Consistency of scaling relations among bedrock and alluvial channels



[1] This paper presents field data on channel geometry and potential control variables from 47 field settings representing a diverse range of environments. These data are used to evaluate existing scaling relationships used in models of the evolution of bedrock channel geometry and to test the hypothesis that channel width (w) increases more slowly and depth (d) more rapidly in relation to discharge (Q) or drainage area (A) as substrate resistance increases. For this data set, wA0.3, wQ0.5, dA0.2, and dQ0.3. The w-A and w-Q relations are close to those found by previous investigators. The d-A and d-Q relations have not previously been reported for bedrock channels. Examination of trends within the data does not support the hypothesis and, instead, suggests that the erosional resistance of channel boundaries is not the primary control on scaling relations for channel geometry.

1. Scaling Relations for Bedrock Channels

[2] The most widely used approach to modeling bedrock channel incision in the context of landscape evolution assumes that incision rate (E) is proportional to stream power [Howard et al., 1994], which is a function of discharge (Q), or the commonly used surrogate drainage area (A), and channel gradient (S)

equation image

where k is a dimensional constant for erodibility that depends on bedrock properties, and m and n are dimensionless exponents. Recent papers on bedrock channel processes in the context of landscape evolution consistently cite the need for further field data to test model assumptions about how bedrock erodibility influences incision rate and about how bedrock channel geometry scales with A, S, and Q in field settings with differing lithology, discharge, sediment supply, and tectonic uplift [Montgomery and Gran, 2001; Whipple, 2004; Wobus et al., 2006]. In this paper, we present new field data that can be used to test assumptions of scaling in bedrock channels. Scaling of bedrock channel geometry is particularly critical in models of bedrock channel evolution through time and space.

[3] The simplest way to scale bedrock channel geometry is to use downstream hydraulic geometry relations established for alluvial channels. Downstream hydraulic geometry relations for width (w) and depth (d) take the form of

equation image
equation image

In their original formulation of downstream hydraulic geometry on the basis of data from 10 rivers, Leopold and Maddock [1953] proposed that average values of exponents for alluvial channels are b = 0.5 and f = 0.4. Subsequent studies drawing on larger data sets [Park, 1977] expanded the range of values for b (0.03–0.89) and f (0.09–0.70), but 0.5 and 0.4 remain accurate averages for alluvial channels worldwide.

[4] The rationale behind downstream hydraulic geometry is that channel shape, expressed via w and d, determines the distribution of velocity (v) and shear stress, which in turn interact with the erodibility of the bed and banks to control cross-sectional shape and boundary roughness as Q and sediment supply change downstream [Leopold and Maddock, 1953]. The rates at which w, d, and v increase downstream are determined by the need to transport sediment load as Q and S change. Numerous interrelated variables thus influence the coefficients and exponents in equations (2) and (3).

[5] Numerical derivations of scaling relations for bedrock channels have been derived from: flow resistance equations and mass conservation principles, producing wQ0.38 and wS−0.2 [Finnegan et al., 2005]; assumptions that erosion rate scales with local shear stress, which results in wQ0.4 and wS−0.2 [Wobus et al., 2006]; and minimization of potential energy, for which wA0.5 [Turowski et al., 2007]. Such derivations produce a small range of values for b. Some numerical derivations have also assumed constant w/d ratios for bedrock channels [Finnegan et al., 2005; Wobus et al., 2006], whereas others assume a relation between w/d and A or Q [Turowski et al., 2007].

[6] Although several investigators have documented field-based downstream hydraulic geometry relations for bedrock channels within the past decade, the number of such data sets remains small relative to alluvial channels (Table 1). Following the lead of Montgomery and Gran [2001], most of the bedrock channels studies substitute the more readily and consistently determined A for Q. Scaling of Q in relation to A depends on site-specific hydroclimatology

equation image

Reported values of e vary from 0.4 to 1.0 [Knighton, 1987; Jennings et al., 1994; Cathcart, 2001; Eaton et al., 2002; Flores et al., 2006], and changes in hydroclimatology within a drainage basin as a function of elevation, for example, can produce differences in the scaling relation [Flores et al., 2006]. The uncertainties introduced by substituting A for Q in downstream hydraulic geometry relations for bedrock channels without specifying the A-Q scaling within a drainage have not yet been rigorously addressed.

Table 1. Summary of Published Data Sets for Downstream Hydraulic Geometry of Bedrock Channelsa
Field AreaDrainage Area (km2)Downstream Hydraulic Geometry RelationSource
  • a

    Multiple relations from one study represent best fit equations for individual river basins.

California, Washington, and Oregon, United States0.1–1000w ∼ 0.007A0.47, w ∼ 0.054A0.32, w ∼ 0.002A0.51, w ∼ 0.015A0.42, w ∼ 0.002A0.53, w ∼ 0.008A0.37Montgomery and Gran [2001]
North California, United States4–19wA0.4Snyder et al. [2003a]
Washington, United States390w ∼ 4.20A0.42, w ∼ 10.82Q0.47Tomkin et al. [2003]
Southeast Australia11,000w ∼ 0.0004A0.55, w ∼ 0.001A0.53, w ∼ 0.012A0.41 (overall)Van der Beek and Bishop [2003]
South California, United States3–8w ∼ 0.08A0.33 (low uplift), w ∼ 0.007A0.42 (high uplift), wA0.4 (overall)Duvall et al. [2004]
Virginia, Maryland, California, Oregon, and Washington, United States, and China0.1–1000w ∼ 3.3A0.36Whipple [2004]
Italy65wA0.35–0.5Cowie et al. [2006]

