# The variability of grain size metrics in gravel-bed rivers

Daniel Vázquez-Tarrío, Alain Recking

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Daniel Vázquez-Tarrío, Alain Recking. The variability of grain size metrics in gravel-bed rivers. CATENA, 2025, 255, pp.108974. 10.1016/j.catena.2025.108974. hal-05450025

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# The variability of grain size metrics in gravel-bed rivers

Daniel Vázquez-Tarrío<sup>a,*</sup>, Alain Recking<sup>b</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Grain-size distribution

Grain-size variability

Gravel-bed rivers

Fluvial geomorphology

Bedload

## ABSTRACT

Grain size in gravel-bed rivers can be highly variable at the reach scale. Quantitative characterisation of the different sources of this spatial variability has many implications. For example, for how grain size sampling is programmed in gravel-bed rivers, or how this variability is considered in the propagation of uncertainties associated with the use of models and equations that employ grain size metrics as input parameters (e.g. bedload models, flow friction equations). In this paper we have undertaken a re-analysis of a large database of 462 grain size distributions (GSDs) of gravel-bed rivers compiled from previous studies. In a first step, we explore this database to identify which distribution model best describes the GSD of the collected data. Five different probability distribution models (PDFs) (lognormal, log-logistic, Weibull, Pareto, and log-raised cosine) and one empirical GSD model (Recking's similarity model) are tested. Weibull and Recking's models are the ones showing a best fit to the general shape of the GSD. This first analysis allows us to identify the minimum parameters we need to focus on to adequately characterise grain-size variability, namely the central tendency (e. g., $D_{50}$), the variability around this central tendency, a parameter characterising the tail towards the fine terms (e.g. % of sediment < 2 mm) and another for the coarse tail (e.g. $D_{84}$). In a second step, we do a statistic analysis on those datasets where more than one GSD sample was collected across the same river reach, with the aim of identifying trends in reach-scale variability in these parameters. For each shape-defining and scaling parameters of the different distribution models we report a good fit to a normal or log-normal distribution at the reach-scale. Finally, we show how the information derived from our analysis can be combined with statistical resampling methods for maximising the information that can be obtained from most conventional grain size characterisations in gravel-bed rivers.

## 1. Introduction

Sampling the grain size distribution (GSD) is an unavoidable step in studies of sedimentology, fluvial geomorphology, sediment transport and river hydraulics of gravel-bed rivers (Wolman, 1954; Bunte and Abt, 2001). Indeed, GSDs based on samples collected in the field are widely used to interpolate simple metrics such as the median ($D_{50}$) or the i-th percentiles (sizes below which the i-th percent of the sediment is finer) (Church et al., 1987; Rice and Church, 1996; Bunte and Abt, 2001). These metrics are then introduced as input parameters into equations and models used to compute flow velocity or sediment transport (Parker, 2004). For example, many friction equations use the 84th or 90th percentile of the surface GSD as an 'equivalent' measure of riverbed roughness (e.g. Hey 1979; Bray 1980; Ferguson, 2007). Similarly, bed-load equations typically employ one individual GSD metric, such as $D_{50}$ (e.g., Meyer Peter and Müller, 1948) or $D_{84}$ (Recking, 2013; Recking

et al., 2016), or even the full GSD (e.g., Parker, 1990; Wilcock and Crowe, 2003; Recking, 2016), as input parameters to calculate bedload rates. In the same vein, morphodynamic models typically use a multiple of $D_{50}$, $D_{84}$ or $D_{90}$ as a proxy for the thickness of the 'active layer' of the streambed (Parker, 1990; DeVries, 2002; Church and Haschenburger, 2017; Vázquez-Tarrío et al., 2021; Recking et al., 2023). However, this practice implicitly ignores the highly heterogeneous nature of the bed in gravel-bed rivers (Church and Kellerhals, 1978; Mosley and Tindale, 1995; Church et al., 1987; Crowder and Diplas, 1997; Bunte and Abt, 2001; Rice and Church, 1998; 2010), i.e., the fact that GSD metrics such as the median or $D_{84}$ should be spatially variable over a given cross-section or across a river reach. Hence, quantifying this variability in GSD statistics is essential for error propagation analysis, and for estimating uncertainties associated with the use of river hydraulic and sediment transport equations (Fernández and García, 2017; Monsalve et al., 2016; Recking et al., 2024).

<footnote><sup>a</sup> Department of Geo-Hazards & Climate Change, IGME, CSIC, Madrid, Spain</footnote>

<footnote><sup>b</sup> Univ. Grenoble Alpes, INRAE, CNRS, IRD, Grenoble INP, IGE, 38000 Grenoble, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: d.vazqueztarrio@igme.es (D. Vázquez-Tarrío).</footnote>
In principle, the GSD variance should be sensitive to two controls. On the one hand, there is the inherent variability in GSD metrics associated with sampling size. That is, each sample represents only a fraction of the sampled population of interest, and sampling necessarily introduces variability and some bias. In this regard, previous work (e.g., Church et al., 1987; Rice and Church, 1996; Ferguson and Paola, 1997; Petrie and Diplas, 2000; Bunte and Abt, 2001; Green, 2003; Eaton et al., 2019) has documented that grain size percentiles are poorly estimated in small and/or poorly sorted samples. On the other hand, some of the variability in the GSD metrics is also related to the inhomogeneous nature of gravel beds. For instance, many previous studies have reported different patterns of grain size sorting in the downstream direction of channels (e.g., Sternberg, 1875; Shaw and Kellerhals, 1981; Rice and Church, 1998; Fedele and Paola, 2007; Singer, 2008), along gravel bars (e.g., Bluck, 1982; Wolcott and Church, 1991; Powell, 1998), within the cross section (e.g. Mosley and Tindale, 1985) or vertically (Vázquez-Tarrío et al., 2020; Recking et al., 2023) in gravel-bed rivers. This spatial heterogeneity in grain size is more evident in braided rivers (Guerit et al., 2014), which are characterised by a variety of complex macroforms and spatial patterns of grain size sorting. The simplest way to characterise the intra-reach variability in GSD metrics is to collect multiple samples over a given reach, but this has not been widely done, probably because it is a time-consuming task. We are aware of only a few studies that have addressed this issue (e.g., Church and Kellerhals, 1978; Mosley and Tindale, 1985; Rice and Church, 2010; Guerit et al., 2014; Misset et al., 2020; Recking et al., 2024), and these studies highlighted that there can be considerable variability in the GSD values at the river-reach scale.

The quantitative characterisation of all these different sources of variability is closely related to the search for the best probability distribution function (PDF) to describe the shape of the GSD of a gravel-bed river. In this regard, the distribution model controls on which specific statistical parameters (mean? median? 84th percentile?) should we focus when characterising and studying spatial variability in grain size (Bunte and Abt, 2001). Since the early days of river research many scientists have observed that particle sizes in fluvial sediments are logarithmically distributed (Udden, 1914; Wentworth, 1922; Krumbein, 1938; Otto, 1939; Inman, 1952) and how GSDs tend to be asymmetric (Pettijohn, 1975). Thus, over time, various researchers have used diverse lines of probabilistic and statistical reasoning to justify the existence of this asymmetry, in an attempt to infer the distribution that should theoretically provide a more ‘adequate’ description of the GSD of riverbed sediments (Middleton, 1970).

One group of authors has suggested that the GSD of clastic sediments is controlled by the processes that generate sediments from parent rock materials by weathering and fragmentation, which in the field of fluvial geomorphology we could relate to particle breakdown and abrasion during transport. In this regard, Kolmogorov (1941) stated the basis of fragmentation theory and derived a log-normal distribution for the GSD of sediments derived from rock crushing, after assuming that the probabilities of particles being crushed into smaller clasts are independent of each other and that fragmentation extends over a long time. Other authors working on coal fragmentation and powdering (Rosin and Rammler, 1933; Bennet, 1936) deduced that a Weibull-type distribution (also called ‘Rosin-Rammler’ law) was most appropriate for fragments produced by coal crushing. On this basis, Krumbein and Tisdel (1941) proposed that many clastic weathering products should follow this distribution and suggested that sediments supplied into a stream channel would evolve from Weibull to log-normal distribution with transport, as also Ibbeken (1983) and Schleyer (1987) later observed. More recently, some authors have reported that the size distribution of fragmentation products exhibits fractional self-similarity over a very wide range of scales (Turcotte, 1986), and have proposed alternative distributions that satisfy this fractal distribution condition, such as Pareto distribution (Allen et al., 2015, 2017). Nevertheless, fluvial GSDs are likely to be related to sediment transport (Inman, 1949), and not only to particle fragmentation. In particular, the coarser limb (i.e., coarser clasts) of the

GSD must somehow reflect transport competence, while the finer limb of the gravel (and sand) component reflects the size and transport behaviour of the fine sediment fractions. Therefore, another group of authors related the characteristics of fluvial GSDs to sorting during downstream transport, often referred to as 'selective transport' in the field of fluvial geomorphology. These groups of works mostly suggested a log-normal distribution as the most appropriate for the GSD of riverbed sediment (Krumbein, 1938; Inman, 1949; Friedman, 1962; Middleton, 1970; Bagnold and Barndorff-Nielsen, 1980; Shih and Komar, 1990; Kothyari, 1995). Nevertheless, Shih and Komar (1990) documented a change from a lognormal to a Weibull distribution in bedload GSDs with increasing discharge, which they associated with a change from size-selective to full-mobility transport conditions with increasing discharge.

