1.  INTRODUCTION

The modeling of surface runoff using physically based methods with the aid of a computer is well established in research and practice. Any such modeling uses either kinematic, diffusion, or dynamic waves, with papers continuing to appear in the literature describing applications of the three models. However, there is still some confusion as to their applicability, as demonstrated by a recent paper by Tayfur et al. (1993), where a comparison is made of results obtained by the three models using a very steep slope (S = 0.086), while disregarding the effect of time-of-rise t_r. The comparison showed essentially no difference between results obtained with any of the three models.

Kinematic and dynamic waves have been established since the 1960's, first in research, and then in practice. Diffusion waves, however, have remained within the realm of research until only recently, when they were incorporated into Version 4.0 of HEC-I: Flood Hydrograph Package (Hydrologic Engineering Center, 1990). Given the current emphasis on computational models of surface runoff, for both water quantity and water quality applications, a review of the three models is warranted at this time. Accordingly, this paper focuses on the following: (1) definition and properties of kinematic, diffusion, and dynamic waves, (2) role of numerical diffusion in kinematic wave modeling, (3) nature of kinematic shock, and (4) role of the Vedernikov number in surface runoff modeling.

2.  KINEMATIC WAVES

The mathematical theory of kinematic waves is attributed to Lighthill and Whitham (1955), although its origin can be traced back to the end of the last century, specifically to Kleitz and Seddon (Chow, 1959). In the past three decades, the application of kinematic wave theory to overland flow and streamflow has gained considerable momentum (Wooding, 1965; Woolhiser and Liggett, 1967; Hydrologic Engineering Center, 1990). There is a substantial body of knowledge about the kinematic wave, and papers continue to appear in the literature describing the model's capabilities and limitations (Hromadka and DeVries, 1988; Ponce, 1991a). There is still, however, some misunderstanding about the precise role of kinematic waves in surface runoff modeling (Woolhiser, 1992).

A kinematic wave may be defined in three ways. First, a kinematic wave transports mass, in contrast to the inertial wave, the so-called "gravity" wave of classical mechanics, which transports energy. Second, a kinematic wave is characterized by a one-to-one relationship between discharge and stage. Third, a kinematic wave accounts for only gravitational and frictional forces, and neglects the forces arising from the flow depth gradient (dy/dx, in which y = flow depth, and x = distance along the channel) and inertia.

The above three definitions are related. To put them in the proper perspective, we note that Lighthill and Whitham (1955), in laying the foundations of kinematic wave theory, subtitled their paper Flood Movement in Long Rivers. Flood waves transport mass; kinematic waves also transport mass. However, while flood waves are kinematic in nature, not all kinematic waves are flood waves. To clearly distinguish between flood waves and kinematic waves, we examine Lighthill and Whitham's subtitle further. The implication is that the kinematic wave is applicable only to long rivers! If this were the case, the kinematic wave could not be applied to steep mountain streams, which are relatively short compared to most alluvial rivers. In practice, however, kinematic waves are seen to be applicable to both 'short' mountain streams and 'long' alluvial rivers,

The resolution of this conflict was made possible by Ponce and Simons (1977), who identified the parameter describing the applicability of kinematic waves. They showed that the latter is controlled, not by the 'length' of the river, or by the length L of the shallow wave, but rather by the dimensionless ratio L_o/L, where L is a reference channel length, defined as:

do Lo  =  _____             So

(1)

with d_o = average flow depth, and S_o = channel bed slope. In general, a wave is kinematic if the dimensionless wave number

2π σ  =  _____ Lo             L

(2)

is sufficiently small for a given Froude number

uo Fo  =  _______           (gdo)1/2

(3)

The reference channel length L_o is readily obtained, provided an average flow depth can be established, which is usually the case. The same does not follow for the wavelength L, which needs to be converted to the temporal domain for practical use. Since:

L  =  cT

(4)

where c is the wave celerity and T is the wave period, the ratio L_o/L can be expressed as:

Lo             do _____  =  ________    L            cTSo

(5)

Seddon's law (Seddon, 1900; Chow, 1959) may be used to express the kinematic wave celerity c in terms of the mean flow velocity:

5 c  =  _____ uo            3

(6)

applicable to Manning friction in hydraulically wide channels. The ratio L_o/L is now expressed as follows:

Lo               3           do/uo _____  =  ( ______ ) ( ______ )     L                5            TSo

