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Estimating the bankfull velocity and discharge for rivers using remotely sensed river morphology information
Journal of Hydrology (2007) 341, 144– 155
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/jhydrol
Estimating the bankfull velocity and discharge for rivers using remotely sensed river morphology information David M. Bjerklie
*
Connecticut Water Science Center, US Geological Survey, 101 Pitkin Street, East Hartford, CT 06108, United States Received 10 July 2006; received in revised form 27 March 2007; accepted 11 April 2007
KEYWORDS Hydrology; Bankfull velocity of rivers; Bankfull discharge of rivers; River flow; River meander; Flow resistance and meandering in rivers; Remote sensing of rivers
A method to estimate the bankfull velocity and discharge in rivers that uses the morphological variables of the river channel, including bankfull width, channel slope, and meander length was developed and tested. Because these variables can be measured remotely from topographic and river alignment information derived from aerial photos and satellite imagery, it is possible that the bankfull state of flow can be estimated for rivers entirely from remotely-sensed information. Defining the bankfull hydraulics of rivers would also provide a reference condition for remote tracking of dynamic variables including width, stage, and slope, and for quantifying relative change in flow conditions of rivers over large regions. This could provide a more efficient method to inventory and quantify river hydraulic attributes and dynamics. Published by Elsevier B.V.
Summary
Introduction Continuous ground measurement of water-surface height (stage) and of discharge in large numbers of rivers (the monitoring network) by the establishment of stage-discharge ratings can be expensive, especially in remote regions and in areas with difficult access. Additionally, ground-based measurements of discharge at relatively high in-bank flows that correspond to the channel-forming or near channelforming discharge can be difficult and dangerous. Definition
* Tel.: +1 860 291 6770; fax: +1 860 291 6799. E-mail address: [email protected]. 0022-1694/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.jhydrol.2007.04.011
of the bankfull hydraulics of rivers would provide a baseline condition for quantification of large-scale regional changes in river morphology through remote tracking of variables including width, stage, and slope, and serve as a reference for remote tracking of hydraulic dynamics. This ability could contribute to the development of a method to use remotesensing techniques to monitor the discharge of rivers on a continuous basis and to fill in spatial and temporal gaps in discharge records using archived and real-time remote observations. Additionally, estimating flow velocity independent of an estimate of the flow resistance would provide a much needed alternative method to estimate energy loss and flow resistance in rivers, which is difficult to estimate otherwise (HEC, 1986).
Estimating the bankfull velocity and discharge Camporeale et al. (2005) suggested that a characteristic meander length could be useful in evaluating river hydraulic conditions as an alternative to depth-based relations because the meander length can be easily measured from a variety of satellite imagers for many rivers (Bjerklie et al., 2003). Alsdorf et al. (2005) have suggested that satellitebased remote sensing of rivers using interferometric synthetic aperture radar can achieve reasonably accurate measures of the water-surface slope, width, and elevation (stage). Various airborne instruments also can provide these measures, thus the potential exists for estimating flow conditions in rivers on the basis of data/information obtained via satellite or aerial remote sensors, provided a reasonable estimate of the depth and flow resistance can be made. Alternatively, if a reference stream velocity could be measured independently, the depth and flow resistance associated with the reference velocity could be estimated, and general scaling functions could then be applied to track the river’s flow at varying stage. Except in specific clear water situations (Legleiter and Roberts, 2005) or using assumptions of a regular geometric cross-sectional shape (Fonstad and Marcus, 2005), remotesensing capabilities may not provide sufficient information to estimate depth or flow resistance directly, a fundamental pre-requisite to near real-time tracking of river discharge from satellites is the development of relations between observed variables and the unknown quantities of depth and velocity (or flow resistance). Development of a general relation between the planform (as shown on a map) channel shape, as indicated by the meander length, and velocity would therefore maximize the use of observed information in an effort to reduce the number of unknown hydraulic variables. General relations between various measures of river morphology, including the channel width, slope, and meander length and pattern, have been associated with the channel-forming discharge, typically thought to be that discharge at or near the bankfull flow condition (Leopold et al., 1964; Leliavsky, 1966; Carlston, 1965). Thus, the potential exists for the channel pattern, which is a variable that can be observed from remote platforms, to be a useful predictor of the hydraulic state of the channel at a reference flow condition associated with the bankfull flow, at least for single thread meandering rivers that are unaffected by backwater. The concept of the channel-forming discharge is based on the understanding that the shape of a river channel tends to be relatively stable over time, and is associated with frequently occurring high-flow events that have sufficient stream power to establish and maintain the channel geometry through bank erosion and bed-load movement, and that the size of these flow events are coincident with the channel capacity (i.e. the bankfull condition). Williams (1978) has shown that the recurrence interval of the bankfull discharge is highly variable, although it is often assumed to be at or near the 1.5–2.3-year recurrence interval flood. In this study, the flood frequency that relates to the bankfull flood is not relevant other than for comparison and calibration of derived velocity relations. As such, it is assumed that, on average, the bankfull discharge is approximated by the mean annual flood (Leopold et al., 1964), and serves pri-
145 marily as a reference flow condition that is at or near the true bankfull discharge and geometry. This paper develops an equation that uses the meander length to estimate the mean velocity of a river and equations to estimate the flow depth at the channel forming (bankfull) discharge that is independent of a specific flowresistance variable. These estimates can then be used to derive the reference bankfull channel dimensions and hydraulic conditions from remotely sensed information. The various equations and hydraulic relations presented in this paper generally assume gradually varied steady flow conditions confined to flow within the banks, and therefore define relatively constant hydraulic characteristics for the channel within the time frame of interest. Additionally, for the applications presented in this paper, the meander length can be more broadly defined as a bend length, construed here to be twice the length between two inflection points, and therefore need not constitute a classic meandering pattern.
Methods – hydraulic relations Bankfull channel velocity On the basis of geomorphologic considerations, an approach to estimating the bankfull mean water velocity of a river is developed using a general relation between the meander length and the ratio of bankfull mean depth (Yb) to bankfull flow resistance (fb). Camporeale et al. (2005) derives the following relation from numerical investigations of meander behavior: k ¼ 13:4
Yb ; fb
ð1Þ
where k is the spatially-averaged meander wavelength and fb is the Darcy–Weisbach friction factor with the subscript b denoting that it is associated with the hydraulic conditions at the channel forming or bankfull discharge. This relation compares well with a geomorphologic relation given by Jansen et al. (1979): k ¼ 14
Yb : fb
ð2Þ
Coupling this relation with the Chezy equation as presented in Henderson (1966) applied to the bankfull flow (given as): rffiffiffiffiffiffiffiffiffiffi gY b S ; ð3Þ Vb ¼ a where Vb = mean water velocity, S = the friction slope (approximated by the water surface slope for uniform flow), a is the dimensionless resistance factor, and g is the acceleration due to gravity; and realizing that fb = 8a (Chow, 1959), the following equation is derived to estimate the bankfull velocity (Vb) as a function of the meander length and the slope, (thus replacing the depth and flow resistance factor): pffiffiffiffiffiffiffiffi V b ¼ 0:8 gkS: ð4Þ A similar relation can be derived from force-balance considerations applied over a characteristic channel length where
146
D.M. Bjerklie
the resisting force is equal to the driving gravitational force for an open channel. Assuming that the mass of water, M, flowing down the slope would reach an equilibrium (constant) velocity V over a time T, such that the force of the flowing water is in balance with the resisting force within the length L defined by V · T, then we can define two equations that describe the state of the flow. The first assumes that the impulse defined by the resisting force, F, over time (T) is the difference between the potential velocity (Vp) that would occur if there were no resistance (given as the fall distance of the mass of water times the gravitational acceleration, g), and the resultant equilibrium velocity such that: FT ¼ MðV p VÞ;
ð5Þ
and via Torricelli’s theorem (Bueche, 1972) pffiffiffiffiffiffiffiffiffiffiffiffi V p ¼ 2gDH;
ð6Þ
where DH is the change in elevation over the reach. Second, the equilibrium velocity can be defined as: 1=2ðV þ V 0 Þ ¼
L ; T
ð7Þ
where V0 is the velocity at the head of the reach, in this case equal to V, and therefore L T¼ : V
Substituting Eq. (6) for Vp and rearranging yields: pffiffiffiffiffiffiffiffiffiffiffiffi FL : V ¼ 2gDH MV
Thus the equilibrium velocity is defined as a net head drop, which would, in principle, be defined as the equilibrium slope, S, times a pertinent downstream length variable, L, over which the energy loss takes place (referred to here as the resistance length) such that: pffiffiffiffiffiffiffiffiffiffi V ¼ 2gSL: ð14Þ
ð8Þ
ð9Þ
Y L¼4 : f
ð10Þ
An interesting aspect of Eqs. (9) and (10) is that the term FL is an expression of the work done by the resisting force to slow down the water that would be accelerated by the fall distance DH. The resisting force, F, can also be defined as M times a deceleration term, i, such that the second term in Eq. (10) is iL/V, and because V = L/T, the term reduces further to iT, which is a velocity of resistance (Vr). Thus the equilibrium velocity can be described by: V ¼ V p V r:
where Hp is the potential head and Hr is the resistance head. Rearranging Eq. (12), the equilibrium velocity can be expressed as a function of total head drop minus resistance head: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð13Þ V ¼ 2gðHp Hr Þ:
Relating Eq. (14) to the Chezy and Darcy Weisbach equations (Chow, 1959), the following relation between the resistance length and the ratio of depth to the total flow resistance is obtained:
Substituting (8) into (5) and rearranging gives: FL ¼ MðV p VÞV:
ply to a channel reach with constant slope and depth such that the equilibrium velocity is attained within a time interval (T) and a distance (L) over which work is done to resist the downstream gravitational acceleration. Thus, the energy equation reduces to the difference between the potential and resisting velocity heads. Eq. (11) therefore becomes: ! V 2p V 2r V2 ¼ ¼ ðHp Hr Þ; ð12Þ 2g 2g 2g
ð11Þ
