The accuracy of hydraulic models depends on the quality of the bathymetric data they are based on, whatever the scale at which they are applied. The along-stream (longitudinal) and cross-sectional geometry of

Hydraulic modeling is based on the description of river morphology (cross-sectional geometry), and this is the essential input of models despite its scarcity and cost of acquisition. The most important aspect to know is the river bathymetric data at the local scale, detailed and specific to the site and local conditions

Morphological units are topographic forms that shape the river corridor

Among the most frequently observed alternating MUs, pools and riffles have been recognized as fundamental geomorphological elements of meandering streams

Different views of pool–riffle sequences.

For many years, many researchers have been trying to develop techniques to identify MUs and especially pools and riffles using hydraulic variables or topographic ones, or both (Table

Review of some methods of morphological units' identification (variable used and MU types).

On the other hand, some studies focused only on hydraulic parameters to identify MUs. For example,

All these methods handle topographic or hydraulic parameters separately. Recently, however, several researchers have improved MU identification through the use of the covariance of several parameters in a multidimensional approach.

Some of the methods presented in the literature have shown limits in calculating the wavelengths of pool–riffle sequences. Others have given results that are often difficult to interpret in terms of bedform amplitude. This amplitude, which varies according to each bedform, involves the use of the pseudo-period. A few methods are developed to extract this pseudo-period from alternating MU rivers. We, therefore, choose to work with wavelet analysis that estimates the local variability strength of a signal and extracts the signal amplitude and wavelength. In this study, we apply continuous wavelet transform (CWT) to calculate the wavelength

In reality, the longitudinal spacing

Some researchers have investigated the variability of longitudinal spacing depending on geometric or hydraulic parameters.

The studies that used wavelet analysis in the geomorphological field consist in extracting components of a given spatial series (e.g.,

The objective is to extract some quantitative properties of these alternating morphological units, such as the mean and the median of their longitudinal spacing, with a continuous vision of the topography instead of a discrete classification. For that, we focus on two numerical criteria computed at reach scale: the distribution of spacings between morphological units (mean, median, etc.) and the evaluation of correlations between all geometrical and flow variables. We use in this work four classical variables (velocity, hydraulic radius, bottom shear stress, and the local channel direction angle) because they respond directly to morphodynamic processes (flow convergence routing or meander migration), and they are independent hydraulic degrees of freedom.

In this study, we focus mainly on alternating alluvial channels, especially pool–riffle sequences, even though the method presented here could be used to analyze any morphology characterized by alternating topographic forms. We first present the dataset of six river reaches in France used for this analysis (Sect. 2). In Sect. 3, we present the wavelet ridge extraction method to identify pool–riffle sequences in the univariate and multivariate cases with the four variables. Section 4 presents results and compares them with the BDT developed by

The rationale behind this approach is to provide a continuous description of geometric and flow patterns along a reach using the wavelength, a description that could be subsequently used to create a synthetic river as in the RiverBuilder

Six reaches of small French rivers are used in this study

The locations of the study reaches in France.

Characteristics of the six reaches and their catchment. The bankfull width

Cross sections were surveyed along the river reaches at the level of hydraulic controls and morphological breaks to describe the major variations in terms of width, height, and slope in the main channel and the floodplain and at the level of pool–riffle sequences. Cross sections and water surface profile measurements were surveyed in 2002–2004, covering the main channel and floodplain and using an electronic, digital, total-station theodolite. Water surface profiles were measured at different flow discharges

velocity

hydraulic radius

bed shear stress

local channel direction angle (planform)

The fourth variable chosen is related to the channel planform: we define

Definition of

Classical mathematical methods, such as Fourier analysis, extract the wavelengths in the frequency domain for stationary signals, but can also be used for non-stationary signals using an “evolutive” methodology based on spectral estimators

The wavelet Morlet mother function; the plot shows the real and imaginary parts of the wavelets in the space domain (distance).

