Measurement of DEM Roughness Using the Local Fractal Dimension Mesure De La Rugosité Des MNT À L'aide De La Dimension Fracta

Géomorphologie : relief, processus, environnement, 2005, n° 4, p. 327-338

Measurement of DEM roughness using the local fractal dimension Mesure de la rugosité des MNT à l’aide de la dimension fractale

Hind Taud* and Jean-François Parrot**

Abstract The relationships between geological features and DEM surface roughness are studied using fractal geometry analysis. The estimate of the fractal dimension in a 3D space is performed locally on DEM surfaces using an adaptive “box-counting” technique. This procedure has been applied to two regions chosen to represent differences in lithological and tectonic conditions. The first one corresponds to a faulted, homoclinal sedimentary sequence and the second to a compound stratovolcano. In both cases, the results obtained show that the local fractal dimension detects distinct morphologic features. In the first case (Vittel, eastern France), the local fractal dimension detects the limits of the geological units as borders of homogeneous zones or transitions between different dissection depths. Fine structural lines underlying the Vittel fault can also be extracted. In the case of Mount Ararat (Eastern Turkey), different classes can be distinguished within the volcanic formations using a statistical analysis of the fractal dimension values. The treatment can also provide information about the local and regional geological structures. These first results show that measuring DEM surface roughness represents a helpful tool for extracting and mapping morphometric features. Key words: morphology, Digital Elevation Model, surface, local fractal dimension, box-counting.

Résumé Les relations entre les traits géomorphologiques et la rugosité de surface des Modèles Numériques de Terrain (MNT) ont été étudiées par l’intermédiaire de la géométrie fractale. La dimension fractale dans l’espace à trois dimensions est estimée localement sur la surface du MNT. Cette mesure se fait à l’aide d’une procédure dérivée de la technique du « comptage de boîtes ». Ce traitement a été appliqué sur deux zones tests choisies pour leurs différences lithologiques et tectoniques. La première région correspond à une série sédimentaire monoclinale faillée. La seconde est un strato-volcan complexe. Dans les deux cas, les résultats obtenus montrent que la dimension fractale locale détecte différents traits morphologiques. Dans le premier cas (Vittel, NE de la France), la dimension fractale locale définit les limites des unités géologiques comme des bords de zones homogènes ou bien la transition entre différentes profondeurs de dissection. Il est également possible d’extraire de fins traits structuraux soulignant ainsi la position de la faille de Vittel. Dans le cas du Mont Ararat (Turquie orientale), différentes classes peuvent être distinguées parmi les formations volcaniques au moyen d’une analyse statistique des valeurs de la dimension fractale. Le traitement est également en mesure de fournir des informations relatives à la structure géologique tant au niveau local que régional. Ces premiers résultats montrent que la mesure de la rugosité de la surface d’un MNT est un outil utile pour extraire et cartographier les traits morphométriques. Mots clés : morphologie, Modèle Numérique de Terrain, surface, dimension fractale locale, comptage de boîtes.

Version abrégée ractériser la texture dans la mesure où la topographie ter- restre est sensée présenter un comportement fractal indé- La rugosité ou la texture des Modèles Numériques de Ter- pendamment de l’échelle d’observation. Cet article concer- rain (MNT) est susceptible de fournir des informations re- ne l’étude de la rugosité de surface à l’aide de la mesure de latives à la géologie régionale. En effet, les MNT étant une la dimension fractale locale dans un espace tridimension- représentation de la surface, différents attributs peuvent les nel, en vue de mettre en évidence ou d’accentuer divers décrire. Entre autres, la dimension fractale permet de ca- traits géomorphologiques.

* Instituto Mexicano del Petróleo, Apto. Postal 14-805, 07730 México D.F., México. E-mail : [email protected]; [email protected] ** Instituto de Geografía, UNAM, Apto. Postal 20-850, 01000 México D.F. México. E-mail : [email protected] Hind Taud, Jean-François Parrot

