Measurement of DEM roughness using the local fractal dimension

Transcription

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
La rugosité ou la texture des Modèles Numériques de Terrain (MNT) est susceptible de fournir des informations relatives à la géologie régionale. En effet, les MNT étant une
représentation de la surface, différents attributs peuvent les
décrire. Entre autres, la dimension fractale permet de ca-
ractériser la texture dans la mesure où la topographie terrestre est sensée présenter un comportement fractal indépendamment de l’échelle d’observation. Cet article concerne l’étude de la rugosité de surface à l’aide de la mesure de
la dimension fractale locale dans un espace tridimensionnel, en vue de mettre en évidence ou d’accentuer divers
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
des formes naturelles. Un objet fractal est trop complexe
pour être décrit dans un espace cartésien. Seule la dimension fractale est à même de mesurer un objet complexe.
Différentes méthodes ont été proposées pour mesurer cette
dimension. L’une d’entre elles est largement utilisée : il
s’agit du « comptage de boîtes » pouvant être appliqué à
tous type de formes, fractales ou non.
Dans le cas présent, nous proposons de mesurer localement au sein d’un cube la dimension fractale en se fondant
sur cette méthode. L’avantage de ce traitement réside dans
le fait que les « voxels » décrivant le volume pris en compte pour faire ce calcul dépendent directement de l’altitude
des pixels décrivant la surface du MNT. Le « voxel » est
lui-même un cube dont la base est un pixel et la hauteur une
tranche d’altitude correspondant à la dimension du coté du
pixel. Une altitude donnée est elle ainsi représentée par un
empilement de voxels ou cubes élémentaires. En fait la variation du coefficient h décrit plus loin permet de modifier
la hauteur de cette tranche d’altitude. La procédure est la
suivante : au sein d’un cube de taille s × s × s centré sur un
pixel décrivant la surface du MNT, le volume correspondant
à la section du MNT prise en compte est un ensemble de
voxels dont le nombre est compris entre 0 et s. Le nombre de
voxels présents dans le cube est égal à:
vs peut se calculer facilement de la manière suivante en se
plaçant dans l’espace bidimensionnel :
et
I correspond à l’image traitée, Ic est la valeur du pixel central, ps la taille du pixel, h un coefficient définissant la
résolution verticale et s la taille du cube. Ce cube est divisé
en boîtes dont la taille varie de 1 à s/2. Chaque boîte est
considérée comme remplie quand elle contient au moins un
voxel. En d’autres termes, la valeur maximale de vs(i,j)/q est
calculée en appliquant l’équation suivante :
La dimension fractale correspond à l’inverse de la pente
P=ln(q)/ln(Ns), q étant la taille de la boîte et Ns le nombre
total de boîtes remplies.
Les traitements effectués à titre d’exemple concernent deux
milieux choisis pour leurs différences tectoniques et lithologiques et s’appliquent à des zones test antérieurement
étudiées, en vue de valider la méthode. Ils illustrent quelquesunes des possibilités qu’offre cette approche au plan
morphologique et structural. Le premier, situé dans la région
de Vittel (France), est une zone sédimentaire faillée. Le
second, situé en Anatolie orientale (Turquie), concerne l’étu328
de d’un strato-volcan complexe, le Mont Ararat. Dans les
deux cas, des études comparative et statistique prenant en
compte les données existantes et les résultats issus du filtrage provenant de la mesure de la dimension fractale locale,
ont été réalisées en vue d’évaluer les résultats que fournit
cette approche.
Dans le premier cas, la relation existant entre la géomorphologie et les résultats obtenus à l’aide du filtre décrit
antérieurement est particulièrement nette. Il convient de
noter que l’on est ici en présence d’une série monoclinale,
la surface de chacune des couches géologiques répondant
en fonction de ses caractéristiques propres. Les différentes
unités stratigraphiques sont mises en évidence et cernées,
soit par le biais des épaulements qui les limitent comme cela
peut s’observer dans le cas du Rhétien, soit par l’importance et la profondeur des incisions qu’engendre le réseau
hydrographique dans ces formations. Dans ce dernier cas,
les formations du Muschelkalk et du Buntsandstein sont
clairement définies en utilisant ce critère, ainsi que l’accident N-S correspondant à la faille d’Esley qui les met en
contact. Par ailleurs, en jouant sur les valeurs du coefficient h, il est possible de détecter les lignes de crête et les
thalwegs et ainsi de souligner la faille de Vittel et son prolongement occidental.
