Catena
The estimation of bedload in poorly-gauged mountain rivers
--Manuscript Draft--
Manuscript Number: CATENA13284R1
Article Type: Research Paper
Keywords: bedload, gravel-bed rivers; sediment transport; modelling Corresponding Author: Daniel Vázquez Tarrío
University of Oviedo Oviedo, Asturias SPAIN
First Author: Daniel Vázquez Tarrío
Order of Authors: Daniel Vázquez Tarrío
Rosana Menéndez Duarte, PhD
Abstract: Bedload transport is one major driver of gravel-bed river morphodynamics, and its quantification is capital for many environmental issues and river engineering applications, as well as for landscape evolution studies. To this point, bedload transport rates and volumes have been classically computed by means of sediment transport formulae. The most used bedload transport equations compute the bulk mass bedload based on section-averaged hydraulic parameters. However, due to the non- linear behavior of sediment transport, bedload formulae are sensitive to the input parameters. Then, some doubts arise when applying bedload equations on poorly gauged river reaches, i.e. rivers where there are no hydrological records and rating curves. In this paper, we assess the application of bedload equations in the case of poorly gauged river reaches, and we test a workflow to follow in such situations. This workflow consists of three steps: i) Reconstructing the flow duration curve, based on gauging records from neighboring river basins; ii) Solving the hydraulic geometry relations of the study-case river-reach, based on a flow friction equation; and finally, iii) Computing bedload with a sediment transport equation. We tested this approach against the bedload information available in the literature for Idaho streams and we found that it could potentially approximate annual bedload volumes in ungauged reaches under certain conditions. To illustrate the potential of this workflow, we also computed bedload volumes for two ungauged river reaches from the Cantabrian mountains (NW Spain).
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FLOW DURATION CURVE
1D HYDRAULICS:
wetted width, depth: shear stress ...
REGIONAL INFORMATION Calibrate flow duration model
BEDLOAD RATES:
Average anual bedload volumes Sediment rating curve Hydrological/
Gauging data
?
Hydraulic geometry:
Rating curves
?
Channel geometry +Rickenmann and Recking (2011)
aelvetr elW Time
Water level
Flow
Bedload
?
equation
Recking (2013)
GRAPHICAL ABSTRACT
Vázquez-Tarrío and Menéndez-Duarte
Graphical Abstract
The estimation of bedload in poorly-gauged mountain rivers 1
Vázquez-Tarrío, D.a,b,c,d* and Menéndez-Duarte, R.b,c 2
a Geological Hazards Division, Geological Survey of Spain (IGME), 28003 Madrid, 3
Spain 4
b Department of Geology, University of Oviedo, c/ Jesús Arias de Velasco, s/n, 33005 5
Oviedo, Spain 6
c INDUROT, University of Oviedo, Campus de Mieres, s/n, 33600 Mieres, Spain 7
d University of Lyon, CNRS UMR 5600 EVS, Site ENS, F-69362, Lyon, France 8
*Corresponding author: [email protected] 9
Revision, Unmarked (including Title page with author details) Click here to view linked References
2 Abstract
10
Bedload transport is one major driver of gravel-bed river morphodynamics, and its quantification 11
is capital for many environmental issues and river engineering applications, as well as for 12
landscape evolution studies. To this point, bedload transport rates and volumes have been 13
classically computed by means of sediment transport formulae. The most used bedload transport 14
equations compute the bulk mass bedload based on section-averaged hydraulic parameters.
15
However, due to the non-linear behavior of sediment transport, bedload formulae are sensitive to 16
the input parameters. Then, some doubts arise when applying bedload equations on poorly gauged 17
river reaches, i.e. rivers where there are no hydrological records and rating curves. In this paper, 18
we assess the application of bedload equations in the case of poorly gauged river reaches, and we 19
test a workflow to follow in such situations. This workflow consists of three steps: i) 20
Reconstructing the flow duration curve, based on gauging records from neighboring river basins;
21
ii) Solving the hydraulic geometry relations of the study-case river-reach, based on a flow friction 22
equation; and finally, iii) Computing bedload with a sediment transport equation. We tested this 23
approach against the bedload information available in the literature for Idaho streams and we 24
found that it could potentially approximate annual bedload volumes in ungauged reaches under 25
certain conditions. To illustrate the potential of this workflow, we also computed bedload volumes 26
for two ungauged river reaches from the Cantabrian mountains (NW Spain).
27
3 1. Introduction
28
Bedload sediment transport is a key control on the physical and ecological functioning of gravel- 29
bed rivers. The coarse fraction of sediment, travelling as bedload, is one major component of the 30
bed material, that defines alluvial channel geometry and planform morphology (Church, 2006;
31
Wilcock et al., 2009; Vázquez-Tarrío et al, 2020). Additionally, bedload motion contributes to 32
shape habitat for benthic macroinvertebrates and defines the conditions making river gravel 33
habitable for several aquatic organisms (Gibbins et al., 2007; Haschenburger, 2017). Thus, the 34
adequate quantification of bedload volumes is essential for a wide range of river engineering and 35
management issues, such as channel design, environmental flow assessment, hazard evaluation 36
and dam conception/operation, amongst others (e.g. Stroffek et al., 1996; Dufour and Piégay, 37
2009; Habersack and Piégay, 2007; Arnaud et al., 2017; Vázquez-Tarrío et al 2019).
38
Additionally, the smart implementation of some river restoration operations, such as gravel 39
augmentation plans or dam removal schemes, require a good estimation of bedload volumes and 40
their distribution through time. Moreover, bedload transport drives river incision, so its adequate 41
quantification is also capital to model and understand landscape evolution (e.g. Slingerland et al., 42
1994; Molnar et al., 2006).
