Testing bedload transport formulae using morphologic transport estimates and field data: lower Fraser River, British Columbia

Transcription

1 Earth Surface Processes and Landforms Testing Earth Surf. bedload Process. transport Landforms formulae 30, (2005) 1265 Published online 16 August 2005 in Wiley InterScience ( DOI: /esp.1200 Testing bedload transport formulae using morphologic transport estimates and field data: lower Fraser River, British Columbia Yvonne Martin 1 * and Darren Ham 2 1 Department of Geography, University of Calgary, Calgary, AB, T2N 1N4, Canada 2 Department of Geography, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada *Correspondence to: Y. Martin, Department of Geography, University of Calgary, Calgary, AB, T2N 1N4, Canada. ymartin@ucalgary.ca Received 11 February 2004; Revised 4 October 2004 Accepted 18 November 2004 Abstract Morphologic transport estimates available for a 65-km stretch of Fraser River over the period provide a unique opportunity to evaluate the performance of bedload transport formulae for a large river over decadal time scales. Formulae tested in this paper include the original and rational versions of the Bagnold formula, the Meyer-Peter and Muller formula and a stream power correlation. The generalized approach adopted herein does not account for spatial variability in flow, bed structure and channel morphology. However, river managers and engineers, as well as those studying rivers within the context of long-term landscape change, may find this approach satisfactory as it has minimal data requirements and provides a level of process specification that may be commensurable with longer time scales. Hydraulic geometry equations for width and depth are defined using morphologic maps based on aerial photography and bathymetric survey data. Comparison of transport predictions with bedload transport measurements completed at Mission indicates that the original Bagnold formula most closely approximates the main trends in the field data. Sensitivity analyses are conducted to evaluate the impact of inaccuracies in input variables width, depth, slope and grain size on transport predictions. The formulae differ in their sensitivity to input variables and between reaches. Average annual bedload transport predictions for the four formulae show that they vary between each other as well as from the morphologic transport estimates. The original Bagnold and Meyer-Peter and Muller formulae provide the best transport predictions, although the former underestimates while the latter overestimates transport rates. Based on our findings, an error margin of up to an order of magnitude can be expected when adopting generalized approaches for the prediction of bedload transport. Copyright 2005 John Wiley & Sons, Ltd. Keywords: bedload transport formulae; morphologic transport estimates; sediment transport; channel morphology; field data Introduction The outstanding data compilation available for Fraser River provides a unique opportunity to thoroughly evaluate the performance of bedload transport formulae over larger spatial and temporal scales than has typically been possible. Of particular note, we introduce updated decadal-scale, gravel transport estimates for a 65-km reach of Fraser River, which are used in the testing of bedload formulae herein (cf. Church et al., 2001; Ham and Church, 2003). The Fraser River is amongst the 50 largest rivers in the world based on mean annual discharge, with management concerns ranging from navigation, fisheries and recreation to flooding and erosion hazards. In response to these concerns, a diverse collection of field data has been assembled over the past five decades. Some of these data have been published or presented in university theses, government documents, consulting reports and peer-reviewed articles. Several major reports have been published by the Fraser River Research Group in recent years to guide management decisions ( Large rivers seldom boast such a wealth of field information, since data collection is difficult, laborious and expensive. Given constraints of data collection, land managers and researchers studying other large rivers may not

2 1266 Y. Martin and D. Ham have access to the range of field data available for Fraser River. For example, they may have to rely on transport predictions based on relatively simple applications of bedload transport formulae to provide baseline information on erosional and transport regimes. However, such predictions are limited by the predictive capabilities of the formulae themselves. For example, Gomez and Church (1989) examined the performance of bedload formulae for much smaller rivers over individual cross-sections and flood events and found that in most situations the formulae did not estimate bedload transport accurately. In recent years, bedload transport formulae have been examined over larger spatial and temporal scales (e.g. Nicholas, 2000; Martin, 2003), although discharges for the rivers in these studies are considerably lower than for Fraser River. The predictive abilities of bedload transport formulae for large rivers remain largely unknown due to a lack of field measurements for testing. Bedload predictions are also limited by the availability and quality of input data, such as discharge, width, depth, slope and grain size. Herein, we attempt to remedy this shortcoming by taking advantage of the field data collection available for Fraser River. In this study, we test bedload formulae over decadal time scales for a 65-km study reach of Fraser River. Particularly relevant for this study is the availability of gravel transport estimates based on the morphologic method (Ham and Church, 2003) for the study reach over the period , as these data allow for critical testing of bedload predictions. We restrict ourselves to bedload formulae and a generalized approach to input variable definition that are easy to implement and are readily employed for purposes such as reconnaissance land management studies or longterm studies of landscape change; in such cases, knowledge of input data is likely to be limited. In the latter case, a generalized approach may be most appropriate to ensure that process specification remains commensurable with the time scale of the problem. We choose herein to examine in significant detail several formulae that are widely used and have been found to perform relatively well in past studies (e.g. Gomez and Church, 1989; Nicholas, 2000; Martin, 2003). Study Area As Fraser River emerges from the Coast and Cascade Ranges near Hope, it flows through a partially confined cobble gravel fan (floodplain) that extends downstream to Sumas Mountain, roughly 10 km above Mission (Figure 1). The deposition of gravel within the fan is a consequence of a reduction in channel gradient near the mountain front. The fan terminus is marked by an additional (order-of-magnitude) decrease in gradient and transition to a singlethread sand-bed river. Within the alluvial fan, the river has formed a wandering gravel-bed channel. These rivers are common throughout mountainous and foothill regions of western Canada (Desloges and Church, 1989) and typically form part of a downstream continuum of channel planform types between meandering and braided reaches in response to varying environmental controls (Brierley, 1989). Wandering channels commonly flow in irregularly Figure 1. Study reach along the lower Fraser River with inset map of location in British Columbia.

