PeakDetection giCentre, City University London, unpublished research paper 1–15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 . 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 con- tour 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 lo- cal 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 rel- ative 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 iden- tify 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 equiva- lent 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 selec- tion criteria used by parts of the mountaineering and 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 surround- ings. In this way, rd can be used to control the scale at which summits may be extracted. Unlike morphometric measures that allow some form of scale control over extraction, it is only vertical scale that is controlled in this instance. This allows summits of both sharp and rounded peaks to be identified by the same process (see Figure 1).

A

relative drop bounding isoline B threshold rd C

Figure 1. Relative drop identification of summits. Of the three candidate locations A, B, and C, only locations A and C are surrounded on all sides by a relative drop of at least rd. The shaded areas indicate the contributing area of the identified summits. Local curvature has no influence over summit classification, but does influence the spatial extent of the contributing area (peak C therefore has a larger surface area than peak A). PeakDetection

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1 2 3 4 5 row col priority 1 2 3 4 5 row col priority 1 6 8 10 13 15 3 2 12 1 6 8 10 13 15 4 2 11 2 4 9 10 11 13 priority queue 2 4 9 10 11 13 2 3 10 3 9 12 10 10 12 3 9 12 10 10 12 3 3 10 4 9 11 9 9 11 4 9 11 9 9 11 2 2 9 5 8 9 7 8 10 5 8 9 7 8 10 4 3 9 raster DEM 4 1 9 3 1 9 maxDrop = 12-12 = 0 maxDrop = 12-11 = 1 2 1 4 (a) (b)

1 2 3 4 5 row col priority 1 2 3 4 5 row col priority 1 6 8 10 13 15 2 3 10 1 6 8 10 13 15 1 4 13 2 4 9 10 11 13 3 3 10 2 4 9 10 11 13 2 4 11 3 9 12 10 10 12 2 2 9 3 9 12 10 10 12 3 3 10 4 9 11 9 9 11 4 3 9 4 9 11 9 9 11 1 3 10 5 8 9 7 8 10 4 1 9 5 8 9 7 8 10 3 4 10 3 1 9 2 2 9 5 2 9 4 3 9 5 1 8 4 1 9 5 3 7 3 1 9 2 1 4 5 2 9 5 1 8 1 2 8 maxDrop = 12-10 = 2 maxDrop = 12-10 = 2 5 3 7 (c) (d) summit at (3,2) stored 2 1 4

Figure 2. Progress of relative drop algorithm. To calculate the relative drop of the cell at row 3, column 2, it is first added to the neighbour priority queue (a). Any of its eight neighbours that have not already been placed in the queue are then added to the priority queue (b). This process is repeated for the cell at the top of the queue (c), until either the cell at the top of the queue is at the edge, or it has a higher elevation than the start elevation (d), or the queue is empty. The relative drop will be the maximum difference between the start height and any cell that reached the top of the queue.

Analogous to the detection of pits in a surface (Jones 2002, Wang and Liu 2006), the following raster-based summit detection algorithm works by calculating the relative drop for each cell in a raster DEM (see Figure 2). By searching for the highest of the neighbours surrounding any given cell, the process guarantees that for any non-zero relative drop, a closed isoline can always be placed around the cell. The majority of cells in a typical surface will have no relative drop since they have a higher immediate neighbour. The calculation of any cell’s relative drop involves visiting neighbouring cells at most once, so the process has an O(n) time complexity. To measure the relative drop of all raster cells, the worst case complexity would therefore be O(n2), but for non-horizontal surfaces, only a tiny proportion of cells are surrounded by lower cells so in practice the algorithm is much closer to an O(n) process. Algorithm 1, describes how relative drop measurement can be implemented in a procedural language. Algorithm 1: int visited[numRows][numCols] // keeps track of processed cells int visitID←0 // uniquely identifies each potential summit’s neighbours

Function detectSummits(float dem[][], float dropThreshold) { for each cell in dem { drop←findDrop(row,col,dropThreshold) if (drop >= dropThreshold) store(row,col) // summit found at location (row,col) } }

Function findDrop(int row, int col, float dropThreshold) { float startHeight←dem[row][col] // elevation of candidate location PriorityTree neighbours // list of summit neighbours in order of their elevation float maxDrop←0 // maximum surrounding drop float drop // drop from start height to neighbouring cell int row1,col1 // location of neighbouring cell

visited[row][col]←(++visitID) // keep track of processed cells neighbours.push({row,col},startHeight) // insert location using its elevation as priority value PeakDetection