43
Ideally, the characterization of bedload at-a-site would be possible with adequate field 44
measurements. To this point, a wide diversity of field methods have been developed and proposed 45
over the last decades, which include the use of (more or less) sophisticated sediment traps and 46
field sampling methods (e.g. Helley and Smith, 1971; Sterling and Church, 2002; Vericat et al., 47
2006; Bergman et al., 2007), particle tracking (e.g. Laronne and Carson, 1976; Hassan et al., 1984;
48
Haschenburger, 1996; Haschenburger and Church, 1998; Hassan and Ergenzinger, 2003; Liébault 49
et al., 2012; Vázquez-Tarrio and Menéndez-Duarte, 2014; Hassan and Bradley, 2017; Vázquez- 50
Tarrío et al., 2019), volume estimation based on geomorphic change detection (Ham, 1996;
51
Ashmore and Church, 1998; Martin and Church, 1995; Ham and Church, 2000; Lane et al., 2003;
52
Brasington et al., 2003; Wheaton et al., 2010; Lallias-Tacon et al., 2014; Vericat et al., 2017) 53
and/or the use of several acoustic and geophysical methods (Tsai et al., 2012; Rickenmann et al., 54
2014; Roth et al., 2016; Goodwiller et al., 2019; Bakker et al., 2020). However, all these different 55
4
methodologies are costly, both in terms of time and field efforts. Additionally, establishing 56
average values of the different bedload parameters needs a large number of field observations in 57
order to compile data series long enough to grasp all the inherent variability in river hydrology 58
and sediment transport processes. This frequently makes the field measurement of bedload 59
unrealistic in view of the time availability and the immediate needs of the more usual river 60
engineering and research projects.
61
Because of the difficulties with the above methodologies, instead of field measures, hydraulic 62
engineers and river scientists have very often turned to the use of sediment transport equations.
63
Sediment transport formulae allow the computation of bedload rates and volumes based on 64
relatively easy to take (at-a-site) measures of grain size and hydraulic parameters. In this regard, 65
several bedload equations have been proposed (Schocklitsch, 1935, 1950; Meyer Peter and 66
Müller, 1948; Einstein, 1950; Bagnold, 1980; Parker et al., 1982; Parker, 1990; Wilcock and 67
Crowe, 2003; Recking, 2013) and some of the more modern developments provide relatively 68
reliable estimations of the bulk bedload volumes in certain situations (Gomez and Church, 1989;
69
Habersack and Laronne, 2002; Recking, 2013; Vázquez-Tarrío and Menéndez-Duarte, 2014;
70
Hinton et al. 2018). The common strategy followed with bedload equations is the 1D approach 71
(Ferguson, 2003): using cross-sectional averaged values of grain-size, channel slope, width and 72
flow discharge, these equations compute the bulk bedload volumes. When one is interested in 73
retrieving annually averaged bedload volumes, this approach involves three clear steps (Recking 74
et al., 2012), which are summarized in Fig. 1: i) First, defining the flow duration curve for the 75
study reach; then ii) Computing the variation in several hydraulic parameters (width, depth, 76
hydraulic radius…) with flow discharge, i.e. defining the hydraulic geometry and rating curves 77
for the study site; and finally, iii) Combining i) and ii) into a bedload equation to compute bedload 78
rates for the different discharges of the flow duration curve. It would then be possible to weight 79
the estimated bedload rates by its frequency of recurrence and estimate average annual bedload 80
volumes.
81
In this regard, a lot of research has focused on the improvement of bedload models (step iii in Fig.
82
1), and some recent bedload equations succeed to formulate and incorporate in a mathematical 83
5
way several bedload mechanisms, such as the effects of sand/gravel mixing, armour breakup, the 84
transition from partial to full mobility, and the effects of macro-bedforms (Parker, 1990; Wilcock 85
and Crowe, 2003; Recking, 2010; 2013; Recking et al., 2016; Piton and Recking, 2017). However, 86
in principle, these equations would only work well when good information about the hydrological 87
regime (allowing the definition of the flow duration curve) and gauging data (establishing 88
hydraulic geometry and rating curves) are available at the given study site (see workflow in Fig.
89
1). Nonetheless, in many applied situations, we are interested in computing bedload rates in 90
reaches that are poorly gauged (or even ungauged). With that being said, locations where there 91
are no hydrological series long enough to establish a reliable flow duration curve (first step in 92
Fig. 1), neither is it possible to establish hydraulic geometry relations or rating curves (second 93
step in Fig.1). Consequently, we are quite limited in our ability to compute bedload in those poorly 94
gauged river reaches. Nevertheless, regional hydrological information and data could be used to 95
fill the gap in gauging data and to approach the flow duration and rating curves in these types of 96
ungauged reaches. In this regard, for instance, Mueller and Pitlick (2005) used this idea to propose 97
a model of bedload transport capacity in a headwater stream in the Rocky Mountains. Similarly, 98
Piton et al. (2016) followed a comparable approach in order to compute bedload rates in an Alpine 99
torrent and obtained results that were adequately validated with field measurements of bedload 100
volumes.
101
In this sense, the main aim of the present paper is to test how the workflows normally followed 102
for the estimation of bedload volumes in gravel-bed rivers (figure 1) could be adapted and perform 103
in the case of poorly gauged, mountain river reaches. The workflow followed is based on the 104
combination of hydrological regional information, channel geometry and grain size 105
measurements taken in the field, and bedload computation with sediment transport formulae. We 106
tested this approach based on two different data sets: i) First, we applied it on the Idaho bedload 107
data set, already published in Whiting et al. (1999) and King et al. (2004); and then ii) We applied 108
the procedure to our own data set of bedload measurements for the Pigüeña and Coto rivers (NW 109
Spain) to illustrate the potential (and uncertainties) of the workflow.
110 111
6 2. Rationale
112
Bedload transport can be characterized at-a-site by two main sediment transport metrics: i) First, 113
the average annual bedload volumes, which provides an idea of the amount of sediment 114
transported by the studied river reach; ii) The bedload sediment duration curve, which provides 115
an idea of bedload variability, or how bedload is distributed throughout a typical hydrological 116
year.