[7] Part of the difficulty in applying downstream hydraulic geometry to bedrock channels, and in distinguishing between Q and A, is that of choosing a consistent and relevant magnitude and recurrence interval for Q. Leopold and Maddock [1953] used mean annual discharge for downstream hydraulic geometry in alluvial channels. The higher erosional thresholds of bedrock channels imply that it may be more appropriate to relate channel characteristics to high-magnitude flows with long recurrence intervals than to flows that recur approximately once a year because only the larger flows generate sufficient hydraulic forces to substantially influence channel geometry through erosion and deposition [Baker, 1988; Wohl, 1992; O'Connor, 1993]. The recurrence interval of such large flows varies widely between sites, however, and is often poorly captured by systematic gage records of short-duration relative to the recurrence interval of the flows. Consequently, although interactions between discharge variability and erosional thresholds likely affect downstream hydraulic geometry scaling of bedrock channels [Dadson et al., 2003; Snyder et al., 2003b; Lague et al., 2005], a consensus does not yet exist on how to choose the most relevant Q parameter for a single site or among multiple sites.

[8] The field-based data summarized in Table 1 indicate a range of 0.32–0.55 for b, which is similar to the range from numerical derivations and narrower than the range documented in the more extensive data set for alluvial channels. This relatively restricted range may be an artifact of the limited field data to date, but it is interesting given the multiple factors that likely influence the rate of change of bedrock channel w in relation to either Q or A. Examination of such factors has thus far focused mainly on tectonic regime and sediment supply.

[9] Estimating w from satellite images for channels draining 20,000–60,000 km2 in the Nepalese Himalaya, Lavé and Avouac [2001] found that w varied inversely with incision rate in larger rivers, whereas both w and channel gradient (S) were affected in smaller rivers, such that w decreased as uplift rate increased. Subsequent field investigations document analogous trends in which: smaller channels respond to uplift with changes in both w and S [Duvall et al., 2004]; higher uplift produces narrower channels until some threshold of differential uplift also requires change in S [Amos and Burbank, 2007]; or w does not scale with A in active uplift zones undergoing transient channel response [Cowie et al., 2006; Whittaker et al., 2007]. Physical experiments [Turowski et al., 2006] and numerical models [Turowski et al., 2007] also support the observation that w decreases to a limit value as uplift rate increases.

[10] Shepherd's [1972] physical experiments indicated that vertical erosion of bedrock channels is most pronounced when all available sediment is entrained and the tool effect predominates, whereas lateral erosion becomes more important when not all sediment is entrained and the cover effect comes into play. Field-based observations also indicate that sediment supply influences the rate and type of erosion of bedrock channel boundaries [e.g., Seidl et al., 1994; Stock et al., 2005]. These observations led Sklar and Dietrich [2004, 2006] to distinguish tool (abrasion of channel boundaries) and cover (protection of channel bed) effects of sediment as a reflection of the balance between sediment supply and transport capacity. The difficulty of quantifying bed load dynamics in actual bedrock channels has resulted in most subsequent work taking the form of physical or numerical models. Physical experiments indicate that transient channel response to a fixed sediment supply produces wider channels at low transport rates [Finnegan et al., 2007], which corresponds to Turowski et al.'s [2008] observations that greater sediment supply produces wider channels and more lateral mobility along bedrock channels in Taiwan.

[11] Less attention has been given to how variations in erosional threshold, as expressed through some measure of bedrock strength or erosional resistance, influence downstream hydraulic geometry. Wohl [1998] conceptualized bedrock channel geometry as a continuum from the fullest expression of flow structure (and potentially the most consistent downstream hydraulic geometry scaling) in channels with the most homogeneous and least resistant substrates, to the strongest substrate influences (and potentially more inconsistent or poorly developed downstream hydraulic geometry scaling) in channels with the most heterogeneous and resistant substrates. Wohl and Merritt [2001] related different types of bedrock channel geometry to relative substrate resistance, and recent field investigations emphasize the influence of substrate on erosional processes and the formation of strath terraces [Montgomery, 2004; Wohl, 2008]. Although many investigators now report Selby rock mass strength [Selby, 1980] for individual field sites, the model developed by Turowski et al. [2007] represents one of the few studies to systematically evaluate channel geometry scaling with respect to erosional thresholds. They found that channel geometry is less sensitive to rock strength (measured as the ratio of Young's modulus to the square of rock tensile strength [Sklar and Dietrich, 2001, 2004]) as A increases, with strong dependencies present only for very hard rocks in which discontinuities such as joints do not dominate processes and rates of fluvial erosion. No one has yet systematically examined bedrock channel scaling relations in relation to rock resistance.

2. Objectives and Hypotheses

[12] As summarized above, work thus far indicates that downstream hydraulic geometry relations for w-A in bedrock channels have b values in the relatively narrow range of 0.3–0.5, with demonstrated variations in scaling partly dependent on tectonic uplift and sediment supply, and likely also reflecting Q-A scaling and erosional thresholds in relation to discharge regime and lithology. The first objective of this paper is to test existing scaling relationships for w-A and w-Q for a new data set of bedrock channels. A second objective is to examine potential effects of erosional thresholds on scaling relations by testing for differences in w/d ratio, and coefficients and exponents of w-A and w-Q relations as a function of bedrock erosional resistance. We address the first two objectives using a data set of 47 bedrock channel sites that include a wide range of Q, A, S, channel geometry, and rock resistance.

