The classical equations of hydraulic geometry are purely empirical, but the widespread similarity of the scaling (downstream) form of them suggests that they express some important underlying regularities in the morphology of stream channels through the drainage network. A successful physical theory of river regime must be able to reproduce and explain this regularity. In this paper we test the regime theory of Eaton et al. (2004) using selected data of hydraulic geometry. We first use data from environments in which bank strength presumably does not vary greatly, such as in anabranched channel systems and deltas. Regime models parameterized by assuming uniform relative bank strength plausibly describe the observed bankfull channel geometries in these systems. We then test a modified bank strength formulation for vegetated gravel bed rivers against downstream hydraulic geometry data sets in which relative bank strength is supposed to vary with channel scale. Assuming a uniform effective cohesion due to riparian vegetation, the regime model is again able to reproduce details of the channel geometry. Both analyses show that the classical hydraulic geometry represents only an approximation of the variation of channel form. If we have confidence in the theory, we may infer information about bank strength and bed material transport. The pattern of variation in these quantities, as well as discharge, through the drainage system lends approximate regularity to stream channel scaling that is summarized in the empirical relations.
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 The hydraulic geometry arose by analogy with “regime relations” established early in the 20th century for the design of unlined irrigation canals. The earliest relations were deliberately empirical [Kennedy, 1895; Lindley, 1919], but they were succeeded by attempts to clothe the relations in the semblance of a theory (hence “regime theory” [see, e.g., Lacey, 1930; Blench, 1969]). These investigators recognized that a critical control over river channel form is exercised by the materials that form the bed and banks of the channel, but they were able to quantify the effect only by an index “silt factor” [Lacey, 1930; Blench, 1969] or by stratifying regime relations according to the character of bed and bank materials [Lane, 1957; Simons and Albertson, 1963]. This factor has mostly been ignored in considering the hydraulic geometry of rivers, although thoughtful investigators have introduced the related factor of vegetation-mediated bank strength [Andrews, 1984; Hey and Thorne, 1986; Huang and Nanson, 1998], again by index methods, while others have restricted investigations to a particular type of bounding sediment [Bray, 1973; Neill, 1973; Charlton et al., 1978; Andrews, 1984; Hey and Thorne, 1986].
 The purpose of this paper is to explore the hydraulic geometry of selected river systems in terms of a rational regime theory, both as a test of the theory and in order to attempt to learn more about the covariation of terrain and hydrological conditions through a drainage system that leads to the familiar empirical equations for the variation of width, depth and velocity.
2. Statement of the Regime Approach
 A rational regime theory specifies a set of physical relations that apply the relevant governing conditions to determine the hydraulic geometry of a self-formed channel. The complete hydraulic geometry includes, in addition to width, depth and mean flow velocity, channel gradient and planform geometry. The principal governing conditions are the imposed flow and sediment load (which includes both caliber and quantity of bed material sediment), the valley gradient down which the flow and sediment must be passed, and the character of the deposits in which the channel is formed.
 We know that, if the discharge (Q), bed material sediment transport rate (Qbm) and sediment caliber (D) are specified, then a channel geometry comprising gradient (S), hydraulic radius (R) and wetted perimeter (P) can be calculated. Applying a bank stability criterion does not isolate a unique solution since a range of channel geometries satisfies the bank stability constraint. In order to isolate a unique solution one must apply some sort of optimality criterion, such as minimum variance of hydraulic quantities [Langbein and Leopold, 1966] (a purely statistical concept), minimum unit stream power [Yang, 1976], maximum sediment transport efficiency [Kirkby, 1977], minimum power expenditure [Chang, 1979], maximum sediment transporting capacity [White et al., 1982], least action [Huang and Nanson, 2000], or minimum energy expenditure [Huang et al., 2004]. Since these optimality criteria all arrive at nearly identical solutions (but from different assumptions), the differences between them lie not in the predictions that they make so much as in the physical implication of the assumption.
 We have recently proposed what we believe to be a plausible physically based solution [Eaton et al., 2004]. The stable state is the one in which resistance to flow is maximized subject to the condition that the imposed water and sediment load – that is, the load delivered from upstream – be passed on the adopted channel gradient. This condition still permits exchange of sediment between the load and the bed, hence lateral movement of the channel, but no net change occurs in the average channel form. Flow resistance is maximized with respect to the fluvial system, which refers to the valley gradient, Sv, as the controlling system gradient (cf. the governing conditions enumerated above). It turns out that this condition is mathematically equivalent to the condition that the imposed sediment load be passed with the smallest expenditure of energy (in a particular sense, “with maximum efficiency” [Millar, 2005] or, more obscurely, “at capacity”). Accordingly, the preferred gradient–if it be accessible to the river–is the minimum channel gradient [Eaton et al., 2004]. One important way in which the maximum resistance solution differs from other proposals is that it is constrained by the relative erodibility of the bed and the banks. A reduction in slope for a constant transport capacity is accomplished by narrowing the channel, thereby increasing the channel depth and mean boundary shear stress. If the channel becomes too narrow and deep, the shear stress acting on the banks will exceed the critical threshold for erosion and the banks will fail, therefore bank strength places a limit on the degree to which a channel can narrow and deepen, and thus places a limit on the degree to which the channel gradient can be reduced.
 The regime models used in the following analyses are based on two separate approaches to the bank strength problem. The first, more general model represents bank strength as a function of the bed erodibility via an apparent friction angle, ϕ′ [after Millar and Quick, 1993], which describes the ratio between bank strength and bed strength [Millar, 2005], as shown in Figure 1. Because of the simple nature of this model, it is not necessary to make any assumptions about the bank stratigraphy or the physical origin of the bank strength, which is expressed merely as relative to the bed strength. While this is an intuitive way to think about bank strength, the parameter ϕ′ is difficult to estimate.
