Logistic Analysis of Channel Pattern Thresholds: Meandering, Braiding, and Incising

Home , Channel pattern, Channel types, Meander

Geomorphology 38Ž. 2001 281–300 www.elsevier.nlrlocatergeomorph

Logistic analysis of channel pattern thresholds: meandering, braiding, and incising

Brian P. Bledsoe), Chester C. Watson 1 Department of CiÕil Engineering, Colorado State UniÕersity, Fort Collins, CO 80523, USA Received 22 April 2000; received in revised form 10 October 2000; accepted 8 November 2000

Abstract

A large and geographically diverse data set consisting of meandering, braiding, incising, and post-incision equilibrium streams was used in conjunction with logistic regression analysis to develop a probabilistic approach to predicting thresholds of channel pattern and instability. An energy-based index was developed for estimating the risk of channel instability associated with specific stream power relative to sedimentary characteristics. The strong significance of the 74 statistical models examined suggests that logistic regression analysis is an appropriate and effective technique for associating basic hydraulic data with various channel forms. The probabilistic diagrams resulting from these analyses depict a more realistic assessment of the uncertainty associated with previously identified thresholds of channel form and instability and provide a means of gauging channel sensitivity to changes in controlling variables. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Channel stability; Braiding; Incision; Stream power; Logistic regression

1. Introduction loads, loss of riparian habitat because of stream bank erosion, and changes in the predictability and vari- Excess stream power may result in a transition ability of flow and sediment transport characteristics from a meandering channel to a braiding or incising relative to aquatic life cyclesŽ. Waters, 1995 . In channel that is characteristically unstableŽ Schumm, addition, braiding and incising channels frequently 1977; Werritty, 1997. . Changes in the magnitude, threaten human infrastructure. relative proportions, and timing of sediment and Alluvial channel patterns form a continuum rather water delivery in these channels may affect water than discrete types and distinctions between morpho- quality via a wide variety of mechanisms. These logic types are fuzzy and complexŽ Ferguson, 1987; mechanisms include changes in channel morphology, Knighton and Nanson, 1993. . Process and historical planform, bed material, and suspended sediment studies of individual streams suggest that switches between channel patterns may be historically contin- gent on how intrinsic variables have ‘primed’ a ) Corresponding author. Tel.: q1-970-491-8410; fax: q1-970- reach for instability and on the state of the channel at 491-8671. Ž. E-mail addresses: [email protected] the time of an impact Brewer and Lewin, 1998 . Ž.B.P. Bledsoe , [email protected] Ž. C.C. Watson . Because channel patterns are frequently transitional 1 Tel.: q1-970-491-8313; fax: q1-970-491-8671. and shaped by complex sequences of disturbance

