Understanding flow resistance is of great interest to river research and practice. The estimation of flow resistance is important for determining flood magnitudes, predicting ecological habitat, estimating rates of sediment transport, and understanding channel morphodynamics. However, the hydraulics of gravel-bed channels, in particular, are relatively poorly understood see. Given that most of the foundational work in fluid dynamics, upon which conventional approaches to predicting flow resistance are based, was conducted using regular e.g. or uni-scalar e.g. bed geometry, the multiscalar topographic characteristics of these rivers present a major challenge. In particular, individual grains and assemblages of grains (“forms”) on the bed surface, spanning orders of magnitude of scale, have variable contributions to the total flow resistance across different channel types. Thus, moving forward, mainstream empirical approaches to estimating flow resistance based solely on grain diameter would ideally be replaced by approaches that explicitly account for multiple spatial scales see. Decomposing roughness lengths into different scales may contribute to an understanding of channel morphodynamics given that energy dissipation is increasingly recognized as a condition governing system behaviour . Also, the partitioning of bed stresses between grain and form scales is an important step in predicting bedload transport .

Inspired by early work in fluid dynamics and subsequent work in fluvial hydraulics , some geomorphologists sought to disaggregate the roughness length into grain and form contributions by correlating bar geometry with flow resistance . However, further work was likely hindered by limitations associated with the collection of topographic data in rivers . Advances in remote sensing and statistics have since allowed researchers to explore detailed scaling characteristics of gravel-bed surfaces using analyses such as variograms and transforms . Topographic analyses have led to multiscalar decompositions of geometric roughness in rivers, although to our knowledge, full decompositions of hydraulic roughness have not yet been presented. The latter approach has been developed for complex aeolian surfaces using transforms , which serves as a proof of concept for a multiscalar roughness length decomposition.

In a review of flow resistance in gravel-bed rivers, identified two relatively recent advancements in the fields of statistics and fluid dynamics that could contribute to a multiscalar roughness length decomposition tool. The first advancement is the wavelet transform, which is generally superior to the Fourier transform when analysing the underlying structure of complex and aperiodic signals. This is due to the use of a finite (rather than a continuous) wavelet function, which gives rise to a family of wavelets that are dilated (stretched and compressed) and translated (shifted) along the signal . There are now various types of wavelet transforms suited to different applications, some of which have been applied in rivers . The second advancement is the development of roughness correlations for irregular surfaces e.g., which estimate the roughness length of a surface based purely on its geometric characteristics.

In this study, we present a novel method of estimating the relative contribution of different physical scales of river bed topography to the total roughness length based on longitudinal profiles. The general approach consists of (1) a wavelet transform in which the channel surface is decomposed into a set of more simple components each at a different wavelength and (2) a roughness correlation that estimates the roughness length associated with each wavelength, which is expressed as the equivalent sand roughness parameter ks . By modifying the specific roughness correlation that is used, the transform-roughness correlation (TRC) approach may be applied across a wide range of channel types and hydraulic conditions. To demonstrate the TRC analysis, we apply it to a series of original laboratory experiments with high-resolution digital elevation models (DEMs), as well as some additional published data.

The transform-roughness correlation approach is a generic tool that should be adapted based on the hydraulic conditions and the purpose of its application. These considerations should span the dataset, the type of wavelet transform, and the specific roughness correlation that is selected. We first discuss these general considerations to provide important context for the TRC approach, prior to introducing the experimental data and the roughness correlation in Sect. .

First, the minimum resolution and spatial extent of the topographic dataset should be informed by the scale of the features of interest. The data should have a sufficiently high spatial resolution such that they can capture the range of in-channel features that produce drag. Also, to capture the characteristic geometry of bed features (notably, height and spacing) and estimate a reach-averaged roughness length, the spatial extent of the dataset should be at least the length of the largest features that influence the flow; for example, it should span a series of dune crests or pool–riffle pairs.

Second, given that the hydraulic roughness of in-channel features is of interest, the channel topography can be reduced to a one-dimensional profile extending along the thalweg, representative of the primary flow path. It is important to note here that this approach ignores resistance elements, such as channel planform, and three-dimensional interactions between flow and in-channel topography. If both hydraulic and topographic data are available, this assumption may be validated by comparing the roughness length estimated using the roughness correlation to a measured roughness length (see Sect. ). If the range of interactions between the flow and the surface is of interest, multiple parallel elevation profiles could be analysed.

Third, the choice between discrete and continuous wavelet transforms (DWT and CWT) is a trade-off between the resolution of the decomposition and the physical resemblance to the original profile. Compared to the DWT, the CWT extracts more intricate structural characteristics from the signal and yields a greater number of wavelengths between which information is shared . However, the redundancy in the CWT generates a more abstract representation of the topographic variation at a given wavelength. In Fig. , we compare wavelengths extracted using a maximal overlap discrete wavelet transform (MODWT) and a CWT using the same elevation profile. At the wavelength corresponding to the spacing of a pool–bar–riffle sequence (λ≈2 m), the oscillations output by the MODWT are aligned with the pool–riffle undulations (i.e. the position of peaks and the general shape are similar), but the CWT oscillations do not appear to align with the original profile. Given that they do not resemble the channel surface, it may be invalid to infer hydraulic behaviour from CWT wavelengths.

