Geomorphology 122 (2010) 39–55
Contents lists available at ScienceDirect
Geomorphology
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / g e o m o r p h
Low-flow hydraulic geometry of small, steep mountain streams
in southwest British Columbia
Donald E. Reid a,⁎, Edward J. Hickin b, Scott C. Babakaiff c
a
b
c
BC Hydro, 6911 Southpoint Drive, Burnaby, British Columbia, Canada V3N 4X6
Department of Geography, Simon Fraser University, Burnaby, British Columbia, Canada
BC Ministry of Environment, Ecosystems Section, Surrey, British Columbia, Canada
a r t i c l e
i n f o
Article history:
Received 5 May 2009
Received in revised form 20 May 2010
Accepted 24 May 2010
Available online 4 June 2010
Keywords:
Hydraulic geometry
Mountain streams
Low-flow
Habitat
a b s t r a c t
This investigation explores the at-a-station hydraulic geometry (AHG) of small, steep mountain streams at
low discharge. Thirteen reaches in five tributaries of Chilliwack River, British Columbia, ranging in size from
12 to 77 km2 are examined. The resulting data set is composed of eight to twelve measurements of watersurface width, mean depth, and mean velocity at each of 61 cross sections or 625 unique combinations of the
three variables. Mean velocity in a given cross section responds most rapidly to changing discharge, and 31 of
the 61 cross sections have velocity exponents that are greater than the water-surface width and mean-depth
exponents combined. The velocity exponent (m) averages 0.51, while the mean water-surface width
exponent (b) and mean-depth exponent (f) average 0.20 and 0.29, respectively. Somewhat surprisingly, the
AHG of steep mountain streams can be reasonably predicted from just a few measurements of the primary
flow variables and stream discharge. While conditions at the cross section appear predictable from a few
measurements, extrapolating the results from one cross section to another in the same reach involves large
errors. The section-to-section variability of the exponents and coefficients, even when they are located in
similar channel units such as riffles, prevents accurate extrapolation to unmeasured cross sections.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
This work explores the hydraulic geometry of small, steep
mountain streams at low discharge (the lower quartile of the
discharge range). In this range of flow, the study of hydraulic
geometry can be thought of as the quantitative description of how
stream discharge fills an essentially non-deformable boundary. The
stream channel is self-formed at relatively high flow (arguably
bankfull discharge) and its size and shape, as described by the
hydraulic geometry, are governed by a set of imposed constraints that
include the stream discharge (Q), sediment supply (Qs), sediment
calibre, and geomorphic history (Hey, 1978; Knighton, 1998, p. 2). In
steep mountain streams the valley slope and boundary materials
(coarse and even non-alluvial in places) impose additional constraints
on the channel morphology that make mountain streams unique and
unlike their lowland counterparts (Jarrett, 1984, 1990; Grant et al.,
1990; Rice and Church, 1996; Montgomery and Buffington, 1997;
Wohl and Wilcox, 2005; Comiti et al., 2007; Wohl, 2007).
Hydraulic geometry has been explored widely and remains a core
technique of river science (Knighton, 1998). Despite this widespread
use, the study of hydraulic geometry remains an essentially empirical
⁎ Corresponding author. Tel.: + 1 604 528 1426, + 1 604 813 8507(Cell); fax: + 1 604
528 2940.
E-mail addresses: donald.reid@bchydro.com, dereid@alumni.sfu.ca (D.E. Reid).
0169-555X/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.geomorph.2010.05.012
enterprise because we lack universal flow resistance and sediment
transport relations (Church, 1980; Bathurst, 2002). The term was
coined by Leopold and Maddock in their seminal 1953 work
quantitatively describing the relationship of the principal hydraulic
variables of water-surface width, mean depth, and mean velocity to
changing stream discharge (Leopold and Maddock, 1953). Simple
power functions remain the principal basis for describing these
relationships (Leopold and Maddock, 1953):
w = aQ
b
ð1Þ
d = cQ
f
ð2Þ
v = kQ
m
ð3Þ
where w = water-surface width (m), d = mean depth (m), v = mean
velocity (m/s), and Q = stream discharge (m3/s). If there is flow
continuity,
ð4Þ
Q = wdv
b
f
Q = aQ cQ kQ
m
= ackQ
ðb + f + mÞ
ð5Þ
40
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
Remarkably, this model has remained the foundation of descriptions of river form and process, for over almost a half century of
modern science (Clifford, 1996).
Since 1953 the technique has been applied worldwide (Park,
1977) largely to determine the degree to which rivers respond to
different sets of imposed constraints in varied geographic settings.
Hydraulic geometry has also often been employed as an environmental and engineering design tool. Recent applications include the
definition of instream flow standards (e.g., minimum environmental
flow) that are set to minimize the impact of water use on fish
populations (Jowett, 1998; Babakaiff, 2004). These applications have
focussed attention on the lack of data in steep mountain settings and
lack of understanding of low-flow hydraulics. Contributing to this data
need constitutes a primary purpose of this work.
Frequently, flow variables of hydraulic geometry are calculated
from data collected during a range of “typical” flows centred near the
middle of the discharge range (Park, 1977). This practice likely reflects
the desire to understand the formation and maintenance of channels
and stream morphology as they relate to sediment entrainment and
erosion of the channel boundary. Yet, it is among the less frequent
flows found at the upper and lower ends of the discharge range where
abrupt changes in channel hydraulics (e.g., resistance at low flows,
channel width at high flows) occur (Leopold and Maddock, 1953;
Hogan and Church, 1989). If present, these abrupt changes (or
discontinuities) are thought to appear as breaks in the slope of the
log-linear relations of hydraulic geometry but are rarely measured.
This study focuses on the implications of this data deficiency at the
lower limits of the flow-measurement range (i.e., low flow) for the
hydraulic geometry of streams in SW British Columbia.
The concept of discontinuities in hydraulic geometry is not new to
the literature. Ferguson (1986) noted that discontinuities separate the
hydraulic geometry of one range of flow from another by physical or
hydraulic differences in the cross section in each flow range. Jowett
(1997) recognized at least two discontinuities in any cross section:
one where the base of the channel is just filled, a second where flow
spills out of the channel at bankfull. He went on to note that such
discontinuities are usually most evident in rivers of moderate gradient
in well-defined channels (Jowett, 1997). Knighton (1998) argued that
the AHG has at least three phases: a residual phase below the
threshold for bed mobilisation, an active phase when the bed is
mobile, and an overbank phase at stream discharges greater than
bankfull when the floodplain becomes inundated. Low-flow hydraulic
geometry describes conditions in the residual phase.