0169-555Xr01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S0169-555XŽ. 00 00099-4 282 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 events, stochastic alternatives to deterministic thresh- investigators have also suggested that bank resis- olds may be more useful and realistic for predictive tance is a confounding factor in the analysis of and postdictive work on the effects of geomorphic channel planformŽ Carson, 1984; Ferguson, 1987; changeŽ. Graf, 1981; Tung, 1985; Ferguson, 1987 . Bridge, 1993. . Probabilistic discrimination of relatively stable and Several theoretical approaches to defining the unstable channel types using basic sedimentary and transition from meandering to braiding have also hydraulic variables offers improved predictions of been developedŽ e.g., Parker, 1976; Fredsøe, 1978; environmental impacts. Such predictions of risk help Fukuoka, 1989. . These techniques have limited pre- establish priorities for management, planning, and dictive value, however, because the techniques rely policy. on parameters such as width, depth, and Froude number that are usually not available for a priori 1.1. Discriminators of braiding and meandering predictions of channel form. The theoretical approaches do seem to suggest that braiding generally In the latter half of the twentieth century, geomor- occurs at channel width-to-depth ratios in excess of phologists and engineers attempted to predict the 50Ž.Ž. Fredsøe, 1978 . Bridge 1993 provides an excel- planform of rivers based on the slope and some lent review and summary of the various empirical measure of discharge or drainage area. The concept and theoretical thresholds that have been proposed of a threshold discharge–slope Ž.Q–S combination for the transition from braiding to meandering. that discriminates braiding rivers from meandering In an attempt to reconcile some of the issues ones has become firmly embedded in the doctrine of associated with the original empirical approaches, fluvial geomorphologyŽ. Carson, 1984 . The Q–S van den BergŽ. 1995 devised a simple parameter discriminator was first suggested by LaneŽ. 1957 and representing potential specific stream power that en- Leopold and WolmanŽ. 1957 . Many other investiga- ables discrimination of braiding and meandering tors have provided support for the notion of a thresh- riversŽ. sinuosityG1.3 in unconfined alluvium. Us- old between meandering and braiding types and have ing a data set of 126 streams and rivers, he arrived at used this approach to interpret a variety of field and a discriminant function of the form: flume dataŽ Schumm and Khan, 1972; Chitale, 1973; g . v( SQ0.5s900D 0.42 Ž.1 Osterkamp, 1978; Begin, 1981; van den Berg, 1995 . a Õ bf 50 Critical examination of the early work on meandering to braiding thresholds has subsequently re- where specific stream power Ž.v is a function of vealed several limitations of the original approach. valley slope, estimated bankfull dischargeŽ m3rs,. Critical stream power for braiding appears to vary and a regression coefficient Ž.a computed for a with bed material size; i.e., the critical slope for particular collection of streams that varies between braiding at a given discharge is higher for gravel sand bed and gravel bed rivers. The discriminate than for sand bed channelsŽ Carson, 1984; Ferguson, function applies to both sand and gravel bed rivers 1987. . The threshold suggested by Leopold and Wol- with median bed material ranging from 0.1 to 100 manŽ. 1957 is too high for sand bed channels and too mm. Potential specific stream power and median low for gravel bed channels because it is based on a particle size Ž.D50 are expressed in Watts per square combination of the two channel types. Other con- meterŽ Wrm2 . and meters, respectively. The result- cerns with the original stream power approach arise ing specific stream power represents the maximum from bias associated with using channel slope and potential specific stream power for a particular val- bankfull discharge parametersŽ. van den Berg, 1995 . ley slope. Channel slope and bankfull discharge are not necessarily independent of planform. Braided channels are 1.2. Stream power and eÕolution of incising sand bed inherently steeper than meandering channels for a streams given valley slope because of generally low sinuosity. Braided channels may also have larger bankfull Channel incision, or bed lowering by erosion, cross-sections as a consequence of braiding. Some results from an imbalance in the power aÕailable to B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 283 move a sediment load and the power needed to such as specific stream power, the size of bed mate- move the sediment loadŽ. Bull, 1979; Galay, 1983 . rial, sediment load, and bank resistance. Logistic Where banks are cohesive and resistant to hydraulic regressionŽ. Menard, 1995; Christensen, 1997 is a forces, bed erosion is initially dominant over channel statistical technique that allows comparison of binary widening. As an incising channel deepens, flow and continuous parameters. The research described events of increasingly greater magnitudes are con- below is targeted at developing a risk-based ap- tained within the channel banks. This process may proach for predicting the occurrence of stable mean- create a positive feedback wherein shear stress on dering, braiding, and incising channel forms as a the bed and toe of the banks during large events is function of simple hydraulic and sedimentary at- magnified through incision. If bed degradation con- tributes. The specific objectives of the proposed tinues until banks reach a critical height for geotech- research were the following. nical failure, radical channel widening may be initi- Ž.i To use a very large data set from actual ated. Such widening can proceed at a staggering rate, streams with logistic regression methodology to with tenfold increases in cross-sectional areas re- identify probabilities of occurrence of braiding or ported in some instancesŽ. Harvey and Watson, 1986 . incising channels versus meandering channels using Conceptual models of the temporal response or simple indices of specific stream power and charac- evolutionary sequence of incised channels have been teristics of bed material. described by various investigatorsŽ Schumm et al., Ž.ii To contrast the logistic regression results with 1984; Harvey and Watson, 1986; Watson et al., van den Berg’sŽ. 1995 previously defined geomor- 1988a,b; Simon, 1989; Simon and Hupp, 1992. . The phic threshold between meandering and braiding models generally describe the same process: excess forms. specific stream power initially results in bed degradation followed by a period of widening and bank failure and further bed adjustment until the channel 2. Methods eventually establishes a new flood plain and quasi- equilibrium channel form within newly deposited 2.1. Logistic regression analysis— basic concepts alluvium. When compared with work on the meandering to Commonly applied regression techniques are ap- braiding threshold, incised channel forms have re- propriate only when the dependent variable and the ceived relatively little attention from researchers. explanatory variables are quantitative and continu- Even less common are investigations of thresholds of ous. To analyze a binary qualitative variableŽ. 0 or 1 stream power associated with the development of as a function of a number of explanatory variables, incised channel forms. special techniques must be used if the analysis is to be performed adequately. The regression problem may be revised so that, rather than predicting a 1.3. ObjectiÕes binary variable, the regression model predicts a continuous variable that stays within the 0–1 bounds. Water resources engineers and fluvial geomor- One of the most common regression models that phologists sometimes need to relate qualitative de- accomplishes this is the logit or logistic regression pendent variables to one or more independent vari- modelŽ. Menard, 1995; Christensen, 1997 . Although ables, which may or may not be quantitative. In the logistic regression is most frequently applied in the case of channel instability and ecosystem changes social and health sciences, TungŽ. 1985 presented a because of excess stream power, the dependent vari- logistic regression methodology to evaluate the po- able might be the occurrence or nonoccurrence of a tential of channel scouring as a function of quantita- rapid and significant geomorphic response in the tive data taken from flume studies. form of widening andror incision. Thus, the depen- In the logistic regression model, the predicted dent variable is binary and qualitative and the inde- values for the dependent variable are never F0or pendent variable may be quantitative and continuous, G1 regardless of the values of the independent 284 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 variables. This is accomplished by applying the fol- of model performance. In logistic regression, it is lowing regression model: common to be more concerned with whether the q q q predictions are correct or incorrect than with how expŽ.b011b x ... Bxnn ys Ž.2 close the predicted values are to the observedŽ 0 or q b qb q qb 1 expŽ.011x ... nnx 1. values of the dependent variables. Therefore, R2 Regardless of the regression coefficients or the mag- has little meaning in logistic regression analysis nitude of the x values, this model will always Ž.Tung, 1985; Christensen, 1997 . One of the most produce predicted values in the range of 0 to 1. meaningful ways of interpreting a logistic regression Suppose the binary dependent variable y represents model is to examine the number of incidences of an underlying continuous probability p, ranging from misclassificationŽ. Tung, 1985 . The number of cor- 0 to 1. The probability p may be transformed as: rect and incorrect classifications may be used to calculate an ‘odds ratio’Ž. Neter et al., 1988 from a X p p slnŽ. 3 2=2 classification table which displays the pre- 1yp ž/ dicted and observed classification of cases for a This transformation is referred to as the logit or X binary dependent variable. Although the odds ratio logistic transformation. Note that p can theoretically provides a useful measure of predictive accuracy, it assume any value between minus and plus infinity. is quite sensitive to sample size. Care must be taken Since the logit transform solves the issue of the 0–1 in comparing across models that have differing num- boundaries for the original dependent variableŽ prob- bers and proportions of dependent variable cases. In ability. , the logit transformed values could be used in this study, the percentages of streams and rivers an ordinary linear regression equation. If Eq.Ž. 3 is correctly classified by the various logistic regression applied to both sides of the logistic regression equa- models was used as a primary measure of model tion stated in Eq.