Universal features of mountain ridge patterns on Earth
We study structure of the mountain ridge systems based on the empirical elevation data collected by Shuttle Radar Topography Mission (SRTM). We consider several prominent mountain ranges from different geological periods and different geographical locations: the Alps, the Pyrenees, the Baetic Mountains, the Scandinavian Mountains, the Southern Alps, the Appalachian Mountains, and a part of the Himalayas. By using a network-based approach, for each mountain range we construct a simple “topographic” network representation (i.e., the ridge junctions as nodes and the ridges connecting them as edges) as well as a “ridge” representation (i.e., the ridges as nodes and the ridge junctions as edges). Then we calculate the main parameters characterizing these networks, like the node degree distribution and the average shortest path length. We observe that the topographic networks inherit the fractal structure of the mountain ranges but do not show any other complex features. In contrast, the ridge networks, while lacking the proper fractality, reveal the power-law cumulative degree distributions (cdfs) with a scaling exponent . By taking into account the fact that the analyzed mountains differ in many properties, like their area, height, age, and geological origin, these values of seem to be universal for the earthly mountainous terrain.
Mountains are among the most common geomorphological structures stahr2015 (). Sometimes they assume a simple form of isolated peaks like the active stratovolcanos or the tepuis, but typically they form whole ranges with a rich internal structure of ridges that form chains of peaks, bifurcate and send side ridges, and disappear in valleys or lowlands. Their complexity stems from the fact that they are subject to the concurrent yet interwoven processes of upthrust, folding, rock-mass displacement along faults, weathering, fluvial and glacial erosion and transport, and the downhill mass movements due to landslides and diffusion, in which an originally flatter but elevated terrain is gradually carved first into shallow rills and then shaped to a combination of more prominent ridges, gullies, and valleys koyama1997 (); babault2012 (); hovius1998 (); korup2007 (); pelletier2003 (); mcguire2013 (); howard1994 (); montgomery2002 (). Since the valleys are associated with the surface drainage systems, except for the rather seldom occurring endorheic areas, an overwhelming majority of the mountain ranges form the hierarchical, tree-like ridge systems complementary to the river networks.
A ridge-valley system of a typical mountain range irrespective of its geographical location assumes a dendritic, parallel or trellis-like structure with a clear hierarchy and self-similarity. It sometimes shows also a considerable amount of regularity that is expressed, e.g., in evenly-spaced parallel ridges/valleys on different hierarchy levels, which amplifies the overall impression of self-similarity perron2009 (). These features put the ridge and drainage systems among the well-recognizable examples of natural fractals. The structural self-similarity has extensively been studied in the drainage systems, for which there exist several well-established power-law relations that functionally link such quantities like stream length and stream basin length, stream length and basin area, and so on tarboton1988 (); dodds1999 (). Probability distributions of some of those quantities, like basin area and stream length also reveal power-law tails. If one reduces mountain ridges to ridge lines, it occurs indeed that their structure mirror the drainage networks on the same territory, so one may expect that the analoguous power-law relations are valid for the ridges.
Like in the case of the river networks, there is some ambiguity in attributing significance to the ridges, thus ordering their hierarchy. In rivers, a significance scheme being the most often used is the Horton-Strahler ordering, in which an th-order stream segment is recognized below a confluence of two th-order stream segments horton1945 (); strahler1957 (). This is the easiest scheme to construct as no additional information on the streams is needed. Other possible schemes that require more data include attributing to a stream below a confluence the order of the subsidiary with a larger discharge or length, and increasing by 1 the order of the other one (together with all its subsidiaries above the considered confluence). In all these schemes, at the beginning of the procedure, the first order is attributed to those streams that start with a source. Obviously, by using each of the schemes, one arrives finally at different orderings.
In the case of the mountain ridges, the situation might even be more diverse, because more possible schemes are conceivable. On the one hand, a direct application of the Horton-Strahler ordering scheme is possible, as well as an application of the ordering scheme based on the ridge lengths. The scheme based on the river discharges can naturally be replaced by schemes based on the average ridge heights or the maximum heights. On the other hand, a largely different scheme may be introduced that is related to the hydrological divisions of the mountainous areas. In this scheme, a top-bottom ordering is started by attributing the first order to a ridge that is the main watershed in some area, then the higher-order ridges are identified according to the successively less important watersheds.
