Data Structures for 2D Representation of Terrain Models

Definition

This entry gives an overview of the main data structures and approaches used for a two-dimensional representation of the terrain surface using a digital elevation model (DEM). A DEM represents the elevation of the earth surface from a set of points. It is used for terrain analysis, visualisation and interpretation. DEMs are most commonly defined as a grid where an elevation is assigned to each grid cell. Due to its simplicity, the square grid structure is the most common DEM structure. However, it is less adaptive and shows limitations for more complex processing and reasoning. Hence, the triangulated irregular network is a more adaptive structure and explicitly stores the relationships between the points. Other topological structures (contour graphs, contour trees) have been developed to study terrain morphology. Topological relationships are captured in another structure, the surface network (SN), composed of critical points (peaks, pits, saddles) and critical lines (thalweg, ridge lines). The SN can be computed using either a TIN or a grid. The Morse Theory provides a mathematical approach to studying the topology of surfaces, which is applied to the SN. It has been used for terrain simplification, multi-resolution modelling, terrain segmentation and landform identification. The extended surface network (ESN) extends the classical SN by integrating both the surface and the drainage networks. The ESN can itself be extended for the cognitive representation of the terrain based on saliences (typical points, lines and regions) and skeleton lines (linking critical points), while capturing the context of the appearance of landforms using topo-contexts.

1. Introduction

How to represent a terrain surface and interpret it has always been a fundamental issue in many disciplines, including geography, surveying and environmental science. Representation techniques have evolved through time with the rise of new technologies. When considering cartography, the objective was to facilitate the user’s interpretation of the terrain portrayed on a map. As mentioned by Morato-Moreno [1], the scientific representation of terrain elevation started around the end of the sixteenth century with the use of contour lines to represent points at the same depth in a riverbed. By the end of the 1950s, the increased power of computers enabled researchers to create and manipulate an electronic version of a map called a digital terrain model (DTM). Miller and Laflamme [2] define the DTM as “a statistical representation of the continuous surface of the ground by a large number of selected points with known X, Y, Z coordinates in an arbitrary coordinate field”. Other terms are also used to describe terrain surface. The most common alternative term is the digital elevation model (DEM). Li et al. [3] consider that the DTM represents the spatial distribution of various types of information on the terrain, not only the elevation, while the DEM only measures the elevation of the bare Earth surface. Other types of information are morphometric data (slope, curvature), terrain features (geomorphological features, rivers, lakes) and environmental data such as soil, geology or climate.

DEMs were constructed mainly for applications in civil engineering. Originally, the set of points were obtained from ground surveys or through photogrammetric processes. The surface was formed by triangulating these points into triangulated irregular networks (TINs). With the development of remote sensing, data became increasingly available, covering larger areas and being easier to process, so that digital models could be created at different scales with increasing precision. These data were often produced or stored as raster images, where each pixel contained an elevation, for ease of processing. Consequently, much research was carried out to create more advanced tools for terrain analysis on regular square grids. More specifically, geomorphometry developed as a discipline on its own [4], providing new techniques for terrain partitioning and classification, including object-based image analysis—OBIA [5]. DEMs were also produced from existing data, as found on topographic maps, such as contour lines and streams. The generated DEM could be either a TIN [6] or a square grid [7].

The creation and development of airborne lidar (Light Detection and Ranging) brought higher resolution, with a resolution around or below one metre and larger volumes of data [8]. As pointed out by Clarke and Romero [9], higher-resolution data brought new sources of error, and algorithms faced new issues. For example, drainage computation methods that were efficient on low-resolution images were no longer accurate with the presence of new features such as roads and culverts, requiring specific drainage enforcement methods [10].

While TIN and square grids are the most common DEM structures, other types of meshes can be used, such as equilateral triangles and hexagonal grids [11]. Data structures are mainly chosen according to the data source format, terrain representation and the type of usage.

In addition to computational analysis, data structures are also built from DEMs for reasoning and interpretation purposes. Interest in describing the terrain through such structures can be traced back to earlier works from Cayley [12]. Cayley reasoned on contour lines to define feature points on the terrain that are summits (local maximums), immits (local minimums) and knots (saddles). He then characterised ridge and course (or thalweg) lines, pointing out that they start at a saddle and end at a maximum or a minimum. Shortly after, Maxwell [13] presented mathematical formulae relating the different types of points. He also introduced hills and dales as districts where lines of slope run to the same maximum or minimum point and he showed that hills are bounded by thalwegs and that dales are bounded by ridges. Most importantly, he demonstrated that hills and dales each form a partition of the terrain.