 The model assumes a trapezoidal channel geometry, which facilitates the use of empirical functions to estimate the shear stress distribution along the channel boundary on the basis of the channel shape and the average boundary shear stress [Millar and Quick, 1993]. The threshold for bank erosion depends on the specified critical dimensionless shear stress (or Shields number) for entrainment of the representative grain size used to characterize the banks (τc*), the relative strength of the banks (ϕ′) and the side slope angle of the trapezoid. The critical shear stress for entrainment of the bank material is selected according to the character of the bed, and the reference grain size used in the analysis. For gravel bed rivers in which the surface median grain size, D50, is used in the bank stability analysis, τc* is set at 0.03; when the surface D84 is used, τc* is set at 0.02, and if the D90 is used τc* is set at 0.015. Since the typical ratio of D90:D50 in gravel bed streams is approximately two, setting τc*(D90) = 0.015 is tantamount to applying an entrainment threshold of τc*(D50) ≈ 0.03, which is cited as an appropriate threshold for gravel bed streams [Neill, 1968; Andrews, 1982]. For sand bed rivers, a set of linear functions that estimate the critical Shields number as a function of the grain Reynolds number (Re*) is used in the bank stability analysis [Van Rijn, 1984].
 Bed material sediment transport capacity (Qbm) is estimated in the model using two different equations. For gravel bed streams, the Parker  equation is used. When the bed material is primarily sand, we use the Van Rijn  equation. In both cases, the calculated sediment transport capacity is interpreted as merely an index of the actual transport capacity since bed load transport equations are not very accurate.
 A regime solution is found using as input parameters values of slope (S), formative discharge (Q), a flow resistance parameter (in this analysis, Manning's n) and surface grain size (often D50). Solution geometries are found that satisfy the constraint that the banks are stable for the range of possible trapezoid bed widths, and the predicted bed material sediment transport capacity (Qbm) is plotted against the bed width. The chosen solution is the one associated with the highest sediment transport capacity (i.e., the most efficient channel), since this is equivalent to the minimum slope for a constant value of Qbm [White et al., 1982] which in turn is equivalent to the maximum flow resistance for the fluvial system [Eaton et al., 2004].
 In the original presentation of the regime model [Eaton et al., 2004], flow resistance was indexed using the Darcy-Weisbach coefficient. Manning's n is adopted here since it is related to powers of the hydraulic variables, which is convenient for scaling arguments. The regime models have been coded in MATLAB™ for numerical solution.
 For gravel bed streams with vegetated floodplains, we use an alternative bank strength model proposed by Eaton . This model parameterizes bank strength for vegetated banks by recognizing a root-laced, semicohesive upper bank section of height H overlying a lower bank with properties similar to those of the streambed, to which a frictional strength proportional to the true friction angle, ϕ, is applied. The parameter H is thought to be roughly equivalent to the rooting depth for the dominant riparian vegetation, and can be interpreted as an effective cohesion (C′) due to riparian vegetation. Since two parameters are required (C′ and ϕ), this model is slightly more complicated than the ϕ′ model. The primary advantage of it is that it explicitly incorporates the scale-dependent nature of bank strength due to riparian vegetation: while the relative bank strength (ϕ′) for a forested stream declines downstream (because the channel steadily grows deeper than the riparian rooting depth), becoming similar to streams with grassy banks for bankfull flows greater than about 400 m3/s [Eaton and Millar, 2004], the absolute vegetation-related bank strength for a forest (C′) remains constant, even though the importance of the vegetation in determining the channel shape declines [Eaton, 2006].
 This model imposes a composite channel cross section, comprising vertical upper banks atop a trapezoidal lower channel. The stability analysis is applied to the sides of the lower trapezoid, which represents the cohesionless gravel toe of the channel. Deriving a regime solution, one proceeds in exactly the same way as above, with one further complication: once the channel depth predicted by the model becomes less than H, there is no cohesionless gravel toe upon which to perform the bank stability analysis. This affects a very small number of streams in the data sets that we analyze, since it applies only in situations where the relative bank strength is very high (i.e., small channels in forested streams where root systems dominate the channel banks). In this circumstance, we bypass the bank stability analysis and simply maximize the sediment transport capacity [after White et al., 1982]. This procedure selects the narrowest and deepest possible channel.
 We face a serious problem in attempting to test these two models against empirical data. We know of no field data set that presents sufficient information to make a rigorous test of a complete regime theory, primarily because there exists no practical measurement procedure to determine the shear strength of channel banks in the field, and no field report of such data is known to us. The general objective of this paper, then, is to obtain an indirect test of our regime models by applying the appropriate formulation to available data sets of the hydraulic geometry of alluvial channels. Specifically, we (1) apply the regime model to data sets in which the relative bank strength (ϕ′) can plausibly be assumed to be uniform, (2) apply the regime model to data sets in which the vegetation-related bank strength (C′) can be assumed to be uniform, and (3) invert the regime model to fit the observed widths by varying the relative bank strength (ϕ′) for the data sets referred to in (2). By applying physically based models, we are able to investigate the physical processes underlying downstream hydraulic geometry relations.
3. Systems With Nominally Uniform Relative Bank Strength
 We found five cases in which it seems reasonable to assume that relative bank strength is approximately uniform. For each data set, the regime formulation using ϕ′ was adopted. The value of ϕ′ used in the model was varied to bring the predicted and the observed bankfull widths and depths as close to agreement as possible. Each data set presented values of Q, S, surface grain size and sometimes n differently and, presumably, with various degrees of accuracy. Therefore we were forced to make a range of assumptions about the input parameters.
 We investigate the model performance by fitting power laws to the modeled channel dimensions and comparing these equations to the empirical power laws. For the sake of consistency, we have calculated the power law relations for both the empirical data sets and the modeled channel dimensions by applying linear regression to the log-transformed variates: this allows us to directly compare the scale relations associated with the regime model with the scale relations associated with the empirical data without encountering the potentially biasing effects of different curve-fitting procedures. We also calculate RMS error and mean bias statistics, both expressed as a percent of the estimate, for the regime model predictions, and RMS errors for the empirical equations. These statistics help us to assess the performance of the regime model and to compare the relative ability of both approaches to explain the observed variance in the data set.
