Elementary forms for land surface segmentation: The theoretical basis of terrain analysis and geomorphological mapping

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Geomorphology 95 (2008) 236 – 259 www.elsevier.com/locate/geomorph

Elementary forms for land surface segmentation: The theoretical basis of terrain analysis and geomorphological mapping Jozef Minár a,c,1 , Ian S. Evans b,⁎ a

Department of Physical Geography and Geoecology, Comenius University in Bratislava, Mlynská dolina, 842 15 Bratislava 4, Slovakia b Department of Geography, Durham University, South Road, Durham City DH1 3LE, England, United Kingdom c Department of Physical Geography and Geoecology, University in Ostrava, Chittussiho 10, 71000 Ostrava, Czech Republic Received 7 July 2006; received in revised form 6 June 2007; accepted 7 June 2007 Available online 27 June 2007

Abstract Land surface morphology is fundamental to geomorphological mapping and many GIS applications. Review and comparison of various approaches to segmentation of the land surface reveals common features, and permits development of a broad theoretical basis for segmentation and for characterization of segments and their boundaries. Within the context of defining landform units that maximise internal homogeneity and external differences, this paper introduces the concept of elementary forms (segments, units) defined by constant values of fundamental morphometric properties and limited by discontinuities of the properties. The basic system of form-defining properties represents altitude and its derivatives, constant values of which provide elementary forms with various types of homogeneity. Every geometric type of elementary form can be characterized by a defining function, which is a specific case of the general polynomial fitted function. Various types of boundary discontinuity and their connections and transformations into other types of morphological unit boundaries are analysed. The wealth of types of elementary forms and their boundaries is potentially unbounded and thus is sufficient to cover the real variety of landforms. Elementary forms in the basic set proposed here have clear potential for genetic and dynamic interpretation. A brief worked example documents the possibility of analytical computation of various models of ideal elementary forms for particular segments of landform. Ideal elementary forms can be considered as attractors, to which the affinity of surface segments can be measured by multivariate statistical methods. The use of the concept of elementary forms in landscape segmentation is promising and it could be adapted for elementary segmentation of various other spatial fields. © 2007 Elsevier B.V. All rights reserved. Keywords: Landform segmentation; Derivative; Altitude; Slope; Curvature; DEM

1. Introduction ⁎ Corresponding author. Tel.: +44 191 334 1877; fax: +44 191 334 1801. E-mail addresses: [email protected] (J. Minár), [email protected] (I.S. Evans). 1 Tel.: +421 2 60296518. 0169-555X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2007.06.003

1.1. Background Geomorphological regionalization and mapping remain fundamental research methods of geomorphology and provide many promising applications (see e.g.

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Cooke and Doornkamp, 1990; Evans, 1990; Voženílek, 2000; Lee, 2001). Traditional geomorphological mapping needs to adapt to challenges for greater precision and objectivity within a GIS environment (Gustavsson et al., 2006). However, the theoretical base for definition, delineation and interpretation of mapping units is not satisfactory. Only a few authors discuss the strict definition of landform segments and the minimisation of subjective factors in the segmentation process. An intuitive approach is usual (see e.g. Crofts, 1974; De Graaff et al., 1987; Urbánek, 1997) and a theoretical synthesis is lacking. Yet the concept of a geomorphological information system (Dikau, 1993; Minár et al., 2005) requires strict definition of basic mapping units. Our aim is to provide this. As Crofts (1974, p. 231) claimed, land classification and evaluation should be made on the basis of geomorphological mapping and “should be a prime aim of geomorphological mapping”. Identification of elementary landform units is important not only in the study of past and present geomorphic processes but also in studies of spatial aspects of interaction among landforms, soil, vegetation, topoclimate and hydrological regime (Beruczashvili and Zuczkova, 1997; Giles and Franklin, 1998; Ventura and Irvin, 2000; Blaschke and Strobl, 2003). Land surface segmentation is a kind of regional taxonomy or regionalization that is generally treated as a specific case of the general classification problem (Bezák, 1993). General classification theory is developed mainly in mathematics (e.g. Shelah, 1990), information (library) sciences (e.g. Bowker and Star, 1999; Tennis, 2005) and philosophy (e.g. Hacking, 1999). Regardless of various measures of acceptance or emphasis on the subjectivity of classification (social constructivism), the stability of classificatory structures can be accepted as a criterion of structural quality (cf. Hacking, 1999). This needs to be applied to landform segmentation. However, land surface segmentation is characterized by a fundamental peculiarity arising from the continuity of the classified objects in space, i.e. the land surface forms a continuous field. Regional taxonomy generally solves the spatial aspect of classification but avoids the problem of definition of elementary geographical objects in such a continuous reality (Bezák, 1993). The basic geomorphological goal of land surface segmentation should be to distinguish segments (elements) that are homogeneous genetically and therefore also morphologically. Using a library analogy, we are not classifying existing books but distinguishing individual books within continuous text. The polygenesis of many

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landform segments, due to overlapping of geomorphic agents in time and space, renders the objective existence of such ‘books’ problematic: an extreme social constructivist attitude would deny the objective existence of forms. We recognise that both construction and identification of forms are problematic: this accounts for some of the subjectivity in geomorphological mapping. Nevertheless there is a demand for land surface segmentation, which we address here. 1.2. Approach Landform mapping is based on four principles: the morphologic, the genetic, the chronologic, and the dynamic. The morphologic principle should be primary, in the sense that defensible application of any of the other three principles depends on an accurate appreciation of morphology (Speight, 1974). Thus geomorphological map unit boundaries should generally follow morphological boundaries (Lee, 2001). Identification of the most specific geometrical (and simultaneously genetic) geomorphic individuals is a central objective here. Such individuals might be described as ‘natural’. Two major considerations in making such an approach relatively objective are: identification of natural geomorphic boundaries, with maximal change of genetic, geometric and process character (as attempted by ‘intuitive’ traditional geomorphological mapping); and specification of a clear algorithm for surface segmentation, with a minimum of subjective decisions. The geomorphological interpretability of landform segments in terms of genesis, dynamics and chronology is a major concern if they are to be seen as elementary geomorphic individuals with the character of subsystems. The segments should have internal associations stronger than external. The internal homogeneity and external contrasts of segments in terms of their geometry should reflect their genesis and recent dynamics. Hence the morphometric variables should be those used in geomorphological theory and models. Geomorphological theory defines genetically and geometrically pure geomorphic individuals — landforms (such as alluvial fans, aeolian dunes and glacial cirques) and elements (such as cliffs, floors, slip faces and channels). To identify these in the landscape is a goal of land surface segmentation. We term these theoretical classificatory categories ‘ideal’ models, contrasted with the actual results of classification termed ‘real’ (landforms or elements). Land surface form (the relief of the Earth's surface) is characterized by a complex structure of nested hierarchies (Dikau, 1992).

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Various analytical taxonomies of relief units reflect this structure (e.g. Crofts, 1974; Dikau, 1989; Dymond et al., 1995; Phipps, 2001; Wielemaker et al., 2001) and they have some common features. Three types of relief unit can be distinguished on the basis of increasing complexity. Elementary forms represent the smallest and simplest units, which are indivisible at the resolution considered. Their geometric simplicity (e.g. linear slope, curved slope or horizontal plain) facilitates their recognition as fundamental units in a system for land surface segmentation. Some traditional landforms are single elementary forms, but most are compounded from several elementary forms. Landforms that are composite forms create the second level of relief complexity. Characteristic patterns created by form associations provide a third level of complexity and are termed land systems. They are equivalent to the relief form associations, terrain systems, landform (landscape, terrain) patterns or types of relief of other authors (Fig. 1). Modern geotechnologies (remote sensing, geographical information systems/GIS, global positioning systems/GPS) have brought a huge methodological impetus to geomorphological mapping (e.g. Dikau, 1989; Voženílek, 2000; Phipps, 2001; Blaschke and Strobl, 2003). On the other hand the theoretical basis of mapping has changed little over decades (e.g. Cooke and Doornkamp, 1990; Lee, 2001). Despite the amount of attention paid to the problem of segmentation of land surfaces at detailed scales, the situation remains unsatisfactory. The definition of geomorphic objects below the level of sub-catchment (e.g. hillslopes, valley bottoms and their components) is an unresolved fundamental problem of theoretical geomorphometry and of geomorphological GIS analysis (Schmidt and Dikau, 1999).