3 Testing bedload transport formulae 1267 Figure 2. Typical annual hydrograph at Hope (1999). Figure 3. Flood frequency analysis for Hope ( ). sinuous single-thread channels, but are frequently split around large, forested islands, even during peak flows. River morphology is dominated by erosion and transport of stored sediments, although tributary streams may contribute significant loads. Coarser particles accumulate in wide, locally unstable sedimentation zones, which can exhibit loworder braiding, and are separated by narrower, stable transport zones. An extensive network of side channels (seasonal, perennial and abandoned) is usually found along larger islands and adjacent floodplain. These channels mark former positions of active channel and are subject to both avulsions and abandonment during floods. Figure 2 illustrates a typical annual hydrograph at Hope. Low flows dominate over most of the winter period. Significant rain and rain-on-snow events may occur locally in parts of the basin during the autumn and winter seasons, but due to the large contributing drainage area of km 2 at Hope, such inputs are attenuated and not pronounced in the hydrograph. The flood regime is dominated by the snowmelt flood, which typically begins in May and continues well into the summer months. The mean annual flood at Hope, based on maximum daily flow data for (study period), is about 8500 m 3 s 1 (Figure 3); the largest flood over this period was m 3 s 1. Significant bedload transport in our study reach begins at about 5000 m 3 s 1 (McLean et al., 1999). For the period , the mean number of bedload transporting days (flows >5000 m 3 s 1 ) per annum at Hope is 65. Bedload Formulae Used in Study Original and rational Bagnold formulae The Bagnold formula is the most well known of stream power formulae and was found to perform well by Gomez and Church (1989). The Bagnold equation has the form (Bagnold, 1980):

4 1268 Y. Martin and D. Ham γ s ω ω o 23 / 12 / ib = ib ref ( d/ dref ) ( D/ Dref ) (1) γs γ ( ω ωo) ref where i b is specific bedload transport rate, γ and γ s are specific gravity of fluid and sediment respectively, ω is specific stream power, ω o is critical specific stream power, d is depth, D is some characteristic particle size, usually D 50, and the subscript ref refers to a reference value obtained from a reliable data set; Bagnold chose to use data of Williams (1970). The term γ s /(γ s γ) is introduced for the conversion from immersed weight to dry weight (Gomez and Church, 1989). Within the equation, stream power is defined by: 32 / ω = ρqs/w (2) where ρ is density, Q is discharge, S is energy slope and w is width. Critical stream power for initiation of bedload transport is obtained from: ω o = 5 75{0 04(γ s γ)ρ} 3/2 (g/ρ) 1/2 D 3/2 log(12d/d) (3) where g is gravity and 0 04 represents Shields entrainment number. The Shields number may be modified to represent different bed conditions/structures. Reference values for Equation 1 are: i b ref = 0 1; (ω ω o ) ref = 0 5; d ref = 0 1; D ref = (4) Martin and Church (2000) reanalysed the Bagnold formula using a much larger data set than was used to develop the original (Bagnold, 1977, 1980) and sought a rationalization of this relation. Dimensional analysis revealed a rational version of the Bagnold equation having the form: Additional terms in the equation collapse to a coefficient of i b (ω ω o ) 1 5 d 1 D 1/4 (5) Meyer Peter and Muller formula The Meyer-Peter and Muller (1948) formula (hereafter referred to as MPM formula) is among the most frequently used bedload formulae (e.g. HEC, 1991; Nicholas, 2000) and was found to perform reasonably well by Gomez and Church (1989). The equation has the form: and: i b 32 / γ s ( QB/ Q)( KB/ KG) ds 0 047{( γs γ)/ γ} D = 13 / γs γ ( 025 / γ)( γ/ g) 32 / (6A) K B = u/d 2/3 S 1/2 1/6 K G = 26/D 90 (6B) where u is velocity and Q B is the portion of the discharge that acts on the bed. Meyer-Peter and Muller defined Q B /Q as equal to 1 in wide channels. The term (K B /K G ) accounts for the presence of form resistance, which reduces shear stress available for transport. Stream power correlation Following Martin (2003), the performance of a simple correlation between stream power and bedload transport rate is tested. This generalized equation requires minimal input data in comparison to other formulae and incorporates no threshold for transport. For these reasons, it is not expected to perform as well as other formulae. Martin (2003) found that despite these constraints, this formula performed relatively well in some instances. This equation is investigated herein as it provides an option when grain size and depth data, perhaps the most difficult variables to acquire in the field, are not available. Bedload data of Gomez and Church (1989) were analysed to define the best-fit relation (R 2 of 0 53): i b = ω (7)