3.1.1. Colorado River
 The first case is a data set on the single-thread, meandering Colorado River published by Pitlick and Cress . We consider only those reaches that are identified as alluvial or quasi-alluvial which, the authors suggest, are generally free to adjust their channel width. All of the reaches are relatively wide and deep (>2.4 m; relative roughness, D50/d, <0.025) with minimal cohesive bank sediment, so we have made the assumption that the relative bank strength is fairly uniform and that it is relatively low. The authors estimated the bankfull flows using the cross sectional areas and estimates of Manning's n from fitting a 1-D flow model to the reach of interest [Pitlick and Cress, 2000]. We use their (unpublished) estimates for bankfull flow and Manning's n in conjunction with the published channel slopes and surface textures of Pitlick and Cress . We developed empirical equations for interpolating S, n and D50 as functions of Q (Figure 2). We then used these functions to generate input parameters for the regime model over the observed range of Q, and compared the predicted channel geometry with the observed geometry. Since we have parameterized the model by choosing a value of Q, then estimating S, n, and D50 as a function of Q, the regime model produces smooth trends. In order to determine the effect of using these functions to smooth the model output, we also ran the model using the raw input data for each study reach.
 One of the reaches (Professor Valley) [Pitlick and Cress, 2002] was associated with a bankfull discharge about twice that of the reaches immediately upstream and downstream, and was excluded from the analysis on the grounds that the identified channel geometry most likely represents the dimensions of a low terrace rather than a bankfull channel.
3.1.2. Fraser River
 A data set from gravel bedded anabranches of the Fraser River was published by Ellis and Church . The bankfull channel dimensions and bankfull discharge were carefully estimated by interpolation amongst measurements to higher than bankfull stage for a number of anabranches and for the main channel of the Fraser River. Since all of the anabranch channels are relatively deep (>2.5 m; D50/d, <0.029), and are formed in materials having similar sedimentological properties and riparian vegetation, we can assume that the relative bank strength is uniform (and relatively low).
 The regime model applied to this system uses the measured values of Q and D50 for each study reach. The channel gradient is small enough to be difficult to measure accurately over the length (∼250 m) of a single study reach. Therefore values of the slope for each reach were estimated from measurements of average velocity and hydraulic radius reported by Ellis and Church , using the Strickler equation to estimate the grain Manning's n for the measured surface D84. Since total flow resistance is typically underestimated by the Strickler equation, the predicted slopes are lower than they ought to be. The average slope for the system is tolerably well known, and when the estimates of the slopes for each study reach are scaled upward by 50%, the average of the estimates agrees with the known average slope for the system. We use these scaled slope values to parameterize the regime model. Since D84 was used to estimate S, we use D50 for the bank stability analysis. Since we have already used the measurements of velocity and depth to estimate S, it would be inappropriate to use them again to estimate n. Therefore we have held n constant (at a value of 0.03) to generate predictions of width and depth: n values back calculated from our adjusted slopes range from 0.024 to 0.036 with a mean value of 0.029.
3.1.3. Columbia River
Tabata and Hickin  reported a data set on sand bed anabranch channels of the upper Columbia River. The banks of these channels are almost entirely composed of silt and clay, which we believe is consistent with a uniformly high relative bank strength. Data reported by Tabata and Hickin  were used to estimate average values of S (0.001 m/m), n (0.028), and D50 (0.5 mm). The formative discharge was estimated from field measurements in each anabranch that was surveyed. The regime model applied to the Columbia held S, n, D50 and ϕ′ constant and varied only Q, generating predictions of width and depth that vary smoothly with Q.
3.1.4. Danube Delta
 The fourth case is a data set on the sand bed delta distributary channels of the Danube delta [Mikhailov, 1970]. These channels are sedimentologically similar to the anabranches in the upper Columbia River, and we have assumed that they are associated with a uniform, high relative bank strength. We used the estimated formative discharges in conjunction with the relations reported by Mikhailov that express S and n as functions of Q to parameterize the model. The data reported support the selection of only an average, representative bed grain size (0.15 mm).
3.1.5. Laitaure Delta
 The last case comprises data from the sand bed channels of the Laitaure delta in northern Sweden, reported by Andrén . The energy gradient for these channels was held constant (0.0002 m/m, consistent with values provided by Andrén [1994, p. 84]), as was Manning's n (0.025). On the basis of the empirical data reported by Andrén, grain size was varied as a function of discharge:
 The regime model was parameterized using the reported formative discharge values, the constant values of S and n, and the function relating the characteristic grain size to Q.
3.2.1. Colorado River
Pitlick and Cress  express their empirically based hydraulic geometry equations using the dimensionless parameters of Andrews , W* = W/D50, d* = d/D50, and Q* = , where s is the specific gravity of the sediment grains. The equations based on their data are
These equations differ from the results typically reported from studies of downstream hydraulic geometry (see Knighton [1998, p. 173] for a summary), in which width ∝ Q0.5, and depth ∝ Q0.4, approximately, such that the width/depth ratio typically increases downstream. Pitlick and Cress's  data set has the peculiar property that the width/depth ratio decreases downstream. The authors speculate that the origin of this feature of the hydraulic geometry is the variation of grain size and slope downstream, possibly driven by a declining supply of bed material along the system.
 The regime model fits the data from the Colorado River when the relative bank strength is set to about 1.25 (ϕ′ = 36°). Figure 3 plots both the regime model predictions and the observed channel geometry against the estimated Q. While the relation between predicted W and Q is fairly well described by a power law, the relation predicted by the regime model between d and Q is not well described by a power function. The empirical data are at least as consistent with the regime-based depth relation, insofar as they suggest no clear preference for a power law fit. The RMS error for estimating width and depth using the regime model is almost identical to the error associated with the empirical power fits (Table 1). Table 1 also presents the RMSE estimates for a regime model based on the raw field data (not shown in Figure 3) rather than the fitted functions of Figure 2: the values are nearly identical to the results discussed above.
Table 1. RMSE Values for Width and Depths Estimated Using Regime Models
Empirical Power Law
RMSE Width, %
RMSE Depth, %
Values in brackets for the Colorado River are based on using the raw field data for S, n, and D50.
Values in parentheses for the Fraser River are based on a data set excluding the two smallest channels.
Values in parentheses are based on data sets excluding the smallest channels (for Columbia and Laitaure, Q < 20 m3/s; for Danube, Q < 200 m3/s).
Reported statistics are based on the 5-year floods.
 While the regime model results are not entirely consistent with power law fits, they can be used to estimate dimensionless hydraulic geometry relations of the same form as equations (2) and (3). The equations are
While these equations are not identical to equations (2) and (3) they are sufficiently similar that we can conclude that the regime model has reasonably reproduced the hydraulic geometry equations for this reach.