[13] The greater erosional resistance of bedrock channel boundaries relative to the boundaries of most alluvial channels suggests at least two scenarios for differences in channel geometry between the two channel types: (1) bedrock channels have a lower rate of change in w in relation to A or Q than alluvial channels or (2) bedrock channels have a similar rate of change but a consistently lesser w for a given value of A or Q than an alluvial channel. Either scenario could result from supply limited conditions, which are more common in bedrock channels than in alluvial channels [Sklar and Dietrich, 2004]. As noted previously, limited sediment supply enhances the tool effect of available sediment and promotes vertical channel erosion. Lesser downstream increases in w, or similar rates of change but consistently smaller w, for bedrock channels could also reflect the confinement associated with most bedrock channels, which allows erosive force to increase more rapidly with increasing Q and to be most effectively exerted against the channel bed [Shepherd and Schumm, 1974; Baker, 1988; Duvall et al., 2004]. The third objective of this paper is to explore these alternatives by comparing downstream hydraulic geometry relations for 12 paired bedrock and alluvial channels. We hypothesize that the greater resistance of bedrock channel boundaries relative to alluvial channel boundaries will result in a slower rate of increase in w with Q or A, such that the w/d ratio of bedrock channels will be lower, and the w-Q exponent will be lower, than those of alluvial channels.

[14] The data presented here represent more diverse field settings, although they do not have a greater range of w or A values, than those reported previously. The data here do have the advantage of consistently applied field criteria for estimating Q, which facilitates evaluation of w-Q as well as w-A relations. The final objective of this paper is to make available to the community field data on bedrock channel geometry from diverse environments. The 47 bedrock channel sites presented in this paper represent a broad range of drainage area (0.2–17,380 km2), lithology, and climate, but differ from much of the other bedrock channel data in the literature in that they are mostly not from regions undergoing tectonic uplift at present.

3. Methods

3.1. Data Types and Sources

[15] All of the data used in these analyses were originally collected for other purposes by Wohl and various colleagues. Many of these data were used in a discriminant analysis of channel morphologic type by Wohl and Merritt [2001] but supplemental data have also been added. The bedrock channel data, including the original citation for each field site, are summarized in Table 2. The bedrock and alluvial channel data are summarized in Table 3. Each row in Tables 2 and 3 represents average values for a study reach, defined as a length of channel at least ten times the width of the channel during peak annual flow, and having consistent morphology and gradient. Channel geometry of each reach was characterized through field surveys of channel dimensions and bed gradient.

Table 2. Characteristics of Each Bedrock Channel Study Reach
SiteA (km2)Q (m3/s)S (m/m)wa (m)d (m)w/dD50, D84 (m)SelbyΩ (W)ω (W/m2)Ω/SelbySHbSource
  • a

    Variables for channel geometry, grain size, substrate resistance, and hydraulics are averaged over multiple surveyed cross sections in a step backwater modeling reach.

  • b

    Substrate heterogeneity is a categorical variable: 1, homogeneous; 2, moderately homogeneous; 3, moderately heterogeneous; 4, heterogeneous. See Wohl and Merritt [2001] for more extensive description.

  • c

    Indicates slot canyon.

