minar, evans - elementary forms for land surface segmentation -- the theoretical basis of terrain...

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

Jozef Minra,c,1, Ian S. Evans b,

aDepartment of Physical Geography and Geoecology, Comenius University in Bratislava, Mlynsk dolina, 842 15 Bratislava 4, Slovakia

b Department of Geography, Durham University, South Road, Durham City DH1 3LE, England, United Kingdomc 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 2007Available online 27 June 2007

Abstract

Land surface morphology is fundamental to geomorphological mapping and many GIS applications. Review and com-parison of various approaches to segmentation of the land surface reveals common features, and permits development of abroad theoretical basis for segmentation and for characterizat ion of segments and their boundaries. Within the context ofdefining landform units that maximise internal homogeneity and external differences, this paper introduces the concept ofelementary forms (segments, units) defined by constant values of fundamental morphometric properties and limited bydiscontinuities of the properties. The basic system of form-defining properties represents altitude and its derivatives, constantvalues 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 ofboundary discontinuity and their connections and transformations into other types of morphological unit boundaries areanalysed.

The wealth of types of elementary forms and their boundaries is potentially unbounded and thus is sufficient to coverthe real variety of landforms. Elementary forms in the basic set proposed here have clear potential for genetic and dynamicinterpretation. A brief worked example documents the possibility of analytical computation of various models of idealelementary forms for particular segments of landform. Ideal elementary forms can be considered as attractors, to which theaffinity of surface segments can be measured by multivariate statistical methods. The use of the concept of elementaryforms in landscape segmentation is promising and it could be adapted for elementary segmentation of various other spatialfields. 2007 Elsevier B.V. All rights reserved.

Keywords: Landform segmentation; Derivative; Altitude; Slope; Curvature; DEM

1. Introduction

1.1. Background

Geomorphological regionalization and mappingremain fundamental research methods of geomorphol-ogy and provide many promising applications (see e.g.

Available online at www.sciencedirect.com

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

Corresponding author. Tel.: +44 191 334 1877; fax: +44 191 3341801.

E-mail addresses: [email protected] (J. Minr),[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
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Cooke and Doornkamp, 1990; Evans, 1990; Voenlek,2000; Lee, 2001). Traditional geomorphological map-ping needs to adapt to challenges for greater precisionand objectivity within a GIS environment (Gustavssonet al., 2006). However, the theoretical base for defi-

nition, delineation and interpretation of mapping units isnot satisfactory. Only a few authors discuss the strictdefinition of landform segments and the minimisation ofsubjective factors in the segmentation process. An in-tuitive approach is usual (see e.g. Crofts, 1974; DeGraaff et al., 1987; Urbnek, 1997) and a theoreticalsynthesis is lacking. Yet the concept of a geomorpho-logical information system (Dikau, 1993; Minr 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 geo-morphological mapping and should be a prime aim ofgeomorphological mapping. Identification of elemen-tary landform units is important not only in the study ofpast and present geomorphic processes but also instudies of spatial aspects of interaction among land-forms, soil, vegetation, topoclimate and hydrologicalregime (Beruczashvili and Zuczkova, 1997; Giles andFranklin, 1998; Ventura and Irvin, 2000; Blaschke andStrobl, 2003).

Land surface segmentation is a kind of regional tax-onomy or regionalization that is generally treated as a

specific case of the general classification problem(Bezk, 1993). General classification theory is devel-oped mainly in mathematics (e.g. Shelah, 1990), infor-mation (library) sciences (e.g. Bowker and Star, 1999;Tennis, 2005) and philosophy (e.g. Hacking, 1999).Regardless of various measures of acceptance oremphasis on the subjectivity of classification (socialconstructivism), the stability of classificatory structurescan be accepted as a criterion of structural quality(cf. Hacking, 1999). This needs to be applied to landformsegmentation.

However, land surface segmentation is characterizedby a fundamental peculiarity arising from the continuityof the classified objects in space, i.e. the land surfaceforms a continuous field. Regional taxonomy generallysolves the spatial aspect of classification but avoids theproblem of definition of elementary geographical ob-jects in such a continuous reality (Bezk, 1993). Thebasic geomorphological goal of land surface segmenta-tion should be to distinguish segments (elements)that are homogeneous genetically and therefore alsomorphologically. Using a library analogy, we are notclassifying existing books but distinguishing individualbooks within continuous text. The polygenesis of many

landform segments, due to overlapping of geomorphicagents in time and space, renders the objective existenceof such books problematic: an extreme social con-structivist attitude would deny the objective existenceof forms. We recognise that both construction and

identification of forms are problematic: this accountsfor some of the subjectivity in geomorphological map-ping. Nevertheless there is a demand for land surfacesegmentation, which we address here.

1.2. Approach

Landform mapping is based on four principles: themorphologic, the genetic, the chronologic, and the dy-namic. The morphologic principle should be primary, inthe sense that defensible application of any of the other

three principles depends on an accurate appreciation ofmorphology (Speight, 1974). Thus geomorphologicalmap unit boundaries should generally follow morpho-logical boundaries (Lee, 2001).

Identification of the most specific geometrical (andsimultaneously genetic) geomorphic individuals is acentral objective here. Such individuals might be de-scribed as natural. Two major considerations in makingsuch an approach relatively objective are: identifica-tion of natural geomorphic boundaries, with maximalchange of genetic, geometric and process character(as attempted by intuitive traditional geomorphologi-

cal mapping); and specification of a clear algorithm forsurface segmentation, with a minimum of subjectivedecisions.

The geomorphological interpretability of landformsegments in terms of genesis, dynamics and chronologyis a major concern if they are to be seen as elementarygeomorphic individuals with the character of subsys-tems. The segments should have internal associationsstronger than external. The internal homogeneity andexternal contrasts of segments in terms of their geometryshould reflect their genesis and recent dynamics. Hence

the morphometric variables should be those used ingeomorphological theory and models.

Geomorphological theory defines genetically andgeometrically pure geomorphic individuals land-forms (such as alluvial fans, aeolian dunes and glacialcirques) and elements (such as cliffs, floors, slip facesand channels). To identify these in the landscape isa goal of land surface segmentation. We term thesetheoretical classificatory categories ideal models, con-trasted with the actual results of classification termedreal (landforms or elements). Land surface form (therelief of the Earth's surface) is characterized by acomplex structure of nested hierarchies (Dikau, 1992).

237J. Minr, I.S. Evans / Geomorphology 95 (2008) 236259


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Various analytical taxonomies of relief units reflect thisstructure (e.g. Crofts, 1974; Dikau, 1989; Dymond et al.,1995; Phipps, 2001; Wielemaker et al., 2001) and theyhave some common features. Three types of relief unitcan be distinguished on the basis of increasing com-

plexity. Elementary forms represent the smallest andsimplest units, which are indivisible at the resolu-tion considered. Their geometric simplicity (e.g. linearslope, curved slope or horizontal plain) facilitates theirrecognition as fundamental units in a system for landsurface segmentation. Some traditional landformsare single elementary forms, but most are compoundedfrom several elementary forms. Landforms that arecomposite forms create the second level of reliefcomplexity. Characteristic patterns created by formassociations provide a third level of complexity and are

termed land systems. They are equivalent to the reliefform associations, terrain systems, landform (land-scape, terrain) patterns or types of relief of other au-thors (Fig. 1).