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
fait, la dimension fractale locale calculée en utilisant des
fenêtres de grande taille se révèle utile pour mettre en évidence 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
SW induisant une réponse de la rugosité de surface différente de celle qui domine sur le reste du massif. Cette
différence se retrouve au niveau de la lithologie des émissions effusives. Un filtrage de grande taille révèle et précise
la nature des édifices (localisation de la ligne de base, position des cônes de cendres et/ou des dômes volcaniques). De
plus, les coulées répondent comme des linéaments caractérisés par une haute valeur au toit de la formation (absence
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 définir au sein des grands ensembles pétrographiques une
partie des éléments qui les composent.
Le résultat de l’application de ce traitement sur ces deux
zones différentes nous conduit aux conclusions suivantes.
Lorsque la zone d’étude correspond à un ensemble tectoniquement homogène, la compréhension de la réponse passe
par une analyse descriptive relativement facile à réaliser et
le traitement peut donc représenter dans ce cas un outil efficace permettant de préciser les traits géomorphologiques.
En revanche, dans le cas d’ensembles complexes, la signification des traits morphologiques mis en évidence par cette
Measurement of DEM roughness using the local fractal
approche méthodologique nécessite une étude de détail de
la réponse de tous les objets mis en jeu, dans le but de réaliser une synthèse objective.
Introduction
As a Digital Elevation Model (DEM) is a representation
of a surface, several algorithms have been developed to
study the surface properties of DEMs and to provide a large
set of descriptor attributes (Wilson and Gallant, 2000).
Given that altitudes correspond to grey levels, it is then possible to describe, quantify, and model rough surfaces using
image-processing techniques in the spatial and frequency
domains and to apply pattern recognition processes. Textural analysis is closely linked to roughness assessment. The
concept of texture is quite difficult to define as it often
includes the notions of roughness, regularity, contrast, and
thinness. Numerous authors have attempted to outline and
clarify this notion. J. C. Russ (1999) proposed that roughness equates with the high frequencies of the 2D signal
constituting a DEM. He pointed out that a DEM contains
three levels of information. The first level, related to the
lower frequencies, coincides with its form; the second level,
related to the middle frequencies, corresponds to its waviness; and the third level corresponds to its roughness.
Some recent techniques in image analysis use fractal
and/or multi-fractal approaches to characterize the texture of
a greyscale image or the roughness of a surface. Concerning
DEM surfaces, several authors have demonstrated that the
topography of the Earth generally exhibits fractal characteristics and that the relief preserves the same statistical
characteristics over a wide range of scales (Huang and Turcotte, 1989; Klinkenberg and Goodchild, 1992). By
describing the terrain as a fractal surface, the local or the
global analysis of DEM roughness reveals interpolation
artefacts, provides an assessment of the DEM quality (Polidori et al., 1991; Datcu et al., 1996), and can assist in
understanding erosional phenomena (Chase, 1992; Cheng et
al., 1999).
As every landscape appears to have a particular fractal
dimension value, it implies that this dimension calculated
locally has to be related with the particular features of the
local landscape units. On the other hand, each local landscape unit presents its own morphologic feature in relation
to the erosion processes and mainly to the nature of the basement. Thus, the purpose of this research is to study the
relationships between geomorphic features and the surface
roughness of a DEM by locally measuring fractal dimension
in 3D space.