5 Testing bedload transport formulae 1269 Morphologic Data and Transport Estimates The morphology-based sediment budget is primarily based on repeat topographic surveys of the channel bed, bars and islands. Comparison of digital elevation models produced from the surveys is used to detect bed level changes associated with sediment erosion, transport and deposition. On Fraser River, it is known that major changes in river morphology evolve over years to decades, with erosion and deposition largely occurring in distinct zones. Repeat surveys of the channel are therefore appropriate for detecting and measuring channel deformation, provided the spatial and temporal resolution is sufficiently high. The first survey of the river was completed in 1952 by Public Works Canada. The river bed was surveyed by sonar, while exposed (above-waterline) surfaces were mapped photogrametrically. These data were acquired by digitizing the soundings points and contour lines published on 1:4800 charts. The most recent survey was completed in 1999 by Public Works and Government Services Canada. Bed elevations were recorded at sub-metre intervals along transect lines spaced every 200 m. The data were thinned using a specially written Fortran program that retained data at 20 m intervals between the start and end point of each transect to produce a working dataset comparable in density to the 1952 survey. Elevations along exposed bars and floodplain surfaces were obtained from a laser profiling topographic survey (Lidar) completed several months prior to the sounding survey. The Lidar data were acquired at 5 to 10 m spacing, also along 200 m transects, but the track lines are not necessarily coincident with the bathymetry. An additional survey was completed in 1984 using a combination of automated hydrographic survey and terrestrial ground survey (McLean, 1990). These data are not included herein, however, since the purpose of our present study is to present a long-term average transport rate, not a varying rate over two time periods. Despite the lengthy (47 years) interval between the bathymetric surveys, no information on net erosion or deposition between survey dates is lost. As there is no known transport of gravels past the downstream limit of the study reach, there is no practical upper limit of temporal survey spacing since all gravels are trapped. Details of active channel deformation during this same period are, of course, limited. A sediment budget approach is used to relate changes in channel morphology to sediment transport. Within a defined length or reach of channel, this can be expressed as: V o = V i (1 p) V (8) where V o is volumetric sediment output and V i is sediment input to the reach over a specified period of time; V is the change in storage, measured as the net difference between scour and fill along the channel bed, adjusted for porosity (1 p) to convert all terms to mineral volumes. The equation can be reduced to a mean transport rate by dividing all terms by the time between successive survey intervals. A complete sediment budget must also account for sediment removed by dredging, and changes in storage associated with the erosion and deposition of island and floodplain sediments. For island and floodplain deposits, we are only interested in the basal sand and gravel component of the deposits and have removed the 1 3 m thick layer of overbank sands and silts from the calculations. Finally, the volumetric changes are converted to gravel volumes by adjusting for the proportion of medium and coarse sand (<2 mm) along the channel. The sand fraction is estimated as 30 per cent from reach 13 to the mid-point of reach 3, but increases rapidly to 95 per cent near Mission Bridge (Figure 1). These proportions were derived from pooled bulk samples collected in 1983 and 2000 and are described in the following section. In this study, the first reach (at Mission) is assigned a transport rate of 0 and calculations are extended upstream on a per-reach basis so that the transport rate can be illustrated for different locations. These reaches (shown in Figure 1) demarcate the major known zones of aggradation (major bar and island deposits) and transport (few bars or islands) along the river as determined from sequences of historic aerial photographs, available for many dates since To calculate erosion and deposition volumes between surveys, it is first necessary to compute a topographic surface for each date. Surfaces were created using the Topogrid model in Arc/Info GIS, an interpolation method that uses a discretized thin plate spline technique, where an exact spline surface is replaced with a locally smoothed average (Burrough and McDonnell, 1998). A 20-m grid cell model was created for each date as this cell size is similar to the point density of the bathymetry surveys. Spline techniques are criticized for producing an unrealistically smooth surface, but smoothing tolerances can be reduced in the geographical information system (GIS) to produce a sharper, less generalized output. It was also found that model output could be improved by incorporating contour lines as breaklines along channel margins. The superior performance (both quantitative and visual) of the Topogrid model for sounding data collected along widely spaced transects versus other interpolation schemes, including the more widely used TIN model (cf. Lane et al., 1994; Brasington et al., 2000), is discussed by Ham and Church (2003). The choice of appropriate interpolation scheme becomes decreasingly important with increased survey density. Following completion of the surface models, each was clipped to eliminate interpolation outside the margins of known channel