1.3. Continuity and discontinuity We can identify three axioms, which create a theoretical base for land surface segmentation: 1) Land surface form can be analysed as a continuum — the geometric field of altitude. 2) At a given scale, the land surface may, nevertheless, exhibit discontinuities; these may be recognised as natural boundaries of geomorphic objects. 3) These discontinuities and other characteristics of the land surface result from morphogenetic processes most of which are influenced by gravity. Exploitation of geological contrasts and lineations often produces discontinuities. The first axiom is basic for many modern geomorphometric analyses (Evans, 1972; Krcho, 1973) and for the universal representation of land surfaces in GIS as DEMs. Altitude (elevation) is evidently a continuous landscape field (Burrough, 1996), yet in many cases the land surface cannot be considered smooth (Shary et al., 2005). Caves, overhangs and boulders are excluded or treated separately. In several subject areas (e.g. genetic geomorphology and geosystem analysis) an object representation is regarded as more appropriate (Minár, 1995; Brown et al., 1998). Cox (1978) concisely termed the first approach the continuous hypothesis (slope profiles are continuous curves without definite breaks) in contrast to the atomistic hypothesis (profiles as sequences of components), where landscapes are essentially a mosaic of discrete units. We suggest that a synthesis of the two is desirable. Segmentation of the land surface can provide a transition from the field model to the object model (Brändli, 1996), and from general geomorphometry to specific geomorphometry (Evans,

Fig. 1. An example of elementary segment representation in a landform hierarchy.

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1972, 1987), thus connecting the continuous and atomistic hypotheses. Specific structural elements of the field provide a natural basis for its segmentation. We can term them singular points and lines. These include for example extremum points and lines (peak, pit and saddle points; ridge and valley lines), inflections and discontinuities of the altitude field and its derived fields (Lastoczkin, 1987). As the land surface is in general locally continuous, it contains only a finite number of singular lines. This contrasts with the infinite number of isolines and slopelines/streamlines. Defining singular lines as boundaries of landform segments (elementary forms) therefore provides a good start in making land surface segmentation as objective as possible. Next, the formal description of segments by smooth mathematical functions may bring considerable advantages to their identification and interpretation. Landform discontinuities and the presence of definable elementary forms are consequences of the spatial differentiation of past and recent geomorphic processes. Most processes are strongly influenced by gravity. The connections between elevation, gravity and geomorphic processes permit genetic interpretations of elementary forms. Therefore the recognition of a physical-geometric analytical system of basic types of elementary landform unit provides the skeleton of land surface segmentation theory. A modern concept of land surface segmentation should also take into account uncertainties in the identification of discontinuities and in the degree of membership of a given individual form to a form type (see e.g. Schmidt and Hewitt, 2004; Schmidt et al., 2005). 2. Review of approaches in land surface segmentation The process of land surface segmentation should arise from a theoretical concept of geomorphic units. Both the interior properties of ideal units and the character of their boundaries have a crucial role in the definition of units. Existing concepts have several common features, but they differ in degree of explicitness, complexity, exactness and formal expression, and in theoretical background or specific methods. Elementary geomorphic units are generally seen as geometrically homogeneous parts of the land surface, indivisible from a geomorphological point of view. Other types of homogeneity (genesis, age, contemporary processes, rocks and soils) have been treated variously. Traditional geomorphological mapping was developed predominantly on a morphogenetic basis (e.g. Demek, 1972; Spiridonov, 1975; De Graaff et al., 1987). Attempts to include the whole set of morphogenetically

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relevant characteristics of landscape (character of ground, soil, surface material, and drainage) in the process of segmentation made formulation of a strict algorithm difficult. Less attention was paid to the morphological character of the elementary relief units, termed facets and segments, elementary forms, relief elements, morphotopes (defined in geoecology as the smallest homogeneous relief units), or genetically homogeneous surfaces. Further approaches developed that were more sophisticated and may be divided into graph-based and classificatory; their synthesis will be discussed in Section 3. 2.1. Graph-based approach A large part of the work related explicitly to land surface segmentation deals with the definition of segment boundaries. Identification of a unit's boundary is the primary goal and the character of the interior may not influence the determination of its limits. This can be termed the graph-based approach (Brändli, 1996). Morphological mapping as defined in Britain by Waters (1958) and Savigear (1965) represents a typical example. More precisely, this is morphographic mapping, based on form rather than process. This atomistic approach started from the simple assumption that the ground surface consists of planes bounded by morphological discontinuities (Waters, 1958). A wider theoretical background appears in the system of morphological units and their boundaries presented by Savigear (1965), which has been widely used with only small modifications (e.g. Cooke and Doornkamp, 1990; Griffiths et al., 1995; Lee, 2001). A morphological unit is either a facet (a plane surface area) or a segment (a smoothly curved, upward-convex or -concave surface area). Facets and segments join at discontinuities: breaks of slope, changes of slope and inflections. Microfacets (microsegments that are very narrow in relation to the map scale, and are thus depicted with linear symbols) represent a transition between areal morphological units and linear boundaries. Savigear's boundaries have a singular character, but they do not guarantee the full demarcation of morphological units. To complete the separation of morphological units, Young (1972) suggested use of boundaries defined by permitted ranges of mean slope angle for facets, or of mean curvature for elements (curved units; Young, 1972, p. 182). They may thus be isolines of slope or curvature; unfortunately, their arbitrary nature reduces the objectivity of segmentation. Although morphological units are defined geometrically, their genetic and dynamic interpretation is possible and desirable, as for example in the hypothetical nine

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unit land surface model for slope profiles (Dalrymple et al., 1968). Morphological units are classified into nine dynamically interpreted units, in sequence downslope as a catena. This model describes each of the nine units by position, slope, profile curvature, actual processes, significant microforms and the specific characteristics of rock, soil and vegetation. Unlike morphological mapping, however, units are delimited also by specification of process transitions at the boundaries. Parsons (1988) recognised that confusion of morphological and morphodynamic criteria is a problem. A more comprehensive and theoretical application of the graph-based approach is the segmentation model of Lastoczkin (1987, 1991). He introduced the set of structural lines and characteristic points defining elementary surfaces. The basic structural lines are singularities representing (on a profile) local extremes of altitude z (ridge lines and valley lines); local extremes of slope gradient z′, the first derivative of altitude (inflection lines of maximum or minimum slope); and local extremes of profile curvature z″, the second derivative of altitude (convex and concave flexures involving breaks and changes of slope). Structural lines thus represent various kinds of discontinuities of morphometric properties (Fig. 2). The basic structural lines are further classified on the basis of the linear, convex and concave shape of the profiles on either side. Characteristic points include peaks and pits as well as the ends or junctions of structural lines. Lastoczkin's system deepens the theoretical background of morphological mapping, but its initial form was too focused on the behaviour of altitude in the direction of slopelines, which is not sufficient for full delineation of elementary

forms in three dimensions. Like Young (1972), Lastoczkin (1987) therefore introduced an extra (asystemic) factor in the bounding of elementary surfaces — “side limitation”, represented by slopelines or cross-lines which do not have a singular character: an infinite number is possible. His classification of the main geometric types of elementary surfaces remained similar to that of morphological mapping. Plains and slopes – linear, convex and concave in profile – are classified further only according to their positional properties. A very important improvement is made in Lastoczkin's (1991) later work where he included the concept of plan curvature, and added its zero isoline to structural lines. This opened the way to a more comprehensive perception of land surface segmentation. 2.2. Classification approach A second major line of thought in land surface segmentation is the classification approach of Brändli (1996). This focuses on definition of the internal properties of elementary forms, from which the definition of boundaries follows. Geometric forms or elementary forms defined on the basis of their curvatures (Richter, 1962; Troeh, 1965; Young, 1972; Krcho, 1973; Dikau, 1989; Shary, 1995) provide an example of supervised classification; the range of each property is pre-ordained. The importance of both profile and plan curvature for earth surface processes has been recognised at least since Aandahl (1948). The simplest model is a 2 × 2 classification based on the signs of profile and plan curvature, giving four basic forms (Table 1; Troeh, 1965). These basic geometric forms have an important dynamic interpretation in terms of gravity flows, with

Fig. 2. Profiles across types of structural lines after Lastoczkin (1987), and their interpretation as lines of discontinuity.