The purpose here does not include demonstrating the fractal behaviour of the DEM nor comparing the performances
of different fractal surface estimators. The first section of
the paper presents a brief overview of fractal geometry and
explains the notion of local fractal dimension. The second
section describes the procedure, while the third section
reports the results obtained in two case studies. These training areas have been chosen according to diverse lithological
conditions. The well known geological features of the sedi-
mentary Vittel region leads to investigate the ability of the
treatment to extract these features in accordance to their
morphology. Taking into account the former results, the
treatments applied to the Mount Ararat illustrate the reliability of the method to emphasize local roughness
differences related to regional tectonism and to detect peculiar and unknown features that can be useful for geological
and geomorphological mapping.
Fractal geometry and local
fractal dimension
Fractal geometry
Fractal geometry, introduced and developed by B. Mandelbrot (1982), provides a mathematical description of a
wide range of natural forms and phenomena. Fractal objects
are defined as scale-invariant (self-similar or self-affine).
This means that the fractal object can be presented as an
assemblage of rescaled copies of itself. Self-similarity occurs
when the rescaling is isotropic or uniform in all directions,
and self-affinity occurs when the rescaling is either
anisotropic or dependent on direction.
Fractal objects exhibit details at arbitrarily small scales,
and they are too complex to be represented in a Euclidean
space. Also known as the Hausdorff-Besicovitch dimension
(Falconer, 1990), the fractal dimension differs from the
more familiar Cartesian or topological dimension. In this
last case, integer values are required: 1 for a line, 2 for a surface, and 3 for a volume. The fractal dimension measures
the complexity of the object. A shape with a higher fractal
dimension is more complicated or irregular than one with a
lower dimension. For example, a shape with a fractal dimension falling within the range [1, 2] fills more space than a
one-dimensional curve and less space than a two-dimensional surface. Self-similarity is defined statistically when it
cannot be tested through an infinite range of scales. The statistical fractal behaviour is then related to a given scale
range. When the fractal dimension of a particular pattern
changes within consecutive ranges of scale, one generally
refers to the notion of multi-fractality.
Several methods are available for estimating the fractal dimension of surfaces, such as the fractional Brownian model
(Mark and Aronson, 1984), triangular prism areas (Clarke,
1986), box-counting (Falconer, 1990), and the projective
covering method (Xie and Wang, 1999). Measurement may
either be direct, when it is applied to grey tones, or indirect
as in the case of the examination of “profiles” (one-dimensional transects) or isolines (conversion of the surface into a
contour map). In this study, the box-counting method was
used because it can be applied to various sets of any dimension and patterns with or without self-similarity (Peitgen et
al., 1992). According to K. Falconer (1990) who has discussed the mathematical aspect of the box-counting method,
the fractal dimension D can be derived from the relation:
1=NssD or D—logNs/log(s) where Ns corresponds to the
number of boxes of size (s) needed to cover the structure.
The D value is calculated with the following formula:
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(1)
It is easier to calculate vs (i,j) in a bi-dimensional space as
follows:
(4)
Local fractal dimension
The local measurement of fractal dimensions has been
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
each point (i,j) corresponds to its fractal dimension FD calculated in a window of size (2w + 1)×(2w + 1):
(2)
Textural features derived from the local measurement of
the fractal dimension have been used for segmentation and
classification purposes. Among these techniques, the Fourier
power spectrum (Pentland, 1984), the blanket method
(Dellepiane et al., 1991), differential box-counting (Sarkar
and Chaudhuri, 1994; Chaudhuri and Sarkar, 1995), and the
fractional Brownian motion (Chen et al., 1989; Toennies and
Schnabel, 1994) are the most common. Using a wavelet
technique, M. Datcu et al. (1996) have estimated the local
roughness of various DEMs and provided some examples
showing that the fractal analysis of these DEMs allows
different roughness classes to be distinguished and some
artefacts, due to the computation of elevation data, to be
detected. For computing the local self-similar properties of a
DEM, Y.C. Cheng et al. (1999) have developed a 3D boxcounting method applying the triangular prism surface
method proposed by K.C. Clarke (1986). They observe that
fractal dimension values vary as a function of altitude and
have interpreted this phenomenon as reflecting spatial
variability in erosional potential. This last method differs
from the previous ones because the fractal dimension is
estimated in unit areas and not at each point of the image. In
our approach, the fractal dimension of each pixel of the DEM
surface is calculated inside a moving window centred on this
pixel, by using a 3D box-counting adaptive method. The
advantage of this treatment is that the volume corresponding
to DEM section observed locally is directly related to the
altitude values of the pixels defining the DEM surface. The
treatment does not require any modelling or interpolation of
the surface in order to calculate this dimension.