These approaches to the best distribution model for the GSD of river bed sediments were essentially deductive (Middleton, 1962). However, as we see, depending on the assumptions used as starting hypothesis, we could infer a different PDF (e.g. log-normal versus Weibull) for the GSD of gravel-bed rivers (Middleton, 1970). One alternative would be a more 'inductive' or 'empirical' approach as, for instance, Recking (2013), who applied similarity analysis to 78 GSDs (from the Idaho dataset; King et al., 2004) to propose a model that seems to describe adequately well the GSD of gravel-bed rivers. This suggests that 'data-driven' approaches, based on the analysis of large amounts of data, can eventually provide interesting information to better understand the shape of the grain size distribution of gravel-bed rivers.

Given the above, the departing research question of this paper is as follows: *What insights do large datasets from gravel-bed rivers provide for improving the characterization of grain size variability across a river-reach?* We think that the information stored in these datasets may ultimately provide some ideas or insights for enhancing reach-scale geomorphological (including hydraulic and sediment transport) assessments of gravel-bed rivers. For example, some recent developments based on 3D point cloud or aerial imagery analysis have focused on correlating roughness metrics or pixel raster values with a median grain size (or coarser percentiles), but they are sometimes limited by the point cloud density or pixel resolution for retrieving information on the fine tails. As a result, these methods in many cases focus on the coarse framework and are not generally used to retrieve the full sub-pixel GSD, which could be of interest for many research topics. In this regard, having a general theoretical PDF may help to maximise the GSD information that can be derived from the estimates of coarse percentiles. Similarly, the identification of the most appropriate PDF will also help to determine which specific statistical parameters we should prioritise when using these 3D point clouds or remote sensing data to describe the reach-scale variability in GSD, or when making comparisons between different sites (Bunte and Abt, 2001).

To address this question and its various implications in this paper we rely on a large dataset of 462 previously published and unpublished GSDs from gravel-bed rivers. We first carry out an empirical analysis of how the field GSDs available from the compiled studies fit the theoretical PDFs usually proposed for gravel-bed rivers. Then, in a second step, we analyse how the parameters defining the central tendency and dispersion of these distributions vary at the reach scale, based on the available datasets where more than one GSD sample was collected in the same reach. Finally, we briefly discuss the geomorphological and practical implications this may have for a better characterisation of the spatial variability in grain size in gravel-bed rivers.

## 2. Materials and methods

In this manuscript, we implicitly make the hypothesis that there is a common PDF that allows us to generalise GSD characteristics in gravel-bed rivers to a reasonable level of significance (e.g. 95 %). Therefore, the answer to our departing research question passes to previously test this hypothesis on the compiled database of GSDs from previous studies.

After this first analysis, the subsequent statistical work on the
database focuses on three specific research questions: i. first, how fine sediment fractions and potential bimodality can influence the fit of the different PDFs to the field data? ii. next, how the statistical parameters defining the shape and scale of the different PDFs are distributed at the reach scale? and iii. finally, which source of variability has the greatest impact on the compiled data of gravel-bed rivers: within-sample or between-sample?

In this methods section, we describe first the characteristics of the database that was collected for this work. We then expose the details of the various statistical analyses carried out to address each of these questions.

## 2.1. Compiled GSD data

A collection of 140 surface and 108 subsurface GSDs for 79 gravel-bed river reaches were compiled from previously published scientific papers. Grain sizes were obtained from tables summarizing GSD statistics and/or graphical reading of GSD curves extracted from the compiled papers. A large amount of this information had been collected and organised over several years by one of the co-authors of this paper and made freely available through the Bedloadweb project (https://en.bedloadweb.com/), as well as supplementary information in several papers (e.g., Recking, 2010, 2013). Table S1, in the document with the Supporting Information (SI), summarises this database and the main characteristics of the rivers to which the compiled GSDs belong.

The database was completed with 24 unpublished GSDs (12 surface and 12 subsurface) from the braided Durance river; and our own field data from 18 gravel-bed river reaches, four located in the French Alps and fourteen in the Cantabrian Mountains (NW Spain). Five of these river reaches are wandering/braided: i. a ~ 1 km reach of the Vénéon River at Plan du Lac (Vázquez-Tarrío et al., 2017), located in the Ecrins massif; ii. a ~ 2-km reach of the Bouinenc close to Digne, located in the Alpes-de-Haute Provence; iii. two ~ 0.5-km reaches of the Severaisse river, at La Chapelle-en-Valgaudemar and Villar de Loubière, located in the Hautes Alpes (France); and iv. a ~ 1-km reach of the Saja river, near Cabezón de la Sal, located in the northern watershed of the Cantabrian mountain range (Spain). The remaining 14 river reaches are single-thread gravel-bed rivers located in the Cantabrian Mountain range (NW

Spain). Table S2 in SI summarises the main characteristics of these sites and the sampling protocols.

Finally, our database is composed of a total of 462 GSDs for gravel-bed rivers: 327 corresponding to surface GSDs and 135 to subsurface. This dataset can be found in Recking and Vázquez-Tarrío (2025). In 85 of the selected river reaches we have information on both surface and subsurface grain size distributions. Most of the surface GSD data compiled are from ~ 100 particle Wolman pebbles, while the subsurface GSD data are in most cases from volumetric bulk samples where the largest clast represented < 1–5 % of the total mass of the sample (Tables S1 and S2 in SI). This database is integrated with GSD data from 98 river reaches covering a wide range of morphologies, bed slopes (0.0002–0.14), channel widths (1–500 m) and drainage areas (1–250,000 km²), which are mainly located in North America (USA and Canada) and Europe (mostly draining the Alps and Cantabrian mountains) (Fig. 1).

## 2.2. Testing of different PDFs

### 2.2.1. Selected distribution models

Numerous probability distributions have been proposed in the statistical literature, each defined by its underlying statistical assumptions. Although there are hundreds of distributions that could easily be applied to the data collected here, testing so many distributions without sound theoretical justification would increase our risk of using inappropriate models and potentially lead us to incorrect conclusions or misinterpretation of the fitted parameters in terms of the physical processes they represent. Since the aim of this paper is to extract the maximum amount of geomorphological information implicit within the GSD of river sediments, rather than to perform an extensive statistical analysis of dozens of probability distributions, we thought it was more valuable to first consider the processes that control the shape of the GSD. After considering these processes, we could then select the appropriate PDFs to test the hypothesis that a common distribution model exits, allowing the generalisation of GSD characteristics to a meaningful significance level (95 % in our case).

Previous field research has identified some general characteristics for the grain size distribution (GSD) of sediments in gravel-bed rivers.

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** A) Geographic location of the GSD compiled for the present research. B) Main morphological characteristics of the selected reaches. C) Differences in the compiled data according to dominant channel morphology.</figcaption></figure>
Having said that, GSDs in gravel-bed rivers tend to be asymmetric, with a short tail for the fine particles and a long tail toward the coarser sizes. The theoretical distributions we test might be able to reproduce this asymmetry and the coarse tail towards larger particles. Additionally, we know there are physical constraints on the upper and lower bounds of particle sizes observed in a gravel-bed river. On the one hand, the larger particle sizes are constrained by sediment production in the catchment and abrasion during transport along the channel network. As a result, particle sizes larger than 512–1000 mm are rare outside the torrent contexts. On the other hand, particle sizes cannot be negative and particles in the mud and silt range are not abundant in gravel-bed rivers. Consequently, the theoretical distributions we can expect to show a best fit to GSDs are likely to have coarse tails, though probably not as heavy as some PDFs used to model rare or extreme events.

The above considerations help in understanding why the probability distribution of sediments and sedimentary rocks has been usually approximated by a lognormal distribution (Krumbein, 1938; Krumbein and Pettijohn, 1938; Fieller et al., 1992; Kondolf and Adhikari, 2000), and the same has been applied in the case of gravel-bed rivers (Bagnold and Barndorff-Nielsen, 1980; Shih and Komar, 1990; Kothyari, 1995; Bunte and Abt, 2001), although some other probability distributions (e. g. Pareto, Weibull) have also been proposed (Ibbeken, 1983; Tsai et al., 2012; Recking, 2013; Allen et al., 2017). Nevertheless, deviations from these theoretical distributions have been frequently reported and, to our knowledge, no general, definitive probability distribution function (PDF) has been found for the GSD of gravel-bed rivers (Rice and Church, 1996).

Based on this previous literature, in this paper we decided to focus mainly on six PDFs or statistical models (see equations and details on these PDFs and models in section S3 in the SI document) with the collected grain size data, namely: i. the log-normal distribution, which is commonly assumed for GSD in gravel-bed rivers; ii. the log-logistic function: the probability distribution of a random variable whose logarithm has a logistic distribution, which has a shape similar to the log-normal distribution (but with heavier tails); iii. the Pareto distribution, a power-law probability distribution used by Allen et al. (2017) to

describe the GSD in rivers, which satisfies the fractal self-similarity conditions hypothesised by Turcotte (1986) for the size distribution produced by weathering; iv. the Weibull distribution, also called the Rosin-Rammler distribution (Rosin and Ramler, 1933), which is commonly used to describe the GSDs of particles generated by grinding, milling and crushing (Kittleman, 1964; Ibbeken, 1983; Bunte and Abt, 2001; Sklar et al., 2017); v. the log-'raised cosine distribution', which is an approximation to the log-normal distribution that has been recently applied to model GSDs by researchers using seismic data as a surrogate measure of bedload transport (e.g., Tsai et al., 2012; Bakker et al., 2020; Antoniazza et al., 2023); and vi. Recking's GSD similarity model, which was empirically defined based on field data for gravel-bed rivers (78 GSDs from Idaho rivers) and is claimed to describe reasonably well the GSD of gravel-bed rivers by Recking (2013). These six distribution models all have in common that they do not take negative values (as the normal distribution does) and that they are asymmetric with positive tails, which makes them a priori well suited to the characteristics of the GSD of gravel-bed rivers reported in field studies.

