# Catchment basin versus Mountain range tessellations from DTMs for islands: Lesvos, Kerguelen, Crete, Cyprus, New-Caledonia, Formosa and Sri Lanka

Christian Depraetere, Serge Riazanoff

► To cite this version:

Christian Depraetere, Serge Riazanoff. Catchment basin versus Mountain range tessellations from DTMs for islands: Lesvos, Kerguelen, Crete, Cyprus, New-Caledonia, Formosa and Sri Lanka. Mediterranean Geosciences Union (MedGU) conference, 27 - 30 November 2023, Istanbul, Turkey., Eurasia Institute of Earth Sciences; Istanbul Technical University Istanbul, Nov 2023, Istanbul, Turkey. [https://doi.org/10.44269/959](https://doi.org/10.44269/959)

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# Catchment basin versus Mountain range tessellations from DTMs for islands: Lesvos, Kerguelen, Crete, Cyprus, New-Caledonia, Formosa and Sri Lanka

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Christian Depraetere<fnref n="1" />, Serge Riazanoff<fnref n="2" />

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Email: corresponding author christian.depraetere@ird.fr

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## Abstract.

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The aim is to compare the geomorphometric signatures of catchment basins (using Steepest Descent lines converging to Outlets, SDO) and mountain ranges ('massifs', using Steepest Ascent lines Toward Summits, SATS) on a set of islands in various topographical, geological and morphoclimatic settings: Lesvos (1656 km<fnref n="2" />, 39.51°N, max elevation 903 meters), Kerguelen (6761 km<fnref n="2" />, 49.19°S, 1792 m), Crete (8325 km<fnref n="2" />, 35.31°N, 2442 m), Cyprus (9230 km<fnref n="2" />, 35.13°N, 1971 m), 4205 m), New-Caledonia (16 469 km<fnref n="2" />, 21.24°S, 1621 m), Formosa (35 901 km<fnref n="2" />, 23.6°N, 3917 m), and Sri Lanka (66 375 km<fnref n="2" />, 7.87°N, 2518 m). The catchment and mountain tessellations provide complementary dual partitions of each landscape. For comparability, the same SRTM 3 arc second DTM is used. While the SDO basin tessellation depends mostly on hydrological processes, the dual fragmentations given by the SATS are conversely more related to the lithology and the tectonics of mountainous areas. The main hypothesis is that there are relationships between environmental factors and pattern signatures for both catchment basins and mountain ranges.

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The tessellation of catchment basins uses the D8 steepest descent method available in most GIS toolboxes. On the other hand, our *ad hoc* stepwise algorithm (MAPAM) delineates elementary hills up to the major mountain ranges associated with highest summits. The outcome of merging SDO and SATS is a combined tessellation in which tiles are landform units (LU2s) collecting water from the same slope within the same catchment and collected by the same section of river (thalweg) between two confluences. The statistical textural properties of the tiles are analyzed for both SDO and SATS tessellations and the resulting LU2s combined patterns on the seven islands are compared using the area-number method also called "Korčak Number-Area law". The size distributions are fitted with the power law functions $N=\alpha \cdot A^{-\beta}$. The parameter $\beta$ varies with geological context and morphogenic climatic factors.

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Coupled with other geostatistical approaches for land or bathymetric surface analysis (i.e. variograms, geomorphometry, etc.), this combined SDO and SATS tessellation contributes to innovative semi-automatic survey and mapping methods for a wide range of environmental applications, for instance topographical habitats, distributed hydrological modeling, geomorphologic risk assessment and transportation planning optimization.

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Keywords: DTM, steepest descent and ascent, tessellation, Korčak exponent, islands, catchments, mountains.

<footnote n="1">¹ Institut de Recherche pour le Développement, Maison de la Télédétection, Montpellier 34000, France</footnote>

<footnote n="2">² VisioTerra, Scientific consulting for earth observation, rue Albert Einstein, 77420 Champs-sur-Marne, France</footnote>
41 **1 Introduction**

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DTM algorithms contribute to deciphering the terrestrial landform patterns resulting from complex and en-
tangled geomorphologic processes. Several pioneering researches on the topology of closed contour line maps
date from the mid-nineteenth century: Reech (1858, [1]), Cayley (1859, [2]) and especially Maxwell (1870, [3]),
considered as the "first quantitative geomorphologist" [4]. Our work with GIS tools builds upon their ideas. By
analogy with physics, the following research deals with properties of the "vector field" as the codomain of the
"scalar field" of the topography given by DTM. Two kinds of topographical "footprints" are hereby extracted
from the DTM (Map 1A) by specific algorithms; the "downprint" (Map 1B) referring to a tessellation of catch-
ment basins related to outlets along the shores (steepest descent to lowest points) and the "upprint" (Map 1C) as
a pattern of converging steepest ascent lines toward highest points such as small bumps, hill tops or mountain
summits, with embedded magnitudes [5]. These landform units are defined as "massifs" (as "anti-catchments"
related to high points) and are complementary to catchments. Both types can be labelled "land units".