These early works established the basic foundations for reasoning on terrain. and they led Pfaltz [14] to formalize terrain data as a graph structure called the surface network. Other structures were also introduced, such as the contour tree and the Reeb graph [15]. These data structures explicitly represent hierarchical and adjacency relationships between terrain elements. They can be used to identify terrain objects that are relevant to a particular analysis, such as the identification of landforms.

This entry gives an overview of the main data structures used for 2D terrain representation and analysis. Section 2 presents some approaches that build a two-dimensional representation directly from data. These structures were mainly defined for computational analysis. Section 3 introduces more advanced structures, especially surface networks, that can characterise terrain features and store hierarchical or topological relationships between the features. Section 3 also presents recent developments for the use of extended surface networks applied to the cognitive representation of the terrain based on topographic saliences, skeletons and topo-contexts. Section 4 concludes the entry and briefly mentions new research avenues opened by the recent developments of surface networks and related techniques for terrain modelling.

2. Surface Representation

2.1. Raster

The most popular form of DEM is the raster, a regular square grid, where each pixel corresponds to an elevation datum. In a raster, each pixel is located by its row and column index. Indexes provide an implicit topology, since adjacent pixels are the previous and next pixels in a column or a row. Two kinds of connectivity are defined. A grid point can have four orthogonal neighbours, located on the same row or column, or eight neighbours by adding four nodes connected diagonally. The eight-connectivity model is more commonly used because it yields distances closer to the reality. However, the distance between adjacent points is not constant, with distances along the diagonals $\sqrt{2}$ times longer than in the orthogonal directions.

The raster is still the most used representation because of its simplicity. It allows for the computation of slope, curvature and other derivatives, using finite differences and is commonly used for terrain analysis and segmentation [16]. For this purpose, one finds many applications in environmental sciences where the terrain can be partitioned into morphometric classes. The level of detail of a raster is defined by its resolution (i.e., the size of the pixel) and can be adjusted through resampling and smoothing techniques.

Terrain analysis and visualisation can also be performed at different levels (or map scales). More specifically, in a DEM different phenomena can be observed at different scales by adjusting the size of the moving window [17]. For visualisation purposes, a pyramid-based approach was proposed to ensure smooth scale change for terrain rendering [18]. The pyramid increases the storage space, since it stores a grid for each defined scale. In areas where terrain variations are small, storing an elevation at each grid node can lead to redundancies. The quadtree is a hierarchical structure based on a regular decomposition of a region into quadrants. Each quadrant can be subdivided into four quadrants, each storing more detailed information [19]. While a quadtree efficiently stores coverage data (e.g., land cover, land use), it is not usual in a DEM to find four grid nodes with an identical height. Hence, building a quadtree often requires a tolerance to consider adjacent points have the same value, leading to some loss of information.

2.2. TIN

The raster grid provides a rigid representation that cannot be easily adapted to the details of the terrain. Another structure, the triangulated irregular network (TIN) was developed for that purpose. A TIN is formed by a triangulation of points on the surface. Originally, TINs were defined to provide a more compact representation of the terrain. Only very important points (VIPs) were kept and were triangulated, providing a piecewise linear representation [20]. Nowadays, TINs are built directly from ground points. Triangulation is usually performed using a Delaunay triangulation and can be constrained to preserve some terrain features. Triangulation is performed in two dimensions, since the elevation is an attribute of the position. The approach is adaptive, keeping more points in areas with important variations and removing most of them in flat areas or in regular slopes. The triangulation can also be constrained by lines, so that feature lines of the terrain, such as fault lines and valley lines, are preserved.

In a TIN, the topology is explicitly stored. This means that adjacency relationships between points (vertices) and triangles are recorded. Different structures were defined for that purpose. The most elementary one consists in storing on one side a table of points with an index and their coordinates, and on the other side a table of triangles in which each triangle is defined by the indexes of its three points (Figure 1). However, such a structure does not provide direct access to adjacent triangles, and one cannot walk through the TIN by following a gradient descent, for example. Therefore, other structures were proposed.