La géométrie fractale est une description mathématique de d’un strato-volcan complexe, le Mont Ararat. Dans les des formes naturelles. Un objet fractal est trop complexe deux cas, des études comparative et statistique prenant en pour être décrit dans un espace cartésien. Seule la dimen- compte les données existantes et les résultats issus du filtra- sion fractale est à même de mesurer un objet complexe. ge provenant de la mesure de la dimension fractale locale, Différentes méthodes ont été proposées pour mesurer cette ont été réalisées en vue d’évaluer les résultats que fournit dimension. L’une d’entre elles est largement utilisée : il cette approche. s’agit du « comptage de boîtes » pouvant être appliqué à Dans le premier cas, la relation existant entre la géomor- tous type de formes, fractales ou non. phologie et les résultats obtenus à l’aide du filtre décrit Dans le cas présent, nous proposons de mesurer locale- antérieurement est particulièrement nette. Il convient de ment au sein d’un cube la dimension fractale en se fondant noter que l’on est ici en présence d’une série monoclinale, sur cette méthode. L’avantage de ce traitement réside dans la surface de chacune des couches géologiques répondant le fait que les « voxels » décrivant le volume pris en comp- en fonction de ses caractéristiques propres. Les différentes te pour faire ce calcul dépendent directement de l’altitude unités stratigraphiques sont mises en évidence et cernées, des pixels décrivant la surface du MNT. Le « voxel » est soit par le biais des épaulements qui les limitent comme cela lui-même un cube dont la base est un pixel et la hauteur une peut s’observer dans le cas du Rhétien, soit par l’importan- tranche d’altitude correspondant à la dimension du coté du ce et la profondeur des incisions qu’engendre le réseau pixel. Une altitude donnée est elle ainsi représentée par un hydrographique dans ces formations. Dans ce dernier cas, empilement de voxels ou cubes élémentaires. En fait la va- les formations du Muschelkalk et du Buntsandstein sont riation du coefficient h décrit plus loin permet de modifier clairement définies en utilisant ce critère, ainsi que l’acci- la hauteur de cette tranche d’altitude. La procédure est la dent N-S correspondant à la faille d’Esley qui les met en suivante : au sein d’un cube de taille s × s × s centré sur un contact. Par ailleurs, en jouant sur les valeurs du coeffi- pixel décrivant la surface du MNT, le volume correspondant cient h, il est possible de détecter les lignes de crête et les à la section du MNT prise en compte est un ensemble de thalwegs et ainsi de souligner la faille de Vittel et son pro- voxels dont le nombre est compris entre 0 et s. Le nombre de longement occidental. voxels présents dans le cube est égal à: L’utilisation de la dimension fractale locale dans le cas du Mont Ararat est d’un usage plus délicat dans la mesure où un strato-volcan est une structure complexe, tant au plan de la nature du matériel volcanique que de la tectonique. En vs peut se calculer facilement de la manière suivante en se fait, la dimension fractale locale calculée en utilisant des plaçant dans l’espace bidimensionnel : fenêtres de grande taille se révèle utile pour mettre en évi- dence les grandes unités structurales qui caractérisent ce massif, ainsi que l’extension des différentes coulées volcaniques qui le constituent. Situé dans un réseau de failles de direction N-S dont l’une passe par le Grand Ararat, l’ensemble présente en premier lieu une zone d’effondrement et SW induisant une réponse de la rugosité de surface diffé- rente de celle qui domine sur le reste du massif. Cette I correspond à l’image traitée, Ic est la valeur du pixel cen- différence se retrouve au niveau de la lithologie des émis- tral, ps la taille du pixel, h un coefficient définissant la sions effusives. Un filtrage de grande taille révèle et précise résolution verticale et s la taille du cube. Ce cube est divisé la nature des édifices (localisation de la ligne de base, posi- en boîtes dont la taille varie de 1 à s/2. Chaque boîte est tion des cônes de cendres et/ou des dômes volcaniques). De considérée comme remplie quand elle contient au moins un plus, les coulées répondent comme des linéaments caracté- voxel. En d’autres termes, la valeur maximale de vs(i,j)/q est risés par une haute valeur au toit de la formation (absence calculée en appliquant l’équation suivante : de rugosité au sein de la fenêtre d’observation), cernés par des valeurs plus faibles correspondant aux flancs latéraux, en raison de l’écart hypsométrique enregistré qui abaisse la valeur de la dimension fractale. Il est ainsi possible de défi- La dimension fractale correspond à l’inverse de la pente nir au sein des grands ensembles pétrographiques une P=ln(q)/ln(Ns), q étant la taille de la boîte et Ns le nombre partie des éléments qui les composent. total de boîtes remplies. Le résultat de l’application de ce traitement sur ces deux Les traitements effectués à titre d’exemple concernent deux zones différentes nous conduit aux conclusions suivantes. milieux choisis pour leurs différences tectoniques et litholo- Lorsque la zone d’étude correspond à un ensemble tectoni- giques et s’appliquent à des zones test antérieurement quement homogène, la compréhension de la réponse passe étudiées, en vue de valider la méthode. Ils illustrent quelques- par une analyse descriptive relativement facile à réaliser et unes des possibilités qu’offre cette approche au plan le traitement peut donc représenter dans ce cas un outil effi- morphologique et structural. Le premier, situé dans la région cace permettant de préciser les traits géomorphologiques. de Vittel (France), est une zone sédimentaire faillée. Le En revanche, dans le cas d’ensembles complexes, la signifi- second, situé en Anatolie orientale (Turquie), concerne l’étu- cation des traits morphologiques mis en évidence par cette