It is possible to conceive the relation described by Eq. (11) in terms of an energy difference between a potential and actual state at a specific location. With this conception, the resultant or actual energy state would be the difference between the potential energy and the total energy loss represented by the velocity of resistance. Because the resulting velocity is the difference between a potential velocity and a resisting velocity, the total energy loss is conceptually equivalent to the total resistance to flow in the context of a uniform flow formula such as the Chezy equation. In uniform flow the total resistance to flow and the total energy loss are equivalent (the friction slope is equal to the energy slope) (Chow, 1959). Uniform flow as described by the Chezy (or Manning) equation uses the flow depth as the pertinent length variable, and a reach length over which uniform flow is established is not specified. In this paper, the concept of uniform flow is understood to ap-
ð15Þ
The length scale presented in Eq. (15) is fully consistent with and equivalent to the downstream decay length referred to by Edwards and Smith (2002) in their analysis of the dynamics of river meanders. This length, as discussed by Edwards and Smith (2002) is the length required for shear stresses perpendicular to the downstream velocity to recover from changes in channel curvature, and therefore sets the scale for the meandering wavelength and the distance between channel inflection points (1/2 of the meander bend length, which is the distance along the channel through the meander). Thus, the resistance length, L, measured along the channel, would be expected to be strongly related to some fraction of the meander wavelength. The bend length, as defined by Williams (1986), is 1/2 of the along-channel distance through a meander wavelength (measured along the valley). The full meander bend length, or meander length, is defined here as twice the bend length. This measure will be used for the analysis because it can be easily measured and digitized and is consistent with the idea of the decay length as discussed by Edwards and Smith (2002). Where meander wavelength is reported (which is usually the case), the meander bend length (or meander length, k*) is estimated by multiplying the wavelength (l) by the sinuosity (P), such that k* = kP. According to Edwards and Smith (2002), the decay length has meaning for both straight and meandering channels, and therefore is purely a function of energy losses through a reach with secondary currents. Thus, it is hypothesized that the resistance length concept should apply to any openchannel flow condition within any morphologic structure,
Estimating the bankfull velocity and discharge including the self-formed alluvial river. It might be expected, however, that the specific fraction of the meander length that constitutes the resistance or decay length is variable, depending on morphologic features of the channel. Fig. 1 illustrates the definitions of meander wavelength and meander length used in this paper. Few contemporaneous data sets of bankfull river hydraulic information combining river cross-section, channel slope, velocity, and meander characteristics that can be used to evaluate the resistance length could be found in the literature. A small set of bankfull flow data representing a range of natural rivers is available from Church and Rood (1983) (Table 1). These data are compiled within a much larger catalogue of alluvial channel regime data. The catalogue consists of rivers located primarily in the United States, the United Kingdom, and Canada. The subset of these data used in this study included those rivers for which associated discharge, cross-sectional area (width and depth), channel slope, meander wavelength, and sinuosity information were available. In some cases, more than one bankfull discharge estimate was associated with the channel hydraulic information depending on the assumption of what event constitutes bankfull. In these cases, the average of the estimated values was used. The data obtained from Church and Rood (1983) was supplemented with a small set of river data derived by crossreferencing bankfull cross-section and discharge data with meander length data for the same rivers presented in two separate reports (Williams, 1978; Williams, 1986). Due to the paucity of the river data (N = 22), flume data developed by Leopold et al. (1960) (Table 1) in a study designed to
645000 Meander Length Measured Along Channel Through the Wavelength
evaluate the effect of meander bends on total energy loss and flow resistance were also used for the analysis. The flume data consists of fixed-bed channels formed to include regular meander patterns with fixed-channel slopes. Variable flow was introduced into each channel and the equilibrium hydraulic conditions, including width, depth, and velocity, were measured. Because the experimental flume channels were fixed, each flow condition in effect represents a unique equilibrium condition comparable to a bankfull discharge that would be associated with the formation of meanders in rivers. More extensive data sets of bankfull river hydraulic data are available (Church and Rood, 1983; Williams, 1978; Van den Berg, 1995), some of which include information on sinuosity and meander radius of curvature; however, information on the meander length for the discharge locations is generally not available. In the interest of consistency, we have used only these data sets that include all of the desired hydraulic information. Although it is anticipated that the flume experiments should reflect the conditions of rivers and open channels, it must be recognized that significant scale differences exist between the flume and actual rivers and that these factors might introduce non-linear comparative results. In addition to potential issues of scale, the flume experiments do not include many forms of flow resistance and energy loss that would be present in natural settings and in larger rivers. These would include, among others, the effects of bedload and sediment movement, channel irregularities, and obstructions. Given the issues of scale and differences between the flume and natural settings, it is expected that the relation between the resistance length and the meander length would be variable, especially between the river and flume data, depending on the specific energy loss conditions in the channel. Thus, it is assumed that the meander length for each flow may not represent the resistance length established for the equilibrium condition, and therefore the following general equation is hypothesized: rffiffiffiffiffiffiffiffiffiffiffiffi 2gSk Vb ¼ m
644500
Y Coordinate
147
644000 Meander Wavelength 643500 Meander Wavelength 643000 995000 996000 997000 998000 999000 1000000 1001000
X Coordinate
Figure 1 Plot of a meandering river reach in an arbitrary X–Y coordinate system showing how the meander wavelength and meander bend length are defined. The wavelength is the straight line distance between consecutive high or low points of the curve (shown by the arrows) and the bend length, referred to in this paper as the meander length, is the along channel distance between the same points. The distance between the coordinate points on the channel path is 100 feet in this example. As can be seen, the defined meander lengths can be highly variable along the channel, and longer distances between points can obscure shorter meanders.
ð16Þ
where m is an arbitrary fraction of the meander length. The use of the variable m, which is comparable to the Manning n in the Manning resistance equation, allows for the downstream meander length, which is observable from remote platforms, to be used to help define the flow conditions as opposed to the depth variable, which is the pertinent length variable in the Chezy and Manning equations. Treating the two data sets independently, a constant value for m was determined by adjusting its value until the slope of the line between the right and left sides of Eq. (16) closely approximates 1. For the river data, m is found to be approximately 8 and for the flume data, approximately 1.2. This indicates that the general flow resistance and energy loss in a reach is established through a reach length on the order of 0.13–0.83 of a meander length, which brackets an expected distance of 0.3 times the meander wavelength (as opposed to the meander length) determined by comparing Eq. (1) with Eq. (15). The wide range of values for m when using the meander length is in part due to the effect of multiplying by the sinuosity. Interestingly, if m is determined for all of the data (both the river
148
Table 1
Hydraulic data and velocity estimates
River reach
Depth, Y (m)
Slope, S (m/m)
Meander length, k (m)
Observed velocity, V (m/s)
56.0 54.0 61.7
2.43 2.74 2.87
0.00011 0.00021 0.00033
1920 1040 1224
0.68 0.85 1.26
42.0 275.0 104.3 120.7 33.0 73.0 72.0
3.77 5.44 3.24 2.81 1.68 1.20 2.01
0.0002 0.00052 0.00094 0.00084 0.00055 0.0033 0.0018
1360 4550 3230 5760 1344 864 2037
121.0 186.0 67.7 32.7 103.7 15.0 54.5
2.14 3.09 1.32 1.45 1.57 0.87 0.87
0.0012 0.00044 0.0037 0.0008 0.002 0.0018 0.0042
67.0
2.43
23.5 13.7 57.9 36.6 0.210 0.210 0.210 0.210 0.210 0.211 0.211 0.210 0.210 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.255
0.7 0.7 1.78 3.38 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027
Optimized length fraction, M
Estimated velocity constant m, V (m/s)
Estimated velocity m from Eq. (17), V (m/s)
Estimated velocity from Eq. (18), V (m/s)
9.0 5.9 5.0
0.51 0.52 0.70
0.87 0.88 1.07
0.91 0.92 1.11
0.94 2.41 3.21 2.25 1.08 1.60 1.51
6.0 8.0 5.8 18.8 12.4 21.9 31.8
0.58 1.70 1.93 2.44 0.95 1.87 2.12
0.94 1.88 2.03 2.36 1.29 1.99 2.16
0.98 1.95 2.10 2.44 1.34 2.03 2.21
3150 3770 1921 816 2880 600 1560
1.86 2.10 2.21 1.29 1.35 1.56 2.25
21.6 7.4 28.6 7.7 61.7 8.7 25.4
2.15 1.43 2.95 0.89 2.66 1.15 2.83
2.18 1.67 2.67 1.24 2.49 1.46 2.60
2.24 1.74 2.71 1.28 2.55 1.50 2.64
0.0016
1921
1.64
22.4
1.94
2.04
2.09
0.0036 0.0025 0.0008 0.003 0.00035 0.00097 0.00166 0.00246 0.00337 0.00448 0.00559 0.00723 0.00853 0.00036 0.00097 0.00156 0.00239 0.00319 0.00448 0.00559 0.00723 0.00853
334 119 1326 814 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25
1.67 1.27 2.19 2.95 0.114 0.167 0.208 0.248 0.281 0.314 0.354 0.396 0.430 0.124 0.175 0.232 0.276 0.317 0.354 0.390 0.439 0.466
8.5 3.6 4.3 5.5 0.7 0.8 0.9 1.0 1.0 1.1 1.1 1.1 1.1 0.6 0.8 0.7 0.8 0.8 0.9 0.9 0.9 1.0
1.21 0.60 1.14 1.73 0.08 0.13 0.17 0.21 0.24 0.28 0.31 0.36 0.39 0.08 0.13 0.17 0.20 0.24 0.28 0.31 0.36 0.39
1.51 0.97 1.45 1.90 0.12 0.17 0.20 0.22 0.25 0.27 0.29 0.32 0.33 0.12 0.17 0.19 0.22 0.24 0.27 0.29 0.32 0.33
1.54 0.99 1.50 1.93 0.12 0.17 0.20 0.23 0.25 0.28 0.29 0.32 0.34 0.13 0.17 0.20 0.23 0.25 0.28 0.29 0.32 0.34
D.M. Bjerklie
Lesser Slave River Beaver River Pembina River below Paddy Creek Swan River Smoky River Little Smoky River McLeod River Wolf Creek Freeman River Pembina River near Entwhistle Red Deer River Oldman River Castle River Willow Creek St. Mary River Kneehills Creek Highwood River at Browns Ranch Highwood River near Aldersyde Tomichi Creek White River Blacks Fork Thomas Creek Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume
Width, W (m)
0.210 0.210 0.210 0.210 0.210 0.210 0.254 0.254 0.254 0.254 0.254 0.254 0.210 0.210 0.210 0.210 0.210 0.210 0.210 0.254 0.252 0.255 0.254 0.255 0.255 0.254 0.210 0.210 0.210 0.210 0.210 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.254
0.020 0.020 0.020 0.020 0.020 0.020 0.027 0.027 0.027 0.027 0.027 0.027 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.020 0.020 0.020 0.020 0.020 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027
0.00124 0.00255 0.0037 0.00544 0.00686 0.00914 0.00124 0.00255 0.0037 0.00544 0.00686 0.00914 0.00112 0.00217 0.00297 0.00382 0.005 0.00654 0.008 0.00122 0.00224 0.00299 0.00396 0.00516 0.00658 0.0081 0.0022 0.0042 0.0058 0.0072 0.0086 0.0022 0.0039 0.0057 0.0072 0.0078 0.0088 0.0102 0.0118