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while (neighbours not empty) { {row1,col1}←neighbours.pop() // find ( highest ) location at top of priority queue drop←startHeight - dem[row1][col1] // calculate drop to neighbour

if (drop < 0) return maxDrop // stop search if highest neighbour is higher than start point

if (drop > maxDrop) maxDrop←drop // update maximum drop

// stop search if required relative drop has been reached if (maxDrop >= dropThreshold) return maxDrop // stop if search has reached the edge of the DEM if ((row1,col1) at edge of dem) return maxDrop;

for each adjacent neighbour of (row1,col1) { if (visited[row2][col2] != visitID) { // store unprocessed neighbours in priority queue visited[row2][col2]←visitID; neighbours.push({row2,col2},dem[row2][col2]) } } } return maxDrop; }

Algorithm 1 can produce multiple summit locations within the same contributing area for flat-topped peaks or elevation models with a low vertical resolution. Sets of such sum- mit locations may be contiguous or separated by an intervening dip (see Figure 3). Given that a set of summits in the same contributing area will be the same height, some other criterion must be used to select a single location from the set. A number of possibilities exist such as using a measure of local morphometry to identify the candidate located on the most ‘peak-like’ shape, or selecting some centroid based on the set of candidates. The former approach is probably best suited to the isolated candidates, while the latter is best suited to contiguous candidates. Here we implement a simple and robust procedure where the distance from each summit location to the peak’s centroid (shaded grey in Figure 3) is calculated. The summit closest to the centroid is selected as the single summit point associated with the peak.

AB

relative drop bounding isoline threshold rd C

Figure 3. Peaks with multiple summit locations. Locations A and B are both contained in the same contributing area as they are located at the same height without a sufficiently large drop separating them. Area C contains contiguous locations that are part of a common contributing area. To represent all summits as single point locations the most representative point from the set [A,B] as well as from the set of locations in [C] must be chosen.

3.2. Peak Detection A peak has some areal expression and will be associated with one or more summits, but unlike a summit, a peak’s spatial extent is not well defined and may be delineated in a number of ways. A simple approach is to define it with respect to a fixed relative drop. As illustrated in Figure 1, this allows us to define the peak by the contributing area of its summit. Peaks thus defined will have crisp boundaries delineated by the isoline rd vertical units below their summit elevation. Figure 4 shows peaks defined this way using a relative drop of 100m. This method can be efficiently implemented in a raster system by modifying Algorithm 1 to keep track of all visited cells (stored in the array Visited[][]) that are within PeakDetection

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Figure 4. Simple peak delineation using the contributing area of relative drop. Dark grey areas are vertically within 100m of the summits (symbolised with dots) with which they are associated. The number next to each summit is its relative drop measured in metres. The DEM is of part of the English Lake District, with a planimetric resolution of 50m (DEM is Crown Copyright Ordnance Survey 2008. An EDINA Digimap/JISC supplied service). rd units of the summit. This generates a set of non-overlapping discrete peak areas, each surrounding a single summit. Given the only parameter that controls the selection of summits and their associated peak is the relative drop rd, the identification process can be modified by measuring the relative drop of all locations within a region. All locations that have a relative drop greater than zero can be regarded as a summit to some extent and their contributing area, an ex- pression of the peak extent. The boundary of the contributing area will also intersect with the pass that separates a summit from a neighbouring higher summit. This is an important step in building a topological surface network (Pfaltz 1976, Wolf 1984, Takahashi 2005). The relative importance of summits on a surface can be quantified by ordering summits by their relative drop. Summits can be arranged as an ordered tree since each summit’s contributing area may itself contain lower peaks each associated with their own summit (see Figure 5). Nodes in the tree represent summits ordered by their relative drop. The peak level of the hierarchy represents the number of peaks upon which any given summit may be located. There is a similarity to ‘mountain order’ defined by Yamada (1999) who used manual processing of contour lines to quantify the nesting of peaks in an analogy with the Strahler ordering (Strahler 1964) of stream links in a drainage network. Following from Yamada (1999), peaks B, D and E in Figure 5 would be labelled ‘first order’, peak C ‘second order’ and peak A ‘third order’. This is not necessarily the most useful way of ordering a connected summit tree since, unlike stream networks, the nature of low order features has relatively little effect on the nature of higher order features. Furthermore, the presence or absence of low order peaks is highly dependent on the resolution and quality of the elevation model as well as the threshold value rd. This in turn would arbitrarily affect the mountain order PeakDetection

J. Wood, 2008 7 of all parent peaks in a terrain. Instead, a top-down peak level is a more useful measure, where the highest peak is labelled level 0, its immediate sub-peaks level 1 etc. Thus the most prominent features in a landscape retain the same peak level largely independent of the quality of the DEM or value of rd. Note that this is not simply a reverse of Yamada’s ordering, as illustrated in Figure 5 where peaks B, C and D would all be given the same peak level but not the same mountain order.