The selected distributions share the characteristic that they have been proposed in previous studies, thus having some empirical or theoretical support. Furthermore, the six models, a priori, allow us to approximate the asymmetry and the coarse tails of the GSD, while also offering the opportunity to test different approaches for fitting the coarse tails. In this regard, these distribution models differ in the length and 'heaviness' of their tails (Fig. 2). For example, the Pareto distribution, which follows a power-law decay, has a heavier tail than both the log-normal and the log-logistic distributions. This means that extreme values (larger particles) are more likely in the Pareto distribution, as the probability of these values decreases more slowly compared to the decaying tails of the log-normal and log-logistic distributions. The log-logistic, in turn, has a heavier tail than the log-normal and the log-raised cosine. The Weibull, on the other hand, is more flexible and can adapt to tails that are more or less "long" or "heavy". In the case of Recking's empirical model, it is based on the definition of three input parameters, one for the central tendency of the GSD, one for defining the tail of fines and one for the tail of coarse particles: this makes it very

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** In this figure we have fitted the six distribution models to a same 100 pebble-count sample (from the Vénéon river, the blue line). As it can be observed in the figure, the six fitted models differ significantly in the way they simulate the coarse tail. Some models have a longer and heavier tail than others (e.g., Pareto, log-logistic). Recking's model, being a more flexible (it has three parameters) and empirically based model, modifies the shape of the distribution over its range of values, adapting to the irregular shape of the empirical grain-size distribution.</figcaption></figure>
flexible and allows it to capture a wide range of different situations in gravel-bed rivers.

Although the manuscript will focus mainly on these six distributions, we also tested other parametric distribution models, including PDFs with light tails (exponential, gamma, Rayleigh), heavy tails (t-student, Cauchy), and more flexible distributions that can fit both light and heavy tails (Tukey-lambda, Generalised Pareto, Generalised Extreme Value). This complementary analysis was carried out with two main objectives: i. to explore whether there could be any parametric model that fits the GSD of gravel-bed rivers better than the more commonly used PDFs; and ii. to indirectly test the assumptions or hypotheses regarding the tail of the GSDs behind the more commonly used PDFs. These models are presented in the SI document (section S.4.8).

### 2.2.2. Statistical testing of the selected models

We fitted these different PDFs to the compiled data and checked how well they describe the available GSDs. To evaluate the performance of the different distribution models, we calculated for each of them the cumulative percentages (or passing fractions, $F_i$) for several Wentworth (1922) grain size classes: 2–8 mm (very fine to fine gravel), 8–16 mm (medium gravel), 16–32 mm (coarse gravel), 32–64 mm (very coarse gravel), 64–128 mm (cobble), 128–256 mm (cobble to boulder) and > 256 mm (boulder). We then compared the modelled passing fractions with the field measured ones. We also assessed the performance in the estimate of several percentiles of the GSD ($D_{16}$, $D_{50}$, $D_{84}$ and $D_{90}$) using the following score ($\delta_i$):

$$ \delta'_i = \frac{D_i}{Df_i} \qquad (1) $$

which is the ratio between the i-percentile $D_i$ of the GSD estimated from a given PDF, and the same percentile estimated from the field data ($Df_i$). We estimated the $\delta'_i$ score for each distribution model fitted to each GSD of the compiled dataset.

Additionally, following Ibbeken (1983), we also estimated the coefficient of determination ($R^2$) for the regression between the empirical GSDs and the cumulative distributions obtained with each distribution model. In doing so, we attempted to characterise the 'goodness of fit' of the different PDFs to the field distributions, i.e., the percentage of variability in the field GSDs explained by the tested models. Furthermore, we also accomplished Kolmogorov-Smirnoff (KS) tests to evaluate the statistical significance of differences between the empirical GSDs and the different distribution models, under the null hypothesis that the field GSD follows the theoretical distribution. A small p-value (here considered < 0.05) in the results of the KS test indicates then that the given field GSD is unlikely to follow the specified theoretical distribution. Finally, we also estimated the root mean square error (RMSE) of the differences between the cumulative frequency distributions estimated with each model and the empirical GSDs (see section S.4.3. in the SI document for more details).

For these analyses we used all 462 compiled GSDs for the five theoretical PDFs (lognormal, log-logistic, log-raised cosine, Pareto and Weibull), but in the case of the sixth distribution model tested (Recking's empirical model) we excluded the 78 GSDs from the Idaho dataset that were used to build the model.

### 2.2.3. Two scenarios

In principle, having a reliable theoretical distribution for the GSD of gravel-bed rivers can be helpful in two different practical situations: assessing the reach-scale variability in GSD; or reconstructing some missing GSD metrics when grain size information is restricted. Therefore, we considered two distinct 'fitting scenarios' with the compiled data. In a first scenario, each of the tested distributions was fitted using the actual field measured parameters that define the shape of the distribution (e.g. mean and standard deviation in the lognormal and logistic distributions; shape and scale parameters in the Weibull; see SI

document for details); i.e., we 'simulated' a situation where the GSD is well documented with the field data. In a second scenario, we tried to reproduce a situation where the only available confident information is the median diameter for each GSD. For this test, we used the measured $D_{50}$ of each GSD to fit the first moment of the tested models and approximated the second moment assuming a predefined value for $D_{84}/D_{50}$. This scenario matches several practical situations where only $D_{50}$ is available or reliable. For instance, the commonly used 100 pebble count method is associated with errors in the lower percentiles that can be important compared to the median or coarser percentiles (Rice and Church, 1996; Green, 2003). Similarly, the estimates obtained from common photosieving tools are close to those obtained from pebble counts for the median sizes, but diverge for the tails of the GSD (Strom et al., 2010; Cislaghi et al., 2016; Carbonneau et al., 2018; Chardon et al., 2021) (see section S3 in SI document for more details on how the distribution models are fitted to the compiled GSD data).

## 2.3. Analysis of GSD tails

One of the greatest difficulties in grain size sampling is accurately describing the tails of the GSD distribution, for which field protocols (e.g., Wolman, bulk sampling) require very large samples. Recent methods based on high resolution topography or aerial photography have similar limitations for those fractions below pixel or resolution size. At the same time, many gravel bed rivers have been described as bimodal: i.e., with two grain size modes, one corresponding to the coarse bed framework and the other corresponding to the finer sediment that often overpasses this coarse framework during transport events, or infiltrates into the pockets and interstices between coarser grains.

This combination of sampling limitations and bimodality can give rise to great divergence between the theoretical PDFs and the tails of the empirical GSDs, and even compromise the hypothesis of a generalisable, theoretical distribution for the GSD of gravel-bed rivers in the case of high fine sediment contents. We then accomplished a specific analysis on the tails (fine and coarse) of the GSDs, considered through the 5th and 95th percentiles, which we use as metrics for the fine and coarse tails respectively. Thus, we computed the $\delta_5$ and $\delta_{95}$ scores to quantify how the tested PDFs approximated these two percentiles.

For this analysis, we classified the data according to their fine sediment content (<2 mm) and according to their dominant channel morphology into plane-bed, riffle-pool, step-pool and multi-thread channels, following Montgomery and Buffington (1997) scheme. We did this because these two characteristics are a proxy for controls on GSD that operate at a scale higher than reach scale. For example, fine sediment content is largely controlled by the supply of sand sediment from the catchment, which in turn depends on the vegetation cover (e.g. tree cover), climate (e.g. glacial sources) and hydrology (e.g. rainfall). Similarly, channel morphology is conditioned by sediment and water budgets, and each type of channel morphology is associated with different patterns of flow and sediment conveyance. Consequently, by analysing how the different PDFs behave according to these two reach characteristics, we tried to explore the potential influence of catchment controls on the local measured grain size information, insofar as these effects could partly explain discrepancies between modelled and measured GSDs.

To further explore the effects of bimodality in the compiled GSDs, we also estimated the Dip-Hartington statistic (Hartigan and Hartigan, 1985). This statistical test aims to determine whether a data series has more than one mode. It is based on measuring the difference between the cumulative frequency distribution of the empirical data and the unimodal distribution that best fits the data. If this difference exceeds a level of statistical significance (in this case, 0.05), it is considered that the data may be multimodal. We applied this test to explore what GSDs could be statistically bimodal.

Having identified the GSD in the database that could be bimodal, we fitted a mixed lognormal distribution to these cases. To do this, we
generated two lognormal distributions: one with the mean and standard deviation of the particle sizes truncated at 8 mm, and another with the mean equal to 2 mm and the 99.9th percentile equal to 8 mm. We then combined both distributions, weighted by the relative proportions of fractions larger and smaller than 8 mm. We finally compared the fit of the mixed lognormal with that of a simple lognormal to assess whether explicitly accounting for bimodality improves the description of the data.

## 2.4. Characterisation of intra-reach GSD variability in gravel-bed rivers

Grain size in gravel-bed rivers shows large spatial variability at the reach scale (Crowder and Diplas, 1997). In this respect, a reach is understood here as a section of a river with dimensions around 1–10 channel widths (typically from tens to hundreds of meters), whose geomorphological conditions (e.g., valley width, channel geometry, planform) and dynamics are relatively consistent and homogeneous; at least in the terms typically used in river assessments in fluvial morphology and river management (Gurnell et al., 2016). At this reach-scale, gravel deposits are typically a heterogeneous mosaic of alternating coarser and finer 'patches' (Dietrich et al., 2006; Nelson et al., 2009, 2015), i.e., they show complex patterns of grain size sorting in plain view. Therefore, we also addressed this issue here and investigated whether it is possible (or not) to extract some information about the reach-scale variability of GSD in gravel-bed rivers from the compiled data.