Shichiri, Japan101500.00916.5116.50.08, 0.155013,2304802653Wohl and Ikeda [1998]
Torii 1, Japan0.340.0240.850.05, 0.10572200180393Wohl and Ikeda [1998]
Rio Puerco, New Mexico6101080.0229., 0.105523,28023504233Wohl and Merritt [2001]
New River1738047700.002265644.2 7693,57035012312Wohl and Merritt [2001]
Anacostia 2, Maryland1204900.002211.911.10.12, 0.306296003601552Wohl and Merritt [2001]
20 Mile Creek, Ontario2961300.007121.58 6789204101333Wohl and Merritt [2001]
Swayze Creek, Ontario660.0254.50.85.6 551470690272Wohl and Merritt [2001]
Deep Run Creek, Maryland161400.00712.7112.70.2, 0.065296001901851Wohl and Merritt [2001]
Paran 2, Israel295025000.018625.112.20.37, 0.8847171,500215036494Wohl et al. [1994]
Piccaninny 3, Australia611000.003321.422.10.11, 0.25602940300491Wohl [1993]
Piccaninny 2, Australia611000.00424., 0.22602940300491Wohl [1993]
Piccaninny 1, Australia611000.004443.512.60.11, 0.26602940300491Wohl [1993]
Paran 1, Israel295025000.008417.75.40.15, 0.5665171,500215026391Wohl et al. [1994]
Futama 2, Japan4560.09462.42.50.03, 0.105645,00013608041Wohl and Ikeda [1998]
Muddy River, Washington345900.036415.37.7 8031,75074103971Wohl and Merritt [2001]
Cheat River, West Virginia260026400.00744., 0.6576181,100576023831Wohl and Merritt [2001]
Wire Pass, Utahc200700.0432., 0.285029,5004305901Wohl et al. [1999]
Upper 40 Mile Creek, Utahc503200.0211.680.20.005, 0.106365,860160010451Wohl et al. [1999]
Lower 40 Mile 2, Utahc753400.0263.812.70.30.003, 0.104849,980160010411Wohl et al. [1999]
Coyote Wash, Utahc65160.0262., 0.49563290290591Wohl et al. [1999]
Buckskin Creek 1, Utahc7901000.0085.2130.40.18, 0.505278405401511Wohl et al. [1999]
Buckskin Creek 2, Utahc10001050.0213., 0.425221,61011704161Wohl et al. [1999]
Pawnee site 1, Coloradoc0.210.0190., 0.105326053051Wohl and Merritt [2001]
Pawnee site 2, Coloradoc0.240.09810.91.10.002, 0.104935503600721Wohl and Merritt [2001]
Lower 40 Mile 1, Utahc753400.0187140.50.003, 0.104449,980160011361Wohl et al. [1999]
Torii 2, Japan0.340.023.50.940.03, 0.10542200180413Wohl and Ikeda [1998]
Futama 1, Japan4560.10372.82.50.05, 0.155745,00013607901Wohl and Ikeda [1998]
Big Box, Arizona72600.1074.21.430.08, 0.258362,92014,6807582Wohl [2000]
Little Box, Arizona4250.1393., 0.288334,06010,6704102Wohl [2000]
Swamp Run, Maryland1130.1614., 0.256320,50033103251Wohl and Merritt [2001]
Difficult Run, Maryland1516800.0251744.20.04, 0.2885166,600588019601Wohl and Merritt [2001]
Tacoma Creek, Washington2.590.107   0.16, 0.4672944048401311Wohl and Merritt [2001]
Forsythe Creek, Colorado1540.19911., 0.41552570760471Wohl and Merritt [2001]
N. Fork Poudre River, Colorado9502900.01121., 0.377831,2608204011Wohl and Merritt [2001]
Anacostia 1, Maryland1204900.009251.417.90.55, 0.137343,22017305921Wohl and Merritt [2001]
Greenbrier River 1, W. Virginia379017720.0021025.518.50.25, 0.587334,7305004761Wohl and Merritt [2001]
Greenbrier River 2, W. Virginia379017700.0018810.58.40.1, 0.324317,3704104041Wohl and Merritt [2001]
St. Charles River, Colorado4302300.01216.22.760.1, 0.216727,05016804041Wohl and Merritt [2001]
Buckhorn Creek, Colorado120130.0106.516.50.14, 0.24751270200171Wohl and Merritt [2001]
N. St. Vrain Creek, Colorado312250.01415.10.818.90.15, 0.55773380220441Wohl and Merritt [2001]
S. St. Vrain Creek, Colorado210210.00810.3110.30.15, 0.487716101602311Wohl and Merritt [2001]
Rio Chagrecito, Panama213700.034341.7200.11, 0.3272123,284378517121Wohl [2005]
Rio Chagres 1, Panama203600.034312.214.10.3, 0.4072119,952457416661Wohl [2005]
Rio Chagres 2, Panama373900.02432.5216.20.24, 0.385491,72822016991Wohl [2005]
Rio Chagres 3, Panama614400.01720.5210.20.19, 0.247273,3043610181Wohl [2005]
Pondicherry 1, India0.680.0223., 0.20471725780371Wohl and Achyuthan [2002]
Pondicherry 2, India0.680.1441.41.410.10, 0.205511,29050802051Wohl and Achyuthan [2002]
Table 3. Characteristics of Paired Bedrock and Alluvial Channel Reaches
SiteA (km2)Q (m3/s)S (m/m)w (m)d (m)D50 (m)SelbyΩ (W)Ω/w
Pawnee 1a0.23.70.0261.60.40.001 943590
Pawnee 1b0.23.70.0190.50.8 536891378
Pawnee 2a0.23.70.0192.00.50.002 689344
Pawnee 2b0.23.70.0981.00.9 4935543554
Pondicherry a0.680.00110.00.40.01 782
Pondicherry b0.680.0223.50.9 551725493
Tacoma a2.290.1034.00.70.26 90852271
Tacoma b2.290.1072.00.6 7294374718
Little Box a410.0703.30.80.16 686208
Little Box b410.1393.21.4 831362432
Forsythe a153.70.0352.70.60.24 1269463
Forsythe b153.70.0713.40.3 552575753
Chagres 1a20.63600.01432.52.20.30 493921520
Chagres 1b203600.03431.02.2 721199523869
Chagrecito a22.33700.00736.22.70.11 25382701
Chagrecito b21.23700.03434.01.7 721232843626
Chagres 2a37.53900.011352.40.24 420421201
Chagres 2b36.63900.02432.52.0 54917282822
Chagres 3a60.64400.011502.60.19 47432949
Chagres 3b60.84400.01720.52.0 72733043576
Big Box a722.50.0099.51.00.09 2073218
Big Box b722.50.1074.20.9 83246425867
Rio Puerco a6101080.00410.51.00.16 4234403
Rio Puerco b6101080.022101.6 55232852328

[16] Discharge was estimated using paleostage indicators of maximum flow and surveyed channel geometry in the step backwater model Hydrologic Engineering Center River Analysis System (HEC-RAS) [Hydrologic Engineering Center, 1995], with roughness values used in HEC-RAS calibrated against gaged discharges where stream gage records existed. The estimated recurrence interval of these discharges varies widely among sites, partly as a function of the climatic regime and the preservation of paleostage indicators and partly in relation to the type of paleostage indicator used. Q values for some of the wetter climates, such as sites in Japan, Ontario, Maryland, and Panama likely have a recurrence interval of a few decades, whereas Q values from the drier sites, such as Israel, Australia, Utah, Colorado, and Arizona have recurrence intervals of many decades to a few centuries. The common factor is that each Q value exceeds erosional thresholds, as indirectly estimated using empirical relations for mobilization of the coarsest clasts in the partial alluvial cover and for initiation of cavitation. Although the recurrence intervals of these discharges are much longer, we interpret them as being analogous to mean annual Q in alluvial channels in that they drive erosional and depositional processes that maintain channel geometry and create channel changes that persist until another flow of similar magnitude recurs. Grain-size distribution at each site in Table 2 was measured using clast counts [Wolman, 1954] where at least partial alluvial cover was present. Grains were measured in riffles or rapids at each site.

[17] Rock mass strength was measured using the index originally developed for hillslopes by Selby [1980]. This index combines intact rock strength, as measured in the field using a Schmidt hammer, with numerical rating categories for visually assessed degree of weathering, spacing, orientation, width, and continuity of joints, and outflow of groundwater, with combined numerical values ranging from 25 to 100. Although not developed explicitly for channels, and thus potentially not the best possible measure of bedrock erosional resistance to fluvial processes [Sklar and Dietrich, 2001], the Selby index remains the most readily and widely used field method of categorizing and comparing substrate resistance in bedrock channels. In addition to the Selby index, we characterized substrates using the categorical variable of substrate heterogeneity, which combines rock mass strength and homogeneity of lithology exposed along the channel at each study site [Wohl and Merritt, 2001].