While using each of these ordering schemes is equally justified, it can lead to largely different topologies if ridges are mapped into a network representation. Essentially, the following two representations seem to be the most natural. The first one may be called “topographic”. Here, the ridges are represented by edges and the ridge bifurcations are represented by nodes. Both the binary and the weighted networks can be constructed in this manner with the weights defined by heights (either maximal or average) of the corresponding ridge segments. The most characteristic property of such networks is their direct visual correspondence to the topography of the areas studied (expressed by, e.g., their ridge maps), especially if the edge lengths are proportional to the respective segment lengths. The other network representation is relatively more abstract from the topographic point of view: each node is associated with a ridge and edges connect this node with the nodes representing the parent ridge and the side-ridges. In this case constructing of the weighted network is also conceivable with the weights reflecting the bifurcation point heights. Obviously, the networks in both representations form trees if the corresponding mountain range does not contain any endorheic territory.
In the present work we shall address two questions: (1) what are the topological characteristics of the networks in both representations? and (2) are these characteristics range-dependent or are there any universal topological features of the networks that can reflect some universality of the mountainous terrain structure? The most interesting problem related is whether its self-similarity leaves any marks on the network topology.
Our analysis is based on the empirical terrain elevation data obtained by the 2000 Shuttle Radar Topography Mission (SRTM) srtm (), which for many selected mountainous areas is available from a dedicated internet site viewfinder (). The resolution of the data used is 3” in both the meridional and the zonal direction, which is equivalent to m along a meridian and m along a latitude circle . This resolution seems to be sufficient for creating the ridge maps since it is better than typical separation of the neighbouring ridges. From the SRTM data covering a given mountain range we extract the exact run of the ridge axes by using a modified version of the profile recognition and polygon breaking algorithm chang1998 (); chang2007 (); zhou2007 () and transform the so-obtained ridge maps into networks in both forementioned representations. Then we study their basic topological characteristics.
Our paper is organized as follows: in Section 2 we present details of the ridge-axis detection algorithm, in Section 3 we show the networks created for several mountain ranges and discuss their topology, and in Section 4 we give a summary of the outcomes of the study.
2 Ridge-axis detection method
In our study the crucial step is to recognize properly the structure of the ridges for a given mountain range from the SRTM data. The data is provided as a set of rectangle arrays of size 12011201 points (11 arc degree). Each data point on a grid represents an elevation of the corresponding terrain point. The number of such arrays to be processed depends on the mountains and ranges from 11 (Pyrenees) to 306 (Scandinavian Mountains). The ridges are approximated by their axes being the series of points where the terrain slopes on both opposite sides. Although determining a single ridge axis is relatively easy and many corresponding algorithms exist, identification of all the ridge axes in some geographical region is computationally a demanding task, especially for the mountains that cover large areas, which implies that highly optimized procedures have to be applied.
Among such procedures there is a particularly widely used one called Profile Recognition and Polygon Breaking Algorithm (PPA) chang1998 (); chang2007 (); zhou2007 (). It consists of several steps. First, by using the moving -point line profiles, one obtains a set of straight-line cross-sections of the elevation grid (see Fig. 1(a)), in which each cross-section has one of 4 orientations: N-S, E-W, NE-SW, and NW-SE (Fig. 1(b)). We used profiles of length following Ref. chang2007 (), which is optimal as it allows us both to preserve continuity of the ridge axes (by surpassing the noise effects) and to reduce the computation burden glowacki2015 (). In each position of the moving profile, its central point is tested for being a ridge-axis candidate. If on both sides of this point there is at least one point that has lower elevation than the central point’s one, this central point is considered to be such a candidate (Fig. 1(a)). The candidate points that are neighbours on a cross-section are then connected by a line segment (Fig. 1(c)).
As the resulting structure is overpopulated by intersecting lines, the second step of the profile recognition consists of reducing the number of the line segments by the following actions: (1) eliminating the diagonal segment with a lower average elevation if two such diagonal segments intersect in a grid square, and (2) eliminating those of the two or three neighbouring parallel segments that have lower average elevation and leaving only the one with the highest average elevation (if such parallel segments exist, the segment left is called a reliable segment). The average elevation of a segment is calculated based on elevations of its ends. However, after this step, the structure still has too many segments and cycles that do not reflect any real ridges.