A Level 0 A3 C 1 B1 C2 D1 E 2 E1

B D

Figure 5. Ordered summit tree. Summits A-E are ordered according to their relative drop. Each summit’s contributing area is shaded according to its peak level with lighter shades indicating higher level nested contributing areas. The Yamada mountain order is given in italics for comparison.

A summit tree can be extracted from a raster DEM as shown in Algorithm 2, which exploits the fact that a child node in the tree can never have a relative drop greater than its parent. Algorithm 2: Function buildSummitTree(float dem[][], float dropThreshold) { int peaks[numRows][numCols] // raster of peakIDs / hierarchy level PriorityTree summits // list of summits in order of their relative drop.

for each cell in dem { drop←findDrop(row,col, dem.max-dem.min) // drop threshold set to maximum possible drop.

// summit found at location (row,col) if (drop >= dropThreshold) summits.push({visitID,row,col,parentID},drop) } // Link each summit with its parent by processing them in relative drop order . while (summits not empty) { summit←summits.pop() if (peaks raster at summit location != 0), set summit parent to peaks raster ID. label contributing area in peaks raster with summit ID } }

An implemented version of this algorithm is used in the software LandSerf 1 and was used to generate the raster figures in this paper. Figure 6 shows an example of the process applied to a 50m DEM of the English Lake District and Figure 7, the underlying network structure of a selection of the summits. The contributing areas of each summit with a relative drop of 50m or greater is shaded in brown. Where a sub-peak is nested within a peak’s contributing area, it is symbolised with a lighter shade of brown. Note that there is evidence of an ‘edge effect’ here where the DEM is not sufficiently wide to surround a summit’s contributing area entirely (Helvellyn and Pike in Figure 6 ), its bounding isoline will be artificially high (at the elevation where the peak intersects the edge of the DEM). This results in smaller discrete contributing areas than might be expected and a series of fragmented trees that would otherwise be part of a single parent associated with the highest summit. However, an advantage of this form of raster processing over contour methods is that the networks can still be calculated from any spatial subset of a surface.

1freely available from www.landserf.org PeakDetection

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Figure 6. Spatial arrangement of peak hierarchy with summits of rd = 50m or greater. Peak Level is indicated by shade of grey. The DEM is of part of the English Lake District, with a planimetric resolution of 50m (DEM is Crown Copyright Ordnance Survey, 2008. An EDINA Digimap/JISC supplied service).

High Pike Knott

Bowscale Fell

Skiddow Little Man

Great Mell Fell Hopegill Head Lorton Lord's Seat Stybarrow Dodd Sheffield Pike High Rigg Whinlatter Beda Head Angletarn Pikes Raise Above Outerside Ullswater High Spy Clough Head Above Crookdale Crag Barrow Hindscarth Catbells High Street High Wether Howe Robinson Stony Pike King's How White Howe Crag Hill Causey Pike Dale Head Ard Crags Helvellyn North Brund Fell High House Bank Tarn Crag Gavel Fell Rannerdale Knotts Ullscarf Mellbreak South Whatshaw Common Helm Crag Steel Fell Red Screes Blake Fell Allen Crags High Raise Green Quarter Starling Dodd High Seat Sallows Great End Baystones Haystacks Above Thirlmere Harrison Stickle St Sunday Crag Kirk Fell Esk Pike Scafell Iron Crag Lingmell Bowfell Broad Crag Fairfield

Haycock Scafell Pike Seat Sandal Glaramara Above Brocklebank Hart Crag Pike O'Blisco Harter Fell Eskdale Crinkle Crags Brown Haw Great How Stickle Pike Green Crag Illgill Head Ling Fell Hard Knott Whin Rigg Knock Murton Eskdale Moor

Wetherlam Silver How

Grey Friar Coniston Old Man Caw Dow Crag

Figure 7. Summit hierarchy network diagram. Summits with rd > 50m for the three main peak systems in the Lake District. Summits of level 0 (those that have no parent peaks) are shown with a dark grey background; those without sub-peaks at the 50m rd level are shown as unbordered text. PeakDetection

