Site specificity of bed load measurement using an acoustic Doppler current profiler

Authors


Abstract

[1] Concurrent measurements of bed load transport velocity (v) from the bottom tracking feature of an acoustic Doppler current profiler (aDcp) and bed load transport rate (gb) from conventional pressure difference samplers are presented. Data sets were collected from both gravel bed and sand bed reaches of the Fraser River, covering a bed material range of 0.25–25 mm. Strong relations, in which v explained >70% of the variability of measured gb, were observed in a gravel bed and a sand bed reach. Differences in correlation between v and gb among the contrasting environments are attributed to variations in both the bed load particle size and aDcp operating parameters. Similar values of v were associated with lower mass transport rates in a sand bed than in a gravel bed reach. A nondimensional data collapse, accounting for differences in bed load particle size, explained 42% of the observed variance of the combined data set. Longer averaging times were required in the gravel bed reach, likely due to the stochastic nature of bed load entrainment in gravels and the resulting heterogeneous bed load velocity field. Bed load transport was modeled using both shear stress models and a kinematic model that utilizes the estimated bed load velocity. The standard shear stress models provided poor matches to the measured bed load transport rates: the sand bed data were overpredicted, and the gravel bed predictions correlated poorly with the measured predictions. Use of the kinematic model yields an estimate for the product of bed load concentration and bed load layer depth. This work highlights the potential of acoustic techniques for estimating bed load.

1. Introduction

[2] Bed material transport is a fundamental aspect of rivers. The spatiotemporal distribution of the transfer of bed material through a reach of river determines river morphology. Understanding of bed material transport is thus important for many aspects of river management, including the production and maintenance of aquatic habitats, stability of engineered structures such as bridges and pipelines, and prediction of channel change. However, the spatiotemporal distribution of bed material transport is a complicated, nonlinear function of sediment supply, bed state, and fluid forcing. Bed material transport remains poorly understood, in part due to the lack of measurement methods that can capture spatial and temporal variability of bed material transport.

[3] Bed material transport involves intermittent entrainment and transport of material derived from the river bed. Bed material transport in most fluvial environments occurs within a few particle diameters of the bed surface by rolling or saltating over the bed, although bed material has been observed to travel in suspension in some high-velocity sand bed rivers [e.g., Kostaschuk and Ilersich, 1995; Kostaschuk and Villard, 1996]. Bed load is a measurement of sediment transport near the bed surface; thus bed load is generally assumed to be a measure of bed material transport.

[4] Conventional measurement involves collection of bed load using a physical trap or sampler. Only time-integrated samples may be collected and only as spot measurements; thus spatiotemporal resolution is limited. Furthermore, mechanical samplers are difficult to deploy during channel-forming flows when bed material is moving, and the physical presence of the sampler disturbs the flow field; thus an estimated sampling efficiency is required [Hubbell et al., 1985]. We have been investigating remote measurement of bed load using the bottom-tracking feature of a commercially available acoustic Doppler current profiler (aDcp) [Rennie et al., 2002; Villard et al., 2004]. The goal is a robust and relatively simple means of bed load measurement with improved spatial and temporal resolution. Rennie and Millar [2004] considered the ability of the method to measure spatial distributions of bed load. Rennie et al. [2002] found that concurrent measurements in a gravel bed reach of bed load velocity (v) from an aDcp and bed load transport rate (gbM) using conventional samplers were correlated. The correlation served as a calibration for measurement of bed load transport using an aDcp, although it was hypothesized that the calibration was site-specific. Similarly, Villard et al. [2004] demonstrated a correlation between v and gbM in a sand bed channel and were able to model bed load transport using v as an input parameter.

[5] In this paper, available data of concurrent v and gbM are brought together, along with new measurements that extend the range of bed materials examined. Our objective is to examine the site specificity of correlations between v and gbM and thus explore the general applicability of bed load measurement using an aDcp. The opportunity for modeling bed load transport using aDcp data is examined, and model estimates of bed load are used for validation of measured bed load. The paper briefly describes the new measurement technique, outlines measurement procedures and analytical methods, and reviews various bed load models. The v versus gbM correlations from the various study sites are presented as are the bed load modeling results. Differences in the correlation between data sets are explained as a result of variations in caliber of bed load as well as differences in aDcp operating procedures. The paper concludes with a discussion of the limitations of bed load measurement with commercially available aDcps, including recommendations to improve measurement of bed load using bottom tracking.

2. Measurement of Bed Load With an Acoustic Doppler Current Profile (aDcp)

[6] The new technique will be briefly reviewed here; more detail is available in the work of Rennie et al. [2002] and Rennie [2002]. The technique utilizes the bottom tracking feature of aDcps. These instruments are usually deployed from a boat to measure spatially averaged three-dimensional velocities throughout a vertical column of water. The water velocities are measured with respect to the boat, and thus boat velocity must be known to determine absolute water velocities. Boat velocities are determined either by Differential Global Positioning System (DGPS) data or by bottom tracking. Bottom tracking involves sending an acoustic pulse along multiple beams to the river bed. The velocity component parallel to each beam can be determined from the Doppler shift in frequency of the backscattered return pulse of each beam. The Doppler shift is assumed to be due to boat velocity, but if the river bed material is moving, bottom tracking is biased by the bed velocity. This bias is a measure of the spatially averaged bed load velocity (v) and can be determined by comparing the boat velocity by DGPS (vDGPS) and by bottom tracking (vBT) (Figure 1):

equation image
Figure 1.