Modern geotechnologies (remote sensing, geograph-ical information systems/GIS, global positioning sys-tems/GPS) have brought a huge methodological impetusto geomorphological mapping (e.g. Dikau, 1989;Voenlek, 2000; Phipps, 2001; Blaschke and Strobl,2003). On the other hand the theoretical basis of map-ping has changed little over decades (e.g. Cooke andDoornkamp, 1990; Lee, 2001). Despite the amount of

attention paid to the problem of segmentation of landsurfaces at detailed scales, the situation remains un-satisfactory. The definition of geomorphic objects belowthe level of sub-catchment (e.g. hillslopes, valley bot-toms and their components) is an unresolved funda-mental problem of theoretical geomorphometry and ofgeomorphological GIS analysis (Schmidt and Dikau,1999).

1.3. Continuity and discontinuity

We can identify three axioms, which create a the-oretical 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 asnatural boundaries of geomorphic objects.

3) These discontinuities and other characteristics of theland surface result from morphogenetic processesmost of which are influenced by gravity. Exploitationof geological contrasts and lineations often producesdiscontinuities.

The first axiom is basic for many modern geomor-phometric analyses (Evans, 1972; Krcho, 1973) and forthe universal representation of land surfaces in GIS asDEMs. Altitude (elevation) is evidently a continuouslandscape field (Burrough, 1996), yet in many cases theland surface cannot be considered smooth (Shary et al.,2005). Caves, overhangs and boulders are excluded ortreated separately. In several subject areas (e.g. geneticgeomorphology and geosystem analysis) an object rep-resentation is regarded as more appropriate (Minr,1995; Brown et al., 1998). Cox (1978) concisely termedthe first approach the continuous hypothesis (slope pro-

files are continuous curves without definite breaks) incontrast to the atomistic hypothesis (profiles as se-quences of components), where landscapes are es-sentially a mosaic of discrete units. We suggest that asynthesis of the two is desirable. Segmentation of theland surface can provide a transition from the field modelto the object model (Brndli, 1996), and from generalgeomorphometry to specific geomorphometry (Evans,

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

238 J. Minr, I.S. Evans / Geomorphology 95 (2008) 236259


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1972, 1987), thus connecting the continuous and at-omistic hypotheses.

Specific structural elements of the field provide anatural basis for its segmentation. We can term themsingular points and lines. These include for example

extremum points and lines (peak, pit and saddle points;ridge and valley lines), inflections and discontinuities ofthe altitude field and its derived fields (Lastoczkin,1987). As the land surface is in general locally con-tinuous, it contains only a finite number of singular lines.This contrasts with the infinite number of isolines andslopelines/streamlines. Defining singular lines as bound-aries of landform segments (elementary forms) thereforeprovides a good start in making land surface segmenta-tion as objective as possible. Next, the formal descriptionof segments by smooth mathematical functions may

bring considerable advantages to their identification andinterpretation.Landform discontinuities and the presence of defin-

able elementary forms are consequences of the spatialdifferentiation of past and recent geomorphic processes.Most processes are strongly influenced by gravity. Theconnections between elevation, gravity and geomorphicprocesses permit genetic interpretations of elementaryforms. Therefore the recognition of a physical-geometricanalytical system of basic types of elementary landformunit provides the skeleton of land surface segmentationtheory. A modern concept of land surface segmentation

should also take into account uncertainties in the iden-tification of discontinuities and in the degree of mem-bership 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 arisefrom a theoretical concept of geomorphic units. Both theinterior properties of ideal units and the character of theirboundaries have a crucial role in the definition of units.

Existing concepts have several common features, butthey differ in degree of explicitness, complexity, exact-ness and formal expression, and in theoretical back-ground or specific methods. Elementary geomorphicunits are generally seen as geometrically homogeneousparts of the land surface, indivisible from a geomor-phological point of view. Other types of homogeneity(genesis, age, contemporary processes, rocks and soils)have been treated variously.

Traditional geomorphological mapping was devel-oped predominantly on a morphogenetic basis (e.g.Demek, 1972; Spiridonov, 1975; De Graaff et al., 1987).Attempts to include the whole set of morphogenetically

relevant characteristics of landscape (character ofground, soil, surface material, and drainage) in theprocess of segmentation made formulation of a strictalgorithm difficult. Less attention was paid to themorphological character of the elementary relief units,

termed facets and segments, elementary forms, reliefelements, morphotopes (defined in geoecology as thesmallest homogeneous relief units), or geneticallyhomogeneous surfaces. Further approaches developedthat were more sophisticated and may be divided intograph-based and classificatory; their synthesis will bediscussed 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 seg-ment boundaries. Identification of a unit's boundary isthe primary goal and the character of the interior may notinfluence the determination of its limits. This can betermed the graph-based approach (Brndli, 1996).

Morphological mapping as defined in Britain byWaters (1958) and Savigear (1965) represents a typicalexample. More precisely, this is morphographic map-ping, based on form rather than process. This atomisticapproach started from the simple assumption that theground surface consists of planes bounded by morpho-logical discontinuities (Waters, 1958). A wider theoret-

ical background appears in the system of morphologicalunits and their boundaries presented by Savigear (1965),which has been widely used with only small modifica-tions (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 andsegments join atdiscontinuities: breaks of slope, changesof slope and inflections. Microfacets (microsegmentsthat are very narrow in relation to the map scale, and arethus depicted with linear symbols) represent a transition

between areal morphological units and linear bound-aries. Savigear's boundaries have a singular character,but they do not guarantee the full demarcation of mor-phological units. To complete the separation of morpho-logical units, Young (1972) suggested use of boundariesdefined by permitted ranges of mean slope angle forfacets, or of mean curvature for elements (curved units;Young, 1972, p. 182). They may thus be isolines of slopeor curvature; unfortunately, their arbitrary nature reducesthe objectivity of segmentation.

Although morphological units are defined geometri-cally, their genetic and dynamic interpretation is possibleand desirable, as for example in the hypothetical nine



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unit land surface model for slope profiles (Dalrympleet al., 1968). Morphological units are classified into ninedynamically interpreted units, in sequence downslope asa catena. This model describes each of the nine units byposition, slope, profile curvature, actual processes,

significant microforms and the specific characteristicsof rock, soil and vegetation. Unlike morphologicalmapping, however, units are delimited also by specifi-cation of process transitions at the boundaries. Parsons(1988) recognised that confusion of morphological andmorphodynamic criteria is a problem.

A more comprehensive and theoretical applicationof the graph-based approach is the segmentation modelof Lastoczkin (1987, 1991). He introduced the set ofstructural lines and characteristic points defining ele-mentary surfaces. The basic structural lines are sin-

gularities representing (on a profile) local extremes ofaltitude z (ridge lines and valley lines); local extremesof slope gradient z, the first derivative of altitude(inflection lines of maximum or minimum slope);and local extremes of profile curvature z, the secondderivative of altitude (convex and concave flexuresinvolving breaks and changes of slope). Structurallines thus represent various kinds of discontinuities ofmorphometric properties (Fig. 2). The basic structurallines are further classified on the basis of the linear,convex and concave shape of the profiles on eitherside.