J. Minár, I.S. Evans / Geomorphology 95 (2008) 236–259 Table 1 Local slope forms classified by curvature, profile first (varies down column), then plan (contour; varies along row), for the scheme of Richter (1962) [also Krcho (1983) and Dikau (1989)], that of Troeh (1965) and that proposed here in Fig. 5 Troeh (1965) VX XX

Richter (1962) etc. VV XV

VX SX XX

VS SS XS

VV SV XV

Proposed (in Fig. 5) Contour shapes Planar

Circular convex

Circular concave

Clothoid

Divergent

SS VS XS vxS xvS

SX VX XX vxX xvX

SV VV XV vxV xvV

S vx/xv V vx/xv X vx/xv vx vx/xv xv vx/xv

SS VS XS vxS xvS

S = straight, V = concave, X = convex, xv = convex-concave, vx = concav-convex. Note that the first three columns and rows of the proposed scheme represent a re-ordering of Richter's scheme.

acceleration or deceleration of flow by slope curvature, and concentration or dispersal of flow by plan curvature. The basic forms are naturally bounded by inflections (that is, by singular lines). Richter (1962) further distinguished straight slopes, in profile and in plan, giving a 3 × 3 classification; this approach was followed by Krcho (1983) and Dikau (1989). Table 1 relates Troeh's model to that of Richter, and to the present extended proposal of 5 × 5 (+ 1) classes. The introduction of straight forms, with linear contours and constant gradient, where zero isolines of profile or plan curvature degenerate into areas (Krcho, 2001), implicitly requires new threshold lines separating these from curved forms. These are modelled as discontinuities of curvature or of change of curvature. As straight forms are, however, defined by one or more non-zero values of curvature (Young, 1972; Ruhe, 1975; Krcho, 1983; Dikau, 1989), the boundaries of such forms are arbitrary isolines of curvature and are not singular lines. For example, Dikau (1989) and other German authors defined ‘straight’ as having a radius of curvature greater than 600 m, so those (+ and −) isolines delineate linear forms. This problem of arbitrary choice extends to the more comprehensive systems with further types of curvature introduced by Shary (1995) and Krcho (2001). Maps of geometric forms are usually created by overlay of maps of zero or other isolines of curvatures. When threshold values are applied on the wider set of morphometric properties (altitude, slope angle, aspect, curvatures, etc.), landform segments defined by the set of morphometric quantities are produced (e.g. Speight,

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1968; Dikau and Jäger, 1995; Pennock and Corre, 2001). Krcho (1983) offered a fully formalised expression of these units and termed them morphotopes. BolongaroCrevenna et al. (2005) used thresholds (‘tolerance values’) of 6° for gradient and 0.0001 (units unspecified) for convexity, in defining cells as plane, ridge, channel, pass, peak, or pit. The advantages of this approach are the clearly defined homogeneity of units and the simple practical interpretation. However, as Reuter et al. (2006) show, this method leads to strong scale dependence of results that can be eliminated only partially by optimisation (scale dependent shifting of threshold values). Moreover as isolines may pass through areas of very gradual change, isoline boundaries may create artificial areas without sufficient respect to the natural structure of landform units with various types of homogeneity. Use of such artificial boundaries will hinder interpretation and confuse the analysis of natural structures. The use of mean values of important variables as threshold values, as in the recent innovative approach by Iwahashi and Pike (2007), eliminates the subjectivity of threshold definition, but the main problem of arbitrary incidence remains. The use of numerical taxonomy (cluster analysis) in the framework of a parametric approach (Speight, 1974) attempts to overcome this artificial character and preserve the synthetic nature of the landform units defined by a set of properties. This statistical approach, unsupervised classification, obtained a big impetus from the adoption of Digital Elevation Models (DEMs) and GIS facilities, and it is still the most widespread tool for land surface segmentation. In the most popular type of cluster analysis, the initial areal units (cells) are grouped (clustered) on the basis of their similarity, that is, their separation in attribute space. These clustering algorithms minimise intraclass and maximise interclass differences, in accordance with the general requirement for land surface segmentation into elements. However, this approach has several problematic aspects related to subjective choice of procedures: input parameters, clustering methods and interpretation of results. The majority of authors (e.g. Speight, 1974; Irvin et al., 1997; Ventura and Irvin, 2000; Blaschke and Strobl, 2003; Adediran et al., 2004) divide the land surface into territorially discontinuous classes – types of landform element – each of which can include numerous spatially discrete landform elements. This raises a fundamental problem: the classification is based on thematic similarity alone, ignoring position. The minimisation of intraclass and maximisation of interclass differences do not relate to individual landform elements but to whole classes, scattered across the map. Local homogeneity becomes less

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likely as the number of landform elements in a class increases. Definition of individual spatially continuous regions by a contiguity criterion can be a solution. This produces relatively homogeneous areas bounded by lines where the homogeneity is disrupted. Friedrich (1998) introduced distance grouping methods for deriving homogeneous relief units by coupling agglomerative procedures with a spatial neighbourhood analysis. Romstad (2001) demonstrated the usefulness of the approach very convincingly by his method of relief classification with contextual merging, and a similar principle was used by Bonk (2002) for delineation of ‘terrain form objects’ by spatial grouping on the basis of local contiguity. However, cell- (pixel-) oriented approaches are limited by their mesh-dependence and neglect of neighbourhood relationships. “The shift from per-pixel-based to objectbased analysis requires a shift from pixels having meaning to user-defined objects having meaning” (Drăguţ and Blaschke, 2006, p. 333). Cluster analysis permits the use of a large number of input parameters, which should provide more synthetic results. Speight (1974) used 21 morphometric properties as “a result of an intuitive selection of those aspects of geometry of terrain that appear relevant either to land use or to geomorphological process”. Other authors have also included non-morphometric properties, giving areas which are relatively homogeneous not only geometrically, but also from the point of view of land cover, land use, and soil (e.g. Irvin et al., 1997; Giles and Franklin, 1998; Blaschke and Strobl, 2003). However, an increase in the number and variety of input parameters can hinder clear geomorphological interpretation of delineated landform elements; also it prevents assessment of how closely landforms relate to soil or land cover. When geomorphic types are not defined prior to mapping, and multivariate statistical methods are used, the reference of terrain units to process or genesis is lost (Brown et al., 1998). Instead of segmenting ‘blindly’, the character of geomorphic units may be defined before segmentation. The specific geomorphometry of Evans (1972, 1987) offers a solution. Statistical characteristics of morphometric properties of a known set of landforms can lead to the identification of other landforms of a given type. This can be termed the signature approach. Pike (1988) demonstrated the effectiveness of this approach for identification of landslide-prone areas. Within this approach Chorowitz et al. (1995) distinguished parametric analysis (centred on the identification of boundaries of landslides from contour geometry), stochastic analysis (on the basis of local fractal dimension and variability, in combination with binary