1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79
0.154 0.201 0.248 0.281 0.308 0.341 0.167 0.223 0.268 0.313 0.341 0.374 0.148 0.187 0.208 0.221 0.241 0.251 0.268 0.122 0.155 0.186 0.207 0.235 0.247 0.268 0.127 0.181 0.194 0.214 0.228 0.134 0.167 0.203 0.219 0.223 0.236 0.250 0.252
1.3 1.5 1.5 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.4 1.6 0.7 0.8 0.9 1.0 1.1 1.4 1.5 1.1 1.2 1.1 1.2 1.2 1.4 1.5 2.1 2.0 2.4 2.4 2.6 1.9 2.2 2.1 2.3 2.4 2.5 2.5 2.9
0.15 0.21 0.25 0.31 0.35 0.40 0.15 0.21 0.25 0.31 0.35 0.40 0.10 0.14 0.17 0.19 0.22 0.25 0.28 0.11 0.15 0.17 0.19 0.22 0.25 0.28 0.16 0.22 0.25 0.28 0.31 0.16 0.21 0.25 0.28 0.29 0.31 0.34 0.36
0.18 0.23 0.25 0.29 0.31 0.34 0.18 0.23 0.25 0.29 0.31 0.34 0.14 0.18 0.20 0.21 0.23 0.25 0.27 0.15 0.18 0.20 0.21 0.23 0.25 0.27 0.19 0.23 0.25 0.27 0.29 0.19 0.22 0.25 0.27 0.28 0.29 0.30 0.32
0.18 0.23 0.26 0.29 0.31 0.34 0.18 0.23 0.26 0.29 0.31 0.34 0.15 0.18 0.20 0.22 0.23 0.25 0.27 0.15 0.18 0.20 0.22 0.24 0.25 0.27 0.19 0.23 0.26 0.28 0.29 0.19 0.23 0.26 0.28 0.28 0.29 0.31 0.32
Estimating the bankfull velocity and discharge
Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume
149
D.M. Bjerklie
and flume data) using the wavelength, its average value is 0.32, which is very similar to the expected value of 0.3. However, for all of the data using the meander length instead of the wavelength, the average value of m is 0.19, which reflects the average sinuosity and the fact that the meander bend length is longer than the wavelength. To evaluate the variability of m, its value was adjusted to match the measured velocity (Table 1), and thus was optimized for each flow. It is hypothesized that the value of m should be related to the total head loss over the meander length scale, and thus would be a function of the meander length times the slope. Regression analysis was used to develop an estimating equation for the optimized value of m as a function of the product of meander length and slope resulting in the following derived regression equation: m ¼ 9:67ðSkÞ
0:36
;
r ¼ 0:86:
ð17Þ
r2 ¼ 0:95:
ð18Þ
Fig. 2 shows the estimated velocity (Table 1), determined from: (1) Eq. (16), assuming constant values for m (8 for the river data and 1.2 for the flume data); (2) m estimated from Eq. (17); and (3) estimated directly from Eq. (18) plotted as a function of the observed velocity. Table 4 shows that the mean estimate uncertainty for a large number of estimates would be expected to be less than 2% for any of the methods used. Using Eq. (17) to estimate m shows a similar range of uncertainty (as indicated by the standard deviation of the residuals) compared to estimating velocity directly with Eq. (18). The largest range of estimate uncertainty results when assuming a constant value for m for the
Table 2
Intercept k*S
Table 3
Intercept S k*
4 3 2
10
0.0 9 8 7 6 5 4 3 2
10
Constant m m from Equation 17 m from Equation 18 Observed1
-1.0 9
10
2
The regression statistics for Eq. (17) are shown on Table 2 and the standard error of the estimate for m is 51%. The strength of this relation is at least in part due to the general correlation between slope and flow resistance (Jarrett, 1984). Eq. (17) shows that m is strongly related to the meander length and slope, indicating that a general multiple regression analysis of velocity versus slope and meander length would result in a robust prediction equation. Completing this analysis results in a good prediction equation (Eq. (18)), as shown by the coefficient of variation and regression statistics (Table 3) below. V ¼ 1:37S0:31 k0:32 ;
Estimated Discharge (cubic meters per second)
150
Regression statistics for Eq. (17) Coefficients
T stat
P-value
Lower 95%
Upper 95%
9.67 0.36
28.24 21.59
<0.0001 <0.0001
8.24 0.32
11.34 0.39
Regression statistics for Eq. (18) Coefficients
T stat
P-value
Lower 95%
Upper 95%
1.37 0.31 0.32
1.99 11.32 36.68
0.05 <0.0001 <0.0001
1.00 0.26 0.30
1.88 0.37 0.34
-1.0
2
3
4
5
6 7 8
10
0.0
2
3
4
Observed Discharge (cubic meters per second)
Figure 2 Observed velocity for the river and flume data plotted against velocity estimated using Eq. (16) with a best fit constant value of m (= 8 for rivers and 1.2 for flume), variable m estimated using Eq. (17), and velocity estimated directly using Eq. (18).
river and flume data separately. It is recommended that Eqs. (16) and (17) be used rather than Eq. (18) because this allows for specific knowledge of m to be incorporated into the estimate if this information is available. There remains an issue of what the length fraction m physically represents. It is anticipated that its value reflects the energy loss condition of the channel through a hydraulically meaningful length. If this is correct, then its length would dictate a characteristic length scale for evaluating hydraulically averaged variables including width, depth, flow resistance, and slope.
Results – application using remotely sensed information The methods developed in this paper are intended for application with remotely sensed information, specifically where ground-based information may not be available. Thus, as a test of the methods, a data set of hydraulic information was developed from remotely sensed sources, including aerial photographs (from the National Aerial Photo Project (NAPP)), topography (from US Geological Survey topographic maps), and river alignment (from the US Geological Survey National Hydrography DataBase (NHD)). From these remote data sources, the bankfull width (NAPP), channel slope (topographic maps), and meander length (NHD) were measured. Ground-based estimates of the bankfull discharge and velocity were available for the rivers selected for analysis and thus remote estimates can be directly compared to an independent ground-based estimate of these variables. The ground-based bankfull discharge estimates were taken to be the mean annual flood (approximate 2-year recurrence interval) derived from USGS gaging station records for the river near the measurement locations. This assumption is made recognizing that the actual bankfull discharge is generally not equivalent to the mean annual flood,
Estimating the bankfull velocity and discharge Velocity estimation uncertainty statistics
Error statistic
Mean Standard deviation
Estimated velocity m from Eq. (17), V
Estimated velocity constant m, V
Estimated velocity from Eq. (18), V
Log residual (%)
Relative residual (%)
Log residual (%)
Relative residual (%)
Log residual (%)
Relative residual (%)
0.06 21.79
1.89 20.47
1.63 29.35
1.68 26.73
1.16 21.64
0.74 19.86
but instead may vary widely with respect to a particular recurrence interval. It is assumed here, however, that the mean annual flood can be considered to be a near-bankfull event, and therefore represents a reasonable approximation for comparative purposes. Because the estimated bankfull discharge is based on measured channel dimensions, it would reasonably represent a reference flow condition that may, in fact, be closer to the actual bankfull flow than the mean annual flood. The measurement of the bankfull width and channel slope from the aerial photos and topographic information are described in Bjerklie et al. (2005). The measurement of the meander bend length is described in the following section.
8
6
Y
Table 4
151
4
2
0 0
1000
2000
3000
4000
5000
6000
7000
X
Measuring the meander length A method to estimate the meander length was developed that uses the channel alignment, referenced to a coordinate system, obtained from the NHD. The sign (positive or negative) of the differential area, dYdX, determined from the coordinates obtained for a grid spacing equal to approximately one channel width is used to determine a change in direction of the river alignment which would correspond to an inflexion point in a meander bend. The differential area is determined from the coordinate spacing by: ðX 1 X 2 ÞðY 1 Y 2 Þ ¼ dYdX;
ð19Þ
where the subscripts 1 and 2 represent adjacent points along the alignment separated by a length of approximately one river width. The average length between periods of either positive or negative values for dYdX is used to determine 1/2 of a meander length along the channel. Because the measurement is made along the channel, it does not require the determination of the valley length, which would be necessary to measure the sinuosity. This procedure was completed for a set of 16 gaged rivers (Table 6) by computing the X and Y coordinates along the river upstream of the gage. As an illustration, a hypothetical river meander pattern (Fig. 3) set in a coordinate system is determined to have a variable meander length averaging 3600 meters. The average length is taken to be the sum of the two adjacent 1/2 meander lengths that best represent the conditions in the reach (in this case excluding the end points which do not include an entire wavelength). In actual rivers, the meander patterns are considerably more complex, and for this reason an average of 5 lengths upstream of the gaging location was used as a representative value for k*. Assuming 5 lengths as representative provided somewhat better results than using a single length immediately upstream of the gage and the average of 3 lengths upstream. The best
Figure 3 Hypothetical meandering river, with mean meander length 3600 meters. The Y coordinate in the figure is in arbitrary units.
method for obtaining the most representative length, however, remains an open question, which must consider the potential for measurement errors in addition to hydraulic factors.
Estimating the bankfull channel depth In the absence of direct knowledge of the bankfull depth, a regression equation that relates the depth to the observed bankfull width and channel slope was developed from a large data base of bankfull discharge, channel cross-section geometry, and channel slope (Barnes, 1967; Church and Rood, 1983; Dingman and Palaia, 1999; Osterkamp and Hedman, 1982 and Schumm, 1960). The range of bankfull data used in the regression analysis are shown on Table 5. The resulting regression equation to estimate the bankfull depth is:
Table 5
Bankfull hydraulic geometry data
Statistic
Maximum
Minimum
Mean
Standard deviation
Discharge (m3/s) Width (m) Depth (m) Velocity (m/s) Channel slope (m/m)
283,170
0.06
1374
12,615
3870 33 6.06 0.081
2 0.04 0.16 0.000013
96 2.34 1.90 0.0054
214 2.83 0.93 0.0096
152
D.M. Bjerklie
Table 6
Regression statistics for Eq. (20)
Intercept Slope Width
Y b ¼ 0:08
W 0:39 b S0:24
;
Coefficients
T stat
P-value
Lower 95%
Upper 95%
0.08 0.24 0.39
27.994 12.0459 16.67337
<0.0001 <0.0001 <0.0001
0.07 0.28 0.34
0.10 0.20 0.43
r2 ¼ 0:73:
ð20Þ
Table 6 shows the regression statistics for this relation. The standard error of the estimate is approximately 58%. It is recognized that there may be significant error in the bankfull depth estimated by using Eq. (20). This relation is employed here as a matter of convenience in order to evaluate the estimates of bankfull velocity as they translate to estimates of bankfull discharge. An alternative estimate of the bankfull depth also can be obtained from the meander length, as reported by Williams (1986). If the meander length is used in the estimates of both the depth and the velocity, however, any error in its measurement would be compounded in the discharge estimate (Williams, 1983). Improved methods to estimate the bankfull depth from observed variables is certainly warranted, and could potentially be obtained from long-term remote observations of water-surface elevation in rivers or from the remotely sensed data itself under appropriate conditions. The banklfull data also were used to derive a regression equation relating the bankfull discharge to the bankfull (observed) width. This relation provides a direct estimate of the bankfull discharge from a single observed variables and is based on the general assumptions of regime channels (Leopold et al., 1964). The regression equation is: Q ¼ 0:24W 1:64 ;
r2 ¼ 0:90:
ð21Þ
The regression statistics for Eq. (21) are shown in Table 7.