To do this, we took advantage of the GSD information available for some study sites where many samples were collected over the same reach. Specifically, in the Vénénon, Bouinenc, Severaisse (at two reaches, La Chapelle-en-Valgaudemar and Villar de Loubière) and Saja wandering rivers, we have 13–35 samples collected over study reaches of 1–3 km length. Practically only two parameters, median size and standard deviation, can be used to fit the shape and scale of many of the distribution models tested in this paper. Some bedload research (Wilcock and Crowe, 2003; Recking, 2013; MacKenzie and Eaton, 2017) suggests that other grain-size metrics, such as the $D_{84}$ or the percent fraction of fine (< 2 mm) sediment ($F_s$), are also important proxy-parameters for characterizing gravel-bed mobility. In this way, as a first approximation, we relied on these four parameters (median, standard deviation, 84th percentile and $F_s$) as an entry point to evaluate the spatial variability in the statistics that define the GSD. Then, for each one of those five study cases, we computed the ratio ($R_j$) between the single value for each individual j-GSD of the reach and the reach-mean for these four parameters:

$$R_{50j} = \frac{D_{50j}}{\langle D_{50} \rangle} \qquad (2)$$

$$R_{84j} = \frac{D_{84j}}{\langle D_{84} \rangle} \qquad (3)$$

$$R_j = \frac{CV_j}{\langle CV \rangle} \qquad (4)$$

$$R_{Fsj} = F_{sj} - \langle F_s \rangle \qquad (5)$$

where $D_{50j}$, $D_{84j}$, $CV_j$ and $F_{sj}$ are the median, the 84th-percentile, the median-centred coefficient of variation (i.e., the ratio between the standard deviation $\sigma$ and the $D_{50}$ of each j-GSD) and the percentage of sediment finer than 2 mm for each j-GSD sample, available in each study reach, respectively; $\langle D_{50} \rangle$, $\langle D_{84} \rangle$, $\langle CV \rangle$ and $\langle F_s \rangle$ are the reach-averaged values. Once estimated, we constructed the cumulative frequency distributions of each of these parameters in the five study cases and analysed how they fit a normal or lognormal distribution.

## 2.5. Comparison of 'between-sample' and 'within-sample' GSD variability

GSD in each identifiable morphological unit or textural patch of a gravel bed is far from being homogeneous and is inherently scattered. One might therefore ask how different are the local ('within-patch') and channel-scale ('between-patch') sources of variability. In other words, what exerts a greater control on the overall reach-scale variability in GSD commonly observed in gravel-bed rivers: differences between patches (or morphological units), or the intrinsic heterogeneity to be expected within each individual patch? This question has some practical implications. For example, when propagating uncertainties in hydraulic or sediment transport computation, or designing a sampling campaign to characterise variability at different scales, it is important to have an idea of the contribution of these two different sources of variability.

In this regard, we could imagine two extreme situations or scenarios of GSD variability in gravel-bed rivers (Fig. 3). In the first scenario 'between-sample' (~'between-patch') is much bigger than 'within-sample' (~'within-patch') variability. Conversely, in the second scenario, 'within-sample' is larger than the 'between-sample' variability. To explore this question, we again took advantage of the GSD information available for each of the five reaches mentioned above (see section 2.4.) where data for more than one GSD sample exists for the same reach.

To approximate 'between-sample' variability in each of these five reaches, we estimated the differences ($e_j$) between the median size of each of the j-GSD samples available, and the reach averaged $D_{50}$ ($\langle D_{50} \rangle$):

$$e_j = \frac{|D_{50j} - \langle D_{50} \rangle|}{\langle D_{50} \rangle} \qquad (6)$$

Then, to characterize 'within-sample' variability for each individual j-GSD, we estimated the median-centered coefficient of variation ($CV_j = \sigma_j / D_{50}$). To reinforce the characterization of 'within-sample' variability, we also estimated the difference ($d_j$) between $D_{84}$ and $D_{50}$ (normalized by $D_{50}$), to specifically focus on the variability of the coarse mode of the GSDs. Once estimated, we visually inspected the estimates of $e_j$, $\sigma_j$ and $d_j$ distributions for the five reaches and applied a Welch's t-test for unequal variances to compare the means of these three parameters, under the null hypothesis that there are no differences between the three sets of values (statistically significance level $\alpha = 0.05$).

To statistically strengthen our comparison of between-sample and within-sample variances, we also estimated the ratio of average 'between-sample' ($\bar{\sigma}_{btw}$) and 'within-sample' ($\bar{\sigma}_{wth}$) standard deviations for each reach:

$$Cd = \frac{\bar{\sigma}_{btw}}{\bar{\sigma}_{wth}} \qquad (7)$$

where:

<figure><img src="image_6.png" /><figcaption>Fig. 3. Conceptual model of the relative differences between 'within-sample' and 'between-sample' sources of grain-size variance in gravel-bed rivers. Two opposite, extreme scenarios could be imagined. One first scenario (left) in which 'between-sample' is larger than 'within-sample' variability. One second scenario (right) in which 'between-sample' is smaller than 'within-sample' variability.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 4. Exploratory analysis on the compiled GSD data. A) Comparison between $D_{84}$ and $D_{50}$ in the compiled data. B) Comparison between $D_{90}$ and $D_{50}$ in the compiled data. C) Interquartile range versus $D_{50}$ in the compiled data. D) Percentage of fine sediment (<2 mm) plotted versus the $D_{50}$ in the compiled data. Blue points: Surface GSDs. Red points: Subsurface GSDs.</figcaption></figure>

$$ \bar{\sigma}_{btw} = \frac{\sum_{1}^{N} |D_{50i} - \langle D_{50} \rangle|}{N} \qquad (8) $$

and:

$$ \bar{\sigma}_{wth} = \frac{\sum_{1}^{N} \sigma_{j}}{N} \qquad (9) $$

where N is the number of GSD samples available for each reach. This ratio is somewhat equivalent to Cohen's d, which is commonly used for quantifying 'effect size'. As a rule of thumb, a value of less than 0.3 is generally considered to be a small effect size, and this is the criterion we follow. Similarly, we also calculated the following ratio:

$$ F = \frac{\bar{\sigma}_{btw}^2/N - 1}{\bar{\sigma}_{wth}^2/n - N} \qquad (10) $$

where n ~ 100, as almost all the GSD samples for the five reaches come from 100 pebble counts. This ratio follows the logic of the Fisher's F-statistic. Our aim was to quantify whether, on average, the differences between 'within sample' and 'between sample' variance are statistically significant at the reach scale in each of the five study cases. We therefore compared our F estimates with the Fisher-Snedecor distribution. That said, we considered this ratio as an F-statistic with N-1 degrees of freedom for the numerator and 100 - N for the denominator. We then compared to the Fisher-Snedecor distribution table to see if its value is significantly different from what would be expected under the null hypothesis that between-sample variance is equal to or less than within-sample variance (at an α = 0.10 level of statistical significance).

## 3. Results

### 3.1. Testing for different distribution models

We can observe some strong and statistically significant correlations ($R^2$ between 0.60 and 0.71; p-value < 0.05) between the $D_{50}$, the $D_{84}$, the $D_{90}$ and the interquartile range (estimated as $[D_{84}-D_{16}]/2$) (Fig. 4). We also report a moderate and significant logarithmic correlation between the percentage of fine sediment (<2 mm) and the $D_{50}$ ($R^2 = 0.61$; p-value < 0.05), which may indicate that the coarser the gravel framework, the greater the infiltration of fine sediment. More interestingly, the observed correlations are similar between the surface and subsurface GSD data.

We then assessed how the tested distribution models capture the cumulative percentages of the different size classes (Fig. 5). In general, the tested models do not provide a good description of the tail of fine sediments (< 2 mm), with an overall tendency for underestimation. As far as the coarser framework of the GSD is concerned, all models approximate the passing percentages for the different size classes similarly well and provide estimates that do not deviate too much from the 1:1 line (Fig. 4), except the Pareto distribution. A visual comparison of the quantile-quantile (QQ) plots provides some additional information (see Fig. S3 in SI document). In general, all models show scatter for the finer quantiles, although this scatter is less for the Recking's model. For the coarser quantiles, the Weibull and Recking's models show the best fit, while the other models show overestimation. Furthermore, the exploratory tests carried out with some other parametric models (e.g. gamma, exponential, t-student, Tukey-lambda) do not show an overall fit to the field GSDs remarkably better than the Weibull or the
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Fractions of sediment finer than 2 mm (A), 8 mm (B), 16 mm (C), 32 mm (D), 64 mm (E) and 128 mm (F), estimated based on the six tested distribution models (lognormal, log-logistic, Pareto, Weibull, log-raised cosine, Recking's empirical model), compared to the field values (compiled GSD data). These results were obtained after directly calibrating the different models with the empirical field data (first modelling scenario, see main text). In the case of the passing fraction below 2 mm ($F_s$), we observe a perfect fit for the Recking's model, since $F_s$ is an input parameter of this distribution model. In the case of Recking's empirical model, we excluded from this plot the 78 GSDs from the Idaho dataset that were used to build the model. For the remaining five models, we used all 462 compiled GSDs. A similar plot for the second calibration scenario can be found in the Supporting Information document.</figcaption></figure>

lognormal, and in some tested distributions the fit is worse (see QQ plots in Fig. S7, in the SI document).