Another structure adds pointers to the three neighbouring triangles (Figure 2). Vertices are ordered in counterclockwise order. Neighbouring triangles are ordered so that the i^th triangle opposes the i^th vertex. Edges in the triangulation are implicitly represented through the adjacency relations of two triangles: the i^th edge is the edge opposing the i^th vertex and also the edge shared by the i^th triangle. This structure is notably implemented in CGAL, the Computational Geometry Algorithms Library [21].

A second kind of structure is centred on vertices, relying on the definition of stars and links of each vertex [22]. The star of a vertex is the set of edges that start at the vertex. The link of a vertex v is the set of vertices that end the edges in the star (Figure 3). Vertices in a link are ordered around the vertex. Hence, edges starting at v are defined by pairs $(v,v_i)$, where $v_i$ is in the link of v. Edges are directed and the edge $(v_j,v_i)$ is in the star of $v_j$. Similarly, given an edge $(v_i,v_j)$, a triangle is obtained by finding $v_j$ in the link of $v_i$ and taking the next vertex. For example, in Figure 3, given the edge $(v,v_1)$, the vertex following $v_1$ in the link of v is $v_2$. An implementation of the structure for TIN is presented in [23].

Other structures using polygonal meshes can also be used for storing a triangulation [24]. These structures offer an explicit definition of vertices, edges and triangles (or, more generally, faces) and store connections between edges and vertices and between edges and triangles. Edges are directed, having an origin vertex and a destination vertex.

The quad-edge data structure [25] stores, for each edge, pointers to the origin and destination vertices and to the left and right triangles (Figure 4, top). Edge functions are also defined to return the previous and next edges when rotating around a vertex or a triangle (Figure 4). They are the functions that allow for navigating through the triangulation. One can obtain all the edges around a vertex (Figure 4, bottom left) or around a face (Figure 4, bottom right). By storing a direct access to faces and points, the quad-edge can also store the dual graph. In the case of a triangulation, the quad-edge can store both the triangulation and its dual, the Voronoi diagram [26].

The DCEL (doubly connected edge list) [24] decomposes each edge into two half-edges. Both half-edges are in opposite directions and are associated with a different triangle (Figure 5). The two half-edges are associated as twins. A half-edge also points to its previous and to its next half-edge. In the other way, a face points at one adjacent half-edge and a vertex points towards a half-edge starting at this vertex. A DCEL does not store the dual graph but offers a more compact representation than the quad-edge. Thus, the DCEL is the structure used in libraries such as the Point Data Abstraction Library (PDAL).

These data structures are more complex to handle than a simple grid. Consequently, grid structures remain more commonly used because few geographical information systems can directly handle TINs, often converting them in a raster form for processing. Nonetheless, terrain parameters such as derivatives and curvatures can also be computed on a TIN. For example, they are used for terrain segmentation [27] and for drainage computation [28,29]. The TIN can also be extended to handle more complex representations and can integrate tunnels and bridges, corresponding to holes in the structure [30].

As mentioned before, TINs have the benefit of being more adaptive. Vertices can be easily added or removed for terrain simplification. For this reason, they are mainly used for 3D rendering and visualisation. To have a smooth rendering, the triangulation is stored in a hierarchical structure where details can be added or removed depending on the point of view and the level of zoom [31]. More recently, the availability of LIDAR (light, detection and ranging) systems, collecting massive point clouds, created a need for new spatial data structures that can handle big data volumes [32]. The terrain tree was developed as a spatial index for the representation and analysis (including slope and curvature computation) of very large TINs [33].

2.3. Hexagonal Grid

In addition to square grids and TINs, other data structures have been investigated. Researchers were mainly interested in regular structures that provide a uniform partitioning of the surface. Apart from squares, the two other grid cells that can be considered are the equilateral triangle and the regular hexagon. The triangle is rarely considered, since triangles can have two orientations, leading to inconsistencies [34]. By contrast, the hexagon tessellation (Figure 6) provides a more meaningful connectivity. Hexagons have a more circular shape than squares and each hexagon is connected to its six neighbours by its edges. The distance between the centres of two adjacent cells is constant, reducing the orientation bias of the square grid in distance measurement.