328 Géomorphologie : relief, processus, environnement, 2005, n° 4, p. 327-338 Measurement of DEM roughness using the local fractal approche méthodologique nécessite une étude de détail de mentary Vittel region leads to investigate the ability of the la réponse de tous les objets mis en jeu, dans le but de réa- treatment to extract these features in accordance to their liser une synthèse objective. morphology. Taking into account the former results, the treatments applied to the Mount Ararat illustrate the relia- Introduction bility of the method to emphasize local roughness differences related to regional tectonism and to detect pecu- As a Digital Elevation Model (DEM) is a representation liar and unknown features that can be useful for geological of a surface, several algorithms have been developed to and geomorphological mapping. study the surface properties of DEMs and to provide a large set of descriptor attributes (Wilson and Gallant, 2000). Fractal geometry and local Given that altitudes correspond to grey levels, it is then pos- fractal dimension sible to describe, quantify, and model rough surfaces using image-processing techniques in the spatial and frequency Fractal geometry domains and to apply pattern recognition processes. Textu- ral analysis is closely linked to roughness assessment. The Fractal geometry, introduced and developed by B. Man- concept of texture is quite difficult to define as it often delbrot (1982), provides a mathematical description of a includes the notions of roughness, regularity, contrast, and wide range of natural forms and phenomena. Fractal objects thinness. Numerous authors have attempted to outline and are defined as scale-invariant (self-similar or self-affine). clarify this notion. J. C. Russ (1999) proposed that rough- This means that the fractal object can be presented as an ness equates with the high frequencies of the 2D signal assemblage of rescaled copies of itself. Self-similarity occurs constituting a DEM. He pointed out that a DEM contains when the rescaling is isotropic or uniform in all directions, three levels of information. The first level, related to the and self-affinity occurs when the rescaling is either lower frequencies, coincides with its form; the second level, anisotropic or dependent on direction. related to the middle frequencies, corresponds to its wavi- Fractal objects exhibit details at arbitrarily small scales, ness; and the third level corresponds to its roughness. and they are too complex to be represented in a Euclidean Some recent techniques in image analysis use fractal space. Also known as the Hausdorff-Besicovitch dimension and/or multi-fractal approaches to characterize the texture of (Falconer, 1990), the fractal dimension differs from the a greyscale image or the roughness of a surface. Concerning more familiar Cartesian or topological dimension. In this DEM surfaces, several authors have demonstrated that the last case, integer values are required: 1 for a line, 2 for a sur- topography of the Earth generally exhibits fractal character- face, and 3 for a volume. The fractal dimension measures istics and that the relief preserves the same statistical the complexity of the object. A shape with a higher fractal characteristics over a wide range of scales (Huang and Tur- dimension is more complicated or irregular than one with a cotte, 1989; Klinkenberg and Goodchild, 1992). By lower dimension. For example, a shape with a fractal dimen- describing the terrain as a fractal surface, the local or the sion falling within the range [1, 2] fills more space than a global analysis of DEM roughness reveals interpolation one-dimensional curve and less space than a two-dimen- artefacts, provides an assessment of the DEM quality (Poli- sional surface. Self-similarity is defined statistically when it dori et al., 1991; Datcu et al., 1996), and can assist in cannot be tested through an infinite range of scales. The sta- understanding erosional phenomena (Chase, 1992; Cheng et tistical fractal behaviour is then related to a given scale al., 1999). range. When the fractal dimension of a particular pattern As every landscape appears to have a particular fractal changes within consecutive ranges of scale, one generally dimension value, it implies that this dimension calculated refers to the notion of multi-fractality. locally has to be related with the particular features of the Several methods are available for estimating the fractal di- local landscape units. On the other hand, each local land- mension of surfaces, such as the fractional Brownian model scape unit presents its own morphologic feature in relation (Mark and Aronson, 1984), triangular prism areas (Clarke, to the erosion processes and mainly to the nature of the base- 1986), box-counting (Falconer, 1990), and the projective ment. Thus, the purpose of this research is to study the covering method (Xie and Wang, 1999). Measurement may relationships between geomorphic features and the surface either be direct, when it is applied to grey tones, or indirect roughness of a DEM by locally measuring fractal dimension as in the case of the examination of “profiles” (one-dimen- in 3D space. sional transects) or isolines (conversion of the surface into a The purpose here does not include demonstrating the frac- contour map). In this study, the box-counting method was tal behaviour of the DEM nor comparing the performances used because it can be applied to various sets of any dimen- of different fractal surface estimators. The first section of sion and patterns with or without self-similarity (Peitgen et the paper presents a brief overview of fractal geometry and al., 1992). According to K. Falconer (1990) who has dis- explains the notion of local fractal dimension. The second cussed the mathematical aspect of the box-counting method, section describes the procedure, while the third section the fractal dimension D can be derived from the relation: D reports the results obtained in two case studies. These train- 1=Nss or D—logNs/log(s) where Ns corresponds to the ing areas have been chosen according to diverse lithological number of boxes of size (s) needed to cover the structure. conditions. The well known geological features of the sedi- The D value is calculated with the following formula:

Géomorphologie : relief, processus, environnement, 2005, n° 4, p. 327-338 329 Hind Taud, Jean-François Parrot

It is easier to calculate vs (i,j) in a bi-dimensional space as (1) follows:

Local fractal dimension (4)

The local measurement of fractal dimensions has been with developed to investigate image texture and surface roughness. When this type of measurement is applied to an image I, the results are reported in an image R where the value of where I is the original image, Ic the value of the central each point (i,j) corresponds to its fractal dimension FD cal- pixel, ps the pixel size, h a coefficient that defines the verti- culated in a window of size (2w + 1)×(2w + 1): cal resolution, and s the cube size. The cube is partitioned into boxes of size q varying between 1 and s/2 (fig. 1). Each (2) of these boxes is considered as filled if at least one voxel is contained in this box. In other words, the maximum of Textural features derived from the local measurement of vs(i,j)/q in each cell is determined. The computation is done the fractal dimension have been used for segmentation and as follows: classification purposes. Among these techniques, the Fourier power spectrum (Pentland, 1984), the blanket method (Dellepiane et al., 1991), differential box-counting (Sarkar (5) and Chaudhuri, 1994; Chaudhuri and Sarkar, 1995), and the where Max is a function calculating the maximum. The fractional Brownian motion (Chen et al., 1989; Toennies and fractal dimension corresponds to the inverse of the slope Schnabel, 1994) are the most common. Using a wavelet P=ln(q)/ln(Ns), where q is the size of the box and Ns the technique, M. Datcu et al. (1996) have estimated the local total number of filled boxes. When calculating the slope, roughness of various DEMs and provided some examples the coefficient of determination, R2, is computed. The esti- showing that the fractal analysis of these DEMs allows mate of the fractal dimension depends on various factors as different roughness classes to be distinguished and some discussed below. As the studied central point is by defini- artefacts, due to the computation of elevation data, to be tion located in the middle of the testing cube, an isolated detected. For computing the local self-similar properties of a point, surrounded by values lower than the base of the cube DEM, Y.C. Cheng et al. (1999) have developed a 3D box- s × s × s, generates a vertical line and a line generates a ver- counting method applying the triangular prism surface tical plane. The slopes are respectively expressed by the method proposed by K.C. Clarke (1986). They observe that following relations: y = -x, y = -2x with R2 = 1. In the same fractal dimension values vary as a function of altitude and way, a horizontal surface fills the half of the testing cube have interpreted this phenomenon as reflecting spatial and the corresponding FD = 3 with R2 = 1 (fig. 2). When variability in erosional potential. This last method differs the volume presents irregularities its fractal dimension from the previous ones because the fractal dimension is decreases. estimated in unit areas and not at each point of the image. In The volume inside the cube depends on the coefficient h. our approach, the fractal dimension of each pixel of the DEM The transformation into voxels can smooth the surface or, surface is calculated inside a moving window centred on this on the contrary, can enhance the surface roughness. When pixel, by using a 3D box-counting adaptive method. The h = 1, the volume depends directly on the resolution of the advantage of this treatment is that the volume corresponding DEM. Varying the coefficient h in Eq. (4) implies a change to DEM section observed locally is directly related to the in the number of voxels vs, and thus the volume occupied altitude values of the pixels defining the DEM surface. The inside the cube (fig. 1). High h values accentuate the surface treatment does not require any modelling or interpolation of roughness because the weight of the first term in this for- the surface in order to calculate this dimension. mula is more important than the second one (s/2). On the contrary, low values produce smooth surfaces because the Procedure second term of the formula is predominant. It is then possible to adapt the h value according to the nature of the DEM Inside a cube of size s × s × s centred on the pixel of the surface and the level of information expected. studied DEM, the volume corresponding to the surface is Determining the most appropriate range of grid sizes is a defined by a set of voxels. A voxel is an elementary cube, common problem for fractal dimension estimation the sides of which are equal to the pixel size. Thus, in the (Foroutan-pour et al., 1999). The size of the frame used for (x,y) space, each point (i,j) of the cube contains a number the computation can be an even or an odd number; the range vs(i,j) of voxels falling between 0 and s. The total number of of grid size can be a power of two (Biswas et al., 1998) or a voxels describing the volume is equal to: succession of integers. In our treatments, based on various tests applied to training zones, the size q of the sliding cube (3) is chosen to be 12 or 24. Using these sizes, the slope P=ln(q)/ln(Ns) is more regular when exact dividers are