Estimation of bed load velocity using a three-beam ADP. A single along-beam velocity component is determined within the sampling volume of each beam (see text).

[7] The aDcp we used was a three-beam 1.5 MHz acoustic Doppler profiler (ADP) made by SonTek, Incorporated. The sampling volume for a three-beam aDcp consists of three independent volumes located at the impingement points of the three beams. The size of the insonified sampling area of the acoustic beams on the bed was described by Rennie et al. [2002]. The sampling volume was briefly described by Rennie et al. [2002], and further detail was provided by Rennie and Millar [2004]. The sampling location is a volume as opposed to an area as suspended scatterers situated within the sample volume can contribute to the bed load velocity signal. Importantly, the height of the sample volume varies during a ping, with the maximum height for each beam occurring at [0.5 lp sin(90 − χ − θ)] above the bed, where χ is the beam deflection angle from vertical, θ is half of the half-intensity beam width, and lp is the bottom track pulse length, which equals the duration of the acoustic pulse times the speed of sound. Pulse lengths of 0.20 m were used in the 2000 measurements, and pulse lengths of 0.60 m were used in the 2001 measurements, with corresponding maximum sampling volume depths of 9 cm and 27 cm, respectively. The longer pulse lengths were used in 2001 in an attempt to reduce measurement variance [Brumley et al., 1991].

[8] As only one velocity component is determined along each beam, resolution of the velocity vector requires the assumption of equal bed load velocity in all three beam footprints. Assuming homogeneity of the velocity field, consideration of the three-beam geometry yields predictors based on the forward (vxi) and transverse (vyi) velocities through each beam [Rennie et al., 2002]:

equation image
equation image

Beams are numbered counterclockwise when the ADP is facing downward, with the horizontal component of beam 1 parallel to the x axis (see Figure 1) [Rennie et al., 2002, Figure 1b]. Owing to the unequal weighting of vxi in the estimated equation imagex, heterogeneous vxi will necessarily produce an incorrect equation imagex. Furthermore, equation imagex and equation imagey are contaminated by heterogeneous velocities in the opposite component [Theriault, 1986]. During sparse transport, when particle velocity differs between the three insonified areas, the direction and magnitude of the measured velocity vector will necessarily be incorrect. It may be possible to account partially for this error by specifying the expected direction of transport.

3. Study Areas

[9] All of the data presented in this paper were collected in the Fraser River. The Fraser River drains 250,000 km2 of southern British Columbia, Canada. At the town of Hope the Fraser River enters the alluvial Fraser Valley, where it proceeds to deposit its load of gravel in the gravel bed reach. Downstream of the town of Mission, the Fraser River is sand bedded until the estuary at the city of Vancouver, where it discharges into the Straight of Georgia. At Hope (Water Survey of Canada gauge station 08MF005), mean annual discharge is 2900 m3 s−1, mean annual flood is 8800 m3 s−1, and maximum freshet-driven flows exceed 11,000 m3 s−1. The sand bed reach is tidally influenced, with maximum tide range in the estuary of 5 m [Kostaschuk et al., 1989], but during freshet, there is no evidence of salt wedge intrusion or flow reversal upstream of the river mouth at Steveston [Thompson, 1981; Villard, 1995]. We took measurements during freshet in the gravel bed reach at Agassiz and in two sand-bedded estuarine channels (Figure 2). Details of the study sites are summarized in Table 1. It is apparent from Table 1 that a wide range of bed materials and flow conditions is covered by the data set.

Figure 2.

(a) Study site locations. (b) Boat used for sand bed measurements. Note that the acoustic Doppler profiler (ADP) was mounted on the port side, the Differential Global Positioning System (DGPS) receiver was mounted above the ADP, and the davit for bed load sampling was mounted on the starboard side. (c) Fraser River, Sea Reach, facing upstream. The channel width was 300 m. The reach was reasonably triangular in section. In this image we are deploying the Helley-Smith bed load sampler. (d) Fraser River at Agassiz, facing upstream. The channel width was 510 m. If bridge spans are numbered 1 through 5 from left to right (facing upstream), the thalweg flowed through span 5, and a shallow bar occurred at span 2. All gravel bed load sampling stations were downstream of spans 3 and 4.

Table 1. Study Reach and Sampling Characteristics
 AgassizSea ReachCanoe Pass
Latitude49.21°N49.10°N49.07°N
Longitude121.78°W123.16°W123.14°W
Width, m510300350
Maximum depth, m∼1110∼10
d50, mm250.250.35
d90, mm800.350.50
Year200020012000, 2001
Maximum dune height, m0.23
Maximum dune length, m2050
Flow at Hope, m3 s−15600–68005100–53006320–7560 (2000)
 5100–5400 (2001)
n9 (composite), 13 (5 min)6849 (2000), 15 (2001)
Sample time, min2–92 (composite), 5 (5 min)55
Bottom track pulse length, cm206020 (2000), 60 (2001)
Sample depths, m2.1–5.12.3–4.55.0–7.4
Sample U, m s−11.83–2.300.36–1.060.37–1.00

3.1. Gravel Bed Agassiz

[10] Measurements were taken during the 2000 freshet at the Agassiz-Rosedale bridge site in the gravel reach [Rennie et al., 2002]. The hydrology, bed material, and sediment transport characteristics of the site were described by McLean et al. [1999] on the basis of a 20 year program of sediment transport measurements carried out by the Water Survey of Canada. The river width is 510 m, and the channel gradient is 4.8 × 10−4. The D50 and D90 of the surface sediment were reported to be 42 and 80 mm, respectively. The d50 and d90 of the subsurface sediment were 25 and 80 mm, respectively. The bed material was bimodal, mostly gravel-cobble with some sand. Further details of the site are provided by Rennie [2002] and by Rennie and Millar [2001].