6 1270 Y. Martin and D. Ham Table I. Morphologic transport estimates for lower Fraser River. Values are average annual transport rates over the period Reach Gravel transport (m 3 a 1 ) change (i.e. beyond the maximum historic extent of outer channel banks). A surface of difference was then created by simply subtracting the two models, then multiplying by cell area to compute volumes. Procedural details are described further in Church et al. (2001). In fact, the 0-gravel assumption leads to negative transport rates for several reaches; one possibility that would lead to this result is if some gravel is transported past Mission. To remove these negative transport rates, all transport rates are adjusted by the value of the highest negative transport rate (cf. Martin and Church, 1995). Morphologic transport estimates for the 13 study reaches are shown in Table I. The morphologic gravel budget estimates are subject to a number of limitations that affect the reliability of our results. The sounding surveys have inherent measurement error associated with the precision by which horizontal and vertical positions can be recorded. However, these errors are largely random, and become incorporated into error associated with the accuracy of the interpolated surface grids. Ham and Church (2003) estimated root mean square errors of roughly ±8000 m 3 per kilometre of channel for the period Further, incomplete knowledge of removal volumes (for which records prior to 1964 are not available, and records since may be incomplete) represents a negative bias in the budget, the magnitude of which is not fully known. There are additional sources of potential bias associated with the construction of the sediment budget. A major assumption of the approach is that there is no compensating scour or fill between successive surveys, as this leads to undetected sediment transport. However, as no known volume of gravel passes through the downstream extent of the study reach, the measurement of gravel flux should not be affected (even if the assumption of 0-transport is incorrect, the actual volume is certainly modest and would have little material impact on our upstream transport estimates). A lack of complete knowledge with respect to the spatial distribution of sand throughout the study reach may also bias our estimates. Our estimate of 30 per cent sand along much of the study area is based on a trend line of the field data, which exhibits considerable scatter. Also, the downstream end of the reach has few emergent bars and remains poorly sampled. While this scatter is reduced over the length of a single morphologic reach, it nevertheless affects the precision with which the actual gravel proportion can be estimated. While the need to collect additional information is desirable, the sampling effort on such a large river represents an onerous and costly exercise. As such, the provided transport values represent our best possible estimate given the limitations of available data. Ham and Church (2003) provide a detailed analysis and review of all these sources of error and suggest that the gravel budget estimates may vary by ±35 per cent. Input Variables Discharge Two significant tributaries, Harrison River and Chilliwack River, drain into Fraser River along the length of our study reach. Therefore, discharge adjustments must be made for assignment of correct flows along the study reach. Discharge measurements are available at several locations along the study reach from Water Survey of Canada archives. The longest discharge series, covering the period from 1912 to the present day, is available at Hope, just

7 Testing bedload transport formulae 1271 upstream of our study reach. Discharge measurements are available since 1965 for Mission, the lowermost boundary of our study reach. In addition, discharge data are available for the Chilliwack River, a large tributary entering our study reach, for the period 1951 to present. Regression was performed for discharges between Hope and Mission for the years 1965 to 1992 (r 2 of 0 98), the period for which discharge values are available at Mission. These results were used to estimate discharge values for years of missing data. The contribution of discharge from the other major tributary, Harrison River, is unknown and is estimated as: Q Harrison = Q Mission Q Chilliwack Q Hope (9) Reaches 8 to 13 are assigned discharge values at Hope; Reaches 5 to 7 are assigned discharge values at Hope plus the input from Harrison River; finally, Reaches 1 to 4 are assigned the discharge values for Mission. The difference in mean annual flow between Hope and Mission is about 600 m 3 s 1. Width and depth The form of the hydraulic geometry relations for both width and depth are based on conventional power functions. Hydraulic geometry relations are established for water surface width from a series of morphologic maps prepared from historic sequences of aerial photographs for 1928, 1949, 1962, 1971, 1983, 1991 and For each of these dates, channel banks, islands and bars were mapped using an analytic stereoplotter, and digitized lines were imported to Arc/Info GIS to create areal (polygon) maps of the same features. For each morphologic reach, water surface width is simply calculated as the area of the water surface divided by reach length. As the flow is known on each date of mapping and encompassed a wide range of flows (varying from 700 m 3 s 1 in 1999 to 5380 m 3 s 1 in 1983), a power function is used to relate width to discharge. An additional point was added to simulate bankfull discharge, where water surface width was defined as the active channel area (total area of water and bars, excluding wooded islands) divided by reach length. Derived values of coefficients and exponents for width hydraulic geometry equations are given in Table II. Significant variations in the value of the width exponent correspond to distinct channel morphologies and chiefly reflect differences in bank strength. Reaches 1 to 3 and 13 are largely confined by non-alluvial materials and have very steep banks, so increases in discharge are mainly accommodated by increases in flow depth and velocity. The calculations required to derive similar relations for water depth are considerably more complex. Although hydraulic geometry relations can be obtained from periodic measurements at hydrometric stations, gauges on Fraser River are located at narrow, stable locations and are not representative of typical river cross-sections. An alternative GIS-based approach was developed wherein depth was directly related to water surface width. Since the relation between width and discharge was already established, the two equations could be collapsed along any reach to produce hydraulic geometry relations for depth and discharge. The first step in the GIS was to extract a series of cross-section profiles at 400 m intervals from the 1999 bed surface model. Over the entire gravel reach, a total of 78 profiles were created; the number in each morphologic reach varied from three to nine based on reach length. Extracted cross-sections were then re-imported to the GIS and Table II. Width and depth coefficients/exponents for hydraulic geometry equations. Results for hydraulic geometry calculations based on these coefficients and exponents are in metres Reach Width coefficient Width exponent Depth coefficient Depth exponent