Cartesian coordinates are no longer suited for hexagonal grids. Different coordinate systems were defined to index the cells. The most common is the offset indexing (Figure 6, left), similar to the Cartesian system. The coordinates of each odd row are simply shifted by half a cell. A limitation of this kind of indexing is that formulas for accessing neighbours of a cell are different for odd and even rows. A two-coordinate system can also be defined by considering non-orthogonal axes (Figure 6, middle). This system does not need to distinguish odd and even rows. The angle between the two axes can be 60° or 120°. A third coordinate system uses three axes (Figure 6, right). While it adds one coordinate, it simplifies many operations [35]. Topographic parameters can then be calculated over the whole terrain [36]. The hexagonal grid can also be used to compute the flow direction and the drainage network, showing a higher capacity than the square grid to preserve the flow direction [37].

Hierarchical hexagonal grids can also be defined. However, a hexagon cannot be partitioned into smaller hexagons, contrary to squares or triangles that can be partitioned into similar shapes. The larger hexagon covers a central hexagon and its six neighbours. Other patterns can also be defined [35].

3. Morphological Structures

3.1. Contour Graphs and Trees

While previous data structures were designed for fast access to TIN elements and computational analysis, other topological structures were developed to study the terrain morphology and to assist in terrain interpretation. Historically, terrain interpretation was carried out on contour maps. This interpretation relies on the inclusion and adjacency relationships between contours. Thus, one of the earliest applications of graph theory to terrain interpretation was proposed by Morse [38] for the analysis of contour maps. Morse defined a contour graph based on contour adjacency: two contours are adjacent if they can be connected by a line that does not intersect any other contour. Areas between contours are called intercontour regions.

However, because contours are closed lines, the most used structure to represent relationships between contours is the contour tree. The hierarchical relationship between contours is defined by containment [39]. Kweon and Kanade [40] used the contour tree to extract feature points on the terrain. Peaks and pits are found in series of enclosed contours; and saddles are found between contours at the same elevation. A similar reasoning can be applied on the intercontour region graph. In [41], a data structure, the feature tree, was proposed to extract elevations and depressions from the intercontour region graph. Depressions and elevations form a hierarchy of terrain features that is also based on inclusions. These structures are represented in Figure 7, where numbers are contour elevations and letters are intercontour regions.

3.2. Surface Networks

As mentioned above, Cayley [12] and Maxwell [13] identified topological relationships between critical points and lines. Relationships are represented by a graph called the surface network (Figure 8), where vertices are the peaks, pits and saddles of the terrain and edges are ridges and thalwegs. A surface network obeys the following properties:

• A ridge connects a peak and a saddle;
• A thalweg connects a pit and a saddle;
• Ridges and thalwegs intersect only at saddles;
• At a saddle, the number of ridges is equal to the number of thalwegs;
• When turning around a saddle, edges alternate between ridges and thalwegs.

As identified by Maxwell, thalwegs delineate hills (the red lines in Figure 8) and ridges delineate dales (the blue lines in Figure 8). Each hill or dale is associated with exactly one peak or one pit. The intersection of a hill and a dale is called a hillslope (yellow in Figure 8). Considering flowlines on the terrain, a hill is also defined as the set of all flowlines starting from a given peak. A dale is the set of all flowlines ending at a pit. Consequently, a hillslope is the set of all flowlines starting at a given peak and ending at a pit.

Different methods have been used to extract critical points and lines. The earliest approaches for raster DEMs were proposed in [20,44]. These methods were used to extract very important points (VIPs) and to build a TIN, as mentioned in Section 2.2, but they do not provide a topological structure. Adopting another point of view, other authors compute the drainage networks using both a DEM and the associated inverted DEM. Thalwegs are the streams in the DEM and ridges are streams in the inverted DEM [45]. However, the result depends on user-defined parameters and both networks are computed separately and remain disconnected.

Pfaltz [14] proposed a definition of surface networks grounded in the Morse theory. The Morse theory provides mathematical definitions and tools for studying the topology of surfaces. The surface network is a Morse–Smale complex, meaning that the terrain is fully partitioned into Morse–Smale cells (the hillslopes). Hills and dales are, respectively, descending and ascending manifolds in the Morse theory. A review of methods computing surface networks from a DEM with their mathematical background is presented in [46].