Average $R^2$ values do not allow us to clearly distinguish between the different distribution models in terms of goodness of fit: again, Pareto stands out as the distribution showing a weaker fit, while Recking's model, log-logistic, lognormal and Weibull give comparable results (Table 1). The same can be said of the results obtained with the KS tests (Table 2): Recking's empirical model, the lognormal, the log-raised cosine and Weibull distribution models are not (on average) significantly different from the compiled GSD data. However, in the case of the Pareto and log-logistic distributions, KS tests in general do not provide support for the similarity between these distributions and the field GSDs. Again, the results derived from the exploratory analysis accomplished with a broader collection of parametric models (e.g., gamma, exponential, GEV) do not provide statistical evidence of a stronger fit than the six PDFs privileged in this manuscript (see Section S.4.8. in the SI document).

Under the second calibration scenario, the different distribution models are fitted to each GSD just based on the $D_{50}$ and a standard deviation estimated by assuming a $D_{84}/D_{50}$ ratio of ~ 2.2 (as reported in Fig. 4A). It is interesting to note that in this second scenario the tested distributions lost accuracy in the fine percentiles ($D_{16}$, Fig. 6) but gained

**Table 1**

Summary of the analysis of the strength of correlation between the modelled GSDs and the field GSDs. The second and third columns show the percent of cases where the $R^2$ of the correlation is between 0.90–1 and 0.95–1, respectively (first modelling scenario). Values in brackets correspond to results obtained when fitting the distribution models in the second scenario (see main text). In the case of Recking's empirical model, we excluded from this analysis the 78 GSDs from the Idaho dataset that were used to build the model. For the remaining five models, we used all 462 compiled GSDs.

<table><thead><tr><th>GSD model</th><th>Mean R<sup>2</sup></th><th>0.90 &lt; R<sup>2</sup> &lt; 1 (%)</th><th>0.95 &lt; R<sup>2</sup> &lt; 1 (%)</th></tr></thead><tbody><tr><td>Lognormal</td><td>0.934 (0.973)</td><td>79 (97)</td><td>58 (91)</td></tr><tr><td>Log-logistic</td><td>0.922 (0.970)</td><td>72 (97)</td><td>49 (88)</td></tr><tr><td>Pareto</td><td>0.822 (0.835)</td><td>24 (26)</td><td>9 (10)</td></tr><tr><td>Weibull</td><td>0.975 (0.968)</td><td>97 (96)</td><td>92 (86)</td></tr><tr><td>Log-raised cosine</td><td>0.934 (0.967)</td><td>83 (97)</td><td>63 (85)</td></tr><tr><td>Recking</td><td>0.955 (0.948)</td><td>92 (91)</td><td>70 (65)</td></tr></tbody></table>

**Table 2**

Average p-values from the Kolmogorov-Smirnov test comparing the different distribution models with each compiled GSD. In the case of Recking's empirical model, we excluded from this analysis the 78 GSDs from the Idaho dataset that were used to build the model. For the remaining five models, we used all 462 compiled GSDs.

<table><thead><tr><th>GSD model</th><th>Average p-value (1st modelling scenario)</th><th>% of data p-value &gt; 0.05</th><th>Average p-value (2nd modelling scenario)</th><th>% of data p-value &gt; 0.05</th></tr></thead><tbody><tr><td>Lognormal</td><td>0.192</td><td>53</td><td>0.166</td><td>49</td></tr><tr><td>Log-logistic</td><td>0.093</td><td>33</td><td>0.067</td><td>22</td></tr><tr><td>Pareto</td><td>0.056</td><td>34</td><td>0.055</td><td>29</td></tr><tr><td>Weibull</td><td>0.170</td><td>54</td><td>0.173</td><td>53</td></tr><tr><td>Log-raised cosine</td><td>0.169</td><td>60</td><td>0.156</td><td>54</td></tr><tr><td>Recking</td><td>0.106</td><td>42</td><td>0.039</td><td>19</td></tr></tbody></table>

it in the coarser percentiles ($D_{84}$, Fig. 6). This could point at the bimodal nature of the GSDs of gravel-bed rivers and support the idea that these PDFs are adequate for the coarse mode or framework of the riverbed, but not so adequate for the fine mode.

## 3.2. Analysis of GSD tails

Fig. 7 summarizes the results of the analysis on how the different distribution models fit the tails of the GSDs. For both the fine and the coarse tails, the Pareto distribution shows a worse performance, with tendency to overestimation for both the 5th and the 95th percentile. On the other hand, the Weibull distribution seems to approximate these two percentiles better than the rest of the PDFs.

For a more in-depth analysis, we have classified the data according to $F_s$ and the dominant channel morphology. This is shown in Figs. 8 and 9. For aims of simplicity, in these figures we focus specifically on the lognormal and Weibull distributions, as these are the most used PDFs for the GSD of gravel-bed rivers.

Results show that the fit of the two PDFs becomes worse as $F_s$ of the sample increases. In Fig. 8 we can see how the two distributions are relatively close to the field values of $D_5$ and $D_{95}$ when the amount of sand sediment is low. However, for high sand sediment contents, the
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Ratios between estimated values and the field measures, for 16th and 84th percentiles of the GSD, based on six different distribution models (lognormal, log-logic, Pareto, Weibull, log-raised cosine, Recking's empirical model). A and C: Ratios for $D_{16}$. B and D: Ratios for $D_{84}$. The different distribution models were directly calibrated with the empirical field data (first modelling scenario, see main text) in figures A and B, while in figures C and D they were calibrated using the field $D_{50}$ and assuming a $D_{84}/D_{50}$ ratio ~ 2.2 (second calibration scenario, see main text). In the case of Recking's empirical model, we excluded from this plot the 78 GSDs from the Idaho dataset that were used to build the model. For the remaining five models, we used all 462 compiled GSDs. Ln: lognormal. Ll: log-logistic. P: Pareto. Wb: Weibull. Cs: log-raised cosine. Rc: Recking's empirical model.</figcaption></figure>

dispersion of $\delta_{5\%}$ and $\delta_{95\%}$ clearly increases in case of lognormal distribution. This scatter increases with sand content without a clear trend for under- or overestimation of the 5th and 95th percentiles. The same happens for the 5th percentile in the case of the Weibull distribution, but not for the 95th percentile, indicating that estimates of coarser GSD percentiles with this distribution are more robust.

In addition, these trends are observed regardless of the dominant channel morphology. Fig. 9 shows the same pattern for all channel morphologies: the way in which the theoretical distributions approximate the tails of the distribution worsens with sand content, but this effect is more pronounced for the fine fraction in multi-thread and riflemen-pool channels (compare $\delta$ values in the figures), i.e., channels with well-developed macroforms and higher sediment supply are those showing a poorer approximation to the GSD tails.

After this descriptive analysis, we evaluated whether explicitly considering the effect of the fine fraction (sand) in PDF fit could improve the description of the compiled GSD data. To do so, we started by applying the Dip-Hartington test, the results of which indicate statistical evidence

that 68 % of the distributions could show bimodality. No covariation is observed between test results and the fine sediment content (Fig. 10A). Instead, some covariation is observed with the size content between 2 mm and 16 mm: potentially bimodal distributions have lower contents in this size classes (Fig. 10A). This suggests that these size fractions globally represent a gap between two grain size two modes.

We then fitted a mixed lognormal distribution to these cases and compared its fit with that of a simple lognormal (Fig. 10B). The change in PDF fit is marginal. This may indicate that the two modes have considerable overlap, which would reduce the impact of considering bimodality on improving model fit. Furthermore, since the proportion of the fine fraction is generally small compared to the coarse fraction, this also can have a limited effect on the overall distribution fit. Therefore, although bimodality exists in a technical manner in a remarkable amount of the compiled GSD, its quantitative influence is small and difficult to capture accurately using a mixed distribution. Nevertheless, in making the adjustment of the mixed lognormal, we have assumed that the distributions have two modes, with the finer centred at 2 mm. It
<figure><img src="image_10.png" /><figcaption>Fig. 7. Ratios between estimated values and the field measures, for the 5th and 95th percentiles of the GSD, based on six different distribution models (lognormal, log-logistic, Pareto, Weibull, log-raised cosine, Recking's empirical model). A: Ratios for D<sub>5</sub>. B: Ratios for D<sub>95</sub>. The different distribution models were directly calibrated with the empirical field data (first modelling scenario, see main text). In the case of Recking's empirical model, we excluded from this plot the 78 GSDs from the Idaho dataset that were used to build the model. For the remaining five models, we used all 462 compiled GSDs.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Ratios between estimated values for the 5th and 95th percentiles of the GSD based on two different distribution models (lognormal and Weibull) and the field measures. Data have been segregated according to sand content. A: Log-normal. B: Weibull.</figcaption></figure>

could be that the particle sizes have more than two modes, or that the second mode is not necessarily centred on the terms below 2 mm in all the GSDs. This could also contribute to explain why the mixed log-normal does not significantly improve the distribution fit in our analysis. Moreover, we explored a relatively simple case, which is the combination of two log-normal distribution. It may be that each of the modes of the GSD fits better to different types of distributions.

### 3.3. Characterizing the intra-reach GSD variability in gravel-bed rivers

In Fig. 11, we explore the probability distribution of $R_{50j}$, $R_{84j}$, $R_{CVj}$ and $R_{FSj}$ in the subset of five reaches where several GSD samples were collected across the same reach. Although there is some variability, the cumulate distributions of $R_{50j}$, $R_{84j}$, $R_{CVj}$ and $R_{FSj}$ show comparable trends in the five reaches.

