Entropy and Intermittency of River Bed Elevation Fluctuations

Abstract

River beds evolve as a result of a complex interaction between strongly nonlinear processes such as near‐bed turbulence, particle‐particle interaction, and particle‐bed interaction. This interaction contributes to the initiation and evolution of extremely variable river bed elevation patterns, commonly known as bedforms that span across a range of spatiotemporal scales. In this paper, we employ a refined definition of entropy, that is, the multiscale entropy (MSE), to characterize the observed variability in the fluctuations of bed elevation time series under variable discharges obtained from a field‐scale laboratory flume. The MSE accounts for the sequence of data points in a series and quantifies the complexity and lack of information in a system. We show that the MSE of bed elevation fluctuations is higher for higher discharges. When compared with surrogates, which are the linearized series of a signal, the MSE provides across‐scale information about the underlying nonlinearity and linear correlation in a signal. The MSE difference between the original and surrogate series is due to the inherent nonlinearity that is higher for higher discharge at the smaller scales and peaks at intermediate scales. These results indicate the presence of a heterogeneous arrangement of extreme fluctuations that enhances the underlying complexity in bed elevation, rendering them less predictable at higher discharges. We further investigate the role of asymmetry of the bed elevation increments in observed complexity. Our results of asymmetry together with entropy suggest that characteristics of both small‐ and large‐scale features should be included for the accurate predictive modeling of sediment transport.

Spectral Properties of Water Hammer Wave

“…Figure 8c-d illustrate the response of discharge on the complexity of a pressure wave induced by the water hammer. Specifically, complexity obtained from both SampEn and ApEn increases linearly with discharge significantly, which is consistent with previous studies [9]. However, complexity obtained from SampEn does not include self-similar patterns as ApEn does [37].…”

“…From the perspective of colors of noise, we can say that both wave sensors exhibit near-flicker or pink noise, rather than purely random behavior. Additionally, this pink noise property supports our assumption about the wave equation presented in Equation (9). Additionally, the absolute value of the slope of PSD for P1 is slightly greater than that for P2, indicating that the P1 wave has a slightly higher frequency variation than the P2 wave, which is consistent with the behavior of the water hammer wave.…”

“…In this study, the values of m, r, etc. were determined based on the multi-scale phenomena of time series confirmed by previous research [9,12,35].…”

“…Theoretical approaches such as power spectral density and entropy of time series are frequently used to quantify spectral properties in many disciplines. Recently, these approaches have been used in the hydrology field to study natural channel networks [9][10][11][12]. However, very little attention has been taken to the application of these approaches in the context of pipe water distribution and their physical phenomena.…”

Controls of the Topological Connectivity on the Structural and Functional Complexity of River Networks

“…Second, the sample entropy (hereafter referred to as entropy) can be computed for each series at different scales as In this equation, n m ( s ) ( r ) and n m +1 ( s ) ( r ) are the total number of matches for all existing m ‐point and m +1‐point patterns, respectively, at scale s (see Supporting Information S1 for more details). A higher entropy (MSE) implies that the signal is more complex and thus less predictable (Costa et al, 2005; Delgado‐Bonal & Marshak, 2019; Golan, 2008; Pincus, 1991; Ranjbar & Singh, 2020).…”