Ž. 2 , the standard linear regression performance. model is obtained: Goodness-of-fit tests may also aid in the interpre- X s q q q p b011b x ... bnnx Ž.4 tation of the results of logistic regression. The likelihood L for the null model, where all slope parame- In logistic regression analysis, maximum likeli- 0 hood techniques may be used to maximize the value ters are zero, may be directly compared with the likelihood L of the fitted model. Specifically, one of a function, the log-likelihood function, which 1 x 2 indicates how likely it is to obtain the observed can compute the statistic for this comparison as: 2 sy y values of the dependent variable given the values of x 2Ž. logŽ.L01log Ž.L Ž.5 the independent variables and parameters, b ,...,b . 0 n x 2 Unlike ordinary least squares regression, which is The degrees of freedom for this value are equal able to solve directly for parameters, the solution to to the number of independent variables in the logistic x 2 the logistic regression model is found through an regression. If the p-level associated with this is iterative process that maximizes the likelihood func- significant, the estimated model yields a significantly tion until the solution converges within tolerance. better fit to the data than the null model and the Maximum likelihood estimation was used for all regression parameters are statistically significant. logistic regression analyses described in this study. A This statistic is equivalent to the F statistic in ordi- quasi-Newton method provided in the statistical nary least squares regression. analysis software Statisticaw Ž StatSoftw , 1999. was Logistic regression diagnostics, like using linear used to optimize the maximum likelihood function. regression diagnostics, may seem more art than sci- Ž. Model results from Statisticaw were verified using ence Menard, 1995 . Diagnostic statistics hint at the SAS software packageŽ. SAS Institute, 1990 . potential problems, but whether remedial action is required is a matter of judgement after close inspec- 2.2. Regression interpretation and diagnostics tion of the data. In a sample of 200–250, random sampling variation alone will produce 10–12 cases In ordinary regression analysis, the coefficient of of residuals with absolute values )2 on standard- determination Ž R2 . is frequently used as a measure ized, normally distributed variables such as the de- B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 285 viance residual or the Studentized residual. Very aggradation, degradation, or changes of width for a large residuals do not necessarily indicate problems period of years. Streams exhibiting dynamic stabil- in the modelŽ. Menard, 1995 . As described below, ity, such as lateral migration without significant only a few independent variables were considered changes in width and depth, were deemed acceptable for inclusion in the logistic regression models of for inclusion. Like meandering, braiding has been channel response. Measures of flow energy and sedi- defined in numerous ways by various investigators. mentary characteristics were either combined as a In this analysis, braiding referred to multiple-thread single parameter or used as two separate, indepen- channels with generally unvegetated bars and islands dent variables. Therefore, issues associated with that were not significantly wider than the adjacent model mis-specification and multi-collinearity of in- channels. Unfortunately, insufficient data precluded dependent variables were kept to a minimum. The attempts to distinguish between actively braiding and residuals of each regression model were examined relatively inactive braided streams. for outliers, influential cases, and non-normality. As By contrast, only streams that were actively and might be expected with a large and diverse data set progressively incising over periods of 1–10 years on river characteristics, some rivers were consis- were included in the analysis. Channels identified as tently misclassified. Such cases were examined for incising were all classified as CEM Type 2 accord- potential errors, but all cases were retained in the ing to the Watson et al.Ž. 1988a,b model of incised data set because no compelling reasons existed to channel evolution. Type 2 channels are in the early suspect that mistakes in reporting or measurement stages of rapid incision and have yet to initiate the had occurred. dramatic widening that occurs in subsequent stages of evolution. Quasi-equilibrium channels were de- 2.3. Field data used in analyses of channel pattern fined as channels that have a stable slope after a full and stability cycle of incisionŽ. CEM Types 4 and 5 according to the Watson et al.Ž. 1988a,b model of channel evolu- A primary objective of this study was to develop tion. an extensive data set on slope, discharge, character- To facilitate the separation of channels with sand istics of bed material, planform, and morphology of beds from the rest of the data set during statistical alluvial rivers from various regions of the world. analyses, the data were also tagged as having one of Data chosen for the analysis were selected from three categories of bed material: sand, mixed, or several scholarly publications including peer-re- gravel. Sand was defined as a D50 from 0.0625–2 viewed articles from scientific journals and govern- mm, mixed was defined as )2–10 mm, and gravel ment reports and documents. Streams and rivers was defined as )10 mm. These distinctions are included in the data set ranged from live bed chan- admittedly arbitrary, but such a separation allows a nels with beds of very fine sand to threshold chan- rough discrimination between live bed channels that nels composed of cobble-sized materials. The data are very sensitive to sediment supply and threshold were limited to self-formed channels in alluvial ma- channels that are somewhat less sensitiveŽ Mont- terial that are free from constraints such as bedrock gomery and Buffington, 1997.Ž. . van den Berg 1995 outcrops and stabilization structures that restrict the reported that relatively few streams occur with a D50 evolution of braided rivers. The resulting data set of between 2 and 10 mm. The mixed channels were 270 streams and rivers from around the world along only included in collective analyses of all size frac- with sources can be obtained from the corresponding tions. author. Five categories of channels were delineated for 2.4. Variables used in the analyses the analyses: stable meandering, braiding, incising, braiding or incising, and quasi-equilibrium channels. The parameters considered for inclusion in the Stable meandering channels were defined as single- logistic regression models were limited to the basic thread channels that have maintained a sinuosity variables of slope, discharge, and some representa- greater than or equal to 1.3 without progressive tion of bed material. Sinuosity was used to relate bed 286 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 slope to valley slope where valley slope data were had explanatory power equaling that of models con- not available. In cases where sinuosity data were not taining slope, discharge, and D50 as separate inde- available for braided rivers, valley slope was as- pendent variables. This was especially true for chan- sumed to be 10% steeper than bed slope. Bed slope nels with sand beds. The inclusion of D50 in the was assumed to be representative of energy slope for denominator simply scales flow energy relative to all streams with the exception of incising and quasi- the size of bed material. The form of this parameter equilibrium streams in the Yazoo River watershed of is also consistent with the theoretical relationships northern Mississippi. For these streams, energy slopes developed by ChangŽ. 1985 and the empirical rela- were estimated using surveyed cross-sectionsŽ Wat- tionship of van den BergŽ. 1995 . This parameter is son et al., 1998a, 1999. in the hydraulic model approximately proportional to unit stream power, VS HEC-RASŽ. USACE, 1997 . Ž.Yang, 1976 , multiplied by a relative submergence Unfortunately, limiting the analysis to streams term: with detailed descriptions of the characteristics of bed materials would have drastically reduced the size Q S Re h S AAVS 7 of the data set. For a great number of streams, only 0.5 Ž. ( D50 (D50 ž/D50 the median particle size Ž.D50 was reported. As a result, only the diameter of the median particle size The ratio of forces represented by this parameter was utilized to maximize sample size. Estimates of suggests that it is an index of erosive power or the median size of bed materials were the result of mobility. Henceforth, it will be referred to as a sieve analyses for sand fractions and Wolman counts Amobility index.B It is worth noting that Eq.Ž. 6 may Ž.Wolman, 1954 for gravel and coarser materials. be made dimensionless by adding kinematic viscos- Sediment information was not available for some ity Ž.Õ to the denominator under the radical: streams in the Yalobusha River watershed in Missis- sippi. Median size of sediment in the basin, however, Q is extremely uniform within a range of 0.3–0.5 mm S Ž.8 ( D50 Õ Ž.Watson et al., 1998b . The median size of particles in unsampled streams was assumed to equal the If one assumes a value of 10y6 m2rs for clear water median size of 0.36 mm for streams that have been at 208C, then the value of Eq.Ž. 8 is simply the value sampled in the Yalobusha watershed. of Eq.Ž. 6 multiplied by a factor of 103 in SI units. The dimensionless value of this parameter is not 2.5. DeÕelopment of model parameters reported in this study because potential differences in kinematic viscosity make application and interpreta- In developing parameters for the logistic regres- tion more difficult. sion models, emphasis was placed on maintaining The mobility index represented in Eq.Ž. 6 was independent variables in forms that clearly represent used extensively in the logistic regression analyses. physical processes having a theoretical foundation. 0.5 In addition, models containing SQ with D50 as Specifically, slope Ž.S and discharge ŽQ .were com- separate independent variables were employed. Anal- bined into one variable representing a surrogate for yses were performed using bedslope and valley slope. specific stream power. A preliminary logistic regres- Annual flood and bankfull discharges were utilized sion analysis was performed using combinations of in combination and separately to allow a comparison this parameter in conjunction with D and fall 50 of accuracy. The log10 transforms of all data were velocity. The results of this analysis suggested that utilized in the logistic regression models. Two- Q parameter logistic regression models were of the S Ž.6 form: ( D50 q q expŽ.b01122b x b x is a simple but robust predictor of channel planform pŽ.instability s Ž.9 q b qb qb and stability. In some cases, this single parameter 1 expŽ.01122x x B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 287