Typically, the so-called polygon breaking algorithm is applied at this phase chang1998 (). It consists of sorting the segments in ascending order according to their average elevation and, by starting from the lowest one, successively removing those segments that are parts of closed polygons. As this step requires much computer resources, certain workarounds are used in order to speed up computation, like, e.g., the dead-end detection that prevents the algorithm from considering those segments that have already been identified as not containing cycles chang2007 (). However, even with those workarounds, the polygon-breaking algorithm significantly bounds the maximum possible region size to be studied and for such mountains like the Alps it demands to allocate too much amount of CPU time even on a supercomputer.
In order to avoid the pitfalls of that algorithm, a more efficient minimum spanning tree approach can be applied bangay2010 (). It requires by up to several orders of magnitude less CPU time. In this approach, the grid of elevation points connected by the line segments, being the output of the profile recognition algorithm, is considered a weighted network, in which the points are the nodes, the segments are the edges. For such a network spanned by nodes, the minimum spanning tree (MST) is a network subset that contains all the nodes, but only edges selected in such a way that the total sum of edge weights is minimum possible barrow1985 (); mantegna1999 (). As mountain ridges are convex structures, in order for them to form an MST, we have to transform elevation into depth relative to some reference level , selected in such a way that for all the grid points with the coordinates . This transformation preserves the metric properties of and . Now we consider the segment average depths as the weights of the corresponding network edges. Out of several different MST-construction algorithms, we choose the Prim’s algorithm prim1957 (). We sort the weights of a network in ascending order and connect the first pair of nodes by the edge with the smallest weight. Then, by considering edges with higher weights one by one, we connect only those node pairs, in which at most one node has already been connected. We proceed with this procedure till there is no unconnected node left.
The output of the MST procedure is a tree consisting of branches representing run of the ridge axes (Fig. 1(d)). However, its structure may still contain spurious double or multiple parallel branches representing in fact the same ridge. It may also contain short branches that are associated with terrain roughness rather than with any real ridge. Therefore, a filtering method called Branch Reduction, being a part of PPA, has to be applied at this step chang2007 (). In this method, all the branches are shortened in order for them to be terminated only with a reliable segment and the branches shorter than half a profile length () are removed together with the segments that are not connected to any larger structure (Fig. 1(e)). We verified on sample maps covering pieces of a few mountain ranges that the ridge-axis trees after the branch reduction step coincide with the actual ridge structure of those mountains with satisfactory precision and, thus, they can be the input to further analysis. As an illustration, in Fig. 2 we show the result of the above ridge-axis-detection procedure applied to the Ligurian Alps.
In order to prepare data for our study, we selected several mountain ranges and obtained the SRTM data for the corresponding geographical regions srtm (). The mountains of interest differ in maximum height and size of the occupied area as well as come from different geological eras. All ranges except one are products of the tectonic plate collision (orogeny), while the Scandinavian Mountains, being a part of the continental passive margin, are believed to originate solely due to the uplift and erosion of the interface between the continental and oceanic litosphere japsen2000 (); chalmers2010 () (see Tab. 1).
|Range||Area (km)||Height (m)||Origin||No. data pts.|
3 Mountain ridge networks
Having decoded the ridge axes in the SRTM grids covering all the mountain ranges of our interest, in the subsequent step we identify the junction points of two or more ridges and associate them with the network nodes of Type 1 (T1). We also associate the ridge loose ends with the nodes of Type 2 (T2). Then we connect each node with its nearest neighbours by a binary edge, provided that two nodes are the nearest neighbours if it is possible to move from one node to the other along a ridge axis without passing through any other node. As a result, a tree graph with nodes and edges is formed representing the complete ridge structure of a given mountain range. We call this network representation a topographic network since the particular ridges can be easily identified by looking at its topology. The topographic networks for the 7 mountain ranges considered in this work are presented in Fig. 3.
In order to present the mountains in the ridge network representation, we have to attribute a hierarchy to the ridges. Out of a number of possibilities, we choose the ridge ordering based on the ridge length. From a network perspective, it is convenient to proxy the metric length of a ridge along its axis with the corresponding network path length. The former is more tedious to calculate since the length of the SRTM grid segments varies from one set of the geographical coordinates to another, while the latter can be calculated straightforward from a network. Such an approach is justified since the side ridges of most ridges are more or less uniformly spaced and the longer the ridge is, the better fulfilled is this correspondence. Therefore, for each mountain range, we calculate the corresponding network diameter and identify the main ridge with one of the paths whose length is equal to the diameter. The main ridge has the ridge order . Then, for each node of the main ridge, we attribute to the longest network path connecting this node with the most distant node in the subnetwork attached to this node. We repeat this procedure with the side ridges of the ridges, attribute them , and so on, until all the ridges are ordered. It is easy to see that, for a given mountain range, the highest ridge order possible is , where denotes floor.