8 1272 Y. Martin and D. Ham intersected with a horizontal line corresponding to the elevation of bankfull discharge to generate a polygon data layer. Bankfull elevations were obtained by superimposing Lidar points on a morphologic overlay map of recently established vegetation (determined as bare bar surfaces in 1983 and continuously vegetated surfaces by 1999). It is assumed that these surfaces remain dry or are just inundated at bankfull stage. A second-order polynomial was fitted to the data (elevation at distance downstream) to interpolate the bankfull elevation at any cross-section location. For each cross-section, area and surface width of polygons below bankfull elevation were tabulated (areas above this elevation correspond to islands and occasional high bar surfaces). Dividing the total area of all polygons by total width yields a value of mean depth. The exercise was repeated by subtracting 1-m increments from the bankfull elevation until five depth values were recorded. For reaches in which the range in flow depth was found to be less than 4m for one or more cross-sections, depth increments were subtracted in 0 5 m increments. Within each morphologic reach, all width and depth values were averaged for each depth increment and a power function was calculated to relate with the width discharge equations. Coefficients and exponents for hydraulic geometry equations for mean channel depth are given in Table II. Scatter in variations for the value of the depth exponent provides no obvious relation to differences in morphology or relative bank strength. However, adding these values to the width exponent demonstrates the degree to which increases in discharge must be accommodated by higher flow velocity. For most reaches, the velocity exponent falls within the range , hence appear reasonable given known values at the Mission and Agassiz gauges of 0 58 and 0 54 respectively (McLean, 1990). In contrast, the velocity exponent exceeds 0 75 along the lower three reaches and approaches unity in Reach 1. There is no physical basis to suggest that these figures are credible, particularly given our adoption of a constant slope for all flows. Indeed, it appears that the weakness in our approach to estimate reach-averaged depth exponents for this section of channel produces the anomaly. While we recognize that sediment transport estimates would be more reliable if we incorporated known hydraulic geometry relations from the Mission gauge, the availability of this information is fortuitous such information may not always be available. Therefore, we have instead chosen to apply the reported values in Table II for consistency along the entire gravel reach. Insofar as these reaches are at the downstream limit of the study area, sediment transport estimates for upstream reaches are unaffected. Our results from the hydraulic geometry calculations give a negative exponent for depth in Reach 10 (the value is close to zero). This implies that flow depth declines with discharge, which of course is not physically reasonable. Similarly, the value of the exponent is small for several reaches (<0 1). This occurs because our calculations are based on mean depth for the entire reach, not for a single cross-section. Although maximum depth should positively increase with discharge, the hydraulic geometry calculations for depth appear to fail in reaches with a prominent deep flow channel. Here, water surface width increases slowly as discharge and depth increase, but width increases rapidly once the flow begins to cover exposed barforms. This demonstrates a problem with using mean values for sediment transport calculations, at least for some channel morphologies. However, the use of a mean value may be a reasonable compromise for wandering channels since maximum depths represent only a small fraction of the entire active bed. Alternatively, it might be reasonable to mask out those additional areas of channel where we do not expect much transport to occur, and limit the analysis to a smaller active channel zone with a larger mean depth. This should correspondingly modify transport rates, especially for those formulae that are sensitive to changes in depth. Gradient Channel gradient was computed for each morphologic reach from a water surface profile of the river collected by Public Works and Government Services Canada in 1999 at a discharge of approximately 7000 m 3 s 1 (Figure 4). The water surface parallels the average bed surface at the reach scale, but also masks local variations in bed topography (Richards, 1982; Rice and Church, 2001). Along the bed of alluvial channels, these variations correspond to riffle pool topography. Over the complete gravel reach, the river surface forms a common concave-up profile that flattens downstream. In total, the water surface drops 24 m in elevation along a sinuous thalweg that measures 74 km in length. For each reach, gradient was calculated from the first derivative of the best-fit equation for the water surface profile (R 2 of 0 98): S = exp ( distance) (10) with units of distance in an upstream direction from Mission (0 m) in metres. Slope remains relatively constant along the first 20 km extending from the upstream limit of the study area, where a single-thread meandering morphology is maintained at all flows. The rate of decline lessens downstream as channel width generally increases within the dominantly wandering section of the study area. The flattening of the gradient roughly 20 km upstream of Mission reflects the initial transition zone where sand becomes the dominant bed material.

9 Testing bedload transport formulae 1273 Figure 4. Water surface data and best-fit exponential equation at Hope. Figure 5. D 50 and D 90 values along the study reach. Grain size Data on grain size were taken from 70 field samples collected in 1983 and McLean (1990) collected bulk samples in 1983 along bar heads and flanks along the entire gravel reach. The more recent samples were collected by the Fraser River Research Group by pooling nine bulk samples from upper-, mid- and lower-bar sites along the reach (yielding three measurements for every bar). Church et al. (2001) found no systematic difference between the data collected for different years, so all data were combined and plotted as grain size versus distance upstream to produce the desired grain size estimates for each reach (Figure 5): D 50 = ( distance)/ D 90 = ( distance)/ (11A) (11B) where grain sizes are in metres, and distance in an upstream direction from Mission is in kilometres (D 50 R 2 of 0 53; D 90 R 2 of 0 50). The pooled data were also used to estimate the division between sand and gravel for each morphologic reach and are used in the sediment budget calculations to present transport estimates of gravel only. Bedload Rating Curves Rating curves versus field measurements Field measurements of bedload were undertaken at Mission (Reach 1 in this study) in 1966, 1967, 1968, 1972, 1974 and 1979 (see McLean et al. (1999) for details on measurement procedures). Rating curves for bedload transport