0.5 where x12is the specific stream power term; x is Assuming that x12and x happen to be log 10SQ the median bed material term; and pŽ.instability is and log10D 50 , respectively, then a power function the probability of braided, incising, or unstable chan- approximately corresponding to ps0.5 is: nel types when regressed against stable meandering bb02 or quasi-equilibrium channels. The single parameter s yy SQ' 10 bb11D50 Ž.12 models were equivalent to Eq.Ž. 9 without the b22x terms. In the single parameter models, the mobility This approach was used in comparisons with the indexŽŽ.. Eq. 6 was used exclusively as the indepen- discrete threshold proposed by van den BergŽ. 1995 . dent variable. Over 200 logistic regression models In addition, the original data of van den BergŽ. 1995 were tested using one- and two-parameter ap- were re-analyzed using a two-parameter logistic re- proaches. The logistic regression results provided gression model. This allows a risk-based interpreta- explicit probabilistic statements that were attached to tion of the discriminant function presented in Eq.Ž. 1 . diagrams of channel stability to clearly associate the risk of a particular channel form with varying levels of excess stream power. 3. Results and discussion

2.6. Relating logistic regression results to other thresholds A comprehensive summary of logistic regression results for 74 models is presented in Table 1. Each logistic regression model was assigned a number in The logistic regression results were compared with Table 1 that is used in referring to specific models thresholds of channel form and stability reported by throughout this discussion. The excellent fit of these van den BergŽ. 1995 . An interesting feature of logis- models underscores the predictive power of the sim- tic regression is the potential for transforming a ple indices used to represent the balance of erosive relationship like Eq.Ž. 9 into a function representing and resistant forces. All log-likelihood x 2 statistics the combination of independent variables corre- of the models were significant at the p-10y5 to sponding to a 50% chance of the binary-dependent p-10y29 levels. variable occurring in either category. For example, In general, the most accurate results for the great- assume that a two-parameter logistic regression est number of cases of all stream types were ob- model has been fit to log10 transformed data on tained by using bedslope in conjunction with annual meandering and braiding streams. The combination flood discharge as first priority with bankfull dis- of independent variables resulting in a probability of charge substituted for those streams without annual braiding of 0.5 represents the circumstances under flood data. Overall, differences in predictive accu- which it is most uncertain whether a stream will be racy, resulting from using combinations of annual braiding or meandering. Recalling that for a two- flood and bankfull discharge data versus using an- parameter logistic model: nual flood or bankfull alone, were generally less than 10%. The most notable exceptions were detected p when using valley slope instead of bedslope to repre- ln sb qb x qb x Ž.10 ž/1yp 01122 sent specific stream power. The predictive accuracy of the logistic regression models was reduced in all it follows that the combination of independent vari- cases when bedslope was replaced with valley slope. ables corresponding to ps0.5 chance of meander- The one- and two-parameter models varied in ing or braiding may be written as: explanatory power depending on the type of channels examined. For example, the single parameter r 0.5 model using the mobility index SQŽ.D50 was bb02 x sy y x Ž.11 extremely robust in analyses of channels with sand 12bb 11 beds using bedslope. In contrast, two-parameter 288 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q bf af bf af bf af bf af bfbf af bf af af bfbf af af bfbf af af bfbf af af bfbf af af af bf af bf af bf af bf afaf bf af bf af bf af bf af bf af bf af bf af bf af bf bf afaf bf bf afaf bf af bf af bf bf afaf bf bf afaf bf bf Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Flow used Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q n Odds ratio 2.22 131.05 46.67 206 1.69 86.04 20.18 206 1.70 91.42 32.61 206 2.18 134.99 68.66 206 2.10 56.89 42.86 127 6.97 39.59 9.486.23 53.36 90 14.03 109 6.062.65 55.11 79.58 250.676.15 57.22.45 37.35 67 58.39 49.83 127 30.32 67 127 8.83 98.39 105.00 109 6.193.01 54.49 83.33 250.676.71 72.312.70 37.15 67 61.15 131.6 127 40.799.79 79 67 127 47.5 90 57.85 33.30 40.25 67 12.12 84.43 66.50 90 y y y y y y y y y y y y y y y y y y y 0.472.06 160.97 39.91 012