The literature contains examples of several other types of AHG
discontinuities. Hogan and Church (1989) in their work in the Queen
Charlotte Islands, British Columbia, reported a discontinuity in their
relationships when flow spilled from a small inset channel onto a
large lateral bar. Leopold and Maddock (1953), in their initial work on
hydraulic geometry, described a discontinuity at an artificial cross
section with bridge abutments. They showed that increases in lowflow discharge filled the bed of the channel until the flow had filled
the available width between the bridge abutments, leading to a new
relation above this point where width remained constant and
subsequent increases in stream discharge led to larger increases in
mean flow depth and velocity. Hickin (1995) described a discontinuity in the plot of the hydraulic parameters for a cross section of the
Fraser River at Marguerite, British Columbia, where general bed
mobilisation and scour above a threshold discharge leads to an abrupt
change in the width, depth, and velocity curves above this value. He
also argued that this discontinuity is obscured in a log–log plot and
that an examination of the relationship in an arithmetic plot is always
an important first step of analysis. Lewis (1966) reported hydraulic
geometry discontinuities at very low discharges where low flows
occupy a smaller inset channel within the larger channel and the
Fig. 1. Chilliwack watershed and study basins.
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
width of the flow expands quickly as flow spills out of the inset
channel across the entire bed of the channel.
A distinction is made between AHG, downstream hydraulic
geometry (DHG), and more recently, interchannel hydraulic geometry
(Tabata and Hickin, 2003). At-a-station hydraulic geometry describes
the relationship between water-surface width, mean depth, and mean
velocity with changing discharge as each varies at a single point (cross
section) in a stream network. Therefore, AHG describes the “temporal
variation in flow variables as discharge fluctuates at a cross section”
(Knighton, 1998, p. 180). More recently Stewardson (2005) has
extended the concept to reach-scale AHG. In contrast, downstream
hydraulic geometry describes the relationship between the variables at
different locations within a single stream (e.g., headwaters, mid basin,
and lower basin) or between streams. To allow comparison among sites
of differing channel size, measurements are referenced to a single
discharge with a known exceedence probability (often taken as bankfull
flow). Thus, DHG can be thought of as the “spatial variations in channel
properties at a reference discharge” (Knighton, 1998, p. 180). Neither of
the preceding terms accurately describes the multi-channelled form of
the anastomosing river system. Tabata and Hickin (2003, p.839) coined
the term interchannel hydraulic geometry to describe “the general
bankfull channel form and hydraulics of primary and secondary
channels in the anastomosing channel system”.
In summary, the purpose of this paper is to present new AHG data at
the lower limits of the flow-measurement range (i.e., low-flow) from
SW British Columbia, to describe the AHG relations in this flow range,
41
and to examine the data for discontinuities that might preclude the
extrapolation of observations made in the typical flow range to low flow.
2. Field sites
The field sites are in the Chilliwack River watershed in southern
coastal British Columbia in the Skagit Range of the Cascade Mountains
where local relief is high and maximum relief approaches 2000 m
(Fig. 1). Five tributary basins draining watershed areas between 12
and 77 km2 were selected for study. Detailed descriptions of these
field sites are available in Reid (2005).
The valley was heavily glaciated during the Pleistocene leaving
thick glacial deposits as tills, kame terraces, moraines, and glaciolacustrine and outwash material on the lower valley and hillslopes
(Clague, 1981; Saunders et al., 1987). In addition to acting as sediment
sources, these glacial deposits act as groundwater reservoirs supplying water to lowland stream channels during periods of drought.
Other sources of baseflow in the watershed are the persistent winter
snowpack over 800 m above sea level (B.C. Ministry of Forests, 1995)
and the numerous areas of permanent ice located on the higher
elevation mountains across the basin. The watershed lies in the
Coastal Western Hemlock (CWH) biogeoclimatic zone (BC Ministry of
Forests and Range, 2008) characterised by a dense coniferous forest
and wet, mild winters.
The hydrologic regime of the Chilliwack watershed is typical of
coastal British Columbia (Fig. 2) in the sense that peak flows are
Fig. 2. Chilliwack River annual daily-flow hydrographs for WSC gauges: Slesse Creek near Vedder Crossing (A) and Chilliwack River above Slesse Creek (B).
42
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
generated in two seasons: in the spring (June and July) by snowmelt
and in winter (October through January) by storms. The largest flows
in the river are produced by infrequent rain-on-snow events in the fall
and winter when large, warm Pacific storms affect the coast. The flood
of record in the valley occurred on 10 November 1990 during these
rain-on-snow conditions.
Like peak flows, low flows occur during two separate periods
throughout the year: the summer and winter low-flow seasons.
Typically, the summer low-flow period from August through October
is the more severe season in any given year in which smaller
minimum flows occur over longer periods. In contrast, winter low
flows are produced during cold, dry periods when the upper portions
of the watersheds are frozen and precipitation falls as snow.
In addition to extreme low-flow conditions, the study period
includes a large flood on 17 October 2003 (Fig. 3A). The storm that
produced this flood delivered 53 mm of rain in a 24-h period on 16
October 16 and an additional 95 mm of rain on 20 October at the
Chilliwack River hatchery climate station (DFO, 2003; Fig. 3B). In all,
the storm supplied 275 mm of rain to the watershed during the nineday storm, generating a 15-year flood in nearby Slesse Creek and
likely a similar response in the study streams.
Five tributaries to Chilliwack River were selected for study (Fig. 1).
This data set includes a combination of purposefully selected and
randomly selected basins. The purposefully selected basins are Frosst
Creek and Liumchen Creek, both previously gauged by the Water
Survey of Canada (2004), so a continuous record of streamflow is
available for these two basins. Random selection yielded Borden, Foley,
and Chipmunk Creeks in basins ranging in size from 11.9 to 78.6 km2.
Thirteen reaches in the five study basins were selected for detailed
study. Stream gradients range from 0.017 to 0.075, and stream
morphologies vary between pool-riffle, plane-bed, step-pool, and
cascade reaches (Table 1). The set of channels chosen for this study
are typical of small, steep streams in SW British Columbia.
3. Methods
A detailed account of the field procedures adopted in this study is
available in Reid (2005), so a brief summary serves the present
purpose. Hydraulic measurements were made over a six-month
period from 19 June 2003 to 16 December 2003 at multiple stream
discharges at set cross section locations within the study basins.
Section layout was carried out in a consistent manner for each study
reach. First, each reach was walked to ensure that it displayed uniform
stream morphology for more than 10 bankfull widths. Only Liumchen
Creek, a stream gauged by the Water Survey of Canada (2004), did not
meet the general length criteria. Two shorter reaches are included
here to ensure that this important gauged basin is included in the
study. After walking the entire reach, an arbitrary starting point near
the downstream end of the reach was selected. Cross sections were
located from the starting point at equally spaced intervals of more
than two bankfull widths. Again, Liumchen Creek was an exception to
this protocol because of its shortened reaches; here, only three cross
sections spaced one bankfull width apart are included in the study.