geological information) and structural syntactic analysis (identification of landforms as a series of more elementary slope patterns). Giles and Franklin (1998) used discriminant analysis for the identification of ten types of slope unit defined a priori, in combination with breaks of slope gradient as unit boundaries. A specific structuredknowledge model was developed by Argialas (1995): an expert system permits recognition and delineation of a landform from user-supplied pattern elements. Recently van Asselen and Seijmonsbergen (2006) used objectoriented classification by eCognition Professional for recognition of eight genetic landforms using only slope, altitude and contributory area. Prima et al. (2006) used a signature approach and tested which morphometric variables distinguished defined landforms such as volcanoes and alluvial plains. All earlier approaches to land surface segmentation produced crisp classifications of land form. Recently continuous (fuzzy) classification has frequently been applied within the cluster analysis approach (e.g. Brändli, 1996; Irvin et al., 1997; MacMillan et al., 2000; Blaschke and Strobl, 2003; Schmidt and Hewitt, 2004). This is a new way of reflecting the complex variation of landscape character in space. Internal homogeneity of elementary landform units is a basic aim of most approaches to land surface segmentation. However, if within-object homogeneity of mapped units is considered a poor or even unrealistic assumption (Burrough, 1996), the fuzzy approach can be an ideal tool for expressing the affiliation of a real landform segment to any ideal type of elementary landform. Likewise it expresses the diffuse nature of many boundaries of elementary landform units. A more specific model approach to classification approximates the shapes of individual landform segments with three-dimensional equations (Troeh, 1965; Young, 1972; Schmidt et al., 2003). Landform representation by fitted functions is widely used in modern morphometric analysis. Fitted functions (of one family) represent regular segments of land surface, usually a submatrix of the square network of the DEM, and they can be used as interpolation functions. But every elementary form can be represented by a specific fitted function (see Parsons, 1988). The characters of fitted functions describing adjacent segments can determine the character of their boundary (Minár, 1998), as discussed in Section 4. Revived interest in problems of land surface segmentation is connected with the recent development of GIS techniques. Use of GIS technology stresses the important requirement for “more detailed and explicit formalisation of the terrain analysis process” (Argialas, 1995, p. 104). Unfortunately, most authors focus on the

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methodological aspect (automation) at the expense of the conceptual background of segmentation. Many authors prefer more readily automated concepts, which are not necessarily the best. Papers often represent a new look at old concepts from the point of view of their feasibility in the GIS environment. Some algorithms combine principles mentioned earlier (Dymond et al., 1995; Giles and Franklin, 1998; MacMillan et al., 2000; Drăguţ and Blaschke, 2006), but a general theoretical unification of the various concepts is still needed. 3. A new system: the concept of elementary forms Existing approaches to land surface segmentation define boundaries and surfaces of elementary landform units in various ways. Connections among them are expressed in summary form on Fig. 3. Boundaries ideally sensed as lines of discontinuity of some characters, and homogeneity ideally defined by uniformity of relevant characters, can provide the link between statistical approaches and basic analytical concepts. Such an approach is developed in the integrating concept of elementary forms (Minár, 1992, 1998). 3.1. The model proposed The concept is based on geometric and physical aspects of field theory (Krcho, 1973). Land surface form is considered here as a geometric field of altitudes that is directly linked with the gravitational field (Devdariani, 1967; Shary, 1995). Altitude and secondary morpho-

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metric variables (defined by derivatives of altitude in various directions) have both geometric and physical significance. Homogeneity of genetic influence is therefore reflected by morphometric homogeneity (uniformity) of landform, and change of genetic influence is connected with morphometric discontinuity. Consequently, constant values of some morphometric characters define the area of an ideal elementary form, and discontinuities define its boundaries. The basic morphometric system introduced by Evans (1972) and Krcho (1973) can be extended formally in the form (Minár, 1999): M¼

n

ð0Þ



 o M ¼ fzg;ð1Þ M ¼ fzi g;ð2Þ M ¼ zij ;ð3Þ M ¼ zijk ; N

ð1Þ where M is the set of all local morphometric variables (definable at every point of the land — cf. Shary et al., 2002). (0)M is a subset of M containing only the element altitude (z), an initial quantity that has the lowest order (zero). (1)M is the subset of morphometric variables (zi) of the first order, defined by the first directional derivative of altitude in direction i. (2)M is the subset of morphometric variables (zij) of the second order, defined by the directional derivative of quantities of subset (1)M in direction j, and so on. Included in subset (1)M are not only slope gradient and aspect (direction), but also any other variable (apparent slope) defined by first derivatives of altitude in any direction. Subset (2)M includes various curvatures (profile, plan, tangential and Gaussian: Schmidt et al., 2003) but also any other variables defined by second derivatives of altitude in any direction (Shary,

Fig. 3. Relationships among various types of morphometrically defined elementary geomorphic areas (segments) and their boundaries.

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1995; Shary et al., 2005). Subset (3)M includes variables expressing change of curvatures, and so on. The set defined by Eq. (1) contains an infinite number of variables, but some are more significant from a geomorphological point of view. The vector field of slopes determines the gravitational flow of matter and energy on the earth's surface. Therefore variables defined by change of altitude in the normal direction of slopelines (direction of maximum slope gradient), or by the complementary tangent direction (direction of contour lines), have a special importance in geomorphology, which all geomorphometric systems respect (e.g. Evans, 1972; Krcho, 1973; Moore et al., 1991; Jenčo, 1992; Pike, 2000; Krcho, 2001). Shary (1995) terms these morphometric variables field-specific (in contrast to field-invariant). Maximal homogeneity of field-specific variables within segments is a generally accepted postulate of elementary segmentation. A constant value of field intensity is the highest possible degree of homogeneity; it is the final state of equilibrium of the field. In the case of the land surface it is represented by a constant value of altitude; horizontal planes thus defined are the most readily interpretable elementary forms (approximated by accumulation plains and surfaces of planation). But the majority of the earth's surface is in slopes with non-zero gradients. A constant value of slope (the first derivative of the field) represents a lower type of homogeneity: homogeneity of change. It is a vector with magnitude and direction (slope gradient and aspect: Evans, 1972). A constant value of both gradient and aspect defines a planar facet, the most homogeneous linear slope (e.g. a fault slope, or an abrasion slope). Steady changes of slope gradient or aspect are frequent, providing curved elements of slopes (segments, form elements, or geometric forms generally; alluvial fans, sink holes or periglacial dells specifically), and segments can be more or less homogeneous in this respect also. In general, homogeneity decreases with the increasing order of derivatives that are constant. Fig. 4 shows the mutual relations and basic geomorphological importance of the field-specific variables, altitude and its first three derivatives. The geomorphological interpretation of uniformity of a variable stems from geomorphological theory, as given in particular process models (e.g. Scheidegger, 1970; Carson and Kirkby, 1972; Rohdenburg, 1989; Moore et al., 1991; Lastoczkin, 1991; Mitášová and Hofierka, 1993; Minár, 1995). Many variables have related equivalents (radius of plan curvature and plan curvature, normal gradient change and profile curvature) that are widely used. Variables of the third order have not generally been used. However, the scheme at the bottom of