bbb x y y MBIBIQ g 94 75 11.87 6.54 g 92.6 70.2 1.87 4.95 – 108.53 29.52 206 g 92.6 61.7 8.34 5.1 g 90.6 59.6g 94.6 64.9 0.68 3.73 – 8.96 68.65 5.64 14.26 206 g 95.3 77.2 12.64 7.14 g 92.6 73.7 2.01 5.06 – 109.25 35.13 206 g 89.9 56.7 0.84 3.78 – 70.85 11.68 206 r r r r r r r r ab 2 c m m m m m m m m r r r r r r r r s sand 89.6sand 92.9 85.4sands 4.63 91.8 14.39 88.1 – 4.51 13.15 82.45 68 111.80 – 4.28 100.06 90 sand 13.95 65.60 – 87.5 109 43.18 90.5 15.73 67 4.35 14.07 – 78.41 66.50 90 s sandsand 72.9sand 77.1 71.4sand 68.8 73.8 70.8s 1.38 82 6.29 9.55 85.2 6.44 – 1.63s 9.57 6.35 35.40 6.18 6.73 –sand 90 49.68 70.8 10.00 109 69 1.22 5.96 – 30.37 5.42 90 sand 87.5sand 90.5 87.5sand 24.08 93.8 14.34 s 90.5 19.94 11.40 68 25.38 13.87sand 6.18 83.3 43.01 20.19 90.5 67 23.93 13.9 s s D D D D D D D D D D Ž. 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 and log and log and log and log and log and log and log and log and log and log D D D D D D D D D D D r r r r r 0.5 0.5 0.5 0.5 r r r r r r 0.5 0.5 0.5 0.5 0.5 0.5 SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. ŽŽ . . ŽŽ .. Ž. Ž. ŽŽŽŽ . . .. Ž. Ž. ŽŽŽŽ . . .. Ž. Ž. ŽŽŽŽ . . . . ŽŽ .. Ž. Ž. ŽŽŽŽ . . .. 1010 50 v 50 1010 v 1010 50 10 v10 1010 50 50 v10 50 10 v1010 50 10 v10 50 10 50 10 50 10 50 10 10 50 10 50 50 v 10 50 10 50 10 v 1010 50 10 v10 10 50 50 50 10 50 9 log 1234 log 567 log 8 sand gravel 97.9 94.6 sand 84.2 73.5 gravel sand 89.6 93.5 gravel 59.1 64.7 95.8 96.8 63.6 58.8 2.33 3.75 10.97 6.50 0.24 2.45 – – 5.44 5.69 55.08 76.38 – 250.67 – 48.89 30.52 67 58.26 127 12.42 26.58 67 127 1215 log 1618 log 19 log 20 log 22 log 23 log 25 log log 28 log log 3437 log 38 log log 17 log 2124 log log 31 log 39 log Table 1 Summary of the results of logistic regression 10111314 sand gravel2627 97.9 94.6 sand 84.2 76.5 gravel2930 95.8 93.5 68.4 67.632333536 18.12 12.53 10.75 6.52 8.53 10.35 sand 6.48 gravel 5.71 97.8 94.6 sand 84.2 79.4 gravel 93.8 95.7 sand 73.7 67.6 gravel 97.9 95.7 sand 84.2 76.5 gravel 97.9 95.7 2.07 73.7 4.13 64.7 10.94 7.01 0.01 2.54 – – 7.46 5.93 17.61 13.81 54.45 83.20 10.66 – 250.67 – 7.01 67.89 9.03 11.00 35.19 67 61.10 127 7 5.93 42.00 46.52 67 127 Model no. Independent variable s Bed % Classified correctly B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 289 Q Q Q Q Q Q q q q q q q bf bf af bf bf bfbf af af af af bf af af afaf bf bf Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q arameters are q q q q q bfbf af bf af bf af bf af af bf bf bf bf bf bf bf bf bf bf bf bf af af af af af af s, and slopes are Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q r 3 used as first priority. VDB refers to the Q with Q 2.23 119.07 82.42 174 6.81 95.65 65.6 109 6.22 43.20 15.73 67 5.81 33.04 5.42 90 5.57 47.25 12.398.62 109 40.69 17.33 77 1.89 84.69 40.593.25 94.60 174 45.001.54 57.03 126 35.332.54 VDB 98.3 892.40 51.20 VDB 92.68 157 29.44 VDB 157 VDB 3.24 78.38 150.862.97 56.73 121 60.901.82 66.913.47 121 71.170.91 7.02 27.555.66 91 – 13.751.21 25.77 VDB 64.874.93 – 47 59.421.38 39 25.23 VDB 59.47 VDB 70.00 95 38.33 52 VDB VDB 52 95 VDB VDB 16.5 84.54 330 77 17.98 45.46 200.0012.57 33.15 44 44.0011.43 23.52 44 68.00 28 VDB and Ž. y y y y y y y y y y y y y y y y y y y y y y y y Q is in millimeters. Discharges are in units of m D refers to combining Q q Q is in meters. In two-parameter models, used as first priority. 50 50 Q D g 90.1 66 1.06 4.09 – 74.61 17.66 174 g 95 81.1 12.03 6.72 g 95g 67.9 93.1 76.9g 89.8 80g 94.8 73.8 8.98g 92.2 5.47 16.21 71.4 10.06 10.46 6.83 13.79 8.78 12.78 8.08 g 91.7 77.4 2.10 4.98 – 105.28 37.93 174 r r r r r r r r with m m m m m m m m r r r r r r r r sand 70.8 69 8.72 5.81 sand 66.7sand 71.4 80.3 81s 1.46 6.07 – 2.09 7.81 44.41 8.17 – 109 36.16 10.63 77 sand 85.4sand 94.3sand 91.8 95.2 22.93 12.64 s 88.1 37.5 21.43 68 25.61 13.99 sand 85.4sand 70.8sandsand 91.8 74.3 83.6 4.21 85.7 12.77 8.90 88.1 – 5.77 95.62 13.16 68s 65.60 8.47 4.28s 109 13.95 –s 43.18s 15.73s 67 sand 88.6s 95.2 7.89 22.11 – 82.61 155.00 77 Q and D D D D D D D D D D D D Q Ž. 0.5 0.5 0.5 0.5 0.5 0.5 0.5 .. .. and log and log and log and log and log and log and log and log and log and log and log D D D and log .Ž. D D D D r r r 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 r r r r 0.5 0.5 0.5 0.5 ( ( SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ SQ Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž ŽŽŽŽ . . .. Ž. Ž. Ž ŽŽ .. Ž. Ž. ŽŽŽ . . ŽŽ .. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. Ž. 10 v1010 v1010 50 v10 50 1010 50 v10 50 10 10 v10 50 10 50 10 50 10 50 10 10 50 50 10 50 v 10 50 10 50 10 v 1010 50 v 1010 50 v 1010 50 v 1010 50 v 10 50 10 50 refers to combining Q g refers to sand, mixed, and gravel channels modeled collectively. r q m af bf af bf af bf af bf af bf r M, B, I, BI, and Q refer to meandering, braided, incising, braided or incising, and quasi-equilibrium channels CEM Types 4 and 5 , respectively. Model p s Q a b c 43 log 4750 log log 57 log exclusive use of van den Berg’s 1995 data set. 40 log 52535556585961626465 sand67 gravel68 96.2 97.8 sand70 88.9 77.4 gravel71 84.6 96.7 sand73 77.8 67.7 gravel74 96.2 97.8 sand 88.9 77.4 graveldefined 84.6 as 96.7 sand in 2.50dimensionless. Eq. 88.9 3.93 67.7 gravel 12 . 13.40 In single-parameter 94.4 models, 6.91 95.3 sand 0.34 80 2.47 – 77.8 gravel – 100 8.00 6.02 17.35 71.4 sand 13.94 42.97 50 84.6 gravel 13.42 78.36 – 200.00 – 100 6.91 150.86 93.9 sand 8.96 11.42 72.7 28.56 79.3 gravel 44 58.73 121 8.31 6.02 19.25 97.6 60.90 90.9 14.10 12.80 63.6 79.3 10.33 10.42 44 121 9.48 8.89 6.86 6.38 12.04 9.34 9.19 7.13 8.52 11.46 6.36 8.38 45 log 51 log 414244 log log 46 log 4849 log log log 54 log 6063 log 66 log 69 log 72 log log 290 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 models were most appropriate for rivers with gravel In channels with sand beds, a greater level of flow beds. energy relative to the caliber of bed materials is associated with a fully braided form when compared to an incised formŽ. Fig. 1 . This seems logical 3.1. Channels with sand beds because a greater shear stress is typically exerted on the bed of the channel than on the banks, and one Diagrams based on the single parameter logistic would suspect that erosion of the bed would often regression models provide a convenient means for precede bank erosion in noncohesive sand. More- envisioning the risk associated with varying levels of over, incision is a process typically occurring at the specific stream power relative to sediment sizeŽ Fig. initiation of channel adjustment as opposed to braid- 1. . In general, systems that are particularly suscepti- ing that, at least in the case of a formerly meandering ble to abrupt shifts between the specified channel channel, is a channel form that results after signifi- patterns produce steep logistic curves. Transitions in cant adjustment. The greater magnitude of r 0.5 channel type were particularly abrupt in the compari- SQŽ.D50 in a braided channel may, in many son of stable meandering and incising channels with situations, reflect increases in slope associated with sand bedsŽ. Fig. 1 . The logistic curve, representing bank erosion and straightening as well as possible the transition from meandering to incisingŽ model decreases in the median size of bed material. These s r 0.5 15, n 90. , classified stable meandering and incis- processes could increase the value of SQŽ.D50 as ing channels with sand beds with 89.6% and 92.9% a channel adjusts to the form of a braided channel. accuracy, respectively. The accuracy of predicting an The mobility index was also effective in predict- incising channel was improved to over 95% in the ing the transition from a stable meandering form to model using only annual flood dischargesŽ. Table 1 . an incising or braiding form of a channel with a

Fig. 1. Results of regression models 2 and 15 comparing stable meandering with braiding and incising streams with sand beds. Discharge is represented by annual flood as first priority then bankfull. B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 291

0.5 s Fig. 2. Probability of incising or braiding as a function of SQ and D50 for streams with sand bedsŽ. model 21, n 109 . Discharge is represented by annual flood as first priority then bankfull.