Characteristic points include peaks and pits as well asthe ends or junctions of structural lines. Lastoczkin'ssystem deepens the theoretical background of morpho-logical mapping, but its initial form was too focused onthe behaviour of altitude in the direction of slopelines,which is not sufficient for full delineation of elementary

forms in three dimensions. Like Young (1972), Las-toczkin (1987) therefore introduced an extra (asystemic)factor in the bounding of elementary surfaces sidelimitation, represented by slopelines or cross-lineswhich do not have a singular character: an infinite num-

ber is possible. His classification of the main geometrictypes of elementary surfaces remained similar to that ofmorphological mapping. Plains and slopes linear,convex and concave in profile are classified fur-ther only according to their positional properties. Averyimportant improvement is made in Lastoczkin's (1991)later work where he included the concept of plan cur-vature, and added its zero isoline to structural lines. Thisopened the way to a more comprehensive perception ofland surface segmentation.

2.2. Classification approach

A second major line of thought in land surface seg-mentation is the classification approach of Brndli(1996). This focuses on definition of the internalproperties of elementary forms, from which the defini-tion of boundaries follows. Geometric forms or ele-mentary forms defined on the basis of their curvatures(Richter, 1962; Troeh, 1965; Young, 1972; Krcho, 1973;Dikau, 1989; Shary, 1995) provide an example ofsupervised classification; the range of each property ispre-ordained. The importance of both profile and plan

curvature for earth surface processes has been recognisedat least since Aandahl (1948). The simplest model is a22 classification based on the signs of profile and plancurvature, giving four basic forms (Table 1; Troeh,1965). These basic geometric forms have an importantdynamic interpretation in terms of gravity flows, with

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



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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 dis-tinguished straight slopes, in profile and in plan, giving a3 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 of55 (+1) classes.

The introduction of straight forms, with linear con-tours and constant gradient, where zero isolines of profileor plan curvature degenerate into areas (Krcho, 2001),implicitly requires new threshold lines separating thesefrom curved forms. These are modelled as disconti-nuities of curvature or of change of curvature. As straightforms are, however, defined by one or more non-zerovalues 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 authorsdefined straight as having a radius of curvature greaterthan 600 m, so those (+ and ) isolines delineate linearforms. This problem of arbitrary choice extends to themore comprehensive systems with further types ofcurvature introduced by Shary (1995) and Krcho (2001).

Maps of geometric forms are usually created byoverlay of maps of zero or other isolines of curvatures.When threshold values are applied on the wider set ofmorphometric properties (altitude, slope angle, aspect,curvatures, etc.), landform segments defined by the set ofmorphometric quantities are produced (e.g. Speight,

1968; Dikau and Jger, 1995; Pennock and Corre, 2001).Krcho (1983) offered a fully formalised expression ofthese units and termed them morphotopes. Bolongaro-Crevenna et al. (2005) used thresholds (tolerancevalues) 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 theclearly defined homogeneity of units and the simplepractical interpretation. However, as Reuter et al. (2006)show, this method leads to strong scale dependence ofresults that can be eliminated only partially by opti-misation (scale dependent shifting of threshold values).Moreover as isolines may pass through areas of verygradual change, isoline boundaries may create artificialareas without sufficient respect to the natural structure oflandform units with various types of homogeneity. Use

of such artificial boundaries will hinder interpretationand confuse the analysis of natural structures. The use ofmean values of important variables as threshold values,as in the recent innovative approach by Iwahashi andPike (2007), eliminates the subjectivity of thresholddefinition, but the main problem of arbitrary incidenceremains.

The use of numerical taxonomy (cluster analysis) inthe framework of a parametric approach (Speight,1974) attempts to overcome this artificial character andpreserve the synthetic nature of the landform unitsdefined by a set of properties. This statistical approach,

unsupervised classification, obtained a big impetus fromthe adoption of Digital Elevation Models (DEMs) andGIS facilities, and it is still the most widespread tool forland surface segmentation. In the most popular type ofcluster analysis, the initial areal units (cells) are grouped(clustered) on the basis of their similarity, that is, theirseparation in attribute space. These clustering algorithmsminimise intraclass and maximise interclass differences,in accordance with the general requirement for landsurface segmentation into elements. However, thisapproach has several problematic aspects related to sub-

jective choice of procedures: input parameters, cluster-ing methods and interpretation of results. The majority ofauthors (e.g. Speight, 1974; Irvin et al., 1997; Venturaand Irvin, 2000; Blaschke and Strobl, 2003; Adediranet al., 2004) divide the land surface into territoriallydiscontinuous classes types of landform element each of which can include numerous spatially discretelandform elements. This raises a fundamental problem:the classification is based on thematic similarity alone,ignoring position. The minimisation of intraclass andmaximisation of interclass differences do not relate toindividual landform elements but to whole classes, scat-tered across the map. Local homogeneity becomes less

Table 1Local slope forms classified by curvature, profile first (varies downcolumn), then plan (contour; varies along row), for the scheme ofRichter (1962) [also Krcho (1983) and Dikau (1989)], that of Troeh(1965) and that proposed here in Fig. 5

Troeh (1965) Richter (1962) etc.

VX VV VX VS VVXX XV SX SS SV

XX XS XV

Proposed (in Fig. 5)

Contour shapes

Planar Circular convex Circular concave Clothoid Divergent

SS SX SV S vx/xv SSVS VX VV V vx/xv VSXS XX XV X vx/xv XSvxS vxX vxV vx vx/xv vxSxvS xvX xvV xv vx/xv 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.



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likely as the number of landform elements in a classincreases.

Definition of individual spatially continuous regionsby a contiguity criterion can be a solution. This producesrelatively homogeneous areas bounded by lines where

the homogeneity is disrupted. Friedrich (1998) intro-duced distance grouping methods for deriving homoge-neous relief units by coupling agglomerative procedureswith a spatial neighbourhood analysis. Romstad (2001)demonstrated the usefulness of the approach very con-vincingly by his method of relief classification withcontextual merging, and a similar principle was used byBonk (2002) for delineation ofterrain form objects byspatial grouping on the basis of local contiguity. How-ever, cell- (pixel-) oriented approaches are limited bytheir mesh-dependence and neglect of neighbourhood

relationships.

The shift from per-pixel-based to object-based analysis requires a shift from pixels havingmeaning to user-defined objects having meaning(Drguand Blaschke, 2006, p. 333).

Cluster analysis permits the use of a large number ofinput parameters, which should provide more syntheticresults. Speight (1974) used 21 morphometric propertiesas a result of an intuitive selection of those aspects ofgeometry of terrain that appear relevant either to land useor to geomorphological process. Other authors havealso included non-morphometric properties, giving areaswhich are relatively homogeneous not only geometri-

cally, but also from the point of view of land cover, landuse, and soil (e.g. Irvin et al., 1997; Giles and Franklin,1998; Blaschke and Strobl, 2003). However, an increasein the number and variety of input parameters can hinderclear geomorphological interpretation of delineatedlandform elements; also it prevents assessment of howclosely landforms relate to soil or land cover.