(7)

Furthermore, the flood wave period T may be expressed in terms of the more familiar time-of-rise t_r, or time-to-peak of the flood hydrograph. Assuming for simplicity

T  =  2tr

(8)

then:

Lo              3            do/uo _____  =  ( ______ ) ( ______ )     L              10            trSo

(9)

In nature, while d_o and u_o are usually restricted within a narrow range, t_r and S_o tend to vary within a broad range. In fact, the flood wave time-of-rise t_r can be as short as 5 to 15 minutes in small steep catchments, and as long as 3 to 6 months in large catchments of mild relief. To give an extreme example, the time-of-rise of the Upper Paraguay River at Porto Murtinho, Brazil (at the outlet of the great Pantanal of Mato Grosso), is approximately 6 months. Ihe channel bed slope S_o typically varies between S_o = 0.1 (or steeper) in some mountain stream situations, and S_o = 0.00001 (or milder) in certain deltaic and tidal settings. Thus, in general, the ratio L_o/L is inversely related to the product t_rS_o. For a given Froude number, the larger the value of (and therefore, the smaller L_o/L), the more kinematic the wave is.

In light of the preceding considerations, the meaning of Lighthill and Whitham's subtitle is now fully elucidated: The adjective 'long' should be construed to refer to a small L_o/L or large t_rS_o. The latter implies that either t_r or S_o, or both, should be large. Experience reveals that in Nature these two parameters are not likely to be large simultaneously. Either the time-of-rise t_r is long (as in a large catchment of mild-relief), or the channel bed slope S_o is steep (as in a mountain stream, or in overland flow in a typical small-catchment setting), but usually not at the same time. This behavior confirms the wide range of field situations in which the kinematic wave is applicable: For both steep and mild catchments, and both fast-rising and slow-rising hydrographs, provided the product is sufficiently large.

Ponce et al. (1978) developed a criterion for the applicability of kinematic waves in surface runoff. The criterion states that for a shallow water wave to be kinematic, it should satisfy the following dimensionless inequality:

uo N  =  trSo ( _____ ) > 85                      do

(10)

The larger the value of N, the more kinematic the wave is. For instance, if t_r = 6 hr, S_o = 0.01, mean velocity u_o = 2 m/s, and flow depth d_o = 1 m, it follows that N = 432 > 85, confirming that this wave is kinematic. According to the definition of kinematic wave, this wave: (1) will transport mass, (2) will describe a one-to-one relationship between discharge and stage at any cross-section, and (3) the forces arising from the flow depth gradient and inertia will be so small so as to be negligible compared to the gravitational and frictional forces.

3.  DIFFUSION WAVES

The specification of a one-to-one relationship between discharge and stage, a key trait of the kinematic wave, imposes a significant physical and mathematical constraint: The wave cannot diffuse. It can convect downstream and transport mass in the process, but it cannot dissipate its discharge or stage. This limitation of the kinematic wave is grounded in its formulation: The neglect of the flow depth gradient and inertia terms results in a first-order partial differential equation governing the motion. This equation cannot describe diffusion, since the latter is a second-order process. From a physical perspective, the one-to-one stage-discharge relationship implies that wave diffusion is clearly out of the problem, since the latter is caused by the presence of a loop (however small!) in the stage-discharge rating.

Since in Nature there exist shallow water waves which do diffuse, although in small amounts, the theory of kinematic waves is incomplete without a means of incorporating this important diffusion mechanism. Lighthill and Whitham (1955) clearly saw this when they suggested the extension of kinematic waves to the realm of diffusion waves, i.e., of kinematic waves that incorporate a small amount of diffusion. To accomplish this, the formulation of kinematic waves is modified to include the flow-depth gradient term, while still excluding the inertia terms. This significant extension allows the description of stage-discharge ratings and, consequently, of the diffusion of kinematic waves, now propery, diffusion waves. To summarize, diffusion waves are still kinematic in nature; they still transport mass; however, unlike kinematic waves, diffusion waves have the capability to undergo small but perceptible amounts of physical diffusion.