 In order to assess the downstream changes in the transport capacity, the Parker  equation was used to calculate the bed material sediment transport concentration, Qbm/Q. Bed material concentration declines systematically downstream at a rate that is faster than if Qbm were constant (Figure 3). This implies that the absolute yield of bed material sediment declines downstream, which is consistent with the speculation by Pitlick and Cress . Since the transport capacity predicted by the model depends on the channel width, depth and surface texture, which the regime model faithfully reproduces, the spatial variation in bed material transport capacity does not depend on the model assumptions at all, but only on the quality of the initial data and the performance of the Parker formula as an index of bed material transport rates.
3.2.2. Fraser River Anabranches
 The empirical hydraulic geometry equations for these anabranches are not typical either, with an increase in bankfull discharge being accommodated by a disproportionately large increase in width. The equations reported by Ellis and Church  are
The regime model reproduces the observed width scaling fairly well (Figure 4) when the relative bank strength is set equal to unity (ϕ′ = 30°), but does a poorer job of predicting the channel depths, particularly for the smaller anabranches in the system. The model predictions are adequately described by power functions. The power laws fitted to the regime predictions are
which are reasonably close to the empirical relations of Ellis and Church . The RMS error for width and depth based on the regime model are similar to those associated with the empirical fits (Table 1): the regime model appears to do a better job of predicting the channel widths, while the empirical equations are superior for predicting the observed depths.
 The relation between bed material transport concentration and Q for the Fraser River anabranches is the reverse of the pattern evident in the Colorado River (Figure 4). Since the anabranch channels do not reflect downstream trends, the increase in bed material concentration with increasing discharge implies that the smaller channels are less efficient at transporting bed material than are the larger ones.
3.2.3. Columbia River Anabranches
 The anabranches of the upper Columbia River exhibit an empirical hydraulic geometry rather similar to that of the gravel bed anabranches in the Fraser River. The empirical equations are
They are fairly well described by our regime model when the banks are nearly 10 times stronger than the bed (ϕ′ = 78°, see Figure 5). The observed widths and depths are predicted about as well as for the Fraser River (see Table 1): again, the empirical equations for depth outperform the regime model. The smallest anabranches seem to be consistently wider and shallower than predicted, just as they were in the Fraser River system.
 The relation between bed material transport concentration (Qbm/Q) and Q is also similar to that of the Fraser River (Figure 5), with progressive division of the channel into smaller and smaller threads being associated with a reduction of bed material transport concentration, albeit less rapidly than for the Fraser River. Since both locations are aggradational settings, the positive correlation between transport efficiency and channel size suggests that the processes producing the channel divisions are closely linked to the occurrence of bed material deposition in these environments.
 Power law relations fit the modeled widths quite well, but not the modeled depths (see Figure 5). The equations are
These power fits are systematically different from the empirical fits presented in equations (10) and (11). The differences are primarily attributable to the poor model performance for the smaller channels.
3.2.4. Danube Delta
 The empirical hydraulic geometry equations for the delta channels investigated by Mikhailov  are similar to the equations typically observed for downstream hydraulic geometry:
The modeling results are shown on Figure 6. The regime model assumes relatively strong banks (ϕ′ = 83°), and produces results similar to the analysis of the Columbia River anabranches. The bed material transport concentration declines as channels are progressively subdivided, and most (but not all) of the smallest channels are, again, wider and shallower than predicted by the regime model. While the regime model predicts widths about as well as the empirical power fits, it predicts depths less well than the empirical equations (Table 1). The RMS errors associated with the empirical fits are also rather high compared to the previous examples.
 The hydraulic geometry equations fit to the channel dimensions predicted by the regime model for the Danube Delta are
Like the Columbia River results, the equations associated with the regime model are quite different from those based on the empirical fit to the data, primarily because of poor model performance for at least some of the smaller distributary channels.
3.2.5. Laitaure Delta
 The empirical equations for the Laitaure Delta, based on Andrén's  data are
While these equations are different from the expected empirical hydraulic geometry, they are also quite poorly known, having the largest uncertainties for the value of the exponents, and the largest RMS errors (Table 1).
 The data for the Laitaure Delta were modeled using the same bank strength value as was used for the Danube delta channels. The power law equations based on the modeled widths and depths are
Once again, the regime equations based on the predicted geometry are not similar to the empirical equations based on the reported data, primarily because of deviations from the model at the lower discharges, resulting in large RMS errors for the modeled depths (Table 1).
4. Systems With Spatially Varying Relative Bank Strength
 The above analyses targeted environments in which relative bank strength appears to be at least approximately constant. In most drainage basins, this is clearly not the case. However, by applying the second formulation of the regime model, in which bank strength is divided into a component due to the influence of vegetation (C′) and one due to the true frictional strength (ϕ), we can make some progress. This approach is limited to gravel bed streams where it is reasonable to assume that the banks comprise a lower, cohesionless gravel toe overlain by a vegetation-reinforced vertical upper bank, characterized by the effective cohesion due to vegetation, C′. We have analyzed in some detail two data sets on mountain streams in Idaho and in Colorado that exhibit fairly typical hydraulic geometry relations.
4.1.1. Salmon River
 The first data set, from the Salmon River area of Idaho (Figure 7), was published by Emmett . Even though the Salmon River is located in a semiarid environment, an inspection of satellite imagery (Google Earth, http://earth.google.com/) reveals that the floodplains have a nearly continuous cover of grass, shrubs, and occasional deciduous trees. The nature and extent of the riparian vegetative cover combined with the range of channel sizes suggest that, in the Salmon River area, the effects of stream bank vegetation dominate bank strength in the upper reaches, but have a much reduced effect along the larger stream channels.