Importantly, the surface network can be computed on any data structure, whether a TIN or a grid. A first group of methods relies on the so-called growing regions approach. Starting from pits, regions are grown by following the gradient direction to build the ascending manifolds. Similarly, regions are grown from peaks following the steepest descent to build the descending manifolds. Both structures are then intersected, which creates the surface network. Critical lines are the boundaries of manifolds and saddles are the intersections between ridges and thalwegs. However, these intersection points do not always match with the location of saddles on the terrain [46].

The second approach is called the boundary-based approach. It is more commonly used on a DEM and is based on the detection of critical points followed by the detection of critical lines, leading to the definition of the boundaries of the Morse–Smale cells (the hillslopes). Critical lines are initiated at saddles, running up to a peak or down to a pit. A method specific to piecewise linear surfaces defined from a triangulated grid was introduced in [47]. Critical lines do not follow the gradient but the steepest edge, so that they always align with the edges of the triangulation.

Bremer et al. [48] presented a method for TINs where critical lines directly follow the gradient. Triangles are split so that critical lines are still aligned with the edges of the triangulation. Two ridges or two thalwegs can merge; but instead of keeping two overlapping lines, a junction point is inserted and lines are split at the junction.

These approaches consider the terrain as a continuous, smooth surface. In practice, a TIN is a piecewise linear surface defined by a discrete set of points. Gradient computation is based on finite differences. Inconsistencies can be generated where thalwegs and ridges intersect outside saddle points. Bremer et al. [48] insert new edges in the triangulation to prevent these intersections. In a recent approach [49], intersections are prevented by computing first the thalwegs and the hills. Ridges are computed next and constrained to remain within their hill.

Forman [50] proposed a discrete Morse theory. In this approach, an elevation is assigned to each DEM element (vertex, edge and triangle) and a discrete vector field is defined between each (vertex–edge) and (edge–triangle) pair. Thus, a gradient path is a sequence of (vertex–edge) and (edge–triangle) pairs. In such a representation, a pit is a vertex lower than its adjacent edges, a saddle is an edge higher than its adjacent vertices but lower than the triangles of both sides and a peak is a triangle higher than its three edges. A thalweg is a succession of (vertex–edge) and a ridge is a succession of (edge–triangle). In this representation, thalwegs and ridges cannot intersect, except at saddles. However, the result depends on the calculation of the discrete vector field. This vector field is not unique, and flat areas in the terrain require specific processing.

Magillo et al. [51] presented a comparison of different computational approaches (growing regions, boundary based and discrete Morse theory) on different TINs and found out that they all provided similar results. The differences stemmed from the boundaries of hills and dales that were modified. Depending on the method, a triangle could be located within a hill (or dale) or in a neighbouring one. However, these differences were generally fewer than 2.5%, bringing very similar terrain partitions. The authors concluded that these approaches were theoretically well-founded.

The Morse theory was first applied to terrain simplification and multiresolution modelling in the computer graphics domain [52]. The TIN was simplified while preserving critical lines and points that represent the main features of the terrain. The surface network was also used for terrain segmentation and landform identification. Sinha and Mark [53] extracted topographic eminences based on the prominence and isolation of peaks. Argudo et al. [54] trained a random forest classifier to recognize mountain landscapes, such as the Alps and the Himalayas. Metrics, including prominence and isolation, were computed on the ridge network. The authors showed that each landscape can be identified by topographic features associated with peaks and ridges and that one can use these features to generate synthetic terrain models.

In [55], the surface network is computed from a bathymetric model to detect submarine canyons. Canyons are characterised by thalwegs. The surface network is first simplified by merging thalwegs that have a similar slope. Canyons are then detected according to their thalweg slope and sinuosity. Sonke et al. [56] make use of the surface network to identify channel networks from a riverbed. Channels form braiding patterns due to shallow shoals or steps in the riverbed. If a shoal (a hill in the surface network) is too small, a thalweg is removed, aggregating the shoals on both sides. The approach provides a hierarchy of channels and allows for the computation of sand volume, thus estimating the sediment transport in the river. Finally, the surface network was also considered to delineate landslide systems [57] where the system is contained in a hillslope delineated by the ridges of the network.