Evaluating the estimates of bankfull velocity and discharge using measured meander length Several studies have indicated that there is a general relation between the meander wavelength and the bankfull width (Williams, 1986; Carlston, 1965; Leliavsky, 1966; Leo-
Table 7
k ¼ 11W b ;
ð22Þ
and k ¼ 10:2W 1:12 b :
ð23Þ
Eq. (22) estimates the meander wavelength, and Eq. (23) estimates the meander (bend) length. The mean and standard deviation of the estimates of the velocity determined from the measured meander length and the meander length estimated from the general relations described by Eqs. (22) and (23) are compared in Table 8. It is evident that the measured meander length provides estimates that more closely resemble the mean and range (standard deviation) of the observed velocities, indicating that measuring the meander lengths would provide a more realistic estimate of the bankfull velocity and discharge as compared to using a width based meander relation. The estimated velocity using Eqs. (22) and (23) are nearly identical. With the measured bankfull widths and estimated bankfull velocity and depth, the bankfull discharge can be estimated and compared with the independently derived estimate of bankfull discharge determined from flow records. The bankfull discharge was estimated from the measured width, estimated depth, and velocity determined using Eqs. (16) and (17), Eq. (23) to estimate k*, and discharge estimated directly from width using Eq. (21). The measured and the estimated bankfull discharges and the log residuals of the estimates are shown on Table 9 and
Regression statistics for Eq. (21)
Intercept Width
Table 8
pold et al., 1964). The estimated values of bankfull velocity using Eq. (16) can thus be derived from these general relations and compared with the velocity estimates derived from the measured values of meander length (see Section ‘‘Estimating the bankfull channel depth’’). Thus, the value of measuring the meander length can be directly evaluated. The relations of Leopold et al. (1964) and Williams (1986) are given respectively as:
Coefficients
T stat
P-value
Lower 95%
Upper 95%
0.24 1.64
14.9909 67.50744
<0.0001 <0.0001
0.20 1.59
0.29 1.68
Comparison of velocity estimates
Comparative statistic
Field estimated
Based on meander length Eqs. (16) and (17)
Based on Eq. (22)
Based on Eq. (23)
Mean Standard deviation
2.17 0.99
1.76 0.46
1.47 0.33
1.47 0.34
Remotely Measured Data and Bankfull Discharge Estimates
River
Pemigewassett R. at Plymouth Pemigewassett R. at Woodstock White R. at West Hartford Ammonoosuc R. at Bethlehem Baker R. at Rumney Smith R. at Bristol Mississippi R. at Thebes Potomac R. at Point of Rocks S. Platte R. near Kersey Missouri R. near Culbertson Kansas R. at Fort Riley Willamette R. at Salem Wenatchee R. at Monitor Pomperaug R. at Southbury Sacramento R. nr. Red Bluff above Bend Bridge Connecticut R. Mean uncertainty Standard deviation uncertainty
Bankfull width (m)
Channel slope (m/m)
Meander length (m)
Observed mean annual flood estimate (m3/s)
Discharge estimate – measured meander length (m3/s)
Discharge estimate – Eq. (21) (m3/s)
Discharge estimate – meander length Eq. (23) (m3/s)
Log residual discharge from measured meander length
Log residual discharge from Eq. (21)
Log residual discharge from meander Eq. (23)
82
0.0017
1480
588
328
330
323
0.254
0.251
0.260
67.1
0.0026
1140
303
236
238
235
0.109
0.105
0.110
83.5
0.0012
1450
488
325
340
324
0.177
0.157
0.177
27.9
0.0075
1150
119
76
56
55
0.195
0.325
0.333
23.5
0.0013
800
145
46
43
36
0.495
0.533
0.610
18.6 801
0.0037 0.000137
720 11600
49 15900
35 12295
29 13873
26 14208
0.144 0.112
0.228 0.059
0.280 0.049
381
0.00027
3600
3500
3178
4101
4091
0.042
0.069
0.068
125
0.00093
980
240
491
659
644
0.311
0.439
0.428
343
0.000156
2650
683
2383
3452
3259
0.543
0.704
0.679
115
0.00049
2370
600
552
575
528
0.037
0.018
0.055
219
0.00032
3680
4500
1503
1654
1575
0.476
0.435
0.456
126
0.0032
1910
550
679
668
720
0.092
0.084
0.117
0.0021
985
74
37
28
24
0.305
0.415
0.487
0.425
0.390
0.405
0.131 0.12 (24%) 0.27 (86%)
0.033 0.10 (21%) 0.33 (114%)
0.029 0.12 (28%) 0.34 (119%)
18.4 163
0.000575
2640
2500
939
1019
985
330
0.0003
2130
3001
2219
3240
3209
Estimating the bankfull velocity and discharge
Table 9
153
Estimated Discharge (cubic meters per second)
154
D.M. Bjerklie 2
10 4 7 5 4 3 2
10 3 7 5 4 3 2
MEASURED EQUATION 23 EQUATION 24 EQUATION 22 QOBS
10 2 7 5 4 3 2
10 1 2
3 4 5 6 78
2
10
2
3 4 5 6 78
3
10
2
3 4 5 6 78
4
10
2
Observed Discharge (cubic meters per second)
Figure 4 Bankfull discharge estimated from remote measures of width, slope and meander length.
Fig. 4. The results suggest that the bankfull discharge can be reasonably estimated from remotely-measured meander length, slope, and bankfull channel width with a mean uncertainty (average error for a large number of estimates) on the order of 24% and the standard deviation of the uncertainty on the order of 86%. The smallest variation of uncertainty is found with the measured meander length, as opposed to the meander length estimated from width (Eq. (23)), and the estimate based solely on width (Eq. (21)). The latter estimate gives a somewhat lower mean uncertainty; however, it does not require estimates of velocity and depth, and therefore provides less information. Because of this, the estimates derived from the measured meander length are considered the most useful overall. In the use of these results, it must be remembered that the observed bankfull discharge itself may have a significant amount of uncertainty due to variable definitions of what constitutes the bankfull discharge (Williams, 1978). Thus, the estimated discharge is considered to be a reference flow condition that approximates the bankfull flow.
Conclusions and discussion In this paper, an equation is developed in which the channel slope and the length of bends in rivers are used to estimate the velocity at the bankfull discharge. A general relation to estimate the bankfull depth from observed width and slope is also presented, enabling an estimate of the bankfull discharge entirely from remotely-measured channel morphology information. It is anticipated that the bankfull flow can be estimated remotely with a mean uncertainty on the order of 24% or less for a large number of estimates, and that improvements in estimates of the depth will reduce both the mean and standard deviation of the uncertainty. The bankfull state of flow can serve as a reference flow condition because it defines the geometry and roughness at a defined flow level. With the bankfull flow serving as the reference point, remote tracking of other hydraulic variables, such as stage, width, and slope of the river channel can be used to quantify changes in flow conditions. The ability to remotely estimate river geometry and the bankfull flow of rivers will improve with advances in remote sensing
technology and additional research. Remotely estimating the bankfull flow and geometry of rivers would enable large-scale inventories of rivers to be developed by assigning specific geometric and hydraulic attributes to the large-scale river network that otherwise would require extensive ground-based field work. This would be a particularly valuable tool for regions where ground access is difficult and costly. Additionally, defining the geometric, hydraulic and discharge characteristics and dynamics of the river network from independent remote-sensing platforms could be very useful for calibration of large scale land-surface runoff models as well as for filling in gaps within a ground-based river monitoring network. The recently proposed WatER satellite mission (Alsdorf et al., 2005) to use a wide swath interferometric synthetic aperture radar (SAR) application to measure water-surface position and elevation from a satellite platform would enable water-surface elevation mapping and identification of the areal extent of water surfaces within an image. This mission would provide information on the alignment of rivers, water-surface area, and elevation flux of water surfaces, which could then be used to define the reference channel and track change in water-surface slope, depth, and width of single thread meandering rivers unaffected by backwater. Thus, the WatER mission could provide sufficient information for the estimation and monitoring of river flow from a single satellite platform for rivers that meet the aforementioned criteria. The issue of observation frequency, however, would remain an important issue with respect to monitoring the dynamics of river flow. Two important technical issues not addressed in this analysis are: (1) the assumed reach length over which the average width, slope and meander bend length would be estimated; and (2) the development of methods that would apply to rivers that are not self-formed, such as those whose slope, depth, width, and bend length are controlled by manmade or natural features that confine or alter the flow. There is an indication that the reach length over which hydraulic variables should be averaged, in order to derive more general relations between those variables, is associated with a unique fraction of the meander wavelength. However, the question as to whether the relations described in this paper apply in rivers with imposed meander bend characteristics, such as those in bedrock controlled reaches, is an important outstanding issue that will need to be addressed in order to develop a fully general methodology. Additionally, defining the appropriate resistance length in non-meandering rivers, including straight and braided rivers, remains an important and open question.
Acknowledgements Elements of this research were funded by NASA grant numbers NAG5 – 7601 and NAG5 – 8683, and an ongoing NASA funded project ‘‘Monitoring Inland Water Bodies Using Terra and Aqua Satellite Sensors’’ (University of New Hampshire/ US Geological Survey, principal investigator Bala ´zs Fekete, NRA-03-EOS-02). I thank Robert Jarrett of the US Geological Survey for his insightful and critical review comments on this work, as well as Edward Bolton, Department of Geology
Estimating the bankfull velocity and discharge at Yale University; S. Lawrence Dingman, Prof. Emeritus Department of Earth Science University of New Hampshire; and Carl Bolster, USDA, Bowling Green, Kentucky, for providing early review and ideas that contributed to this work.
References Alsdorf, D., Ernesto Rodriquez, Dennis Lettenmeier, Jay Famiglietti, 2005. Water: The water elevation and recovery satellite mission, Response to the National Research Council Decadel Survey Request for Information, NASA Surface Water Working Group, http://www.geology.ohio-state.edu/swwg. Barnes, H.H., 1967. Roughness characteristics of natural channels, USGS Water-Supply Paper 1849, 213p. Bjerklie, D.M., Lawrence Dingman, S., Vorosmarty, Charles J., Bolster, Carl H., Congalton, Russell G., 2003. Evaluating the potential for measuring river discharge from space. Journal of Hydrology 278 (1–4), 17–38. Bjerklie, D.M., Moller, D.K., Smith, L., Dingman, S.L., 2005. Estimating discharge in rivers using remotely sensed hydraulic information. Journal of Hydrology 309, 191–209. Bueche, F., 1972. Principles of Physics. McGraw-Hill Book Co, New York. Camporeale, C., Perona, P., Proporato, A., Ridolfi, L., 2005. On the long-term behavior of meandering rivers. Water Resources Research 41, W12403. Carlston, C.W., 1965. The relation of free meander geometry to stream discharge and its geomorphic implications. American Journal of Science 263, 864–885. Chow, Ven Te, 1959. Open-Channel Hydraulics. McGraw-Hill Book Co, New York. Church, M., Rood, K., 1983. Catalogue of alluvial river channel regime data, first ed. Department of Geography, University of British Columbia, Vancouver, Canada. Dingman, S.L., Palaia, K., 1999. Comparison of models for estimating flood quantiles in New Hampshire and Vermont.