Procedure
Inside a cube of size s × s × s centred on the pixel of the
studied DEM, the volume corresponding to the surface is
defined by a set of voxels. A voxel is an elementary cube,
the sides of which are equal to the pixel size. Thus, in the
(x,y) space, each point (i,j) of the cube contains a number
vs(i,j) of voxels falling between 0 and s. The total number of
voxels describing the volume is equal to:
(3)
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with
where I is the original image, Ic the value of the central
pixel, ps the pixel size, h a coefficient that defines the vertical resolution, and s the cube size. The cube is partitioned
into boxes of size q varying between 1 and s/2 (fig. 1). Each
of these boxes is considered as filled if at least one voxel is
contained in this box. In other words, the maximum of
vs(i,j)/q in each cell is determined. The computation is done
as follows:
(5)
where Max is a function calculating the maximum. The
fractal dimension corresponds to the inverse of the slope
P=ln(q)/ln(Ns), where q is the size of the box and Ns the
total number of filled boxes. When calculating the slope,
the coefficient of determination, R2, is computed. The estimate of the fractal dimension depends on various factors as
discussed below. As the studied central point is by definition located in the middle of the testing cube, an isolated
point, surrounded by values lower than the base of the cube
s × s × s, generates a vertical line and a line generates a vertical plane. The slopes are respectively expressed by the
following relations: y = -x, y = -2x with R2 = 1. In the same
way, a horizontal surface fills the half of the testing cube
and the corresponding FD = 3 with R2 = 1 (fig. 2). When
the volume presents irregularities its fractal dimension
decreases.
The volume inside the cube depends on the coefficient h.
The transformation into voxels can smooth the surface or,
on the contrary, can enhance the surface roughness. When
h = 1, the volume depends directly on the resolution of the
DEM. Varying the coefficient h in Eq. (4) implies a change
in the number of voxels vs, and thus the volume occupied
inside the cube (fig. 1). High h values accentuate the surface
roughness because the weight of the first term in this formula is more important than the second one (s/2). On the
contrary, low values produce smooth surfaces because the
second term of the formula is predominant. It is then possible to adapt the h value according to the nature of the DEM
surface and the level of information expected.
Determining the most appropriate range of grid sizes is a
common problem for fractal dimension estimation
(Foroutan-pour et al., 1999). The size of the frame used for
the computation can be an even or an odd number; the range
of grid size can be a power of two (Biswas et al., 1998) or a
succession of integers. In our treatments, based on various
tests applied to training zones, the size q of the sliding cube
is chosen to be 12 or 24. Using these sizes, the slope
P=ln(q)/ln(Ns) is more regular when exact dividers are
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.
331
employed. For a flat surface and a volume, the slopes P
obtained are -2 and -3 respectively. Even if these proportions are respected and do not produce a different result after
normalisation, odd sliding windows provide a slight deviation of the slope P for theoretical forms. The exact dividers,
which define the range of grid sizes q, never correspond to
half of the window side and then only a portion of the slope
is taken into account in calculating the fractal dimension.
When applying a 12 × 12 × 12 cubic pattern, five exact
dividers (1, 2, 3, 4, and 6) can be found, excluding any border effect. In the second case (24 × 24 × 24), one can find
seven exact dividers (1, 2, 3, 4, 6, 8, and 12). Treatments that
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
the second produces no bias. As the second term in Eq. 4
depends on the window size s, the use of the smaller window displays small irregularities; the larger window exhibits
the general features of the DEM studied.
