### Refine

#### Keywords

- Graph theory (3)
- Complexity (1)
- Digital elevation model (1)
- Digital terrain analysis (1)
- Drainage networks (1)
- Fuzzy (1)
- Geomorphic coupling (1)
- Geomorphic systems (1)
- Geosciences (1)
- Modelling (1)

Through their relevance for sediment budgets and the sensitivity of geomorphic systems, geomorphic coupling and (sediment) connectivity represent important topics in geomorphology. Since the introduction of the systems perspective to physical geography by Chorley and Kennedy (1971), a catchment has been perceived as consisting of landscape elements (e.g. landforms, subcatchments) that are coupled by geomorphic processes through sediment transport. In this study, we present a novel application of mathematical graph theory to explore the network structure of coarse sediment pathways in a central alpine catchment. Numerical simulation models for rockfall, debris flows, and (hillslope and channel) fluvial processes are used to establish a spatially explicit graph model of sediment sources, pathways and sinks. The raster cells of a digital elevation model form the nodes of this graph, and simulated sediment trajectories represent the corresponding edges. Model results are validated by visual comparison with the field situation and aerial photos. The interaction of sediment pathways, i.e. where the deposits of a geomorphic process form the sources of another process, forms sediment cascades, represented by paths (a succession of edges) in the graph model. We show how this graph can be used to explore upslope (contributing area) and downslope (source to sink) functional connectivity by analysing its nodes, edges and paths. The analysis of the spatial distribution, composition and frequency of sediment cascades yields information on the relative importance of geomorphic processes and their interaction (however regardless of their transport capacity). In the study area, the analysis stresses the importance of mass movements and their interaction, e.g. the linkage of large rockfall source areas to debris flows that potentially enter the channel network. Moreover, it is shown that only a small percentage of the study area is coupled to the channel network which itself is longitudinally disconnected by natural and anthropogenic barriers. Besides the case study, we discuss the methodological framework and alternatives for node and edge representations of graph models in geomorphology. We conclude that graph theory provides an excellent methodological framework for the analysis of geomorphic systems, especially for the exploration of quantitative approaches towards sediment connectivity.

The assessment of uncertainty is a major challenge in geomorphometry. Methods to quantify uncertainty in digital elevation models (DEM) are needed to assess and report derivatives such as drainage basins. While Monte-Carlo (MC) techniques have been developed and employed to assess the variability of second-order derivatives of DEMs, their application requires explicit error modeling and numerous simulations to reliably calculate error bounds. Here, we develop an analytical model to quantify and visualize uncertainty in drainage basin delineation in DEMs. The model is based on the assumption that multiple flow directions (MFD) represent a discrete probability distribution of non-diverging flow networks. The Shannon Index quantifies the uncertainty of each cell to drain into a specific drainage basin outlet. In addition, error bounds for drainage areas can be derived. An application of the model shows that it identifies areas in a DEM where drainage basin delineation is highly uncertain owing to flow dispersion on convex landforms such as alluvial fans. The model allows for a quantitative assessment of the magnitudes of expected drainage area variability and delivers constraints for observed volatile hydrological behavior in a palaeoenvironmental record of lake level change. Since the model cannot account for all uncertainties in drainage basin delineation we conclude that a joint application with MC techniques is promising for an efficient and comprehensive error assessment in the future.

Applications of graph theory have proliferated across the academic spectrum in recent years. Whereas geosciences and landscape ecology have made rich use of graph theory, its use seems limited in physical geography, and particularly in geomorphology. Common applications of graph theory analyses of connectivity, path or transport efficiencies, subnetworks, network structure, system behaviour and dynamics, and network optimization or engineering all have uses or potential uses in geomorphology and closely related fields. In this paper, we give a short introduction to graph theory and review previous geomorphological applications or works in related fields that have been particularly influential. Network-like geomorphic systems can be classified into nonspatial or spatially implicit system components linked by statistical/causal relationships and spatial units linked by some spatial relationship, for example by fluxes of matter and/or energy. We argue that, if geomorphic system properties and behaviour (e.g., complexity, sensitivity, synchronisability, historical contingency, connectivity etc.) depend on system structure and if graph theory is able to quantitatively describe the configuration of system components, then graph theory should provide us with tools that help in quantifying system properties and in inferring system behaviour. (C) 2015 Elsevier B.V. All rights reserved.

Graph theory has long been used in quantitative geography and landscape ecology and has been applied in Earth and atmospheric sciences for several decades. Recently, however, there have been increased, and more sophisticated, applications of graph theory concepts and methods in geosciences, principally in three areas: spatially explicit modeling, small-world networks, and structural models of Earth surface systems. This paper reviews the contrasting goals and methods inherent in these approaches, but focuses on the common elements, to develop a synthetic view of graph theory in the geosciences. Techniques applied in geosciences are mainly of three types: connectivity measures of entire networks; metrics of various aspects of the importance or influence of particular nodes, links, or regions of the network; and indicators of system dynamics based on graph adjacency matrices. Geoscience applications of graph theory can be grouped in five general categories: (1) Quantification of complex network properties such as connectivity, centrality, and clustering; (2) Tests for evidence of particular types of structures that have implications for system behavior, such as small-world or scale-free networks; (3) Testing dynamical system properties, e.g., complexity, coherence, stability, synchronization, and vulnerability; (4) Identification of dynamics from historical records or time series; and (5) spatial analysis. Recent and future expansion of graph theory in geosciences is related to general growth of network-based approaches. However, several factors make graph theory especially well suited to the geosciences: Inherent complexity, exploration of very large data sets, focus on spatial fluxes and interactions, and increasing attention to state transitions are all amenable to analysis using graph theory approaches. (C) 2015 Elsevier B.V. All rights reserved.