The wavelet transform uses a whole family of “daughter” wavelets generated by scaling and translating the mother wavelet

Wavelet analysis is prevalent in many fields, such as fluid mechanics (e.g.,

In this study, we use another application of the wavelet analysis called the wavelet ridge extraction method

Few methods in the literature have been trying to identify river characteristics with wavelets. For example,

In the present study, we test a new wavelet ridge analysis on spatial series with the Morlet mother basis function represented in Fig.

Given a spatial series

The complex wavelet transform can be classically visualized using a scalogram, i.e., a colored map of the modulus

In Sect.

In the univariate case, we choose a single variable

There may be several curves that verify Eq. (14); in practice, we choose curves that cross the domain of the wavelet transform (from one cone of influence to another) continuously and belong to the region where the maximum power of the wavelet is. This curve

The phase function

In the end, we can extract the wavelength function of pool–riffle sequences, which corresponds to a pseudo-period function of the signal

In the next section, we extend the definition of phase ridge points and ridges to the case where several variables are sampled along the reach, all of them potentially correlated and embedding information about the pseudo-periodicity of the channel's hydraulic behavior.

The multivariate case is the extension of the univariate case to a set of

Power of the wavelet of the four variables: velocity, hydraulic radius, bed shear stress, and local channel direction angle. The black curve

As a result, the phase shift of every variable is calculated by

In our case, after calculating the phase and amplitude, we modeled each variable as in Eq. (

Variation of the modeled function

Figure

As already mentioned in Sect.

In this section, we present the results of the analysis on the six reaches presented in Sect. 2. We compare the univariate to the multivariate approach and also the multivariate to the benchmark method. First, the methods are compared in terms of the statistics (mean, median, etc.) they yield. Second, we present the benchmark method called the BDT and compare its results for the six reaches with the multivariate case.

First, both approaches are employed on all reaches to extract statistics such as the mean, median, and standard deviation wavelengths of morphological units (pool–riffle sequences). The wavelet method extracts the wavelength for an assessed length

The assessed length provided by the wavelet analysis for all reaches in the univariate case using the velocity, hydraulic radius, bed shear stress, or local channel direction angle and in the multivariate case using all these four variables.

Moreover, the multivariate approach takes into account all the variables and therefore looks for a single pseudo-periodicity between the four variables, and then we are going to have a pseudo-periodicity that represents the reach and not the chosen variable.

Figure S1 in the Supplement shows the comparison between the univariate and multivariate results for the six reaches (from 1 to 6) using the four variables (velocity, hydraulic radius, bed shear stress, and cosine of local channel direction angle. Table

Summary of results for all reaches in the univariate case using the velocity, hydraulic radius, bed shear stress, or local channel direction angle and in the multivariate case using all these four variables. For each variable, we compute the mean, median, and standard deviation

We compare the methods in terms of longitudinal spacing (

in the Graulade River, the longitudinal spacing identified using the multivariate approach matches closely the one associated with the hydraulic radius (in the mean and the median with a deviation of

in the Semme River, it matches those of the local channel direction angle (in the mean and the median with a deviation of

in the Olivet River, it matches the bed shear stress (in the mean with a deviation of

in the Ozanne River, it matches those of the hydraulic radius and the velocity (in the mean and the median with a deviation less than

in the Avenelles, it matches those of the velocity, hydraulic radius, and bed shear stress (in the mean with a deviation less than

in the Orgeval River, it matches those of the hydraulic radius (in the mean with a deviation of

Consequently, the multivariate estimates of

the distribution of

the dispersion of this multivariate distribution, measured by

In the following section, we compare the wavelet method with a benchmark method using talweg elevation.

In this section, we compare our method's results with a selected benchmark method from the literature (i.e., BDT). This method shows good results in the identification of these bedforms according to some studies (e.g.,

The technique of

This procedure is applied to all rivers, and the results are depicted in Fig.