When geomorphic types are not defined prior tomapping, 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 beforesegmentation. The specific geomorphometry of Evans(1972, 1987) offers a solution. Statistical characteristicsof morphometric properties of a known set of landformscan lead to the identification of other landforms of agiven type. This can be termed the signature approach.Pike (1988) demonstrated the effectiveness of thisapproach for identification of landslide-prone areas.Within this approach Chorowitz et al. (1995) distin-guished parametric analysis (centred on the identifica-tion of boundaries of landslides from contour geometry),stochastic analysis (on the basis of local fractal di-mension and variability, in combination with binary

geological information) and structural syntactic analysis(identification of landforms as a series of more ele-mentary slope patterns). Giles and Franklin (1998) useddiscriminant analysis for the identification of ten types ofslope unit defined a priori, in combination with breaks of

slope gradient as unit boundaries. A specific structured-knowledge modelwas developed by Argialas (1995): anexpert system permits recognition and delineation of alandform from user-supplied pattern elements. Recentlyvan Asselen and Seijmonsbergen (2006) used object-oriented classification by eCognition Professional forrecognition of eight genetic landforms using only slope,altitude and contributory area. Prima et al. (2006) used asignature approach and tested which morphometric var-iables distinguished defined landforms such as volca-noes and alluvial plains.

All earlier approaches to land surface segmentationproduced crisp classifications of land form. Recentlycontinuous (fuzzy) classification has frequently beenapplied within the cluster analysis approach (e.g. Brndli,1996; Irvin et al., 1997; MacMillan et al., 2000; Blaschkeand Strobl, 2003; Schmidt and Hewitt, 2004). This is anew way of reflecting the complex variation of landscapecharacter in space. Internal homogeneity of elementarylandform units is a basic aim of most approaches to landsurface segmentation. However, if within-object homo-geneity of mapped units is considered a poor or evenunrealistic assumption (Burrough, 1996), the fuzzy

approach can be an ideal tool for expressing the affiliationof a real landform segment to any ideal type of elementarylandform. Likewise it expresses the diffuse nature ofmany boundaries of elementary landform units.

A more specific model approach to classificationapproximates the shapes of individual landform seg-ments with three-dimensional equations (Troeh, 1965;Young, 1972; Schmidt et al., 2003). Landform repre-sentation by fitted functions is widely used in modernmorphometric analysis. Fitted functions (of one family)represent regular segments of land surface, usually a sub-

matrix of the square network of the DEM, and they canbe used as interpolation functions. But every elementaryform can be represented by a specific fitted function (seeParsons, 1988). The characters of fitted functionsdescribing adjacent segments can determine the char-acter of their boundary (Minr, 1998), as discussed inSection 4.

Revived interest in problems of land surface seg-mentation is connected with the recent development ofGIS techniques. Use of GIS technology stresses theimportant requirement for more detailed and explicitformalisation of the terrain analysis process (Argialas,1995, p. 104). Unfortunately, most authors focus on the



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methodological aspect (automation) at the expenseof the conceptual background of segmentation. Manyauthors prefer more readily automated concepts, whichare not necessarily the best. Papers often represent a newlook at old concepts from the point of view of their

feasibility in the GIS environment. Some algorithmscombine principles mentioned earlier (Dymond et al.,1995; Giles and Franklin, 1998; MacMillan et al., 2000;Drgu and Blaschke, 2006), but a general theoreticalunification of the various concepts is still needed.

3. A new system: the concept of elementary forms

Existing approaches to land surface segmentationdefine boundaries and surfaces of elementary landformunits in various ways. Connections among them are

expressed in summary form on Fig. 3. Boundaries ide-ally sensed as lines of discontinuity of some characters,and homogeneity ideally defined by uniformity ofrelevant characters, can provide the link between sta-tistical approaches and basic analytical concepts. Suchan approach is developed in the integrating concept ofelementary forms (Minr, 1992, 1998).

3.1. The model proposed

The concept is based on geometric and physicalaspects of field theory (Krcho, 1973). Land surface form

is considered here as a geometric field of altitudes that isdirectly linked with the gravitational field (Devdariani,1967; Shary, 1995). Altitude and secondary morpho-

metric variables (defined by derivatives of altitude invarious directions) have both geometric and physicalsignificance. Homogeneity of genetic influence is there-fore reflected by morphometric homogeneity (uniformity)of landform, and change of genetic influence is connected

with morphometric discontinuity. Consequently, constantvalues of some morphometric characters define the area ofan ideal elementary form, and discontinuities define itsboundaries. The basic morphometric system introducedby Evans (1972) and Krcho (1973) can be extendedformally in the form (Minr, 1999):

M 0M zf g;1M zif g;2M zij

;3M zijk

; N

n o1

where M is the set of all local morphometric variables

(definable at every point of the land

cf. Shary et al.,2002). (0)Mis a subset ofMcontaining only the elementaltitude (z), an initial quantity that has the lowest order(zero). (1)Mis the subset ofmorphometric variables (zi) ofthefirst order, defined by the first directional derivative ofaltitude in direction i. (2)Mis the subset ofmorphometricvariables (zij) of the second order, defined by thedirectional derivative of quantities of subset (1)M indirectionj, and so on. Included in subset(1)Mare not onlyslope gradient and aspect (direction), but also any othervariable (apparent slope) defined by first derivatives ofaltitude in any direction. Subset (2)M includes various

curvatures (profile, plan, tangential and Gaussian:Schmidt et al., 2003) but also any other variables definedby 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 variablesexpressing change of curvatures, and so on.

The set defined by Eq. (1) contains an infinitenumber of variables, but some are more significant froma geomorphological point of view. The vector field

of slopes determines the gravitational flow of matterand energy on the earth's surface. Therefore variablesdefined by change of altitude in the normal direction ofslopelines (direction of maximum slope gradient), or bythe complementary tangent direction (direction ofcontour lines), have a special importance in geomor-phology, which all geomorphometric systems respect(e.g. Evans, 1972; Krcho, 1973; Moore et al., 1991;Jeno, 1992; Pike, 2000; Krcho, 2001). Shary (1995)terms these morphometric variables field-specific (incontrast to field-invariant).

Maximal homogeneity of field-specific variables with-in segments is a generally accepted postulate of ele-mentary segmentation. A constant value of field intensityis the highest possible degree of homogeneity; it is thefinal state of equilibrium of the field. In the case of the landsurface it is represented by a constant value of altitude;horizontal planes thus defined are the most readily inter-pretable elementary forms (approximated by accumula-tion plains and surfaces of planation). But the majority ofthe earth's surface is in slopes with non-zero gradients. Aconstant 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 (slopegradient and aspect: Evans, 1972). A constant value ofboth gradient and aspect defines a planar facet, the mosthomogeneous linear slope (e.g. a fault slope, or anabrasion slope). Steadychanges of slope gradient or aspectare 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 thisrespect also. In general, homogeneity decreases with theincreasing order of derivatives that are constant.