[18] We classified bedrock channels as those along which at least half of the boundary is exposed bedrock, or is covered by an alluvial veneer that is largely mobilized during high flows such that underlying bedrock geometry strongly influences patterns of flow hydraulics and sediment movement [Tinkler and Wohl, 1998]. We classified alluvial channels as those with limited, if any, bedrock exposure along the channel banks and bed, and with alluvial bed and bank cover sufficiently thick to limit the influence of bedrock on channel form and process during even high discharges.

[19] With the possible exception of the five sites from Japan, two from Israel, and nine of the Utah and Colorado sites that are slot canyons, these study reaches likely represent steady state rather than transient channel responses. The Japanese sites could potentially represent transient responses because uplift rates near the field sites average 30 cm/kyr [Wohl and Ikeda, 1998]. Similarly, incision rates at the sites in Israel average 10 cm/kyr [Wohl et al., 1994]. The slot canyons, defined here as having w/d values < 1.5 and identified in Table 2 by the superscript c, might also represent transient channel response to localized uplift or relative baselevel fall. These two categories of sites were examined separately in the analyses because of the possibility that they represent transient channel response.

3.2. Analyses

[20] Comparison of downstream hydraulic geometry relations are based on standard log-log plots of relevant variables. Coefficients and exponents of best fit regression lines, confidence intervals for the coefficients and exponents, and coefficients of correlation were calculated using these plots. We also used ANOVA and multiple regression analyses in the software R [R Core Development Team, 2007] to evaluate the best predictors of the response variables of w, d, w/d, and S. Variables used in the multiple regressions are listed in Table 2. All regressions were checked for homoscedasticity and normality of residuals. After log transformation of the variables w, d, Q, S and ω, standard diagnostics [Kutner et al., 2005] did not indicate any violations of the assumptions for the regression models presented in this paper. Collinearity among the independent variables was analyzed using a Pearson correlation matrix. Because interrelated variables cannot be used in the same regression model, the collinear variable exhibiting the greatest influence on the response variable was kept in the model and the remaining collinear variable(s) were eliminated. Residual plots were analyzed to check model assumptions and to examine potential outliers. Analyses were computed with and without potential outliers to determine whether the outlier has a significant effect on the model selection process.

4. Results

4.1. Bedrock Channels: Downstream Hydraulic Geometry Relations

[21] The first group of analyses was designed to evaluate the exponents, coefficients, and significance of scaling relationships of bedrock channel geometry and the independent variables A, Q, and S. The data set presented here supports earlier analyses of w-A relations, with b = 0.32 (Figure 1a), and w-Q relations, with b = 0.5 (Figure 1b). The rate of change for w-S relations, with b = −0.63 (Figure 1c), is steeper than indicated in previous studies. A and Q each explain approximately half of the variability in w. S is less consistently related to w, although still significant at α = 0.05. Plots of w-A for the bedrock channel data set with the sites potentially representing transient channel geometry (slot canyon and uplift) removed also resulted b = 0.32, and plots of w-Q with these sites removed resulted in b = 0.5. The inclusion of sites potentially representing transient channel response thus does not significantly change scaling relations.

Figure 1.

Scaling of channel width in relation to (a) drainage area, (b) discharge, and (c) bed gradient for the bedrock channel reaches in Table 1. All correlations are significant at α = 0.05. Sample size is 47 for all analyses. Values for 95% confidence limits are in Table 4.

[22] The results in Figure 1 suggest the question of whether differences in substrate resistance correspond to different scaling relationships. The bedrock channel reaches reported here have values of rock mass strength from 43 to 85. We arbitrarily subdivided them into five groups and replotted the w, A, Q, and S data with these subdivisions (Figure 2). None of these plots show any consistent trend in either the slope of the best fit line (b values) or the degree of correlation (R2 values) with respect to substrate resistance, and thus do not provide any particular insight into potential causes of scatter in the plots of Figure 1, in which scatter could also reflect sediment supply, uplift rate, and other parameters not quantified in this analysis. Because the 95% confidence limits for coefficients and exponents of regression models from individual subsets overlap (Table 4), these results also do not support the hypothesis that w increases more slowly with A or Q in more resistant substrates. The lack of change in scaling relations for w may reflect a data set too small for five subdivisions of rock mass strength, but it also suggests, in connection with the identical exponent of w-Q for bedrock and alluvial channels, that the erosional resistance of channel boundaries is not the primary control on scaling relations for channel geometry.

Figure 2.

Scaling of channel width in relation to independent variables, as in Figure 1, but with data subdivided into categories of Selby rock mass strength. Sample size for Selby = 40–49 is 6, sample size for Selby = 50–59 is 16, sample size for Selby = 60–69 is 9, sample size for Selby = 70–79 is 12, and sample size for Selby = 80–89 is 4. Values for 95% confidence limits are in Table 4.