117
Based on the workflow already outlined in figure 1, we could easily identify which are the three 118
major sources of uncertainty when computing these bedload metrics in poorly gauged river 119
reaches: i) The establishment of the flow duration curve; ii) The definition of how the different 120
hydraulic parameters (wetted width, hydraulic radius, shear stress) vary with flow discharge; and 121
iii) The choice of an adequate bedload formula. Concerning the first point, it has been suggested 122
that the complete range of mean daily flow discharges could be approximated by an inverse 123
gamma probability density function (Crave and Davy, 2001; DiBiase and Whipple, 2011; Lague, 124
2014; Lague et al., 2005; Molnar et al., 2006; Croissant et al., 2016) of the form:
125
𝑝𝑑𝑓(𝑄) = 𝑘𝑘+1
𝛤(𝑘+1)∙ exp(−𝑘
𝑄∗) ∙ 𝑄∗−(2+𝑘) Eq.1
126
where the daily discharge Q is normalized by the annually averaged daily discharge Qm (Q* = Q 127
/ Qm). The symbol Γ represents the gamma function and k is a parameter that controls the shape 128
of the probability density function capturing the variability of the hydrological regime. Different 129
values of k were proposed, from 0.2 (in case of a high variability of daily discharges, typical of 130
arid or monsoon dominated areas) to ~4 (low variability). The inverse gamma distribution exhibits 131
an exponential tail toward low discharges and a power law tail for large discharges. For an inverse 132
gamma pdf, the return time of a specific daily discharge, Qsp, can be computed from (Croissant et 133
al., 2016):
134
𝑡𝑟(𝑄𝑠𝑝) = 𝛤 (𝑘 𝑄⁄ 𝑠𝑝, 𝑘 + 1)−1 Eq.2
135
where tr is the return period.
136
In case that some regional information is available about the daily discharge distribution in 137
neighboring basins, this information could be used to calibrate the value of k in eq. 1.
138
7
Nevertheless, the estimation of the daily discharge distribution from an inverse gamma function 139
still relies on knowing the annually averaged daily discharge (Qm in eq. 1), which is not known a 140
priori in non-gauged river reaches. However, it is well established in hydrology how discharge 141
correlates to drainage area (A) across a region (Hack, 1957; Thomas and Benson, 1970;
142
Slingerland et al., 1994), providing that hydroclimatic conditions are homogeneous:
143
𝑄𝑚 = 𝑎 ∙ 𝐴𝑏 Eq. 3
144
Again, if regional information was available, the b parameter in equation 3 could be calibrated, 145
and then the eq. 3 could be inverted to compute the annually averaged daily discharge based on 146
the area draining the considered ungauged reaches. Thus, eq 1 and 3 could be combined with the 147
available regional information to build flow duration curves for the ungauged reaches. A more 148
complex situation would arise if no regional information was available. In such cases, some 149
assumptions should be made for the intercept and the exponent in eq. 3. In this regard, a common 150
pattern observed in humid-temperate regions is a linear relationship between mean annual 151
discharge and drainage area (Hack, 1957; Slingerland et al., 1994; Solyom and Tucker, 2004), i.e.
152
b-exponent~1 and a-intercept ~0.01-0.1. However, this may differ in other hydro-climatic settings 153
(Solyom and Tucker, 2004).
154
Once the flow duration curve is established, a second source of uncertainty would be the 155
computation of the different hydraulic parameters (wetted width, hydraulic radius, shear stress).
156
In gauged rivers, this information is provided by the rating curves built based on gauging records.
157
In ungauged river reaches, we are forced to build the rating curves based on a flow friction 158
formula, which allows the quantification of the links between flow discharge, stream velocity and 159
water depth. Ferguson (2007) proposed a flow resistance equation that accounts for differences 160
in relative roughness between shallow and deep flows. This equation was validated against a 161
broad database of the field data of gravel bed rivers by Rickenmann and Recking (2011), who 162
also provided an equivalent expression for the Ferguson (2007) flow friction model, that allows 163
the estimation of average flow depth (d) directly from flow discharge:
164
𝑑 = 0.015 ∙ 𝐷84∙𝑞∗2𝑝
𝑝2.5 Eq. 4
165
8 where:
166
𝑞∗= 𝑞 𝑤 ∙ √𝑔𝑆𝐷⁄ 843 Eq. 5
167
and q is the water discharge, S the channel slope, p = 0.24 if q*<100 and 0.31 otherwise, D84 is 168
the 84th percentile of the surface grain size distribution and w is the active channel width. The 169
last two parameters are easy to measure in the field. Therefore, eq. 4 could be used together with 170
the flow duration curve (computed based on the set of eqs. 1 to 3) and relatively simple field 171
measurements to build theoretical rating curves for ungauged study reaches.
172
The last source of uncertainty is which bedload formula we choose. Several bedload transport 173
equations have been proposed in scientific literature. It is not the aim of this paper to test different 174
bedload formulae but to illustrate how bedload rates could be estimated (or not) in comparable 175
conditions in ungauged and gauged gravel-bed rivers. Therefore, we proposed to work with only 176
one single bedload formulae, the Recking (2013) bedload transport equation:
177
𝑞𝑠= 14 ∙ √𝑔 ∙ 1.65 ∙ 𝐷843 ∙ 𝜏∗2.5
1+(𝜏𝑚𝜏∗∗)4
∙ 𝑤 Eq.6
178
where qs are the specific bedload rates, g is the gravity acceleration, τ* is the Shields shear stress 179
and τm* is the reference Shields stress separating partial, from full mobility conditions. Shields 180
stress is estimated from:
181
𝜏∗=(𝜌 𝜏
𝑠−𝜌)∙𝑔∙𝐷84 Eq. 7
182
where ρs is the density of sediment, ρ is the water density and τ is the bed shear stress, which can 183
be estimated from the depth (d)-slope (S) product:
184
𝜏 = 𝜌 ∙ 𝑔 ∙ 𝑆 ∙ 𝑑 Eq. 8
185
We decided to choose Recking’s (2013) equation because it does not require calibration, and 186
compared to other bedload models it considers explicitly the transition from partial to full mobility 187
conditions (through the τm*
parameter) and it seems to work rather well in gravel-bed rivers 188
(Hinton et al., 2018). Additionally, this equation has already been tested against one of the datasets 189
used in the present research (the Idaho dataset) (Recking, 2013). Furthermore, the more recent 190
development of this equation (Recking et al., 2016) incorporates the effects of macro-bedforms, 191
9
based on accounting for the dominant channel style when computing τm*
from the following 192
formulae:
193
𝜏𝑚∗ = 1.5 ∙ 𝑆0.75 Eq. 9
194
in the case of plane-bed and step-pool streams, and in the case of riffle-pool rivers:
195
𝜏𝑚∗ = (5 ∙ 𝑆 + 0.06) ∙ (𝐷𝐷84
50)4.4∙√𝑆−1.5 Eq. 10
196
where D50 is the median size of the surface grain-size distribution.
