" and high" being taken from the relevant portion of the data to be gridded (for example within the 10 degree cell extending from 10 to 20 degrees, 10 is treated as low and 19 as high). This leading digit in a cycle is then followed simply by the next applicable digit for first latitude and then longitude: thus an input value of latitude +11.0, longitude +12.0 degrees will be encoded as the 5 degree c-square code 1101:1 and the 1 degree code 1101:112. Inspection of this code will show that the input latitude value can be recovered directly from the digits 1101:112 while the longitude is included as 1101:112; the sign for these is both positive, as indicated by the first digit of the leading 4 (1 in this case, indicating the north east global quadrant).

From 2002 onwards (still current at 2020), an online "latlong to c-squares conversion page" is available at the website of CSIRO Marine Research (now CSIRO Oceans and Atmosphere) which will convert input values of latitude and longitude to the equivalent c-square code at user selectable resolutions from 10 to 0.1 degree cell size. Alternatively it is a comparatively simple task to program from first principles (or construct as, for example, a Microsoft Excel worksheet) according to the c-squares specification;[17] an example is available here.