Table 4. The 95% Confidence Limits for Regressions Shown in Selected Figures
FigureDescriptorEstimated Value95% Confidence Limits
1acoefficient3.061.87, 5.01
1aexponent0.320.22, 0.42
1bcoefficient1.120.60, 2.09
1bexponent0.500.37, 0.62
1ccoefficient0.840.29, 2.45
1cexponent−0.63−0.88, −0.38
2a (Selby = 40s)coefficient2.040.60, 6.76
2a (Selby = 40s)exponent0.390.17, 0.62
2a (Selby = 50s)coefficient3.631.78, 7.41
2a (Selby = 50s)exponent0.13−0.06, 0.32
2a (Selby = 60s)coefficient4.790.65, 36.30
2a (Selby = 60s)exponent0.25−0.15, 0.66
2a (Selby = 70s)coefficient5.010.93, 26.90
2a (Selby = 70s)exponent0.310.03, 0.58
2a (Selby = 80s)coefficient1.150.13, 104.70
2a (Selby = 80s)exponent0.52−0.49, 1.53
2b (Selby = 40s)coefficient0.520.05, 6.17
2b (Selby = 40s)exponent0.560.13, 0.99
2b (Selby = 50s)coefficient1.350.52, 3.39
2b (Selby = 50s)exponent0.400.14, 0.65
2b (Selby = 60s)coefficient4.980.15, 165.96
2b (Selby = 60s)exponent0.22−0.44, 0.87
2b (Selby = 70s)coefficient2.090.72, 6.03
2b (Selby = 70s)exponent0.470.29, 0.64
2b (Selby = 80s)coefficient1.020.00004, 26,302.68
2b (Selby = 80s)exponent0.49−1.65, 2.63
2c (Selby = 40s)coefficient0.170.003, 10.00
2c (Selby = 40s)exponent−0.95−1.89, −0.02
2c (Selby = 50s)coefficient3.180.40, 25.70
2c (Selby = 50s)exponent−0.13−0.70, 0.44
2c (Selby = 60s)coefficient1.000.07, 14.13
2c (Selby = 60s)exponent−0.57−1.11, −0.03
2c (Selby = 70s)coefficient1.460.06, 35.48
2c (Selby = 70s)exponent−0.64−1.32, 0.03
2c (Selby = 80s)coefficient0.270.0007, 104.71
2c (Selby = 80s)exponent−1.28−3.33, 0.77
3coefficient1.020.72, 1.45
3exponent0.200.13, 0.27
4 (Selby = 40s)coefficient0.04144.54, 0.28
4 (Selby = 40s)exponent−0.26−0.61, 0.08
4 (Selby = 50s)coefficient0.040.02, 0.08
4 (Selby = 50s)exponent−0.08−0.28, 0.11
4 (Selby = 60s)coefficient0.040.004, 0.42
4 (Selby = 60s)exponent−0.35−0.82, 0.11
4 (Selby = 70s)coefficient0.110.06, 0.22
4 (Selby = 70s)exponent−0.42−0.53, −0.31
4 (Selby = 80s)coefficient0.250.008, 7.94
4 (Selby = 80s)exponent−0.34−1.12, 0.43
8a (alluvial)coefficient4.511.91, 10.72
8a (alluvial)exponent0.320.05, 0.58
8a (bedrock)coefficient2.340.98, 5.62
8a (bedrock)exponent0.420.14, 0.69
8b (alluvial)coefficient2.091.07, 4.07
8b (alluvial)exponent0.460.29, 0.63
8b (bedrock)coefficient1.120.49, 2.51
8b (bedrock)exponent0.520.31, 0.73
8c (alluvial)coefficient1.200.08, 18.20
8c (alluvial)exponent−0.47−1.08, 0.14
8c (bedrock)coefficient0.660.02, 28.84
8c (bedrock)exponent−0.68−1.85, 0.48

[23] Although other investigators have not used depth (d) for analogous scaling relations, we investigated the depth relation for downstream hydraulic geometry using this data set. These analyses indicate that dA0.2 (Figure 3), dQ0.3, and dS−0.21. Subdividing these data by rock mass resistance does not reveal any consistent trends, and thus also does not support the hypothesis. Plots of w/d ratio in relation to rock mass strength and a nondimensional ratio of driving force to substrate resistance, total stream power/rock mass strength, might also be expected to correlate if channel geometry scales differently as a function of differing substrate resistance. The low R2 values (0.09 and 0.01, respectively) for these relations, however, do not support the hypothesis. Similarly, although S correlates with A, there are no consistent differences in this relationship as a function of differences in rock mass strength (Figure 4 and Table 4).

Figure 3.

Scaling of flow depth in relation to drainage area. Correlation is significant at α = 0.05. Sample size is 47.

Figure 4.

Channel gradient versus drainage area, with data subdivided into categories of Selby rock mass strength. Overall correlation is significant at α = 0.05. Sample sizes are as in Figure 2.

[24] Another way to consider the influence of rock strength on channel geometry is to compare Selby values and indices of stream power. If bedrock channel adjustments occur through some combination of changes in w and S, rather than changes in w alone, then total stream power Ω (=γQS, where γ is specific weight of water) and stream power per unit area ω (=γhSv, where h is flow depth, and v is average velocity) provide a means of testing the influence of substrate resistance on channel geometry. Total stream power does not correlate with Selby values (Figure 5a), but an exponential growth curve approximates the relation between stream power per unit area and Selby values, in which more resistant substrates have higher values of unit stream power (Figure 5b). This relation is significant, although Selby value explains less than half of the variability in unit stream power. Because unit stream power represents energy expenditure per m2 of channel bed, Figure 5b suggests that channels in more resistant bedrock are slightly narrower and steeper, although the decrease in w as rock resistance increases is not sufficiently large to be significant in Figure 2.

Figure 5.

(a) Total stream power versus Selby rock mass strength and (b) stream power per unit area versus Selby rock mass strength.

[25] We also examined the scaling of w/d ratio. This ratio varies substantially among bedrock channels (Figure 6a). Plots of w/d ratio versus Q or A have no significant trend, and designation of bedrock channel subsets on the basis of rock mass strength does not improve the correlations. An exponential decay curve can be fit to the plot of w/d ratio versus S, but there is a lot of scatter (Figure 6b). These results do not support the assumption of a constant w/d ratio for bedrock channels [Finnegan et al., 2005; Wobus et al., 2006], but they also do not support the assumption of a weak scaling between w/d ratio and Q or A [Turowski et al., 2007].