<figure><img src="image_2.png" /><figcaption>**Map 1** Altitudes from DTM 3 arc second (A); tessellations of catchment basins (B) and massifs (C) on Lesvos island</figcaption></figure>

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The tessellations will be applied to 7 islands of different shapes (Map 2A and B), with areas ranging from 1600 to 63 000 km², located in various morphoclimatic and geologic environments: Mediterranean (Lesvos, Crete and Cyprus), tropical (New-Caledonia and Formosa), equatorial (Sri Lanka) and subpolar (Kerguelen). For statistical analysis the Korčak number-area law (or “rule”) providing the synthetic Korčak exponent is used to quantify the fragmentation patterns.

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<figure><img src="image_3.png" /><figcaption>**Map 2.** Location and main characteristics of the 7 selected islands (Source BIM [16])</figcaption></figure>

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(This crisscrossing of lines for instance on Lesvos island (Map 1C) tends to be associated with faults and weak rocks. For obvious reasons, these minimum elevation lines are often closely related to the transportation networks because trails, roads (Map 2D) and railways tend to follow the valley bottoms and go through the saddle connecting them.)

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66 2 Data and Methods

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Data: the results are calculated from the same SRTM 3 arc second DTM for all 7 islands, with a resolution of about 100 meters and a vertical accuracy of the order of few meters, suitable for hilly and mountainous areas but not for gently undulated plains. A DTM with better landform rendering (i.e. Lidar) would give the same tessellation on large landform units but more details on flattish bumpy areas.

Methods for catchment basin and massif tessellations: geoscientists are familiar with the D8 method calculat-
ing Steepest Descent lines converging to Outlets (SDO) that allows the extraction of catchment basins and their
watersheds shaping the topographic “downprint” of a region (i.e. Map 1B). Conversely, the delineation of
Steepest Ascent lines Toward Summits (SATS) are not often computed despite being the dual graph of SDO and
providing the complementary “upprint” of the topography (Map 1C): the landform units correspond to bumps,
hills and mountains. These upprint units will be defined as “massifs” whatever their size and relief amplitude.
The ad hoc stepwise algorithm MAPAM (MAssifs PArtitioning Method [6] [7] [8]) for massif tessellation ap-
plies the same D8 method [9] on the inverted DTM with elevation turned upside down, as a first step (Initial or
elementary massifs of order 0). At further steps, the massifs are merged step by step (Massifs of order 1, 2, and
so on) according to the pathways through saddle points presenting the most amplitude from surrounding high
points. At the final stage, all the massifs are merged into one unit associated with the highest summit. As clearly
shown on Rasemann's diagram of Figure 1, it is worth noticing that the SATS are not an inversion of SDO de-
spite it seems counter intuitive at first row.

<figure><img src="image_4.png" /><figcaption>**Figure 1** Slope lines derived up (SA) and down (SD) the slope take different courses when plan curvature occurs
(After from [10] and [5])</figcaption></figure>

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(Note that the MAPAM algorithm was conceived in 2004 [6] without knowing that James Clerk Maxwell had
the same idea in 1870 [3]: according to him, the secondary summits around a main summit could be topological-
ly considered as vertical "saddle points" - which is exactly what MAPAM is doing.)

The properties of size (Area A) distributions (Number N) of both catchments and massifs are summarized by
the Korčák Number (N) - Area (A) empirical law [11]

$$
N(A > A_0) = \alpha \cdot A_0^{-\beta}, \text{ with } N \text{ the number of objects with area } A \text{ greater than area } A_0 \quad (1)
$$
By convention, the parameter $\beta$ is called Korcak's exponent of the power function fitted to the size distribution. This exponent is compared for the tessellation patterns of catchments and massifs on the 7 islands. For instance on Lesvos island (Figure 2), the parameter $\beta$ is equal to 0.623 for catchments and 0.839 for massifs, with the SRTM 3 arc second DTM. For this island, large catchments with $A>15 \text{ km}^2$ do not conform to the distribution functions and small catchments ($A<0.1 \text{ km}^2$) and massifs ($A<1 \text{ km}^2$) are under-sampled. It has to be mentioned that the Korcak exponent is a non-fractal descriptor for topographic patchiness and geographic features [12] [13] contrariwise with one of Mandelbrot's hypothesis [14].