10 1274 Y. Martin and D. Ham Figure 6. Rating curves for bedload equations versus field measurements at Mission. formulae at Mission are compared to field measurements in Figure 6. Input variables for the formulae are obtained according to procedures outlined in the previous section. Calculations are performed for discharges ranging from 500 to m 3 s 1 in 500 m 3 s 1 increments. Field measurements exhibit a significant degree of scatter, often up to an order of magnitude. Field data indicate that initiation of bedload transport occurs between 1000 and 2000 m 3 s 1, which is in close agreement with results of the original Bagnold formula (and the rational version, which uses the same equation for threshold transport). The MPM formula predicts the onset of bedload transport at a discharge of about 6000 m 3 s 1, a much higher discharge than field data indicate. The stream power formula incorporates no threshold for transport. As previously mentioned, the threshold for significant gravel transport occurs at approximately 5000 m 3 s 1. Bedload transport rates measured at this discharge range from about 70 to 700 tonnes/day; only transport predictions for the original Bagnold formula lie within this range of transport values. At a discharge of m 3 s 1, measured transport rates range from 300 to 3000 tonnes day 1 ; again, only results for the original Bagnold formula fall within this range of values. Sensitivity analysis of input variables Since input variables may be defined in a variety of ways (e.g. point estimates versus reach-averaged values), or using completely different approaches from those adopted herein, sensitivity analyses are undertaken to explore how such variations may affect transport predictions. Rating curves are recalculated for an extended range of values for grain size, slope, width and depth. Results are shown for two representative reaches, Reach 1 and Reach 7. The former is a confined channel reach, with calculated widths showing relatively constant values of about 500 m and depth values of approximately 10 m. Widths for Reach 7 are more variable, ranging from 500 m at lower discharges through to >1100 m at higher discharges; depths range from about 3 to 4 m over this same discharge range. Grain size. Grain size is a key variable in the calculation of threshold discharge for transport in the Bagnold-type and MPM formulae and is also found within the main part of these equations. Values of D 50 and D 90 for Reach 1, based on best-fit lines, are respectively 4 5 mm and 15 2 mm. Results are shown for these best-fit values of grain size and for grain sizes ±2 mm of best-fit values (Figure 7A). For Reach 7, we implemented grain size values greater and less than 1 standard error from the best-fit values (Figure 7B; D 50 SE is 4 5 mm; D 90 SE is 16 mm). For Reach 1, threshold discharges for the Bagnold-type formulae decrease by about 1000 m 3 s 1, from an original value of 2000 m 3 s 1, when grain size is decreased; an increase in grain size increases threshold discharge by about 500 m 3 s 1. Threshold discharges for the MPM formula change by ±1500 m 3 s 1 when grain sizes are increased and

11 Testing bedload transport formulae 1275 Figure 7. Sensitivity analysis for grain size: (A) Reach 1; (B) Reach 7. Figure 8. Sensitivity analysis for slope: (A) Reach 1; (B) Reach 7. decreased. Threshold discharge for the Bagnold-type formulae is more sensitive at Reach 7, with its values changing by 3000 m 3 s 1 and m 3 s 1 for the decrease and increase in grain size respectively. Changes in threshold discharge for the MPM formula are approximately the same for the two reaches. For Reach 1, the range between transport estimates using increased and decreased values of grain size varies from well below an order of magnitude at higher discharges, while for lower discharges the range is approximately an order of magnitude. The original Bagnold formula demonstrates the greatest sensitivity to grain size. Overall, transport estimates are more sensitive to grain size at Reach 7 than Reach 1. Slope. Slope gradients are decreased and increased by 10 per cent and 20 per cent from the best-fit values (Figure 8). Effects of slope changes on threshold discharges are relatively low for both Reach 1 and Reach 7. Changes in slope modify values of stream power or shear stress for a given discharge, although slope is not contained within the threshold terms themselves. The initiation of transport for the Bagnold-type formulae is modified by up to several thousand cubic metres per second; changes are minimal for the MPM formula. Sensitivity of transport estimates to slope is overall greater at Reach 7 than Reach 1 for the Bagnold-type formulae. Maximum variation between the highest and lowest transport estimates approaches an order of magnitude at Reach 7 in some instances. The MPM and stream power formulae are not as sensitive to changes in slope as the Bagnold-type formulae. Width. Values of width are decreased and increased by amounts equal to 25 per cent and 50 per cent of the original values (Figure 9). The discharge at which transport first occurs varies by relatively low amounts (generally <<1000 m 3 s 1 ) at Reach 1 for the Bagnold-type formulae; changes for the MPM formula are up to ±3000 m 3 s 1. When