197
Fig. 2 summarizes all the previous considerations and resumes the workflow proposed here to 198
compute bedload transport in non-gauged gravel-bed rivers. As previously stated, we are going 199
to test this approach against the Idaho dataset of bedload data (King et al., 2004).
200 201
3. Materials and methods 202
3.1. Preliminary information 203
3.1.1. Idaho dataset (USA) 204
To test the workflow presented above (and summarized in figure 2), we needed a dataset where 205
field information on bedload fluxes, rating curves and hydraulic/sediment parameters was 206
available for a broad collection of mountain rivers, ideally in a wide variety of conditions. Our 207
aim is to take advantage of such a dataset to see how information is (or is not) lost along the data- 208
treatment chain as we reduce data-availability (figure 2) and to check how this may affect whether 209
sediment estimates deviate (or not) from the actual field measures. This will allow us to tease out 210
some lessons on how well the workflow presented in the previous section could be adapted and 211
perform in poorly gauged mountain rivers. In this regard, the exceptional database of bedload 212
measurements for Idaho rivers represented the ideal dataset to test the workflow described above 213
(section 2; Fig. 2). This exceptional database includes bedload measurements for 33 Idaho rivers, 214
representing one very complete and freely available dataset of bedload measurements for gravel- 215
bed rivers. This information was made freely accessible online by the USDA Forest Service at 216
the website of the Boise Adjudication Team (BAT,
217
https://www.fs.fed.us/rm/boise/AWAE/projects/sediment_transport/sediment_transport_BAT.sh 218
10
tml) and these data have already been analyzed in several papers (e.g. Whiting et al., 1999; King 219
et al., 2004; Barry et al. 2004; Mueller et al., 2005; Barry et al., 2008; Muskatirovic, 2008; Pitlick 220
et al., 2008; Recking, 2010; Mueller and Pitlick, 2014).
221
These streams are part of the Northern Rocky Mountain Province in Idaho (Fenneman, 1931) 222
(with the exception of Rapid River) and all (except Cat Spur Creek) are within the Snake River 223
basin. Frontal systems coming from the Pacific Ocean are the source of most of the precipitation 224
for these basins. Snow accumulates from fall through spring at higher elevations, accounting for 225
over half the annual precipitation. Consequently, the hydrology of these streams is snowmelt- 226
dominated, reaching peak flows from April to June, in association with spring snowmelt runoff 227
and rain-on-snow events. Low flows are typically reached in September or October (King et al., 228
2004).
229
Data on bed slope, channel geometry and grain-size for these 33 streams was obtained from King 230
et al. (2004), Mueller and Pitlick (2005) (supplementary files) and the bedload web project 231
(https://www.bedloadweb.com/). Bankfull depths and widths range from 0.2 to 5 m, and from 2 232
to 200 m, respectively. On the other hand, bed slope at the study sites spans from 0.0003 to 0.07.
233
Finally, D50 ranges from 23 to 207 mm and D84 from 62 to 558 mm. Bedload transport was 234
measured with Helley-Smith samplers, having 76.2 mm and 152.4 mm square orifices and a 0.25 235
mm mesh on the collecting bag, and the results of these measurements are also freely accessible 236
online through the BAT official website. Nevertheless, Recking (2010) (in its supplementary 237
files) has already made a great effort homogenizing and putting together all the bedload 238
information, and we benefited from this previous work of compilation.
239
There are streamflow gauge stations managed by the USGS in 22 of the 33 selected reaches. We 240
downloaded mean daily discharges for these sites from the US National Water Information 241
System (https://waterdata.usgs.gov/nwis). In the case of the remaining 11 streams, some 242
discharge information could be retrieved at the website of the BAT, managed by the USDA Forest 243
Service. Based on the discharge information, we made a flow duration analysis for each one of 244
the selected streams. To do so, we ranked discharge data in descending order of values, and we 245
assigned to each value its respective sorting order value (m). Then, we estimated the empirical 246
11
relative frequency of being equaled or exceeded for each ranked flow as the ratio m / N+1, where 247
N is the total number of discharge data available in the series. In this way, we built the empirical 248
Flow Duration Curve (FDC) for each one of the selected streams.
249
3.1.2. Cantabrian rivers 250
A second dataset was selected to test the potential of the workflow presented in section 2, 251
integrated with our own field data on bedload transport in Cantabrian rivers. The Cantabrian range 252
is a chain of mountains with ~500 km length and ~100 km width, running parallel to the 253
northwestern coastline of the Iberian Peninsula. The proximity to the coast (~50-70 km) 254
determines intense rainfall and high regional slopes, and consequently a drainage network 255
consisting of steep-sloped channels installed on the northern face. This set of mountain basins 256
constitute the Cantabrian Fluvial System (Prego et al., 2008). The Cantabrian fluvial system is 257
integrated by 28 river basins of relatively small drainage areas (28-4900 km2). The climatic 258
conditions are homogeneous throughout the region. The mean annual rainfall is ~1100 mm and 259
precipitations are well distributed the whole year round. Vegetal cover is continuous and consists 260
of an alternation of bush areas, deciduous forests, and meadows.
261
Stream gauging information is available for 27 river reaches across the Cantabrian region and is 262
freely accessible online (https://ceh.cedex.es/anuarioaforos/default.asp). Conversely, to date there 263
has been an almost total lack of quantitative data about sediment transport in the region. In 264
previous research (Vázquez-Tarrío, 2013 and Vázquez-Tarrío and Menéndez-Duarte, 2014;
265
2015) we made the first attempt to quantify bedload transport in two river reaches from this 266
region, using a tracer-based field experiment. We will use these previous measurements in order 267
to illustrate how the workflow presented in section 2, and tested in this research, could be 268
generalized to characterize bedload transport in other poorly gauged river reaches different to the 269
Idaho dataset. The two selected reaches (Fig. 3) are two tributaries of the Narcea river, one of the 270
main rivers in the region: i) River Pigüeña (405 km2 catchment area), we focused on a reach 2 km 271
upstream of the confluence with the River Narcea; and ii) River Coto (120 km2 catchment area), 272
we focused on a river reach 1 km upstream of the confluence with the Narcea river. Both rivers 273
have a coarse streambed, composed primarily of gravel and cobbles and can be comparable to the 274
12
33 rivers used to ‘fine-tune’ the workflow presented in section 2. Descriptive information on both 275
reaches could be obtained at Vázquez-Tarrío and Menéndez-Duarte (2014; 2015).