330 Géomorphologie : relief, processus, environnement, 2005, n° 4, p. 327-338 Measurement of DEM roughness using the local fractal

Fig. 1 – Local fractal calculation by using a 3-dimensional box counting. Example with s= 12 and different grid size q. A: initial topography equivalent to q= 1; B: q= 2; C: q= 3; D: q= 6. Fig. 1 – Calcul de la dimension fractale locale par la technique tridimensionnelle du « comptage de boîtes ». Exemple avec s = 12 et différentes tailles de maille q. A : topographie initiale équivalente à q = 1 ; B : q = 2 ; C : q = 3 ; D : q = 6.

Fig. 2 – Local 3D fractal dimension of a line and a plan with s = 12 and q = 1, 2, 3, 6. Fig. 2 – Dimension fractale locale d’une ligne et d’un plan avec s = 12 et q = 1, 2, 3, 6.

Géomorphologie : relief, processus, environnement, 2005, n° 4, p. 327-338 331 Hind Taud, Jean-François Parrot

employed. For a flat surface and a volume, the slopes P rounding this centre. Using one of these points leads to the obtained are -2 and -3 respectively. Even if these propor- deviation of the result that favours one orientation. tions are respected and do not produce a different result after These remarks imply that the procedure must either cal- normalisation, odd sliding windows provide a slight devia- culate the local fractal dimension on each of these four tion of the slope P for theoretical forms. The exact dividers, pixels with the risk of blurring the response when calculat- which define the range of grid sizes q, never correspond to ing the mean, or define the value of the centre by taking the half of the window side and then only a portion of the slope average value of the four pixels located at the centre of the is taken into account in calculating the fractal dimension. window. Ic in Eq. (4) is calculated according to the second When applying a 12 × 12 × 12 cubic pattern, five exact alternative in order to avoid blurring and to minimize the dividers (1, 2, 3, 4, and 6) can be found, excluding any bor- computation cost. der effect. In the second case (24 × 24 × 24), one can find seven exact dividers (1, 2, 3, 4, 6, 8, and 12). Treatments that Test areas and results employ these cubic sizes show that using the dividers 1, 2, 3, and 6 in the first case and the dividers 1, 2, 3, 6, and 12 in In order to test its ability to extract morphologic features the second produces no bias. As the second term in Eq. 4 according to different lithological and tectonic conditions, depends on the window size s, the use of the smaller win- the local fractal dimension was applied in two regions. The dow displays small irregularities; the larger window exhibits first example, located in the region of Vittel (NE France), the general features of the DEM studied. characterizes a sedimentary area affected by faults. The sec- The fractal dimension is calculated locally for each surface ond example, located in eastern Anatolia (Turkey), is a pixel in a sliding window centred on this pixel. As discussed volcanic region. In order to illustrate the performance of the above, the quality of results is greater with an even size than procedure, the treatments concerning the Vittel region show with an odd one. However, in the first case, the concept of mainly the variation of coefficient h and range of grid sizes ‘centre’ must be defined because the studied point cannot be effect, whereas the variation of the window size has been located exactly in the centre but in one of the four points sur- employed in the second example.

Fig. 3 – Vittel region. A: shaded relief map; B: Vittel geological sketch map: Symbols; B = Buntsandstein; M = Muschelkalk; K = Keuper; R = Rhetian; C: local fractal with s = 12, h = 100; D: local fractal with s = 12 and h = 2. Fig. 3 – Région de Vittel. A : MNT ombré ; B : carte géologique de Vittel ; symboles : B = Buntsandstein ; M = Muschelkalk ; K = Keuper ; R= Rhétien ; C : dimension fractale locale avec s = 12 et h = 100 ; D : dimension fractale locale avec s = 12 et h = 2.

332 Géomorphologie : relief, processus, environnement, 2005, n° 4, p. 327-338 Measurement of DEM roughness using the local fractal

Fig. 4 – Distribution of fractal dimension according to sliding window size. Fig. 4 – Distribution de la dimension fractale en fonction de la taille de la fenêtre mobile.