This physical diffusion is confirmed by theory and experience. As long as the flow depth gradient is not negligible, it will produce a looped stage-discharge rating for every shallow wave, which will in turn cause the wave to dissipate as it travels downstream. In practice, as the channel bed slope S_o, decreases (as the flow moves from mountain streams to alluvial rivers), the friction slope S_f decreases accordingly (as channel roughness typically decreases in a downstream direction), and the flow depth gradient becomes increasingly too important to be disregarded. Intuitively, while kinematic waves are seen to apply to mountain streams, diffusion waves are seen to apply to valley streams and aluvial rivers. A rule of thumb validated by experience, says that if the channel bed slope is greater than 1 percent (S_o > 0.01), the wave will most likely be kinematic, will feature a one-to-one relationship between discharge and stage, and will not diffuse appreciably. If the channel bed slope is less than 1 percent, the wave may not be kinematic; it may be a diffusion wave. If so, it will feature a looped stage-discharge rating and show a small but perceptible amount of diffusion.

Ponce et al. (1978) have presented a criterion for the applicability of diffusion waves. The criterion states that for a shallow wave (whether flood wave or overland flow wave) to be a diffusion wave, it should satisfy the following dimensionless inequality:

g M  =  trSo ( ____ )1/2 > 15                      do

(11)

For instance, if t_r = 6 h, S_o = 0.001, and flow depth d_o =1 m, it follows that M = 67.6 > 15, confirming that this wave is a diffusion wave. This wave will have the following properties: (1) it will transport mass, like the kinematic wave; (2) it will diffuse appreciably, unlike the kinematic wave; (3) it will describe a looped stage-discharge relationship at any cross-section; and (4) the force arising from the flow depth gradient can no longer be neglected.

It is noted that in the example of the previous section, had the channel bed slope been S_o = 0.001, then N = 43.2, and the wave would not have qualified as a kinematic wave. However, in the example of this section, if the slope is S_o = 0.01 instead, then M = 676, and the wave would still qualify as a diffusion wave. It is concluded that while the kinematic wave model does not apply to diffusion waves, the diffusion wave model does apply to kinematic waves. In other words, the theory of diffusion waves can properly describe both kinematic and diffusion waves.

4.  DYNAMIC WAVES

Dynamic waves approach the problem of shallow water wave propagation in its most general form, i.e., by considering all forces, including gravitational, frictional, flow-depth gradient, and inertial force. This leads to a set of two partial differential equations of continuity and motion, the Saint Venant equations, for which there is no known complete analytical solution. There are, however, several incomplete analytical solutions of the Saint Venant equations (Lighthill and Whitham, 1955; Dooge, 1973; Ponce and Simons, 1977). In particular, the linear solution of Ponce and Simons is significant because it gives great insight into the behavior of shallow water waves, including kinematic, diffusion, dynamic, and inertial waves. (It is noted that in classical mechanics, the inertial waves are commonly referred to as "gravity" waves. This is the source of some confusion, since the gravity force is not included in the formulation of inertial waves).

The work of Ponce and Simons (1977) is summarized as follows:

1. The dynamic wave lies towards the middle of the spectrum of dimensionless wave numbers (10⁰ < σ < 10²), while kinematic/diffusion waves lie to the left (10^-2 < σ < 10⁰), and inertial waves to the right (10¹ < σ < 10⁴).

2. In the stable flow regime, i.e., for Vedernikov number V < 1, the dynamic wave shows very strong diffusive tendencies. The Vedernikov number is the ratio of relative kinematic wave speed to relative Lagrange wave speed (Craya, 1952; Ponce, 1991b).

3. At the threshold of flow instability (V = 1), the Seddon and Lagrange speeds (Chow, 1959) are the same, and kinematic, dynamic, and inertial waves have the same celerity and lack diffusion.

4. In the unstable flow regime (V > 1), kinematic, dynamic, and inertial waves have a tendency to amplify during propagation.

The findings of Ponce and Simons (1977) pose an interesting question which helps place the nature of shallow water waves in the proper perspective. Kinematic waves, and by extension diffusion waves, lying to the left of the wave number spectrum, transport mass. On the other hand, the inertial waves, lying to the right of the wave number spectrum, transport energy. What, then, do dynamic waves transport, since they lie in the middle of the wave number spectrum? Mass or energy? The logical answer is that they transport both. Therein lies the reason for the markedly strong dissipative tendencies of dynamic waves. Shallow water waves can transport mass and energy simultaneously only at the expense of wave diffusion. In the stable flow regime (V < 1), the more dynamic a wave is, the more strongly dissipative it is (Ponce et al. 1978). At the threshold of flow instability (V = 1), dynamic waves lose their ability to dissipate, and their properties coalesce with those of kinematic and inertial waves.