However, it has to be noted that our definition of the main ridge makes it topologically unique with two T2 nodes (as both its ends are loose) unlike for any other ridge with only a single T2 node. In reality, the main ridge typically coincides largely with the axis of a mountain range that goes roughly parallel to the border of the colliding tectonic plates. Thus, the main ridge length is determined by a particular plate size or a fault length, while the length of its side-ridges is typically bounded by a much smaller width of the thrust zone. In many cases this may cause that the main ridge is not only the longest ridge by definition, but also it is incomparably longer than any side ridge. To make the main ridge resembling topologically other ridges, in the topographic representation we split it into two parts at the node where the longest side ridge is attached. After the splitting we obtain two ridges with a single T2 node each that are always the longest ones in the topographic networks. We let both the parts inherit the order. This division is illustrated in Fig. 3 where the parts of the main ridge are distinguished by different colours (blue and red) together with the longest ridge (green).
In the ridge networks each node represents a ridge and each edge connects two nodes in such a way that the corresponding ridges have a common junction point. As a result, each network of this type has a central hub representing the longer part of the main ridge and a secondary hub representing the shorter part, both linked by an edge. Their remaining nearest neighbours represent the ridges of order , their second-nearest neighbours (except for the paths passing through their common edge) represent the ridges of order , etc. By construction, the number of nodes in a ridge network and the number of the Type-2 nodes in the related topographic network are equal: . No similar relation exists that would link with except for the inequality: originating from the fact that about 8% of the junction nodes have a degree . (Note that in the ridge-axis detection procedure we use, the maximum degree possible is (Fig. 1(b)), but this value does not imply that in reality there is no terrain point where more than 6 child ridges detach from their parent ridge). There is also a condition that relates the maximum ridge order with the diameter of a ridge network: . Fig. 4 presents the ridge networks corresponding to the mountain ranges considered here.
By comparing Fig. 4 with Fig. 3, we see that even though both network representations are acyclic and connected, their topologies are significantly different from each other. The topographic networks are distributed networks with small maximum node degree and large average shortest path length , while the ridge networks are highly centralized with a clear hierarchy of nodes, high maximum node degree and the small-world property: . On the other hand, the different mountain ranges set out in the same representation can show strong similarity irrespective of their type, height, and geographical location. For example, there is a simple relation between the maximum ridge order and the number of nodes in the topographic representation: . It is fulfilled for all the complete mountain ranges analyzed here. The sole exception is the Himalayas, only a part of which is considered, and we suppose that their incompleteness might be the cause for this deviation.
The most interesting observation regarding the ridge networks comes from Fig. 5 showing the cumulative node degree distributions. It occurs that for each mountain range the best approximation of their shape is a power-law model: with the scale-free region spanning 2 or even 3 orders of magnitude , depending on the range area and arborescence. (We tested other possible models that resemble partially the scale-free behaviour, like the stretched exponential and the log-normal ones, but the power law model seems to be optimal in this case.) There are some deflections from an ideal power law in some networks for highly connected nodes, but they are rather moderate. For a comparison, we also show the joint cumulative node degree distribution constructed from all 7 mountain groups (Fig. 5(H)) and observe a similar behaviour to any individual case.
The distribution type observed here is not surprising, however, because the power-law relations are typical for fractal objects the mountain ranges are examples of gagnon2006 (). What is surprising actually are the values of the scaling exponents that are common across different mountains: . As the selected mountain groups differ considerably among themselves in many properties, such similarity might suggest that we observe an effect of the geometric packaging of the ridges within some area, which is governed by the properties of the rock material and the geophysical processes that different mountains are universally subject to, limiting the possible ridge spacing pelletier2003 (); mcguire2013 (); tucker1998 (); roering2007 (). Thus, we expect that, for different mountain groups that are not included in our study, the node degree distributions for the ridge networks can reveal similar power-law tails.