After locating the appropriate number of cross sections within each
reach, the bed and banks were surveyed by total station. To ensure that
every subsequent survey reproduced the same cross section, survey
pins were installed on both banks as part of the initial survey. These
survey pins were used as position, elevation, and distance control for
the remainder of the study. The initial survey of the cross sections
included all topographic breaks across the section and included any
individual large boulders located on the cross section line.
At each cross section, several measurements were made of the key
hydraulic parameters at a variety of stream discharges. Water-surface
widths and depths were measured using one of two methods: a level
survey of the water-surface or multiple direct measurements of depth
along the survey line. Method one, usually employed at higher stages
when wading the entire cross section was difficult, used a combination of the surveyed cross section boundary and a surveyed watersurface elevation and width at the observation date. The watersurface elevation was plotted on the surveyed cross section boundary
and the total flow area (A) calculated in a computer aided drafting
program. Mean depth (d) was calculated from
d = A= w
ð6Þ
and mean velocity (v) was calculated from
v = Q=A
Fig. 3. Streamflow in Slesse Creek (A) and total precipitation at Chilliwack River
hatchery (B) during the study period.
ð7Þ
Method two, used more often than method one when the section
could be waded, used a technique similar to stream discharge gauging
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
43
Table 1
Study reach geomorphology.
Stream
Reach
name
Dominant
morphologya
Channel planform/
coupling to
hillslopes
Location — distance
upstream from
mouth
(km)
Number
of cross
sections
Average
cross section
spacing
(m)
Reach
length
(m)
Drainage
area to head
of reach
(km2)
Mean basin
elevation
(m)
Bed
slope
(%)
Mean
bankfull
width
(m)
Frosst Creek
Lower
Upper
Lower
Upper
Lower
Middle
Upper
Lowest
Lower
P-R
PB
PB
C
F(P-R)
PB
S-P
C
PB/P-R
1.2
2.1
0.9
1.0
0.0
0.2
0.5
4.5
4.8
5
5
3
3
5
5
5
5
5
25
35
25
25
20
15
20
50
45
215
188
74
79
100
112
105
250
290
30.2
27.1
54.5
54.5
17.8
17.8
17.7
33.5
33.0
760
790
1090
1090
1260
1260
1270
1320
1320
1.9
3.1
3.5
4.9
3.5
2.9
7.5
3.4
1.7
10.0
8.6
24.4
17.7
9.9
9.4
10.2
15.5
15.9
Upper
Tributary
Lower
PB/C
PB
C
5.5
0.1
0.6
5
5
5
25
25
75
134
119
367
12.8
11.9
76.9
1280
1400
1300
2.3
2.5
2.5
9.6
8.4
22.8
Upper
PB/F(P-R)
Straight/buffered
Sinuous/buffered
Sinuous/buffered
Straight/buffered
Sinuous/buffered
Sinuous/buffered
Sinuous/buffered
Sinuous/coupled
Sinuous/intermittently
coupled
Straight/buffered
Sinuous/buffered
Sinuous/intermittently
coupled
Sinuous/intermittently
coupled
1.0
5
75
392
75.9
1310
1.9
23.6
Liumchen Creek
Borden Creek
Chipmunk Creek
Foley Creek
a
P-R, Pool-riffle; PB, plane-bed; S-P, step-pool; C, cascade; F(P-R), forced pool-riffle (Montgomery and Buffington, 1997).
where multiple direct measurements of the flow depth were made
across the section using a stadia rod. Distance across the section was
measured on an overhead tape for each depth measurement and for
measurements of the edges of rocks protruding through the surface of
the flow. From the depth and distance measurements, the area of flow
for individual cells was calculated and summed across the entire cross
section to compute the total flow area. As in method one, relations
among continuity, stream discharge, water-surface width, and total
flow area were used to compute the mean depth and mean velocity.
Independent measurements of stream velocity were not made at
these cross sections.
Stream discharge was measured using standard wading-techniques and summing the individual discharge cells (Rantz, 1982). The
average velocity in the vertical was measured at the standard 0.6
depth. Although some researchers have found that the mean velocity
in the vertical is more likely to be found at 0.5 depth (0.5 d) in steep
mountain streams because of the S-shaped vertical velocity distribution (Jarrett, 1990), this was not the case here. No statistical difference
exists between the velocity at 0.6 d, 0.5 d, or the calculated mean for
the vertical (Reid, 2005). In addition, the gauging sites for this study
were located in optimal reaches in the tailouts of pools or other deep,
slow moving sections of the stream where the flow area is large and
velocities correspondingly low and flow is fairly uniform across the
section. Without clear evidence that it was inappropriate, the
standard gauging depth of 0.6 d was adopted.
To avoid the problems of flow-metering in sections not well suited
to measurement, discharge in each reach was measured at an optimal
gauging section and the discharge through the reach assumed
constant. These optimal flow-metering sections are located in large,
channel-spanning pools that are located within or close to the study
reach so that significant gains or losses of streamflow between the
gauging location and the study cross sections are unlikely. To test the
accuracy of the discharge measurements, and to test the assumption
that standard gauging techniques could produce a precise discharge
estimate using a Swoffer current meter, a single cross section was
repeatedly gauged on 10 June and 11 June 2003. To ensure a steady
flow state, the test was conducted during a dry period and the stage at
the measuring section was noted at the start and end of the test; no
stage change was noted. The three discharge estimates made on each
day were observed to vary b5% from the mean of that day and b5%
from independent repeated discharge measurements made with a
separate Price pygmy-style current meter. While the test does not
confirm the accuracy of the measurements, the close agreement of
independent meters and multiple measurements with each meter
provides some confidence in the calculated discharge estimates and
the gauging techniques employed.
Several supplementary observations that allow the hydraulic
geometry and resistance to flow to be related to the stream reaches
and the conditions during the observations were recorded. These
include overhead photographs; oblique photographs during a high,
moderate and low stage measurement; bed material measurements
(after Kellerhals and Bray, 1971); longitudinal profile surveys;
morphologic bankfull-stage surveys; and observations of gravel bars
and large-woody debris frequency. These supplementary observations are especially useful when relating flow resistance to possible
form roughness contributions (Reid and Hickin, 2008).
4. Results
At-a-station hydraulic geometry was determined from stream
discharge and flow area measurements made at the 61 study cross
sections during 8 to 14 individual stream discharges over a period of
seven months from June to December 2003. The complete core data
set will not be reproduced here but is available in a set of appendices
in Reid and Hickin (2008). From these measurements, 61 individual
sets of hydraulic geometry relationships were derived, a set of
relationships for each cross section (Figs. 4–10). The data cover five
orders of magnitude range of stream discharge from 0.0006 to
5.52 m3/s, with most values in the 0.1 to 5 m3/s range. This range of
flow is considered low to moderate in these steep streams.