Figure 6.

(a) Width versus depth and (b) w/d ratio versus channel gradient for the bedrock channel data.

4.2. Bedrock Channels: ANOVA and Multiple Regression Analyses

[26] Of the variables listed in Table 2, elimination of collinear variables removed Ω/Selby (correlated with Ω) and D50 and D84 (correlated with each other). Predictor variables were those significant at the level α = 0.05. In the model using w as the response variable, Q, S and Selby value are the best predictors in order of descending importance (Adjusted R2 = 0.71; Table 5a). The model for d indicates Q and Selby value as significant predictors (Adjusted R2 = 0.54; Table 5b). The model for w/d indicates S and Selby value as significant predictors (Adjusted R2 = 0.33; Table 5c). The model for S indicates ω and Q (Adjusted R2 = 0.71; Table 5d). These results support the downstream hydraulic geometry analyses in selecting Q as the most significant predictor of bedrock channel w and d. By indicating a correlation between w and S (Figure 7), the results also support previous work on the effects of uplift on bedrock channel geometry. Finally, the selection of Selby value as a significant predictor of w, d, and w/d suggests that, although scaling relations for w do not change significantly or systematically with increasing Selby value (Figure 2), erosional threshold does exert an influence on bedrock channel geometry across a range of environments and drainage areas, as also suggested in Figure 5b.

Figure 7.

Channel width versus channel gradient, illustrating the inverse linear correlation in which width increases as gradient decreases.

Table 5a. Significant Predictors in Multiple Regression Model for logwa
  • a

    Predictor variables were those significant at the level α = 0.05. Remaining variables in analysis were found to be not significant predictors and are therefore not shown here. Here multiple R2 = 0.74, adjusted R2 = 0.71, and F-statistic is 22.74 with 39 df.

logQ0.38 ± 0.19<0.001
logS−0.16 ± 0.280.01
Selby0.025 ± 0.0150.03
Table 5b. Significant Predictors in Multiple Regression Model for logda
  • a

    Predictor variables were those significant at the level α = 0.05. Remaining variables in analysis were found to be not significant predictors and are therefore not shown here. Here multiple R2 = 0.59, adjusted R2 = 0.54, and F-statistic is 11.33 with 39 df.

logQ0.27 ± 0.10<0.001
Selby−0.015 ± 0.0150.02
Table 5c. Significant Predictors in Multiple Regression Model for logw/da
  • a

    Predictor variables were those significant at the level α = 0.05. Remaining variables in analysis were found to be not significant predictors and are therefore not shown here. Here multiple R2 = 0.40, adjusted R2 = 0.33, and F-statistic is 5.44 with 40 df.

logS−0.38 ± 0.320.02
Selby0.03 ± 0.020.003
Table 5d. Significant Predictors in Multiple Regression Model for logSa
  • a

    Predictor variables were those significant at the level α = 0.05. Remaining variables in analysis were found to be not significant predictors and are therefore not shown here. Here multiple R2 = 0.75, adjusted R2 = 0.71, and F-statistic is 38.24 with 42 df.

logω0.65 ± 0.16<0.001
logQ−0.37 ± 0.09<0.001

4.3. Bedrock and Alluvial Channels

[27] Montgomery and Gran [2001] found no systematic difference in w-A relations for bedrock and alluvial channel reaches in sedimentary lithologies in Oregon and Washington, but they did find that bedrock channels were narrower and deeper than alluvial channels with equivalent drainage areas in granite and limestone terrains of California. They also found that w decreases substantially when channels cross from limestone into more resistant granite, although S does not change appreciably at these lithologic contacts. These results form part of the basis for the hypothesis posed in this paper, so we also used a data set of paired alluvial and bedrock channel reaches to examine scaling of w and d in relation to A, Q, and S (Tables 5a–5d). These are much smaller data sets than those analyzed by Montgomery and Gran [2001], although they cover an equivalent range of w and A. Each pair of study reaches are located a relatively short distance (typically less than 10 times active channel width) upstream or downstream from one another. Because this was a smaller data set, we did not subdivide bedrock channel reaches on the basis of Selby rock mass strength values.

[28] Analysis of w-A relations found similar exponents for bedrock (b = 0.42) and alluvial (b = 0.32) channels, but different coefficients (Figure 8a). Although alluvial channels are wider for a given A in smaller drainages, the best fit lines converge as A approaches 1000 km2. It is worth noting that the four bedrock channel sites with w greater than 10, which appear to drive this convergence of best fit lines, all come from the Rio Chagres drainage in Panama (Tables 5a5d). A similar plot with these data removed still has best fit lines that converge between A of 100 and 1000 km2, however, with wider alluvial channels at A less than 100 km2. The 95% confidence limits (Table 4) indicate that there is no significant difference between the regression models for alluvial and bedrock channels. By comparison, Montgomery and Gran [2001] found that coefficients for w-A relations were significantly different along the Yuba River, California, indicating wider alluvial reaches, whereas w-A relations along Knowles Creek, Oregon did not distinguish bedrock and alluvial reaches at drainages greater than 1 km2.

Figure 8.

Scaling of channel width in relation to (a) drainage area, (b) discharge, and (c) channel gradient for paired bedrock and alluvial channel reaches. Correlations for w-A and w-Q for bedrock and alluvial channel data are significant at α = 0.05. Correlations for w-S for bedrock and alluvial data are not significant at α = 0.05 or 0.10. Sample size for alluvial channels and for bedrock channels is 12 each.