<figure><img src="image_5.png" /><figcaption>Figure 2 Korcak number-area laws for catchment basins (Map 1B) and massifs (Map 1C) on Lesvos island (NB: the results show minor discrepancies between the 1 arc and 3 arc second SRTM DTM)</figcaption></figure>

## 3 Results: The number-area law and the Korcak exponents on the 7 islands

Figure 3 shows that according to the number-area method the size distributions of catchments and massifs (with $A>1 \text{ km}^2$ for both) are quite similar despite the differences of size, shape and environmental contexts of each island. The main points about the fitted power functions (See (1)) are as follows:

* The parameter $\alpha$ is the number $N$ of land units with $A>1 \text{ km}^2$ (which depends mostly on the size of islands);

* The parameter $\beta$ (or Korcak exponent) is the “slope” of the size distribution on a log-log graph and reflects the homogeneity of patchiness over a class of areas (Figure 3A): i.e. on Lesvos, $\beta=0.62$ for the catchments with $1<A<15 \text{ km}^2$, $\beta=0.93$ for the massifs with $1<A<120 \text{ km}^2$ (Figure 3B);

* The size distributions are “steeper” for massifs ($0.86<\beta<0.93$) compared to catchments ($0.45<\beta<0.75$). Fits are best for land units $1<A<100 \text{ km}^2$;

* Above a threshold value of $A$, the largest land units depart from the distribution functions calibrated on smaller land units: i.e. on Lesvos, for large catchments with $A>15 \text{ km}^2$, and massifs with $A>120 \text{ km}^2$ (Figure 3B).
<figure><img src="image_6.png" /><figcaption>**Figure 3** Size distribution properties for catchments (Blue) and massifs (Red) with the Korcak number-area method applied to the 7 islands. A./ Comparison between the 7 islands (B, C, D, E, F, G) B./ Lesvos island (idem **Figure 2**)</figcaption></figure>
127 **4** Discussion

128 For comparison, **Figure 4** plots the Korcak exponents combining both catchments ($\beta_c$) and massifs ($\beta_m$) from the empirical results of Figure 3A. Depending on various factors, each island has a different ($\beta_c$, $\beta_m$) “signature”. 129 130 The sharpest contrast for catchment signatures is between Sri Lanka ($\beta_c$ =0.45, equatorial, oval shape) and Kerguelen ($\beta_c$ =0.75, sub polar, coastline with fjords). The massif signatures tend to be similar with an average value $\beta_m$=0.89, except for lower values for Crete ($\beta_m$=0.86, Mediterranean), Kerguelen ($\beta_m$=0.86) and higher for Lesvos ($\beta_m$=0.93, Mediterranean). 131 132 133 134

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<figure><img src="image_7.png" /></figure>

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<caption>**Figure 4** Combined ($\beta_c, \beta_m$) signatures of catchments and massifs according to the Korcak exponents</caption>

137 These empirical findings from a limited set of 7 islands cannot assume to be representative of all land patterns 138 but they give insights to more theoretical works. For instance, Figure 5 could be a theoretical prototype for the 139 most frequent landform pattern signatures ($\beta_c, \beta_m$) characterizing islands. It could apply also to mountain ranges 140 but not to non-islands where border effects cause distortions. It is not suitable for karstic or endorheic regions. 141

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<figure><img src="image_8.png" /></figure>

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<caption>**Figure 5** Theoretical prototype of landform patterns ($\beta_c, \beta_m$) on islands with Korcak exponent $\beta$ from Figure 4.</caption>

144 The outcome of overlapping SDO with SATS is a combined tessellation in which tiles are landform units 145 (LU2s) collecting water from the same slope within the same catchment and collected by the same section of 146 river (thalweg) between two confluences. Notice that the LU2 tesselation is more accurate if catchments are split 147 into sub-catchments according to a pre defined river network.
148 **5 Conclusions**

149 As a dual pattern of catchment basin tessellations, the massif patches give complementary information
150 on the relationships of hydrological processes with geology and tectonics.

151 Topographical downprint and upprint power functions are relevant to estimate size distributions of
152 catchment basins and massifs on islands.

153 This quantitative approach to landforms should be combined with other geomorphometric methods
154 such as variograms, hypsometric curves, geomorphological hydrographs, etc.