Fig. 4 shows the mutual relations and basic geo-morphological importance of the field-specific var-iables, altitude and its first three derivatives. Thegeomorphological interpretation of uniformity of avariable stems from geomorphological theory, as givenin particular process models (e.g. Scheidegger, 1970;Carson and Kirkby, 1972; Rohdenburg, 1989; Mooreet al., 1991; Lastoczkin, 1991; Mitov and Hofierka,1993; Minr, 1995). Many variables have related equiv-alents (radius of plan curvature and plan curvature,normal gradient change and profile curvature) that arewidely used. Variables of the third order have not gen-erally been used. However, the scheme at the bottom of

Fig. 4 explains possible long-term dynamic stability ofthe constant values. Erosional and depositional effectsgenerally depend on change of energy and mass ofagents. These depend on profile and tangential curva-tures (cf. Scheidegger, 1970; Carson and Kirkby, 1972;

Mitov 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) (lowerright of Fig. 4). Hence radius of plan curvature andnormal gradient change are highly significant, togetherwith their derivatives. The variables in Fig. 4 thereforedefine the basic system of elementary forms suggestedhere. Minr (1999) published equations defining thesevariables, most of which were derived earlier by variousauthors (e.g. Evans, 1972; Krcho, 1973; Jeno, 1992;

Shary, 1995). Planar slopes (including horizontal andvertical planes) have been used in land surface seg-mentation for a long time. The interval determination ofcurvature for individual segments by Young (1972), andthe postulation of homogeneous plan and profile curva-ture of form elements by Dikau (1989), also point to therepresentation of curved slope elements by constantvalues of second-order morphometric variables.

Parsons (1977) expressed a similar idea for profilerepresentation, with criteria of constant inclination, con-stant curvature and constant change of curvature definingsimple components of a slope profile. Theoretically we

can model land surface form with elements defined byconstant values of morphometric properties within the fullsetM. The use of more types of element allows a betterapproximation of reality, but their interpretability is alimitation. As yet, properties up to the third order seem tobe interpretable and thus effectively usable now, but thesystem is open to further development.

3.2. Elementary forms

We can generalize the proposed model as follows.

Land surface form consists of segments characterized byvarious types and degrees of homogeneity. These canideally be expressed by constant values of altitude or itsderived morphometric properties. Discontinuities of theseproperties provide logical boundaries to the segments.Then we can define ideal elementary forms as landformelements with a constant value of altitude, or of two or

more readily interpretable morphometric variables,

bounded by lines of discontinuity. The constant valueswhich define the elementary form are termed the form-defining properties.

Elementary forms are individual units of the landsurface, which can be classified into typological systems



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



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on the basis of various criteria (geometric, dynamic,topological and genetic). Separation by type (but notsign or value) of form-defining properties is fundamen-tal and defines the basic geometric types of elementaryforms. The number of basic types is limited by the

compatibility of form-defining properties. For exampleconstant aspect is compatible with constant (non-zero)profile curvature or change of curvature, but constantslope gradient is not compatible with constant non-zeroplan or tangential curvature.

Distinguishing forms defined by positive andnegative values of form-defining properties has a specialand universal significance (Fig. 4). Therefore, within thebasic geometric types, basic geometric subtypes ofelementary forms can be distinguished by positive andnegative values of form-defining variables. The number

of basic subtypes rises with the order of elementaryform. Basic subtypes can be defined not only by thesigns of form-defining variables, but also by signs of allrelated variables of lower order; for example, the sub-types of a linear slope can be terrestrial or marine, andregular or inverse (overhangs: in theory, although theseare not represented in DEMs).

Only local, not regional variables are used here forthe definition and delimitation of elementary forms.Regional variables in the sense ofShary et al. (2002) areproperties defined by spatial relationships, e.g. open-ness, flow path length, upslope contributing area and

dispersal area. They are not used in the definition ofgeomorphological individuals because they describerelations beyond a single elementary form and areincompatible with the condition of spatial uniformity.However, regional variables should be used forcharacterization and further classification of elementaryforms, after they have been defined. Types of elemen-tary form that take position into account may be termedland elements, afterSchmidt and Hewitt (2004).

Relations between form-defining properties deter-mine relations among basic types of elementary form.

Every property(r)

zi of order r (r1) is defined by thederivative of some property (r 1)zj of order r 1. Theproperty (r 1)zj can thus be termed a parent property ofproperty (r)zi and the following relation exists betweenconstant values of the parent and derived properties:

r1zj const Zrzj 0: 2

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

constant (see the last column ofFig. 5). Forms share theproperties of lower-order forms only on specific linessuch 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 higherorder. For rectilinear segments and curved elements ofprofile this was already stated by Cox (1978). Con-sequently, a mathematical expression defining a formwith a constant value of the property (r 1)zj is only aspecific case of that defining a form with a constantvalue of the property (r)zi (see Fig. 5). Thus a basicsystem of elementary forms defined by only one fun-damental equation can be created.

3.3. Equations for ideal elementary forms

Fig. 5 represents the proposed system, based on thethree principles:

1) The functional dependence of altitude (z) of a pointwith map coordinates (x, y) on distance from anystart point with altitude H and coordinates (I, J):

z H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia x I 2b y J 2

q3

where a and b are constants and (H, I, J) can becoordinates 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 parallelcontours. All contours are either concentric, or parallelto the defined starting contour (linear, or parallelcurves that are represented here by clothoidmodels).Parallel contours generally determine a constant valueof the normal change of radius of contoursRn on everyindividual slopeline (i.e. Rnn=0).2) A polynomial function of i-th order represents aconstant value of the i-th derivative of altitude in

the normal direction. Then the function:

z H Bn Cn2 Dn3 4

where (x,y) expresses the shape of contours, gen-erally describes forms with constant normal changeof downslope curvature (Gnn) in models with parallelcontours.3) Forms with divergent contours can be defined byfunctional dependence of altitude on angular coeffi-cient:

z H F arctan

y q

x p 5



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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 columnof 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 interpretability. From left to right, the columns represent linear, circular positive, circular negative, parallel contour (clothoid) and divergent elementary formDouble columns or rows distinguish basic subtypes, convex and concave. From the top down, the rows are straight (S), constantly curved (V conca

profiles of slopelines (vx concaveconvex, xv convexconcave). Because of low interpretability, horizontally curved variants of divergent mode


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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 toexpress various basic types and subtypes of elementaryforms by one general function. Specific values ofconstants and variables simplify the function into shapesdescribing thirteen basic types of elementary forms(boxes in Fig. 5), and dozens of subtypes.

The proposed concept of elementary forms is inprinciple independent of scale, and the methods pro-posed here can be applied to data at any resolution.Multi-scale landform characterization (Schmidt and

Andrew, 2005) could provide a basis for application ofthe concept at various scales. Many aspects of the landsurface, however, are scale-specific (Evans, 2003), andthe value of applying this approach at broader scalesrequires further investigation.

The definition of elementary forms enables us (inharmony with Troeh, 1965) to represent every form by a

specific mathematical function (fitted function). The sys-

tem includes and specifies in more detail the various typesof morphological units (Savigear, 1965), elementary sur-faces (Lastoczkin, 1987), geometric forms (Troeh, 1965;Krcho, 1973) and form facets or form elements (Dikau,1989). A different typology of elementary forms coulddefine types by interval values of morphometric proper-ties (analogous to the morphotopes ofKrcho, 1983),orbyelements of such models as the hypotheticalnine unit landsurface model (Dalrymple et al., 1968). The system ofelementary forms is fully open; adding to or modifying itdepends only on our ability to formally express andeffectively interpret more complex (higher order) types of

elementary forms. Forms characterized by homogeneousderivatives of altitude in non-standard directions (Fig. 6)provide an example. They are analogous to the flat (lowgradient) elements ofSchmidt and Hewitt (2004)becausethey are not defined by gravitational field-specific var-iables (Shary, 1995). They can be formed by the dominantinfluence of geological structure or of a directional

Fig. 6. Selected elementary forms defined partially or fully by a constant value of field-specific morphometric variables dependent on non-gravitational geomorphological fields (e.g. morphotectonic) one or two derivatives of altitude (z) are constant in the direction of axis of anisotropyof 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 ofy, zxx, zxxx second and third derivative of altitude in direction of x, R radius of plan curvature, H, B, m constants.