155 Journal of the American Water Resources Association 35, 1233– 1243. Edwards, B.F., Smith, D.H., 2002. River meandering dynamics. Physical Review 65, 046303-1–046303-12. Fonstad, Mark A., Marcus, W.A., 2005. Remote sensing of stream depths with hydraulically assisted bathymnetry (HAB) models. Geomorphology 72, 320–339. Henderson, F.M., 1966. Open Channel Flow. Macmillan, New York. Hydrologic Engineering Center (HEC), 1986. Accuracy of computed water-surface profiles, US Army Corps of Engineers Hydrologic Engineering Center Research Document 26. Jansen, P., van Bendegom, L., van den Berg, J., de Vries, M., Zanen, A. 1979. Principles of River Engineering, Pitman, p. 509. Jarrett, R.D., 1984. Hydraulics of high gradient streams. ASCE Journal of Hydraulic Engineering 110, 1519–1539. Legleiter, C.J., Roberts, D.A., 2005. Effects of channel morphology and sensor spatial resolution on image-derived depth estimates. Remote sensing of Environment 95, 231–247. Leliavsky, Serge, 1966. An Introduction to Fluvial Hydraulics. Dover Publications, New York, p. 257. Leopold, L.B., Bagnold, R., Wolman, M.G., Brush, L.M., 1960. Flow Resistance in Sinuous or Irregular Channels, US Geological Survey Prof. Paper 282-D. Leopold, L.B., Wolman, M.G., Miller, J.P., 1964. Fluvial Processes in Geomorphology. W.H. Freeman Press, New York. Osterkamp, W.R., Hedman, E.R., 1982. Perennial-streamflow characteristics related to channel geometry and sediment in Missouri River Basin, USGS Professional Paper 1242, p. 37. Schumm, S.A., 1960. The shape of alluvial channels in relation to sediment type, USGS Professional Paper 352-B. Van den Berg, Jan H., 1995. Prediction of alluvial channel pattern of perennial rivers. Geomorphology 12, 259–279. Williams, G.P., 1978. Bankfull discharge in rivers. Water Resources Research 14 (6), 1141–1154. Williams, G.P., 1983. Improper use of regression equations in earth sciences. Geology 11 (4), 195–197. Williams, G.P., 1986. River meanders and channel size. Journal of Hydrology 88, 147–164.
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Estimating the bankfull velocity and discharge for rivers using remotely sensed river morphology information David M. Bjerklie
*
Connecticut Water Science Center, US Geological Survey, 101 Pitkin Street, East Hartford, CT 06108, United States Received 10 July 2006; received in revised form 27 March 2007; accepted 11 April 2007
KEYWORDS Hydrology; Bankfull velocity of rivers; Bankfull discharge of rivers; River flow; River meander; Flow resistance and meandering in rivers; Remote sensing of rivers
A method to estimate the bankfull velocity and discharge in rivers that uses the morphological variables of the river channel, including bankfull width, channel slope, and meander length was developed and tested. Because these variables can be measured remotely from topographic and river alignment information derived from aerial photos and satellite imagery, it is possible that the bankfull state of flow can be estimated for rivers entirely from remotely-sensed information. Defining the bankfull hydraulics of rivers would also provide a reference condition for remote tracking of dynamic variables including width, stage, and slope, and for quantifying relative change in flow conditions of rivers over large regions. This could provide a more efficient method to inventory and quantify river hydraulic attributes and dynamics. Published by Elsevier B.V.
Summary
Introduction Continuous ground measurement of water-surface height (stage) and of discharge in large numbers of rivers (the monitoring network) by the establishment of stage-discharge ratings can be expensive, especially in remote regions and in areas with difficult access. Additionally, ground-based measurements of discharge at relatively high in-bank flows that correspond to the channel-forming or near channelforming discharge can be difficult and dangerous. Definition
* Tel.: +1 860 291 6770; fax: +1 860 291 6799. E-mail address: [email protected]. 0022-1694/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.jhydrol.2007.04.011
of the bankfull hydraulics of rivers would provide a baseline condition for quantification of large-scale regional changes in river morphology through remote tracking of variables including width, stage, and slope, and serve as a reference for remote tracking of hydraulic dynamics. This ability could contribute to the development of a method to use remotesensing techniques to monitor the discharge of rivers on a continuous basis and to fill in spatial and temporal gaps in discharge records using archived and real-time remote observations. Additionally, estimating flow velocity independent of an estimate of the flow resistance would provide a much needed alternative method to estimate energy loss and flow resistance in rivers, which is difficult to estimate otherwise (HEC, 1986).
Estimating the bankfull velocity and discharge Camporeale et al. (2005) suggested that a characteristic meander length could be useful in evaluating river hydraulic conditions as an alternative to depth-based relations because the meander length can be easily measured from a variety of satellite imagers for many rivers (Bjerklie et al., 2003). Alsdorf et al. (2005) have suggested that satellitebased remote sensing of rivers using interferometric synthetic aperture radar can achieve reasonably accurate measures of the water-surface slope, width, and elevation (stage). Various airborne instruments also can provide these measures, thus the potential exists for estimating flow conditions in rivers on the basis of data/information obtained via satellite or aerial remote sensors, provided a reasonable estimate of the depth and flow resistance can be made. Alternatively, if a reference stream velocity could be measured independently, the depth and flow resistance associated with the reference velocity could be estimated, and general scaling functions could then be applied to track the river’s flow at varying stage. Except in specific clear water situations (Legleiter and Roberts, 2005) or using assumptions of a regular geometric cross-sectional shape (Fonstad and Marcus, 2005), remotesensing capabilities may not provide sufficient information to estimate depth or flow resistance directly, a fundamental pre-requisite to near real-time tracking of river discharge from satellites is the development of relations between observed variables and the unknown quantities of depth and velocity (or flow resistance). Development of a general relation between the planform (as shown on a map) channel shape, as indicated by the meander length, and velocity would therefore maximize the use of observed information in an effort to reduce the number of unknown hydraulic variables. General relations between various measures of river morphology, including the channel width, slope, and meander length and pattern, have been associated with the channel-forming discharge, typically thought to be that discharge at or near the bankfull flow condition (Leopold et al., 1964; Leliavsky, 1966; Carlston, 1965). Thus, the potential exists for the channel pattern, which is a variable that can be observed from remote platforms, to be a useful predictor of the hydraulic state of the channel at a reference flow condition associated with the bankfull flow, at least for single thread meandering rivers that are unaffected by backwater. The concept of the channel-forming discharge is based on the understanding that the shape of a river channel tends to be relatively stable over time, and is associated with frequently occurring high-flow events that have sufficient stream power to establish and maintain the channel geometry through bank erosion and bed-load movement, and that the size of these flow events are coincident with the channel capacity (i.e. the bankfull condition). Williams (1978) has shown that the recurrence interval of the bankfull discharge is highly variable, although it is often assumed to be at or near the 1.5–2.3-year recurrence interval flood. In this study, the flood frequency that relates to the bankfull flood is not relevant other than for comparison and calibration of derived velocity relations. As such, it is assumed that, on average, the bankfull discharge is approximated by the mean annual flood (Leopold et al., 1964), and serves pri-
145 marily as a reference flow condition that is at or near the true bankfull discharge and geometry. This paper develops an equation that uses the meander length to estimate the mean velocity of a river and equations to estimate the flow depth at the channel forming (bankfull) discharge that is independent of a specific flowresistance variable. These estimates can then be used to derive the reference bankfull channel dimensions and hydraulic conditions from remotely sensed information. The various equations and hydraulic relations presented in this paper generally assume gradually varied steady flow conditions confined to flow within the banks, and therefore define relatively constant hydraulic characteristics for the channel within the time frame of interest. Additionally, for the applications presented in this paper, the meander length can be more broadly defined as a bend length, construed here to be twice the length between two inflection points, and therefore need not constitute a classic meandering pattern.
Methods – hydraulic relations Bankfull channel velocity On the basis of geomorphologic considerations, an approach to estimating the bankfull mean water velocity of a river is developed using a general relation between the meander length and the ratio of bankfull mean depth (Yb) to bankfull flow resistance (fb). Camporeale et al. (2005) derives the following relation from numerical investigations of meander behavior: k ¼ 13:4
Yb ; fb
ð1Þ
where k is the spatially-averaged meander wavelength and fb is the Darcy–Weisbach friction factor with the subscript b denoting that it is associated with the hydraulic conditions at the channel forming or bankfull discharge. This relation compares well with a geomorphologic relation given by Jansen et al. (1979): k ¼ 14
Yb : fb
ð2Þ
Coupling this relation with the Chezy equation as presented in Henderson (1966) applied to the bankfull flow (given as): rffiffiffiffiffiffiffiffiffiffi gY b S ; ð3Þ Vb ¼ a where Vb = mean water velocity, S = the friction slope (approximated by the water surface slope for uniform flow), a is the dimensionless resistance factor, and g is the acceleration due to gravity; and realizing that fb = 8a (Chow, 1959), the following equation is derived to estimate the bankfull velocity (Vb) as a function of the meander length and the slope, (thus replacing the depth and flow resistance factor): pffiffiffiffiffiffiffiffi V b ¼ 0:8 gkS: ð4Þ A similar relation can be derived from force-balance considerations applied over a characteristic channel length where
146
D.M. Bjerklie
the resisting force is equal to the driving gravitational force for an open channel. Assuming that the mass of water, M, flowing down the slope would reach an equilibrium (constant) velocity V over a time T, such that the force of the flowing water is in balance with the resisting force within the length L defined by V · T, then we can define two equations that describe the state of the flow. The first assumes that the impulse defined by the resisting force, F, over time (T) is the difference between the potential velocity (Vp) that would occur if there were no resistance (given as the fall distance of the mass of water times the gravitational acceleration, g), and the resultant equilibrium velocity such that: FT ¼ MðV p VÞ;
ð5Þ
and via Torricelli’s theorem (Bueche, 1972) pffiffiffiffiffiffiffiffiffiffiffiffi V p ¼ 2gDH;
ð6Þ
where DH is the change in elevation over the reach. Second, the equilibrium velocity can be defined as: 1=2ðV þ V 0 Þ ¼
