The physical terrain over which the search is conducted requires a representation which is computationally manageable for real-time path planning and which is able to realistically capture the nature of the topography. As O'Rourke
[128,
page270],
we desire a model where "details are abstracted away from a real-life application to produce mathematically 'cleaner' versions of the problem" under the criterion that "if the abstraction is performed intelligently, the theoretical explorations have practical import."To this end we turn to developments in Geographical Information Systems. 4 . 1 G eographical Information Systems
The rapidly expanding and influential area of Geographical Information Systems (GIS) is forging new links among a diverse range of disciplines, including Operations Research
(OR) and Management Science (MS). GIS are versatile and powerful tools which "focus on spatial entities and relationships, together with specific attention to spatial analyt ical and modelling operations" ,
[111,
page17].
The systems can maintain storage and retrieval for not only geographical data but historical, statistical, economic and social data, related to specific geographic regions. Hence, they provide a platform for many diverse applications, including problems requiring multiattribute decisions.Potvin et al.
[139]
designed a computer system named ALTO which allows the user to test and evaluate heuristic solution strategies for complex vehicle routing and scheduling problems. ALTO is built upon a flexible 'general heuristic' which is capable of generating specific heuristic procedures - including those existing in the literature as well as userdefined approaches. This framework enables a dynamic approach to developing heuristic methods for problems where there is "an inability to deduce the kind of strategy that
will perform well given the (problem's) characteristics" ,
[139,
page451].
ALTO provides a graphical, interactive interface where the user is able to explore the effects of different approaches to identify promising avenues for future development. The authors consider ALTO to be the first step towards the creation of an expert system for this field of problems.GIS can be linked with fundamental OR/MS techniques, including shortest path, ver tex and arc routing algorithms, to develop powerful routing systems as seen in GeoRoute
[101].
GeoRoute, inspired by the ALTO system[139]
and designed by the Centre de Recherche sur les Transports de l'Universite de Montreal in cooperation with their cliEmts, provides software to solve complex routing problems over street networks. It incorpo rates an application-dependent and graphically interactive approach, and has since been released by Kositzky & Associates Inc. for use on the Windows platform[3, 85].
The designers[101,
page83]
state that the "main innovation of GeoRoute is the use and manipulation of the structure of the network" , the idea being "quite new in the GIS and operations research community" , where a trade-off between visual and structural prop erties is often seen. Additional recent vehicle routing software that utilize GIS interfaces are surveyed by Hall and Partyka[85].
Emergency Information System (EIS) software is a leading crisis management tool developed in the United States to manage information in crisis situations. Situations that EIS has been used in include such events as toxic chemical spills, flooding, aircraft crashes, bombings, earthquakes and terrorism. EIS offers real-time decision support, powerful communication and visualization capabilities, and integration of geographical data. In particular the EIS/InfoBook brought out in
1995
combines EIS with GIS as a joint result of EIS International collaborating with the Environmental Research Systems Institute.EIS is linked with greater terrain mapping capabilities in a specific Search and Res cue (SAR) operation application by Ketcham and Ketcham
[98].
Ketcham and Ketcham linked EIS with digital orthophotographic map sets of high risk wilderness areas in What corn County, Washington, USA. They also incorporated aerial photographs to enable eas ier visualization of specific terrain features, particularly any that may have altered since the map's production. This inclusion enabled: the maps to be annotated with search in formation as it came to hand; map sections to be printed and distributed to searchers on the ground; searched areas to be visually classified; and searchers' locations to be identi fied with the aid of GPS units on the ground and plotted on the maps. The incorporation of the maps with the EIS software allowed search coordinators to track resources, and redirect them as necessary, from laptops at the scene. Ketcham and Ketcham state that "this application is extremely versatile so it can be adapted to fit any of the unique4.2. Digital Terrain Models 83
circumstances posed by each SAR mission."
"Traditionally, SAR management has been of a more reactive nature, modi fying its approach days into a mission after consolidating notes and opinions. Now, SAR coordinators can take a more preemptive approach as well as track up-to-the-minute search progress" , [98, page 16] .
As geographic information and data bases become more readily available - especially via the information highway - the full potential of such powerful systems will become a reality. In the USA there is already movement in this direction with the current TIGER files1 and the "national spatial data infrastructure" [52, page 45] called for by Vice President Gore in 1994. Currently New Zealand terrain elevation data is available in a 20m contour set owned by Land Information New Zealand and distributed by Terralink. The copyright restrictions on this data are due to be lifted by the government in the near future. Pairman, of Landcare Research New Zealand, states the organization's intention to generate a nationwide digital elevation model, at a detail level of a 25 x 25m grid, from this data set early in the year 2000 [129].
Fischbeck [52, page 45] predicts that
"within the next few years, a flood of data will be available that will allow for even more creative research opportunities for those merging OR/MS with the GIS technology."
It is this form of geographical data stored in spatial network structures that lends itself to the SAR application. By enabling OR techniques to be performed upon data structures which permit the handling of real-time routing requirements and yet also accurately represent reality, the realistic modelling of search terrain and the coordination of effort allocation can occur.