Fig. 7. Basic discontinuity lines.



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geomorphic agent, giving an axis of form symmetry. Thussimilar expressions of elementary forms by fitted func-tions (e.g. the polynomial models ofSchmidt 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 withconstant values of specified morphometric properties hasstrict theoretical consequences for the demarcation ofelementary landform units. In the geometrically idealcase, the boundary of two elementary forms defined bydifferent constant values of some morphometric proper-ties must be a line of discontinuity a sudden discrete

change of value of some property (Fig. 7).

The property value which jumps at the boundary canbe termed the boundary-defining property. If the line ischaracterized by discontinuity of only one property, wecall it a simple discontinuity line; otherwise, it is acompound discontinuity line. The character of a

boundary-defining property is limited by the characterof the form-defining properties of adjacent forms, whichare defined in the same direction as a boundary-definingproperty. If a defining property (r)zi of one form isderived from (or identical with) a defining property (s)ziof an adjacent form, then for the order of the boundary-defining property (t)zi we have:

tV rzs: 6

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 bothsides, it cannot be a discontinuity of curvature or changeof curvature. Identity of the boundary-defining propertywith a form-defining property of higher order ((r)zi=

(t)zi)is the most stable situation, which we term a form-

determined boundary (Fig. 8). Constant values of aboundary-defining property within each adjacent form arecharacteristic of form-determined boundaries.

Every line of discontinuity has an interpretationrelated to the constant values of boundary-definingproperties (Fig. 4). The importance of a discontinuityline rises with the value of discontinuity. Differences inthe values of boundary-defining properties on both sidesof a boundary give the specific sharpness of boundary.In the case of slope we can separate aspect and gradientsharpness; alternatively we can define a 3-D synthetic

sharpness as the angular difference between slope vec-tors. Depending on the number of boundary-definingproperties, the boundary can be characterized by one ormore specific sharpnesses. Further properties such as thehorizontal extent of adjacent forms determine the spatialimportance of a boundary, and the genetic interpretationof a boundary can be derived from the genesis of adjacentforms. Unlike the rule concerning constant values of

parent properties determining constant values of derivedproperties (Fig. 4), discontinuity of parent propertiesdoes not underlie discontinuity of derived properties;altitude discontinuity does not determine slope or cur-vature discontinuity.

Equations describing forms of the third order inspecific conditions (b N a+ c and sign b sign cfor a part of the domain of definition) give specificsolutions. Three elementary forms that are subtypes ofthe same basic type can be distinguished in terms ofaspect (AN) and normal change of contour curvature(Rn). In Fig. 9, discontinuities are regular boundaries ofelementary forms, so one equation can describe threeelementary forms. But the inflection line is not a dis-continuity and so it should not be the boundary of anelementary form; an homogeneous convexconcave

element extends on both sides of the inflection.

4.2. Contrast, smoothness and resolution

There are further relations between discontinuitylines and other singular lines. Discontinuity is an idealgeometric category and its relevance to real terraindepends 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

constant value of normal change of radius of plan curvature (Rn), resulting from the equation: z 350ffiffiffiffiffiffiffiffiffiffiffi

x2y2p

40 x2 y2 x2 y2 3=2

.



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a discontinuity of one property into an extremum line(maximum or minimum) of another property. The modelof land surface form with elementary forms describedby individual fitted functions is termed the model of landform contrast. It consists of homogeneous elementswith distinct heterogeneity across boundaries. Thesmooth landform model (represented ideally by only

one smooth mathematical interpolation function describ-ing the modelled area) is an alternative, sometimes usedin geomorphometry for computation of morphometricproperties.

A discontinuity line of property (r)zi in the contrastmodel is transformed into an extremum line of derivedproperties (r+ 1)zi and the zero isoline of property

(r+ 2)ziin the smooth model. That means the local extremesidentified by regular morphometric analysis (and thusthe zero isolines of a derived property) can be theboundaries sought for elementary forms. However, the

smooth model also generates at least one extremum line(the zero isoline) within each elementary form, whichmay be interpolated as being central although the wholeform has a zero value of the property. Horizontal planes

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.

Fig. 11. A brief algorithm for delimitation of elementary forms. Protoforms represent preliminary delimited units that may or may not be elementaryforms. 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 Devnska Kobyla Mountain, Slovakia. Eight models, D1L3 arefitted 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 ofthe form. Results are given in Table 3. Contour interval 5 m. For comparison the corresponding section of the geomorphological map ofMinr andMiian (2002) is displayed.



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will thus be divided by zero isolines of slope, linearslopes by zero isolines of curvature, and so on. These

misleading zero isolines (those not corresponding todiscontinuities in the contrast model) are of lower orderthan transformed discontinuities (zero isolines which docorrespond to discontinuity lines). The number of lineshas a tendency to rise with the order of the elementaryform (Fig. 10), complicating the situation.

Another type of boundary transformation connectedwith change of resolution (Fig. 10C) is where discon-tinuity lines change into elementary forms when viewedin greater detail. This arises because, at some dis-tinguishing level, every discontinuity can be consideredas a microsegment in the sense of Savigear (1965).Some form-defining properties of the microsegment

elementary form are derivatives of boundary-definingproperties 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 formwith constant altitude, discontinuity of gradient by aform with constant gradient, and so on.

5. A worked example

We introduced a theoretical concept of land surfacesegmentation that on the one hand reflects the intuitiveapproach of geomorphological mapping, yet on the otherhand increases objectivity by the development of precisealgorithms. Although the problems of internal homoge-neity and boundary discreteness were considered

Table 2Derivation 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

Normalorder

L linear models C circular models D divergent models

n a b gx hy

c gx hy 2

d gx hy 3

n a b ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix I 2 y J 2qc x I 2 y J 2

h id x I 2 y J 2

h i3=2n a b arctan

y q

x pc arctan

y q

x p

2

darctany q

x p

3

a =meanzmeanz(a=0)

1st g medzx

h medzy

b

medzAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 h 2

p 1

I

med

x jRjd tan ANffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan 2AN 1

p !

J F

med

yF

jRjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitan 2AN 1

pd

b FmeanG;

p

med

xFjRRjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tan 2 90 AN 1p

!

q

med

y RR tan ANffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

tan 2 90 AN 1p

b medG RR

2ndc

medzAA

2 g2 h2

b medzA

medzA b0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g2 h 2p

c medGn

2

b medG medG b0

c

medGn R

2R

2

b medG RR med

G b0 RR

3rd d medzAAA

6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g2 h23q

c medzAA

medzAA c0

2 g2 h2

d medGnn6

c medGn

medGn c0

2

d med Gnn R3R6

c

med

Gn R2R

med Gn c0 R2R2

x, y, z are map coordinates of points of real segment (z=altitude), zx, zy are partial derivatives ofz=f(x,y), G gradient (tangent), Gn normalchange 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 forcomputation of the models in Fig. 12.



