RESEARCH ARTICLE Identification of Peaks and Summits in Surface Models

PeakDetection giCentre, City University London, unpublished research paper 1–15 RESEARCH ARTICLE Identification of peaks and summits in surface models J. WOOD∗ the giCentre, School of Informatics, City University London, EC1V 0HB, UK (Last modified January 2009) A new set of methods for identifying point-based summits and area-based peaks from a DEM is proposed that avoids morphometric measurement. Relative drop provides a more robust basis for surface topological modelling for DEM generalisation and characterisation. When traversing from any given point on a surface to a higher one, relative drop is defined minimum height lost in completing that traversal. If the traversal minimises height lost, the low point on that traversal represents the pass separating a summit from a neigh- bouring higher summit. Peak extents can be defined by the contour passing through that pass, fully enclosing the summit. An algorithm is presented that identifies such features as well as the nested structure of peaks within peaks. The method is adapted to account for fuzzy peak boundaries by using height from summit to define fuzzy membership. Methods are evaluated by comparing with named mountain features in the En- glish Lake District. Quantitively defined ‘Hewitts’ are used to assess the effect of DEM and interpolation accuracy on the process, demonstrating an improvement from 88% to 95% accuracy when comparing 50m and 10m resolution DEMs. Named mountain features in the ‘Wainwright’ walking guides are used to assess the degree to which relative drop characterises more subjective judgements of mountain significance. These are found to be less sensitive to DEM error and features with a relative drop of at least 34m and an absolute summit height of at least 350m are found to produce the closest match with a 61% accuracy. Keywords: peaks; summits; DEM; surface networks; Lake District. 1. Introduction The detection of local maxima in surface models has an important role in the automated processing and analysis of terrain. Applications include the explicit identification and de- limitation of mountain features in a terrain model for geomorphological analysis (e.g. Yamada 1999, Miliaresis 2006); as part of an ethnophysiographic approach to landscape classification (e.g. Smith and Mark 2003, Fisher et al. 2004); or as part of more general surface processing in the topological restructuring of an elevation model (e.g. Peucker and Douglas 1975, Fowler and Little 1979, Takahashi 2005). Existing methods tend to suffer from several limitations. Firstly, many are concerned with identifying maxima only as point features, so are not appropriate for applications that consider the spatial extent of peak-like features. Secondly, many techniques have relied on manual processing of contour lines and so are not readily amenable to automatic processing in a GIS environment. Thirdly, many automated techniques are highly dependent on the spatial resolution and data quality of the elevation model and are therefore limited in their robustness and scale of analysis. This paper presents a new method for the efficient detection of local summit points and peak areas, and demonstrates how their identification can be used to improve our understanding of a landscape represented by a digital elevation model. Given the vagueness with which the concept of hill or mountain is defined (Fisher and Wood 1998, Smith and Mark 2003, Fisher et al. 2004), it is not surprising that within the terrain processing literature, various terms are used to define and label local surface ∗Email: [email protected] PeakDetection 2 maxima. These include summit and hill (Cayley 1859, Maxwell 1870), peak (Peucker and Douglas 1975, Evans 1979, Wolf 1984); mountain (Yamada 1999); high point (Wentz et al. 2001); and hills and ranges (Chaudhry and Mackaness 2008). For the purposes of this paper, two key concepts are distinguished, termed here as summit and peak. While both terms are associated with landscapes, they can be applied to any continuous surface with the form z = f(x; y). A summit is defined here as a zero-dimensional point of local maximum on a surface. There is no implied scale inherent in this definition and will be considered equally valid for defining local maxima at any height or over any extent. This definition is equivalent to that used by Wentz et al. (2001) of a ‘high point’. Summit is used here as it has a clear association with the notion of a local maximum and is usually considered a point location rather than area on the ground. The point location of a summit can be distinguished from the areal location and extent of a peak which is defined here as a two-dimensional portion of the surface that is proximal to, and associated with, a summit. While a surface measure, a peak area can easily be used to construct a three-dimensional volume when integrated over an elevation model. The rules determining the spatial extent of a peak are deliberately left undefined at this stage as it allows us to accommodate the vagueness inherent in the definition of hills and mountains (Fisher and Wood 1998, Smith and Mark 2001, 2003, Fisher et al. 2004). 