12 1276 Y. Martin and D. Ham Figure 9. Sensitivity analysis for width: (A) Reach 1; (B) Reach 7. Figure 10. Sensitivity analysis for depth: (A) Reach 1; (B) Reach 7. width is increased by 50 per cent at Reach 7, this value increases by up to 4000 m 3 s 1 for the Bagnold-type formulae and 8000 m 3 s 1 for the MPM formula. The MPM formula is more sensitive to width than the Bagnold-type formulae at Reach 1. Sensitivity to width change is greater at Reach 7 than Reach 1 for all formulae. Depth. Depths are decreased and increased by 25 per cent and 50 per cent of the original values (Figure 10). The discharge at which bedload transport begins to occur at Reach 1 does not change for the Bagnold-type formulae and changes by up to ± several thousand cubic metres per second for the MPM formula. At Reach 7, this value changes by up to ±1000 m 3 s 1 for the Bagnold formulae and up to ± several thousand cubic metres per second for the MPM formula. The MPM formula is more sensitive to changes in depth than the Bagnold formulae for both Reach 1 and Reach 7. Gravel Transport Predictions Because morphologic estimates represent gravel transport only, this must be accounted for in calculations. Hence, transport rates are calculated only for discharges exceeding the threshold for significant gravel transport of 5000 m 3 s 1. Average annual gravel transport rates are calculated for the four bedload formulae over the study period Transport estimates are compared to morphologic transport estimates in Figure 11A. Calculated/observed (C/O) values for the four transport formulae are given in Table III.

13 Testing bedload transport formulae 1277 Figure 11. (A) Gravel transport rate (average annual) along the Fraser River (B) Close-up of Bagnold-type formulae. Table III. Calculated/observed values for gravel transport estimates. Average annual transport rates based on the morphologic approach are shown in the second column Reach Transport rates (m 3 a 1 ) Original Bagnold Rational Bagnold MPM Stream Power N/A N/A N/A N/A

14 1278 Y. Martin and D. Ham Transport patterns calculated using the four bedload formulae differ notably from one another. The original Bagnold formula replicates the overall decrease in transport from Reach 13 to Reach 4 found in the field data, and also replicates the increase from Reach 4 to Reach 1 (Figure 11B is scaled to allow for closer examination of the Bagnold results). This formula underestimates gravel transport rates along the study reach, in most cases by less than an order of magnitude. The rational Bagnold formula does not replicate the field transport patterns as well as the original formula and, moreover, it underestimates the magnitude of transport rates by a greater amount (by nearly two orders of magnitude in many cases), making the original Bagnold formula the preferred version. The MPM formula overestimates transport rates significantly, although most C/O values indicate that transport predictions are within an order of magnitude of field-derived values. The stream power correlation shows a steadily decreasing transport rate in a downstream direction, a pattern in large part produced by the downstream decrease in gradients found along the study reach, reflecting that gradient is a key variable in this equation. This equation does not capture the more complex downstream pattern evident in the field-based transport estimates. Furthermore, gravel transport is overpredicted by a significant amount. Apart from the Bagnold equation, the bedload formulae predict significant gravel transport past Mission even though field evidence indicates that the bed is mainly sand, and no gravel moves downstream past Mission. Practically, there is no physical means to examine the bed here there are no emergent bars in this essentially confined section of river, and water depths are very high up to 25 m in places. In fact, there may be a small volume of gravel transported along the bed but a general lack of gravel on downstream bars means the volume must be modest. The results show that only the Bagnold formula actually predicts zero transport. This is a reflection of the relative performances/inadequacies of the different formulae, since they should be predicting zero based on our limited field knowledge of the study area. Effects of changes in bed structure were explored for the original Bagnold and MPM formulae by modifying the value of the Shields parameter found in these equations. Critical shear stress for entrainment is defined by: τ cr = cg(ρ s ρ)d (12) where τ cr is critical shear stress for entrainment, c is Shields parameter, g is gravity, ρ s is density of sediment, ρ is density of water and D is grain size. Although the Bagnold and MPM formulae define the Shields parameter as constant (0 04 and respectively), in reality the parameter is expected to vary with bed structure and composition. In fact, Shields (1936) originally defined a value for this coefficient of Since that time, values ranging from as low as 0 01 to greater than 0 1 have been reported in the literature (Church, 1978; Williams, 1983), with often being adopted as a representative value (e.g. Komar, 1988). Gravel transport calculations are recomputed using a range of values for the Shields parameter between 0 02 and 0 1 to assess the sensitivity of equations to this parameter (Figure 12). When the Shields parameter is decreased to 0 02, a situation that implies a looser bed structure, results for the Bagnold formula converge towards field transport values. If the Shields parameter is increased, implying a more resistant bed structure, results for the Bagnold formula become unusually small. When the value of the Shields parameter is decreased for the MPM formula, transport estimates increase even higher than their already overall high values. An increase in the Shields parameter to values of 0 06 to 0 08 brings some of the transport estimates more in line with field transport rates; however, a value of zero transport is now found for six of the 13 reaches, which is not a realistic result. Cumulative Distributions of Bedload Transport Cumulative distributions of bedload transport with discharge provide information regarding the importance of various discharges in transporting bedload. Cumulative percentages of bedload transport over the entire study period are calculated for the four formulae (Figure 13). For presentation purposes, results are shown for two reaches, Reach 1 and Reach 7. Results for Reach 1 show markedly different curves for the various formulae. The rating curve for the stream power formula, the only equation with no threshold for transport, results in a convex-upward curve; the strong contribution of lower discharges to total bedload transport is very evident. The Bagnold-type formulae have a much lower threshold for bedload transport than the MPM formula. The curves for the Bagnold-type and MPM formulae show upward concavity at low discharges, indicating that as discharge increases, significantly greater transport occurs. However, the curves level off as discharge increases to high values and show a convex-upward profile, reflecting the decreasing contribution of bedload transport by very high discharges. Results for Reach 7 are similar to Reach 1 for the MPM and stream power formulae. The curves for the Bagnoldtype formulae shift dramatically to the right of the plot at Reach 7. The discharge at which transport is initiated changes significantly, by about 4000 m 3 s 1.

