A fuzzy c-means classification of elevation derivatives to extract the morphometric classification of landforms in Snowdonia, Wales
Introduction
The characterisation and classification of land surfaces to improve our understanding of the processes that have acted, or are acting, upon them, has long been a goal of geomorphological research. Both traditional geomorphometric measures (Evans, 1972) and statistical measures employing spectral (Pike and Rozema, 1975), geostatistical (Mulla, 1988), entropy-based (Culling, 1988) and fractal methods (Chase, 1992) have been employed in this context. Although there is evidence that statistical measures such as fractal dimension describe different components of the land surface more effectively than traditional geomorphometric measures (Klinkenberg, 1992), arguably, the use of geomorphologically significant measures is more clearly linked to process. A range of geomorphometric measures can be extracted from a surface, the usefulness of each of these measures being dependent on the type of surface and the specific objectives of the study. However, the first and second derivatives of elevation (slope or gradient, aspect, and plan and profile curvature) are the most commonly used (Evans, 1972; Fisher et al., 2004; Wood, 1996a). Slope determines the effectiveness of gravity for geomorphological work and curvature controls the acceleration and convergence of flow processes. It has been suggested that higher order derivatives do not provide useful information (Evans, 1990; Skidmore, 1989; Wood, 1996a). Slope and curvature are easily extracted from a DEM within a geographical information system (GIS). Typical raster-based GIS facilitate the manipulation of DEMs, and have provided valuable environments for the automated analysis of surface form, in contrast to early manual approaches by Fenneman (1946) and Hammond (1964). Geomorphic or morphometric classes such as ridges, peaks and passes have been widely extracted from DEMs (Wood, 1996a). As with much other environmental information (Burrough, 1989; Moraczewski, 1993; Luo and Dimitrakopoulos, 2003) the clear definition of morphometric classes has been questioned recently, and morphometric classes have been reconceptualised as vague or fuzzy sets (Fisher et al., 2004, Fisher et al., 2005; Irvin et al., 1997; Lagacherie et al., 1997). Fuzzy set theory was introduced by Zadeh (1965). Fuzzy set theory allows soft boundaries and vague landform classes to be explicitly represented, providing more information on landscape structure than discrete and often spatially disjointed Boolean landform classes.
The two principal approaches to fuzzy set recognition of landforms are by deductive or inductive classification (using predetermined and self-selecting landform classes, respectively). The first deductive approach examines the DEM with a template of the arrangement of relative elevations within a fixed area (window) and allocates the centre point of that area to one of a limited set of morphometric classes. The number of classes varies depending on the implementation, from the minimal six classes recognised by Evans (1980), Peucker and Douglas (1974) and Wood (1996a) to the 11 classes included in the work of Pellegrini (1995). This deterministic approach leaves no doubt over the allocation of the pixel to a class, but Wood, 1996a, Wood, 1996b, Wood, 2002 has shown that many parameters, including morphometric class, are subject to variation with the resolution of the DEM on which analysis is executed. Classification is therefore consistent with the conception of the land surface as a multiscale composite of scale-dependent and scale-independent elements (Tate and Wood, 2001) where specific morphometric classes that are identifiable as discrete objects are comparatively rare (Fisher and Wood, 1998). Fisher et al., 2004, Fisher et al., 2005 exploited this observation by using this uncertainty as a method for populating fuzzy memberships of morphometric classes. The simplest inductive method is that reported by Roberts et al. (1997), who took a number of different elevation derivatives and formulated a fuzzy set membership from steepest to least steep and most to least concave etc., and then combined them to give a fuzzy classification which they related to risk of soil salinisation. Other researchers using inductive methods have employed unsupervised classification with a variety of morphometric derivatives as input to an automated morphometric classification of landforms (Burrough et al., 2000; de Bruin and Stein, 1998; Irvin et al., 1997; Lagacherie et al., 1997; Macmillan et al., 2000; Ventura and Irvin, 2000). Burrough et al. (2001) have gone on to use the same approach in mapping the topo-climatic structure of forest. All these studies of automated classification have employed the fuzzy c-means classification (Bezdek et al., 1984), and they have also worked only at the resolution of the original DEM, ignoring the uncertainty associated with the change of resolution.
The aim of the research reported here is to fuse the fuzzy identification of morphometric landform classes from DEMs by inductive methods with the recognition that the classes identified at one DEM resolution may be different from those identified at another resolution. This paper describes a fuzzy c-means analysis of a DEM representing an area in Snowdonia, Wales, and builds on the studies mentioned above by examining the change in morphometric classes identified at different resolutions. After a description of the study area and data, we outline the methods of derivative extraction and fuzzy classification, and provide three methods of interpreting the resultant clusters of morphometric classes.
Section snippets
Study area and data collection
An Ordnance Survey 50 m resolution gridded DEM tile of Snowdonia, Wales, provided the initial data for this investigation and was used to extract the morphometric classes present in the study area by examination of the first and second derivatives of elevation. The geographical location of the study was chosen as it contained a varied collection of morphometric classes and a variable topography with a significant elevation range. The study area characteristics are summarised in Table 1.
Derivative extraction
A modification of the Zevenbergen and Thorne (1987) partial quartic surface equation was chosen to calculate the first and second derivatives of elevation (slope and curvature respectively) from the DEM. The nine coefficients A–I in this locally fitted nine-term partial quartic equation (Eq. (1)) can be determined by Lagrange polynomials. The equation has a degree of flexibility as it alters to fit the surface with the relevant coefficients equalling zero:
Results
There are four stages to interpreting the geomorphological significance of the clusters generated from the analysis. Initially their average (centre) attributes are examined to gain a geomorphometric understanding of each cluster and how it fills the attribute space in which it lies. The clusters can then be visualised in the context of the landscape draping them over the DEM surface. The third stage is the examination of the spread of membership values through the surface, which provides
Discussion and conclusions
This work has confirmed the observations of Burrough et al. (2000) and Irvin et al. (1997) that performing a fuzzy c-means classification on a landscape is both possible and sensible, allowing the extraction of geomorphologically significant classes. The clusters created provide a classification with high information content, allowing fine-scale and subtle landform delineations to become apparent. Examination of the fuzzy memberships provided information on the distribution of morphometric
Acknowledgements
We would like to thank Kate Moore for her help with Java programming and Prof Bezdek for sharing the code of the fuzzy c-means classifier. The comments of Ian Evans, Steve Wise and anonymous reviewers have all assisted with improving the quality of the paper, but the contents remain the responsibility of the authors. Data were collected from the Ordnance Survey/EDINA Digimap service. The data are © Crown copyright/database Right 2007 and Digimap is an Ordnance Survey/EDINA supplied service.
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