The complexity of gravel bed river topography examined with gradual wavelet reconstruction

This study analyzes the nonlinear nature of gravel bed elevation data series obtained in a large scale flume facility for four different discharges, including spatial transect series and temporal series at a fixed location. The goals are to infer the degree of complexity, to assess the scales and features in the signals that mostly contribute to this complexity, and to discern the difference in the manner in which this complexity is expressed in time and space. The asymmetry series (magnitude of the first-order derivative of the signal over a separation distance, Δ, raised to the third power) of the bed elevation data forms the basis of our nonlinearity analysis. An important and novel dimension is the adoption of a recently introduced approach to surrogate data generation, gradual wavelet reconstruction. This produces partially linearized counterparts of the original series that preserve an increasing degree of the underlying nonlinear structure in the original data, while randomizing the rest. This allows us to discern the difference in the manner in which the nonlinear asymmetry is expressed in the temporal and spatial data series. Comparison in the real and phase space of the original and surrogate series is performed, and the analysis reveals that the complexity of both spatial and temporal series increases with discharge and that the nature of nonlinearity is quantitatively and qualitatively different. For the spatial series, asymmetry is expressed at large scales and is shown to result from the organization of intermediate scale features. In contrast, asymmetry in the temporal series is a smaller-scale phenomenon. This is a consequence of the scale-dependent propagating velocity of topographic features. Our results have implications for understanding the complexity of geomorphic processes as a function of the space and time scale considered: the complexity over the "bed form Lagrangian time scale" is different in nature to that over the "flow velocity Lagrangian time scale".