The Vittel region

The Vittel area is located in northeastern France, 105 km west of the Rhine graben, between 48° and 49° N and 5° and 6° E. In the Vosges massif, outcrops of homoclinal Triassic and Liassic units overlie Variscan metamorphic and intrusive rocks. The east- west-trending Vittel fault is a major composite fault zone of complex geometry, resulting from the reactivation of a Palaeo- zoic discontinuity. In the region located between this fault and the N45-trending Esley fault, different groups of faults favour the water cir- the Rhetian border is strong enough to induce in the moving culation that supplies the commercial Vittel spring water window high roughness values codified with low grey tone (Sykioti, 1994). values. Thus, the limit of the Rhetian appears clearly and The DEM of this area is an IGN (Institut Géographique corresponds exactly to the limits reported in the geological National) product with a horizontal resolution of 50 m and a map. Concerning the Muschelkalk (limestones about 50 m conical projection (fig. 3A). The corresponding geological in thickness) and the Bundsandstein (Vosgian sandstones), map (Fig. 3B) has been geometrically corrected according to the incision depth of the drainage network is closely related this projection in order to be overlaid on the DEM and com- to the respective nature of these two formations. For this pared with the local fractal results. The zone size is about reason, these units are clearly individualized and the contact 13.75 × 23.55 km (275 lines × 471 columns). Several treat- corresponding to the N45-trending Esley fault is obvious, as ments were applied to the DEM using a cubic size of well as the stratigraphical contact located at the base of 12 pixels and modifying only the value of the coefficient h. Muschelkalk escarpment. The results are improved by According to Eq. (4), decreasing coefficients contribute to superposing the geological map and the image resulting smooth the roughness because a voxel inside the cube cor- from the local fractal treatments. The frequency of the local responds to a bigger hypsometric interval (ps/h). Then, the fractal dimension depends on the sliding window size procedure detects only high altitude variations. On the con- (fig. 4). Using a great size, the histogram is unimodal and trary, when coefficient h increases, the surface roughness is becomes multi-modal with decreasing sizes, allowing us to accentuated and therefore small altitude variations are obtain various classes corresponding to each mode. detected as well as large altitude ones. Thus, the position and extent of the Vittel fault (fig. 3C) The Ararat volcano are precisely highlighted and its prolongation westwards remains clear. When using high h coefficients (i.e., 100), With an elevation of 5123 m a.s.l., Mount Ararat is the every elevation difference is taken into account. Then, a largest volcanic centre and the highest point of eastern Ana- thalweg feature in the testing cube corresponds to a filled tolia. It corresponds to a compound stratovolcano formed by cube with a gully whatever the altitude variation. The frac- the ‘Greater’ and the ‘Lesser’ Ararat. This N150-trending, tal dimension obtained is close to 3. On the other hand, a elongated massif lies inside a pull-apart basin with a similar crest line is represented by a “wall” as formerly described trend (Pearce et al., 1990). This fault system is a horsetail (see Fig. 2) and its fractal dimension is close to 2 (fig. 3C). splay structure and in addition to the right-lateral slip com- These results are obtained using a window size s = 12 and a ponent, these faults have a normal component. The horsetail range of grid sizes varying between 1 to s. These ensure the splay fault system cuts through the Greater and Lesser continuity of the features by producing a standardization of Ararat volcanoes and controls the position of the main erup- the estimated fractal dimension and increase the detection of tive centres of Ararat, as well as the alignment of parasitic crest and thalweg features. cones (Karakharian et al., 2002). The volcanic activity of In contrast, the major units encountered in the studied this region continued without interruption until historical zone were underlined using a low h coefficient but greater times, possibly reaching its climax during the late Miocene than one, because the region is relatively flat. With a coeffi- and Pliocene (6 to 3 Ma). During Quaternary times, the vol- cient equal to 2 (fig. 3D), the main structural and geological canism appears to have been restricted to a few localised units are detected. A high value is obtained when, at the centres (Yilmaz et al., 1998). The youngest eruptions from observation scale, the studied zone inside the moving win- the Ararat volcano are older than 10 000 years, and isotopic dow is relatively flat. The escarpment that corresponds to dating of the youngest lavas yields an age of 20 000 years