3.2. Sand Bed Canoe Pass

[11] Data were collected in both 2000 [Villard et al., 2004] and 2001 in Canoe Pass, a sand-bedded estuarine distributary. Maximum flow appears to occur at the beginning of ebb tide. The portion of the reach where measurements were conducted has been the site of several hydraulic and sand transport studies [e.g., Villard, 1995; Kostaschuk, 2000]. The channel was ∼350 m wide, and the deepest sections were <10 m deep at high tide. The d50 of the bed material was ∼0.35 mm, and the d90 was ∼0.5 mm. Large, steep, asymmetric dunes reach heights of 3 m and lengths of 50 m [Kostaschuk, 2000].

3.3. Sand Bed Sea Reach

[12] Measurements during the maximum flow at the end of ebb tide were taken in 2001 in Sea Reach, another sand-bedded estuarine distributary of the Fraser River. Maximum depth in Sea Reach at low tide during moderate freshet was about 5 m. The reach was 300 m wide. On channel margins, bed material d50 was 0.20 mm, whereas in the center of the channel, d50 was 0.25 mm and d90 was 0.35 mm. Dunes were relatively small; heights and lengths were not observed to exceed 0.2 and 20 m, respectively.

4. Methods

4.1. Bed Load Velocity

[13] ADP measurements of bed load velocity (v) were collected from stationary positions. Data were collected as five-ping ensemble averages every 5 s (0.2 Hz) in 2000 and as single-ping ensemble averages approximately every 2 s (0.5 Hz) in 2001. Flow in the gravel bed reach was steady during sampling periods, which ranged between 2 and 92 min. In the gravel reach, boat position was maintained by either tethering to the bridge (sample durations >5 min) or by motoring (2 min sample durations). Rennie et al. [2002] provided further details on the gravel bed measurements. In the sand bed reaches, boat position was achieved by anchoring. Most of the samples in the sand bed reaches were taken at positions on the stoss side of dunes, where the bed load was presumed to be reasonably spatially homogeneous [see Villard et al., 2004]. Flow in the sand bed reaches was unsteady, due to the variable stage of ebb tide and migrating bed forms. Sand bed sample durations were thus 5 min, except for six 2.5 min samples in the Canoe Pass 2001 campaign. In the sand bed reaches, several samples were collected in series from the same position. These samples were assumed to be statistically independent, due to the unsteady nature of the flow and bed and the stochastic nature of bed load transport. The ADP was incapable of bottom tracking in depths less than 2 m and greater than ∼8 m, which resulted in gravel reach and Canoe Pass samples that had to be eliminated due to poor data quality. In the sand bed reaches, good data in >70% of the bottom track ensembles were required. Rennie et al. [2002] provided the data quality criteria for the gravel bed reach.

4.2. Bed Load Transport Rate

[14] Bed load transport rate was measured using conventional pressure difference samplers at locations in close proximity to the ADP. In the gravel bed reach, both a VuV sampler for gravel and a Helley-Smith (HS) sampler for sand were used [Rennie et al., 2002]. In the sand bed reaches the HS sampler was used exclusively. The bed load samplers were deployed from the bridge in the gravel reach, with the shipboard ADP deployed within a few meters downstream of the sampler. As a result, the bed load sampler was situated close to the edge of the sampling circle defined by the ADP beams [see Rennie et al., 2002], depending on local depth. In the sand bed reaches, there was a constant displacement of ∼3 m between the bed load sampling and the centroid of the bottom track measurements; that is, the bed load sampler and the ADP were displaced by 3 m at the water surface. Thus the bed load sampler was deployed near the edge or within the ADP sampling circle. Displacement of the bottom track sampling from the bed load sampling may have introduced an unknown degree of decorrelation between v and gbM, although we assumed that the sampling locations were statistically equivalent.

[15] In the sand bed reaches, bed load transport sample durations corresponded to ADP sample durations. Two samples from Sea Reach appeared to be outliers, presumably due to dragging of the bed during sampler deployment, and were eliminated. In the gravel bed reach, several 5 min gbM samples were collected and averaged during a single ADP sample [see Rennie et al., 2002]. In this paper, individual 5 min gbM samples from the VuV for which sand was not in transport are also compared to the corresponding 5 min v from the ADP.

[16] It is customary to correct bed load transport measurements for sampler efficiency. HS sampling efficiency is likely operator- and site-specific. Laboratory studies of the HS sampler suggest a sampling efficiency, dependent on particle size, of up to 150% [Glysson, 1993]. Gaudet et al. [1994] observed a drop in sampling efficiency when the sampler was misaligned with respect to the mean flow, which may occur during field deployments in complex flows. The sampling efficiency of the HS sampler was assumed to be 100% in the sand bed reaches, as observed in the field by Emmett [1980]. Sampling efficiency assumptions in the gravel bed reach were described by Rennie et al. [2002].