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separately, effective application requires both to bejoined in the process of delimitation. This can be aniterative process in which the changing degree of affinityof real segments to the given models is measured forvarious degrees of boundary discontinuity (Fig. 11). In

practice, the most distinctive discontinuities are definedfirst, then the fit of expressions from Section 3.3 ismeasured. If none fit adequately, it may be necessary tosubdivide into areas bounded by less distinctivediscontinuities. Alternatively, a form that has not reachedthe required threshold of homogeneity may be acceptedas an exception; e.g. a landslide area composed ofmicroforms on a finer scale. Rarely, as a last resort, a newkind 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 thegeomorphological map of Devnska Kobyla Mountain,Slovakia (Minr and Miian, 2002) is interpreted interms of the elementary form model. Field research andvisual analysis of a DEM were used for the delimitationof elementary forms.

The boundaries fit with lines traditionally used ingeomorphological mapping, interpreted here as differenttypes of discontinuity line. Examination of the geometriccharacter of areas is more straightforward. For the com-putation of constants in specific equations for selectedtypes of elementary forms (Table 2), we have started from

average values of key morphometric properties. Proper-ties of first order are completely accurate, those of secondorder are approximate, and properties of third order arecurrently uncertain because of problems of computationof derivatives of higher order (see e.g. Krcho, 2001).Multiplication of properties has a similar effect, influenc-

ing computation of the divergent models: hence theiterative computation of constants b and c is based onapproximation to median values of gradient and gradientchange.

The third order divergent model (D3) was not used

becauseof these complications in the computation of thirdorder properties. Clothoid-based models (the penultimatecolumn in Fig. 5) were also excluded, considering similarproblems and their lower interpretation value. Includingthe 0-model (horizontal plane), nine models of geomet-rically ideal elementary forms were computed for thecentral parts of three elementary forms (Sl, Sa and Po inFig. 12) from a DEM with a square grid mesh of 3.333 m(i.e. 11.111 m2 per cell). These models represent ideallyinterpretable elementary forms and the affinity of real landsegments to them expresses the applicability of an ideal

geometric and genetic interpretation. Average volumetricdivergence per unit area was determined, and themembership functions of real segments to classes ofideal elementary forms were computed as a criterion ofaffinity. This also documents the efficiency of a fuzzyapproach to elementary forms.

The results in Table 3 permit evaluation of the degreeof affinity of these real segments to nine ideal elementaryforms. Affinity cannot be greater for a higher-ordermodel than for a component lower-order model. A rela-tively low affinity to higher-order and divergent formscan result from the computational unreliability of higher-

order morphometric properties. The statistical signifi-cance of reductions in deviation is difficult to test, as thealtitudes are positively autocorrelated; if they were not,meaningful derivatives could not be calculated.

Slovinec (Sl) and Podhorsk (Po) have the character-istics of degraded planation surfaces with relatively high

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

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

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

Sandberg(Sa: 7200 m=2724=648 cells)

Mf Mf Mf

0 plain 1.02 m 0.59 2.30 m 0.54 4.94 m 0L1 0.75 m 0.70 0.33 m 0.93 0.79 m 0.80L2 0.66 m 0.76 0.33 m 0.93 0.68 m 0.83L3 0.66 m 0.76 0.32 m 0.94 0.66 m 0.84C1 0.43 m 0.83 0.28 m 0.94 0.68 m 0.83C2 0.43 m 0.83 0.27 m 0.95 0.64 m 0.84C3 0.40 m 0.84 0.26 m 0.95 0.64 m 0.84D1 0.95 m 0.62 0.27 m 0.95 0.88 m 0.78D2 0.82 m 0.67 0.26 m 0.95 0.81 m 0.80

Absolute mean deviation () 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=14 / (o tan c), where o is mean length of the form in the direction of slopelinesand c is a critical angle for distinguishing plain and slope (steeper segments must not be considered as plains; tan c=0.20 in this case).



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affinity to horizontal planes. Slovinec is an exhumederosion surface with more resistant limestone in thecentre and high affinity to circular (domal) models:linear and divergent models are markedly less appro-priate. Podhorsk is a cryoplanation surface (a gently

deformed glacis), and deviations for all non-linearmodels are adequately low. The good results for di-vergent models reflect the real tendency of lateralchange of gradient conditioned by a diffuse lithologicboundary. The considerable difference in betweenlinear and non-linear models is surprising, implying ahigh geometric homogeneity and a truly non-linearform. Sandberg (Sa) is a young fault slope withlandslides on sand and sandstone. Hence it has thehighest absolute deviations in spite of having thesmallest area. All the Sandberg models show better fits

for higher-order forms, confirming the non-linearity inprofile 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. 12depends on fuller automation. Further work is needed onthe improvement of computational accuracy for higher-order morphometric properties.

6. Discussion

A major difference between this concept of elemen-

tary forms and some dynamically and genetically basedapproaches is the strict separation of elementary formdelimitation and its subsequent dynamic and geneticcharacterization. This is necessary if we want to dis-tinguish the presentation of geomorphological facts(morphology) from their interpretation (genesis, poten-tial processes), so as to make geomorphological researchmore objective (Savigear, 1965; Urbnek, 1997). How-ever, the ability to interpret elementary forms can in-fluence the choice of geometric type of elementary formby which we approximate a concrete segment of the real

land surface. The same principle may limit the choice ofbasic geometric types of elementary forms usable ingeomorphology generally.

6.1. Exclusion of non-morphometric and positional

variables from initial form definition

Most concepts of land units are not confined topurely morphometric variables but are based on visualinterpretation from field survey or air photos. In landsystem analysis, aerial images are the main tools forlandform recognition by numerous researchers (Argia-las, 1995; Chorowitz et al., 1995; Giles and Franklin,

1998). Consideration of non-morphometric characters inimages can lead to more detailed genetic distinction ofgeometrically similar landforms. But mixing variouscriteria in the process of segment delimitation is prob-lematic without deeper theoretical reasoning, because

the homogeneity of definition of geomorphic indivi-duals may be violated. Our definition of elementaryforms relies on the geometric signature only and can beconsidered a pure digital approach (Brndli, 1996).Naturally, classification or internal division of elemen-tary forms on the basis of various other criteria can bringinteresting applications.

The next question is whether we need to insert anypositional (regional of Shary et al., 2002) variablesinto the segmentation algorithm. According to Brndli(1996), land units are not individually determined but are

related to surrounding units (the context). Many otherauthors include regional variables in their concept oflandform units (Dalrymple et al., 1968; Dymond et al.,1995; Schmidt and Dikau, 1999; MacMillan et al.,2000). The importance of regional variables for landclassification is beyond doubt. But their mixing withlocal variables in algorithms for identification ofboundaries can be problematic for segmentation. Basins(watersheds) are the most widely recognised landformunits determined by a regional (integrating) approach.Shary et al. (2002, 2005) document the use of regionalvariables for delimitation of depressions, hills and sad-

dles. The boundaries are slopelines and contour linesrespectively. They can be singular lines but are notrequired to be. Their morphogenetic significance istherefore questionable (a flat water divide is a geneticallyhomogeneous surface although it belongs to two dif-ferent basins). Some regional variables (e.g. catchmentarea, altitude percentile, and UPNESS of Summerellet al., 2005) cannot in principle be constant over anyarea. Although regional variables could be added to theset of form-defining properties, in principle they reflectbroader dynamic and not local genetic relations. They

should be used rather for determination of higher-ordergeomorphic units, or for classification of delimited ele-mentary forms.