On 17 October 2003, a large flood altered cross sections in five of
the thirteen study reaches (lower Frosst, upper Frosst, lower Borden,
middle Borden, and Chipmunk Tributary reaches). These changes are
reflected in the exponents and coefficients of the hydraulic geometry
when the data are analysed separately for the two periods, pre and
post-flood. Data from the pre-October period are only presented in
the results section. A complete analysis of the flood and its effects are
included in the discussion section.
After examining the hydraulic geometry of the 61 cross sections,
three main relationships became evident. First, water-surface width
(w), mean water depth (d), and mean flow velocity (v) all increased
with increasing discharge at the section. This is described by the
positive exponents of the relationships that range from 0.05 to 0.45
for b, 0.09 to 0.47 for f, and 0.17 to 0.84 for m (Figs. 4–10).
Second, the power function describes remarkably well the
observed relationships at each cross section with the strength of the
relationships measured by R2 (Figs. 4–10). R2 values range from 0.26
to 1.00 but average 0.90; a large majority (95%) are N0.7. Only eight of
Fig. 4. Hydraulic geometry of Frosst Creek — lower and upper reaches.
44
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
45
Fig. 5. Hydraulic geometry of Liumchen Creek — lower and upper reaches.
173 R2 values are b0.70. It should be noted that the relationships
presented include spurious correlation because Q, by definition, is not
independent of w, d or v. Spurious correlation can yield correlations of
0.7 (Schlager et al., 1998) and as high as 0.90 (Benson, 1965) for
random data, and as a result, no statistical inference can be made from
the R2 terms.
Third, the coefficients and exponents of the relationships vary
considerably between the cross sections in most of the study reaches
(Figs. 4–10). This variability is measured by the 95% confidence
interval (approximately equal to ±two standard errors about the
mean). Standard errors range from 2% to 39% of the mean values, and
Upper Chipmunk reach is the least variable while upper Liumchen
reach is the most variable. The least variable single value is the velocity
coefficient of Upper Chipmunk Creek where the standard error is 2% of
the mean. The most variable single value is the width exponent of
upper Liumchen Creek where the standard error is 39% of the mean.
The study of hydraulic geometry typically concentrates on the
analysis of the exponents because these provide a basis for
comparison between sites and between rivers. In contrast, the
coefficients can simply be thought of as the absolute value of w, v
and d for unit stream discharge. The frequency distribution of the AHG
exponents in this study is shown in Fig. 11. The width exponent (b)
varies from 0.05 to 0.45 and has a mean value of 0.20 (the lowest of
the three exponents). The distribution of the width exponents,
although positively skewed, has a strong modal value between 0.1
and 0.2. The depth exponent (f) ranges from 0.09 to 0.47 and is less
variable than the width exponent; it also has a slightly higher mean
value of 0.29. The distribution of the depth exponent is quite
symmetrical about the mean value with a strong modal value at the
mean. The velocity exponent (m) ranges from 0.17 to 0.84, has a mean
value of 0.51, and the distribution is also symmetrical about the mean.
The distribution of the velocity exponent has much lower kurtosis
than either the width or depth exponents with a similar frequency of
occurrence between 0.4 and 0.7. Few values occur outside this range.
The mean values of the velocity exponent are greater than the mean
width and mean depth exponents combined.
Another way of examining the three principal exponents of
hydraulic geometry is to consider all three values simultaneously on
46
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
Fig. 6. Hydraulic geometry of Borden Creek — lower reach.
a ternary plot (Graham and Midgley, 2000). Each data point on the
ternary plot in Fig. 12 represents the three principal exponents of
hydraulic geometry for each study cross section, each plotted on its
own axis. As an interpretive aid, each creek is represented by a
different symbol and shading is used to differentiate between the
various reaches of any given creek.
Most of the data plot in the top right portion of the ternary diagram
suggesting that m N f N b for most of the cross sections in the study.
Indeed, m is the largest of the three exponents at 53 of the 61 cross
sections, with 44 of the 61 cross sections having m N f N b and 9 having
m N b N f. At 31 cross sections, the velocity exponent N0.5, and therefore
greater than both the width exponent and depth exponent combined.
Fewer cross sections plot with other ratios of b to f to m (Fig. 12).
Three cross sections have b N m N f, one cross section has b N f N m, and
four cross sections have f N m N b.
5. Discussion
5.1. Exponent comparison
The mean exponents of 0.20, 0.29, and 0.51 for the water-surface
width (b), mean depth (f), and mean velocity (m), respectively, are
similar to values reported in the literature for gravel-bed and British
Columbia rivers (Table 2). All three means and modes fall within the
reported range. With the exception of Park's (1977) analysis, the
studies included in Table 2 represent the highest mean-velocity
exponents and correspondingly low water-surface width and meandepth exponents found in the literature.
In the context of the published exponents, two characteristics of
the present data stand out. First, the range of exponents calculated for
the 61 study sections approaches the range calculated from a set of
worldwide rivers (Park, 1977). Thus, the study exponents are highly
variable in the global context. Second, the highest velocity exponents
in the present data set appear to be unique. Values as high as 0.73
were reported from Bonanza Creek (Hogan and Church, 1989), but
none higher. Velocity exponents as high as 0.84 seem rare, likely
because of the focus on large rivers with established gauging sections
reported in the world literature. Elsewhere we have reported that
velocity increases rapidly as low-flow increases in these small
mountain channels because of the sensitive dependence on declining
relative roughness of the channel (Reid and Hickin, 2008). Interestingly, the Bonanza Creek study, with similar data collection methods
applied to small ungauged creeks, reports the next highest velocity
exponents reported in the literature.
In 53 of the 61 study cross sections, the velocity exponent (m)
exceeds the water-surface width exponent (b) and mean-depth
exponent (f) and is greater than the sum of b and f in 31 of the 53 cross
sections. This result is noteworthy because it demonstrates the
dominant role of velocity in accommodating changing stream
discharge in these steep settings.
Fig. 7. Hydraulic geometry of Borden Creek — middle and upper reaches.
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
47
Fig. 8. Hydraulic geometry of Chipmunk Creek — lowest and lower reaches.
48
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
Fig. 9. Hydraulic geometry of Chipmunk Creek — upper and tributary reaches.
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
49
Fig. 10. Hydraulic geometry of Foley Creek — lower and upper reaches.
50
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
Fig. 11. Histograms of the AHG exponents (b, f, and m).
Fig. 12. Ternary plot of the AHG exponents (b, f, and m) with analysis subdivisions (after Rhodes, 1977).
51
52
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
Table 2
Comparison of the study AHG exponents to other research with special focus on gravel-bed and British Columbia rivers.