0.5 s Fig. 3. Probability of incising or braiding as a function of SQv50and D for streams with sand bedsŽ. model 22, n 109 . Discharge is represented by annual flood as first priority then bankfull. 292 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 sand bed. The relationship resulting from model 19 presented in Figs. 2 and 3. The predictive accuracies is: of the two-parameter models using bedslope are essentially equal to the accuracies of the single pa- Q rameter models. This reflects the fact that even in the exp 4.51q13.15log S 10 D two-parameter models channel pattern varied with ž/( 50 ps Ž.13 D0.5. When using valley slope, however, the accura- Q 50 1qexp 4.51q13.15log S cies of the two-parameter models are superior when 10 D compared to the mobility index models. ž/( 50 Exclusive use of bankfull discharges in conjunc- where S is the dimensionless bedslope; Q is the tion with bedslopes resulted in the most accurate 3r annual flood or bankfull discharge in m s; and D50 predictions of braiding in channels with sand beds. is the median size of particle in meters. Eq.Ž. 13 Unfortunately, the limited number of annual flood classified stable meandering versus incising or braid- data from braiding channels precluded any meaning- ing streams with sand beds with 85.4% and 91.8% ful analysis based solely on annual flood data. Mod- accuracy, respectively Ž.ns109 . The logistic curve els utilizing annual flood data in conjunction with represented by Eq.Ž. 13 is essentially equivalent to bankfull discharge data for missing values Ž.ns67 the incision curve presented in Fig. 1, with a 50% were 96% accurate for classifying meandering chan- risk corresponding to a mobility index of 0.45 mrs0.5. nels, but only managed to classify braiding channels Again, this indicates that the incision data define the with 82% and 68% accuracy using bedslope and r 0.5 lower limit of SQŽ.D50 values associated with valley slope data, respectively. Accuracy for classifi- instability of sand bed channels. The results of the cation of braiding channels increased to 89% for two-parameter models for predicting an incising or bedslope and valley slope models when only bank- braiding channel form in sand bed channels are also full values were used Ž.ns44 . In contrast, the ex-

0.5 s Fig. 4. Probability of braiding as a function of SQ and D50 for streams with sand bedsŽ. model 10, n 67 . Discharge is represented by annual flood as first priority then bankfull. B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 293 clusive use of bankfull data reduced accuracy for exerted much less influence over the transition from prediction of meandering channels. The models de- meandering to braiding in gravel channels as com- veloped with only bankfull discharge data were more pared to sand channels. This is evidenced by the strongly influenced by the D50 term and suggested relatively low values of b2 in some of the two- that very high values of stream power were neces- parameter logistic regression models for gravelŽ Ta- sary to achieve braiding in coarse sand. These results ble 1. . These low values indicate that holding the were strongly influenced by a few meandering chan- exponent of the D50 term at a constant value of 0.5, nels in coarse sand occurring on relatively steep as in the mobility index, reduces the accuracy of valleys. The models using annual flood as first prior- some models. The best-fitting two-parameter models ity were based on 23 more samples and were less suggest that the stream power associated with a influenced by these extreme values as evidenced by certain risk of a braided channel varies with D50 to higher x 2 valuesŽ. Figs. 4 and 5 . the exponent 0.40 to 0.49. In general, regression relationships for stable rivers with gravel beds re- 3.2. Channels with graÕel beds ported by other investigators suggest that bed materi-

als in the range of D65to D 90 exert more control on Braiding in channels with gravel beds apparently channel slope than D50 Ž.Knighton, 1998 . Unfortu- is more difficult to predict accurately than incising or nately, data regarding coarser fractions of bed mate- braiding in channels with sand beds, given the mod- rials were not available for a large number of streams els examined. Bank resistance, flood plain resistance used in this study. to the development of chute cutoffs, the rate of The two-parameter models developed using a mix sediment delivery from eroding banks, and other of 127 annual flood and bankfull data classified factors may be more influential in controlling the stable meandering channels with over 93% accuracy form of gravel bed channels. In many instances, D50 when using either bedslope or valley slope. By con-

0.5 s Fig. 5. Probability of braiding as a function of SQv50and D for streams with sand bedsŽ. model 13, n 67 . Discharge is represented by annual flood as first priority then bankfull. 294 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 trast, the same models classified braiding channels rate predictions of channel form for each type. At- with only 77% and 68% accuracy when using beds- tempts to develop a robust predictor of braiding for lope and valley slope, respectively. Diagrams depict- sand and gravel streams, taken collectively, produced ing these models for gravel streams are presented in a relatively high percentage of misclassified streams. Figs. 6 and 7. Exclusive use of bankfull discharge Re-analysis of the data used to develop van den data did not substantially improve the accuracy of Berg’s discriminant function also indicated that sepa- the two-parameter models. rating sand and gravel yielded slightly more accurate resultsŽ. Table 1 . Part of van den Berg’s success in developing a 3.3. Comparisons with the meandering–braiding single discriminator for the transition from meander- thresholds of Õan den Berg() 1995 ing to braiding in streams with a D50 of 0.1–100 mm may be attributed to the regime widths assumed When compared to the data set compiled by van for estimating specific stream power. By selecting a den BergŽ. 1995 , the expanded data set developed regime equation from the design of an irrigation for this study resulted in a 1–10% lower rate of canal, streams with sand beds were assumed to be correct classification in the logistic models when 57% wider than streams with gravel beds in van den using valley slope as an independent variableŽ Table Berg’s analysis. It follows that estimates of specific 1. . Nonetheless, the logistic models extend van den stream power for sand bed streams were biased 57% Berg’s original data set and assign risks of braiding lower than the estimates for streams with gravel as opposed to a discrete threshold. In contrast to van beds. In reality, streams with sand beds transporting den Berg’s combined analysis of streams with D50 substantial sediment loads may require more specific between 0.1 and 100 mm, these results suggest that stream power than threshold channels composed of separating sand and gravel streams yields more accu- fine to medium gravelŽ. Howard, 1980 . van den

0.5 s Fig. 6. Probability of braiding as a function of SQ and D50 for streams with gravel bedsŽ. model 11, n 127 . Discharge is represented by annual flood as first priority then bankfull. B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 295

0.5 s Fig. 7. Probability of braiding as a function of SQv50and D for streams with gravel bedsŽ. model 14, n 127 . Discharge is represented by annual flood as first priority then bankfull.