Emmett's  data were used to investigate the variation in relative bank strength in the downstream direction that is required to produce the observed empirical hydraulic geometry using a rational regime model. Since bank strength is one of the key variables affecting the channel dimensions predicted by our regime model, we selected only those stream channels that, according to the map of geology and surficial deposits provided by Emmett , formed in modern alluvium or unconsolidated glaciofluvial deposits. Another important factor affecting the regime model predictions is the coarse sediment supplied to the stream channel. No data exist to allow us to predict which areas have a high bed material sediment yield and which have low sediment yield, so the selected data were further classified on the basis of the dominant lithologies in their drainage basins. The result is to divide Emmett's data set into two groups (Figure 7). The first group contains 13 stations along the main stem of Salmon River, which is dominated by granitic rock from the Idaho Batholith, as well as rock units mapped as undifferentiated Paleozoic rock [Emmett, 1975]. The second group contains 7 stations in the drainage basin of the East Fork Salmon River, which is dominated by rocks from the Challis Volcanics.
 In order to apply the regime model to the two data sets, we used the reported values of Q, S, D50 and D84 for each group to generate functions relating S, n, D50 and D84 to Q, as described above for Colorado River (see Figure 8 and Table 2). These functions were then used in the regime model to estimate S, n, D50 and D84 for various specified values of Q.
Table 2. Data-Based Trends and Correlation Coefficients Used to Reconstruct the Hydraulic Geometry for Two Groups of Streams, Salmon River Areaa
 The estimates of the surface D50 were used in the Parker  equation to calculate the bed material sediment transport capacity, Qbm, and the estimates of the surface D84 were used in the bank stability analysis on the grounds that bank stability is most likely to be controlled by the larger grains found in the bed and therefore in the lower part of the channel bank.
 According to the analysis presented by Eaton  the riparian vegetation type in the Salmon River area is consistent with C′ between 1.5 and 3.8 kPa, which corresponds to Hey and Thorne's  type II and III channels. We adjusted C′ for each group to produce the best agreement between the modeled channel geometry and the observed channel dimensions.
4.1.2. Mountain Streams in Colorado
 The second data set (Figure 9), on gravel bed rivers in Colorado, was published by Andrews . He classified channels into ones with “thin” and ones with “thick” riparian vegetation, which should roughly correlate with the relative erodibility of the stream banks. In this case, the data come from streams in different environments with different bank vegetation, so we cannot generate reasonable empirical relations between discharge and the other required input data (S, n and grain size). We therefore applied the regime model to each study reach and evaluated model performance using 1:1 plots of the expected and observed channel dimensions. We used the D90 reported by Andrews for the bank stability analysis and the reported values of D50 to calculate an index of the sediment transport capacity via the Parker  equation.
 The initial analysis using the reported bankfull flows suggested that, for many of Andrews's stream channels, the bed was actually more resistant to erosion than the banks, which is at odds with previous findings [Eaton and Millar, 2004; Eaton, 2006]. A comparison of the Andrews hydraulic geometry equations against empirical data collected from similarly vegetated gravel bed channels suggests that the formative discharge is probably higher than the reported bankfull flows (see Appendix A). This conclusion is supported by the fact that, for the majority of the streams in the data set, the reported bankfull discharge is less than the mean annual instantaneous peak flow reported by the USGS, on average about 25% less. Some researchers have claimed that, in semiarid regions like Colorado, the formative discharge may occur less frequently than in humid environments; in truly arid regions, possibly as rarely as once in 200 years [Baker, 1977; Friedman et al., 1996].
 In our analysis, we use the 5-year and 20-year floods to parameterize the regime model: the bankfull depths reported by Andrews were adjusted upward to reflect the fact that we are invoking larger formative discharges (see Appendix A for details). Since neither the geometry nor, consequently, the mean velocity for the 5-year or 20-year events are accurately known, we have applied a constant Manning's n of 0.03 to all of the channels.
4.1.3. Inversion of the Relative Bank Strength Model
 For both of the foregoing data sets, we have estimated the relative bank strength, ϕ′, associated with the observed channel geometry by inverting the regime model. To do this, we applied the regime model using the reported data for each study reach. In the computer program, we varied the parameter ϕ′ for each study reach until the predicted channel width matched the observed width. The purpose of this analysis is to link the patterns in sediment transport concentration evident in the C′ regime model to the more easily interpretable changes in relative bank strength, ϕ′.
4.2.1. Salmon River
 Salmon River exhibits fairly typical empirical hydraulic geometry (Figure 7). Power law fits for group 1 and group 2, respectively, are
The exponents of the width equations in the two groups of data (equation (22) and (24)) are significantly different at α = 0.002 (i.e., the 99.8% confidence level), and the coefficients are significantly different at α = 0.17. The exponents in the depth equations (equations (23) and (25)) are significantly different at α = 0.14, while the coefficients are different at α = 0.20. While the confidence levels for these differences are generally modest at best, there appear to be systematic differences between the two groups. These differences may be due to differences in the yield of bed material sediment from the drainage basin, to differences in the typical bed material grain shape and surface arrangement or, possibly to systematic differences in the bank conditions.
 The results of the regime modeling are shown in Figure 10. The effective cohesion value that produced the best fit to the data for group 1 was 1.9 kPa, and for group 2 it was 3.0 kPa. Both values are consistent with the range exhibited in the Hey and Thorne  data set when analyzed by Eaton . As in the Colorado River data set analyzed above, the bed material transport concentration declines downstream in the Salmon River. The predicted channel dimensions are not related to the formative discharge by a simple power function. The analysis produced trends that are clearly not power laws, but the modeled trends are about as consistent with the data as are empirical power fits, particularly for group 2 (Table 1).
4.2.2. Mountain Streams in Colorado
 For Andrews'  streams with thin riparian vegetation, we obtained the following empirical equations for width and depth as functions of the originally reported bankfull flows:
For thick vegetation, the empirical equations are
The results of the regime model analysis assuming a formative discharge with a return period of between 5 years and 20 years are shown in Figure 11. The downstream variations in width and depth are successfully reproduced by the model assuming, for the 5-year flood, that C′ equals 1.7 kPa and 1.9 kPa for the “thin vegetation” and “thick vegetation” channels, respectively. Substantially similar results were obtained for the 20-year flood by assuming C′ values of 2.2 kPa and 2.6 kPa for the thin and thick data sets. The pattern of declining bed material concentration is also evident in both of these analyses. The RMS errors for the fitted regime models using the 5-year flood are similar to (albeit slightly greater than) the RMS errors for the empirical equations using the originally reported bankfull flows (Table 1).