4.3. Modeling of Bed Load Transport Rate

[17] Two different classes of bed load transport models were used to estimate transport rate for comparison to the measured gbM. The shear stress models of van Rijn [1984] (gbVR) and modified Ackers White [White and Day, 1982] (gbAW) were employed. These models estimate gb without using v as an input. The van Rijn [1984] model was used because it is a standard model for transport of sand. The modified Ackers White model was selected because it provided the best fit to previous bed load transport data from the Agassiz gravel bed reach [McLean et al., 1999]. Model inputs include the particle size distribution of the bed material and the shear velocity, which was determined using the Keulegan equation (see equation (13)), with the depth-averaged velocity from the ADP. The Keulegan equation was employed because this was the method used by van Rijn [1984] to develop his model. Furthermore, measured velocity profiles in the sand bed reaches were insufficiently coherent to determine grain shear from near-bed profiles. In the gravel reach the Keulegan equation yielded similar u* values as a near-bed log law fit. Refer to Appendix A for the solution algorithm of each model.

[18] Alternatively, bed load transport rate can be estimated kinematically (gb,kin):

equation image

where the summation occurs over all surface particles i in the sampling area (A), vi is particle velocity, dai is the depth of transport of particle i, λai is the porosity of the bed load transport layer at particle i, Ai is particle planar area, and ρs is the density of the sediment particles. Note that v measured by the ADP is assumed to be the spatially averaged bed load velocity, including immobile areas of the bed:

equation image

In fact, the velocity of each surface particle is weighted by backscatter intensity rather than simply planar area. Application of the kinematic model using v is relatively straightforward in the sand bed reaches, where the bed load transport layer can be assumed to be spatially homogeneous, with constant values of particle velocity and depth and porosity of the active transport layer:

equation image

However, the kinematic model using v is not applicable in the gravel bed reach, where transport is spatially heterogeneous over a wide range of particle sizes and particle velocities, including large proportions of the bed surface that are instantaneously immobile. Essentially, it is not possible to separate vi from the other terms in the summation of equation (3).

[19] Kinematically modeled estimates of bed load transport rate (gbVRkin) were determined for the sand bed reaches using v and flume-based empirical formulae estimates of da [van Rijn, 1984, equation (10)] and the volumetric void ratio cb = (1 − λa) [van Rijn, 1984, equation (21)]. For each reach we also determined the product of da(1 − λas that produced the best fit between kinematically modeled (gbBFkin) and measured bed load transport (gbM). The v used in the best fit kinematic model was adjusted for the influence of suspended scatterers by subtracting the intercept of the v versus gbM regression [see Rennie et al., 2002]. This adjustment only affected the intercept of gbBFkin versus gbM, allowing the intercept to be zero.

4.4. Functional Relations

[20] Both linear regressions and functional relations are presented in this paper. Linear regression minimizes the sum of square errors vertically from the regression line, assuming that all measurement error occurs in the y variable. Use of the regression equations is appropriate for site-specific prediction of gb from ADP measurements of v, whereas the functional relations best describe the relation between gb and v. Villard et al. [2004] described functional analysis in the context of geomorphic measurements, and Davies and Goldsmith [1972, p. 209] provided confidence limits on the functional slope. The functional slope is

equation image

where br is the regression slope, r2 is the coefficient of determination, and λ is the ratio of gb (y variable) and v (x variable) measurement error variance:

equation image

Note that if r2λ/br2 > 25, then the functional slope equals the regression slope, and if r2λ/br2 < 0.10, then the functional slope equals the slope of an x-upon-y regression [Mark and Church, 1977].

[21] The parameter λ was determined using results of v error modeling [Rennie and Millar, 2002] and the assumption of a linear relation between gb and v. Specifically, we pooled the real variability and measurement errors to create the observed variance in the data:

equation image
equation image

Primes indicate true fluctuations, and E indicates measurement error. Further, we assumed that gb is linearly related to v:

equation image

where bf is the functional slope. Then

equation image

Combining equations (7), (8), and (10) yields

equation image

We have σgb2 and σv2 from the observations. We require σEv, which was estimated to be 0.012 m s−1 using error modeling for 5 min samples at a gravel reach station with a long (91 min) steady flow [Rennie and Millar, 2002]. The functional slope was found by iteration using equations (6) and (11). Note that this method was likely more successful in the gravel bed reach than in the sand bed reach as the estimate of σEv2 was derived from gravel bed data. The ratio of bed load estimated measurement variability to total variability (σEgbgb) was 0.5 for Agassiz, suggesting that sampling errors accounted for 50% of the observed variability in bed load transport rates. However, the ratio was 0.8 for Canoe Pass 2000 and 1.2 for Canoe Pass 2001. Obviously, σEgb should not have actually exceeded σgb. The ratio for Sea Reach (0.4) was more reasonable. It would be better to use an estimate of σEv2 specific for each site. However, this was not possible in the unsteady, tidally influenced sand bed reaches because the error modeling required stationary data.