The fractal dimension is calculated locally for each surface
pixel in a sliding window centred on this pixel. As discussed
above, the quality of results is greater with an even size than
with an odd one. However, in the first case, the concept of
‘centre’ must be defined because the studied point cannot be
located exactly in the centre but in one of the four points sur-
rounding this centre. Using one of these points leads to the
deviation of the result that favours one orientation.
These remarks imply that the procedure must either calculate the local fractal dimension on each of these four
pixels with the risk of blurring the response when calculating the mean, or define the value of the centre by taking the
average value of the four pixels located at the centre of the
window. Ic in Eq. (4) is calculated according to the second
alternative in order to avoid blurring and to minimize the
computation cost.
Test areas and results
In order to test its ability to extract morphologic features
according to different lithological and tectonic conditions,
the local fractal dimension was applied in two regions. The
first example, located in the region of Vittel (NE France),
characterizes a sedimentary area affected by faults. The second example, located in eastern Anatolia (Turkey), is a
volcanic region. In order to illustrate the performance of the
procedure, the treatments concerning the Vittel region show
mainly the variation of coefficient h and range of grid sizes
effect, whereas the variation of the window size has been
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
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 eastwest-trending Vittel fault is a major
composite fault zone of complex geometry,
resulting from the reactivation of a Palaeozoic discontinuity. In the region located
between this fault and the N45-trending
Esley fault, different groups of faults favour the water circulation that supplies the commercial Vittel spring water
(Sykioti, 1994).
The DEM of this area is an IGN (Institut Géographique
National) product with a horizontal resolution of 50 m and a
conical projection (fig. 3A). The corresponding geological
map (Fig. 3B) has been geometrically corrected according to
this projection in order to be overlaid on the DEM and compared with the local fractal results. The zone size is about
13.75 × 23.55 km (275 lines × 471 columns). Several treatments were applied to the DEM using a cubic size of
12 pixels and modifying only the value of the coefficient h.
According to Eq. (4), decreasing coefficients contribute to
smooth the roughness because a voxel inside the cube corresponds to a bigger hypsometric interval (ps/h). Then, the
procedure detects only high altitude variations. On the contrary, when coefficient h increases, the surface roughness is
accentuated and therefore small altitude variations are
detected as well as large altitude ones.
Thus, the position and extent of the Vittel fault (fig. 3C)
are precisely highlighted and its prolongation westwards
remains clear. When using high h coefficients (i.e., 100),
every elevation difference is taken into account. Then, a
thalweg feature in the testing cube corresponds to a filled
cube with a gully whatever the altitude variation. The fractal dimension obtained is close to 3. On the other hand, a
crest line is represented by a “wall” as formerly described
(see Fig. 2) and its fractal dimension is close to 2 (fig. 3C).
These results are obtained using a window size s = 12 and a
range of grid sizes varying between 1 to s. These ensure the
continuity of the features by producing a standardization of
the estimated fractal dimension and increase the detection of
crest and thalweg features.
In contrast, the major units encountered in the studied
zone were underlined using a low h coefficient but greater
than one, because the region is relatively flat. With a coefficient equal to 2 (fig. 3D), the main structural and geological
units are detected. A high value is obtained when, at the
observation scale, the studied zone inside the moving window is relatively flat. The escarpment that corresponds to
the Rhetian border is strong enough to induce in the moving
window high roughness values codified with low grey tone
values. Thus, the limit of the Rhetian appears clearly and
corresponds exactly to the limits reported in the geological
map. Concerning the Muschelkalk (limestones about 50 m
in thickness) and the Bundsandstein (Vosgian sandstones),
the incision depth of the drainage network is closely related
to the respective nature of these two formations. For this
reason, these units are clearly individualized and the contact
corresponding to the N45-trending Esley fault is obvious, as
well as the stratigraphical contact located at the base of
Muschelkalk escarpment. The results are improved by
superposing the geological map and the image resulting
from the local fractal treatments. The frequency of the local
fractal dimension depends on the sliding window size
(fig. 4). Using a great size, the histogram is unimodal and
becomes multi-modal with decreasing sizes, allowing us to
obtain various classes corresponding to each mode.