276
3.2. Methodology 277
3.2.1. Idaho rivers 278
For each one of the selected streams, first we estimated the two metrics already outlined above:
279
i) The average annual bedload volumes; and ii) The Bedload sediment Duration Curve (BDC).
280
To achieve this, we used the available (field) bedload information for the selected rivers. Then, 281
for each case study, we searched for the best power law fitting bedload rates to water discharge 282
and shear stress (i.e. ‘sediment-rating’ curve). These regression equations were later applied to 283
the different bins of discharge in each site’s empirical FDCs and we computed the corresponding 284
BDCs. The calculated values of sediment discharge were then weighted by the annual frequency 285
of each discharge and summed to estimate the annual bedload volumes. In this regard, it should 286
be noted that the Fourth of July was finally excluded from the dataset due to the existence of water 287
diversion in this site, which made not possible to adequately define a flow duration curve for this 288
reach. Hawley creek was also excluded from further analysis, in this case because a weak 289
correlation (R2~0.22) between flow discharge and bedload rates was observed in this reach (King 290
et al., 2004), i.e. it was not possible to achieve a robust estimation of the mean annual bedload 291
volumes or the BDC based on the available field data.
292
As long as our estimations of the bedload metrics (annual load, BDCs) were based on the available 293
field measurements, we considered them as the actual or ‘field-based’ values against which we 294
would test the workflow previously described (Fig. 2). These ‘field-based’ estimations are 295
hereinafter called ‘scenario 0’, and we assumed that they represent an ‘ideal’ situation where there 296
is a perfect availability of all the required at-a-site gauging, discharge and bedload information.
297
Later, we proceeded to estimate bedload parameters for four different situations or modelling 298
scenarios, which we assumed to illustrate different conditions of data availability and are 299
summarized in Table 1. Comparing the bedload estimations obtained under these four scenarios, 300
among them and with scenario 0, we aimed at understanding the influence of the different steps 301
13
of the workflow (Figs. 1 and 2) on the bedload estimations and how the performance of bedload 302
equations may be affected by different data-availability conditions.
303
The first scenario (scenario 1) mimics a situation where the study reach could be totally ungauged.
304
Consequently, no hydrological information, neither rating curves would be available, the only 305
strategy for computing bedload being represented by the workflow outlined above (Fig. 2).
306
Scenario 1 represents the extreme opposite situation to the departing scenario 0, in terms of data 307
availability. Scenario 1 involves that FDCs must be computed based on the set of equations 1 and 308
3, which require calibration: more specifically, k-parameter in eq. 1, and the intercept (a) and the 309
exponent (b) in eq. 3 must be calibrated. Then, we randomly split the dataset into two groups:
310
‘calibration’ group (15 rivers) and ‘validation’ group (14 rivers). As has just been advanced, the 311
first group of data was used for two goals: i) To calibrate the intercept and the exponent in eq. 3, 312
which links annually averaged daily discharge to the drainage area. This was done using classical 313
regression analysis; and ii) To search for the best value of k in eq. 1, explaining the temporal 314
distribution of mean daily discharges. To do so, we searched for the value of k minimizing the 315
root-mean-square deviation between the empirical quantile function of daily discharges and that 316
derived from eq. 1. Once calibrated, we applied eq. 3 for the estimation of annually averaged daily 317
discharges for each stream in the second group of data (validating group of data) under the 318
modelling scenario 1. That said, the estimated discharges were combined with the previously 319
calibrated value of k in eq. 1 to build an estimated FDC for each river in this validation group.
320
Then, we applied eqs. 4 and 5 to each bin of discharge in the estimated FDCs and we estimated 321
flow depths and bed shear stress using the depth-slope product (Eq. 8). The obtained shear stresses 322
were introduced into Recking’s (2013) equation (eq. 6) and we computed the BDC and the annual 323
average bedload volumes. The ‘estimated’ (scenario 1) values for the second group were 324
compared to the ‘field-based’ values (scenario 0), to see how our workflow arrives, or not, to 325
approach reliable estimates of bedload transport.
326
In truth, there are three main sources of uncertainty in the computing workflow (Fig. 1 and 2), as 327
we have already pointed out in section 2. First, the uncertainty linked to the computation of the 328
FDCs. On the other hand, the uncertainty linked to the specific set of equations used to solve flow 329
14
resistance (eqs. 4 and 5). Finally, the uncertainty linked to the choice of the bedload equation.
330
Indeed, the comparison between scenarios 0 and 1 does not allow to discern between the different 331
sources of uncertainty in bedload computation in non-gauged reaches. Therefore, we decided to 332
consider three more modelling scenarios, to weight the relative influence of each source of bias 333
in bedload computation (Table 1). In scenario 2, we repeated the bedload estimations (for the 334
validation group of data) using the Recking’s (2013) equation, but this time based on the actual 335
empirical FDCs and the actual flow and sediment rating curves available for the study site.
336
Moreover, in the third scenario considered (scenario 3), we repeated the bedload computations 337
based on the truth empirical FDCs (again, for the validation group of data), but this time we solved 338
flow resistance using eqs. 4 and 5, rather than the actual rating curves. Then, the bedload was 339
estimated based on the ‘truth’ sediment rating curves relating shear stress to bedload rates and 340
available for each study site. Finally, we repeated the estimation of BDC and bedload volumes 341
(for the validation data group), based on the estimated FDCs, but using the actual rating curves 342
(linking flow depth, width and bedload rates to flow discharge) available for each study case 343
(scenario 4).
344
Consequently, the comparison between the different modelling scenarios and the field-truth 345
bedload data will provide an idea of the reliability of the workflow presented above (section 2, 346
figure 2) for bedload computation in case of poorly gauged river reaches. For instance, the 347
comparison of scenario 1 (no data available) with truth-field data (all data available: scenario 0) 348
will provide an overall idea of how well the bedload volumes could be estimated in an ungauged 349
situation in the selected dataset. Likewise, the comparison of scenario 2 (gauging records + rating 350
curves available, bedload information not available) with the field-truth data will help to 351
overcome the inherent uncertainties linked to the choose of a bedload formula. Similarly, the 352
comparison of scenario 3 (gauging records + bedload available, but rating curves not available) 353
with the truth-field data (scenario 0) will make it possible to get close to the uncertainty in bedload 354
estimation linked to the use of the flow resistance eq. 4, rather than the actual rating curves.