5. Results

5.1. Measured v Versus Measured gb

[22] All available concurrent gbM versus v data are plotted in Figure 3. Linear regression and functional relations are provided in Table 2, in which v is the independent variable. It appears that the bed load velocity resolution is ∼1 cm s−1, based on the range of observed v at low gbM. Strong relations, in which v explains >70% of the variability of measured bed load transport rate, were observed for the long average Agassiz data and the Sea Reach data. The correlations were less good for Canoe Pass 2000 and 2001 data. This was likely due to poor ADP data quality because the water depth in Canoe Pass was >5 m for all samples, and many samples had depths >7 m, which approached the limit of feasible bottom tracking. For the gravel bed reach, data are shown for both the ∼1 hour averaged v and gbM and the 5 min averaged v and gbM. Only the long average v and gbM were well correlated (the correlation in the 5 min average data was heavily influenced by one sample); thus it appears that long averaging times are required to achieve good estimates of v and gbM in the gravel reach.

Figure 3.

Site-specific measured bed load transport rate versus measured bed load velocity. Symbols are as follows: Agassiz (gravel bed) long averages (open squares); Agassiz (gravel bed) 5 min samples (hatched squares); Canoe Pass 2000 (sand bed) (crosses); Canoe Pass 2001 (sand bed) (plusses); Sea Reach (sand bed) (triangles).

Table 2. Linear Regression and Functional Relations for Measured gbM Versus Measured va
Locationnr2RegressionFunctional Relation95% CLbr2λ/br2
  • a

    Values of gbM are in kg s−1 m−1; values of v are in m s−1. The averaging time for most samples was 5 min. Six of the Canoe Pass 2001 samples had 2.5 min durations. For Agassiz, ADP samples ranged from 2 to 92 min, although all had at least three 5 min bed load samples [see Rennie et al., 2002].

  • b

    Lower and upper 95% confidence limits on the functional slope (the exponent for the nondimensional data).

Agassiz90.89gb = 1.2v − 0.037gb = 1.2v − 0.0410.91–1.70.96
Agassiz 5 min130.52gb = 2.0v − 0.059gb = 2.6v − 0.0880.60–7.81.3
Sea Reach680.76gb = 0.057v − 0.0007gb = 0.062v + 0.00050.062–0.0622.1
Canoe Pass 2000490.38gb = 0.23v + 0.001gb = 0.36v − 0.000080.34–0.380.97
Canoe Pass 2001150.42gb = 0.090v − 0.0003gb = 0.14v − 0.00040.0043–0.181.0
Nondimensional1270.42gb* = 0.043(v/u*)0.85gb* = 0.045(v/u*)0.900.74–2.612

[23] Canoe Pass 2001 (pulse length 60 cm) had significantly less gbM for similar v than Canoe Pass 2000 (pulse length 20 cm). The regression intercepts were significantly different (α = 0.05) [Zar, 1996, p. 357]. This was expected due to the greater influence of suspended sediment on estimated bed load velocity with increased pulse length [see Rennie et al., 2002; Rennie and Millar, 2004]. We can estimate the contribution of suspended sediment to v based on the pulse length, range to the bed of the aDcp, near-bed suspended sediment concentration and size distribution, and the flow velocity near the bed [see Rennie and Millar, 2004]. For Canoe Pass 2001 we expect suspended sediment to contribute ∼1 cm s−1 to v, which conforms well to the observed difference between Canoe Pass 2001 and Canoe Pass 2000.

[24] It is immediately evident that the relation between v and gbM differs for the gravel bed Agassiz data and the sand bed Sea Reach data. Sea Reach had a significantly smaller gb for similar v. This is reasonable as gravel particles being transported with a similar spatially averaged velocity will convey more mass than sand. The Canoe Pass data, due to increased scatter, fell within the range of both the Agassiz and Sea Reach data sets. Note that the difference between Agassiz and Sea Reach may also be due to differences in the influence of suspended scatterers. However, the contribution to v of suspended sediment was estimated to be 3 cm s−1 at Agassiz [Rennie et al., 2002] and only <1 cm s−1 at Sea Reach [Rennie and Millar, 2004]; thus this should not have been a factor. These estimates were based on near-bed suspended sediment concentrations typical at each site over the same range of flows as the v and gb measurements. The difference due to particle size can be accounted for by nondimensionalizing gbM and v. The bed load transport rate was nondimensionalized as suggested by Einstein [1950]:

equation image

where g is gravitational acceleration, Ss is the specific gravity of the sediment, and d50 is the median size of the bed load (assumed to be the median bed material size in the sand bed reaches). The bed load velocity was nondimensionalized using the shear velocity (u*), as estimated by the Keulegan equation [van Rijn, 1984]:

equation image

where U is the depth-averaged water velocity and Y is the flow depth. The nondimensional data collapse was reasonably successful (r2 = 0.42) (Figure 4). On the basis of analysis of error propagation, the estimated functional relation between gbM* and v/u* assumed that all the measurement error was due to gbM and v (Figure 4 and Table 1). The estimated λ for the nondimensional log-transformed combined data equaled 20 (λ ranged from 11 to 28 for the individual data sets).

Figure 4.

Nondimensional measured bed load transport rate versus nondimensional measured bed load velocity. Symbols are as in Figure 3. The solid line is an estimated functional relation (see text).