3.3. Extended Surface Networks

As mentioned above, the surface network provides a structure for reasoning on a terrain and identifying landforms. However, the SN was mostly used in Computer Sciences and geometrical modelling, and it received limited consideration in environmental sciences. The main reason is that most tools rely on image segmentation techniques. They are handled by experts who can interpret the maps generated by these tools. Another noteworthy explanation is that the drainage network cannot be computed directly from the surface network. While the surface network can extract thalwegs and ridges, it cannot compute the flow accumulation that is indispensable to defining a drainage network with its main streams and tributaries. Ridges in the surface network represent lines of divergent flows, which correspond to drainage divides. However, all drainage divides are not included in the surface network: if two thalwegs join and run down to the same pit, the junction node is not a saddle and, thus, no ridge is initiated to separate the drainage basin of each stream.

A new structure called the extended surface network (ESN) was proposed in [43]. It integrates both the surface network and the drainage network (Figure 9). In the ESN, ridges are initiated not only at saddle points but also at every point (called confluence) where two thalwegs connect. Hence, ridges correspond to drainage divides and a flow accumulation can be computed for each thalweg. Streams are thalwegs whose flow accumulation has a sufficient area (according to a predetermined threshold).

In the ESN, thalwegs are identical to those computed in a classical surface network. Thus, the descending manifold remains the same. However, there are twice as many ridges because a ridge is initiated at each node where a thalweg starts or arrives. Importantly, dales are no longer centred on pits, but on thalwegs: each dale is delineated by ridges and contains exactly one thalweg. The dale represents the area where all the flows end directly in the thalweg. The ascending manifold of the surface network is obtained by merging all the dales of the thalwegs that are directed towards the pit. Thus, the ESN contains all the features of the surface network.

Considering that the flow always runs from the upper end of a thalweg to its lower end, it is possible to direct the flow in the whole network. For each thalweg, the accumulation is given by the area of its dale added to the accumulation of all upstream thalwegs. However, many pits in the surface network are caused by the lack of precision of the data used to generate the surface network, especially in relatively flat areas. If the flow is supposed to go down the thalwegs, some spurious pits may interrupt the flow. This is a common problem of the existing methods for drainage computation. The common approach consists in removing spurious depressions by altering the DEM, either by filling the depressions or by carving a breach to a lower point [10].

Instead of altering the DEM, the approach proposed in [43] consists in directing the flow in each depression towards its outlet, defined by the lowest saddle on the boundary of the depression. Thus, this technique can compute a whole drainage network without altering the DEM. Moreover, the drainage basin of every thalweg can also be extracted directly from the ESN as the union of all the dales located upstream of the thalweg. The authors in [43] also compared their approach with the traditional D8 approach [58] coupled with breaching [10], and they obtained a positional accuracy above 93% without altering the DEM.

3.4. The Cognitive Use of Extended Surface Networks

Geomorphological mapping plays an essential role in earth sciences. On a topographic map, the terrain is modelled as an elevation field, while topographic objects are represented by geometrical shapes or symbols. This representation is efficient for digital processing using a DEM. However, it does not fit well with the human perception of topography, since people perceive and reason about the geographic space mostly qualitatively [59]. Geomorphologists are trained to recognize landforms and to identify their prominent characteristics (i.e., saliences) in the field, as well on maps. The notion of salience has been investigated in cognitive research on visual attention [60]. Here, a salience is defined as ‘an emphasis of an element or of a set of elements distinguishable from the whole’ [61], characterized by ‘relatively distinct, prominent or obvious features compared to other features’ [62].

The extended surface network (Section 3.3) is composed of elementary topographic saliences (i.e., critical points) and their salient relations captured by the thalweg and the ridge networks [63]. Moreover, geographers can identify more global saliences that are typical of a terrain/landscape shape. This is the case for terrain skeleton lines (such as valley and ridge lines) that are deemed to be important and inseparable parts of topographical maps [64].

In their approach, [63] consider skeleton lines to be important salient features of topographic maps. Considering the ESN data structures, a skeleton line is defined as a portion of either the thalweg network or the ridge network. In accordance with the cognitive principles advocated by Gestalt Theory [65], Guilbert and Moulin hypothesize that many landforms can be characterized by such skeleton lines. The ESN data structures allow for geometrically representing landforms characterized by a skeleton, a ‘salient region’ and a ‘boundary region’ as proposed in [66]. The salient and boundary regions are used to handle vagueness and indeterminacy of the landform location in a similar way as in [67]. Skeleton lines can also provide a topological structure connecting the landforms together, and, hence, they can be used to characterize and identify an ensemble of landforms. As an example, a drainage basin is a hydrological area that is characterised by a skeleton, the network of rivers, and it is delimited by drainage divides enclosing the basin. These elements are determined by the thalweg network and the ridge network of the ESN.