[29] The results for w-Q relations are mixed. Analysis of w-Q relations indicated very similar exponents for bedrock (b = 0.52) and alluvial (b = 0.46) channels, different coefficients, and a lack of convergence in the best fit lines (Figure 8b). This suggests that alluvial channels are consistently wider than bedrock channels with the same discharge, an interpretation supported by a paired sample t-test. This test is used to test the null hypothesis that the average of the differences between a series of paired observations is zero. The paired sample t-test for log w (two-sided test with alpha is 0.05, t-value is 3.203, degrees of freedom (df) is 11, p-value is 0.008, and mean of differences is 0.19) rejects the null hypothesis. The 95% confidence limits (Table 4), however, indicate no significant difference between the regression models for bedrock and alluvial channels. In addition, a multiple regression model using a binary dummy variable (1 is alluvial and 0 is bedrock) and log Q as predictors of log w, indicated no significant differences between the bedrock and alluvial channels. These apparently conflicting results among the different tests indicate that, although the alluvial channel of an individual pair tends to be wider for a given value of Q, the population of alluvial channel segments used in this analysis is not significantly wider than the population of bedrock channel segments. Analysis of w-S relations (Figure 8c) indicates that these relations are not significant and the 95% confidence limits indicate no significant differences between the regression models.

[30] Comparison of w/d ratios for the paired bedrock-alluvial channel data set indicates no significant differences in the means of the two populations (Figure 9). The results summarized in Figures 8 and 9 thus do not support the hypothesis posed earlier; within the 95% confidence limits, w-A, w-Q, and w-S models do not differ significantly between bedrock and alluvial channels for the limited data set analyzed here.

Figure 9.

(a) Box and whisker plots of w/d ratios for alluvial and bedrock channel data. The line within each box indicates the median value, box ends are the upper and lower quantile, whiskers are the 10th and 90th percentiles, and solid dots are outliers. Mean and standard deviation for each population is shown below the box plot. Mean w/d ratios are not significantly different at α = 0.05 or 0.10. (b) Individual w/d ratios for bedrock and alluvial channel pairs.

5. Discussion and Conclusions

[31] It is appropriate to reiterate the limitations and uncertainties of the data set analyzed here. First, the magnitude and recurrence interval of Q used in these analyses varies widely among sites, so that scaling relations with A may be more consistent. Second, these study sites are predominantly steady state channels, so the observed similarities in scaling relations relative to alluvial channels may be less surprising than if the data set included more transient channel geometries from tectonically active sites.

[32] As explained in the introduction, the greater erosional resistance of bedrock channel boundaries relative to the boundaries of most alluvial channels could reasonably be expected to produce lower w/d ratios in bedrock channels, or greater values of exponents or coefficients for w-A or w-Q relations. The empirical data presented here indicate no significant differences in most w/d ratios for a given alluvial-bedrock channel pair. Although w appears to increase more rapidly with A in bedrock channels (Figure 8a), converging with the values for alluvial channels between 100 and 1000 km2, the w-A regression models are not significantly different for bedrock and alluvial channels (Table 4). Similarly, although alluvial channels are consistently wider than bedrock channels for a given Q (Figure 8b), the w-Q regression models are not significantly different for bedrock and alluvial channels (Table 4). For any given bedrock-alluvial pair, the bedrock reach is typically steeper, and has greater total and unit stream power values than adjacent alluvial channels (Table 3), so bedrock channels are more powerful for a given Q because they are both steeper and narrower. For the bedrock-alluvial channel pairings, the more erosionally resistant channel is associated with higher power.

[33] Subdivision of bedrock channel data by categories of Selby rock mass strength does not indicate any consistent trends in scaling relations as a function of greater rock mass strength, but relations between unit stream power and rock mass strength and the selection of Selby value as a predictor variable in multiple regression analyses of controls on bedrock channel width suggest that substrate erosional resistance does exert some influence on bedrock channel geometry. These results thus partly support the hypothesis that differences in erosional resistance between bedrock and alluvial channels result in consistent differences in channel geometry, although these differences are not statistically significant in simple linear regressions. Alluvial channels tend to be slightly wider than bedrock channels, but scale at similar rates as A and Q increase. The lack of consistent scaling in various channel parameters in relation to rock mass strength is particularly important in terms of the stream power erosion law (equation (1)), which assumes that channels formed in more resistant rocks are systematically steeper than those formed in less resistant rocks. The lack of relations in the linear regression analyses presented here does not refute this assumption, because Selby rock mass strength may not be the most appropriate metric of bedrock erosional resistance and because multiple regression analyses indicate that rock mass strength does influence scaling of channel geometry. The lack of simple, systematic linear scaling of bedrock channel geometry in relation to rock mass strength likely does reflect the complex interactions among sediment supply, flow regime, and erosional thresholds, and thus suggests caution in applying the stream power erosion law.

[34] The results presented here also generally support the conclusions of previous empirical and theoretical studies regarding appropriate values of exponents for scaling bedrock channel w in relation to A and Q. The lack of differences in scaling relations indirectly supports the inferences of Finnegan et al. [2007] that other factors, such as bed load supply, exert a fundamental control on width in bedrock channels. What may be the most significant finding of the analyses presented here is that both alluvial and bedrock channels scale such that wA0.3 and wQ0.5.


[35] The field research that produced the data set analyzed here was funded by grants from the Geological Society of America Gladys W. Cole Award, the Japan Society for the Promotion of Science, the National Council of the Pulp and Paper Industry for Air and Stream Improvement, the National Geographic Society, the National Science Foundation (EAR-0000725, INT-9814154), the U.S. Army Research Office (DAAD 19-00-1-0474), and the US-Israel Educational Foundation. We thank Rob Ferguson, Jens Turowski, Cameron Wobus, and an anonymous reviewer for insightful reviews that substantially improved the manuscript.