355
Finally, the comparison of scenario 4 (rating curves + bedload information available, FDCs not 356
available) with the scenario 0 will help to overcome the inherent uncertainties linked to the 357
15
estimation of the FDCs based on regional information. Two different scores are used during our 358
assessment and comparisons: i) One is the ratio between the estimated and the field-based values 359
(r); and ii) The second one is the percentage of data for which r lays between 0.5 and 2, and 360
between 0.2 and 5 (less than one order of magnitude of difference between the estimated and the 361
field-based values).
362
3.2.2. Cantabrian rivers 363
Mean daily discharge data was obtained from the available gauging records for Cantabrian rivers.
364
Based on this data, we searched for a regional curve, fitting the average annual daily discharge to 365
the drainage area (eq. 3). Mean daily discharge data were also used to get the empirical FDC for 366
each gauging station, and the obtained FDCs were used to calibrate the k parameter in equation 367
1. Yet again, to do so, we searched for the value of k minimizing the root-mean-square deviation 368
between the empirical quantile function of daily discharges and that derived from eq. 1. Then, we 369
used eqs. 1 and 3 (with the previously calibrated value for k), and we built the FDC for River 370
Pigüeña and River Coto. Later, we applied Rickenmann and Recking (2011 flow resistance 371
equation (eq.4) to estimate shear stresses for the different bins of discharge in the BDC, and 372
Recking et al. (2016) bedload formulae to compute bedload rates. The simulated bedload-rating 373
curves were finally compared to our own field measures (based on tagged stones) of bedload rates 374
in River Pigüeña and River Coto (Vázquez-Tarrío et al., 2014).
375 376
4. Results 377
4.1. Idaho rivers 378
4.1.1. Flow duration curves 379
We found a strong and statistically significant power correlation between the drainage area and 380
the mean average daily discharge in the calibration group of data, as expected (Fig. 4A). It is 381
interesting to note that the observed trend is not far from what we would expect (a linear trend) 382
for rivers in humid-temperate regions (e.g. Hack, 1957; Slingerland et al., 1994; Solyom and 383
Tucker, 2004). The obtained (regression) power law was employed to estimate the average annual 384
16
discharge in the validation group of data, and we observed how the real values were reliably 385
estimated (Fig. 4B).
386
We also used the calibration data group in order to search for the better value of k in eq. 1, and 387
we obtained an optimum value of k = 2.9 (Fig. 5A), a value expectable for low variable 388
hydrological regimes in perennial streams. When applied to the validation group of data, we 389
observed a good estimation of higher flows, but still a slight overestimation of low flow 390
discharges (Fig. 5B). The origin of this overestimation may relate to some of the implicit 391
assumptions when using the inverse-gamma distribution model (eq. 1) to approach streamflow 392
variability. This model involves a power-law right tail of the probability distribution of daily 393
discharges, which may not be as accurate as previously thought in geomorphological studies 394
(Rossi et al., 2016).
395
4.1.2. Bedload parameters 396
Following the workflow proposed in fig. 2 (scenario 1), 35% and 71% of our bedload estimates 397
are between 0.5:2 and 0.2:5 of the real values respectively, with an average r of ~2.5. Scores are 398
slightly lower than those obtained with modelling scenario 4 (estimated FDCs + truth rating 399
curves + truth bedload rating relations), but quite similar to those obtained using Recking’s (2013) 400
equation and flow resistance equations 4 and 5 on the ‘truth’ empirical FDC (Fig. 6) (scenario 2 401
and 3). This suggests that differences between the observed and estimated average bedload 402
volumes more likely relate to the performance of the flow resistance and bedload equations, rather 403
than to the bias introduced when modelling the FDC. According to these results, if we choose the 404
adequate flow friction and bedload formula, we would obtain comparable predictions of annually 405
averaged bedload volumes in a gauged and an ungauged hypothetical situation (according to the 406
comparison between modelling scenarios 1 and 2) in the selected study cases.
407
To better understand these results, we looked with more detail at the BDCs (Fig. 7). The computed 408
BDC shows differences in precision according to the probability of non-exceedance. In modelling 409
scenario 1, bedload discharges for rare and high-magnitude events are better predicted than the 410
bedload discharges for lower magnitude and more frequent floods, with differences that can 411
eventually achieve two orders of magnitude. This contrasts with our previous results (Fig. 6), 412
17
showing a reasonably good prediction of the annually averaged bedload volumes. However, 413
bedload rates for low magnitude, frequent episodes are considerably lower than those recorded 414
for higher magnitude flows, i.e. their contribution to the annual bedload volumes is not 415
particularly important (Fig. 8). This may explain why the time-integrated bedload volumes are in 416
general well-estimated, despite the strong overestimation of bedload rates for low flows 417
Indeed, we also observed overestimation and higher scatter in bedload predictions at low- 418
magnitude flows with modelling scenario 2 (Fig. 7), despite using the actual FDCs and rating 419
curves. This points out at the inherent limitations of the bedload formula at low flow rates. In 420
contrast, we observed underestimation of bedload rates at the more frequent, low-magnitude flows 421
in case of modelling scenario 3. This agrees with Rickenmann and Recking (2011) who already 422
pointed out how flow resistance eq. 4 underestimates average flow velocity at low depth- 423
roughness ratios, which may in turn affect the outcomes of the bedload equation. Finally, in the 424
case of modelling scenario 4, we again observed the overestimation of bedload rates at low- 425
magnitude, frequent flows, which should be related to the underestimation of flow duration for 426
low-magnitude discharges with eq. 1 (see Fig. 5B).