L ; T
ð7Þ
where V0 is the velocity at the head of the reach, in this case equal to V, and therefore L T¼ : V
Substituting Eq. (6) for Vp and rearranging yields: pffiffiffiffiffiffiffiffiffiffiffiffi FL : V ¼ 2gDH MV
Thus the equilibrium velocity is defined as a net head drop, which would, in principle, be defined as the equilibrium slope, S, times a pertinent downstream length variable, L, over which the energy loss takes place (referred to here as the resistance length) such that: pffiffiffiffiffiffiffiffiffiffi V ¼ 2gSL: ð14Þ
ð8Þ
ð9Þ
Y L¼4 : f
ð10Þ
An interesting aspect of Eqs. (9) and (10) is that the term FL is an expression of the work done by the resisting force to slow down the water that would be accelerated by the fall distance DH. The resisting force, F, can also be defined as M times a deceleration term, i, such that the second term in Eq. (10) is iL/V, and because V = L/T, the term reduces further to iT, which is a velocity of resistance (Vr). Thus the equilibrium velocity can be described by: V ¼ V p V r:
where Hp is the potential head and Hr is the resistance head. Rearranging Eq. (12), the equilibrium velocity can be expressed as a function of total head drop minus resistance head: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð13Þ V ¼ 2gðHp Hr Þ:
Relating Eq. (14) to the Chezy and Darcy Weisbach equations (Chow, 1959), the following relation between the resistance length and the ratio of depth to the total flow resistance is obtained:
Substituting (8) into (5) and rearranging gives: FL ¼ MðV p VÞV:
ply to a channel reach with constant slope and depth such that the equilibrium velocity is attained within a time interval (T) and a distance (L) over which work is done to resist the downstream gravitational acceleration. Thus, the energy equation reduces to the difference between the potential and resisting velocity heads. Eq. (11) therefore becomes: ! V 2p V 2r V2 ¼ ¼ ðHp Hr Þ; ð12Þ 2g 2g 2g
ð11Þ
It is possible to conceive the relation described by Eq. (11) in terms of an energy difference between a potential and actual state at a specific location. With this conception, the resultant or actual energy state would be the difference between the potential energy and the total energy loss represented by the velocity of resistance. Because the resulting velocity is the difference between a potential velocity and a resisting velocity, the total energy loss is conceptually equivalent to the total resistance to flow in the context of a uniform flow formula such as the Chezy equation. In uniform flow the total resistance to flow and the total energy loss are equivalent (the friction slope is equal to the energy slope) (Chow, 1959). Uniform flow as described by the Chezy (or Manning) equation uses the flow depth as the pertinent length variable, and a reach length over which uniform flow is established is not specified. In this paper, the concept of uniform flow is understood to ap-
ð15Þ
The length scale presented in Eq. (15) is fully consistent with and equivalent to the downstream decay length referred to by Edwards and Smith (2002) in their analysis of the dynamics of river meanders. This length, as discussed by Edwards and Smith (2002) is the length required for shear stresses perpendicular to the downstream velocity to recover from changes in channel curvature, and therefore sets the scale for the meandering wavelength and the distance between channel inflection points (1/2 of the meander bend length, which is the distance along the channel through the meander). Thus, the resistance length, L, measured along the channel, would be expected to be strongly related to some fraction of the meander wavelength. The bend length, as defined by Williams (1986), is 1/2 of the along-channel distance through a meander wavelength (measured along the valley). The full meander bend length, or meander length, is defined here as twice the bend length. This measure will be used for the analysis because it can be easily measured and digitized and is consistent with the idea of the decay length as discussed by Edwards and Smith (2002). Where meander wavelength is reported (which is usually the case), the meander bend length (or meander length, k*) is estimated by multiplying the wavelength (l) by the sinuosity (P), such that k* = kP. According to Edwards and Smith (2002), the decay length has meaning for both straight and meandering channels, and therefore is purely a function of energy losses through a reach with secondary currents. Thus, it is hypothesized that the resistance length concept should apply to any openchannel flow condition within any morphologic structure,
Estimating the bankfull velocity and discharge including the self-formed alluvial river. It might be expected, however, that the specific fraction of the meander length that constitutes the resistance or decay length is variable, depending on morphologic features of the channel. Fig. 1 illustrates the definitions of meander wavelength and meander length used in this paper. Few contemporaneous data sets of bankfull river hydraulic information combining river cross-section, channel slope, velocity, and meander characteristics that can be used to evaluate the resistance length could be found in the literature. A small set of bankfull flow data representing a range of natural rivers is available from Church and Rood (1983) (Table 1). These data are compiled within a much larger catalogue of alluvial channel regime data. The catalogue consists of rivers located primarily in the United States, the United Kingdom, and Canada. The subset of these data used in this study included those rivers for which associated discharge, cross-sectional area (width and depth), channel slope, meander wavelength, and sinuosity information were available. In some cases, more than one bankfull discharge estimate was associated with the channel hydraulic information depending on the assumption of what event constitutes bankfull. In these cases, the average of the estimated values was used. The data obtained from Church and Rood (1983) was supplemented with a small set of river data derived by crossreferencing bankfull cross-section and discharge data with meander length data for the same rivers presented in two separate reports (Williams, 1978; Williams, 1986). Due to the paucity of the river data (N = 22), flume data developed by Leopold et al. (1960) (Table 1) in a study designed to
645000 Meander Length Measured Along Channel Through the Wavelength
evaluate the effect of meander bends on total energy loss and flow resistance were also used for the analysis. The flume data consists of fixed-bed channels formed to include regular meander patterns with fixed-channel slopes. Variable flow was introduced into each channel and the equilibrium hydraulic conditions, including width, depth, and velocity, were measured. Because the experimental flume channels were fixed, each flow condition in effect represents a unique equilibrium condition comparable to a bankfull discharge that would be associated with the formation of meanders in rivers. More extensive data sets of bankfull river hydraulic data are available (Church and Rood, 1983; Williams, 1978; Van den Berg, 1995), some of which include information on sinuosity and meander radius of curvature; however, information on the meander length for the discharge locations is generally not available. In the interest of consistency, we have used only these data sets that include all of the desired hydraulic information. Although it is anticipated that the flume experiments should reflect the conditions of rivers and open channels, it must be recognized that significant scale differences exist between the flume and actual rivers and that these factors might introduce non-linear comparative results. In addition to potential issues of scale, the flume experiments do not include many forms of flow resistance and energy loss that would be present in natural settings and in larger rivers. These would include, among others, the effects of bedload and sediment movement, channel irregularities, and obstructions. Given the issues of scale and differences between the flume and natural settings, it is expected that the relation between the resistance length and the meander length would be variable, especially between the river and flume data, depending on the specific energy loss conditions in the channel. Thus, it is assumed that the meander length for each flow may not represent the resistance length established for the equilibrium condition, and therefore the following general equation is hypothesized: rffiffiffiffiffiffiffiffiffiffiffiffi 2gSk Vb ¼ m
644500
Y Coordinate
147
644000 Meander Wavelength 643500 Meander Wavelength 643000 995000 996000 997000 998000 999000 1000000 1001000
X Coordinate
Figure 1 Plot of a meandering river reach in an arbitrary X–Y coordinate system showing how the meander wavelength and meander bend length are defined. The wavelength is the straight line distance between consecutive high or low points of the curve (shown by the arrows) and the bend length, referred to in this paper as the meander length, is the along channel distance between the same points. The distance between the coordinate points on the channel path is 100 feet in this example. As can be seen, the defined meander lengths can be highly variable along the channel, and longer distances between points can obscure shorter meanders.
ð16Þ
where m is an arbitrary fraction of the meander length. The use of the variable m, which is comparable to the Manning n in the Manning resistance equation, allows for the downstream meander length, which is observable from remote platforms, to be used to help define the flow conditions as opposed to the depth variable, which is the pertinent length variable in the Chezy and Manning equations. Treating the two data sets independently, a constant value for m was determined by adjusting its value until the slope of the line between the right and left sides of Eq. (16) closely approximates 1. For the river data, m is found to be approximately 8 and for the flume data, approximately 1.2. This indicates that the general flow resistance and energy loss in a reach is established through a reach length on the order of 0.13–0.83 of a meander length, which brackets an expected distance of 0.3 times the meander wavelength (as opposed to the meander length) determined by comparing Eq. (1) with Eq. (15). The wide range of values for m when using the meander length is in part due to the effect of multiplying by the sinuosity. Interestingly, if m is determined for all of the data (both the river
148
Table 1
Hydraulic data and velocity estimates
River reach
Depth, Y (m)
Slope, S (m/m)
Meander length, k (m)
Observed velocity, V (m/s)
56.0 54.0 61.7
2.43 2.74 2.87
0.00011 0.00021 0.00033
1920 1040 1224
0.68 0.85 1.26
42.0 275.0 104.3 120.7 33.0 73.0 72.0
3.77 5.44 3.24 2.81 1.68 1.20 2.01
0.0002 0.00052 0.00094 0.00084 0.00055 0.0033 0.0018
1360 4550 3230 5760 1344 864 2037
121.0 186.0 67.7 32.7 103.7 15.0 54.5
2.14 3.09 1.32 1.45 1.57 0.87 0.87
0.0012 0.00044 0.0037 0.0008 0.002 0.0018 0.0042
67.0
2.43
23.5 13.7 57.9 36.6 0.210 0.210 0.210 0.210 0.210 0.211 0.211 0.210 0.210 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.255
0.7 0.7 1.78 3.38 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027
Optimized length fraction, M
Estimated velocity constant m, V (m/s)
Estimated velocity m from Eq. (17), V (m/s)
Estimated velocity from Eq. (18), V (m/s)
9.0 5.9 5.0
0.51 0.52 0.70
0.87 0.88 1.07
0.91 0.92 1.11
0.94 2.41 3.21 2.25 1.08 1.60 1.51
6.0 8.0 5.8 18.8 12.4 21.9 31.8
0.58 1.70 1.93 2.44 0.95 1.87 2.12
0.94 1.88 2.03 2.36 1.29 1.99 2.16
0.98 1.95 2.10 2.44 1.34 2.03 2.21