Study
Study data
Park (1977)a
Brandywine Creekb
Coast Mountain streams
of British Columbiac
Green and Birkenhead riversc
River Bollin-Deand
Bonanza and Hangover Creekse
Pacific Northwest streamsf
a
b
c
d
e
f
Number of
Width exponent (b)
observations
Range
Modal class Mean
Depth exponent (f)
Velocity exponent (m)
Range
Range
61
139
7
–
0.05 to 0.45 0.1 to 0.2
0.00 to 0.59 0.0 to 0.1
0.00 to 0.08 0.0 to 0.1
–
–
0.20
Not reported
0.04
0.21
0.09 to 0.47 0.2 to 0.3
0.06 to 0.73 0.3 to 0.4
0.32 to 0.46 0.4 to 0.5
–
–
0.29
not reported
0.41
0.32
0.17 to 0.84 0.4 to 0.5
0.07 to 0.71 0.4 to 0.5
0.46 to 0.69 0.5 to 0.6
–
–
0.51
Not reported
0.55
0.50
8
12
2
76
0.06 to 0.22 0.1 to 0.2
0.01 to 0.33 0.0 to 0.1
0.08 to 0.35 –
–
–
0.11
0.11
0.22
0.49
0.26 to 0.47 0.3 to 0.4
0.26 to 0.63 0.4 to 0.5
0.09 to 0.19 –
–
–
0.37
0.40
0.14
0.38
0.42 to 0.62 0.4 to 0.5
0.24 to 0.68 0.4 to 0.5
0.56 to 0.73 –
–
–
0.50
0.48
0.65
0.13
Modal class Mean
Modal class Mean
Values based on worldwide analysis performed by Park (1977) on many researchers' work.
From Wolman (1955).
From Ponton (1972).
From Knighton (1975).
From Hogan and Church, 1989.
From Castro and Jackson (2001).
This set of AHG relations has significant implications for fish
(salmonids) transiting, spawning, and rearing, in these streams.
Relatively little change in the available habitat occurs as stream discharge
increases because mean depth and water-surface width are relatively
insensitive to changing discharge. As a result, fish must be able to
(i) move to sheltered areas of the stream when velocity exceeds their
sustained swimming capacity (i.e., seek refuge habitats), (ii) inhabit areas
of the channel that remain wetted but are sheltered from the velocity
fluctuations (i.e., inhabit voids in the substrate), or (iii) be adapted to
withstand the velocity fluctuations that are typical in these settings.
Subdividing the ternary diagram into a series of cells aids the
visual analysis of the exponents in this m–f–b space (Rhodes, 1987).
Each of the 10 cells is interpreted as a channel type; the study data
plot in cells 1 to 6 and 10 in Fig. 12.
The first division of the ternary diagram is made horizontally,
separating cells 1 and 2 from the rest of the diagram (Fig. 12). Thirtyone of the 61 sets of hydraulic geometry plot above this line where the
velocity exponent m is greater than the sum of the width exponent (b)
and depth exponent (f). In these cells, the stream velocity increases
faster than the width and depth combined.
The next set of conditions produces a set of obliquely angled lines
that cut the diagram from right to left. These conditions are m = f,
m = 2/3f, and m = 1/2 f. When m N f (53 of the 61 cross sections), the
mean velocity is increasing faster than the mean depth. This is
interpreted as implying an increase in flow competence with
increasing discharge (Wilcock, 1971). When m N 2/3f, resistance to
flow as measured by Manning's n decreases with increasing discharge
(Rhodes, 1977). All but cross section CHK-Trib4 meet this criterion.
This lone cross section likely has a different set of exponents from the
other study sections because of flow constriction and backwater
effects related to an adjacent log jam in the creek. The remaining
condition is m N 1/2 f. When this condition is met, the Froude number
increases with increasing discharge, and critical flow is therefore
possible (Rhodes, 1977). This condition is met by all but one cross
section: CHK-Trib4 plots below this line again.
The ternary diagram is split vertically by the condition b = f
(Fig. 12). To the right of the line (cells 2, 4, 6, 8, and 10), f N b and mean
depth makes a proportionally larger contribution to the increasing
flow area, relative to the water-surface width, as discharge increases.
Because the depth of flow increases faster than the width, the widthto-depth ratio decreases as the discharge increases. This condition is
thought to reflect stable, narrow channels where the banks are
relatively resistant to erosion (Rhodes, 1977). As expected, this
condition corresponds to a large majority of the cross sections (47 of
the 61). The banks of most of the study reaches are composed of
boulders, cobbles, and bedrock and are well vegetated. Little bank
erosion is evident in any of the reaches.
In the study reaches, the b = f division does not separate wide,
laterally unstable reaches from more stable reaches; instead the division
separates the individual cross sections in a single reach. Five reaches
(upper Liumchen, lower Borden, upper Borden, lowest Chipmunk, and
lower Chipmunk) all have two or more cross sections where b N f. What
separates many of these cross sections from the others of the same reach
is the presence of bars in the channel that are emergent at low-flow and
submerged at moderate-flow. As the bars submerge, the water-surface
width expands quickly as discharge increases. Twelve of the fourteen
cross sections where b N f have significant bars on the cross section. The
other two cross sections (LCH-U1 and BRDN-U2) have very large
boulders that are emergent at low-flow yet submerged at high flow. At
these cross sections, the large boulders act like bars at the other cross
sections (i.e., they are emergent at low-flow and are submerged at
higher flow). Thus, the criterion of b N f seems to be a good indicator of
bar inundation in the range of discharges observed. This explanation is
not universal, however, because an additional four cross sections do not
meet the b N f criterion but have bars that are inundated in the range of
flow observed. At each of these sections, however, the width and depth
exponents are nearly equal, suggesting that the bars have some small
affect on the hydraulic geometry.
5.2. Low-flow hydraulic geometry discontinuities
In general, the data do not exhibit low-flow hydraulic geometry
discontinuities. Instead, the power curves fit the data remarkably well
throughout the data range. This well-defined fit is reflected in the R2
values of which 95% are N0.70, leaving only eight of 173 where
R2 b 0.70. With this level of accordance between the power function
and the data, it is difficult to argue that discontinuities are common in
these cross sections.
To test for the presence of discontinuities, stream discharge was
measured to extreme low-flow conditions on lower Borden Creek in
summer of 2003 when the river partly dried. Despite observations to
near zero discharge in this reach, no systematic low-flow graphdiscontinuity was observed in the data (Fig. 7).
Perhaps the best candidate for a reach-wide AHG discontinuity is
lower Frosst Creek (Fig. 4). In this reach, a transition from a low-flow
geometry to a moderate-flow geometry occurs in all five cross sections
at about 0.2 m3/s. Apparently, lower Frosst Creek is unique because in
no other reach is a low-flow discontinuity so clearly evident although
there are hints of such inflections in individual sections on other
channels (e.g., see the depth and velocity trends in sections FST-U1,
FST-U3, FST-U5, BRDN-L4 and BRDN-L5 in Figs. 4 and 6).