427
In summary, the analysis of the BDCs highlights several things: i) The estimation of bedload rates 428
is reliable at moderate and high-magnitude flows in all the modelling scenarios; ii) With low- 429
magnitude flows, uncertainties linked to the choice of the bedload formula and the estimation of 430
the FDCs result in overall overestimation of bedload rates; iii) The use of flow friction equation 431
4, lead to a general underestimation of bedload rates at low flows, as expected from the original 432
Rickenmann and Recking (2011) analysis on flow resistance; and iv) Underestimation linked to 433
flow resistance eq. 4 is in the same order of magnitude as the overestimation related to the 434
uncertainty in simulating the FDCs. The previous four points clarify the results shown in Fig. 6 435
concerning the estimation of annually averaged bedload volumes: underestimation linked to using 436
eq. 4 to solve flow resistance is balanced by the overestimation linked to the uncertainties in 437
building the FDCs. Consequently, almost all the remaining uncertainty in bedload computation is 438
that inherent to the use of the bedload equation; thus, the results obtained using the workflow 439
adapted here for ungauged reaches should be comparable to those expected applying the same 440
18
bedload formulae in case of the actual FDCs and gauging data being available (scenario 2). That 441
said, bedload rates could be estimated in similar conditions for an ungauged and a gauged 442
situation, and the precision of the results would be (mostly) solely controlled by the performance 443
of the bedload formulae, at least in the selected case studies.
444
In summary, the results suggest that following the workflow described in section 2 (scenario 1), 445
we succeeded in approaching reasonably well the average annual bedload volumes and the 446
relative contribution of each individual discharge to the annual bedload, in spite of a large 447
overestimation of bedload rates for low-magnitude flow discharges.
448
4.2. Cantabrian rivers 449
We obtained a good regional curve fitting the average annual daily discharge to the drainage area 450
in Cantabrian rivers (Fig. 9A). Again, it can be seen that the observed trend does not deviate much 451
from the linear relationship expected for rivers in humid-temperate (e.g. Hack, 1957; Slingerland 452
et al., 1994; Solyom and Tucker, 2004). Additionally, we obtained an optimum k = 3.5 for 453
equation 1 with the dataset of 27 Cantabrian rivers (Fig. 9B).
454
We used the drainage area of River Pigüeña and River Coto to estimate mean daily discharge 455
(based on Fig. 9A). Using these estimates, and applying eq. 1 (with a k=3.5, fig. 9B), we built the 456
FDC for River Pigüeña and River Coto. Then, we followed the workflow presented in section 2:
457
we applied the friction equation 4 and the bedload equation 6 on the obtained FDCs to derive the 458
BDCs and estimate the average annual bedload volumes in both streams. Comparison with field 459
measured bedload rates (based on tracer data) reported in Vázquez-Tarrío and Menéndez-Duarte 460
(2014) suggests that bedload estimates could not be far from the actual values (Fig. 10).
461
Integrating the BDC, we estimated the annually averaged bedload volumes being 26000 and 900 462
m3/year in River Pigüeña and River Coto, respectively. This data represents the first estimations 463
of annual bedload volumes for Cantabrian rivers (to our knowledge).
464 465
5. Discussion 466
As we have previously stated, for many applied situations, river engineers are faced with river 467
reaches where there is a total lack of gauging records, or the available data do not extend for a 468
19
long enough period of time. Even having access to an accurate and trustable bedload equation, 469
some doubts would arise on how reliable bedload computation could be in such cases where there 470
is no hydrological information, neither any rating curve linking water discharge to flow depth or 471
velocity. This situation is reminiscent of the classical distinction in clinic trials between efficacy 472
(i.e. how a medical treatment behaves in an ideal or controlled setting) and effectiveness (i.e. how 473
a medical treatment behaves in a real-world, clinical daily setting), and it has constituted the leit 474
motiv triggering and guiding the present work. Previous work has already demonstrated that (more 475
or less) reliable functional relations can be found between flow strength and bedload rates (i.e.
476
efficient bedload formulae; Parker, 1990; 2004; Wilcock and Crowe, 2003; Recking, 2013;
477
Recking et al., 2016; Hinton et al., 2018). However, like any other mathematical model, bedload 478
equations are largely sensitive to the way the input parameters (such as water discharge, flow 479
duration, hydraulic geometry or flow friction) are defined (Fernandez and García, 2017) and few 480
of these studies have analyzed how these models behave in applied situations, where we could 481
not have a good control on the hydrologic and hydraulic information needed for bedload 482
computation, i.e. how effective really are the available efficient bedload formulae?
483
In the initial scheme shown in Fig. 1, there were three potential sources of uncertainty in bedload 484
estimation. First, the method we used to reconstruct FDCs, which fits well to moderate and high- 485
magnitude flow discharge, but overestimates the more frequent, low-magnitude flows. The bias 486
introduced by this overestimation means that bedload rates tend to be overestimated in 1-2 orders 487
of magnitude at low flows due to this effect. Second, the use of Rickenmann and Recking (2011) 488
flow resistance law, means that shear stresses are underestimated at low-flows, and this effect 489
introduces a bias towards an ~1-2 order of magnitude underestimation of bedload at low flows.
490
Consequently, the overestimation at low flows linked to ‘reconstructing’ the FDC balances the 491
overestimation related to the Rickenmann and Recking (2011) flow resistance model, so the final 492
uncertainty in bedload estimation is mostly related to the choice of the bedload equation. In this 493
regard, our results show how the use of the bedload formulae (Recking et al., 2016) introduces a 494
large scatter in sediment transport computation at low flows. This could be related to the fact that 495
the armour layer is not destabilized at these low-flow conditions, so bedload transport is more 496
20
controlled by the availability of mobile sediment (supply limited conditions, travelling bedload;
497
sensu Piton et al., 2017), rather than the capacity of the channel to move the bed sediment 498
(competent or capacity limited conditions).
499
Nevertheless, the three sources of uncertainty are largely reduced at moderate and high-magnitude 500
flows. The influence of FDC reconstruction and the flow resistance equation become minimized, 501
and similarly the bedload equation provides more reliable bedload estimates. Consequently, 502
bedload rates at moderate and high-magnitude flows seem to be computed in ungauged rivers 503
with comparable accuracy and precision to gauged reaches (compared Scenario 1 and 2 in Fig.
504
6). To the extent that the weight of low magnitude flows on the average annual bedload volumes 505
is very weak, this also means that the annual bedload volumes can be computed in an ungauged 506
situation with a similar accuracy to totally gauged conditions . In summary, our analysis shows 507
how the outcomes of the bedload formulae obtained in a gauged situation are comparable to those 508
obtained in a situation where gauging records are lacking, i.e. bedload equations may show a 509
comparable behavior in ungauged and gauged river reaches, at least in the analyzed data of Idaho 510
and Cantabrian rivers.