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3.3. Fuzzy Peak Detection Relative drop methods identify summits at a given vertical scale along with the set of peaks associated each of them. However in identifying peaks in this way there is a mis- match between the crispness of the implied boundaries between ‘peak’ and ‘not-peak’, and the vagueness of a peak or mountain (e.g. Fisher et al. 2004, Smith and Mark 2003). We may be sure that a summit is definitely part of a peak or set of peaks, but less sure about locations that are further away from that summit at a lower elevation. Therefore, a more meaningful definition of a peak is proposed that uses relative drop to define a fuzzy peak membership value for any location on a surface. Peak membership is constrained to have a value of 1 at its summit location, and a value of 0 at the boundary of its contributing area (its bounding isoline). As an initial step, membership between these two bounds can be assigned as follows:

rds − (zs − zxy) µpeak(x, y) = (1) rdmax

where µpeak(x, y) is the peak membership value at any location (x, y), rds is the relative drop of the peak’s summit constrained between 0 and rdmax; zs is the elevation of the peak’s summit; zxy is the elevation at location (x, y) and rdmax is the maximum relative drop used to set the scale of analysis. In other words, peak membership declines linearly with elevation below the summit. There is an ambiguity in this definition because any given location (x, y) may be asso- ciated with more than one summit s (as shown by the nested peaks in Figure 6). Therefore we need to be precise about which values rds and zs are used in assigning membership. Because a sub-peak can never have a relative drop greater than its parent, taking the max- imum peak membership value for all possible summits associated with a given location will ensure that the final peak membership is the one associated with the highest peak of which it forms a part:

n rds − (zs − zxy) µpeak(x, y) = max (2) s=1 rdmax

where n is the number of summit contributing areas that contain the location (x, y). By changing the value of rdmax, membership of different scales of peak can be found. Setting rdmax to the full range in relief for the entire study area gives a scaled map of elevation and is of comparatively little use. Figure 8 illustrates a more useful scale of analysis where the maximum relative drop permitted is about 30% of the global elevation range. By setting different values of rdmax , processing can be filtered to the desired scale of analysis (see Figure 9).

A (100) C (35) 1 100 E (5)

Peak membership 0 B (45) D (15)

50 Vertical scale Vertical

0

Figure 8. Fuzzy peak identification using relative drop of 30 vertical units. Summits A, B and C all have a peak member- ship of 1. Location D has a peak membership of 0.5 (15/30) and location E has a membership of 0.6 (it is a sub-peak of its neighbour with summit C that is 12 units higher). PeakDetection

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Figure 9. Fuzzy peak membership using the contributing areas up to a maximum relative drop (50m). The darker the red, the higher the peak membership value. The right-hand image shows an enlarged portion of the central Lake Distrct, with the field of view of the photo of Ullscarf indicated between A and B. Named peaks are those that contain summits with memberships of 1.The DEM used has a planimetric resolution of 50m and vertical resolution of 1m (DEM is Crown Copyright Ordnance Survey, 2008. An EDINA Digimap/JISC supplied service).

4. Validation of Peak Detection

The validity and utility of the relative drop approach to peak and summit detection was assessed by comparing those features identified using the methods described above with two datasets of known peaks in the Lake District region of the UK. The first was a list of ‘Hewitts’ that are themselves defined by their relative drop; the second, the ‘Wainwrights’, were selected and described by in his Pictorial Guide to the Lakeland between 1955 and 1966. The Hewitts were used to assess the impact of accuracy on the feature identification, Wainwrights to assess the match between the automated approach and a subjective selection of ‘significant’ peaks.

4.1. Lake District Hewitts A Hewitt, representing the acronym “Hill in England, Wales or Ireland over Two Thou- sand feet” (Dawson 1997), is defined as any peak with a summit elevation of at least 610m and a relative drop of at least 30m. A list of Hewitts within the Lake District region along with their summit elevation (1m vertical precision), relative drop (1m vertical precision) and location (100m horizontal precision) is provided by Dawson (1997) who manually identified spot heights from Ordnance Survey 1:10000, 1:25000 and 1:50000 scale topo- graphic maps. Where data from these map sources conflicted, the larger scale maps were used in preference. Relative drops were calculated by comparing spot heights with either manually interpolated contours or spot heights identified at passes (Dawson 1997). By adapting Algorithm 1 to include the additional constraint of selecting only summits above a given threshold, it is possible to provide an automated selection of Hewitts from a DEM. Any differences between those summits identified automatically and those listed in the Hewitt tables (Dawson 1997) should be due to (a) error in the elevation model; PeakDetection

