Graph theory—Recent developments of its application in geomorphology

Heckmann, Tobias; Schwanghart, Wolfgang; Phillips, Jonathan D.

doi:10.1016/j.geomorph.2014.12.024

Cited by 131 publications

(115 citation statements)

References 141 publications

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“…We represent the river network as a directed acyclic graph G(V E) (Heckmann et al, 2015). The set of graph nodes V i are pixel centers connected by directed edges E(ij ).…”

Section: Appendix Amentioning

confidence: 99%

Bumps in river profiles: uncertainty assessment and smoothing using quantile regression techniques

2017

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Abstract. The analysis of longitudinal river profiles is an important tool for studying landscape evolution. However, characterizing river profiles based on digital elevation models (DEMs) suffers from errors and artifacts that particularly prevail along valley bottoms. The aim of this study is to characterize uncertainties that arise from the analysis of river profiles derived from different, near-globally available DEMs. We devised new algorithmsquantile carving and the CRS algorithm -that rely on quantile regression to enable hydrological correction and the uncertainty quantification of river profiles. We find that globally available DEMs commonly overestimate river elevations in steep topography. The distributions of elevation errors become increasingly wider and right skewed if adjacent hillslope gradients are steep. Our analysis indicates that the AW3D DEM has the highest precision and lowest bias for the analysis of river profiles in mountainous topography. The new 12 m resolution TanDEM-X DEM has a very low precision, most likely due to the combined effect of steep valley walls and the presence of water surfaces in valley bottoms. Compared to the conventional approaches of carving and filling, we find that our new approach is able to reduce the elevation bias and errors in longitudinal river profiles.

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“…We represent the river network as a directed acyclic graph G(V E) (Heckmann et al, 2015). The set of graph nodes V i are pixel centers connected by directed edges E(ij ).…”

Section: Appendix Amentioning

confidence: 99%

Bumps in river profiles: uncertainty assessment and smoothing using quantile regression techniques

2017

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“…Another promising field of research is the application of graph theory, which provides a robust mathematical framework for describing networks such as sediment cascades (Heckmann and Schwanghart, 2013;Heckmann et al, 2015;Cossart, 2016). Graph theory is applied to model a network structure as nodes (representing sediment sources, sediment stores and the outlet) connected by edges (representing linkages by a geomorphological process).…”

Section: Graph Theory Applications To Structural Connectivitymentioning

confidence: 99%

“…Both spatial and topological configurations of the network and the fluxes associated with the respective edges are responsible for the sediment delivery at the outlet (Heckmann and Schwanghart, 2013;Heckmann et al, 2015). The goal is thus to obtain a pattern that can be described by algebraic tools (typology of linkages, identification of local sinks, etc.)…”

Section: Graph Theory Applications To Structural Connectivitymentioning

confidence: 99%

“…One promising field of research has been opened up by the application of graph theory, which offers mathematical tools to analyze statistically the assemblages of all the components of a sediment cascade. It is a spatially explicit analysis since nodes and edges are spatial objects and since distance plays a role in the modeling (Heckmann and Schwanghart, 2013;Heckmann et al, 2015). This mathematical approach is based on a graph representation of the hydrological network that has been used for decades to describe watercourses shape and organization (Strahler, 1957;Chorley and Kennedy, 1971;Shreve, 1974).…”

Section: Introductionmentioning

confidence: 99%

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Assessment of structural sediment connectivity within catchments: insights from graph theory

Cossart

Fressard

2017

Earth Surf. Dynam.

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Abstract.To describe the sedimentary signal delivered at catchment outlets, many authors now refer to the concept of connectivity. In this framework, the sedimentary signal is seen as an emergent organization of local links and interactions. The challenge is thus to open the black boxes that remain within a sediment cascade, which requires both accurate geomorphic investigations in the field (reconstruction of sequences of geomorphic evolution, description of sediment pathways) and the development of tools dedicated to sediment cascade modeling. More precisely, the development of tools devoted to the study of connectivity in geomorphology is still in progress, although graph theory offers promising perspectives (Heckmann and Schwanghart, 2013). In this paper, graph theory is applied to abstract the network structure of sediment cascades, keeping only the nodes (sediment sources, sediment stores, outlet) and links (linkage by a transportation agent), represented as vertices and edges. From the description of the assemblages of sedimentary flows, we provide three main indices to explore how small-scale processes may result in significant broad-scale geomorphic patterns. The main hypothesis guiding this work is that the network structure dictates how sediment inputs from various sources interact at tributary junctions and finally at the outlet of a cascading system. First, we use the flow index to assess the potential contribution of each node to the sediment delivery at the outlet. Second, we measure the influence of each node regarding how it is accessible from both sediment sources and the outlet (using the Shimbel index). Third, we propose a new connectivity index named "Network Structural Connectivity index" (NSC) revealing whether the potential contribution of a node is lower or higher than expected from its location within the network. These indices are first computed for a conceptual sediment cascade network and then applied to a catchment located in the southern French Alps. We demonstrate that this index may be used to simulate sediment transfer and help in identifying the hotspots of geomorphic change. In the present case, we try to predict how a sediment cascade may be impacted by an edge disruption or a reconnection.

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“…In the following we briefly review some basic terminology, algorithms, and assumptions pertinent to our application of graph theory and road networks. For a more detailed introduction into graph theory we refer to Gross and Yellen (2005) and Heckmann et al (2015). A graph G(V E) is the mathematical representation of a network defined by two disjoint sets of vertices V and links E. A link is defined by two vertices u and v, and two vertices are adjacent to each other when a link {u, v} connects them.…”

Section: Graph Theorymentioning

confidence: 99%

Roads at risk: traffic detours from debris flows in southern Norway

Meyer

Schwanghart

Korup

et al. 2015

Nat. Hazards Earth Syst. Sci.

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Abstract. Globalisation and interregional exchange of people, goods, and services has boosted the importance of and reliance on all kinds of transport networks. The linear structure of road networks is especially sensitive to natural hazards. In southern Norway, steep topography and extreme weather events promote frequent traffic disruption caused by debris flows. Topographic susceptibility and trigger frequency maps serve as input into a hazard appraisal at the scale of first-order catchments to quantify the impact of debris flows on the road network in terms of a failure likelihood of each link connecting two network vertices, e.g. road junctions. We compute total additional traffic loads as a function of traffic volume and excess distance, i.e. the extra length of an alternative path connecting two previously disrupted network vertices using a shortest-path algorithm. Our risk metric of link failure is the total additional annual traffic load, expressed as vehicle kilometres, because of debris-flow-related road closures. We present two scenarios demonstrating the impact of debris flows on the road network and quantify the associated path-failure likelihood between major cities in southern Norway. The scenarios indicate that major routes crossing the central and north-western part of the study area are associated with high link-failure risk. Yet options for detours on major routes are manifold and incur only little additional costs provided that drivers are sufficiently well informed about road closures. Our risk estimates may be of importance to road network managers and transport companies relying on speedy delivery of services and goods.

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Graph theory—Recent developments of its application in geomorphology

Cited by 131 publications

References 141 publications

Bumps in river profiles: uncertainty assessment and smoothing using quantile regression techniques

Bumps in river profiles: uncertainty assessment and smoothing using quantile regression techniques

Assessment of structural sediment connectivity within catchments: insights from graph theory

Roads at risk: traffic detours from debris flows in southern Norway

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