3150 3770 1921 816 2880 600 1560
1.86 2.10 2.21 1.29 1.35 1.56 2.25
21.6 7.4 28.6 7.7 61.7 8.7 25.4
2.15 1.43 2.95 0.89 2.66 1.15 2.83
2.18 1.67 2.67 1.24 2.49 1.46 2.60
2.24 1.74 2.71 1.28 2.55 1.50 2.64
0.0016
1921
1.64
22.4
1.94
2.04
2.09
0.0036 0.0025 0.0008 0.003 0.00035 0.00097 0.00166 0.00246 0.00337 0.00448 0.00559 0.00723 0.00853 0.00036 0.00097 0.00156 0.00239 0.00319 0.00448 0.00559 0.00723 0.00853
334 119 1326 814 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25
1.67 1.27 2.19 2.95 0.114 0.167 0.208 0.248 0.281 0.314 0.354 0.396 0.430 0.124 0.175 0.232 0.276 0.317 0.354 0.390 0.439 0.466
8.5 3.6 4.3 5.5 0.7 0.8 0.9 1.0 1.0 1.1 1.1 1.1 1.1 0.6 0.8 0.7 0.8 0.8 0.9 0.9 0.9 1.0
1.21 0.60 1.14 1.73 0.08 0.13 0.17 0.21 0.24 0.28 0.31 0.36 0.39 0.08 0.13 0.17 0.20 0.24 0.28 0.31 0.36 0.39
1.51 0.97 1.45 1.90 0.12 0.17 0.20 0.22 0.25 0.27 0.29 0.32 0.33 0.12 0.17 0.19 0.22 0.24 0.27 0.29 0.32 0.33
1.54 0.99 1.50 1.93 0.12 0.17 0.20 0.23 0.25 0.28 0.29 0.32 0.34 0.13 0.17 0.20 0.23 0.25 0.28 0.29 0.32 0.34
D.M. Bjerklie
Lesser Slave River Beaver River Pembina River below Paddy Creek Swan River Smoky River Little Smoky River McLeod River Wolf Creek Freeman River Pembina River near Entwhistle Red Deer River Oldman River Castle River Willow Creek St. Mary River Kneehills Creek Highwood River at Browns Ranch Highwood River near Aldersyde Tomichi Creek White River Blacks Fork Thomas Creek Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume
Width, W (m)
0.210 0.210 0.210 0.210 0.210 0.210 0.254 0.254 0.254 0.254 0.254 0.254 0.210 0.210 0.210 0.210 0.210 0.210 0.210 0.254 0.252 0.255 0.254 0.255 0.255 0.254 0.210 0.210 0.210 0.210 0.210 0.254 0.254 0.254 0.254 0.254 0.254 0.254 0.254
0.020 0.020 0.020 0.020 0.020 0.020 0.027 0.027 0.027 0.027 0.027 0.027 0.020 0.020 0.020 0.020 0.020 0.020 0.020 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.020 0.020 0.020 0.020 0.020 0.027 0.027 0.027 0.027 0.027 0.027 0.027 0.027
0.00124 0.00255 0.0037 0.00544 0.00686 0.00914 0.00124 0.00255 0.0037 0.00544 0.00686 0.00914 0.00112 0.00217 0.00297 0.00382 0.005 0.00654 0.008 0.00122 0.00224 0.00299 0.00396 0.00516 0.00658 0.0081 0.0022 0.0042 0.0058 0.0072 0.0086 0.0022 0.0039 0.0057 0.0072 0.0078 0.0088 0.0102 0.0118
1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.68 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79 0.79
0.154 0.201 0.248 0.281 0.308 0.341 0.167 0.223 0.268 0.313 0.341 0.374 0.148 0.187 0.208 0.221 0.241 0.251 0.268 0.122 0.155 0.186 0.207 0.235 0.247 0.268 0.127 0.181 0.194 0.214 0.228 0.134 0.167 0.203 0.219 0.223 0.236 0.250 0.252
1.3 1.5 1.5 1.7 1.8 1.9 1.1 1.2 1.3 1.4 1.4 1.6 0.7 0.8 0.9 1.0 1.1 1.4 1.5 1.1 1.2 1.1 1.2 1.2 1.4 1.5 2.1 2.0 2.4 2.4 2.6 1.9 2.2 2.1 2.3 2.4 2.5 2.5 2.9
0.15 0.21 0.25 0.31 0.35 0.40 0.15 0.21 0.25 0.31 0.35 0.40 0.10 0.14 0.17 0.19 0.22 0.25 0.28 0.11 0.15 0.17 0.19 0.22 0.25 0.28 0.16 0.22 0.25 0.28 0.31 0.16 0.21 0.25 0.28 0.29 0.31 0.34 0.36
0.18 0.23 0.25 0.29 0.31 0.34 0.18 0.23 0.25 0.29 0.31 0.34 0.14 0.18 0.20 0.21 0.23 0.25 0.27 0.15 0.18 0.20 0.21 0.23 0.25 0.27 0.19 0.23 0.25 0.27 0.29 0.19 0.22 0.25 0.27 0.28 0.29 0.30 0.32
0.18 0.23 0.26 0.29 0.31 0.34 0.18 0.23 0.26 0.29 0.31 0.34 0.15 0.18 0.20 0.22 0.23 0.25 0.27 0.15 0.18 0.20 0.22 0.24 0.25 0.27 0.19 0.23 0.26 0.28 0.29 0.19 0.23 0.26 0.28 0.28 0.29 0.31 0.32
Estimating the bankfull velocity and discharge
Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume Flume
149
D.M. Bjerklie
and flume data) using the wavelength, its average value is 0.32, which is very similar to the expected value of 0.3. However, for all of the data using the meander length instead of the wavelength, the average value of m is 0.19, which reflects the average sinuosity and the fact that the meander bend length is longer than the wavelength. To evaluate the variability of m, its value was adjusted to match the measured velocity (Table 1), and thus was optimized for each flow. It is hypothesized that the value of m should be related to the total head loss over the meander length scale, and thus would be a function of the meander length times the slope. Regression analysis was used to develop an estimating equation for the optimized value of m as a function of the product of meander length and slope resulting in the following derived regression equation: m ¼ 9:67ðSkÞ
0:36
;
r ¼ 0:86:
ð17Þ
r2 ¼ 0:95:
ð18Þ
Fig. 2 shows the estimated velocity (Table 1), determined from: (1) Eq. (16), assuming constant values for m (8 for the river data and 1.2 for the flume data); (2) m estimated from Eq. (17); and (3) estimated directly from Eq. (18) plotted as a function of the observed velocity. Table 4 shows that the mean estimate uncertainty for a large number of estimates would be expected to be less than 2% for any of the methods used. Using Eq. (17) to estimate m shows a similar range of uncertainty (as indicated by the standard deviation of the residuals) compared to estimating velocity directly with Eq. (18). The largest range of estimate uncertainty results when assuming a constant value for m for the
Table 2
Intercept k*S
Table 3
Intercept S k*
4 3 2
10
0.0 9 8 7 6 5 4 3 2
10
Constant m m from Equation 17 m from Equation 18 Observed1
-1.0 9
10
2
The regression statistics for Eq. (17) are shown on Table 2 and the standard error of the estimate for m is 51%. The strength of this relation is at least in part due to the general correlation between slope and flow resistance (Jarrett, 1984). Eq. (17) shows that m is strongly related to the meander length and slope, indicating that a general multiple regression analysis of velocity versus slope and meander length would result in a robust prediction equation. Completing this analysis results in a good prediction equation (Eq. (18)), as shown by the coefficient of variation and regression statistics (Table 3) below. V ¼ 1:37S0:31 k0:32 ;
Estimated Discharge (cubic meters per second)
150
Regression statistics for Eq. (17) Coefficients
T stat
P-value
Lower 95%
Upper 95%
9.67 0.36
28.24 21.59
<0.0001 <0.0001
8.24 0.32
11.34 0.39
Regression statistics for Eq. (18) Coefficients
T stat
P-value
Lower 95%
Upper 95%
1.37 0.31 0.32
1.99 11.32 36.68
0.05 <0.0001 <0.0001
1.00 0.26 0.30
1.88 0.37 0.34
-1.0
2
3
4
5
6 7 8
10
0.0
2
3
4
Observed Discharge (cubic meters per second)
Figure 2 Observed velocity for the river and flume data plotted against velocity estimated using Eq. (16) with a best fit constant value of m (= 8 for rivers and 1.2 for flume), variable m estimated using Eq. (17), and velocity estimated directly using Eq. (18).
river and flume data separately. It is recommended that Eqs. (16) and (17) be used rather than Eq. (18) because this allows for specific knowledge of m to be incorporated into the estimate if this information is available. There remains an issue of what the length fraction m physically represents. It is anticipated that its value reflects the energy loss condition of the channel through a hydraulically meaningful length. If this is correct, then its length would dictate a characteristic length scale for evaluating hydraulically averaged variables including width, depth, flow resistance, and slope.
Results – application using remotely sensed information The methods developed in this paper are intended for application with remotely sensed information, specifically where ground-based information may not be available. Thus, as a test of the methods, a data set of hydraulic information was developed from remotely sensed sources, including aerial photographs (from the National Aerial Photo Project (NAPP)), topography (from US Geological Survey topographic maps), and river alignment (from the US Geological Survey National Hydrography DataBase (NHD)). From these remote data sources, the bankfull width (NAPP), channel slope (topographic maps), and meander length (NHD) were measured. Ground-based estimates of the bankfull discharge and velocity were available for the rivers selected for analysis and thus remote estimates can be directly compared to an independent ground-based estimate of these variables. The ground-based bankfull discharge estimates were taken to be the mean annual flood (approximate 2-year recurrence interval) derived from USGS gaging station records for the river near the measurement locations. This assumption is made recognizing that the actual bankfull discharge is generally not equivalent to the mean annual flood,
Estimating the bankfull velocity and discharge Velocity estimation uncertainty statistics
Error statistic
Mean Standard deviation
Estimated velocity m from Eq. (17), V
Estimated velocity constant m, V
Estimated velocity from Eq. (18), V
Log residual (%)
Relative residual (%)
Log residual (%)
Relative residual (%)
Log residual (%)
Relative residual (%)
0.06 21.79
1.89 20.47
1.63 29.35
1.68 26.73
1.16 21.64
0.74 19.86
but instead may vary widely with respect to a particular recurrence interval. It is assumed here, however, that the mean annual flood can be considered to be a near-bankfull event, and therefore represents a reasonable approximation for comparative purposes. Because the estimated bankfull discharge is based on measured channel dimensions, it would reasonably represent a reference flow condition that may, in fact, be closer to the actual bankfull flow than the mean annual flood. The measurement of the bankfull width and channel slope from the aerial photos and topographic information are described in Bjerklie et al. (2005). The measurement of the meander bend length is described in the following section.
8
6
Y
Table 4
151
4
2
0 0
1000
2000
3000
4000
5000
6000
7000
X
Measuring the meander length A method to estimate the meander length was developed that uses the channel alignment, referenced to a coordinate system, obtained from the NHD. The sign (positive or negative) of the differential area, dYdX, determined from the coordinates obtained for a grid spacing equal to approximately one channel width is used to determine a change in direction of the river alignment which would correspond to an inflexion point in a meander bend. The differential area is determined from the coordinate spacing by: ðX 1 X 2 ÞðY 1 Y 2 Þ ¼ dYdX;
ð19Þ
where the subscripts 1 and 2 represent adjacent points along the alignment separated by a length of approximately one river width. The average length between periods of either positive or negative values for dYdX is used to determine 1/2 of a meander length along the channel. Because the measurement is made along the channel, it does not require the determination of the valley length, which would be necessary to measure the sinuosity. This procedure was completed for a set of 16 gaged rivers (Table 6) by computing the X and Y coordinates along the river upstream of the gage. As an illustration, a hypothetical river meander pattern (Fig. 3) set in a coordinate system is determined to have a variable meander length averaging 3600 meters. The average length is taken to be the sum of the two adjacent 1/2 meander lengths that best represent the conditions in the reach (in this case excluding the end points which do not include an entire wavelength). In actual rivers, the meander patterns are considerably more complex, and for this reason an average of 5 lengths upstream of the gaging location was used as a representative value for k*. Assuming 5 lengths as representative provided somewhat better results than using a single length immediately upstream of the gage and the average of 3 lengths upstream. The best
Figure 3 Hypothetical meandering river, with mean meander length 3600 meters. The Y coordinate in the figure is in arbitrary units.
method for obtaining the most representative length, however, remains an open question, which must consider the potential for measurement errors in addition to hydraulic factors.