When the lower Frosst Creek data are separated into two ranges at
about 0.2 m3/s, very different component AHGs apply (Fig. 4). In the
lower discharge range (pre-discontinuity), the width and depth
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
exponents are very small (even negative) reflecting the fact that the
water-surface width and mean depth vary little with stream
discharge. In contrast, the velocity exponents (m) are very high,
ranging from 0.89 to 1.07, indicating that changes in velocity account
for nearly all of the increasing stream discharge. In the higher
discharge range (post-discontinuity), the response changes and the
width exponent (b) increases, the depth exponent (f) generally
increases, and the velocity exponent (m) decreases relative to those
observed in the pre-discontinuity data range. Post-discontinuity, the
exponents approach the values calculated for the other study reaches.
One explanation for the abnormally high velocity exponents and
low width and depth exponents in the pre-discontinuity range is that
the flow is confined within an inset channel and spills out over a
greater portion of the stream bed at or near the discontinuity. This
explanation, however, is not supported by the surveys of the cross
sections (Reid, 2005). While cross sections FST-L4 and FST-L5 do have
large bars that are inundated during the observed flow range, the
discontinuity does not occur at a point where flow leaves an inset
channel and spills across the bar. Instead, bar inundation begins at a
mean depth of 0.15 and 0.20 m, respectively, (0.12 and 0.50 m3/s),
and no change to the hydraulic geometry is evident at this stream
discharge. In addition, cross sections FST-L1, FST-L2, and FST-L3 do not
have large bars on the cross section yet their hydraulic geometries do
include a discontinuity. Also, this explanation does not explain the
abnormally low depth exponents in the pre-discontinuity data. If flow
were confined within a small inset channel, both the depth of flow
and velocity likely would increase rapidly as the inset channel is filled.
An alternative explanation for the observations is that the
discontinuity is due to measurement error. The contracting watersurface width at cross section FST-L2 and decreasing depth at cross
section FST-L1 provide support for this argument because it is unlikely
that either variable would decrease as more flow is conveyed through
the cross section. These two negative exponents are likely due to
measurement error. The changes in the hydraulic geometry observed
about the discontinuity, however, are larger than those that can be
attributed to measurement error alone. In addition, error is likely
random, yet the discontinuity occurs at the same stream discharge in
all five of the study sections. Thus, it is likely that the discontinuities
do indeed reflect a physical property of the cross section and not
simply measurement error.
Another explanation may be that the discontinuity occurs at a
stream discharge where the bed of the creek is just filled and further
width expansion is limited by steeper banks. Hogan and Church
(1989) observed this type of discontinuity in small creeks on the
Queen Charlotte Island of British Columbia. However, like the inset
channel explanation, this hypothesis is not supported by the cross
section surveys. The discontinuity occurs at very low stream discharge
when only a small portion of the stream bed is wet. This leaves a
considerable area for the flow to widen across the bed before the
banks are encountered. In contrast to Hogan and Church's data, the
width exponent increases after the discontinuity.
The hydraulic discontinuities may be internal to the flow (Wohl,
2007; Reid and Hickin, 2008). For example, the onset of eddy shedding
may be abruptly increasing resistance to flow as discharge approaches
0.2 m3/s. But we have no direct evidence of this effect so the hydraulic
geometry discontinuities in this reach remain unexplained.
Despite the apparent consistency to the changes in the hydraulic
geometry at stream discharge less than and greater than 0.2 m3/s,
very few of these changes are significant at the 95% confidence level
because of the small and variable data set. More data are required to
clarify these results.
5.3. Predictability of the hydraulic geometry in steep mountain streams
Somewhat surprisingly, the AHG of steep mountain streams can be
reasonably predicted from just a few measurements of the primary
53
flow variables and stream discharge. The general lack of hydraulic
geometry discontinuities in the range of conditions observed,
combined with the good fit of the power function to the data,
suggests that the relationship developed in this low-flow range can be
reliably defined with fewer measurements than we have used in this
study. However, like most curve-fitting exercises, the greater the
range of flow observed and number of observations made, the better
the confidence in the predictions.
While conditions at the cross section appear predictable from a
few measurements, extrapolating the results from one cross section to
another in the same reach involves large errors. The section-tosection variability of the exponents and coefficients, even when they
are located in similar channel units such as riffles, prevents accurate
extrapolation to unmeasured cross sections (Reid and Hickin, 2008).
Thus, to predict the conditions at any particular cross section,
apparently measurements must be made on that cross section.
Hence, for fisheries studies, the focus of study shifts to the limiting
cross section for a particular life stage or species. In other words,
rather than conducting a regular survey of channel cross sections such
as that employed here, it likely is more effective for environmental
design purposes simply to identify in reaches critical sections that are
likely to limit fish survival (because of constraints on passage, food
production or spawning). Once identified, the mean hydraulic
conditions at those sections can be adequately described by the
methods used in this study.
5.4. Temporal changes in hydraulic geometry — the effect of the October
2003 storm
On 17 October 2003, a large Pacific storm on the British Columbia
coast produced floods ranging in size from 200-year events in the
Squamish–Pemberton corridor to 15-year events in the Chilliwack
River tributaries (Fig. 2). Surprisingly, the 15-year event in the
Chilliwack watershed had very little impact in most of the study
reaches. Only one reach was significantly changed, while four reaches
were altered at some of the sections; the remaining eight reaches
appeared unchanged. The most significantly affected reach was upper
Frosst Creek, which doubled in width from 8 to 15 m, a change that led
us to exclude the site from further observation after the flood because
the measurements would not have been part of a consistent data set.
The four moderately affected reaches were lower Frosst, lower
Borden, middle Borden, and Chipmunk Tributary reaches. Of the 61
cross sections in the survey, 20 showed some signs of flood alteration
of the AHG and these were analysed further.
Not only did the October 2003 flood produce varied results
between the study reaches but it also produced varied results
between the sections in the affected reaches. A complete range of
results is evident from increase, decrease, or no change to the
coefficients and exponents (Figs. 4–10). In spite of the varied
response, however, several trends are evident.
In general, the flood produced incision in the affected reaches. The
incision is minor but can be seen in the AHG relationships as a drop in
the water-surface width coefficient (a) and a corresponding increase
in either the depth coefficient (c) or the velocity coefficient (k). The
decline in the width coefficient is expected because incision in the
cross section tends to narrow the section in the absence of
concomitant bank erosion. Sixteen of the 20 cross sections in the
affected reaches exhibit this decline in the width coefficient.
The decline in width coefficient does not necessarily correspond to
an increase or decrease in the width exponent (b). Indeed, the
direction of the coefficient change is unrelated to that of the exponent
change. This suggests that while incision has created a narrower cross
section at the intercept value of 1 m3/s, it has not necessarily
steepened the banks. The sideslopes of the cross section (elevated
marginal bed surfaces and lower banks) control the rate of change of
water-surface width as the discharge increases. Incision, apparently,
54
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
while enough to alter the coefficient, is not great enough to alter the
banks of the channel. This result is consistent with field observations
made following the flood.