Berg’s selection of ws4.7Q0.5 and ws3Q0.5 for the inclusion of more data, the 50% risk level for the regime width of sand and gravel channels, re- gravel is essentially equivalent to van den Berg’s spectively, would tend to remove this offset in the original discriminator. If van den Berg’s data on estimated stream power and facilitate fitting a single sand channels are combined with the gravel data function for sand and gravel. without assuming a 57% difference in width, the By applying the regime width that van den Berg results are significantly different. In this case, the assumed for gravel channels, his original discrimina- 50% risk of braiding for sand and gravel corresponds tor based on 28 sand and 98 gravel bed streamsŽ Eq. to: Ž..1 may be restated as: s 0.32 s 0.42 SQv50' 0.0245D Ž.16 SQv50' 0.0151D Ž.14 0.5 where valley slope is dimensionless; Q is the bank- This constant of 0.0245 indicates a value of SQv 3r full discharge in m s; and D50 is the median size required for a 50% risk of braiding in 1 mm sand of particle in millimeters. For comparison, the logis- that is substantially higher than the value 0.0151 tic regression results for 127 gravel streams utilizing suggested by Eq.Ž.Ž 14 van den Berg’s original dis- annual flood data in combination with bankfull discriminator. , which is biased toward lower powers for chargeŽ. model 14, Fig. 7 may be reduced to: the occurrence of braiding in channels with sand beds. SQs0.0153D0.43 15 v50' Ž. In general, logistic models of the transition to with units corresponding to Eq.Ž. 14 . This equation braiding in streams with sand beds differed substan- represents a 50% probability of braiding and is al- tially from van den Berg’s original discriminant most exactly the same result that van den Berg function. This resulted from a few influential data on proposed for D50 between 0.1 and 100 mm. The braiding channels with D50 between 1 and 2 mm, similarity of these relationships indicates that despite despite logistic models being developed in this study 296 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 being based on 67 streams with sand beds as op- channel forms are clearly associated with specific posed to the 28 used by van den Berg. Meandering combinations of slope, discharge, and characteristics channels in coarse sand exhibited very steep valley of bed materials. Inferences may be drawn from slopes and strongly influenced the nature of the these associations, but the models do not reveal the regression relationship. As a result, the predicted trajectories of specific stream power, size of bed 0.5 level of SQv that is associated with a certain risk material, and roughness characteristics as streams r of braiding is extremely sensitive to D50 . Until respond to erosive forces. Coarsening and or armor- further data on streams with beds of coarse sand are ing of beds may be particularly confounding. In 0.5 obtained, the relationship between SQv50and D addition, gravel bed channels with mobility indices will remain unclear. The logistic regression models suggesting a low risk of braiding may still be later- resulting from van den Berg’s original data set are ally unstable. presented in Figs. 8–10. Another limitation of the logistic regression models is the absence of variables representing sediment supply and the ratio of bank resistance to bed resis- 3.4. Some limitations of the approach tance. Channels with sand beds are extremely sensitive to the inflowing sediment load delivered from Statistical analysis is a tool for associating obser- upstream reaches. Even if a meandering channel with vations with measurements of variables that are a sand bed has a low risk of incision according to the thought to represent underlying controls. But the results of the logistic regression analysis, it might associations between favored variables and observed still incise if the inflowing sediment load is substan- outcomes do not necessarily imply causation. Specif- tially reduced. The limits of stability presented in ically, these analyses do not demonstrate how chan- this study might be more accurately portrayed as nels develop an incising or braiding form. Instead, thresholds of catastrophic channel change. The re- the logistic regression models indicate that certain sults should not be interpreted to suggest that inci-

Fig. 8. Logistic regression analysis of van den Berg’s original data set using bankfull discharge and D50 between 0.1 and 100 mm. B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 297

Fig. 9. Logistic regression analysis of van den Berg’s original data for streams with sand beds using bankfull discharge.

Fig. 10. Logistic regression analysis of van den Berg’s original data for streams with gravel beds using bankfull discharge. 298 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 sion or widening will not occur in channels with In particular, the relative flatness of logistic curves r 0.5 sand beds at values of SQŽ.D50 less than about produced by the meandering–braiding transition re- 0.3 mrs0.5. flects the omission of key factors that control lateral The logistic regression models presented herein adjustment processes and the preponderance of inter- should be considered a starting point. The models mediate channel patterns. The probabilistic diagrams should be refined for general and regional applica- resulting from these analyses provide users with a tion as more data on unstable channels become more realistic assessment of the uncertainty associ- available. In particular, regional relationships for ated with previously identified thresholds of channel channels with sand beds should be developed in an instability. In addition, the mobility index r 0.5 effort to include the effects of sediment supply and SQŽ.D50 proved highly effective in predicting differing bank conditions. Because data on incision the form of channels with sand beds with limited by the stage of channel evolution are rare for areas information. This parameter is also quite useful for outside the Yazoo River watershed of northern Mis- predicting the form of streams with gravel beds and sissippi, the incision data used in this analysis were in scaling stream slopes across a broad range of necessarily limited to that region. These streams are watershed scales and geologic conditions. quite similar and generally have highly cohesive The logistic regression models suggest that chan- banks and medium sand beds. Inclusion of other nels with sand beds with cohesive banks may be incision data could have biased estimates of incipient particularly susceptible to modest increases in spe- incision because many observations of instability are cific stream power. Although incising and braiding, recorded after a channel response has resulted in a streams with sand beds exhibit similarly high levels decrease in stream powerŽ. i.e., slope . An incising of specific stream power relative to stable meander- stream that is in CEM stage 3 or 4, for example, ing streams, incision apparently initiates at levels of typically looks severely unstable. But the process of specific stream power that are less than those found channel incision, having exceeded the critical height in fully braided channels. For channels with gravel for bank failure, may have already resulted in a beds, the logistic regression results provide a proba- significant decrease in channel slope. It is likely that bilistic version of van den Berg’sŽ. 1995 threshold some streams, that are assumed to be rapidly incis- for the transition from meandering to braiding. Al- ing, could have previously lowered values of though data on certain size fractions remain sparse, r 0.5 SQŽ.D50 if slope is measured after incision has the logistic models suggest that the transition to progressed for an adequate period of time. Thus, it braiding in channels with sand beds may be more will be necessary to identify streams with sand beds dependent on the caliber of bed material than in by the stage of channel evolution if regional analyses channels with gravel beds. are to accurately reflect levels of stream power that In general, methods for systematic evaluation of result in incision. potential instability in alluvial channels at the watershed scale are still primitive. Such an evaluation might be performed in a particular basin by estimating available stream power using relations of re- 4. Conclusions gional flow and a geographic information system to determine whether excess stream power is available The high predictive accuracies of the resulting to initiate channel instability. BoothŽ. 1991 and Si- statistical models suggest that logistic regression mon and DownsŽ. 1995 suggest that the likelihood of analysis is an appropriate technique for associating channel instability can be assessed in a general man- basic hydraulic data with various stable and unstable ner for study sites in a given drainage basin and that channel forms. The utility of this approach for identi- such an approach should be pursued. The logistic fying streams that are particularly susceptible to regression approach developed in this study provides changes in the controlling variables is also clear. an alternative tool for risk-based management of Traditional discriminators of channel pattern do not fluvial systems in the context of channel modifica- adequately portray the fuzziness of these transitions. tions and watershed disturbances. B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300 299