Estimating the bankfull channel depth In the absence of direct knowledge of the bankfull depth, a regression equation that relates the depth to the observed bankfull width and channel slope was developed from a large data base of bankfull discharge, channel cross-section geometry, and channel slope (Barnes, 1967; Church and Rood, 1983; Dingman and Palaia, 1999; Osterkamp and Hedman, 1982 and Schumm, 1960). The range of bankfull data used in the regression analysis are shown on Table 5. The resulting regression equation to estimate the bankfull depth is:
Table 5
Bankfull hydraulic geometry data
Statistic
Maximum
Minimum
Mean
Standard deviation
Discharge (m3/s) Width (m) Depth (m) Velocity (m/s) Channel slope (m/m)
283,170
0.06
1374
12,615
3870 33 6.06 0.081
2 0.04 0.16 0.000013
96 2.34 1.90 0.0054
214 2.83 0.93 0.0096
152
D.M. Bjerklie
Table 6
Regression statistics for Eq. (20)
Intercept Slope Width
Y b ¼ 0:08
W 0:39 b S0:24
;
Coefficients
T stat
P-value
Lower 95%
Upper 95%
0.08 0.24 0.39
27.994 12.0459 16.67337
<0.0001 <0.0001 <0.0001
0.07 0.28 0.34
0.10 0.20 0.43
r2 ¼ 0:73:
ð20Þ
Table 6 shows the regression statistics for this relation. The standard error of the estimate is approximately 58%. It is recognized that there may be significant error in the bankfull depth estimated by using Eq. (20). This relation is employed here as a matter of convenience in order to evaluate the estimates of bankfull velocity as they translate to estimates of bankfull discharge. An alternative estimate of the bankfull depth also can be obtained from the meander length, as reported by Williams (1986). If the meander length is used in the estimates of both the depth and the velocity, however, any error in its measurement would be compounded in the discharge estimate (Williams, 1983). Improved methods to estimate the bankfull depth from observed variables is certainly warranted, and could potentially be obtained from long-term remote observations of water-surface elevation in rivers or from the remotely sensed data itself under appropriate conditions. The banklfull data also were used to derive a regression equation relating the bankfull discharge to the bankfull (observed) width. This relation provides a direct estimate of the bankfull discharge from a single observed variables and is based on the general assumptions of regime channels (Leopold et al., 1964). The regression equation is: Q ¼ 0:24W 1:64 ;
r2 ¼ 0:90:
ð21Þ
The regression statistics for Eq. (21) are shown in Table 7.
Evaluating the estimates of bankfull velocity and discharge using measured meander length Several studies have indicated that there is a general relation between the meander wavelength and the bankfull width (Williams, 1986; Carlston, 1965; Leliavsky, 1966; Leo-
Table 7
k ¼ 11W b ;
ð22Þ
and k ¼ 10:2W 1:12 b :
ð23Þ
Eq. (22) estimates the meander wavelength, and Eq. (23) estimates the meander (bend) length. The mean and standard deviation of the estimates of the velocity determined from the measured meander length and the meander length estimated from the general relations described by Eqs. (22) and (23) are compared in Table 8. It is evident that the measured meander length provides estimates that more closely resemble the mean and range (standard deviation) of the observed velocities, indicating that measuring the meander lengths would provide a more realistic estimate of the bankfull velocity and discharge as compared to using a width based meander relation. The estimated velocity using Eqs. (22) and (23) are nearly identical. With the measured bankfull widths and estimated bankfull velocity and depth, the bankfull discharge can be estimated and compared with the independently derived estimate of bankfull discharge determined from flow records. The bankfull discharge was estimated from the measured width, estimated depth, and velocity determined using Eqs. (16) and (17), Eq. (23) to estimate k*, and discharge estimated directly from width using Eq. (21). The measured and the estimated bankfull discharges and the log residuals of the estimates are shown on Table 9 and
Regression statistics for Eq. (21)
Intercept Width
Table 8
pold et al., 1964). The estimated values of bankfull velocity using Eq. (16) can thus be derived from these general relations and compared with the velocity estimates derived from the measured values of meander length (see Section ‘‘Estimating the bankfull channel depth’’). Thus, the value of measuring the meander length can be directly evaluated. The relations of Leopold et al. (1964) and Williams (1986) are given respectively as:
Coefficients
T stat
P-value
Lower 95%
Upper 95%
0.24 1.64
14.9909 67.50744
<0.0001 <0.0001
0.20 1.59
0.29 1.68
Comparison of velocity estimates
Comparative statistic
Field estimated
Based on meander length Eqs. (16) and (17)
Based on Eq. (22)
Based on Eq. (23)
Mean Standard deviation
2.17 0.99
1.76 0.46
1.47 0.33
1.47 0.34
Remotely Measured Data and Bankfull Discharge Estimates
River
Pemigewassett R. at Plymouth Pemigewassett R. at Woodstock White R. at West Hartford Ammonoosuc R. at Bethlehem Baker R. at Rumney Smith R. at Bristol Mississippi R. at Thebes Potomac R. at Point of Rocks S. Platte R. near Kersey Missouri R. near Culbertson Kansas R. at Fort Riley Willamette R. at Salem Wenatchee R. at Monitor Pomperaug R. at Southbury Sacramento R. nr. Red Bluff above Bend Bridge Connecticut R. Mean uncertainty Standard deviation uncertainty
Bankfull width (m)
Channel slope (m/m)
Meander length (m)
Observed mean annual flood estimate (m3/s)
Discharge estimate – measured meander length (m3/s)
Discharge estimate – Eq. (21) (m3/s)
Discharge estimate – meander length Eq. (23) (m3/s)
Log residual discharge from measured meander length
Log residual discharge from Eq. (21)
Log residual discharge from meander Eq. (23)
82
0.0017
1480
588
328
330
323
0.254
0.251
0.260
67.1
0.0026
1140
303
236
238
235
0.109
0.105
0.110
83.5
0.0012
1450
488
325
340
324
0.177
0.157
0.177
27.9
0.0075
1150
119
76
56
55
0.195
0.325
0.333
23.5
0.0013
800
145
46
43
36
0.495
0.533
0.610
18.6 801
0.0037 0.000137
720 11600
49 15900
35 12295
29 13873
26 14208
0.144 0.112
0.228 0.059
0.280 0.049
381
0.00027
3600
3500
3178
4101
4091
0.042
0.069
0.068
125
0.00093
980
240
491
659
644
0.311
0.439
0.428
343
0.000156
2650
683
2383
3452
3259
0.543
0.704
0.679
115
0.00049
2370
600
552
575
528
0.037
0.018
0.055
219
0.00032
3680
4500
1503
1654
1575
0.476
0.435
0.456
126
0.0032
1910
550
679
668
720
0.092
0.084
0.117
0.0021
985
74
37
28
24
0.305
0.415
0.487
0.425
0.390
0.405
0.131 0.12 (24%) 0.27 (86%)
0.033 0.10 (21%) 0.33 (114%)
0.029 0.12 (28%) 0.34 (119%)
18.4 163
0.000575
2640
2500
939
1019
985
330
0.0003
2130
3001
2219
3240
3209
Estimating the bankfull velocity and discharge
Table 9
153
Estimated Discharge (cubic meters per second)
154
D.M. Bjerklie 2
10 4 7 5 4 3 2
10 3 7 5 4 3 2
MEASURED EQUATION 23 EQUATION 24 EQUATION 22 QOBS
10 2 7 5 4 3 2
10 1 2
3 4 5 6 78
2
10
2
3 4 5 6 78
3
10
2
3 4 5 6 78
4
10
2
Observed Discharge (cubic meters per second)
Figure 4 Bankfull discharge estimated from remote measures of width, slope and meander length.
Fig. 4. The results suggest that the bankfull discharge can be reasonably estimated from remotely-measured meander length, slope, and bankfull channel width with a mean uncertainty (average error for a large number of estimates) on the order of 24% and the standard deviation of the uncertainty on the order of 86%. The smallest variation of uncertainty is found with the measured meander length, as opposed to the meander length estimated from width (Eq. (23)), and the estimate based solely on width (Eq. (21)). The latter estimate gives a somewhat lower mean uncertainty; however, it does not require estimates of velocity and depth, and therefore provides less information. Because of this, the estimates derived from the measured meander length are considered the most useful overall. In the use of these results, it must be remembered that the observed bankfull discharge itself may have a significant amount of uncertainty due to variable definitions of what constitutes the bankfull discharge (Williams, 1978). Thus, the estimated discharge is considered to be a reference flow condition that approximates the bankfull flow.
Conclusions and discussion In this paper, an equation is developed in which the channel slope and the length of bends in rivers are used to estimate the velocity at the bankfull discharge. A general relation to estimate the bankfull depth from observed width and slope is also presented, enabling an estimate of the bankfull discharge entirely from remotely-measured channel morphology information. It is anticipated that the bankfull flow can be estimated remotely with a mean uncertainty on the order of 24% or less for a large number of estimates, and that improvements in estimates of the depth will reduce both the mean and standard deviation of the uncertainty. The bankfull state of flow can serve as a reference flow condition because it defines the geometry and roughness at a defined flow level. With the bankfull flow serving as the reference point, remote tracking of other hydraulic variables, such as stage, width, and slope of the river channel can be used to quantify changes in flow conditions. The ability to remotely estimate river geometry and the bankfull flow of rivers will improve with advances in remote sensing
technology and additional research. Remotely estimating the bankfull flow and geometry of rivers would enable large-scale inventories of rivers to be developed by assigning specific geometric and hydraulic attributes to the large-scale river network that otherwise would require extensive ground-based field work. This would be a particularly valuable tool for regions where ground access is difficult and costly. Additionally, defining the geometric, hydraulic and discharge characteristics and dynamics of the river network from independent remote-sensing platforms could be very useful for calibration of large scale land-surface runoff models as well as for filling in gaps within a ground-based river monitoring network. The recently proposed WatER satellite mission (Alsdorf et al., 2005) to use a wide swath interferometric synthetic aperture radar (SAR) application to measure water-surface position and elevation from a satellite platform would enable water-surface elevation mapping and identification of the areal extent of water surfaces within an image. This mission would provide information on the alignment of rivers, water-surface area, and elevation flux of water surfaces, which could then be used to define the reference channel and track change in water-surface slope, depth, and width of single thread meandering rivers unaffected by backwater. Thus, the WatER mission could provide sufficient information for the estimation and monitoring of river flow from a single satellite platform for rivers that meet the aforementioned criteria. The issue of observation frequency, however, would remain an important issue with respect to monitoring the dynamics of river flow. Two important technical issues not addressed in this analysis are: (1) the assumed reach length over which the average width, slope and meander bend length would be estimated; and (2) the development of methods that would apply to rivers that are not self-formed, such as those whose slope, depth, width, and bend length are controlled by manmade or natural features that confine or alter the flow. There is an indication that the reach length over which hydraulic variables should be averaged, in order to derive more general relations between those variables, is associated with a unique fraction of the meander wavelength. However, the question as to whether the relations described in this paper apply in rivers with imposed meander bend characteristics, such as those in bedrock controlled reaches, is an important outstanding issue that will need to be addressed in order to develop a fully general methodology. Additionally, defining the appropriate resistance length in non-meandering rivers, including straight and braided rivers, remains an important and open question.
Acknowledgements Elements of this research were funded by NASA grant numbers NAG5 – 7601 and NAG5 – 8683, and an ongoing NASA funded project ‘‘Monitoring Inland Water Bodies Using Terra and Aqua Satellite Sensors’’ (University of New Hampshire/ US Geological Survey, principal investigator Bala ´zs Fekete, NRA-03-EOS-02). I thank Robert Jarrett of the US Geological Survey for his insightful and critical review comments on this work, as well as Edward Bolton, Department of Geology
Estimating the bankfull velocity and discharge at Yale University; S. Lawrence Dingman, Prof. Emeritus Department of Earth Science University of New Hampshire; and Carl Bolster, USDA, Bowling Green, Kentucky, for providing early review and ideas that contributed to this work.
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