Perhaps the most orderly changes occurred in middle Borden
Creek where all of the width coefficients declined (Fig. 7). This pattern
is consistent with reach-wide channel incision. In this case, the drop in
the coefficient corresponds with a rise in the water-surface width
exponent (b) (i.e., the water-surface width is increasing faster with
increasing discharge than before the flood). Observations in the reach
following the storm indicate that the channel banks were unchanged.
This suggests that incision produced a narrower low-flow channel
(lower coefficient) and general flattening or decrease in concavity
across the streambed.
The statistical significance of the changes in the coefficients and
exponents at the 20 flood-affected sections were tested using a
standard t-test. In general, most of the changes in coefficients (55–75%
of 20 tested sections) and exponents (in 55–90% of 20 tested sections)
of the AHG were not large enough to be statistically significant. This
statistical indifference is due to two reasons. First, the difference
between the pre- and the post-flood values are small because of the
minor changes in most of the cross sections. Indeed, the changes in the
cross sections are considered so minor that they did not warrant a
new survey. Second, even when the change in the values is
considerable, the lack of data in each period (given the variability of
the data about the relationships) precludes a statistical difference.
This is especially true for the Chipmunk Tributary reach where only
two observations were made after the October 2003 flood (Fig. 9).
Despite the lack of statistical differences in the pre- and post-flood
AHG relationships, temporal analysis of the data remains important.
Analysing the entire study period without considering channelaltering events can lead to false results. For example, the width
exponent for the entire study period at cross section BRDN-M4,
although not statistically different from zero, is −0.03 (Fig. 7). A
negative exponent means that the water-surface width decreases as
the stream discharge increases — an unlikely result. Instead, breaking
the analysis into pre-flood and post-flood periods based on knowledge of the history of flow at the site and channel conditions before
and after a large event yields a more rational result (Fig. 7). The width
exponent for the pre-flood and post-flood periods are 0.08 and 0.16,
respectively. These values are low but positive and significantly
different from zero, and there is a distinct shift in the data for the two
periods.
Also of note is the improvement of the regression fit when
considering pre- and post-flood periods. The width R2 term improves
from a low 0.04 for the entire study period to 0.97 and 0.96 for the
pre- and post-flood periods, respectively. Less dramatically, the depth
R2 improves from 0.90 to 0.97 and 1.00. The velocity R2 terms stay
high in all of the analyses because there was very little difference in
the velocity data when the study period was split into pre-flood and
post-flood periods.
more variable than typical hydraulic geometry studies with a
variability approaching the differences observed among rivers
globally.
Despite the general good fit between the power function and the
at-a-station observations, some variability about the relationships still
exists. Some of this variability can be attributed to temporal changes
in the cross section as large floods alter the section. Splitting the
analysis into pre- and post-flood time periods is an effective way of
dealing with this issue provided an adequate number of observations
are made in each time period. Floods do not affect all reaches and
cross sections equally, however, and care should be taken to relate any
changes in the calculated geometry to observations of change at the
cross section.
A second type of variability in the AHG is hydraulic geometry
discontinuities. Splitting the analysis about these discontinuities may
improve AHG, but it is important to relate the inferred discontinuities
to changes in the channel geometry or hydraulics of the section. These
discontinuities seem uncommon in the conditions measured, but may
be more common at higher stream discharge.
These findings suggest that hydraulic geometry can be considered
a reasonable technique for determining minimum environmental
flows and for ramping rates of mountain streams on stream habitat if
information about the limiting cross section conditions for a particular
species or life stage is available. In other words, if the critical
conditions for a species can be quantified and the limiting cross
section identified (e.g., minimum mean depth of 10 cm for rearing
juvenile salmon over riffles), then hydraulic geometry is a reasonable
approach to answering the question: what stream discharge must be
maintained to provide these conditions? At a minimum, three visits to
a properly monumented cross section spanning a reasonably wide
range of flows are required to obtain an estimate of at-a-section
hydraulic geometry parameters, although confidence in the estimates
would be improved by additional measurements. At-a-station
hydraulic geometry is not, however, generally transferable between
the cross sections of a reach. Therefore, measurements are required at
each cross section of interest.
Acknowledgements
We wish to thank field assistants Amada Denney, Emily Huxter,
and Jane Bachman of Simon Fraser University, and Kaz Shimamura for
his cartographic assistance. We are especially indebted to John
Faustini, Robert Jarrett and an anonymous reviewer whose thoughtful
reviews of the original manuscript significantly improved the paper.
This research was funded in part by a Natural Sciences and
Engineering Research Council (NSERC) grant to EJH and by research
grants to DER from the Province of British Columbia (Ministry of
Environment) and from Simon Fraser University.
References
6. Summary and conclusions
In general, simple power functions describe remarkably well the
relationship between stream discharge and the primary flow variables
of water-surface width, mean depth, and mean velocity in these small
steep mountain channels. Of the 173 R2 values calculated for the
study, 165 were N0.7. All of the primary hydraulic variables increase
with increasing discharge, but velocity increases faster than the other
two variables. In 31 of the 61 cross sections, the velocity exponent (m)
is greater than both the width exponent (b) and depth exponent (f)
combined. In 53 of the 61 cross sections, the velocity exponent is the
largest of the three exponents.
While the power functions describe well the relationships at a
cross section, considerable variation exists among the cross sections of
a single reach. The exponents and coefficients in these streams are
B.C. Ministry of Forests, 1995. Coastal Watershed Assessment Procedure Guidebook.
Forest Practices Branch, Ministry of Forests, Victoria, BC.
B.C. Ministry of Forests and Range, 2008. Biogeoclimatic Zones of British Columbia.
http://www.for.gov.bc.ca/hre/becweb/resources/maps/map_download.html.
Babakaiff, S., 2004. At-a-station Hydraulic Geometry: Useful for Considering Instream
Flows or Simply a Bit of Outdated Empirical Science? Draft, unpublished report for
the B.C. Ministry of Water, Land and Air Protection, Victoria, BC.
Bathurst, J.C., 2002. At-a-site variation and minimum flow resistance for mountain
rivers. Journal of Hydrology 269, 11–26.
Benson, M.A., 1965. Spurious correlation in hydraulics and hydrology. Journal of the
Hydraulics Division ASCE 91 (4), 35–42.
Castro, J.M., Jackson, P.L., 2001. Bankfull discharge recurrence intervals and regional
hydraulic geometry relationships: patterns in the Pacific Northwest, USA. Journal of
the American Water Resources Association 37, 1249–1262.
Church, M., 1980. On the Equations of Hydraulic Geometry. Department of Geography,
University of British Columbia, Vancouver, BC.
