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Terrain Maps Displaying Hill-Shading with Curvature Geomorphology 102 (2008) 567–577 Contents lists available at ScienceDirect Geomorphology journal homepage: www.elsevier.com/locate/geomorph Terrain maps displaying hill-shading with curvature Patrick J. Kennelly ⁎ Department of Earth & Environmental Science, C.W. Post Campus of Long Island University, 720 Northern Blvd., Brookville, NY 11548, USA ARTICLE INFO ABSTRACT Article history: Many types of maps can be created by neighborhood operations on a continuous surface such as provided by Received 8 January 2008 a digital elevation model. These most commonly include first derivatives slope or aspect, and second Received in revised form 30 May 2008 derivatives planimetric or profile curvature. Such variables are often used in geomorphic analyses of terrain. Accepted 31 May 2008 First derivatives also provide subtle enhancements to hill-shaded maps. For example, some maps combine Available online 8 June 2008 oblique and vertical illumination, with the latter reflecting variations in slope. Keywords: This study illustrates how second derivative maps, in conjunction with hill-shading, can cartographically Planimetric curvature enhance topographic detail. A simple conic model indicates that image-tone edges where slope or aspect Profile curvature varies by less than 0.5° are visible on curvature maps. Hill-shaded images combined with curvature enhance Illumination the continuity of naturally occurring tonal edges, especially in strongly illuminated areas. Variations in Shaded relief planimetric and profile curvature seem to be especially effective at highlighting convergent and divergent Cartography drainages and variations in erosion rate between or within sedimentary units, respectively. Shading Geographic visualization curvature with consideration given to illumination models can add detail to hill-shaded terrain maps in a manner similar to cognitive models employed by map viewers. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Contours can be used to estimate slope, aspect, and curvature alone. Elevation can be read directly from contour labels, and the spacing or the Geomorphologists and cartographers both employ variables change in spacing among contours indicates changes in slope and profile derived from elevation data to quantify the shape or structure of curvature, respectively. An orthogonal vector defined by two nearby topographic surfaces (Robinson et al., 1995; Wilson and Gallant, 2000; contours on a map yields aspect, and the curve of the contour itself Slocum et al., 2004; Li et al., 2004). This study focuses on the shows the change in aspect or planimetric curvature. cartographic representation of variables computed from local (neigh- Contours, however, can require added information if the viewer borhood) operations directly from the topographic data, the “primary is to translate them into a cognitive model of a topographic surface. attributes” discussed in Wilson and Gallant (2000). Also addressed are For example, are two concentric, closed contours a depression or a operations that treat the terrain as a continuous function f (x,y,z) and hilltop? Determination would require reading contour values, compute first and second order derivatives. looking for hachures, or viewing the contours in the context of the First order derivatives include slope gradient and slope aspect. overall terrain. Also, having aprioriknowledge of an area's Slope, the rate of change in elevation (the z-value), quantifies the geomorphology may be helpful (e.g. karst vs. fluvially carved steepness of terrain. Aspect is the compass direction of steepest slope. terrain). One visual aid frequently applied in discriminating high Second order derivatives include profile and planimetric curvature, vs. low is layer or hypsometric tinting (Imhof, 1965; Robinson et al., which are measured along and across the direction of maximum slope, 1995), in which background colors change with elevation. Color respectively (Evans, 1972; El-Sheimy et al., 2005; Chang, 2006). schemes are designed to be intuitive, reflecting common variations Geomorphology focuses on the patterns of these attributes and why associated with changes in elevation. For example, tints may they are significant to Earth surface processes (Table 1, column 2). Contour progress from green to brown to white with increasing elevation lines (Table 1, column 3) represent the intersection of topography with a in mountainous areas. series of planes parallel to a datum separated by a constant z-interval, Contours create image tones that mimic maps with vertical while tonal contrasts and elevation derivatives, the focus of cartographic illumination (Imhof, 1965). Viewers have a difficult time distinguishing research, help visualize the terrain (Table 1,column4). shapes under such lighting. The Tobacco Root mountains of Montana are shown in Fig. 1 with contours (a) and “slope shading” from vertical illumination (b). Although some hand- and computer-contouring techniques overcome this visualization problem (e.g. Tanaka, 1932, ⁎ Tel.: +1 516 299 2652; fax: +1 516 299 3945. 