4.3. Inversion of the Relative Bank Strength Model
Figure 12 displays the results of the analysis of Emmett's  data set using the regime model with relative bank strength (ϕ′). The observed channel widths at each observation point were fit by varying ϕ′. The estimated relative bank strength declines systematically downstream for both groups. The uppermost reaches have relatively erosion-resistant banks that are about 5 to 7 times stronger than the channel bed, while the largest streams have banks that are between 1.5 and 2 times stronger than the bed (see Figure 1).
Figure 13 presents the same analysis applied to the Andrews  data set: the choice of formative discharge has little effect on the general trends predicted by the model. The bed material transport concentration again declines as formative discharge increases, as does the relative bank strength. The range of relative bank strengths is slightly lower than in Figure 12, with the thick vegetation channels being associated with higher relative bank strengths than the thin vegetation channels.
 It is worth noting that the regime model selects the narrowest possible stream channel that would be stable given the governing conditions of Q, S, and D84 so that, while it is possible that the banks are stronger than indicated by the apparent friction angle shown in Figure 1, they cannot be weaker since this would imply that they would be unstable at the identified bankfull discharge.
 Physically based regime models reasonably fit downstream hydraulic geometry data in a range of environments, reproducing the typical relations in which widths vary approximately with Q0.5 and depths vary with Q0.4 [Emmett, 1975; Andrews, 1984] as well as relations in which there is an unusually rapid increase in width [Tabata and Hickin, 2003; Ellis and Church, 2005] or depth [Pitlick and Cress, 2002] with discharge. The goodness of fit of the predictions is measured using the root mean squared error (RMSE) for the calculated widths and depths, as well as a mean bias (for the regime model only). Table 1 summarizes the model performance for all of the data sets examined. Generally, the RMSE for width estimates using the regime model are about the same as (and sometimes smaller than) the RMSE for the empirical equations, and the bias in the width estimates is low (i.e., <10%). The same is true for the RMSE for modeled depths, except in the anabranch and delta channels. For those data sets, the empirical equations perform substantially better and the modeled depths exhibit a significant mean bias that is about the same magnitude as the RMSE: this is attributable to poor model performance for the smallest anabranch channels (see Figures 4, 5, and 6). When the regime model is tested against only the larger channels, the RMS errors for the modeled depths are similar to those for the empirical equations, and the mean bias values are significantly reduced (Table 1).
 Overall, the regime models summarized in Table 1 perform about as well as empirical equations (particularly for the single-thread gravel bed channels), since the models generally explain about as much variance as do the empirical equations. However, since it might be expected that the empirical equations would outperform the physically based theory, because they are based on finding a relation that minimizes the RMSE of the observations while regime models yield predictive relations based on the physical processes supposed to underlie the observed patterns, the regime model performance is judged to be quite good. Since the regime models are physically based, they are transferable from one basin to another, whereas the empirical fits are not. Furthermore, the invocation of a physically based model leads to further insights about the structure and function of stream channel networks, and is a suitable tool for predicting their response to environmental change.
 The indicated decline in bed material sediment concentration with formative discharge in the single-thread channel systems examined herein (all of our cases of true downstream hydraulic geometry) suggests a common scale effect. Headwaters, in which Qbm/Q predicted by our model is high, appear to transport a relatively large proportion of the total sediment supplied to the channel as bed load (we draw a rough parallel between bed material and bed load transport for the purposes of this discussion). As sediment is transported farther from its original source, it weathers and wears into smaller and smaller sizes. In addition to direct abrasion during transport [Pizzuto, 1995; Sklar et al., 2006] floodplains, where sediment is stored in a wet environment, may be particularly effective at weathering bed material sediment into finer materials [Jones and Humphrey, 1997]. If the grain size is reduced sufficiently, the material is transported in suspension, moving longer distances and in a wider range of flows and bypassing storage as bed material. Continuity of mass in a nonaggrading system implies that a downstream decline in bed material concentration must be accompanied by a downstream increase in either the suspended sediment concentration or the dissolved material concentration. Sklar et al.  suggest that, beyond some critical length scale set by the relative resistance of the basin lithology, bed load transport rates cease to scale with drainage area and become effectively constant, a behavior that is reproduced by our model. In contrast, this sort of downstream change in Qbm/Q is not evident in our data sets from anabranching and delta systems, which are not, in fact, instances of downstream hydraulic geometry at all.
 Anabranch and delta systems exhibit scale relations in the true sense: they represent systematic variation in channel geometry with changing channel size, while gradient and channel boundary materials remain relatively similar within each data set. The regime model consistently predicts that bed material concentration increases with discharge: the relation between Qbm/Q and Q is interpreted to be evidence that, when single thread channels divide into multiple threads, they become less efficient and can transport less sediment. This observation is consistent with a well-known principle of hydraulic engineering that, to remain stable, small distributary channels must exclude bed material transport; that is, they must carry a much reduced sediment load. Amongst the four data sets, there is a remarkable inverse correlation between the exponents for the equations of width and depth (R2 = 0.93) which reveals that, together, width and depth accounts for a about 82% of the variation in channel capacity in every case. The covariation is presumably introduced by the degree to which channels flowing in similar materials must adjust their sediment transporting capacity to remain in equilibrium.
 This result is contrary to the speculation made by other researchers that channel division increases the bed material sediment concentration and that division in anabranch systems occurs in order to accommodate an increase the sediment supply [Nanson and Knighton, 1996; Jansen and Nanson, 2004]. Obviously, our choice of bed material transport law has influenced the model predictions so that the validity of the underlying transport laws, and the way in which we have used them in the model, must be considered when commenting on larger geomorphic issues such as channel pattern and sediment transport efficiency.
 The regime model performs poorly for the smallest anabranch and distributary channels across all four data sets. It overpredicts the depths of the smallest channels, and therefore overpredicts the transport concentration, indicating that the effect of channel division on the transport capacity in the smallest anabranches is in fact greater than predicted by the model. That this result is common to all four data sets suggests that either the assumptions about the parameters used to specify the model are systematically wrong, or that the fundamental premise of the model (i.e., that the channels are in regime) is inaccurate.