5.2. Modeled gb Versus Measured gb

[25] Functional relations between the various modeled versus measured estimates of gb are provided in Table 3. The parameter λ is unknown, and thus the principal axis is shown (assuming λ = 1), but the 95% confidence limits on the slope encompass the 95% confidence intervals for regression slopes of x upon y and y upon x. Also provided is a mean absolute difference,

equation image

which is the average decadal scatter around the line of perfect agreement. The standard shear stress models provided poor matches to the measured bed load transport rates: the sand bed data were overpredicted, and the gravel bed predictions were not significantly related to the measured values (Figure 5). However, the shear stress modeled gb showed trends with v similar to measured gbM (Figures 6 and 7). In particular, strong correlations between modeled gb and v in the sand bed reaches suggest that higher values of v were observed when hydraulic conditions were favorable for transport (Table 4). The functional relations in Table 4 assume that λ is the same as calculated for measured gbM versus v. However, given the unreliability of bed load transport models [Gomez and Church, 1989], modeled gb error may have been larger than measured gbM error, and thus the stated 95% confidence limits on the slope include the confidence limits for regression of y (modeled gb) upon x (measured gbM).

Figure 5.

Modeled bed load transport rate (modified Ackers White, 1982) versus measured bed load transport rate. Symbols are as in Figure 3. The dashed line is perfect agreement.

Figure 6.

Modeled bed load transport rate (modified Ackers White, 1982) versus measured bed load velocity. Symbols are as in Figure 3.

Figure 7.

Nondimensional modeled bed load transport rate (modified Ackers White, 1982) versus nondimensional measured bed load velocity. Symbols are as in Figure 3. The solid line is an estimated functional relation (see text).

Table 3. Functional Relations (Principal Axis) for Modeled gb Versus Measured gbMa
Locationnr2MADbFunctional Relation95% CLcda (1 − λas
  • a

    Units are kg m−1 s−1.

  • b

    Mean absolute difference of modeled data from the measured.

  • c

    Lower and upper 95% confidence limits on the functional slope (see text).

Agassiz
gbVR versus gbM90.0630.38y = 1.0x + 0.0065not significant 
gbAW versus gbM90.0980.70y = 6.0x − 0.085not significant 
Agassiz 5 min.
gbVR versus gbM130.0500.82y = 0.093x + 0.071not significant 
gbAW versus gbM130.361.13y = 0.59x + 0.16not significant 
Sea Reach
gbVR versus gbM680.421.03y = 17x − 0.0145.1–24 
gbAW versus gbM680.490.73y = 20x − 0.0287.4–27 
gbVRkin versus gbM680.650.52y = 4.7x − 0.00582.6–5.80.0035–0.33 (mean 0.11)
gbBFkin versus gbM680.760.38y = 1.0x − 4.0e−40.75–1.30.066
Canoe Pass 2000
gbVR versus gbM490.470.99y = 5.8x − 0.00191.9–8.7 
gbAW versus gbM490.450.66y = 5.1x − 0.00461.6–7.9 
gbVRkin versus gbM490.360.99y = 0.40x + 1.3e−40.20–1.50.0017–0.22 (mean 0.068)
gbBFkin versus gbM490.381.04y = 1.0x − 0.00120.38–2.60.37
Canoe Pass 2001
gbVR versus gbM150.451.20y = 16x + 0.00222.5–46 
gbAW versus gbM150.420.77y = 8.3x − 0.00551.0–29 
gbVRkin versus gbM150.480.72y = 1.4x − 0.000530.34–4.80.0023–0.21 (mean 0.089)
gbBFkin versus gbM150.420.67y = 1.0x − 5.3e−40.19–5.20.14
Table 4. Functional Relations for gbAW by the Modified Ackers White (1982) Bed Load Model Versus Measured va
Locationnr2Functional Relation95% CLbr2λ/br2
  • a

    Values of gbAW are in kg s−1 m−1; values of v are in m s−1.

  • b

    Lower and upper 95% confidence limits on the functional slope (see text).

Agassiz90.007gbAW = 27v − 1.5not significant0.19
Agassiz 5 min130.37gbAW = 2.6v − 0.0500.17–8.00.39
Sea Reach680.71gbAW = 0.89v − 0.00530.67–1.01.8
Canoe Pass 2000490.76gbAW = 1.3v − 0.00030.95–1.50.92
Canoe Pass 2001150.43gbAW = 0.61v − 0.0014not significant2.7
Nondimensional1270.48gbAW* = 0.2(v/u*)1.41.1–1.65.7

[26] The kinematic model using estimated values of da and the volumetric void ratio cb = (1 − λa) from van Rijn [1984] yielded inconsistent results, with overprediction in Sea Reach and underprediction in Canoe Pass 2000 (Figure 8). Note also that r2 increased from the fit between v and gbM in Canoe Pass but decreased in Sea Reach, suggesting that individual estimates of da and cb for each sample do not necessarily improve the fit. The best fit kinematic model yields a similar fit as the relation between v and gbM as the product da(1 − λas is a constant coefficient (Figure 9). The best fit constant value of da(1 − λas was less than expected by van Rijn [1984] on average in Sea Reach but greater than expected in Canoe Pass.

Figure 8.

Kinematic model bed load transport rate with estimates of da and cb = (1 − λ) from van Rijn [1984] versus measured bed load transport rate. Symbols are as in Figure 3. The dashed line is perfect agreement. Note that Sea Reach and Canoe Pass tend to plot above and below the line of perfect agreement, respectively, particularly for higher transport rates.

Figure 9.