Results of the BDT method using a tolerance equal to the standard deviation on the total length (blue) and the assessed one (red) for all reaches (1 to 6). Round points are pools or riffles: pools are high, and riffles are low points.

Results of BDT and the multivariate wavelet methods for all reaches.

Figure

The work of the wavelet analysis is done on the assessed length

For the wavelet method (Fig.

In Table

In this study, we consider the BDT method to be a benchmark method. We do not consider a specific method to be the “true” one, and we only apply these methods to have a general idea on the uncertainties in the identification of morphological units. This method was chosen not because it is the “best” method for pool–riffle identification, but because it does not use thresholds (except for the tolerance

For a long time, researchers have found a common interval of longitudinal spacings that vary between 5 and 7 times the channel width

In this study, the longitudinal spacing varies in the mean and the median from

We worked with a dataset that contains cross sections spaced 0.46 to 2.9 times the bankfull width. Other studies have used much shorter spacings (e.g.,

The wavelet ridge analysis is powerful in identifying pseudo-periods, amplitude, and phase while preserving the correlations between parameters. We can thus identify alternating morphological units more objectively in terms of frequency/wavenumber.

This wavelength can be used to represent the variability of the bathymetry in hydraulic models in cases where we do not have full access to the geometry of the channel (e.g., remote sensing data such as the overcoming Surface Water and Ocean Topography Mission) and the morphology can be modeled by pseudo-periodic functions. Furthermore, it can be implemented in synthetic geometry generators (e.g., River Builder,

On the other hand, it presents drawbacks compared to other methods. First, the cone of influence ignores a large part of the river and sometimes biases the results in the case of small total lengths (the Graulade (1) and Semme (2) reaches). Similarly for reach length and number of morphological units as for the number of cross sections, the larger it is, the more robust the results are, and the smaller the relative portion of “unassessed length” is. Still, the method remains a powerful tool for non-stationary analysis. Another problem is the amplitude, which is sometimes overestimated in some regions of the topography. We visualized this in several cases in our study since we used the Neperian logarithm to avoid negative values, and therefore the inverse function (exponential) gives slightly larger values. However, this does not bias the identified wavelength of the reach.

In this study, we present an automatic procedure based on wavelet ridge extraction to identify some characteristics of alternating morphological units (MUs), such as their longitudinal spacing and amplitude. The method does not rely on any a priori thresholds to identify MU sequences. It was applied to six rivers with a maximum length of 500 m. We chose to work with classical hydro-morphological variables (velocity, hydraulic radius, bed shear stress) in addition to the planform channel direction angle that evaluates the impact of river sinuosity in the determination of the wavelength.

As a result, identified wavelengths are consistent with the values of the literature (mean in 3–

Given the short length of several reaches, the relatively small number of cross sections for each reach, and the possible impacts of artificial modifications, this paper is mainly a proof-of-concept of the wavelet approach. We foresee many perspectives for this work, such as the possibility of extending the work to other rivers with other types of MUs or other longer reaches with a large number of cross sections.

The conjugate form of the mother wavelet is

In the multivariate case, we should resolve Eq. (

Steps of determining the local wavenumber

The script is available upon request from the authors.

The supplement related to this article is available online at:

NLM, MM, and RM designed the study. ON provided the dataset. MM and NLM built the methodology and analyzed the results. MM wrote the paper with contributions from NLM, PR, and RM.

The authors declare that they have no conflict of interest.

The authors acknowledge financial support from the French National Space Agency (Centre National d'Etudes Spatiales, CNES) and Sorbonne University through the PhD grant of Mounir Mahdade. We would also like to thank Gregory B. Pasternack, the second anonymous reviewer, and the editor for the valuable suggestions and comments that greatly helped improve this paper.

This research has been supported by the Centre National d'Etudes Spatiales (CNES; grant no. CNES5100018525) and Sorbonne University (grant no. C18/1749).

This paper was edited by Theresa Blume and reviewed by Gregory Pasternack and one anonymous referee.