Fourth, the specific roughness correlation that is used should match the regime of the channel's boundary Reynolds number Re*=U*k/v, where U* is shear velocity, k is a representative roughness scale, and v is kinematic viscosity. For example, given that gravel-bed rivers tend to be within the fully rough regime where Re*≥70 (e.g. ), it may only be valid to apply roughness correlations obtained for that regime specifically. Also, the flow should be turbulent, and it should be two-dimensional, which may be indicated (although not guaranteed) by flow aspect ratios (w/h, where w is the wetted width and h is flow depth) greater than 5 .

Last, roughness correlations in fluid dynamics tend to be developed for flows sufficiently deep to have logarithmic velocity profiles, which should be considered when they are applied to flows with less developed profiles. suggested that logarithmic layers develop where relative submergence h/k is greater than 40, although observed a logarithmic layer in rough open-channel flow at submergences as low as 1.9. During most flow conditions, it is common for gravel-bed rivers to have relative submergences of less than 10 and, in some cases, as low as 0.1 , where no logarithmic layer can develop because roughness elements are not submerged. However, if one is interested in channel-forming flows capable of reworking the bed surface where relative submergence may be 2 orders of magnitude higher , the logarithmic assumption should be satisfied for most rivers.

In this section, we first seek to validate the TRC approach, and then focus on the multiscalar roughness length decomposition of Experiment 1a, which features a well-developed pool–bar–riffle sequence under a formative discharge. First, we compare the topographically and hydraulically based estimates of ks. Second, we demonstrate the relationship between estimates of ks with and without the wavelet transform. Third, we show how the key parameters of the roughness correlation (standard deviation, effective slope, skewness) vary across each wavelength. Fourth, we estimate the relative contribution of different scales of bed topography to the total roughness length and explain how the estimated values relate to the key parameters and the characteristics of the experiments. Fifth, we compare the performance of different roughness lengths in estimating flow resistance. Finally, we discuss the significance, limitations, and potential applications of the TRC approach.

Recently proposed roughness correlations in fluid dynamics e.g. incorporate information regarding both the height of the roughness elements (a vertical roughness scale, e.g. σz) and the arrangement or spacing of roughness elements (a horizontal roughness scale, e.g. ES). In isolation, either one of these roughness metrics may contribute to an incomplete – and potentially misleading – estimate of flow resistance. It is important to recognize that, depending on the surface of interest, the total roughness length is usually a compromise between vertical and horizontal roughness scales of the bed surface.

In gravel-bed rivers, which are typically ungauged, and where measurement of hydraulic variables is subject to practical limitations , flow resistance is usually estimated using only a vertical roughness scale such as grain diameter . However, the relationship between grain diameter and flow resistance breaks down in natural channels for two main reasons see: (1) grain diameter does not account for larger and often more dissipative roughness elements, and (2) it does not consider the horizontal spacing of these larger roughness elements, which has a systematic effect on the flow . In recent years, the increased availability of high-resolution topographic data has led to the adoption of σz as a roughness metric in gravel-bed rivers, on the basis that it includes information regarding larger-scale bed structures . However, σz only improves upon the first deficiency of grain-based roughness metrics and, consequently, it has inherent limitations. The roughness correlation presented by may improve upon existing roughness metrics used in gravel-bed rivers, and it may be applied to most datasets where σz is calculated.

The TRC analysis has direct applications across geomorphology. Quantification of scale-dependent patterns of channel topography and roughness length may contribute to form- and process-based classifications of channel morphology and dynamics. There have been numerous attempts to classify channels based on in-channel features and their associated processes e.g.; however, analysis of bed topography is typically qualitative. We expect that different channel types exhibit distinctive scale-based patterns of σz and ks, which would enable a quantitative and heuristic classification index.

The scale-based decomposition of ks may assist in identifying and forecasting the hydraulic influence of specific roughness elements in channels. For example, through the manipulation of spatial datasets by the addition or removal of features, the role of natural in-channel features (e.g. large wood) and engineering designs (e.g. rock chutes) could be isolated and determined for flood conditions. Also, multiscalar roughness length decomposition may contribute to an understanding of bedload transport processes, where accurate predictions rely on partitioning bed stresses between grain and form scales .

However, in its current form, there are some conditions in which the TRC approach is limited. The discrepancy between topographic and hydraulic estimates of ks for step–pool channels highlights the potential limitations of the roughness correlation in steep gravel-bed rivers where slope and relative submergence have a greater hydraulic influence. In channels with significant planform resistance, the approach may require modification to account for the slope and curvature of the channel. In multi-thread channels, several profiles may need to be employed and the results weighted according to the size of the channel. Even under such conditions, multiscalar roughness length decomposition may still have considerable value with appropriate research questions and interpretation.

The transform-roughness correlation approach estimates the relative contribution of various scales of in-channel topography to the total roughness length. By modifying the roughness correlation to suit the hydraulic conditions, multiscalar roughness length decomposition may be achieved in virtually any type of river or numerical model and perhaps boundary layers in other environments. The only requirement is that the topographic data are of a sufficient resolution and spatial extent to capture the scales over which the roughness elements occur, and data of this quality are only becoming more available to geomorphologists. In particular, we expect that given the continual advances in methods for collecting bathymetric data in both shallow and deep channels , applying the TRC approach will become increasingly practical in natural rivers.

Given that the TRC approach provides novel and detailed information regarding the interaction between surface topography and fluid dynamics, it may contribute to advances in hydraulics, channel morphodynamics, and bedload transport. Estimates of ks from roughness correlations may provide more immediate benefits by improving upon representative roughness values in estimating flow resistance. We are currently conducting experiments to further develop and apply these ideas.