We consider now the specific constructs, known as Digital Terrain Models, which are used within GIS to model terrain.
4.2 Digital Terrain Models
A Digital Terrain Model (DTM) is differentiated from a Digital Elevation Model (DEM) in that it not only digitally represents the elevation of the land surface but also repre sents other topographical features. The DTM allows visualization and data storage of the shape, and attributes of the land surface to facilitate: interpolation of the terrain ITopologically Integrated Geographic Encoding and Reference (TIGER) files that consist of USA Censor data containing geographical information on the entire country.
behaviour not explicitly represented; interpretation of available information; and iden tification of features such as water networks. DTMs are also useful for classifying the terrain into multivariate characteristics, for example, terrain density
[171].
A DTM is suited to a variety of applications due to its flexibility and adaptability, however, the usefulness of any DTM is limited by its accuracy. This is dependent upon the sampling method used to determine the initial data points, as well as the quality of the terrain data utilized. In most cases DTMs are generated and then examined for their accuracy of fit to the land surface in question. If the fit is not considered to be "good" , i. e.,
the approximated surface does not fall within a specified tolerance level, further points are sampled from regions of ill-fit until the error of the fit meets the tolerance level. A computationally more expensive approach is to model all available points, selectively removing points until the tolerance level will be violated by any additional removals
[106].
DTMs are generally formed of simply2 connected surface models
[171]
and are de scribed as being 2.5-D[171, 162]'
as there exists only one z coordinate for each (x,y) coordinate pair.Weibel and HelIer
[171,
page 27
2]
state that:"A variety of data structures for DTMs has been in use over time. Today, however, the overwhelming majority of DTMs conform to one or other of two data structures: rectangular grid (or elevation matrix), or TIN (Triangulated Irregular N etwor k)."
We now compare the characteristics of these two DTMs to determine the one best suited to modelling terrain for SAR operations.
4.2.1 Triangulated Irregular Networks and Rectangular Grids
Rectangular Grids (or Uniform Sampling Grids) approximate the topography by over laying a grid structure onto the terrain. Each grid square is assigned a height value equal to the elevation value of the contour of the surface region enclosed by that grid square. Hence, the grids may be discontinuous at their perimeters, and approximations are necessary to smooth these inconsistencies when forming a path that moves from one grid square to an adjacent grid square.
The Triangulated Irregular Network (TIN) is a DTM where "2-D objects (triangles) are, strictly speaking, embedded in a 3-D space" ,
[67,
page 125].A TIN geometrically partitions the terrain into triangles by a triangulation generated over a set of representative data points. These points are given as an (x,y) coordinate pair
4.2. Digital Terrain Models 85
with an associated spot height and are selected to provide an economical representation,
i. e., one which utilizes the least number of points yet still retains the required accuracy
level of terrain information for the application at hand. To form such an accurate rep
resentation, a TIN must be created from points sampled from critical physical features, with areas of varying terrain being sampled more heavily. Critical physical features in clude local minima and maxima, ridge lines and watersheds. As the selected points are irregularly positioned, an irregular network is created.
Lee
[105,
page414]
defines a TIN as follows:"A TIN model approximates a topographic surface by connecting a set of irregularly spaced elevation vertices into triangular facets. The triangles share edges and vertices to exhaust the space as if they were a triangular mosaic."
3000 2500 2000
�1SOO
1000
500
500 1000 1500
2000 2500 3000 x (m)Figure
4.1:
Triangulated Irregular Network (viewed from above).It is often advantageous to construct TINs from regular square grid DEMs as the elevation data of the gridded DEMs is generally widely available at reasonable cost. Hence, TINs can be constructed for specific terrains for particular applications, based on either a required tolerance level in the elevation difference of the approximated surface to the original, or a pre-specified TIN size.
A number of procedures exist to construct TINs in this way, four methods in particular are examined by Lee
[106]
- the skeleton, filter, hierarchy and drop heuristic methods.The methods all select 'surface-significant' points from the grid DEMs to form the vertices and edges of the TIN but each differ in their definition of 'surface-significant'.
