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10.1 TIMEAREA METHOD
Catchment Routing
Catchment routing refers to the calculation of flows in time and space within a catchment. The objective of catchment routing is to transform effective rainfall into streamflow. This is accomplished either in a lumped mode (e.g., timearea method) or in a distributed mode (e.g., kinematic wave method). Methods for catchment routing are similar to those of reservoir and stream channel routing. In fact, many techniques used in reservoir and channel routing are also applicable to catchment routing. For instance, the concept of linear reservoir is used in both reservoir and catchment routing. Kinematic wave techniques were originaIly developed for river routing [9], but later were applied to catchment routing [20, 23]. Methods for catchment routing are of two types: (1) hydrologic and (2) hydraulic. Hydrologic methods are based on the storage concept and are spatially lumped to provide a runoff hydrograph at the catchment outlet. Examples of hydrologic catchment routing methods are the timearea method and the cascade of linear reservoirs. Hydraulic methods use kinematic or diffusion waves to simulate surface runoff within a catchment in a distributed context. Unlike hydrologic methods, hydraulic methods can provide runoff hydrographs inside the catchment. Catchment routing models can use parametric, conceptual, andlor deterministic components. For instance, the hydrograph obtained by the timearea method can be routed through a linear reservoir using a storage constant derived by empirical (i.e., parametric) means. The cascade of linear reservoirs is a typical example of a conceptual model used in catchment routing. Kinematic and diffusion models are examples of deterministic methods used in catchment routing. The concepts of translation and storage are central to the study of flow routing, whether in catchments, reservoirs, or stream channels. They are particularly important in catchment routing, because they can be studied separately. Translation may be interpreted as the movement of water in a direction parallel to the channel bottom. Storage may be interpreted as the movement of water in a direction perpendicular to the channel bottom. Translation is synonymous with runoff concentration; storage is synonymous with runoff diffusion. In reservoir routing, storage is the primary mechanism, with translation almost nonexistent. In stream channel routing, the situation is reversed, with translation being the predominant mechanism and storage playing only a minor role. This is the reason why kinematic and diffusion waves are useful models of stream channel routing. In catchment routing, translation and storage are about equally important and, therefore, they are often accounted for separately. The translation effect can be related to runoff concentration, whereas the storage effect can be simulated with linear reservoirs. TimeArea Method
The timearea method of hydrologic catchment routing transforms an effective storm hyetograph into a runoff hydrograph. The method accounts for translation only and does not include storage. Therefore, hydrographs calculated with the timearea method show a lack of diffusion, resulting in higher peaks than those that would have been obtained if storage had been taken into account. If necessary, the required amount of storage can be incorporated by routing the hydrograph obtained by the timearea method through a linear reservoir. The required amount of storage is determined by calibrating the linear reservoir storage constant K with measured data. Alternatively, suitable values of K can be estimated based on regionally derived formulas. The timearea method is essentially an extension of the runoff concentration principle used in the rational method (Chapter 4). Unlike the rational method, however, the timearea method can account for the temporal variation of rainfall intensity. Therefore, the applicability of the timearea method is extended to midsize catchments. The timearea method is based on the concept of timearea histogram, i.e., a histogram of contributing catchment subareas. To develop a timearea histogram, the catchment's time of concentration is divided into a number of equal time intervals. Cumulative time at the end of each time interval is used to divide the catchment into zones delimited by isochrone lines, i.e., the loci of points of equal travel time to the catchment outlet, as shown in Fig. 101 (a). For any point inside the catchment, the travel time refers to the time that it would take a parcel of water to travel from that point to the outlet. The catchment subareas delimited by the isochrones are measured and plotted in histogram form as shown in Fig. 101 (b).
The time interval of the effective rainfall hyetograph must be equal to the time interval of the timearea histogram. The rationale of the timearea method is that, according to the runoff concentration principle (Section 2.4), the partial flow q at the end of each time interval is equal to the product of effective rainfall i times contributing subarea a, i.e., q = ia. The lagging and summation of the partial flows results in a runoff hydrograph for the given effective rainfall hyetograph and timearea histogram. While the timearea method accounts for runoff concentration only, it has the advantage that the catchment shape is reflected in the timearea histogram and, therefore, in the runoff hydrograph. The procedure is illustrated by the following example.