In the case of $R_{50j}$, the cumulate frequency distributions for the five

reaches seem to be well described by a normal distribution with mean equal to 1 and standard deviation ~ 0.31. We applied a KS-test and the obtained p-value > 0.050 does not provide evidence to reject the null hypothesis that the compared distributions are the same.

Similarly, the cumulate distribution of $R_{84j}$ values is visually well described by a normal distribution with mean equal to 1 and standard deviation ~ 0.34 (KS-test, with a p-value > 0.050). Furthermore, the cumulate distribution of $R_{CVj}$ values is well described by a log-normal distribution with mean equal to 0 and standard deviation ~ 0.29 (KS-test with a p-value > 0.050). Also, $R_{FSj}$ visually appears to follow a normal distribution with mean equal to -0.55 and standard deviation equal to 5.32, although in this case the p-value obtained with a KS test is < 0.050, which does not support null hypothesis that this distribution and the composite cumulate distribution of $R_{FSj}$ values for the five reaches are the same.

### 3.4. Comparing 'between-sample' and 'within-sample' variability

In Fig. 12, we compare the estimates of $e_j$ (between sample variability), $CV_j$ (within sample variability) and $d_j$ (within sample difference between $D_{84}$ and $D_{50}$) for the Wolman samples available in five multi-thread river reaches. In the five reaches, we observe the same trend: $e_j$ is smaller than $CV_j$ and $d_j$. These differences are statistically significant at the 95 % confidence level (p-value < 0.050, Welch's t-test for unequal variances). Moreover, in all five cases, our estimates of $Cd$ are between 0.1 and 0.3, suggesting a small effect of 'between-sample' differences in median grain size. Again, this points to 'between-sample' differences that are on average small compared to 'within-sample' differences. Similarly, the estimates of $F$ suggest statistically significant 'between sample' differences only in Saja and Vénéon, while in the remaining study cases the $F$ estimates do not allow us to reject the null hypothesis and confirm our previous observations that the 'between group' variance is not statistically significantly different from the 'within group' variance.

## 4. Discussion

### 4.1. The probability distribution model for grain-size in gravel-bed rivers

#### 4.1.1. Is there a distribution model that is well suited to the GSD in gravel bed rivers?

Our descriptive analysis of the collected data revealed some strong correlations between different percentiles of the compiled GSDs (Fig. 4). These trends highlight two interesting points. First, it appears that there
<figure><img src="image_12.png" /><figcaption>Fig. 9. Ratios between estimated values for the 5th and 95th percentiles of the GSD based on two different distribution models (lognormal and Weibull) and the field measures. Data have been segregated according to sand content and dominant channel morphology.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. A) Differences in < 2 mm and 2–16 mm sediment size content between unimodal samples and those for which the Dip-Hartigan test provided statistical evidence of bi- or multimodality. B) Ratios between estimated values and the field measures, for the 5th, 16th, 84th and 95th percentiles of the GSD, based on an ordinary lognormal and a mixed (bimodal) lognormal distribution.</figcaption></figure>

are quasi-constant relationships between the different parameters describing the GSD of gravel-bed rivers, as already reported by Rick- enmann and Recking (2011) or Recking (2013). Secondly, the general shape or distribution for the GSD in gravel-bed rivers appears to be similar for both surface and subarmour GSDs, as already reported by Rice and Haschenburger (2004). In principle, these trends would justify the search for a general distribution model capable of providing a gen- eral description of the shape of the GSD of gravel-bed rivers. To inves- tigate this, six distribution models were applied to the compiled data. Based on the results, we found that the Pareto and log-logistic distri- butions do not provide in general a good fit for the compiled data, whereas the lognormal and Weibull provide a reasonable fit (Fig. 5; see also Table S4 and Fig. S3 in SI document). These results suggest that the

GSD of the compiled gravel bed rivers is more adequately described by a distribution model with moderately heavy tails, such as lognormal or Weibull, rather than by models with long and heavy tails, such as the Pareto. However, we cannot exclude the possibility that other untested distribution models could fit the compiled GSDs better than those tested in this paper. Nevertheless, the exploratory analysis carried out with a wider collection of parametric models does not show any model that provides a significantly better description of the GSD than the Weibull or lognormal models (see section S.4.8. in SI document).

Our analysis is in line with previous literature in that it shows that both the lognormal and Weibull distributions reasonably approximate the general shape of the GSD of gravel-bed rivers. The historical choice of Weibull and lognormal distribution summarizes two different lines of
<figure><img src="image_14.png" /><figcaption>Fig. 11. Analysis of intra-reach variability in the parameters used as input to the different GSD models. CFD: Cumulative frequency distribution.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 12. Comparison between the ‘between-sample’ variability in $D_{50}$ ($e_i$), the ‘within-sample’ coefficient of variation (median centred, $CV_i$) and the ‘within-sample’ differences between $D_{84}$ and $D_{50}$ ($d_i$), for five river reaches where several GSD samples are available for the same reach: A) Vénéon. B) Saja. C) Bouinanc. D) Severaisse (Villard Loubière). E) Severaisse (La Chappelle). A table with the estimated values of $C_d$ and $F$ for each reach, calculated using the equations (7) and (10) (respectively), is included in the figure.</figcaption></figure>
thought when trying to find the best statistical description of the frequency distribution of grain sizes in riverbeds: one based on the theory of particle fragmentation and the other based on the sorting of particles by selective transport, respectively. That said, these two different lines of thought reflect that, in any individual river reach, the size distribution is constrained first by the size and character of the material delivered to the stream from the landscape; and secondly by the range of hydraulic conditions experienced by that sediment during transport. Depending on which aspect they emphasised, different authors have then assumed the log-normal or the Weibull distribution as the most appropriate to capture the characteristics of riverbed GSDs.

However, the shapes of the log-normal and Weibull distributions can resemble when applied to the same GSD data, and empirical analysis of field data of gravel bed rivers has not been able to discriminate in a conclusive manner which one best fits the field-measured GSDs (Kondolf and Adhikari, 2000). In addition, although these models resemble the general shape of the field GSD curves, analyses trying to fit empirical data to these theoretical distributions in gravel-bed rivers have shown some disparity between the theoretical curves and the empirical ones (Bagnold and Barndorff-Nielsen, 1980; LeRoy, 1981; Rice and Church, 1996; Bunte and Abt, 2001), particularly at the distribution tails. This has led some authors to state that no distribution could be assumed a priori for the GSD of gravel-bed rivers (Church et al., 1987; Rice and Church, 1996).

Our analysis of the GSD tails suggests that the Weibull distribution may be slightly more robust than the log-normal, particularly for fitting the coarse tails (Figs. 7, 8 and 9). The Weibull distribution is quite flexible in adapting to some variations in the length and heaviness of the distribution tails, which may explain why it provided a better fit to the field data. This flexibility allows this distribution to accommodate the variability in the coarse tail characteristics of the compiled data, such as the presence of relatively longer or shorter tails in the coarser sizes. The slightly better fit of the Weibull model could therefore be attributed to its ability to model a wider range of tail behaviours compared to the log-normal distributions.

Our results also suggest that Recking's empirical model provides a first-order approximation to the GSD of gravel-bed rivers comparable to the log-normal or the Weibull distributions (Fig. 5). Because it is easy to implement and parameterise, it can be a reasonable alternative to theoretical PDFs for numerical purposes, such as error propagation in bedload or hydraulic calculations. There is even some potential for model improvement; in this regard, Recking's model parameters (table S3 in section S3.6 of SI) were recalibrated in this paper and refined with the total of 462 samples compiled (table S3 in SI). Results derived from this empirical model are promising in that they illustrate the unexplored potential stored in data from new GSD sampling strategies (e.g., LiDAR, photogrammetry, high-resolution aerial imagery), coupled with big data analysis, to address the long-standing question of how best to describe the shape of the GSD.

### 4.1.2. Is there a single, general distribution model for the GSD of gravel-bed rivers?

The results of the Kolmogorov-Smirnoff tests do not provide statistical evidence that any of the tested distributions (e.g. Weibull, log-normal) are appropriate for all the compiled data (Table 2). In particular, we observed significant deviations in the tails when sand content is important (Figs. 8 and 9). In this respect, it could be argued that rivers across different geographical contexts may be subject to different environmental conditions (e.g., sediment supply, hydrology) that could influence the GSD (e.g., Whittaker et al., 2010). For example, a pristine gravel-bed river may exhibit a GSD resulting from natural fluvial processes, while a dammed river will reflect altered flow regimes and sediment supply, potentially requiring different distributions to best describe its specific conditions. Given this variability, it could be expected that different river scenarios would require their own distributions, rather than assuming a single best distribution across all

contexts. However, while this is true, it is also true that the mechanisms by which these adjustments to different catchment conditions operate are the same (i.e., sediment transport processes, particle abrasion). Therefore, if the statistical assumptions behind these probability distributions are more related to the physical mechanisms that explain grain size, rather than the specific conditions under which these mechanisms operate, there would not necessarily be a change in the distribution. It would only change if different catchment conditions involve different sediment-transport processes, thus invalidating the statistical assumptions behind certain models.

Our analysis provides limited insight in this regard. However, the fit errors of the different models (measured by the RMSE) show some positive covariation, suggesting that when one model fits poorly, the others tend to do so as well (see Fig. S4 in SI document). We also observed no trend in the RMSE of the different probability distributions based on drainage area, bed slope or dominant channel morphology (see Figs. S5 and S6 in SI document). Thus, our data do not provide support to the idea that the shape of the grain-size distribution changes according to channel conditions. However, we are aware that the evidence is very indirect. Further research exploring into how these environmental and geomorphological conditions influence the GSD and the appropriateness of different statistical models will be welcomed.