The Ararat volcano
With an elevation of 5123 m a.s.l., Mount Ararat is the
largest volcanic centre and the highest point of eastern Anatolia. It corresponds to a compound stratovolcano formed by
the ‘Greater’ and the ‘Lesser’ Ararat. This N150-trending,
elongated massif lies inside a pull-apart basin with a similar
trend (Pearce et al., 1990). This fault system is a horsetail
splay structure and in addition to the right-lateral slip component, these faults have a normal component. The horsetail
splay fault system cuts through the Greater and Lesser
Ararat volcanoes and controls the position of the main eruptive centres of Ararat, as well as the alignment of parasitic
cones (Karakharian et al., 2002). The volcanic activity of
this region continued without interruption until historical
times, possibly reaching its climax during the late Miocene
and Pliocene (6 to 3 Ma). During Quaternary times, the volcanism appears to have been restricted to a few localised
centres (Yilmaz et al., 1998). The youngest eruptions from
the Ararat volcano are older than 10 000 years, and isotopic
dating of the youngest lavas yields an age of 20 000 years
333
(Yilmaz et al., 1998). Recently, geomorphic criteria applied
to Mount Ararat have been used in order to study the tectonics of eastern Anatolia (Adiyaman et al., 2003). These
criteria refer to the morphometric method developed by
F. Garcia-Zuniga and J.-F. Parrot (1998) in order to define
the base line and measure its elongation, and to the “Voxel
wall” procedure (Baudemont and Parrot, 2000). The latter
consists in parameterizing the DEM surface in a real 3D
space using a vertical wall of voxels.
The DEM of the Ararat zone (fig. 5A) is generated using
the method developed by H. Taud et al. (1999). The pixel
size equals 30 m. The size of this area is about 9.57 ×
9.42 km (319 lines × 314 columns). The surface roughness
of the different lava flows has been studied by means of different window sizes. A small window size (s = 12 with h = 4)
reveals that the roughest surfaces mainly correspond to the
more recent features, i.e., to the extent of the lava flows
erupted from the recent parasitic cones located on the western flank. The extent of the eastern gully coming from the
Greater Ararat is clearly observed, as well as the total base
line of the whole edifice recently described by O. Adiyaman
et al. (2003). A greater window size (s = 24 with h = 4) takes
into account the regional structural features. As the local
fractal values decrease in relation to brutal hypsometric
variation as well as the character of the studied shape, a
great window size reveals for instance the presence of isolated volcanic cones (low values), the presence of domes
(high values surrounded by a crown of low values). The
local fractal results depend on the size of the moving window as the frequency of the 2D signal is related to the
surface roughness characterizing the type of studied lava
flow. On the other hand, the value of the general roughness
wavelength is an indicator of the regional tectonics. As illustrated by figures 5C and 5D, the SW quarter of the
stratovolcano presents a globally lower roughness signature,
the eastern limit of which emphasizes the presence of a large
fault zone passing through the Greater Ararat crater. This
fault corresponds to a branch of the horsetail splay structure
described by A. Karakharian et al. (2002). In addition to the
right-lateral slip, these authors assume that these faults have
a normal component. The SW zone that has a weaker roughness response could indicate a huge collapse structure. In
fact, the volcanic material (hypersthene andesite) observed
in this zone (fig. 5B) is globally different from the material
outcropping in the eastern part of the stratovolcano (mainly
basalts and hypersthene basalts).
Moreover, in order to study the effect induced by different
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 Ararat. A) MNT ombré ; D) carte géologique (Yilmaz et al. 1998). 1 :
basalte ; 2 : basalte à hypersthène ; 3 : hyalobasalte ; 4 : andésite
et roches pyroclastiques associées ; 5 : andésite à hypersthène ; 6 : hyaloandésite, 7 : moraines ; 8 : cône de déjection ; 9 :
alluvion ; 10 : complexe de base ;
I. neiges permanentes, Q :Quaternaire ; C : dimension fractale locale avec s = 12 ; D : dimension
fractale locale avec s = 24.