Fig. 4 explains possible long-term dynamic stability of the constant values. Erosional and depositional effects generally depend on change of energy and mass of agents. These depend on profile and tangential curvatures (cf. Scheidegger, 1970; Carson and Kirkby, 1972; Mitášová and Hofierka, 1993, Shary et al., 2005). Long-term stability of morphometric uniformity implies not a constant slope-normal change (dhN/dT), but a constant temporal change of altitude (dz/dT) (lower right of Fig. 4). Hence radius of plan curvature and normal gradient change are highly significant, together with their derivatives. The variables in Fig. 4 therefore define the basic system of elementary forms suggested here. Minár (1999) published equations defining these variables, most of which were derived earlier by various authors (e.g. Evans, 1972; Krcho, 1973; Jenčo, 1992; Shary, 1995). Planar slopes (including horizontal and vertical planes) have been used in land surface segmentation for a long time. The interval determination of curvature for individual segments by Young (1972), and the postulation of homogeneous plan and profile curvature of form elements by Dikau (1989), also point to the representation of curved slope elements by constant values of second-order morphometric variables. Parsons (1977) expressed a similar idea for profile representation, with criteria of constant inclination, constant curvature and constant change of curvature defining simple components of a slope profile. Theoretically we can model land surface form with elements defined by constant values of morphometric properties within the full set M. The use of more types of element allows a better approximation of reality, but their interpretability is a limitation. As yet, properties up to the third order seem to be interpretable and thus effectively usable now, but the system is open to further development. 3.2. Elementary forms We can generalize the proposed model as follows. Land surface form consists of segments characterized by various types and degrees of homogeneity. These can ideally be expressed by constant values of altitude or its derived morphometric properties. Discontinuities of these properties provide logical boundaries to the segments. Then we can define ideal elementary forms as landform elements with a constant value of altitude, or of two or more readily interpretable morphometric variables, bounded by lines of discontinuity. The constant values which define the elementary form are termed the formdefining properties. Elementary forms are individual units of the land surface, which can be classified into typological systems

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245

Fig. 4. Relations and implications of constant values of the altitude, slope and curvature properties. n = normal direction (of slopelines), t = tangential direction (of contour lines), const = constant.

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on the basis of various criteria (geometric, dynamic, topological and genetic). Separation by type (but not sign or value) of form-defining properties is fundamental and defines the basic geometric types of elementary forms. The number of basic types is limited by the compatibility of form-defining properties. For example constant aspect is compatible with constant (non-zero) profile curvature or change of curvature, but constant slope gradient is not compatible with constant non-zero plan or tangential curvature. Distinguishing forms defined by positive and negative values of form-defining properties has a special and universal significance (Fig. 4). Therefore, within the basic geometric types, basic geometric subtypes of elementary forms can be distinguished by positive and negative values of form-defining variables. The number of basic subtypes rises with the order of elementary form. Basic subtypes can be defined not only by the signs of form-defining variables, but also by signs of all related variables of lower order; for example, the subtypes of a linear slope can be terrestrial or marine, and regular or inverse (overhangs: in theory, although these are not represented in DEMs). Only local, not regional variables are used here for the definition and delimitation of elementary forms. Regional variables in the sense of Shary et al. (2002) are properties defined by spatial relationships, e.g. openness, flow path length, upslope contributing area and dispersal area. They are not used in the definition of geomorphological individuals because they describe relations beyond a single elementary form and are incompatible with the condition of spatial uniformity. However, regional variables should be used for characterization and further classification of elementary forms, after they have been defined. Types of elementary form that take position into account may be termed land elements, after Schmidt and Hewitt (2004). Relations between form-defining properties determine relations among basic types of elementary form. Every property (r)zi of order r (r ≥ 1) is defined by the derivative of some property (r − 1)zj of order r − 1. The property (r − 1)zj can thus be termed a parent property of property (r)zi and the following relation exists between constant values of the parent and derived properties: ðr1Þ

zj ¼ const Z

ðrÞ

zj ¼ 0:

ð2Þ

For example, a constant value of altitude determines a zero value of gradient, and a constant value of gradient determines a zero value of profile curvature. A specific elementary form can be defined by the zero value of its form-defining property while its parent property is not

constant (see the last column of Fig. 5). Forms share the properties of lower-order forms only on specific lines such as slopelines or contours in this case. As zero is only a specific case of constant value, forms defined by properties of a lower order are only specific cases of those defined by properties of higher order. For rectilinear segments and curved elements of profile this was already stated by Cox (1978). Consequently, a mathematical expression defining a form with a constant value of the property (r − 1)zj is only a specific case of that defining a form with a constant value of the property (r)zi (see Fig. 5). Thus a basic system of elementary forms defined by only one fundamental equation can be created. 3.3. Equations for ideal elementary forms Fig. 5 represents the proposed system, based on the three principles: 1) The functional dependence of altitude (z) of a point with map coordinates (x, y) on distance from any start point with altitude H and coordinates (I, J): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ H þ að x  I Þ2 þbð y  J Þ2 ð3Þ where a and b are constants and (H, I, J) can be coordinates of a single start point or a point on a start contour lying on the same slopeline as the point (z, x, y), generally describes forms with parallel contours. All contours are either concentric, or parallel to the defined starting contour (linear, or parallel curves that are represented here by clothoid models). Parallel contours generally determine a constant value of the normal change of radius of contours Rn on every individual slopeline (i.e. Rnn = 0). 2) A polynomial function of i-th order represents a constant value of the i-th derivative of altitude in the normal direction. Then the function: z ¼ H þ Bn þ Cn2 þ Dn3

ð4Þ

where ξ(x,y) expresses the shape of contours, generally describes forms with constant normal change of downslope curvature (Gnn) in models with parallel contours. 3) Forms with divergent contours can be defined by functional dependence of altitude on angular coefficient: z ¼ H þ F arctan

yq xp

ð5Þ

J. Minár, I.S. Evans / Geomorphology 95 (2008) 236–259 Fig. 5. The unified system of basic elementary forms expressed by variations in the general fitted function: in plan (along rows) and in profile (down columns). The horizontal plain (defined by absence of contours) is the most homogeneous type of form and is a special case of all other models. Geometric complexity of forms increases from top left corner to bottom right corner, inversely to interpretability. From left to right, the columns represent linear, circular positive, circular negative, parallel contour (clothoid) and divergent elementary forms. Contours are parallel except in the latter. Double columns or rows distinguish basic subtypes, convex and concave. From the top down, the rows are straight (S), constantly curved (V — concave, X — convex) and constantly changing profiles of slopelines (vx — concave–convex, xv — convex–concave). Because of low interpretability, horizontally curved variants of divergent models are not presented.

247

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Fig. 6. Selected elementary forms defined partially or fully by a constant value of field-specific morphometric variables dependent on nongravitational geomorphological fields (e.g. morphotectonic) — one or two derivatives of altitude (z) are constant in the direction of axis of anisotropy of the field, coincident in these examples with the coordinate system x and y. zy, zyy, zyyy — first, second and third derivative of altitude in direction of y, zxx, zxxx — second and third derivative of altitude in direction of x, R — radius of plan curvature, H, B, m — constants.

where F is a constant and p and q can be either constants or functions p(x,y) and q(x,y). Combination of these three principles enables us to express various basic types and subtypes of elementary forms by one general function. Specific values of constants and variables simplify the function into shapes describing thirteen basic types of elementary forms (boxes in Fig. 5), and dozens of subtypes. The proposed concept of elementary forms is in principle independent of scale, and the methods proposed here can be applied to data at any resolution. Multi-scale landform characterization (Schmidt and Andrew, 2005) could provide a basis for application of the concept at various scales. Many aspects of the land surface, however, are scale-specific (Evans, 2003), and the value of applying this approach at broader scales requires further investigation. The definition of elementary forms enables us (in harmony with Troeh, 1965) to represent every form by a

specific mathematical function (fitted function). The system includes and specifies in more detail the various types of morphological units (Savigear, 1965), elementary surfaces (Lastoczkin, 1987), geometric forms (Troeh, 1965; Krcho, 1973) and form facets or form elements (Dikau, 1989). A different typology of elementary forms could define types by interval values of morphometric properties (analogous to the morphotopes of Krcho, 1983), or by elements of such models as the hypothetical nine unit land surface model (Dalrymple et al., 1968). The system of elementary forms is fully open; adding to or modifying it depends only on our ability to formally express and effectively interpret more complex (higher order) types of elementary forms. Forms characterized by homogeneous derivatives of altitude in non-standard directions (Fig. 6) provide an example. They are analogous to the flat (low gradient) elements of Schmidt and Hewitt (2004) because they are not defined by gravitational field-specific variables (Shary, 1995). They can be formed by the dominant influence of geological structure or of a directional

Fig. 7. Basic discontinuity lines.