511
However, the limitations of the approach presented here (figure 2) still need to be examined in 512
more extensive and comprehensive way. Although we have used one of the more extensive 513
datasets on bedload measurements for gravel-bed rivers, the explored data have potential bias.
514
First, the used dataset come from just two geographical regions (Northern Rocky Mountain in 515
Idaho and Cantabrian Mountains in NW Spain). It would be ideal to have done the same exercise 516
in rivers from other regions, but this kind of data are not common. Indeed, bedload data available 517
in the literature are lacking from dryland, tropical regions, and/or cold environments (Phillips and 518
Jerolmack, 2019; Vázquez-Tarrío et al., 2020). Different hydro-climatic settings involve 519
contrasted patterns of streamflow variability, so we could also expect differences in the 520
probability distribution of daily discharges, i.e. in k-parameter in eq.1. This may eventually 521
nuance some of the conclusions here outlined; e.g. the fact that the overestimation of low flows 522
linked to ‘reconstructing’ the FDC based on eq.1 balances the underestimation of flow friction 523
linked to eq. 4. In addition, some authors pointed out how the inverse-gamma distribution of daily 524
21
discharge could be not so common as normally assumed (Rossi et al., 2016; Deal et al., 2018), 525
and using another kind of probability distribution function for daily discharges may change some 526
of the trends reported here.
527
Moreover, it should be noted that most of the rivers where we have tested the procedure presented 528
here are relatively narrow plane-bed or riffle-pool rivers. Large-amplitude bar-pool, and multi- 529
thread braided and wandering rivers are misrepresented in the used dataset. These river settings 530
feature a wide diversity of channel-morphologies and a large cross-sectional variability in shear 531
stresses, patterns of flow and sediment conveyance and grain-size sorting (Ferguson, 2003;
532
Francalanci et al., 2012; Recking, 2010; Recking et al., 2016; Vázquez-Tarrío et al., 2019). In 533
these conditions, we are limited by the 1D-approach followed here and classically used in bedload 534
studies (Ferguson, 2003). The large cross-sectional variability in shear stresses and form-drag 535
imposed by macro-forms in these particular settings likely involve a poor performance of both 536
the flow-friction and bedload formulae, so probably the results outlined here would differ from 537
those we would observe in multi-thread rivers.
538
Consequently, some doubts could be posed as to whether some of the conclusions presented here 539
are generalizable to all gravel-bed rivers or other hydro-climatic settings. Nonetheless, in this 540
regard, we would like to outline that it is not the aim of the present manuscript to propose an easy 541
and universal procedure for the estimation of bedload in any situation, which could substitute the 542
field measurement or the use of more sophisticated 1D and 2D hydraulic/morphodynamic 543
modelling approaches. Bedload transport is a very complex phenomenon, which depends on many 544
drivers acting at several time and spatial scales, from the local control exerted by structural 545
arrangements at the clast-scale, to the general sediment supply conditions at the catchment scale.
546
When possible, field measurements of bedload should be prioritized over any attempt at 547
estimating bedload based on bedload formulae. However, field measurement is complex, 548
expensive and it could take some time before the collection of data good enough as to characterize 549
bedload at-a-site is acquired, which could contrast with the more immediate needs of many river 550
engineering or research projects. In this regard, any kind of information on bedload volumes could 551
be better than nothing. We believe that the procedure here presented could be used: i) first, as a 552
22
base to provide some first-order ideas on bedload volumes in poorly gauged reaches; and ii) as a 553
base to further research exploring the efficiency of bedload estimates and the performance of 554
bedload equations under different scenarios of data availability.
555
6. Conclusions 556
In this paper we investigated the potential for the application of bedload formulae in poorly 557
gauged reaches. According to our work:
558
- It is possible to approach the values for the mean annual discharge and the general shape 559
of the flow-discharge duration curve for an ungauged river-reach, based on the 560
information available for neighboring catchments. Then, the flow duration curves 561
obtained in that way can potentially be used for bedload computations.
562
- We tested different modelling scenarios, and the results suggest how accuracy in the 563
estimation of annual bedload volumes is mostly controlled by the performance of the 564
flow-resistance and bedload equation, rather than to the use of a modelled flow duration 565
curve.
566
- Bedload rates computed for low-magnitude, frequent events are far from the ‘real’ ones, 567
but for moderate and high-magnitude flows are comparable to the field measured bedload 568
fluxes. Those high-magnitude flows are those with a more relevant weight in the total 569
annual volumes, so reliable estimations of annual bedload volumes are possible despite 570
the poor estimations for low discharges.
571
- We have illustrated the potential of this approach and we have estimated bedload volumes 572
for two ungauged river reaches from NW Spain, and we have obtained values which seem 573
coherent with the field measurements available for these streams.
574
- In summary, our results suggest that a first-order estimation of annual bedload volumes 575
is possible in poorly gauged mountain rivers, providing that some hydrologic information 576
is available for neighboring catchments.
577
- However, our observations could be potentially biased by the scarce dataset here used.
578
Further research using data from other hydro-climatic environments and 579
23
geomorphological settings will allow to examine our findings in a more extensive and 580
comprehensive way.
581
Acknowledgements.
582
The present work has been possible thanks to the financial support provided by the grant ACB17- 583
44, co-funded by the post-doctoral ‘Clarín Program-FICYT’ (Government of the Principality of 584
Asturias) and the Marie Curie Co-Fund, as well as support from the project RIVERCHANGES- 585
CGL2015-68824-R (MINECO/FEDER, UE). First author was also supported by the Spanish 586
National R&D+i Plan research project entitled “Advanced methodologies for scientific-technical 587
analysis of flood risk for the improvement of resilience and risk mitigation” (DRAINAGE-3-R 588
under Grant CGL2017-83546-C3-3-R AEI/FEDER, UE), funded by the Spanish Ministry of 589
Economy and Competitiveness (now Ministry of Science and Innovation). Finally, we also thank 590
the Editor and the two anonymous reviewers for their valuable comments and remarks.
591
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