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(b) error in the paper map data used by Dawson; (c) error in the selection process. For the purposes of this paper, the 9 tiled Ordnance Survey 50m ‘Panorama’ DEMs and 144 tiled 10m ‘Profile’ DEMs were processed. Elevation accuracy figures are not provided on a tile-by-tile basis, but the general metadata provided with the DEMs claims a vertical accuracy “better than half the contour interval”, which equates to a maximum vertical error of about 5m. Evaluation of elevation accuracy of both data sources by comparison with photogrammetrically derived spot heights suggests a much lower RMSE but with considerable spatial variability. Since the manually detected summits were generally de- rived from larger scale mapping, it is assumed that differences between the automated and manual identification will be largely due to DEM error.

Table 1. Comparison of automated and manual identification of Hewitts using DEMs of 50m and 10m planimetric resolution. False positives are summits identified by the automated process that are not listed by Dawson (1997). False negatives indicate Hewitts not identified by automated detection.

Hewitts Automated detection 50m DEM Automated detection 10m (Dawson, 1997) min height: 610m, rd: 30m min height: 610m, rd: 30m Number of Number of False False Accuracy Number of False False Accuracy summits summits positives negatives summits positives negatives 114 102 1 13 88% 111 3 3 95%

Processing the 50m DEM yielded a classification accuracy of 88% with the majority of mis-classifications due to a failure to detect summits with relative drops close to the 30m threshold defining Hewitts (see Table 1). This is likely to be due to lost vertical ac- curacy from interpolation to a 50m grid. Interpolation of contours to produce the DEM will have rounded peaks and filled passes resulting in a consistent underestimation of the numbers of summits in the region. This effect was examined by using an elevation model with higher accuracy that preserved more of the full range of relief. An improved accuracy of 95% was achieved by processing the 10m DEM of the region. The 3 ‘false positives’ identified using this DEM (see Table 1), were High Spying How (Helvellyn), Nethermost Pike (Helvellyn), and Great Carrs (Consiton) with relative drops of 32.6m, 30.6m and 30.2m respectively. High Spying How is a local maximum on the well known areteˆ Striding Edge and is sufficiently narrow for paper map interpretation from contours to be unreliable. The other two mis-identified peaks were within the margin of declared accuracy of the elevation model and paper map interpretation. The 3 ‘false negatives’ not identified by the automated process are Rough Crag (High Street), Green Side (Helvel- lyn) and (Skiddaw) with measured relative drops of 27.9m, 29.8m and 27.2m respectively. These are likely to be due to minor interpolation errors.

4.2. Wainwrights Analysis of the Hewitt detection provides insight into the sensitivity of peak detection to elevation accuracy, but it says little about how well relative drop acts as a measure of user-perceived significance of mountain features. To provide that context, automated detection was compared with named mountain features in Wainwright’s Pictorial Guides to the Lakeland Fells written between 1955-1966. Unlike Hewitts, Munros and other more quantitative selections of mountain features, Wainwritght’s selection of peaks represents a subjective sample influenced by aesthetics, access, prominence and experience. This selection does not necessarily represent a more universal view of mountain significance, although the success of the series, selling over 2 million copies, will have had an influence on many hill walkers’ perception of that mountain environment. Locations of all 214 published Wainwright summits were taken from GPS measure- ments originally supplied by Newby (2008). These were recorded with a horizontal preci- sion varying between 1 and 100m and a variable horizontal accuracy depending on GPS PeakDetection

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Table 2. Comparison of automatic Wainwright identification using DEMs of 50m and 10m planimetric resolution. False positives are summits identified by the automated process that are not listed by Wainwright. False negatives are listed by Wainwright but not identified by automated detection.

Wainwrights Automated detection 50m DEM Automated detection 10m min height: 350m, rd: 34m min height: 350m, rd: 29m Number of Number of False False Accuracy Number of False False Accuracy summits summits positives negatives summits positives negatives 214 174 27 67 61% 210 47 51 62% reception. GPS were not used to derive height information directly. All locations were checked and corrected if necessary to a horizontal accuracy of 10m by comparison with 1:25000 Ordnance Survey topographic mapping. These summits were then compared to those produced using Algorithm 1, varying both the minimum height at which a summit can be identified, and the threshold rd. The accuracy of automated summit detection was calculated as:

#W accuracy = rd,z % (3) #W + #Srd,z − #Wrd,z

where #W is the number of Wainwright summits (214), #Wrd,z is the number of Wainwright summits that meet the given rd threshold and minimum height criteria and #Srd,z is the total number of summits identified by Algorithm 1 that meet the same cri- teria. Figure 10 shows the accuracy of identification as a function of the two parameters when using the 50m DEM of the study area.