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geomorphic agent, giving an axis of form symmetry. Thus similar expressions of elementary forms by fitted functions (e.g. the polynomial models of Schmidt et al., 2003) can be integrated into the system. 4. Boundaries of elementary forms 4.1. Discontinuities Representation of a full land surface by elements with constant values of specified morphometric properties has strict theoretical consequences for the demarcation of elementary landform units. In the geometrically ideal case, the boundary of two elementary forms defined by different constant values of some morphometric properties must be a line of discontinuity — a sudden discrete change of value of some property (Fig. 7).

249

The property value which jumps at the boundary can be termed the boundary-defining property. If the line is characterized by discontinuity of only one property, we call it a simple discontinuity line; otherwise, it is a compound discontinuity line. The character of a boundary-defining property is limited by the character of the form-defining properties of adjacent forms, which are defined in the same direction as a boundary-defining property. If a defining property (r)zi of one form is derived from (or identical with) a defining property (s)zi of an adjacent form, then for the order of the boundarydefining property (t)zi we have: t V r z s:

ð6Þ

For example, the boundary between a horizontal plane (z = const) and a linear slope (G = const) can be an altitude

Fig. 8. Examples of compounded boundaries of elementary forms. Legend as in Fig. 5; D = form-determined boundary.

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or slope discontinuity line but, as curvature is zero on both sides, it cannot be a discontinuity of curvature or change of curvature. Identity of the boundary-defining property with a form-defining property of higher order ((r)zi = (t)zi) is the most stable situation, which we term a formdetermined boundary (Fig. 8). Constant values of a boundary-defining property within each adjacent form are characteristic of form-determined boundaries. Every line of discontinuity has an interpretation related to the constant values of boundary-defining properties (Fig. 4). The importance of a discontinuity line rises with the value of discontinuity. Differences in the values of boundary-defining properties on both sides of a boundary give the specific sharpness of boundary. In the case of slope we can separate aspect and gradient sharpness; alternatively we can define a 3-D synthetic sharpness as the angular difference between slope vectors. Depending on the number of boundary-defining properties, the boundary can be characterized by one or more specific sharpnesses. Further properties such as the horizontal extent of adjacent forms determine the spatial importance of a boundary, and the genetic interpretation of a boundary can be derived from the genesis of adjacent forms. Unlike the rule concerning constant values of

parent properties determining constant values of derived properties (Fig. 4), discontinuity of parent properties does not underlie discontinuity of derived properties; altitude discontinuity does not determine slope or curvature discontinuity. Equations describing forms of the third order in specific conditions (∣bξ ∣ N ∣ aξ + cξ∣ and sign b ≠ sign c for a part of the domain of definition) give specific solutions. Three elementary forms that are subtypes of the same basic type can be distinguished in terms of aspect (AN) and normal change of contour curvature (Rn). In Fig. 9, discontinuities are regular boundaries of elementary forms, so one equation can describe three elementary forms. But the inflection line is not a discontinuity and so it should not be the boundary of an elementary form; an homogeneous convex–concave element extends on both sides of the inflection. 4.2. Contrast, smoothness and resolution There are further relations between discontinuity lines and other singular lines. Discontinuity is an ideal geometric category and its relevance to real terrain depends on resolution. Greater resolution may transform

Fig. 9. Three elementary forms; various geometrical subtypes of the type defined by constant value of normal change of gradient change (Gnn) and pffiffiffiffiffiffiffiffiffiffiffi 3=2 constant value of normal change of radius of plan curvature (Rn), resulting from the equation: z ¼ 350 x2þ y2  40ðx 2 þ y 2 Þ þ ðx 2 þ y 2 Þ .

J. Minár, I.S. Evans / Geomorphology 95 (2008) 236–259

251

Fig. 10. Two belt profiles showing correlation of discontinuity lines of the contrast model of landform (A) and zero isolines of the smooth model (B); (C) illustrates transformation of a discontinuity to an elementary form by change of resolution.

a discontinuity of one property into an extremum line (maximum or minimum) of another property. The model of land surface form with elementary forms described by individual fitted functions is termed the model of landform contrast. It consists of homogeneous elements with distinct heterogeneity across boundaries. The smooth landform model (represented ideally by only one smooth mathematical interpolation function describing the modelled area) is an alternative, sometimes used in geomorphometry for computation of morphometric properties.

A discontinuity line of property (r)zi in the contrast model is transformed into an extremum line of derived properties (r + 1)zi and the zero isoline of property (r + 2)zi in the smooth model. That means the local extremes identified by regular morphometric analysis (and thus the zero isolines of a derived property) can be the boundaries sought for elementary forms. However, the smooth model also generates at least one extremum line (the zero isoline) within each elementary form, which may be interpolated as being central although the whole form has a zero value of the property. Horizontal planes

Fig. 11. A brief algorithm for delimitation of elementary forms. ‘Protoforms’ represent preliminary delimited units that may or may not be elementary forms. Threshold affinity is a user-defined minimal acceptable degree of accordance between the real and ideal elementary form. The most distinctive discontinuities are also defined by a user-defined threshold affinity — degree of accordance between the real and a geometrically ideal (sharp) discontinuity.

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Fig. 12. Model application of the elementary form concept to three segments on Devínska Kobyla Mountain, Slovakia. Eight models, D1…L3 are fitted to the three rectangular areas (bottom left of each set; Sl, Po and Sa). L = linear, C = circular, D = divergent; the number gives the slope order of the form. Results are given in Table 3. Contour interval 5 m. For comparison the corresponding section of the geomorphological map of Minár and Mičian (2002) is displayed.

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253

Table 2 Derivation of constants of form-defining equations (a, b, c, d, g, h, I, J, p, q — see Fig. 5) the best-fitting ideal elementary forms from median (med) or mean (mean) values of morphometric characters of real segments of georelief Normal order

L — linear models n ¼ a þ bðgx þ hyÞ 2

þ cð gx þ hyÞ

þ d ðgx þ hyÞ3

C — circular models qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ a þ b ð x  I Þ2 þð y  J Þ2 h i þc ð x  I Þ2 þð y  J Þ2 h i3=2 þd ð x  I Þ2 þð y  J Þ2

D — divergent models yq n ¼ a þ b arctan p  x  yq 2 þc arctan xp   yq 3 þd arctan xp

a = meanz − meanz(a = 0) 1st

g ¼ med zx

med

I¼

h ¼ med zy med zA b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 2 g þh2

! jRjd tan AN x  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan 2 AN þ 1

med

J ¼F

ð

jRj yFpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d tan 2 AN þ 1

Þ

b ¼ Fmean G; 2nd

c¼

med

2

ðg2

b¼

zA  med zAðb¼0Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2þh2

d¼

med zAAA qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 ðg2 þ h2 Þ3

med

3rd

zAA þ h2 Þ

med

c¼

zAA  med zAAðc¼0Þ 2 ðg 2 þ h2 Þ

c¼

med

med

med

Gn 2

Gn  R2R 2

c¼

med

Gnn 6

med

Gn  med Gnðc¼0Þ 2

c¼

ð

RR  tan AN y  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan 2 ð90  AN Þ þ 1

q¼

Þ

b ¼ med ðG  RR Þ

b ¼ med G  med G ðb¼0Þ

d¼

! jRR j xFpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan 2 ð90  AN Þ þ 1

med

p¼



b ¼ med G  RR  med G ðb¼0Þ  RR

d¼

med

med

c¼

Gnn  R3R 6



  med ðG 2 nðc¼0Þ  RR ðGn  R2R  2

x, y, z are map coordinates of points of real segment (z = altitude), zx, zy are partial derivatives of z = f(x,y), G — gradient (tangent), Gn — normal change of gradient, Gnn — normal change of change of gradient, AN — aspect, R — radius of plan curvature and RR — radius of rotor curvature, zA, zAA, zAAA — 1st, 2nd and 3rd directional derivatives of z = f(x,y) in direction of slope (aspect, AN) of linear model. The relations were used for computation of the models in Fig. 12.