Kinematic model bed load transport rate with best fit estimate of da(1 − λ)ρs versus measured bed load transport rate. Symbols are as in Figure 3. The dashed line is perfect agreement.

6. Discussion

6.1. Site Specificity of v Versus gb

[27] The relation between bed load velocity measured by the ADP and bed load transport rate is a site-specific function of bed material size, with similar values of v being observed for much smaller transport rates in sand-bedded Sea Reach than in gravel-bedded Agassiz. It appears that bottom track pulse length also influenced the degree to which suspended scatters influence v, as seen in the comparison between Canoe Pass 2000 (20 cm pulse length) and Canoe Pass 2001 (60 cm pulse length). We suspect that the mode of bed material transport (rolling, saltating, intermittently suspended) will also influence the relation between v and gb, although we do not have evidence to support this speculation. The mode of transport may influence the degree of acoustic penetration of the bed load transport layer, with the resulting estimated velocity depending on the depth of penetration, assuming a velocity and concentration gradient in the bed load layer.

[28] A strong correlation between v and gbM was observed with 5 min samples in a sand bed environment with depths <5 m (Sea Reach), but longer averaging times were required in gravel-bedded Agassiz. The sporadic nature of bed load in gravel reduces the quality of the signal. Recall from equation (2) that correct estimation of v requires a spatially homogeneous velocity field. In a gravel bed it is unlikely that the bed load velocity is instantaneously the same in all three beam footprints. On the basis of running averages of stationary time series, Rennie et al. [2002] estimated that 25 min of data were required for reliable estimates of mean bed load velocity in the gravel bed reach. Note that this was due to variance introduced by both real temporal variability of bed load and measurement error. Using a running coefficient of variation on flume measurements of sand gravel bed load transport, Kuhnle and Southard [1988] observed that >20 min of data were required for reliable estimates of mean values, and this was entirely due to real temporal variability of transport. Using our error model for gravel bed data [Rennie and Millar, 2002], ∼7 min of data were required to reduce the measurement error in v (i.e., σEv) to 1 cm s−1. Clearly, this limits the temporal resolution of the technique. A similar analysis is difficult in the sand bed reaches due to unsteady conditions. However, applying the running average technique to quasi-stationary sand bed time series suggests that between 10 and 15 min of data were required for reliable estimates of mean bed load velocity. Thus it appears that bed load velocity measurement with an aDcp is more reliable in sand bed than in gravel bed environments, although long averaging times are still required.

[29] The relation between v and gb will also be affected by the specifics of the aDcp. In particular, aDcps with lower operating frequencies will have less signal attenuation and may be able to penetrate through the bed load transport layer to the stable bed [see Rennie, 2002, p. 140]. The signal processing employed by the aDcp may also influence results.

6.2. Bed Load Modeling

[30] The inability of the standard shear stress-based bed load models to match observed transport rates is not surprising [cf. Gomez and Church, 1989]. Bed load models tend to be based on equilibrium transport in flumes, whereas bed load transport in rivers depends on spatial and temporal variations and on nonlinear interactions between sediment supply, bed state, and forcing flow. Part of the motivation of the present study is to develop an improved method for collecting representative bed load transport data from rivers for bed load modeling efforts.

[31] It could be argued that the shear stress-based models overpredicted transport rates in the sand bed reaches because we used the Keulegan equation to predict u*, which determines a total shear rather than a grain shear. However, as by van Rijn [1984], the roughness height ks was assumed to be 3D90 (see equation (13)), and thus this form of the Keulegan equation should have approximated shear on a flat bed. Furthermore, the van Rijn [1984] model was calibrated with experimental data of sand bed transport over dunes using this method to estimate u*, so this was the appropriate estimate of u* to use with the model.

[32] The best fit kinematic model yielded a constant value estimate for da(1 − λas. Of course, it is difficult to isolate da from λa. Assuming da = 2d50 [Einstein, 1950], values of λa were estimated to be 0.95, 0.80, and 0.92 for Sea Reach, Canoe Pass 2000, and Canoe Pass 2001, respectively. Except for Canoe Pass 2000, these values are consistent with estimates from van Rijn [1984] and suggest that the bed load transport layer is relatively dilute. A typical value of λa for static sediments is 0.4.

[33] An alternative interpretation of da(1 − λas is that the measured transport layer is an extremely dilute near-bed suspended sediment region. Assuming for convenience that da equals 0.076 m (the height of the Helley-Smith aperture), estimated values of volumetric void ratio (1 − λa) were 3.3e-4, 1.8e-3, and 6.9e-4 for Sea Reach, Canoe Pass 2000, and Canoe Pass 2001, respectively. These values correspond to concentrations of 0.87, 4.8, and 1.8 g L−1, respectively. The Sea Reach and Canoe Pass 2001 values are reasonable for suspended sediment concentration in Fraser River. The higher value for Canoe Pass 2000 could reflect bed material transport by intermittent suspension.

[34] It thus remains unclear whether ADP bottom tracking in the sand bed reaches is reflecting off the near-bed suspension layer or the bed load saltation layer. However, as described in section 5.1, the contribution of suspended scatterers to the measured bed load velocity was estimated to be ≤1 cm s−1 in the sand bed reaches, which suggests that ADP bottom tracking was largely reflecting off the bed load saltation layer. Furthermore, it appears that signal attenuation in the turbid Fraser River limited the maximum depth of bottom tracking, and thus the ADP signal processing likely only produced bottom tracking data for signals that reflected off the bed. It is reasonable to suggest that the depth of penetration of the near-bed layer depends on the concentration of suspension. Higher concentrations will be observed when the bed material is being transported in intermittent suspension; thus it is likely that ADP bottom tracking is measuring bed material transport in either case. It is worth noting that physical samplers collect all the material within several centimeters of the bed and thus also measure near-bed suspension.