 A reasonable explanation for poor model performance is suggested by a sedimentological reconstruction of the Columbia River anabranch dynamics published by Makaske et al. . They reported that channels on the Columbia River floodplain initially form on top of crevasse splay deposits (stage 1) and are bounded primarily by sand. As a result, they tend to be fairly wide and shallow, and they would be poorly described by a regime model that assumes the existence of strong, cohesive channel banks. As the channels abstract more water and mature, they deepen by eroding the crevasse splay sands and by depositing cohesive sediments along their banks. Once they attain depths on the order of 2 m, they typically become capable of eroding laterally and maintaining a dynamic equilibrium (stage 2): these channels are likely in regime, and may be well described over a range of formative discharges by a regime model assuming a constant, high relative bank strength. At some point, often because of the development of log jams, the flow into the anabranch is reduced (stage 3), and the channels undergo either rapid vertical aggradation, producing wide, shallow channels similar to stage 1, or a more gradual combination of vertical and lateral aggradation, producing narrow, deeper channels that may or may not be in regime with the fluid and sediment fluxes supplied to them.
 Therefore it appears that the larger channels (in the Columbia system at least) may be in regime with respect to the fully developed cohesive banks (represented by ϕ′ = 78°), while the smaller channels may be in regime with respect to a range of lower relative bank strengths. These channels are covered in Figure 5 by a shaded region bounded by an upper (ϕ′ = 78°) and lower (ϕ′ = 68°) bank strength value. It is possible that a similar sequence of channel initiation and development is responsible for the range of channel shapes characteristic of the smallest distributary channels on both the Danube and Laitaure deltas as well (see Figure 6). Other small channels may be rapidly shrinking as they aggrade, ultimately forming part of the floodplain: these channels are not likely to be at equilibrium with respect to the supply of sediment, and therefore should not be well described by a regime model. Nonequilibrium behavior is likely at least part of the explanation of the poor model performance, and seems to be the most likely explanation for the Fraser River anabranches.
 While the first five cases analyzed in the paper represent carefully chosen situations for which we could make the assumption that the relative banks strength is approximately uniform, one cannot apply such assumptions to most downstream hydraulic geometry sets. In order to fully explain the variations in channel geometry expressed in the data sets presented by Emmett  and Andrews , we would require data that are seldom collected. However, we can reasonably speculate on the physical processes that underlie these variations.
 The ability of the modified regime model described by Eaton  to replicate the general downstream hydraulic geometry of both the Emmett  data set (see Figure 10) and the Andrews  data set (see Figure 11) by adopting reasonable values of vegetation-related bank strength, C′, suggests that the declines in relative bank strength and bed material concentration with increasing Q are driven primarily by vegetation-related bank strength. The effect of vegetation is limited to the zone influenced by the tensile strength of the root systems. For small streams, the rooting depth may be similar to the average channel depth, and the banks may be many times more erosion-resistant than the bed. However, as the channel depth increases (holding the rooting depth constant), the relative importance of the vegetation declines and the relative erodibility of the banks approaches that of the bed [see Eaton and Millar, 2004]. This scale effect is likely to be ubiquitous in channels where vegetation dominates the bank stability, and the results of our analyses are consistent with these plausible conjectures (see Figures 12 and 13). Variations in vegetation type and riparian land use along stream systems, as appears to be the case for the streams in Colorado [Andrews, 1984], are likely to further modify the scale effect.
 The analysis of the Andrews  data set also indicates that the channels with the highest potential bed material transport concentration tend to be those with thick riparian vegetation. These channels, probably laterally stable, are deeper than channels with less dense riparian vegetation. The channels with less dense vegetation and weaker channel banks are more likely to be laterally unstable, but they are less efficient in conveying bed material downstream.
 Our regime model also provides insight into the downstream variations in the Shields number (τ*) along the reach of the Colorado River studied by Pitlick and Cress . The authors explain the unusual hydraulic geometry along this reach as the result of a decrease in Qbm which, they point out, may have produced a downstream increase in the armor ratio, since the armor ratio is related to sediment supply [Dietrich et al., 1989]. However, they also claim that τ* remains essentially constant, which would seem to be at odds with declining transport capacity. The regime model supports the conjecture that bed material transport rates are declining, since Qbm/Q declines by almost two orders of magnitude as one moves down the system, and also suggests that, as the ratio W/d decreases downstream, τ* also decreases. The decrease is relatively small (from just 0.046 to 0.039), and remains close to the τ* value of 0.049 reported by Pitlick and Cress . It should be noted that the scatter in estimates of τ* of Pitlick and Cress is much greater than the range of variation in the model, implying that physically reasonable relations between the parameters W/d and τ* may be obscured by limited measurement resolution.
 We draw the following conclusions from our analyses of downstream hydraulic geometry using rational (physically based) regime models.
 1. Fully elaborated regime models of alluvial channel form are not simple scaling relations, as has been widely supposed, because of the variable influence of several governing factors. However, power law scaling relations furnish very good approximations of the underlying relations in many cases when one or more of the governing relations remains essentially constant.
 2. Systematic variations in relative bank strength and in bed material transport capacity are subsumed within the typically observed empirical hydraulic geometry. By fitting a physically based model to the data of hydraulic geometry, we approach the underlying physical structure of the stream network organization (with respect to grain size variation, transport capacity, boundary resistance and channel gradient), as well as the scaling of channel dimensions with formative discharge. That is, we obtain a glimpse of how stream networks function.
 3. In stream networks where riparian vegetation influences bank strength, relative bank strength declines downstream with increasing discharge. This scale effect is consistent with the notion that the contribution of riparian vegetation to bank strength declines as the channel depth becomes large relative to the rooting depth of the riparian species. The hydraulic geometry of stream networks can be modeled by assuming an appropriate (and constant) value for the absolute bank strength, represented as an effective cohesion, C′.
 4. The bed material sediment concentration in stream networks generally appears to decline with discharge in conjunction with relative bank strength which, in nonaggrading systems, is presumably counterbalanced by an increase in the suspended sediment (and/or dissolved material) concentration in order to maintain continuity of mass.
 5. For large stream channels, like the Fraser River and the Colorado River, the channel geometry can be reasonably modeled by assuming a uniform relative bank strength, ϕ′. The atypical hydraulic geometry relations reported by Pitlick and Cress  and Ellis and Church  are reproduced using such models.