The preceding discussion raises the following question: If the dynamic wave is so strongly dissipative in most cases of practical interest, is it worth attempting to compute it? Would it not dissipate shortly after it is generated, with its mass going to join the underlying larger, kinematic/diffusion, wave? Or, can it be tracked downstream as it propagates? If so, at what characteristic speed? Seddon's or Lagrange's? A more practical question is: If the dynamic wave is so strongly dissipative, could it be properly construed as a flood wave? Lighthill and Whitham (1955) put it very aptly when they stated (op. cit., p. 293): "Under the conditions appropriate for flood waves... the dynamic waves rapidly become negligible, and it is the kinematic waves, following at lower speed, which assume the dominant role."

In summary, dynamic waves do not apply to floods in steep mountain streams. There is still the unresolved question of whether dynamic waves apply to the routing of flood waves in a typical valley setting. Dam-break flood waves notwithstanding, perhaps the only clear statement that can be made at this juncture is that the dynamic wave applies to tidal flow and similar situations where there is a significant downstream control of the flow.

5.  ROLE OF NUMERICAL DIFFUSION IN KINEMATIC WAVE MODELING

If kinematic waves cannot diffuse, why is it that numerical models of kinematic waves are able to show some wave diffusion? The resolution of this paradox lies in the conversion of a partial differential equation into a finite difference equation in a computer model.

This conversion can only be done at the expense of introducing an error. This error is a function of the grid size (Δx and Δt) and tends to disappear as the grid size is progressively refined. In flood routing, the error that creeps into a typical computation using finite differences manifests itself as numerical diffusion and numerical dispersion effects. These effects are the direct result of specifying a discrete space-time domain, and are not necessarily related to the physical diffusion and dispersion which are inherent in the nature of flood waves.

Numerical diffusion arises because the calculated wave amplitude is smaller than the physical wave amplitude. Numerical dispersion arises when the calculated wave celerity is different from the physical wave celerity (Ferrick et al., 1984). In conventional finite difference shallow wave models, the aim is to minimize numerical diffusion and dispersion by choosing a grid size sufficiently small to drive these errors to inconsequential amounts. Then, the convection and diffusion of the shallow wave may be properly described by the numerical model.

Unfortunately, not all finite difference kinematic wave models have sought to minimize numerical diffusion and dispersion. Often, a finite difference kinematic wave model has inadvertently used the numerical diffusion as a way of showing a certain amount of "physically realistic" diffusion in the calculated results (Li et al., 1975; Curtis et al., 1978). Cunge (1969) demonstrated that finite difference schemes of the kinemnatic wave equation introduce varying amounts of numerical diffusion and dispersion. The latter interfere with the physical effects, modifying them (Abbott, 1976; Ponce, 1991a). Thus, a finite difference kinematic wave model may show some diffusion, the amount being a function of the grid size and weighting factors used in discretizing the terms of the kinematic wave equation. The fact that this diffusion is artificial and intrinsically related to the grid size can be readily demonstrated by solving the same problem several times, each time halving the spatial increment Δx and temporal increment Δt. Carried to the practical limit, this test leads to the eventual disappearance of the numerical diffusion in question, with the result approaching the analytical solution of the kinematic wave, which is nondiffusive.

In summary, if the kinematic wave is solved properly, achieving the complete elimination of numerical diffusion and dispersion, the model user can only hope to describe kinematic waves, but not diffusion waves. If the problem does have some physical diffusion, the latter would be entirely missing from this approach. Conversely, if the kinematic wave is solved improperly, introducing numerical difusion and dispersion by the choice of grid size, there is no guarantee that these will be related to the diffusion and dispersion, if any, of the physical problem. Any arbitrary choice of grid size will cause some numerical diffusion and/or dispersion, and since the latter are unrelated to the physical problem, the solution degrades accordingly.

Fortunately, there is a way out of this difficulty. Cunge (1969) and others (Natural Environment Research Council, 1975; Ponce and Yevjevich, 1979) have shown that the numerical diffusion and dispersion of finite difference kinematic wave models can be managed. There is a way to optimize the numerical diffusion while minimizing the numerical dispersion, to make the method and its inherent errors work for us instead of against us.