Clague, J.J., 1981. Late Quaternary geology and geochronology of British Columbia. Part
2: Summary and discussion of radiocarbon-dated Quaternary history. Geologic
Survey of Canada Paper No. 80-35.
D.E. Reid et al. / Geomorphology 122 (2010) 39–55
Clifford, N.J., 1996. Classics in physical geography revisited: Leopold, L.B., Maddock, T.M.
Jr., 1953, The Hydraulic Geometry of Stream Channels and Some Physiographic
Implications, USGS Professional Paper 252. Progress in Physical Geography 20,
81–87.
Comiti, F., Mao, L., Wilcox, A., Wohl, E.E., Lenzi, M.A., 2007. Field-derived relationships
for flow velocity and resistance in high-gradient streams. Journal of Hydrology 340,
48–62.
Department of Fisheries and Oceans (DFO), 2003. Daily Rainfall at the Chilliwack River
Hatchery. Data file from Chilliwack Hatchery staff, Chilliwack, BC.
Ferguson, R.I., 1986. Hydraulics and hydraulic geometry. Progress in Physical
Geography 10, 1–31.
Graham, D.J., Midgley, N.G., 2000. Graphical representation of particle shape using
triangular diagrams: an Excel spreadsheet method. Earth Surface Processes and
Landforms 25, 1473–1477.
Grant, G.E., Swanson, F.J., Wohlman, M.G., 1990. Pattern and origin of stepped-bed
morphology in high-gradient streams, western Cascades, Oregon. Geological
Society of America Bulletin 102, 340–352.
Hey, R.D., 1978. Determinate hydraulic geometry of river channels. Journal of the
Hydraulics Division American Society of Civil Engineers 104, 869–885.
Hickin, E.J., 1995. Hydraulic geometry and channel scour, Fraser River, British Columbia,
Canada. In: Hickin, E.J. (Ed.), River Geomorphology. John Wiley and Sons Ltd.,
Chichester, UK, pp. 155–167.
Hogan, D.L., Church, M., 1989. Hydraulic geometry in small, coastal streams — progress
toward quantification of salmonid habitat. Canadian Journal of Fisheries and
Aquatic Sciences 46, 844–852.
Jarrett, R.D., 1984. Hydraulics of high-gradient streams. Journal of Hydraulic
Engineering, ASCE 110 (11), 1519–1539.
Jarrett, R.D., 1990. Hydrologic and hydraulic research in mountain rivers. Water
Resources Bulletin 26, 419–429.
Jowett, I.G., 1997. Instream flow methods: a comparison of approaches. Regulated
Rivers-Research & Management 13, 115–127.
Jowett, I.G., 1998. Hydraulic geometry of New Zealand rivers and its use as a
preliminary method of habitat assessment. Regulated Rivers-Research & Management 14, 451–466.
Kellerhals, R., Bray, D.I., 1971. Sampling procedure for coarse fluvial sediments. Journal
of Hydraulic Engineering-ASCE HY 8, 1165–1179.
Knighton, A.D., 1975. Variations in at-a-station hydraulic geometry. American Journal of
Science 275, 186–218.
Knighton, A.D., 1998. Fluvial Form and Processes. Oxford University Press Inc., New
York.
Leopold, L.B., Maddock Jr., T., 1953. The hydraulic geometry of stream channels and
some physiographic implications. U.S. Geological Survey Professional Paper 252, US
Government Printing Office, Washington, DC. (57 pp.).
Lewis, L.A., 1966. The adjustment of some hydraulic variables at discharges less than
one CFS. The Professional Geographer 18, 230–234.
55
Montgomery, D.R., Buffington, J.M., 1997. Channel-reach morphology in mountain
drainage basins. Geological Society of America Bulletin 109, 596–611.
Park, C.C., 1977. World-wide variations in hydraulic geometry exponents of stream
channels: an analysis and some observations. Journal of Hydrology 33, 133–146.
Ponton, J.R., 1972. Hydraulic geometry in the Green and Birkenhead Basins, British
Columbia. In: Slaymaker, O.H., McPherson, H.J. (Eds.), Mountain Geomorphology:
Geomorphological Processes in the Canadian Cordillera. Tantalus Research Ltd.,
Vancouver, BC, pp. 151–160.
Rantz, S.E., 1982. Measurement and computation of streamflow; Volume 1, Measurement of stage and discharge; Volume 2, Computation of discharge. U.S. Geological
Survey Water Supply Paper 2175. US Government Printing Office, Washington, DC.
Reid, D.E., 2005. Low-flow hydraulic geometry of small, steep streams in southwest
British Columbia. M.Sc.Thesis, Simon Fraser University, Burnaby, BC.
Reid, D.E., Hickin, E.J., 2008. Flow resistance in steep mountain streams. Earth Surface
Processes and Landforms 33, 2211–2240.
Rhodes, D.D., 1977. The B–F–M diagram: graphical representation and interpretation of
at-a-station hydraulic geometry. American Journal of Science 277, 73–96.
Rhodes, D.D., 1987. The B–F–M diagram for downstream hydraulic geometry.
Geografiska Annaler Series A—Physical Geography 69, 147–161.
Rice, S., Church, M., 1996. Bed material texture in low order streams on the Queen
Charlotte Islands, British Columbia. Earth Surface Processes and Landforms 21,
1–18.
Saunders, I.R., Clague, J.J., Roberts, M.C., 1987. Deglaciation of Chilliwack River valley,
British Columbia. Canadian Journal of Earth Sciences 24, 915–923.
Schlager, W., Marsal, D., van der Geest, P.A.G., Sprenger, A., 1998. Sedimentation rates,
observation span, and the problem of spurious correlation. Mathematical Geology
30, 547–556.
Stewardson, M., 2005. Hydraulic geometry of stream reaches. Journal of Hydrology 306,
97–111.
Tabata, K.K., Hickin, E.J., 2003. Interchannel hydraulic geometry and hydraulic efficiency
of the anastomosing Columbia River, southeastern British Columbia, Canada. Earth
Surface Processes and Landforms 28, 837–852.
Water Survey of, Canada, Environment, Canada, 2004. Water Survey of Canada
Streamflow Data. Water Survey of Canada, Environment Canada (Retrieved
November 2005 from http://www.wsc.ec.gc.ca).
Wilcock, D.N., 1971. Investigations into the relations between bedload transport and
channel shape. Geological Society of America Bulletin 82, 2159–2176.
Wohl, E.E., 2007. Channel-unit hydraulics on a pool-riffle channel. Physical Geography
28, 233–248.
Wohl, E.E., Wilcox, A., 2005. Channel geometry of mountain streams in New Zealand.
Journal of Hydrology 300, 252–266.
Wolman, M.G., 1955. The natural channel of Brandywine Creek Pennsylvania.
Geological Survey Professional Paper 271. US Government Printing Office,
Washington, DC (56 pp.).