240 10% 220

200

180

160

140 20%

120 30% 100

Relative drop (m) 80 40% 60 50% 40 60% 20

0 0 100 200 300 400 500 600 700 800 900 Minimum height (m)

Figure 10. Accuracy of Wainwright identification as a function of minimum height and relative drop threshold rd. The parameters with the highest accuracy for the 50m DEM are minimum height of 350m, and relative drop of 34m (61% accuracy). A similar trend was observed with the 10m DEM, showing a maximum classification accuracy of 62%.

This allowed the optimal minimum height value for identified summits to be set at 350m. Holding this value constant allows the impact of false positive and false negative summit identification to be examined (see Figure 11). A relative drop rd of 34m was found to produce the optimum balance between identification of false positives and false negatives, yielding an overall accuracy of 61%. While classification accuracy is not particularly high, analysis of false positives and false negatives provide insight into the influences of perceived mountain feature promi- nence. Unlike analysis of the Hewitts, increasing accuracy from the 50m DEM to 10m, no significant increase in classification accuracy was achieved (see Figure 11 and Table 2). 240 10% 220 200 180 160 140 20% 120 100 30%

Relative Drop 80 40% 60 40 50% 60% 20 0 0 100 200 300 400 500 600 700 800 900 PeakDetection

13

1000 70% Misclassification of Wainw right Summits Accuracy of Wainw right Summit Identification 900 Using Relative Drop from 50m DEM 60% Minimum Elevation 350m 800 Minimum Elevation: 350m. 700 50% 50m DEM False 600 10m DEM positives 40% 500 400 30% Accuracy Number of 300 20% misclassifications 200 False negatives 10% 100 0 0% 0 100 200 300 400 0 200 400 600 800 Relative Drop Relative Drop

Figure 11. Wainwright identification accuracy as a function of relative drop. Minimum height of peaks was set at 350m.

This suggests that characteristics other than relative drop have an effect on Wainwright’s assessment of peak significance. Three classes of peak mis-identification where recog- nised. Firstly, the algorithm fails to identify highly asymmetric peaks with a large relative dropSummits on one side,summits but onlypositives a small negatives drop to higher land on the other (e.g. sharp cliffs on the214 edge of a plateau).174 Secondly,27 67 a number61% of peaks210 with relatively47 high51 rd values62% were not included in the Wainwright guides due to their inaccessibility by road or foot, or be- cause they fell outside of the region Wainwright’s included in his guides. Thirdly, some peaks were not separately identified in Wainwright’s guides because they form sub-peaks of more well known larger peaks in close proximity.

5. Conclusions and Further Work

This paper has presented a set of new methods for identifying and characterising peak- like features on a surface. By processing raster DEMs directly, a relatively fast and robust set of routines has been developed that does not depend on contour representations of surface form. By using a vertical measure of relative drop, both sharp and rounded peaks can be identified at scales determined by a single threshold value rd. This has potential for generating topological models of surface form, and could be the basis for improved surface generalisation and partitioning methods. The use of relative drop by hill walkers to identify summits worth ‘bagging’ suggests there may be greater accord with general perceptions of mountain features than morpho- metric methods. This has been explored by examining the relative drops of peaks identi- fied in hill walking guides in the Lake District. There is scope for future work to conduct empirical examination of the match between human-preceived mountain features and var- ious methods for automated identification of peaks and summits. The association of a summit with its contributing area provides a basis for areal peak delineation, both of crisp boundaries and fuzzy regions. This in turn allows the nested structure of peaks to be identified and characterised. Further work is required to determine the most discriminating measures of summit network structure that are able to usefully characterise hill and mountain assemblages. A potentially significant weakness in defining contributing area is that the bounding line surrounding a summit is constrained to be horizontal. While this aids computational efficiency, there is evidence from considering named peaks with low relative drops that a more appropriate peak boundary may well vary in elevation. Future work will consider how the peak identification methods described here can be adapted to allow non-horizontal peak boundaries. This in turn may lead to more robust methods for identifying full surface networks (the topological connection between summits, passes and pits via ridge and valley networks). It may also allow more human- centred measures of peakedness to be constructed where views from particular directions preferentially affect the degree of fuzzy peak membership. PeakDetection

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