It is readily seen that the timearea method and the rational method (Chapter 4) share a common theoretical basis. However, since the timearea method uses effective rainfall and does not rely on runoff coefficients, it can account only for runoff concentration, with no provision for runoff diffusion. Diffusion can be provided by routing the hydrograph calculated by the timearea method through a linear reservoir with an appropriate storage constant. Time of concentration The timearea method leads to an alternate way of calculating time of concentration. Provided there is no runoff diffusion as would be the case of a hydrograph calculated by the timearea method, time of concentration can be calculated as the difference between hydrograph time base and effective rainfall duration. Intuitively, as rainfall ceases, the farthest parcels of water concentrate at the catchment outlet at a time equal to the time of concentration. Therefore:
in which t_{c} = time of concentration, T_{b} =
time base of the translatedonly hydrograph, and Equation 101 can also be expressed in a slightly different form.
Assuming that the point of inflection (i.e., the point of zero curvature) on the receding
limb of a measured (i.e., translated and diffused) hydrograph coincides with the end of the translatedonly hydrograph,
the time to point of inflection of the measured hydrograph can be used in Therefore time of concentration can be defined as the difference between the time to point of inflection I and the effective rainfall duration (see Fig. 103):
in which t_{i} = time to point of inflection on the receding limb of a measured hydrograph. The advantage of Eq. 102 over Eq. 101 is that, unlike the time base of the translatedonly hydrograph, the point of inflection on the receding limb of a measured hydrograph can be readily ascertained.
10.2 CLARK UNIT HYDROGRAPH
The procedure to derive a Clark unit hydrograph parallels that of the timearea method [2]. First, it is necessary to determine the catchment isochrones. In the Clark method, however, a unit effective rainfall is used in lieu of the effective storm hyetograph used in the timearea method. This leads to an outflow hydrograph corresponding to a unit runoff depth, that is, a unit hydrograph. Since the unit hydrograph calculated in this way lacks (runoff) diffusion, Clark suggested that it be routed through a linear reservoir. As with the timearea method, an estimate of the linear reservoir storage constant is required. This can be obtained either from the tail of a measured hydrograph or by using a regionally derived formula. In the latter case, the Clark unit hydrograph can be properly regarded as a synthetic unit hydrograph. Like the timearea method, the Clark unit hydrograph method has the advantage that the catchment's properties (shape, hydraulic length, surface roughness, and so on) are reflected in the timearea histogram and, therefore, on the shape of the unit hydrograph. This feature has contributed to the popularity of the Clark unit hydrograph in engineering practice [7]. When using the Clark or timearea methods, the storage constant can be estimated from the tail of a measured hydrograph. For this purpose, the differential equation of storage (Eq. 84) is evaluated at a time for which inflow equals zero (I = 0), i.e., past the end of the translatedonly hydrograph. Alternatively, it can be evaluated at the point of inflection on the receding limb of a measured hydrograph (Fig. 103). This leads to:
and since S = KO, the following expression for K is obtained:
in which O and dO/dt are evaluated past the end of the translatedonly hydrograph or at (the time to) the point of inflection on the receding limb of a measured hydrograph. The derivation of the Clark unit hydrograph is illustrated by the following example.
By using Eq. 104, the linear reservoir storage constant can be calculated directly from the tail of a measured hydrograph.