will thus be divided by zero isolines of slope, linear slopes by zero isolines of curvature, and so on. These ‘misleading’ zero isolines (those not corresponding to discontinuities in the contrast model) are of lower order than transformed discontinuities (zero isolines which do correspond to discontinuity lines). The number of lines has a tendency to rise with the order of the elementary form (Fig. 10), complicating the situation. Another type of boundary transformation connected with change of resolution (Fig. 10C) is where discontinuity lines change into elementary forms when viewed in greater detail. This arises because, at some distinguishing level, every discontinuity can be considered as a “microsegment” in the sense of Savigear (1965). Some form-defining properties of the “microsegment”

elementary form are derivatives of boundary-defining properties of the discontinuity (Gn is the derivative of G in Fig. 10C), but they cannot be the same properties. Discontinuity of altitude cannot be replaced by a form with constant altitude, discontinuity of gradient by a form with constant gradient, and so on. 5. A worked example We introduced a theoretical concept of land surface segmentation that on the one hand reflects the intuitive approach of geomorphological mapping, yet on the other hand increases objectivity by the development of precise algorithms. Although the problems of internal homogeneity and boundary discreteness were considered

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separately, effective application requires both to be joined in the process of delimitation. This can be an iterative process in which the changing degree of affinity of real segments to the given models is measured for various degrees of boundary discontinuity (Fig. 11). In practice, the most distinctive discontinuities are defined first, then the fit of expressions from Section 3.3 is measured. If none fit adequately, it may be necessary to subdivide into areas bounded by less distinctive discontinuities. Alternatively, a form that has not reached the required threshold of homogeneity may be accepted as an exception; e.g. a landslide area composed of microforms on a finer scale. Rarely, as a last resort, a new kind of interpretable elementary form might be sought, possibly leading to improvement of the theory. A brief example of the application of part of this concept is presented in Fig. 12. A section of the geomorphological map of Devínska Kobyla Mountain, Slovakia (Minár and Mičian, 2002) is interpreted in terms of the elementary form model. Field research and visual analysis of a DEM were used for the delimitation of elementary forms. The boundaries fit with lines traditionally used in geomorphological mapping, interpreted here as different types of discontinuity line. Examination of the geometric character of areas is more straightforward. For the computation of constants in specific equations for selected types of elementary forms (Table 2), we have started from average values of key morphometric properties. Properties of first order are completely accurate, those of second order are approximate, and properties of third order are currently uncertain because of problems of computation of derivatives of higher order (see e.g. Krcho, 2001). Multiplication of properties has a similar effect, influenc-

ing computation of the divergent models: hence the iterative computation of constants b and c is based on approximation to median values of gradient and gradient change. The third order divergent model (D3) was not used because of these complications in the computation of third order properties. Clothoid-based models (the penultimate column in Fig. 5) were also excluded, considering similar problems and their lower interpretation value. Including the 0-model (horizontal plane), nine models of geometrically ideal elementary forms were computed for the central parts of three elementary forms (Sl, Sa and Po in Fig. 12) from a DEM with a square grid mesh of 3.333 m (i.e. 11.111 m2 per cell). These models represent ideally interpretable elementary forms and the affinity of real land segments to them expresses the applicability of an ideal geometric and genetic interpretation. Average volumetric divergence per unit area was determined, and the membership functions of real segments to classes of ideal elementary forms were computed as a criterion of affinity. This also documents the efficiency of a fuzzy approach to elementary forms. The results in Table 3 permit evaluation of the degree of affinity of these real segments to nine ideal elementary forms. Affinity cannot be greater for a higher-order model than for a component lower-order model. A relatively low affinity to higher-order and divergent forms can result from the computational unreliability of higherorder morphometric properties. The statistical significance of reductions in deviation is difficult to test, as the altitudes are positively autocorrelated; if they were not, meaningful derivatives could not be calculated. Slovinec (Sl) and Podhorské (Po) have the characteristics of degraded planation surfaces with relatively high

Table 3 Results of comparative fit analysis of models from Fig. 12 Model

0 — plain L1 L2 L3 C1 C2 C3 D1 D2

Slovinec (Sl: 8089 m2 = 28 × 26 = 728 cells)

Podhorske (Po: 12,000 m2 = 36 × 30 = 1080 cells)

Sandberg (Sa: 7200 m = 27 × 24 = 648 cells)

μ

Mf

μ

Mf

μ

Mf

1.02 m 0.75 m 0.66 m 0.66 m 0.43 m 0.43 m 0.40 m 0.95 m 0.82 m

0.59 0.70 0.76 0.76 0.83 0.83 0.84 0.62 0.67

2.30 m 0.33 m 0.33 m 0.32 m 0.28 m 0.27 m 0.26 m 0.27 m 0.26 m

0.54 0.93 0.93 0.94 0.94 0.95 0.95 0.95 0.95

4.94 m 0.79 m 0.68 m 0.66 m 0.68 m 0.64 m 0.64 m 0.88 m 0.81 m

0 0.80 0.83 0.84 0.83 0.84 0.84 0.78 0.80

Absolute mean deviation (μ) is computed from the volume difference between a ‘real segment’ (represented by DEM) and an ideal elementary form (computed by relations from Table 2), divided by segment area. Affinity of the real segment to an ideal elementary form model is expressed by the value of a membership function Mf defined by the relation Mf = 1 − 4μ / (o · tan γc), where o is mean length of the form in the direction of slopelines and γc is a critical angle for distinguishing plain and slope (steeper segments must not be considered as plains; tan γc = 0.20 in this case).

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affinity to horizontal planes. Slovinec is an exhumed erosion surface with more resistant limestone in the centre and high affinity to circular (domal) models: linear and divergent models are markedly less appropriate. Podhorské is a cryoplanation surface (a gently deformed glacis), and deviations for all non-linear models are adequately low. The good results for divergent models reflect the real tendency of lateral change of gradient conditioned by a diffuse lithologic boundary. The considerable difference in μ between linear and non-linear models is surprising, implying a high geometric homogeneity and a truly non-linear form. Sandberg (Sa) is a young fault slope with landslides on sand and sandstone. Hence it has the highest absolute deviations in spite of having the smallest area. All the Sandberg models show better fits for higher-order forms, confirming the non-linearity in profile that is characteristic for landslides. This worked example shows only some aspects of the implementation and use of the elementary form model. Future development of the methods suggested in Fig. 12 depends on fuller automation. Further work is needed on the improvement of computational accuracy for higherorder morphometric properties. 6. Discussion A major difference between this concept of elementary forms and some dynamically and genetically based approaches is the strict separation of elementary form delimitation and its subsequent dynamic and genetic characterization. This is necessary if we want to distinguish the presentation of geomorphological facts (morphology) from their interpretation (genesis, potential processes), so as to make geomorphological research more objective (Savigear, 1965; Urbánek, 1997). However, the ability to interpret elementary forms can influence the choice of geometric type of elementary form by which we approximate a concrete segment of the real land surface. The same principle may limit the choice of basic geometric types of elementary forms usable in geomorphology generally. 6.1. Exclusion of non-morphometric and positional variables from initial form definition Most concepts of land units are not confined to purely morphometric variables but are based on visual interpretation from field survey or air photos. In land system analysis, aerial images are the main tools for landform recognition by numerous researchers (Argialas, 1995; Chorowitz et al., 1995; Giles and Franklin,