 6. The scaling relations for anabranched channels and delta distributary channels, where true friction and cohesion dominate bank strength, are reasonably reproduced using regime models that assume a uniform relative bank strength, ϕ′. For these channels, bed material concentration increases with formative discharge, indicating that division of a channel into multiple threads is associated with reduced sediment transport efficiency and likely with aggradation, as well.
 7. The smallest channels examined in the sand bed anabranch and distributary channels in this study conform to a regime model that assumes a lower bank strength. In the case of the Columbia River, there is a sedimentological basis for believing that this variation is due to real changes in the bank strength. The same argument may apply to the Laitaure and Danube deltas, though there are no published reports to support this assertion. Some of the small channels are probably shrinking as they aggrade: they would not therefore be in equilibrium with their governing conditions and should not be expected to agree with the model predictions.
 Physically based models relating channel geometry to the governing conditions of discharge, sediment supply, grain size and slope can be constructed in a wide variety of environments using appropriately configured regime models, but only if the relative erodibility of the channel boundary is adequately represented. A key consideration is the choice of bank stability analysis method and the choice of bank stability parameter values. Direct knowledge of the bed material discharge would seem to be of secondary importance, since the reach-averaged slope and characteristic grain size are set in response to the long-term average bed material discharge, and since channels have the capacity to respond to small year-to-year variations in sediment discharge without modifying the gross channel morphology simply by varying the bed surface texture and structure.
 While application of a regime model requires knowledge of more than just the formative discharge, the additional parameters are unarguably important in determining the channel morphology, so the increased information requirements appear to be reasonable and necessary for the construction of a physically based theory to explain the observed hydraulic geometry of rivers. In addition to providing insight into hydraulic geometry relations, we believe that regime models can usefully be incorporated into numerical models of drainage network dynamics and even landscape development. They represent a computationally effective method for relating channel morphology to governing conditions and, insofar as they escape unqualified empiricism, they are superior to the empirical hydraulic geometry equations commonly used in drainage network models.
 The initial analysis of the Andrews  data set using a relative bank strength formulation indicated that many of the channels identified as having thin riparian vegetation and one channel with thick vegetation have banks that are more erodible than the bed. This is surprising, since previous analyses using the regime model [Eaton and Millar, 2004; Eaton, 2006] indicate that channel banks with riparian vegetation similar to that of the Colorado streams still remain about 50% stronger than the channel bed and, on average, exhibit apparent friction angles of 40°. Only a handful of the smaller channels with thick vegetation and a single channel with thin vegetation exhibit widths consistent with even the modest bank strength index of 40°. This unexpected result led us to question whether or not the reported bankfull discharges actually correspond to the formative flow responsible for the observed widths and depths. (A previous analysis of these data by Millar  did not consider this possibility.)
 Andrews estimated bankfull flows by projecting surveyed indicators of the bankfull elevation along the stream channel parallel to the water surface slope to a nearby gauge section, where the recorded stage-discharge relation was used to estimate the bankfull Q. Since bankfull depth is difficult to measure in the field, and since the stage relations at gauge cross sections are not necessarily the same as the stage relations elsewhere in a river reach, it is at least possible that formative discharges were systematically underestimated.
 It is also possible that the observed channel widths are associated with flows that are larger than the bankfull flows. Since, in semiarid regions, the rate at which vegetation can colonize an eroded surface is substantially slower than it is in a humid environment, it is likely that stream channels in arid environments are influenced to a greater degree by less frequent flood events [Baker, 1977; Friedman et al., 1996].
 In order to assess the possibility that the widths reported by Andrews  were formed during flows higher than the reported bankfull value, we first used Andrews's dimensionless hydraulic geometry equations to predict the widths of stable gravel bed streams with similar bank vegetation information [Hey and Thorne, 1986]. Hey and Thorne  classified bank vegetation into four classes. We have assumed that the dimensionless equations for thin vegetation reported by Andrews  should describe Hey and Thorne's types I and II channels, which are the least densely vegetated, and that Andrews's equations for thick vegetation should correspond to Hey and Thorne's types III and IV channels. When Hey and Thorne's discharges are entered into the Andrews equations, they consistently overpredict the observed channel widths in Hey and Thorne's data set. We reduced the discharge used in the equations until the observed and predicted widths agreed, on average (Figure A1). The Andrews equations predict widths that are, on average, consistent with the channel widths observed by Hey and Thorne when their Q estimates are reduced to approximately 66% of their original values. This, combined with the initial results using the regime model, suggests that the widths observed by Andrews  were probably formed by flows consistently greater than the reported bankfull flows. We therefore surmise that the bankfull widths observed in the field by Andrews were likely formed by flows less frequent than those identified as bankfull flows in the original data set.
 On the basis of the existing data for each USGS gauging station, we estimated the peak flows with return periods of 5 and 20 years (Table A1) and used these values in the regime model. Streams for which the period of record was not sufficient to accurately estimate the 20-year peak flow have been excluded from the analysis. The 5-year flood frequency best corresponds to the empirical result from the test using the Hey and Thorne  data set: these data were used in regime models. Since there is a large degree of uncertainty in this post hoc analysis, we have also run the regime analysis using the 20-year floods. In order to estimate the water depth associated with the 5-year and 20-year floods, we used recorded water stages reported by the USGS for the historic peak flows at each gauging station. We compared the 5-year and 20-year water stage values to the water stage for a flow close to the reported bankfull stage. In order to avoid the effects of net aggradation or degradation at the gauge section, water stage values were chosen as close together in time as possible. The estimated difference in the stage between the 5-year and 20-year flood and the flood magnitude reported by Andrews  was added to the bankfull depths that Andrews originally reported. The result was an average increase in water depth of about 33% for the 5-year event and 55% for the 20-year event (Table A1).
Table A1. Discharge Data for 5-Year and 20-Year Floods in the Andrews  Gravel Bed Channels
Discharge Data, m3/s
Channel Geometry, m
Ratio to Qbf/Qi
 Rudy Slingerland generously made the data of H. Andrén available to us, and John Pitlick kindly provided unpublished data on the Colorado River. We also extend our thanks to Alex Densmore, Rob Ferguson, Tom Lisle, and an anonymous reviewer for their extensive and very helpful comments on the paper; all of their comments have resulted in a much improved paper.