155 The empirical Korcak number-area rule is valuable for a theoretical approach to these landforms.

156 Beyond GIS, geomorphological mapping and theoretical research, the methods are relevant for applica-
157 tions in several scientific domains such as hydrology, geomorphology, potamology and structural geol-
158 ogy but also transportation operational research and even geoarchaeology [8][15].

159 References

160 1. Reech, M.: Propriété générale des surfaces fermées. Ecole Polytech. Jour. Paris, 1858, v. 37, p. 169–178 (1858).

161 2. Cayley, A.: On contour lines and slope lines. Phil. Mag., 1859, v. 18, p. 264–268 (1859).

162 3. Maxwell, J. C.: On hills and dales. Phil. Mag., 1870, v. 40, p. 421–427 (1870)

163 4. Cockbain, A. E.: James Clerk Maxwell, the first quantitative geomorphologist?: Math. Geol., 1980, v. 12, p. 615–616.

164 5. Depraetere C., Riazanoff S.: Downprint and upprint of landforms from DEM: the case of the volcanic Acores islands. 10th International Conference of the International Association of Geomorphologists (IAG), Coimbra, Portugal, 12-16 September 2022. (2022). https://meetingorganizer.copernicus.org/ICG2022/ICG2022-631.html

167 6. Depraetere C., Riazanoff S.: The new Digital Elevation Model data set from the Shuttle Radar Topography Mission: hydrogeomorphological applications in the Ohrid region (Albania, Greece and Macedonia). Conference on Water Observation and Information System for the Balkans Countries BALWOIS, Ohrid, Northern Macedonia, 25-29 May 2004 (2004). https://igm.univ-mlv.fr/~riazano/publications/20040525_BALWOIS_Conference.pdf

171 7. Depraetere, C.: Introduction à l'hydrogéomorphométrie : études des relations entre le modelé des formes de terrain et les processus hydrologiques dans des contextes insulaires. Habilitation à diriger des recherches (HDR), 2013, Université Paris-Diderot PVI. (2013). https://www.espace-dev.fr/wp-content/uploads/2020/11/HDR_Christian_Depraetere_Document_Inedit_Novembre_2013_Part1.pdf.

174 8. Depraetere, C., Riazanoff, S.: A stepwise massif partitioning algorithm based on steepest ascent method from DEM: Application to potential prospection of new archaeological sites. AIP Conf. Proc. 2611, 090002 (2022), 4–7 September 2021, Heraklion, Greece https://doi.org/10.1063/5.0128461

177 9. Shin, S., Paik, K.: An improved method for single flow direction calculation in grid digital elevation models. Hydrol. Proc. 31 (8), 1650-1661. (2017) https://doi.org/10.1002/hyp.11135

179 10. Rasemann, S.: Geomorphometrische Struktur eines mesoskaligen alpinen Geosystems. Dissertation zur Erlangung des Doktorgrades (PhD) der Mathematisch-Naturwissenschaftlichen, Friedrich-Wilhelms-Universität Bonn, Bonn 2003, 339 pages. (2003). http://tolu.giub.uni-bonn.de/grk/download/phd_rasemann.pdf

182 11. Korčák, J.: Deux types fondamentaux de distribution statistique, Bull. Inst. Int. de Statistique III: 295-299. (1940).

183 12. Imre, A.R., Novotný, J., Rocchini, D.: The Korcak-exponent: A non-fractal descriptor for landscape patchiness. Ecolo-
184 gical complexity. 12. 70-74. https://doi.org/10.1016/j.ecocom.2012.10.001 (2012).

185 13. Imre, A., Novotný, J.: Fractals and the Korcak-law: a history and a correction. EPJ H 41, 69–91 2016 (2016). https://doi.org/10.1140/epjh/e2016-60039-8

187 14. Mandelbrot, B.B.: Stochastic models for the Earth’s relief, shape and fractal dimension of coastlines, and number-area rule for islands, Proc. Natl. Acad. Sci. USA 72: 3825-3838. (1975)

189 15. Depraetere, C., T. Moutsiou, T.: Potential location of early Neolithic sites in Crete and Cyprus: A GIS-based regional potamological approach. 17th International Conference on Environmental Science and Technology, Athens, Greece, 1-4 September 2021. (2021) https://cms.gnest.org/sites/default/files/Proceedings/cest2021_00483/cest2021_00483.pdf

192 16. BIM (Base de données Insulaires Mondiale) Homepage: https://doi.org/10.23708/T37S0K