334
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 specific 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 except 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
values are related to alluvium, basement and fans; median
values to basalt, hyalobasalt and hypersthene andesite (the
two formers are strongly connected); and high values to the
remaining items. The same distribution is observed using
window sizes of 24 where different values of h do not generate any variation (fig. 6C). On the contrary, with a 12 window size (fig. 6B) the items are classified differently. For instance, andesites are characterized by low values when s = 12
and high values when s equals 6 or 24. Therefore, using the
mean value, statistical classifications can be obtained by taking into account the results produced by different values of
coefficients s and h.
The items corresponding to large zones seem to be formed
by a heterogeneous ensemble corresponding to different
erupted materials and the presence of different volcanic
flows. High values are registered at the bulged top of the
lava flows because this top appears as a flat surface at the
observed scale, while the lateral flanks present lower values.
This is the case for the isolated flows such as the hypersthene basalt from the Lower Ararat (fig. 5C). Hence, the
high roughness alignments detected by the treatment, which
are not present in the different geological units mapped by
Y. Yilmaz et al. (1998), could be related to different volcanic
flows running from the eruptive centres.
Conclusion
This paper aims at examining the relation between the surface roughness and the geological and geomorphologic
features. It is well known that the fractal dimension can be
used as a parameter characterizing the surface roughness
and the landscape shape. The treatment proposed here measures locally this dimension by an adaptive box counting
3-dimensional method. The algorithm is based on the use of
two parameters: the size s of the moving window and a scaling factor h defining the hypsometric interval taken into
account. This treatment has been applied to a DEM surface
of two regions characterized by a difference concerning
their lithological and structural conditions. The first example is related to the study of a sedimentary and faulted
region in the Vittel area. In this case, two main results have
been obtained according to the different configurations of
the parameters involved in the calculation. As a first result,
the limits of the geological units detected by the local fractal dimension correspond to these limits as they have been
335
mapped on the field. The position of these limits match
either to the border of a homogeneous morphologic feature
or to the transition zone between two different incision
depths of the drainage network. On the other hand, by using
low hypsometric intervals by means of the coefficient h, the
procedure can detect fine features. We can therefore underline the Vittel fault as well as its western prolongation. The
second example focuses on the volcanic region of Mount
Ararat. In this case, the relation between the local fractal
measurements and the different volcanic formations which
have built up this stratovolcano has been studied by using a
statistical approach. Different volcanic classes can be distinguished by using their mean fractal dimension value. On the
other hand, the results provided directly by the local fractal
dimension show the extent of the volcanic flows according
to their surface roughness. Furthermore, a collapse structure
related to the regional strike slip faulting has been detected.
The results obtained with the two training DEMs corresponding to lithological conditions that are completely
different, corroborate the efficiency of the procedure. The
latter allows the retrieval of information about small structural features in flat zones as well as general morphologic
characteristics of rugged areas.
Based on the first results, one can conclude that the
roughness of a surface is strongly correlated with the nature
of the material that forms the geology of a studied region. In
the case of the Vittel region mainly formed by a sedimentary
sequence and studied as a training set, the application of the
treatment by means of a descriptive analysis shows that this
technique is efficient to extract and precise the meaning of
the different morphometric features. In the case of more
complex region such as the Mount Ararat that presents
unknown structures, the extraction and the meaning of the
morphometric features resulting from the treatment, needs
to understand the nature of the filtering result; with such a
goal, it is necessary to analyse precisely the results provided
by using different values of the coefficients s and h, in order
to realize an objective synthesis.
Actually, the local fractal dimension provides useful
information about geological and geomorphologic features.
The proposed method detects different types of structures
according to the observation scale and provides useful
information concerning the geological and geomorphologic
mapping.
Acknowledgements
We thank the reviewers for their helpful criticism and suggestions, the Mexican Petroleum Institute (IMP) and the
Geographical Institute (UNAM) for their support.
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Article reçu le 20 septembre 2004, accepté le 24 août 2005
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Measurement of DEM roughness using the local fractal dimension

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