6.2. Attractors

Cox (1978) pointed out that linear segments are onlyspecial cases of constantly curved elements; thus theycannot approximate reality better than elements. Shary(1995) regarded linear forms (planar, in 3-D) as rare, andhe argued that their real occurrence (e.g. cells with exactlyzero value of curvature) is minimal. Every form definedby a constant value of some morphometric property is



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only a special case of a more general form, and shouldthus be expected to be rare. This may seem to contradictthe elementary forms concept, but if elementary forms areattractors, quasi- or relatively stable states to whichdevelopment tends, they will be both more frequent and

more interpretable. Apart from purely theoretical con-siderations, histograms of morphometric propertiesprovide clear empirical evidence. As a constant value ofa morphometric variable gives a zero value for derivedvariables, low values of a variable suggest affinity to aconstant value of its parent variable. The global dis-tribution of land altitudes and of gradients is asymmetricwith a predominance of low values. Curvature values tendto peak strongly at zero, showing that linear slopes are anattractor. Appropriate statistical analysis could confirmthe hypothesis that certain curved elementary forms are

also attractors. This requires analysis not only of uni-variate distributions but also of the relations betweenmorphometric properties, as presented by Evans and Cox(1999). However, inaccuracies in computing properties ofhigher orders may be a limiting factor.

Understanding elementary forms as attractors canprovide a bridge between analytical approaches to land-form delimitation starting from mathematical analysis,and statistical approaches using mainly multivariatestatistical methods. The characterization of topographyis a problem requiring a statistical methodology (Evans,1972; Pike, 1988). But statistical analysis can be more

fruitful if based on preliminary knowledge or hypothe-ses. The land surface is not random or fractal, but it ischaracterized by the presence of many structures andpatterns (Evans and McClean, 1995; Evans and Cox,1999). The concept of elementary forms can express theideal structure of the lowest hierarchical order of tax-onomic complexity of landform (in the sense of Dikau,1992), and so create a background for the selection andinterpretation 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 sta-tistical similarity. Identifying composite forms directlyfrom a statistical data set is hindered where structuralinformation from the elementary level is missing. Pre-liminary identification of the elementary units (elemen-tary forms), which can subsequently be clustered intocomposite forms (landforms), provides this information.Moreover, ideal elementary forms may themselvesprovide the examples sought as cores of taxonomicalclasses defined in advance. Supervised classificationon the basis of agglomerative distance procedures witha spatial contiguity constraint can thus be used for de-limitation of elementary forms.

Each ideal elementary form is characterized by zerovalues of one or more form-defining properties; hencelow values signify affinity of a real surface segment tothe relevant elementary form. Individual elementaryforms can be defined by contiguous clustering from

form-defining properties (or their derivatives). Integra-tion of graph-based approaches with preliminaryidentification of discontinuities into a classificationalgorithm (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 toan ideal elementary form can be expressed effectively bycontinuous (fuzzy) classification (Fig. 12; Table 3). This

solves the classification problem of hierarchically inferiorelementary forms; as a simpler elementary form is only aspecific case of a more comprehensive higher-order form,it cannot approximate reality any better. The membershipfunction expresses the affinity of a segment to variousgeometric types of elementary form. Continuous classi-fication radically improves the interpretational potentialof elementary forms. The fuzzy character of real terrainunits is a consequence of the combination of variousfactors that have influenced them. A membershipfunction can express the relative validity of ideal geneticand dynamic interpretations of individual elementary

forms.Segments can be related to several interpretations;

this should be adequate to represent the complexities ofreality. Differences can arise from generalization, wherethe fitted function represents a larger form (such as afault scarp) but the real surface also contains smallerforms (such as landslides or gullies). Alternatively,differences may reflect a transition toward another typeof elementary form; for example the denuded peripheryof a planation surface will differ more from an idealhorizontal plane than will the well preserved central

part. The creation of ideal interpretation schemes for theindividual types of elementary forms and their bound-aries should be a next step, together with interpretationsof various values of their defining properties.

In reality, the boundaries of elementary forms alsohave a fuzzy character sudden change of a boundary-defining variable is a geometric idealization. The deri-vative of a boundary-defining variable provides thebasis for construction of a membership function for theboundary. The desired genetic and dynamic interpreta-tion of boundaries of elementary forms is thus possiblefrom a fuzzy approach based on geometrically idealkinds of discontinuity.



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7. Conclusions

The concept of elementary forms, introduced here,extends and integrates the theoretical background of landsurface segmentation. Various types of area and

boundary considered in previous approaches can berelated to areas and boundaries defined in the concept ofelementary forms (Fig. 3). Moreover the concept re-spects important properties of the land surface includingthe varying homogeneity of its segments and thedifferent types of discontinuity at natural boundariesbetween geomorphic units. The set of interpretable typesof elementary forms (Figs. 5 and 6) contains exactlydefined elementary units as used by previous authors, butit also provides formal descriptions of models previouslyignored or inadequately defined: concentric and diver-

gent models as well as models of the third order. Thedynamic and genetic interpretation of the geometry of anelementary form is important (Fig. 4). Except for theright-hand column, the system of elementary formspresented in Fig. 5 is based on forms characterized byparallel (linear, concentric or freely curved) contours only these are generally stable geometrically anddynamically, and thus clearly interpretable genetically.Non-parallel contours represent the dominance of effectsother than the gravitational principle of organisation ofthe earth's surface. This is consistent with the endeavourto combine geometric and dynamic principles (e.g.

Dalrymple et al., 1968; Lastoczkin, 1987, 1991; Gilesand Franklin, 1998). Fuller dynamic and geneticinterpretations of combinations of form- and boundary-defining properties need additional work, as doesdefinition of types given by field-independent variables.

The concept of elementary forms exactly defines botharea and boundary properties of basic landform segmentsthat can serve as elementary geomorphic individuals. Itenables expression of different ratios of the homogeneityand boundary contrasts of units, and of relations betweenideal interpretations and real morphogenesis and mor-

phodynamics. The concept can be applied not only togeomorphological mapping, but also to synthetic land-scape mapping, evaluation of natural hazards, carryingcapacity, susceptibility and so on. It should be a basicpart of a modern DEM-based geomorphological infor-mation system (cf. Minr et al., 2005), unifying thesetopics at a more synthetic level of scientific knowledge.

Acknowledgements

This study was supported by the Scientific GrantAgency of the Ministry of Education of the SlovakRepublic and Slovak Academy of Science (1/1037/04

and 1/4042/07). We are grateful to Nicholas J. Cox,Peter Shary, the reviewers Tom Farr and Robert I.Inkpen, and the editor Takashi Oguchi for some veryhelpful comments.

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