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1998). Consideration of non-morphometric characters in images can lead to more detailed genetic distinction of geometrically similar landforms. But mixing various criteria in the process of segment delimitation is problematic without deeper theoretical reasoning, because the homogeneity of definition of geomorphic individuals may be violated. Our definition of elementary forms relies on the geometric signature only and can be considered a ‘pure digital approach’ (Brändli, 1996). Naturally, classification or internal division of elementary forms on the basis of various other criteria can bring interesting applications. The next question is whether we need to insert any positional (regional of Shary et al., 2002) variables into the segmentation algorithm. According to Brändli (1996), land units are not individually determined but are related to surrounding units (the context). Many other authors include regional variables in their concept of landform units (Dalrymple et al., 1968; Dymond et al., 1995; Schmidt and Dikau, 1999; MacMillan et al., 2000). The importance of regional variables for land classification is beyond doubt. But their mixing with local variables in algorithms for identification of boundaries can be problematic for segmentation. Basins (watersheds) are the most widely recognised landform units determined by a regional (integrating) approach. Shary et al. (2002, 2005) document the use of regional variables for delimitation of depressions, hills and saddles. The boundaries are slopelines and contour lines respectively. They can be singular lines but are not required to be. Their morphogenetic significance is therefore questionable (a flat water divide is a genetically homogeneous surface although it belongs to two different basins). Some regional variables (e.g. catchment area, altitude percentile, and ‘UPNESS’ of Summerell et al., 2005) cannot in principle be constant over any area. Although regional variables could be added to the set of form-defining properties, in principle they reflect broader dynamic and not local genetic relations. They should be used rather for determination of higher-order geomorphic units, or for classification of delimited elementary forms. 6.2. Attractors Cox (1978) pointed out that linear segments are only special cases of constantly curved elements; thus they cannot approximate reality better than elements. Shary (1995) regarded linear forms (planar, in 3-D) as rare, and he argued that their real occurrence (e.g. cells with exactly zero value of curvature) is minimal. Every form defined by a constant value of some morphometric property is

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only a special case of a more general form, and should thus be expected to be rare. This may seem to contradict the elementary forms concept, but if elementary forms are attractors, quasi- or relatively stable states to which development tends, they will be both more frequent and more interpretable. Apart from purely theoretical considerations, histograms of morphometric properties provide clear empirical evidence. As a constant value of a morphometric variable gives a zero value for derived variables, low values of a variable suggest affinity to a constant value of its parent variable. The global distribution of land altitudes and of gradients is asymmetric with a predominance of low values. Curvature values tend to peak strongly at zero, showing that linear slopes are an attractor. Appropriate statistical analysis could confirm the hypothesis that certain curved elementary forms are also attractors. This requires analysis not only of univariate distributions but also of the relations between morphometric properties, as presented by Evans and Cox (1999). However, inaccuracies in computing properties of higher orders may be a limiting factor. Understanding elementary forms as attractors can provide a bridge between analytical approaches to landform delimitation starting from mathematical analysis, and statistical approaches using mainly multivariate statistical methods. The characterization of topography is a problem requiring a statistical methodology (Evans, 1972; Pike, 1988). But statistical analysis can be more fruitful if based on preliminary knowledge or hypotheses. The land surface is not random or fractal, but it is characterized by the presence of many structures and patterns (Evans and McClean, 1995; Evans and Cox, 1999). The concept of elementary forms can express the ideal structure of the lowest hierarchical order of taxonomic complexity of landform (in the sense of Dikau, 1992), and so create a background for the selection and interpretation of statistical tests. The signature approach (Pike, 1988; Chorowitz et al., 1995; Giles and Franklin, 1998) usually starts from empirically defined examples, with further cases subsequently identified by their statistical similarity. Identifying composite forms directly from a statistical data set is hindered where structural information from the elementary level is missing. Preliminary identification of the elementary units (elementary forms), which can subsequently be clustered into composite forms (landforms), provides this information. Moreover, ideal elementary forms may themselves provide the examples sought as cores of taxonomical classes defined in advance. Supervised classification on the basis of agglomerative distance procedures with a spatial contiguity constraint can thus be used for delimitation of elementary forms.

Each ideal elementary form is characterized by zero values of one or more form-defining properties; hence low values signify affinity of a real surface segment to the relevant elementary form. Individual elementary forms can be defined by contiguous clustering from form-defining properties (or their derivatives). Integration of graph-based approaches with preliminary identification of discontinuities into a classification algorithm (e.g. Dymond et al., 1995; Giles and Franklin, 1998) is therefore promising. 6.3. Fuzzy classification The degree of affiliation of a real surface segment to an ideal elementary form can be expressed effectively by continuous (fuzzy) classification (Fig. 12; Table 3). This solves the classification problem of hierarchically inferior elementary forms; as a simpler elementary form is only a specific case of a more comprehensive higher-order form, it cannot approximate reality any better. The membership function expresses the affinity of a segment to various geometric types of elementary form. Continuous classification radically improves the interpretational potential of elementary forms. The fuzzy character of real terrain units is a consequence of the combination of various factors that have influenced them. A membership function can express the relative validity of ideal genetic and dynamic interpretations of individual elementary forms. Segments can be related to several interpretations; this should be adequate to represent the complexities of reality. Differences can arise from generalization, where the fitted function represents a larger form (such as a fault scarp) but the real surface also contains smaller forms (such as landslides or gullies). Alternatively, differences may reflect a transition toward another type of elementary form; for example the denuded periphery of a planation surface will differ more from an ideal horizontal plane than will the well preserved central part. The creation of ideal interpretation schemes for the individual types of elementary forms and their boundaries should be a next step, together with interpretations of various values of their defining properties. In reality, the boundaries of elementary forms also have a fuzzy character — sudden change of a boundarydefining variable is a geometric idealization. The derivative of a boundary-defining variable provides the basis for construction of a membership function for the boundary. The desired genetic and dynamic interpretation of boundaries of elementary forms is thus possible from a fuzzy approach based on geometrically ideal kinds of discontinuity.

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7. Conclusions The concept of elementary forms, introduced here, extends and integrates the theoretical background of land surface segmentation. Various types of area and boundary considered in previous approaches can be related to areas and boundaries defined in the concept of elementary forms (Fig. 3). Moreover the concept respects important properties of the land surface including the varying homogeneity of its segments and the different types of discontinuity at natural boundaries between geomorphic units. The set of interpretable types of elementary forms (Figs. 5 and 6) contains exactly defined elementary units as used by previous authors, but it also provides formal descriptions of models previously ignored or inadequately defined: concentric and divergent models as well as models of the third order. The dynamic and genetic interpretation of the geometry of an elementary form is important (Fig. 4). Except for the right-hand column, the system of elementary forms presented in Fig. 5 is based on forms characterized by parallel (linear, concentric or freely curved) contours — only these are generally stable geometrically and dynamically, and thus clearly interpretable genetically. Non-parallel contours represent the dominance of effects other than the gravitational principle of organisation of the earth's surface. This is consistent with the endeavour to combine geometric and dynamic principles (e.g. Dalrymple et al., 1968; Lastoczkin, 1987, 1991; Giles and Franklin, 1998). Fuller dynamic and genetic interpretations of combinations of form- and boundarydefining properties need additional work, as does definition of types given by field-independent variables. The concept of elementary forms exactly defines both area and boundary properties of basic landform segments that can serve as elementary geomorphic individuals. It enables expression of different ratios of the homogeneity and boundary contrasts of units, and of relations between ideal interpretations and real morphogenesis and morphodynamics. The concept can be applied not only to geomorphological mapping, but also to synthetic landscape mapping, evaluation of natural hazards, carrying capacity, susceptibility and so on. It should be a basic part of a modern DEM-based geomorphological information system (cf. Minár et al., 2005), unifying these topics at a more synthetic level of scientific knowledge. Acknowledgements This study was supported by the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and Slovak Academy of Science (1/1037/04

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