Related to all the above, some research has already suggested that a change in the shape of the GSD might be expected in relation to changes during source-to-sink transport of sediment, and/or differences in hydraulic conditions or sediment transport stage. For example, Krumbein and Tisdel (1940) and Ibbeken (1983) suggested that the GSD of weathered rocks evolves from a Rossin (Weibull) distribution to a lognormal distribution with fluvial transport. Similarly, Shih and Komar (1990) reported a change in bedload GSD from a lognormal to a Weibull distribution with increasing discharge, which they attributed to a change from size-selective to full-mobility transport conditions. However, we could not go here very far in this respect, as the information available did not allow us to clearly link the shape of the PDF to the bedload transport conditions for each GSD. Nevertheless, our observations suggest that as sand content increases, the coarse tails of the GSDs are better approximated by a Weibull than a log-normal distribution (Fig. 8). Similarly, at low sand contents, the log-normal distribution shows a large discrepancy for both the coarse and fine tails compared to the Weibull distribution in plane-bed and step-pool channels (Fig. 9). To the extent that sand content is somewhat a proxy for bedload conditions, and that plane-bed and step-pool morphologies tend to be preferentially located in the upper reaches of catchments compared to riffle-pool (and multi-thread) (Fig. 1C), these results may indicate a change in the shape of the GSD with the position of the reach in the drainage network and with sediment transport conditions. However, these trends are weak and deserve further analysis, probably with data collected specifically to address these issues.

## 4.2. Fine GSD tails

The tested distribution models tend to not adequately reproduce the fine tails of the GSD when there is a significant amount of sand in the sample (see Fig. 5A, 8 and 9), in line with previous studies (Bagnold and Barndorff-Nielsen, 1980; Rice and Church, 1996; Bunte and Abt, 2001). Actually, the GSDs in gravel-bed rivers are not only controlled by abrasion and selective transport, but also by the methods of sampling and measurement (Middleton, 1970). Having said that, sampling methods typically used in gravel-bed rivers require sample sizes much larger than those used in many of the collected data (Table S1 and S2 in SI) for reliable estimates of the tails of the GSD (Church et al., 1987; Rice and Church, 1996; Bunte and Abt, 2001). This could explain some of the bias observed in the fine size ranges when comparing the theoretical PDFs and the field GSDs. In this regard, gravel-bed rivers have been reported to show a general paucity of 1–8 mm particle sizes, the so-called, ‘grain-size gap’ (Yatsu, 1955; Shambrook Smith and Ferguson,
1955; Venditti and Church, 2014), and our results are congruent in that the samples for which we report statistical evidence of bimodality tend to have a lower 2–16 mm sizes content (Fig. 10A). This gap could therefore be partly explained by sampling biases in the methods commonly used for GSD. For instance, it is difficult to detect smaller grains in a pebble count. Only sand is usually detected because this size fraction is organised in patches between cobbles, transported at the end of a flood or by secondary currents, making it easier to sample.

Nevertheless, observations in flumes and theoretical models also suggested that this gap may emerge from sediment transport mechanics (Parker, 2008; Lamb and Venditti, 2016; Dingle et al., 2021; Church and Hassan, 2023; Dingle and Venditti, 2023). It is true that the 1–5 mm range establishes a boundary between a fine size range where the transport mode is dominated by suspension, and a coarser size range for which sediment suspension becomes vanishingly small under normal flood conditions. Additionally, sediment mobility of fine gravels increases when overlying a sand bed (Parker et al., 2024). Consequently, the GSD of gravel-bed rivers is usually bimodal (Wilcock, 1998; Deng et al., 2024), i.e., it is composite of two distributions governed by their different response to fluid forces, the boundary of which is precisely determined by the limit between the sand and gravel size ranges (2 mm). Together with sampling biases, this distinct modes of transport in gravel and sand ranges can also help explaining the grain-size gap and why the PDFs tested do not perform well in these fine (<2 mm) size fractions (Fig. 5A).

### 4.3. Reach scale variability in grain-size metrics: Practical implications

Our analysis shows some consistent patterns of reach-scale variability in grain size metrics when combining data from several case studies where more than one GSD sample was collected in each reach (Fig. 11). These are interesting results that suggest how certain statistical parameters of the GSD of gravel-bed rivers are approximately normally (or lognormally) distributed at the river reach scale (see section 3.3). This finding is consistent with the central limit theorem, which states that the distribution of the means of a series of samples should be normally distributed, regardless of the underlying distribution of the individual samples. Based on this, Recking (2013) has already hypothesised that $D_{50}$ should vary randomly (at a reach-scale) according to a normal distribution. Our results are therefore in good agreement with these previous assumptions.

Some previous research pointed out that bedload transport quantification and prediction could be improved by explicitly accounting for the variability in the input parameters of bedload equations and models (Wilcock et al., 2009; Monsalve et al., 2016; Recking et al., 2024). GSD statistics are key parameters in all bedload transport equations, and uncertainties in their definition has been reported as a major source of uncertainty in bedload estimation (Recking, 2013; Fernández and García, 2017). We therefore believe that our observations could be used to improve the characterisation of grain size variability at the river reach scale. As such, the distributions shown in Fig. 11 could be combined with some assumed distribution model to estimate the total variability in GSD at the river reach scale using statistical resampling methods.

To better illustrate this approach, we applied this idea to the Vénéon River dataset. In the Vénéon, we collected 35 Wolman samples (100 pebble counts per each sample) in a 2 km river reach, which is a remarkable number of samples for a single study reach and allows us to test this approach. The workflow we followed involved several steps. First, we randomly selected one GSD from the 35-sample dataset and estimated $D_{50}$, and $F_s$; $D_{84}$ was estimated assuming that $D_{84}$ covariates with $D_{50}$, so it is ~ 2.2$D_{50}$ (Fig. 4A). We assumed that the variability of these three parameters follows the distributions shown in Fig. 11, using the sample $D_{50}$ and $F_s$ as reference parameters. We then randomly generated an artificial population of $10^6$ values for $D_{50}$, $D_{84}$ and $F_s$ (< 2 mm) and randomly sampled these three 'populations' and constructed the full GSD using Recking's empirical model. We repeated this step

100,000 times to obtain more robust results and integrated the results of these 100,000 runs. Finally, we estimated the distribution of frequencies for the cumulated GSD after these 100,000 runs and compare it to the original 35-sample dataset.

The median GSD resulting from the 100,000 runs is very similar to the composite sample constructed from the 35 samples (Fig. 13), i.e., our approach allows us to approximate the reach-averaged GSD based on a single sample. Also, the 5th and 95th frequency bounds resulting from the 100 000 runs are generally consistent with the 5th and 95th centiles of the 35 field samples, although there is some underestimation in the case of finer fractions.

Overall, we can conclude that our approach is successful in retrieving the general shape of the GSD and roughly grasps the reach-scale variability in grain size in the Vénéon River, based on a single sample. In principle, the performance of this method would depend on the $D_{50}$ and $F_s$ used as input parameters being close to the average values of the river reach; otherwise, resampling can lead to significant deviations (Fig. 14). It is therefore important that the field sampling is truly representative of the average bed conditions. This requires a detailed visual inspection in the field to do an adequate selection of the area to be sampled (e.g., using Wolman, photosieving), excluding patches with a more pronounced 'exotic' texture, such as large patches of sand produced by secondary flows (Recking et al., 2024). Also, in practice, more than one single sample is often taken in a reach and averaged, so it is to be expected that in practice the input values in the statistical simulation will be more robust than those used here (where only a single sample has been taken). A possible sampling strategy to improve this procedure for characterizing grain size variability would be to first map the different macroforms or textural facies present in the bed and collect a grain-size sample representative of each identified morphology (or facies), and then integrate them into a composite GSD (Crowder and Diplas, 1997; Kondolf, 1997). We believe that the method is robust and can serve to account for natural variability in numerical works when the data are limited (which is often the case) and to improve the propagation of uncertainties.

### 4.4. Sources of grain size variability in gravel-bed rivers

Our analysis documents that the variability in grain size that we can observe within a single GSD sample appears to be (on average) more important than the variability in grain size metrics (e.g. median size) that we could report when comparing different GSD samples collected across a river reach (Fig. 12). In this regard, Church and Kellerhals (1978) analysed replicate Wolman samples collected from several bars in the Peace River (Canada) and found that the 'between-sample' variance was significantly greater than the 'within-sample' variance, even in relatively homogeneous sample locations, which is an opposite trend to that globally observed in our data (Fig. 12). Thus, it is possible that rivers in different areas or contexts may show different contrasts between 'within-patch' and 'between-patch' variability.

Some of the 'within-sample' GSD variability in gravel-bed rivers may be explained by the inherent uncertainty in percentile estimates associated with sample size (for instance, 100 vs 400 pebble counts). In this respect, Bunte and Abt (2001) reported that uncertainties in the estimation of $D_{50}$ ranged from +/-10% to +/-16% for a 100-particle pebble count (Wolman, 1954), depending on whether the error bounds were estimated using a binomial (Fripp and Diplas, 1993) or multinomial (Petrie and Diplas, 2000) approach. Ferguson (2007) equally pointed out that the uncertainty in $D_{84}$ estimates in step-pool channels can be in the range of +/-10% and +/- 20%. Similarly, Rice and Church (1996) suggested that a minimum of 400 measured grains is required to reduce errors in $D_{50}$ estimates to less than +/- 10% using the pebble count method. Those values are lower than the average 'within-sample' coefficients of variation estimated for the GSDs ($e_j$) compiled here, but comparable to the average 'between-sample' variability ($CV_j$) (Fig. 12).

