Delta channel networks: 1. A graph‐theoretic approach for studying connectivity and steady state transport on deltaic surfaces

Abstract

River deltas are intricate landscapes with complex channel networks that self-organize to deliver water, sediment, and nutrients from the apex to the delta top and eventually to the coastal zone. The natural balance of material and energy fluxes, which maintains a stable hydrologic, geomorphologic, and ecological state of a river delta, is often disrupted by external perturbations causing topological and dynamical changes in the delta structure and function. A formal quantitative framework for studying delta channel network connectivity and transport dynamics and their response to change is lacking. Here we present such a framework based on spectral graph theory and demonstrate its value in computing delta's steady state fluxes and identifying upstream (contributing) and downstream (nourishment) areas and fluxes from any point in the network. We use this framework to construct vulnerability maps that quantify the relative change of sediment and water delivery to the shoreline outlets in response to possible perturbations in hundreds of upstream links. The framework is applied to the Wax Lake delta in the Louisiana coast of the U.S. and the Niger delta in West Africa. In a companion paper, we present a comprehensive suite of metrics that quantify topologic and dynamic complexity of delta channel networks and, via application to seven deltas in diverse environments, demonstrate their potential to reveal delta morphodynamics and relate to notions of vulnerability and robustness.

Delta channel networks: 2. Metrics of topologic and dynamic complexity for delta comparison, physical inference, and vulnerability assessment

“…The graph-theoretic framework presented in the companion paper [Tejedor et al, 2015] allowed us to identify upstream (contributing) and downstream (nourishment) subnetworks for any given delta vertex (node), including the apex-to-shoreline subnetworks, referred to also as outlet subnetworks. It also allowed us to compute the steady-state flux propagation in the delta channels and to construct vulnerability maps that quantify how a change in any upstream delta link would affect the shoreline Having established the mathematical machinery based on spectral graph theory that efficiently allows to perform the above computations, we now ask the question as to what quantitative metrics one can build that summarize the topologic complexity of delta networks (reflecting their channel connectivity), as well as their dynamic complexity (reflecting how flux dynamic exchanges happen within the network).…”

“…These subnetworks can be topologically very simple (a straight path of channels) or very complex (multiple splitting and merging paths); see Figures 1a and 1b for an example for Wax Lake and Niger deltas as presented in Tejedor et al, [2015] which marks such outlet subnetworks. The topologic structure of each of the subnetworks is embedded within the whole delta channel network topology to result in various degrees of ''dependence'' among the subnetworks.…”

“…This topologic and dynamic structure of delta networks directly determines how disturbances in an upstream link will propagate downstream and to the coastal outlets. The propagation of this disturbance from upstream links to the shoreline was quantified in our earlier paper [Tejedor et al, 2015] by a common-sense vulnerability metric that depicts the degree to which the flux at the coastal vertices is affected by local flux changes at all upstream links of the subnetworks [see Tejedor et al, 2015, Figure 11]. …”

Controllability of Surface Water Networks

“…In previous studies, Mossy and Wax Lake deltas have been compared to inverted tributary networks based purely on their visual resemblance (Tejedor et al, 2015a). The analysis of control profiles provides a quantitative tool to substantiate these comparisons.…”

“…Typically, a network is represented as a collection of nodes symbolizing junctions, inlets and outlets, and links symbolizing stream paths. Topology-based metrics have been developed to quantitatively analyze and compare surface water networks in three structurally distinct categories: (i) tributary rivers, where the flux converges from several upstream inlets to a single downstream outlet (Bertuzzo et al, 2007;Rodriguez-Iturbe & Rinaldo, 1997), (ii) braided rivers, where a single stream undergoes multiple branching/merging and eventually joins into a single stream (Foufoula-Georgiou & Sapozhnikov, 2001;Howard et al, 1970;Marra et al, 2014;Sapozhnikov & Foufoula-Georgiou, 1996, 1999, and (iii) deltas, where the flux is distributed from a single upstream inlet to several downstream outlets (Edmonds et al, 2011;Passalacqua, 2017;Smart & Moruzzi, 1971;Tejedor et al, 2015aTejedor et al, , 2015b. These studies have significantly advanced our understanding of rivers and streams as networks.…”