By a careful match of physical and numerical difusion, the finite difference kinematic wave model can reproduce both kinematic and diffusion waves, in the methodology referred to as the Muskingum-Cunge method. This is a variant of the Muskingum method (Chow, 1959) in which the parameters K and X are calculated directly, based on hydraulic data (channel friction, bed slope, and cross-sectional characteristics), instead of indirectly, based on hydrologic data (storage-weighted flow relations). The Muskingum-Cunge method was first applied to open-channel flow, and later to overland flow (Ponce, 1986). Extensive tests have shown that the method holds promise for overland flow, since unlike conventional finite difference kinematic wave models, the Muskingum-Cunge model is essentially independent of the grid specification.

6.  NATURE OF KINEMATIC SHOCK

Kinematic waves lack physical diffusion. However, kinematic waves are nonlinear, a property which gives them the inherent tendency to change their shape upon propagation: Either steepen or flatten, depending on the stage relative to the channel cross-section (flow inbank or overbank). Under the right set of circumstances, a kinematic flood wave can steepen to the point where it becomes for all practical purposes a "wall of water." (In overland flow situations, the "wall of water" would be a small discontinuity in the water surface profile). This is the kinematic shock, i.e., a kinematic wave that has steepened upon propagation to the point of being nearly discontinuous.

Contrary to conventional wisdom, there is no physical unreality about the kinematic shock. If the steepening tendency is allowed to continue unchecked, the kinematic shock will form in due time. Diffusion, however, acts to counteract the steepening tendency. Therefore, in cases where diffusion, either physical or numerical, is present, the development of the kinematic shock is likely to be arrested. This explains the pervasive presence of kinematic shocks in analytical solutions of the kinematic wave. On the other hand, kinematic shocks are often absent from finite difference kinematic wave models, particularly from those that have appreciable amounts of built-in numerical diffusion.

Ponce and Windingland (1985) have clarified the conditions under which the kinematic shock is likely to develop. Based on theoretical considerations, supported by extensive numerical experiments, they established the following conditions for kinematic shock development:

1. The wave must be kinematic, i.e., it must have negligible physical diffusion. Diffusion tends to counteract the development of the shock.

2. The ratio of base-to-peak flow Q_b/Q_p, must be small, with zero as the lower limit, such as in the case of an ephemeral stream (recall the flash floods occurring on dry river beds).

3. The channel is: (a) hydraulically wide, i.e., of nearly constant wetted perimeter, to allow the wave steepening to progress unchecked by the cross-sectional shape; and (b) sufficiently long to allow enough time for the shock to develop.

4. The flow is at high Froude number, within the stable flow regime (V < 1). The higher the Froude number within the stable flow regime, the smaller the physical diffusion, and the more likely the shock can continue to develop unchecked. In the limit, as the Vedemikov number approaches 1 (and the Froude number approaches 1.5, for hydraulically wide channels with Manning friction), diffusion vanishes as the flow reaches the threshold of instability.

In practice, all four conditions may prevail at the same time in a given situation. Whether a kinematic shock will form will depend on the strength of any one condition, or, if more than one is present, on their combined strength. For instance, an analytical solution of the kinematic wave in an overland flow plane satisfies conditions 1 and 3 (a), and maybe even 3 (b) if the plane is long enough. The case of a flash flood in an ephemeral stream in an arid or semiarid region satisfies condition 2, and probably even 3 (a), 3 (b), and 4. The fact that kinematic shocks are not a common sight in Nature points to the practical difficulty of satisfying all or most of these conditions at the same time.

Condition 1 is satisfied in channels where the product t_rS_o is large. Condition 2 is satisfied in ephemeral streams. Condition 3 (a) is satisfied in inbank flow in wide rectangular channels, but not if the flow goes overbank, since in this case the wetted perimeter would cease to be nearly constant. Condition 3 (b) is dependent of the catchment's physiography, geology, geomorphology, and drainage density. The longer a stream, uninterrupted by lateral inflow at tributary confluences is, the better the chances for the shock to develop. Condition 4 is dependent on the channel aspect ratio, boundary friction, and presence or absence of riparian vegetation. Jarrett (1984) has noted that high-Froude-number flows are rare in natural streams. Therefore, Condition 4 is more likely to be the exception rather than the rule.