Therefore, our analysis suggests three things: i. the observed 'within-
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Results from the test done with the field samples from the Vénéon river to illustrate how the results from the present work can improve the characterization of GSD variability at the reach-scale in gravel-bed rivers. In the Vénéon river, 35 Wolman samples were collected in a 2-km length reach. In this work, we randomly select one of this 35 samples. We estimated $D_{50}$, $D_{84}$ and $F_s$ (% < 2 mm) for this sample. Three artificial populations of $10^6$ values of these three parameters were generated using a normal distribution (based on the results shown in Fig. 11). Then, these population were randomly sampled 100 000 times and introduced into Recking's GSD model to generate 100 000 simulated GSDs for the Vénéon river. Fig. 13A shows the results of these 100 000 simulations compared to the original field data. In Fig. 13B we summarize the results from Fig. 13A. Interval marks around points in Fig. 13B correspond to the 5-th and 95-th percentiles of field samples.</figcaption></figure>

sample' variability is larger than that simply expected from sample size, i.e., the observed variability cannot be solely attributed to sampling biases; ii, the variability in grain size at the patch scale (~'within-sample') may be greater than the variability that we could observe between patches (~'between-sample'); and iii. the observed 'between-sample' variability in percentile estimates is in the same order of magnitude as that expected from sampling errors, i.e., cannot be easily isolated or discriminated from sample biases.

Consequently, apart from sampling bias, 'within-patch' variability is also explained by the random arrangement of poorly sorted, water-worked gravel. In this respect, 'between-patch' and 'within-patch' grain-size variability in gravel-bed rivers are likely to result from mechanisms operating at two different scales. On the one hand, 'between-patch' variability is mostly associated to metric- and decametric-scale morphologies (e.g., macroforms), whose development is mostly controlled by the dominant patterns of flow and sediment conveyance in the channel. On the other hand, 'within-patch' variability is mostly related to grain-scale entrainment and deposition processes (Powell, 1998) that are dependent on the spatial variability in the local characteristics of flow turbulence (Bluck, 1987; Clifford et al., 1993). Therefore, our results point at two things: i. the local effects of turbulent fluctuations can partially blur the grain-size 'signal' linked to sediment and flow circulation patterns; and ii. the magnitude of grain size

differences expected between different morphological units or patches is close to the detection threshold of classical GSD sampling methods. Nevertheless, we must be cautious in our previous interpretations, as our results may be conditioned by the particular database used here, which is somewhat geographically restricted (Fig. 1).

We believe that these observations have some implications for the use of GSD data in hydraulic and sediment transport models, as well as for grain-size sampling strategies. That said, if the local or intrinsic GSD variability of each patch is the most dominant control on the overall riverbed variability at the reach scale, then the determination of the 'within-patch' variability is the most relevant parameter for the propagation of uncertainties and errors when using flow friction or sediment transport equations. Field sampling efforts should then focus on characterising local, 'within-patch' variability rather than 'between-patch' variability. In this case, few but large and well selected samples (i.e., representative of average bed conditions) should be preferred to many but small samples. Furthermore, if one's research interest is to characterise the grain size differences between different morphological units and areas within a reach to link these differences to hydraulic or sediment transport controls, then several samples distributed across the reach should be preferred. However, the sampling design must be very careful to ensure that we can characterise the differences between patches and that these are not masked by strong local variability or
<figure><img src="image_17.png" /><figcaption>Fig. 14. Results of the statistical resampling test carried out on field data from the Vénéon river, obtained with different starting samples. A) The percent difference (% diff) between the *D*<sub>50</sub> of the departing sample and the reach-average *D*<sub>50</sub> is 5%. B) The percent difference (% diff) between the *D*<sub>50</sub> of the departing sample and the reach-average *D*<sub>50</sub> is 6%. C) The percent difference (% diff) between the *D*<sub>50</sub> of the departing sample and the reach-average *D*<sub>50</sub> is 16%. D) The percent difference (% diff) between the *D*<sub>50</sub> of the departing sample and the reach-average *D*<sub>50</sub> is 51%.</figcaption></figure>

sampling errors. In both situations, the commonly implemented 100 pebble count design seems limited, especially if interested in the fine tail of the distribution, as already noted by Rice and Church (1996).

## 4.5. Future perspectives

The 'Wolman' pebble count method (Wolman, 1954) for the surface GSD and the volumetric bulk sampling for the subsurface GSD (Church et al., 1987) have been two widely used field protocols for sampling GSD by river scientists for many decades (Bunte and Abt, 2001; Kondolf, 1997). Although these methods have provided (and continue to provide) very valuable information about the characteristics of the grain size distribution in gravel-bed rivers, they require an enormous amount of effort and time for obtaining spatially distributed characterisations in gravel-bed rivers. This very likely explains the paucity of information available on the spatial and temporal variability of GSD in gravel-bed rivers. In many studies, grain-size characterization in gravel-bed rivers have been based on sampling designs where few samples were usually taken as representative of whole study reaches. In view of the great heterogeneity of gravel beds, some doubts could arise when extrapolating the GSD parameters obtained from few samples to entire river reaches.

Fortunately, several alternative methods (e.g., photo-sieving from aerial images; or roughness estimates from 3D point clouds) with the potential to provide spatially distributed grain size characterisations have been developed in recent decades. In this respect, a large body of research has explored the use of 3D point clouds obtained from high-resolution topographic methods (e.g., TLS, aerial LiDAR, Sfm photogrammetry from drone imagery) to obtain roughness metrics and grain-scale topographic information from gravel-bed rivers (e.g., Heritage and Milan, 2009; Brasington et al., 2012; Pearson et al., 2017; Vázquez-Tarrío et al., 2017; Woodget et al., 2018; Chardon et al., 2020; Wong

et al., 2024). Some recent research has even developed algorithms capable of automatically segmenting grains from 3D point clouds to obtain information on grain size, orientation or protrusion (e.g., Steer et al., 2022; Yager et al., 2024). Although these methods still have some general limitations (e.g. material cost; complex processing; limitation to exposed areas; complications in rivers with a dense canopy of vegetation; do not provide information on subsurface GSD; need of field data for training/calibration), they have great potential to provide maps and spatially distributed characterisations of GSD in gravel-bed rivers (Díaz-Gómez et al., 2022). This gives them an advantage over classical methods (such as Wolman) to better study the issue of spatio-temporal variability of GSD in gravel-bed rivers, as well as to perform more robust analyses with a larger volume of data on the statistical distribution of grain-size parameters. However, up to our knowledge, few studies have applied these methods to explore the issue of reach scale variability in GSD metrics and surface bed texture (e.g., Storz-Peretz and Laronne, 2013; Storz-Peretz et al., 2016; Rogers et al., 2025). Further research on GSD variability in gravel-bed rivers using these novel methods would be welcomed in the near future. In this respect, we think that our results could provide interesting support or keys for guiding and applying these new methods.

## 5. Conclusions

In the present work, we analysed a dataset of 462 GSD data to investigate which probability distribution model best describes the cumulative grain-size distribution of gravel-bed rivers. Five different theoretical distribution models were tested: lognormal, log-logistic, Weibull, Pareto, and log-raised cosine. As an alternative to these five probability distributions, we also explored and discussed some other parametric models (e.g., gamma GEV) and tested the potential offered by an empirical or 'data-driven' model (proposed by Recking, 2013).
Our results document how some distribution models (e.g., Pareto) fail to adequately describe the thin and coarser tails of GSD in gravel-bed rivers, as shown by previous work. Conversely, some models (lognormal, Weibull, Recking) appear to be robust in approximating the shape of the GSD in the collected data, although it is difficult to discriminate well between these different models when approximating the GSD of gravel-bed rivers.

We then investigated the variability of different probability distribution parameters (e.g., median, standard deviation) at the river reach scale. To do this, we took advantage of the available data for several study sites where more than one sample was collected in the same reach. This dataset shows some patterns in reach-scale variability in grain size metrics, which can be adequately described with a normal and log-normal distribution.

To finish we illustrated the potential of these approaches when grain-size data are incomplete or lacking. That is, we combined these results with statistical resampling methods and Recking's empirical model, and we showed how the results of our analysis potentially help improving the characterisation of grain size variability in a given gravel-bed river.

## CRediT authorship contribution statement

**Daniel Vázquez-Tarrío:** Writing - review & editing, Writing - original draft, Visualization, Validation, Resources, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Alain Recking:** Writing - review & editing, Writing - original draft, Visualization, Validation, Supervision, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

We want to acknowledge M. Vivier, F. Liébault, R. Menéndez and E. Fernández for helping in the collection of GSD data in the field. We also want to acknowledge M. Tal, F. Liébault, C. Le Boutelier and C. Misset for sharing their GSD data. We are also grateful to M. Church, Adrien L. Collins and two anonymous reviewers, whose comments and suggestions helped strongly improving the manuscript. Additionally, the work done by the first author (D. Vázquez-Tarrío) was supported by the 2023-2026 grant signed between the Spanish Directorate General for Water (DGA-MITERD; Government of Spain) and the Spanish National Research Council (CSIC-Ministry of Science, Innovation and Universities), which includes action "Sedimentary Morphodynamics" (20233TE012: IGME-CSIC; Tarquín 2 Project); this author also had the support of the French Ambassade in Spain. Finally, the first author would like to dedicate a few final words to the memory of Raúl Y.G., who can finally roll and rest freely over the riverbed of deep time. I will always regret that we missed our last conversation about Riglos' conglomerates. Thanks, Raúl, for the time we spent together.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.catena.2025.108974.

## Data availability

Grain size database (Original data) (Zenodo).

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