Gestalt Theory also shows that objects are not observed in isolation, but in context [65]. Notably, this principle is exploited in image analysis approaches for the detection of so-called ‘salient regions’ [68] and for the identification of object shapes [69]. Indeed, geomorphologists are trained to consider the global context (i.e., topographic, geologic) when identifying landform shapes and typical (‘salient’) regions where the landforms can be found [70]. In [63], such salient regions are called the ‘context of appearance of landforms’.

In addition, these authors introduce the notion of topo-context, which provides a definition of the context of appearance based on the terrain topography. Using the ESN data structure (Section 3.3), a topo-context (TContext for short) is an area defined by a dale (resp. by a hill) or an aggregation of adjacent dales (resp. adjacent hills) in the ESN. The aggregation criterion depends on the type of topo-context [63]. Interestingly, the aggregation of adjacent dales (resp. hills) preserves the ESN topological properties. Two important components of a TContext, the ring and the skeleton, are needed to formally express the characteristics of TContexts, considering that they are portions of the ESN. The ring of a TContext is defined as the closed thalwegline, or ridgeline, that marks the periphery of the TContext’s spatial area. The skeleton of a TContext is defined as the portion of the ridge network or of the thalweg network contained within the area delimited by the ring of the TContext. In [63], the authors formally define and discuss the properties of different categories of TContexts useful for terrain modelling. They show that the ESN can not only integrate efficiently the surface network and the hydrographic network in a common data structure, but also that it can be used to partition the geographic space using the TContexts that provide the topographic context of appearance of landforms. It is also shown that all these structures are topologically sound. As an illustration, these authors show how to process the streams of the hydrographic network (specified as thalwegs in the ESN) to determine their catchment areas, as well those of their tributaries. A catchment area is defined as the union of all the dales whose flow converges towards a portion of the stream. Considering all the tributaries of the streams of the study area, the ESN provides the set of the drainage basins, as well as the drainage network. The drainage basins are TContexts, whose skeletons are portions of the drainage network, all being computed using the ESN data structure that ensures the topological soundness.

4. Conclusions

The objective of this article was to present an overview of the main data structures that have been proposed for the two-dimensional representation of a terrain, using digital elevation models. It also shows all the ground that has been covered since the early definitions of the surface network around the 1860s until the recent advances provided by the extended surface network and its use for the cognitive representation of the terrain based on topographic saliences, skeletons and topo-contexts. These new approaches grounded on the ESN are built upon a theoretical framework that offers a salience-based representation of the terrain and that can capture the context of appearance of landforms using topo-contexts. Moreover, this recent research provides software tools that integrate the topographic and the hydrologic networks, providing a network representation of elementary saliences (peaks, pits, saddles, ridges and thalwegs) and compound saliences (using skeletons) in relation to topographic data made compatible with the hydrological network.

New avenues are opened for research in earth sciences, such as in hydrology [71], in geomorphology [70], in landscape modelling [72] and in terrain representation [16]. Interestingly, in 2013 a group of researchers advanced the vision of Linked Topographic Data, an approach to make topographic datasets accessible and interoperable using semantic web technologies [73]. They advocated for the use of a data structure based on the surface network that could be efficiently stored and shared on the web and that could be linked to other data (URIs, gazetteers, etc.). An ontology design pattern (ODP), the surface network ODP (SNODP), was proposed to capture the semantics needed to create a topologically consistent surface network. Later, another ODP, the surface water feature ODP (SWFODP), was proposed to capture the main concepts associated with surface water features [74]. These ODPs have been integrated in the Landform Reference Ontology (LFRO), which was proposed as a domain reference ontology for knowledge representation and reasoning about landforms [75]. Unfortunately, we are not aware of any implementations of these ontologies. The recent ESN-based proposal may provide a good foundation for implementing the vision of the Linked Topographic Data and to support the implementation of the SNODP and SWFODP ontology design patterns.