It is worth mentioning that the pdf scaling exponent has the values similar () to many other empirical scale-free networks. For example, there are reports that for the WWW link networks ( for the outcoming links is larger than for the incoming ones) albert1999 (), for the Internet connections at the router level faloutsos1999 (), for the actor co-ocurrence in films barabasi1999 (); amaral2000 (), for the scientific collaboration networks newman2001 (); drozdz2017 (), for the scientific paper citation networks redner1998 (), for the word-coocurrence networks ferrer2001 (), for the protein interaction networks jeong2001 (), for the biochemical cellular pathway networks jeong2000 (), and for the currency comovement networks kwapien2009 (). It should be kept in mind, however, that such a similarity might be superficial only since the mechanisms leading to the power-laws can be distinct in each case.
As the mountain ranges have fractal structure, we are curious whether this property can also be observed in their network representations. These representations are defined in an abstract space, in which the position of a given node is determined by its relation to other nodes in the ridge configuration rather than by any system of spatial coordinates. Therefore, the most adequate method of quantifying network fractality in this case is the renormalization-based approach, in which a network is divided into node clusters (“boxes”) and subsequently coarse-grained on various “length” scales song2005 (). In this method one first defines the distance parameter that bounds the path length between the nodes belonging to the same cluster. Next, a seed node that is considered as a center of the first cluster is randomly chosen. Then, the number of the clusters is calculated after partitioning the network into the node clusters in such a way that the minimum path length between the nodes belonging to the same cluster is not longer than . The same is done independently for different draws of the seed node and the average number of clusters is determined. In a subsequent step, the renormalization, the clusters are replaced by single nodes (linked if there existed at least one connection between the nodes belonging to the corresponding clusters) and a new network consisting of such nodes is formed and subject to the analogous partitioning and replacement procedure. These steps can be repeated until the network is reduced to a single cluster.
However, since the consecutive renormalized networks have similar topological properties to the original network (for example, the node degree distributions show the same scale-free behaviour) song2005 (), it is sufficient to calculate for different values of the parameter . The network is fractal if the following power-law relation holds:
where is the number of nodes in the network. In this case the parameter is considered the fractal dimension of the network. It was documented in literature that the empirical networks can show either the fractal scaling (1) or a non-fractal behaviour of , e.g., its exponential decay song2005 (). Here we calculate this quantity for both the topographic and the ridge network representations for the 7 mountain groups. The results for the topographic networks showing the scale-free dependence with the scaling exponents for being similar for each mountain group are displayed in Fig. 6(a). In contrast, the ridge networks do not present the power-law dependence (1). Instead, the exponential relation can approximate the empirical data for all the values of in this case (Fig. 6(b)). It is interesting that although each representation has different properties: fractal in the case of the topographic networks and non-fractal in the case of the ridge networks, different mountain ranges show similar behaviour if the same representation is considered. These results are not surprising, though. On the one hand, as the topographic networks inherit basic features of the ridge structure of the real mountains, the network fractality comes as its consequence. On the other hand, the scale-free character of the ridge networks implies the existence of massive hubs, and this, by forming highly populated node clusters, limits the total number of clusters in the network even for the moderate values of and produces the exponential decay of song2005 ().
In this work, we considered the ridge structure of several mountain ranges. We chose a network-based approach and focused on the following network representations: the topographic representation, in which the ridge junctions and the ridge ends were depicted by the network nodes, while the edges linked those nodes whose corresponding terrain points were directly connected with a ridge, and the ridge representation, in which the ridges were depicted by nodes and the ridge junctions were depicted by edges. Based on the SRTM data with 3” resolution and by using the ridge-axis identification algorithms, we transformed the following mountain ranges: the Alps, the Baetic Mountains, the Pyrenees, the Scandinavian Mountains, the Himalayas, the Southern Alps, and the Appalachian Mountains into both types of networks. We found that the topographic networks are distributed networks with the long average shortest paths and a fractal structure, probably inherited from the topographic fractality of the mountains, and that the ridge networks are scale-free with the power-law exponent irrespective of the height, area, age, and type of the mountains. This result is especially interesting if one takes a look on the topography of such morphologically different ranges as the Alps and the Appalachian Mountains. The former consist mainly of the dendritic systems of ridges and valleys, while the latter in a large part have a form of the long, alternating ridges and valleys with a trellis-like drainage structure. Therefore, we believe that the topological similarity of the ridge networks for all the mountain ranges considered here might stem from the universality of the mountain-shaping processes, like erosion, and also from the properties of the rocks that the mountains are built of, which set limits on the possible hillslopes and the distances between the neighbouring ridges, thus defining in this way the fractal properties of any mountainous terrain.
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