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(Yilmaz et al., 1998). Recently, geomorphic criteria applied variation as well as the character of the studied shape, a to Mount Ararat have been used in order to study the tec- great window size reveals for instance the presence of iso- tonics of eastern Anatolia (Adiyaman et al., 2003). These lated volcanic cones (low values), the presence of domes criteria refer to the morphometric method developed by (high values surrounded by a crown of low values). The F. Garcia-Zuniga and J.-F. Parrot (1998) in order to define local fractal results depend on the size of the moving win- the base line and measure its elongation, and to the “Voxel dow as the frequency of the 2D signal is related to the wall” procedure (Baudemont and Parrot, 2000). The latter surface roughness characterizing the type of studied lava consists in parameterizing the DEM surface in a real 3D flow. On the other hand, the value of the general roughness space using a vertical wall of voxels. wavelength is an indicator of the regional tectonics. As illus- The DEM of the Ararat zone (fig. 5A) is generated using trated by figures 5C and 5D, the SW quarter of the the method developed by H. Taud et al. (1999). The pixel stratovolcano presents a globally lower roughness signature, size equals 30 m. The size of this area is about 9.57 × the eastern limit of which emphasizes the presence of a large 9.42 km (319 lines × 314 columns). The surface roughness fault zone passing through the Greater Ararat crater. This of the different lava flows has been studied by means of dif- fault corresponds to a branch of the horsetail splay structure ferent window sizes. A small window size (s = 12 with h = 4) described by A. Karakharian et al. (2002). In addition to the reveals that the roughest surfaces mainly correspond to the right-lateral slip, these authors assume that these faults have more recent features, i.e., to the extent of the lava flows a normal component. The SW zone that has a weaker rough- erupted from the recent parasitic cones located on the west- ness response could indicate a huge collapse structure. In ern flank. The extent of the eastern gully coming from the fact, the volcanic material (hypersthene andesite) observed Greater Ararat is clearly observed, as well as the total base in this zone (fig. 5B) is globally different from the material line of the whole edifice recently described by O. Adiyaman outcropping in the eastern part of the stratovolcano (mainly et al. (2003). A greater window size (s = 24 with h = 4) takes basalts and hypersthene basalts). into account the regional structural features. As the local Moreover, in order to study the effect induced by different fractal values decrease in relation to brutal hypsometric window sizes and different values of h, the units of the geological sketch map drawn by Y. Yilmaz et al. (1998) have been reordered according to their petrographic nature. Six volcanic items (types of lava) are present: andesite,

Fig. 5 – Case study of Mount Ararat. A) shaded relief map; B) geological map (after Yilmaz et al. 1998). 1: basalt, 2: hypersthene basalt 3: hyalobasalt, 4: andesite and associated pyroclastic rocks, 5: hypersthene andesite, 6: hyaloandesite, 7: moraines, 8: alluvial and glacial fans, 9: alluvium, 10: basement. I: Permanent ice cap; Q: Quaternary; C) local fractal with s = 12; D) local fractal dimension with s = 24. Fig. 5 – Exemple du Mont Ara- rat. A) MNT ombré ; D) carte géo- logique (Yilmaz et al. 1998). 1 : basalte ; 2 : basalte à hypersthè- ne ; 3 : hyalobasalte ; 4 : andésite et roches pyroclastiques asso- ciées ; 5 : andésite à hypersthè- ne ; 6 : hyaloandésite, 7 : moraines ; 8 : cône de déjection ; 9 : alluvion ; 10 : complexe de base ; I. neiges permanentes, Q :Qua- ternaire ; C : dimension fractale locale avec s = 12 ; D : dimension fractale locale avec s = 24.

334 Géomorphologie : relief, processus, environnement, 2005, n° 4, p. 327-338 Measurement of DEM roughness using the local fractal

Fig. 6 – Evolution of mean fractal value of the studied items as a function of window size and coefficient h. Fig. 6 – Evolutions de la moyenne de la dimension fractale des thèmes étudiés en fonction de la taille de la fenêtre et du coefficient h.

hyaloandesite, hypersthene andesite, basalt, hyalobasalts and hypersthene basalt. On the other hand, alluvium, fluvial and glacial fans, moraines and basement have been also taken into account in the statistical study. This geological map has been geometrically corrected in order to be overlain on the DEM and compared with the local fractal results. The mean of the local fractal dimension measured on the surface of each item is reported in the figure 6. The value of the fractal dimension depends on these two coefficients. Some items present specif- ic values that allow us to characterize or to group them in different classes. When the window size is equal to 6 (fig. 6A), all the items ex- cept the alluvial formations are comprised in an unique class while the h value is equal to 1. For h values greater than 1, three classes can be observed: low fractal Y. Yilmaz et al. (1998), could be related to different volcanic values are related to alluvium, basement and fans; median flows running from the eruptive centres. values to basalt, hyalobasalt and hypersthene andesite (the two formers are strongly connected); and high values to the Conclusion remaining items. The same distribution is observed using window sizes of 24 where different values of h do not gener- This paper aims at examining the relation between the sur- ate any variation (fig. 6C). On the contrary, with a 12 win- face roughness and the geological and geomorphologic dow size (fig. 6B) the items are classified differently. For in- features. It is well known that the fractal dimension can be stance, andesites are characterized by low values when s = 12 used as a parameter characterizing the surface roughness and high values when s equals 6 or 24. Therefore, using the and the landscape shape. The treatment proposed here mea- mean value, statistical classifications can be obtained by tak- sures locally this dimension by an adaptive box counting ing into account the results produced by different values of 3-dimensional method. The algorithm is based on the use of coefficients s and h. two parameters: the size s of the moving window and a scal- The items corresponding to large zones seem to be formed ing factor h defining the hypsometric interval taken into by a heterogeneous ensemble corresponding to different account. This treatment has been applied to a DEM surface erupted materials and the presence of different volcanic of two regions characterized by a difference concerning flows. High values are registered at the bulged top of the their lithological and structural conditions. The first exam- lava flows because this top appears as a flat surface at the ple is related to the study of a sedimentary and faulted observed scale, while the lateral flanks present lower values. region in the Vittel area. In this case, two main results have This is the case for the isolated flows such as the hypers- been obtained according to the different configurations of thene basalt from the Lower Ararat (fig. 5C). Hence, the the parameters involved in the calculation. As a first result, high roughness alignments detected by the treatment, which the limits of the geological units detected by the local frac- are not present in the different geological units mapped by tal dimension correspond to these limits as they have been