2. Existing Methods Automated methods of peak detection tend to produce discrete point sets identifying local maxima. This is due in part to the motivations for detection, which often require only point-based identification, such as the selection of surface critical points (e.g. Takahashi 2005, Bremer et al. 2004), TIN vertices (Fowler and Little 1979), or hydrological and other surface flow measures (e.g. McCormack et al. 1993). It may also be due to the relative simplicity of well defined summit points in comparison to the vagueness associated with peak extents. The few published methods of areal peak identification tend to be used for the mapping of user-perceived delineation of hills and mountains (e.g. Fisher et al. 2004); the partition of surfaces into regions for further analysis (e.g. Cayley 1859, Wood 2000); cartographic generalisation (Chaudhry and Mackaness 2008); or, when used for geomorphological study, extracted manually from contour lines (e.g. Yamada 1999). Identification of both summits and peaks tend to follow two broad approaches. Morpho- metric methods attempt to quantify the shape of local portions of a surface and segment those that are identified as convex-up. Examples include Evans (1979) who classified zero-gradient points from quadratic interpolations of local surface patches; Wood (1996, 1998), Fisher et al. (2004) who used quadtratic interpolation and conic sections to identify convex-up shapes over a range of scales and Miliaresis and Argialas (1999), Miliaresis (2006) who used combined measures of gradient and aspect as part of a region growing segmentation of peak features. Morphometric methods have the advantage of being more easily mapped onto user-perceived models of surface characteristics, but require some in- termediate arbitrary locally continuous model of surface shape. They are also dependent on one or more user-defined thresholds that define how convex-up a patch must be before it is regarded as representing a peak feature. Relative height methods identify peaks and summits largely by comparing heights of different parts of an elevation model while ignoring shape. The earliest approaches did this manually by examining topological arrangements of contour lines (e.g. Cayley 1859, Maxwell 1870). These methods have been extended using a graph theoretic approach (e.g. Pfaltz 1976, Wolf 1984) or by examining nested topological relationships between peaks (Yamada 1999, Chaudhry and Mackaness 2008). Relative height methods that process contour lines have the advantage of making them amenable to topological analysis and PeakDetection J. Wood, 2008 3 visual communication. They have the significant disadvantage of being dependent on the quality of the contour interpolation and generalisation process as well as involving com- putational complexity. Raster-based relative height methods (e.g. Peucker and Douglas 1975, Fowler and Little 1979, Wentz et al. 2001) overcome some of these problems by processing a gridded elevation model directly, but are highly dependent on the resolution of the elevation model used. Relative height methods have been more widely applied in pit detection as a pre- processing stage to hydrological analysis. For example, Jones (2002) uses a height- ordered region growing approach to identify pits and associated least-cost paths to create channels from each pit to some outflow location. Wang and Liu (2006) use an equivalent relative height method to fill pits up to the level of their surrounding ‘spill elevation’ before applying hydrological analysis to a DEM. 3. An Algorithm for efficient user-perceived summit and peak detection A new method of summit and peak detection is proposed here that uses relative height processing of a gridded elevation model while allowing topological peak and summit structures to be built as well as incorporating fuzzy boundaries inherent in peak extents. It was constructed in order to model user-percieved notions of peak features in a landscape, but has more general application as a robust method of region partition around surface maxima. 3.1. Summit Detection The method of summit identification proposed here implements the relative drop selection criteria used by parts of the mountaineering and fell walking community to identify summits worth ‘bagging’ (Stevens 2008): (1) A summit is a point located within a continuous contributing area of a surface that is no higher than the summit elevation itself. (2) A contributing area is being bounded by a closed isoline at a height of at least rd vertical elevation units below the elevation of the summit. The identification of summits will depend on the threshold value rd. Small values of rd result in larger numbers of summits that rise only a little way above their surroundings. Larger values of rd will identify only those summits that dominate their local surroundings.

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