Acknowledgements Knighton, A.D., Nanson, G.C., 1993. Anastomosis and the continuum of channel pattern. Earth Surf. Processes Landforms 18, 613–625. We are very grateful to E.E. Wohl, J.D. Phillips, Lane, E.W., 1957. A study of the shape of channels formed by and an anonymous reviewer for constructive reviews natural streams flowing in erodible material. Missouri River of the manuscript. R.L. Kelley and P.L. Chapman Division Sediment Series No. 9. U.S. Army Engineer Divi- provided invaluable assistance with logistic regression, Missouri River Corps of Engineers, Omaha, NE. Leopold, L.B., Wolman, M.G., 1957. River channel patterns: sion analyses. braided, meandering and straight. U.S. Geol. Surv. Prof. Pap. 282-B, 85 pp. Menard, S.W., 1995. Applied Logistic Regression Analysis. Sage Publications, Thousand Oaks, CA, 98 pp. References Montgomery, D.R., Buffington, J.M., 1997. Channel reach morphology in mountain drainage basins. Geol. Soc. Am. Bull. Begin, Z.B., 1981. The relationship between flow shear stress and 109Ž. 5 , 596–611. stream pattern. J. Hydrol. 52, 307–319. Neter, J., Wasserman, W., Whitmore, G.A., 1988. Applied Statis- Booth, D.B., 1991. Urbanization and the natural drainage system tics. Allyn and Bacon, Boston, MA, 1006 pp. mpacts, solutions and prognoses. NW Environ. J. 7Ž. 1 , 93– Osterkamp, W.R., 1978. Gradient, discharge and particle size 118. relations of alluvial channel in Kansas, with observations on Brewer, P.A., Lewin, J., 1998. Planform cyclicity in an unstable braiding. Am. J. Sci. 278, 1253–1268. reach: complex fluvial response to environmental change. Parker, G., 1976. On the cause and characteristic scales of mean- Earth Surf. Processes Landforms 23, 989–1008. dering and braiding in rivers. J. Fluid Mech. 76, 457–480. Bridge, J.S., 1993. The interaction between channel geometry, SAS Institute, 1990. SASrSTAT User’s Guide, version 6. 4th water flow, sediment transport and deposition in braided rivers. edn. SAS Institute, Cary, NC, 1686 pp. In: Best, J.L., Bristow, C.S.Ž. Eds. , Braided Rivers. Special Schumm, S.A., 1977. The Fluvial System. Wiley-Interscience, Publication of the Geological Society of London, vol. 75, pp. New York, 338 pp. 13–71. Schumm, S.A., Khan, H.R., 1972. Experimental study of channel Bull, W.B., 1979. Threshold of critical power in streams. Geol. patterns. Geol. Soc. Am. Bull. 83, 1755–1770. Soc. Am. Bull. 90, 453–464. Schumm, S.A., Harvey, M.D., Watson, C.C., 1984. Incised Chan- Carson, M.A., 1984. The meandering-braided river threshold: a nels: Morphology, Dynamics, and Control. Water Resour. reappraisal. J. Hydrol. 113, 1402–1421. Publ., Littleton, CO, 200 pp. Chang, H.H., 1985. River morphology and thresholds. J. Hydraul. Simon, A., 1989. A model of channel response in disturbed Eng. 111, 503–519. alluvial channels. Earth Surf. Processes Landforms 14Ž. 1 , Chitale, S.V., 1973. Theories and relationships of river channel 11–26. patterns. J. Hydrol. 19, 285–308. Simon, A., Downs, P.W., 1995. An interdisciplinary approach to Christensen, R., 1997. Log-Linear Models and Logistic Regres- evaluation of potential instability in alluvial channels. Geo- sion. Springer-Verlag, New York, 483 pp. morphology 12, 215–232. Ferguson, R.I., 1987. Hydraulic and sedimentary controls on Simon, A., Hupp, C.R., 1992. Geomorphic and Vegetative Recov- channel pattern. In: Richards, K.S.Ž. Ed. , River Channels: ery Processes along Modified Stream Channels of West Ten- Environment and Process. Blackwell, Oxford, pp. 129–158. nessee. U.S. Geological Survey Open-File Report 91-502, 142 Fredsøe, J., 1978. Meandering and braiding of rivers. J. Fluid pp. Mech. 84, 609–624. StatSoft w , 1999. Electronic Statistics Textbook. StatSoft, Tulsa, Fukuoka, S., 1989. Finite amplitude development of alternate OK. bars. In: Ikeda, S., Parker, G.Ž. Eds. , River Meandering. Am. Tung, Y., 1985. Channel scouring potential using logistic analysis. Geophys. Union, Water Resources Monographs, vol. 12, pp. J. Hydraul. Eng. 111Ž. 2 , 194–205. 237–265. USACE, 1997. HEC-RAS River Analysis System. Version 2.2. Galay, V.J., 1983. Causes of river bed degradation. Water Resour. U.S. Army Corps of Engineers, Hydrologic Engineering Cen- Res. 19Ž. 5 , 1057–1090. ter, Davis, CA. Graf, W.L., 1981. Channel instability in a braided sand-bed river. van den Berg, J.H., 1995. Prediction of alluvial channel pattern of Water Resour. Res. 17, 1087–1094. perennial rivers. Geomorphology 12, 259–279. Harvey, M.D., Watson, C.C., 1986. Fluvial processes and morpho- Waters, T.F., 1995. Sediment in Streams: Sources, Biological logical thresholds in incised channel restoration. Water Re- Effects, and Control. American Fisheries Monograph, vol. 7, sour. Bull. 22Ž. 3 , 359–368. American Fisheries Society, Bethesda, MD, 251 pp. Howard, A.D., 1980. Thresholds in river regimes. In: Coates, Watson, C.C., Harvey, M.D., Biedenharn, D.S., Combs, P.G., D.R., Vitek, J.D.Ž. Eds. , Thresholds in Geomorphology. 1988a. Geotechnical and hydraulic stability numbers channel George Allen and Unwin, Boston, MA, pp. 227–258. rehabilitation: Part I. The approach. In: Abt, S.R., Gessler, J. Knighton, A.D., 1998. Fluvial Forms and Processes: A New Ž.Eds. , Proc. ASCE Hydr. Div. Nat. Conf. ASCE, New York, Perspective. Wiley, New York, 383 pp. pp. 20–125. 300 B.P. Bledsoe, C.C. WatsonrGeomorphology 38() 2001 281–300

Watson, C.C., Harvey, M.D., Biedenharn, D.S., Combs, P.G., D.M., Haas, A.M., 1999. Demonstration Erosion Control 1988b. Geotechnical and hydraulic stability numbers channel Monitoring Sites 1999 Evaluation. Report for USACE, Engi- rehabilitation: Part II. Application. In: Abt, S.R., Gessler, J. neer Research and Department Center, Vicksburg, MS, 157 Ž.Eds. , Proc. ASCE Hydr. Div. Nat. Conf. ASCE, New York, pp. pp. 126–131. Werritty, A., 1997. Short-term changes in channel stability. In: Watson, C.C., Thornton, C.I., Kozinski, P.B., Bledsoe, B.P., et al., Thorne, C.R., Hey, R.D., Newson, M.D.Ž. Eds. , Applied Flu- 1998. Demonstration Erosion Control Monitoring Sites 1997 vial Geomorphology for River Engineering and Management. Evaluation. Report for USACE, Waterways Experiment Sta- Wiley, Chichester, UK, pp. 47–65. tion, Vicksburg, MS, 129 pp. Wolman, M.G., 1954. A method of sampling coarse bed material. Watson, C.C., Bledsoe, B.P., Kozinski, P.B., 1998. Yalobusha Am. Geophys. Union Trans. 35, 951–956. Basin Geomorphic Assessment. Report for the USACE, Wa- Yang, C.T., 1976. Minimum unit stream power and fluvial hy- terways Experiment Station, Vicksburg, MS, 64 pp. draulics. J. Hydraul. Div., ASCE, Proc. Pap. 12238, 102, Watson, C.C., Bledsoe, B.P., Holmquist-Johnson, C.L., Mooney, Ž.HY7 , 919–934.