Beyond the Raster Model: How GIS Represents Continuous Variables over Geographic Space

Although many variables are continuous in space (e.g. temperature, solar radiation, topography, etc.), their digital representation in operational software systems imposes several constraints. These are mainly driven by the limits of our representations, in particular those imposed by the so-called useful information and those imposed by the limits of our computers and software systems. These limits impose the use of intermediate representations, in particular logical and physical models for the representation of these data that take into account the finite and quantified nature of the digital representation. To this end, we distinguish two models or implementation strategies that will be presented through this article: the model based on a tessellation of space and the model based on sampling & interpolation.

Tesselation Model

A tessellation consists in dividing space into contiguous cells, whose boundaries between two cells are deterministic (i.e., sharp). Depending on the shape of the cells, tessellation can be classified into : (1) irregular tessellation or (2) regular tessellation

A regular tesselation is characterized by the attribution of an identical and fixed geometrical shape in terms of size for all the cells composing the space. In the case where this shape is rectangular and regular, it will be the raster model (see figure 1). More precisely, the raster model consists of an assembly or grid of cells, often called pixels, each of which is associated with a pair of coordinates \$(x,y)\$, as well as a value of a variable (e.g. temperature).

The models discussed (i.e., voronoi and raster) are represented in a two-dimensional space, however they can be extended to larger space dimensions. For instance, the basic unit of three-dimensional regular tessellation is a voxel, while three-dimensional Voronoi diagrams can also be generated.

When representing a continuous field by the tessellation mode, one loses all details about the spatial variations within the tesserae, because usually only one value is assigned for each tessellation.

Model based on sampling & interpolation

GISs implement other strategies that are essentially based on the concepts of sampling and interpolation. As a consequence there two steps :

1) sampling : It consists in selecting a finite set of point or polyline geometries whose attributes are known, forming the spatial reference of the continuous variable, and then determining for the other points in space the target value by interpolation.

These geometries (i.e. the spatial reference) are not intended to represent spatial entities in reality, and if they are used, it is only for purposes of representation and discretization of the geographical domain of the field.

2) Interpolation : defined as the means of estimating the value of variable for locations where the value under study is not recorded.

Images A, B, E and F in Figure 2 show four types of spatial reference. A) shows sampling on a set of regularly spaced points; B) shows sampling on a set of irregularly distributed points; E) shows sampling on a set of irregularly distributed points and textured in the form of an irregular triangular model (TIN). In the latter case, the values of the points of the field are estimated by linear or cubic polynomial spatial interpolation using the values of the vertices of the triangle ; finally, F) represents a sampling corresponding to the isohypses (contour lines).

The arrangement and choice of an interpolation method in this type of process is important to obtain more reliable information. For example, the IDW (Inverse distance weighting) method is based on the assumption that the value at an unsampled point can be approximated by a weighted average of values at points that are spatially close to each other.

The majority of interpolation methods are based on the notion of distance. This link comes from Tobler’s first law, called the first law of geography, which states that :

“everything is related to everything, but things close are more related than things far away (Tobler 1970)”.

According to (Goodchild 2004), without this law and principle, any spatial interpolation would be unreliable.

The difference between models based on interpolation and those based on tessellation can be summarized as follows: in the case of tessellation, in order to determine the value of a field at a point x it is necessary to identify the tessellation that contains x, whereas in the case of interpolation, it is necessary to determine the samples of the spatial reference to be used and an interpolation method to be applied.

February 1, 2021

Younes Hamdani

Hi, I'm Younes Hamdani founder of this platform.

I've enjoyed many progressive roles throughout my career as a software engineer, consultant, instructor, and a research scientist in the field of GIS and computer science. I love the idea of combining both to solve real world problems. I am always passionate and curious about how geographic information systems and science can transform and improve the lives of people around the world.

My drive and enthusiasm to find creative solutions, solve real-world problems and share knowledge
has been leading to the development of several challenging and valuable projects, the conduct and teaching of numerous training courses and the production of many scientific publications.

I enjoy meeting new people and hearing new perspectives. Reach out if you want to talk to me about emerging GIS technologies, new startup ideas or soccer :)