15 Testing bedload transport formulae 1279 Figure 12. (A) Sensitivity of Bagnold formula to Shields parameter. (B) Sensitivity of MPM formula to Shields parameter. Figure 13. Cumulative bedload transport: (A) Reach 1; (B) Reach 7.

16 1280 Y. Martin and D. Ham Discussion and Conclusions Bedload transport predictions are frequently made in both applied and theoretical studies. Nonetheless, the lack of field measurements of bedload transport for large rivers means that uncertainty associated with such estimates remains largely unquantified. The morphologic approach for the estimation of gravel transport rates provides a means of acquiring transport estimates over larger scales. The morphologic transport estimates for Fraser River provide a rare opportunity to test bedload formulae for large rivers. The extensive data compilation available for this river allows input variables to be defined in a rigorous manner; hence, we believe our transport predictions are as reliable as are likely to be found for a river of this magnitude when adopting a generalized approach, as outlined herein. We introduce a means for developing at-a-station hydraulic geometry relations for width and depth along our study reach. The approach to define width equations relies on the availability of repeat aerial photography for dates of known discharges. Because such data may be available for other large rivers, this approach may be replicable in many instances. The approach followed to define hydraulic geometry equations for depth is more complex and relies on bathymetric survey data, which are less likely to be available for many rivers and therefore represent a limiting factor in the replicability of the procedure. Alternative approaches, such as the use of cross-section survey data or the Manning equation, may be required to define depth in other studies. This would likely decrease the strength of transport predictions. Grain size and slope data were collected according to standard procedures and are considered replicable for other large rivers. It is acknowledged that the availability and quality of input data for other rivers will, in large part, determine the predictive capabilities of bedload transport equations. The sensitivity analysis undertaken herein allows for exploration of this issue. Field measurements of bedload transport at Mission demonstrate scatter of up to an order of magnitude across most of the discharge range for Fraser River. Many factors may contribute to this scatter, such as measurement error and changes over time and space in flow conditions, bed structure and sediment movement. Therefore, it cannot be expected that bedload equations will always compare closely with field data. Comparison of transport measurements at Mission with estimates for the four formulae shows that results for the original Bagnold formula demonstrate the greatest correspondence, in terms of overall accuracy, over the largest range of discharges. The Bagnold-type formulae also provide a good estimate of the discharge at which bedload transport is initiated. Results for the MPM formula show particularly significant divergence from field measurements above m 3 s 1, which corresponds to approximately a 5-year discharge. In addition to significant uncertainty associated with bedload formulae themselves, the strength of transport predictions may be affected by a lack of accuracy and/or precision in the values of key input variables. Sensitivity analyses of transport estimates to input variables provide information on how such uncertainties may affect predictions. The Bagnoldtype formulae are more sensitive than the MPM formula to changes in grain size and slope, but the MPM formula is more sensitive to changes in width and depth. Depending on which input variables are associated with greatest uncertainty for a particular study, a particular formula may be preferred over another to reduce uncertainty in transport estimates. Average annual transport estimates for the four formulae over the study period differ considerably from one another and with the morphologic transport estimates. Overall, the original Bagnold formula and the MPM formula show the most promising results but have characteristic weaknesses. Both of these formulae generally capture the downstream pattern of transport rates displayed in the field data. The original Bagnold formula underestimates transport by within an order of magnitude in most cases, while the MPM formula overestimates transport within an order of magnitude in most cases. Although other sources of error in transport predictions may be more important, varying the value of the Shields parameter provides interesting insights into the possibility that transport predictions can thereby become more compatible with field measurements. Arbitrary adjustments of the Shields parameter are, of course, not advisable. However, this analysis highlights the possible effects that constant values of parameters embedded within transport relations can potentially have on results. For example, there has been periodic gravel mining along our study reach, which would decrease the strength of the bed structure and could justify some lowering of the Shields parameter in particular locations. It was shown that with what may be plausible adjustments to the Shields parameter, significant improvements in transport predictions can occur. The cumulative bedload transport curve for the stream power formula, the only equation with no threshold for transport, results in a convex-upward curve; the strong contribution of lower discharges to total bedload transport is evident and is not an acceptable result. The Bagnold-type formulae have a much lower and more realistic threshold for bedload transport than the MPM formula at Reach 1. The MPM curve shows a greater correspondence with the Bagnold-type curves for Reach 7. Our approach has significant limitations, as it does not account for details of process mechanisms and spatial/ temporal variability of flow, channel morphology and bed conditions found in natural rivers, which have been shown