6.3. Improvements to the Method

[35] The difficulty with the measurements in the gravel bed reach is violation of the assumption of homogeneity of transport. This assumption is required to resolve a Cartesian velocity from the three beams. The assumption would be less violated if the beam footprints were larger as a larger area of the bed would be integrated for each beam measurement. Alternatively, the beam footprints could be brought closer together, although some beam deflection angle is required for a horizontal velocity component to be recorded. At the limit, all three beams could measure the same location, although this would require substantially different instrument geometry.

[36] Another difficulty with gravel transport is that there is a great range of particle sizes, and much of the bed surface is instantaneously immobile. As a result, there is a broad spectrum of bed load velocities. Signal processing in incoherent (“narrowband”) aDcps usually determines mean velocity by finding a peak in the return spectrum using the slope at zero lag of the autocorrelation function [Millar and Rochwarger, 1972]. The wideband echo from a mobile bed would not be processed well by such a technique. Signal processing that considers the entire spectrum may produce more accurate velocity estimates [e.g., Hansen, 1986]. Bed load velocity error would also be reduced by an increased bottom track pinging rate, which could be achieved if water column pinging was reduced or eliminated. It would also be helpful if a statement of signal quality, such as a signal-to-noise ratio, was output for each ping.

[37] In sand bed environments the primary difficulty with the method is determination of the depth of penetration of the beam and thus the location of the measurement in the near-bed concentration gradient. It may be possible to utilize multiple operating frequencies. Lower frequencies will have less attenuated signals and will thus penetrate deeper. This could allow for determination of the depth of the transport layer and the penetration depth of individual frequencies.

7. Conclusions

[38] Concurrent measurements of bed load velocity from an aDcp and bed load transport rate from conventional samplers from different fluvial environments have been shown to display site-specific calibrations. Measurement of bed load with an aDcp overcomes the problems of deployment and flow disturbance associated with conventional mechanical samplers. Variations in the calibrations between sites were explained as a result of differences in caliber and mode of bed material transport as well as aDcp operation. A nondimensional data collapse, accounting for differences in bed load caliber, successfully explained 42% of the observed variance of the combined data set. While encouraging, accounting for caliber did not explain all the observed variance, and thus aDcp measurements of bed load velocity must be calibrated with site-specific bed load data for estimation of bed load transport rate. Attempts to model bed load were ineffective, highlighting problems with bed load modeling. The primary difficulties with the method depend on the particular fluvial environment; in gravel beds the assumption of spatial homogeneity of transport is violated, and in sand beds the precise location of the measurement within the concentration gradient is indeterminate. Recommendations were made for the development of new instrumentation to overcome these difficulties. This work highlighted the potential of acoustic techniques for estimating bed load, particularly in deep channels, where deployment of mechanical samplers is difficult.

Appendix A:: Bed Load Models

[39] The van Rijn [1984] model is

equation image
equation image
equation image

where ρs is the sediment density, D50 is the 50th percentile particle size by weight (in this case, assumed to be of the surface sediment), Ss is the sediment specific gravity, g is gravitational acceleration, T is the transport stage parameter, D* is the scaled particle parameter, ν is the kinematic viscosity, u* is the grain shear velocity (from the Keulegan equation, see equation (13)), and u*cr is the critical shear velocity (from the Shields diagram).

[40] The modified Ackers White model is a fractional transport model, wherein transport of each grain size fraction is summed to yield the total transport:

equation image
equation image
equation image
equation image
equation image
equation image
equation image

for 1 ≤ Dgri ≤ 60,

equation image
equation image
equation image
equation image

for Dgri > 60,

equation image
equation image
equation image
equation image

where Di is the geometric mean particle size of size fraction i, gbi is the fractional bed load transport rate, fi is the fractional proportion in the bed (in this case assumed to be of the surface sediment), γs is the sediment specific weight, Ggri is the fractional transport parameter, Fgri is the fractional mobility number, Dgri is the fractional dimensionless grain diameter, and all other parameters are defined above.

Notation
gb

bed load transport rate, kg s−1 m−1.

gbM

gb measured with physical samplers.

gbM*

nondimensional measured gb.

gbVR

gb modeled using van Rijn [1984].

gbAW

gb modeled using modified Ackers White [White and Day, 1982].

gbAW*

nondimensional modified Ackers White modeled gb.

gbi

gb for size fraction i.

gb,kin

gb modeled using kinematic model.

gbVRkin

gb modeled using kinematic model with van Rijn [1984] estimates of da and λ.

gbBFkin

gb modeled using kinematic model with best fit estimates of da and λ.

Acknowledgments

[41] This work was supported by the Natural Sciences and Engineering Research Council (Canada) through scholarships to both authors and research grants to Mike Church, Rob Millar, and Ray Kostaschuk. We wish to thank Bob Land and Arjoon Ramnarine for piloting the boats. The manuscript was improved by suggestions from Peter Traykovski and two other reviewers.

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