The skeleton method is a two phase approach of Fowler and Little (cited in
[106]),
which begins by selecting significant structural points to create an initial TIN. The TIN then has support points added to it, to obtain a surface approximation within the given tolerance bounds. The initial points are selected relative to their neighbouring points as being locally minima, maxima, potential ridge or channel points. The method is recognized as a complex approach, requiring line-thinning methods to reduce the points required to model ridge and channel lines. Significant storage is required. The skeleton method may also find peaks in flat areas where noise is present, and will select a large number of significant points in rugged areas.The filter method of Chen and Guevara (cited in
[106])
defines 'very important points' as those whose elevation when approximated by their eight grid neighbours are the most different from their actual elevation. The least significant points are then discarded in such a way that the process terminates when the point set remaining is the desired size or the surface approximation falls within the tolerance target. These points form the vertices of the TIN. The method visits each point only once and is therefore both fast and relatively uncomplicated in its approach. Only local information is considered and the method does not guarantee a fit to the terrain which is globally best.The method of De Floriani et al. (cited in
[106]),
the hierarchy method, is a trian gulation approach which uses triangular patches to approximate the terrain surface in successively finer detail. The method has the advantage of a hierarchical data structure and guarantees that a prespecified elevation precision will be met. The approach begins with the grid being subdivided into two triangles; these triangles are then recursively subdivided into three triangles on the point within the triangle which has the largest elevation difference between its actual elevation and that interpolated from the triangle. The recursion ends when the precision level is achieved or a TIN of required size exists. The main drawback of the hierarchy method is that it tends to create long-thin triangles.The final method, which Lee
[106]
considers, is the drop heuristic method which Lee himself developed. This method is the only one of the four approaches which is based upon optimization. The objective is to determine the best set of points from the grid DEM to triangulate, so that the resulting TIN has the minimum elevation difference to the real surface. All the points are initially used to form the TIN with points then being dropped if they result in the least elevation difference when removed. This procedure continues, using the Delaunay triangulation to reconstruct the TIN after the deletion of the point, until the tolerance level or number of required points is achieved. As the method assesses all remaining points at each iteration, it is computationally expensive.Lee tests the latter three methods against one another to assess their individual and relative performance. He concludes that the drop heuristic method generally performs
4.2. Digital Terrain Models 87
better than the other two approaches, but has the greatest computational requirements. The filter method was the fastest approach of the three and the most suitable for local elevation changes, while the hierarchy method produced the most efficient data structure. Lee noted some distortion occurring in the TINs created by the hierarchy method due to its sampling approach. All methods resulted in better surface approximations when there was a greater number of points in the final TIN.
As a TIN is formed by a number of carefully selected coordinates, it is more economical in its representation than the large number of fine Rectangular Grids needed to accurately represent the same varying topography. This is due to the uniform sampling structure of the grids.
In a TIN the faces of the triangles give an approximate portrayal of the terrain fea tures by capturing the slope of the terrain. The triangulation maintains the relational accuracy of the sampled points explicitly within the computational storage of the struc ture. Proximity information such as neighbouring triangles and vertices are recorded, allowing straightforward retrieval of this information for algorithmic queries. TINs there fore require greater storage requirements than the Rectangular Grids, whose proximity information is implicitly represented by their structure.
Jones et al.
[94]
introduce a storage scheme termed the Implicit TIN which is compu tationally more efficient than the traditional TIN storage structure, in that it stores only the vertices of the triangles and any linear constraints such as roads, ridge lines, fences or rivers. The authors apply the Implicit TIN to the application of a multiscale database where TINs of varying levels of detail are retrieved and reconstructed for specific queries. A spatially indexed, hierarchical quad tree structure is used to store the vertices and linear features. While the Implicit TIN structure is efficient in storage terms, there remains a significant computational effort to retrieve and reconstruct the TINs in response to each query. Hence, a trade-off between storage costs and retrieval time exists.While computationally more complex to handle than Rectangular Grids, TINs have a number of advantages over such a matrix structure. One advantage is that the TIN model contains no vertical discontinuities between neighbouring triangles, and paths between edges and
/
or faces in the TIN can therefore be found without another level of approximation. TINs are also advantageous in that triangles, by their very nature, are more effective at representing non-horizontal planes occurring naturally in the terrain, than are fiat grids. The TIN is able to model non-convex surfaces more realistically as "most interesting terrain is not convex, . . . since it will usually have many mountains and valleys" ,[119,
page180].
DTMs allow data structures to be associated with each triangle face, recording ter rain attribute information such as bush coverage, and natural or man-made features.
The accuracy of the TIN model can be improved by incorporating additional terrain phenomena. Laurini and Thompson
[102,
page247]
state that:"a more realistic representation will be achieved if the spatial data units recognize natural surface changes in slope, at peaks, pits, passes, ridge lines, saddle points and course lines or discontinuities, rather than just be fitted arbitrarily. "
In particular, the explicit modelling of structural features such as roads, rivers, ridges, tramping tracks and fence lines can be represented. Such features, and changes in terrain morphology and vegetation, can be modelled in the TIN by means of a constrained triangulation. In a constrained triangulation these features are represented as edges of the triangles, with the triangulation being formed around these prescribed edges. Modelling such features by Rectangular Grids is not possible without extending the initial model
[171].
A TIN provides a means of interpolating characteristics of those points not sampled by fitting a suitable linear or polynomial function to each triangle face. Hence discontinuities between triangle faces are easily incorporated. Goodchild and Lee
[76]
show that a TIN is considered a good model for interpolation as there is a uniquely appropriate way of fitting a plane to the triangle face, unlike a Rectangular Grid where uniqueness does not exist. It is also possible to string contours through the triangles based on spot height information and subsequent interpolation within each triangle, as outlined in Gold and Cormack[71]
who achieve this by utilizing a contour tree. The tree is found by taking advantage of the structure of the triangulation and its spatial ordering properties which allow it to be traversed in a tree order.3 The authors[71 ,
page148]
state that "the intrinsic concept of ordering within a triangular network should be of value in many other applications."Telcik