1950; Peucker et al., 1975; Yoeli, 1983; Kennelly and Kimerling, 2001; E-mail address: [email protected]. Kennelly, 2002), such techniques are not widespread. 0169-555X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2008.05.046 568 P.J. Kennelly / Geomorphology 102 (2008) 567–577 Table 1 left) as the reason for this psychological effect. Many map viewers see fi Basic attributes of topography, their geomorphic signi cance, and their cartographic inversion of terrain when illuminated from the opposite direction. representation Properly illuminated, such hill-shaded or shaded relief maps are popular for their realistic look and detail (e.g. Thelin and Pike,1991). Hill-shading is calculated as: BV ¼ 255 cos θ ð1Þ where BV is the 8-bit (0 to 255) grayscale brightness of each hill-shaded surface element (0 is black and 255 is white), and θ is the angle between a constant illumination vector and a normal vector to the terrain at each given locale. As such, hill-shading is a function of both slope and aspect, and its formula can also be written as: BV ¼ 255 cos I sin S cosðÞA−D þ sin I cos S ð2Þ where I and D are the inclination from horizontal and azimuthal Excerpted and elaborated from Table 1.1, p. 7 of Wilson and Gallant (2000). declination of the illumination vector, and S and A are the slope and aspect values. Although hill-shaded maps are rich in visual information, and useful Map users find that oblique illumination, a light source shining from for solving specific geomorphological problems (Oguchi et al., 2003; Smith a moderate angle between the horizon and zenith, and from the and Clark, 2005; Van Den Eeckhaut et al., 2005), default computer northwest, provides more intuitive images of the shape of the terrain applications may limit cartographic freedom to enhance particular (Imhof, 1965; Horn, 1981; Robinson et al., 1995; McCullagh, 1998; features of importance (Thelin and Pike, 1991; Allan, 1992; Patterson, Slocum et al., 2004). Imhof (1965) cites westerners' preference for 2004). Some cartographers such as those with the Swiss Institute of general room lighting when writing and drawing (from above and the Cartography, elevated manual hill-shading to an art form, including Fig. 1. Four maps of the Tobacco Root mountains of southwestern Montana: a) unlabeled contours, b) slope shading following “the steeper the darker”, c) conventional hill-shading with oblique northwestern illumination, and d) combined hill-shading from the northwest and vertical illumination. P.J. Kennelly / Geomorphology 102 (2008) 567–577 569 combining hill-shading with slope shading to highlight steep terrain I show how these second derivative maps can be used to highlight (Imhof, 1965). Computer-assisted cartographers combine the two in a physical features of real world data on hill-shaded maps. similar manner to provide additional information about the shape of the terrain in a visually harmonious manner (Patterson and Herman, 2004). I 2. Methods and models include in Fig.1 examples of the Tobacco Root Mountains with hill-shading (c), and hill-shading combined with slope shading (d). The latter adds To demonstrate that second derivative maps enhance patterns not some visual emphasis to ridge lines and valleys, especially those oriented obvious on hill-shaded maps or first derivative maps combined with parallel to the direction of illumination (i.e. northwest to southeast). hill-shading, I chose the simple geometric model of a cone. It is created Representing aspect on a map is a challenge because aspect is a from concentric contours with an interval of 50 m and with cyclical variable; north has a value of both 0° and 360°. Cartographic subsequent variations in radius of 100 m. The lowest contour is at a researchers have addressed this problem by devising new aspect color z-value of zero and the apex has a value of 1000 m. schemes, where luminosity or lightness of colors approximates or From these contours, the cone everywhere would slope at 26.57°. complements tonal variations associated with hill-shading. Moeller- To include a slope variation, all contours below 500 m were offset ing and Kimerling (1990) chose easily discriminated colors, and varied vertically by −1 m. The result is a band around the cone between them with aspect such that the color luminance approximates the hill- elevations of 500 m and 449 m of slightly steeper slope (27.02°) than shading brightness. Brewer and Marlow (1993) varied colors' lightness above or below, a difference in slope of 0.45°. The shape of the cone with aspect and chroma with slope. Kennelly and Kimerling's (2004) provides changes at all aspects. The contours are then converted into a aspect color scheme was designed to complement hill-shading. Subtle triangulated irregular network (TIN), a series of adjacent triangular variations in the luminosity of aspect colors highlight linear terrain faces with each approximating the slope and aspect of the terrain in features oriented parallel to the direction of illumination; these that area (Longley et al., 2005). The TIN in this study includes features are difficult to discern in traditional hill-shaded maps.
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