Géomorphologie : relief, processus, environnement, 2005, n° 4, p. 327-338 335 Hind Taud, Jean-François Parrot

mapped on the field. The position of these limits match Baudemont F., Parrot J.-F. (2000) – Structural analysis of DEMs either to the border of a homogeneous morphologic feature by intersection of surface’s normals in a 3D accumulator space. or to the transition zone between two different incision IEEE Transactions on Geoscience and Remote Sensing, 38, depths of the drainage network. On the other hand, by using 1191-1198. low hypsometric intervals by means of the coefficient h, the Biswas M. K., Ghose T., Guha S., Biswas P. K. (1998) – Fractal procedure can detect fine features. We can therefore under- dimension estimation for texture images: A parallel approach. line the Vittel fault as well as its western prolongation. The Pattern Recognition Letters, 19, 309-313. second example focuses on the volcanic region of Mount Chase C. G. (1992) – Fluvial landsculpting and the fractal dimen- Ararat. In this case, the relation between the local fractal sion of topography. Geomorphology, 5, 39-57. measurements and the different volcanic formations which Chaudhuri B. B., Sarkar N. (1995) – Texture segmentation using have built up this stratovolcano has been studied by using a fractal dimension. IEEE Transactions on Pattern Analysis and statistical approach. Different volcanic classes can be distin- Machine Intelligence, 17, 72-77. guished by using their mean fractal dimension value. On the Chen C. C., Daponte J. S., Fox M. D. (1989) – Fractal feature other hand, the results provided directly by the local fractal analysis and classification in medical imaging. IEEE Transac- dimension show the extent of the volcanic flows according tions on Medical Imaging, 8, 133-142. to their surface roughness. Furthermore, a collapse structure Cheng Y. C., Lee P. J., Lee T. Y. (1999) – Self-similarity dimen- related to the regional strike slip faulting has been detected. sions of the Taiwan Island landscape. Computers and The results obtained with the two training DEMs corre- Geosciences, 25, 1043-1050. sponding to lithological conditions that are completely Clarke K. C. (1986) – Computation of the fractal dimension of different, corroborate the efficiency of the procedure. The topographic surfaces using the triangular prism surface area latter allows the retrieval of information about small struc- method. Computers & Geosciences, 12, 713-722. tural features in flat zones as well as general morphologic Datcu M., Luca D., Seidel K. (1996) – Wavelet-Based Digital characteristics of rugged areas. Elevation Model Analysis. 16th EARSeL (European Association Based on the first results, one can conclude that the of Remote Sensing Laboratories Symposium), Rotterdam, roughness of a surface is strongly correlated with the nature Brookfield, 283-290. of the material that forms the geology of a studied region. In Dellepiane S., Giusto D. D., Serpico S. B., Vernazza G. (1991) – the case of the Vittel region mainly formed by a sedimentary SAR image recognition by integration of intensity and textural sequence and studied as a training set, the application of the information. International Journal of Remote Sensing, 12, 1915- treatment by means of a descriptive analysis shows that this 1932. technique is efficient to extract and precise the meaning of Falconer K. (1990) – Fractal Geometry Mathematical Founda- the different morphometric features. In the case of more tions and Applications. J. Wiley, Chichester, 288 p. complex region such as the Mount Ararat that presents Foroutan-pour K., Dutilleul P., Smith D. L. (1999) – Advances unknown structures, the extraction and the meaning of the in the implementation of the box-counting method of fractal morphometric features resulting from the treatment, needs dimension estimation. 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