It is noted that kinematic shocks, particularly those associated with flash floods, are very difficult to document precisely, given the obvious likelihood of bodily harm and possibly even death for those daring enough to attempt it. Hjalmarson (1985) has documented the flash flood of July 26, 1982, in Tanque Verde Creek, east of Tucson, Arizona, in which the lives of eight unsuspecting bathers were claimed. This flood was in all likelihood a kinematic shock. Kinematic shocks and flash floods are associated with one or mored bursts, (2) a semiarid or a of the following: (1) intense clourid region, (3) a steep, ephemeral stream, (4) a low-friction channel (in both bed and banks), and (5) a catchment with relatively long stream channels.

7.  ROLE OF VEDERNIKOV NUMBER IN SURFACE RUNOFF MODELING

As pointed out by Hayami (1951) in his classical paper on diffusion waves, the hydraulic diffusivity is the controlling physical parameter in diffusion waves. The hydraulic diffusivity is:

qo v  =  ( _____ )              2So

(12)

in which q_o = reference (average) unit-width discharge, and S_o = channel slope. Therefore, the amount of diffusion that a flood wave undergoes during propagation is directly proportional to the unit-width discharge and inversely proportional to the channel bed slope. In other words, the steeper the channel slope, the lesser the amount of flood wave diffusion. In the limit, for a sufficiently steep channel, the diffusion disappears and the flood wave becomes a kinematic wave. Hayami’s hydraulic diffusivity is properly a kinematic hydraulic diffusivity:

qo vk  =  ( _____ )               2So

(13)

because it lacks inertia altogether. It is strictly applicable to flow well within the stable regime, i.e., for small Vedernikov numbers, in the range 0 < V < 0.25.

By including inertia in the formulation, Dooge (1973) and Dooge et al. (1982) have extended the concept of hydraulic diffusivity to the realm of dynamic waves. This leads to the concept of dynamic hydraulic diffusivity (Ponce, 1991a; 1991b):

qo vd  =  (1 - V 2) ( _____ )                             2So

(14)

Unlike its kinematic counterpart, the dynamic hydraulic diffusivity is also a function of the Vedernikov number. As the Vedernikov number V → 0, as with low-Froude-number flows: v_d → v_k. Conversely, as V → 1: v_d → 0, and wave diffusion vanishes. The latter process can not be simulated with Eq. 13. It follows that v_d applies through a wider range of flow conditions than v_k (in the range 0 < V < 1). Since v_d does not significantly complicate the expression for hydraulic diffusivity, it should be the preferred way of modeling surface runoff with diffusion waves. In practice, since wave diffusion is usually small, the dynamic contribution to wave diffusion tums out to be also small (Perumal, 1992).

The inclusion of the Vedernikov number in the expression for hydraulic diffusivity has the additional advantage that it can account for channels of arbitrary cross-sectional shape, i.e., those other than hydraulically wide. Taken to the limit, i.e., for the inherently stable channel (Liggett, 1975; Ponce, 1991b), the Vedemikov number is zero (V = 0), and the dynamic hydraulic diffusivity (Eq. 14) reduces to its kinematic counterpart (Eq. 13). It is seen that in this case, the flood wave attenuation is governed by the kinematic hydraulic diffusivity, for all values of discharge or stage.

7.  SUMMARY

Kinematic waves are useful tools in applied hydrology. With the applicability issue now clearly settled, they should continue to be used in the future. When the extension is made to diffusion waves, the applicability issue is no longer a serious roadblock. The strongly diffusive dynamic wave has yet to show that it is needed in a typical application of surface runoff modeling.

Caution should be exercised when applying the kinematic wave to overland flow and channel flow in the context of a numerical computer model, since the presence of uncontrolled numerical diffusion and dispersion may degrade the accuracy of the computation.

The Muskingum-Cunge method holds particular promise, given its demonstrated grid independence. The use of a dynamic hydraulic diffusivity in lieu of its kinematic counterpart will ensure that the dynamic component of surface runoff, however small, is being accounted for.

Research is needed into the nature of kinematic shock and its applicability to the study of flash floods. The conditions for shock development having been clearly identified, the focus should now shift to the development of a regional hazard rating for flash floods. This should be based on: (1) climatic and precipitation patterns; (2) geology, geomorphology, and catchment drainage density; and (3) stream channel slope, boundary friction, and cross-sectional shape.

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