To illustrate the procedure, in Table 102, Col. 9, the two lines for t = 6 h and Therefore, Eq. 104 can be applied between t = 6 h and t = 7 h. The average outflow (Col. 9) is: (46.19 + 27.72)/2 = 36.955 m^{3}/s. The rate of change of outflow is: (27.72  46.19)/(1 h) = 18.47 (m^{3}/s)/h. Therefore, the storage constant (Eq. 104) is: K = Likewise, between t = 7 h and t = 8 h: K =  [(27.72 + 16.64) / 2] / [(16.64  27.72) / (1)] = 2 h
In other words, Eq. 104 applies at the tail of the outflow hydrograph, after the translatedonly (inflow) hydrograph has receded back to zero. When using the Clark (or timearea) method, the time base of the translatedonly hydrograph is equal to the sum of concentration time plus the unit hydrograph (or effective storm) duration (See Eq. 101). With the help of regional analysis (Chapter 7), the Clark parameters (time of concentration and linear reservoir storage constant) can be estimated based on catchment characteristics. This effectively qualifies the Clark unit hydrograph as a synthetic unit hydrograph. The Eaton [4], O'Kelly [11], and Cordery [3] models are examples of this approach. Singh [18] has a review of Clark synthetic unit hydrograph parameters. 10.3 CASCADE OF LINEAR RESERVOIRS
As seen in Section 8.2, a linear reservoir has a diffusion effect on the inflow hydrograph. If an inflow hydrograph is routed through a linear reservoir, the outflow hydrograph has a reduced peak and an increased time base. This increase in time base causes a difference in the relative timing of inflow and outflow hydrographs, referred to as the lag. The amount of diffusion (and associated lag) is a function of the ratio Δt/K, a larger diffusion effect corresponding to smaller values of Δt/K. The cascade of linear reservoirs is a widely used method of hydrologic catchment routing. As its name implies, the method is based on the connection of several linear reservoirs in series.
For N such reservoirs, the outflow from the first would be taken as inflow to the second, the outflow from the second as inflow to the third, and so on, until the outflow from the (N  1)^{th} reservoir, is taken as inflow to the
The outflow from the Admittedly, the cascade of reservoirs to simulate catchment response is an abstract concept; nevertheless, it has proven to be quite useful in practice. Each reservoir in the series provides a certain amount of diffusion and associated lag. For a given set of parameters Δt/K and N, the outflow from the last reservoir is a function of the inflow to the first reservoir. In this way, a oneparameter linear reservoir method (Δt/K) is extended to a twoparameter catchment routing method. Moreover, the basic routing formula (Eq. 815) and routing coefficients (Eqs. 816 to 818) remain essentially the same. The addition of the second parameter (N) provides considerable flexibility in simulating a wide range of diffusion and associated lag effects. However, the conceptual basis of the method restricts its general use, since no direct relation between either of the parameters to the physical problem can be readily envisaged. Notwithstanding this apparent limitation, the method has been widely used in catchment simulation, primarily in applications involving large gaged river basins. Rainfallrunoff data can be used to calibrate the method, i.e., to determine a set of parameters Δt/K and N that produces the best fit to the measured data. The analytical version of the cascade of linear reservoirs is referred to as the Nash model [10]. The numerical version is featured in several hydrologic simulation models developed in the United States and other countries. Notable among them is the SSARR model (Chapter 13), which uses it in its watershed, stream channel routing, and baseflow modules [19]. To derive the routing equation for the method of cascade of linear reservoirs, Eq. 815 is reproduced here in a slightly different form:
in which Q represents discharge, whether inflow or outflow and j and n are space and time indexes, respectively (Fig. 104).
As with Eq. 815, the routing coefficients C_{0}, C_{1} and C_{2}
are a function of the dimensionless ratio
This ratio is properly a Courant number In terms of Courant number, Eqs. 816 to 818 are expressed as follows:
For application to catchment routing, it is convenient to define the average inflow as follows:
Substituting Eq. 106 to 109 into Eq. 105 gives the following:
or, alternatively, through algebraic manipulation:
Equation 1011 is the routing equation of the SSARR model [20]. Equations 1010 and 1011 are in a form convenient for catchment routing because the inflow is usually a rainfall hyetograph, that is, a constant average value per time interval. Smaller values of C lead to greater amounts of runoff diffusion. For values of C > 2, the behavior of Eq. 1010 (or Eq. 1011) is highly dependent on the type of input.
For instance, in the case of a unit impulse (rainfall duration equal to the time interval), Eq. 1010
For this reason,
The method of cascade of linear reservoirs is illustrated by the following example.
The cascade of linear reservoirs provides a convenient mechanism for simulating a wide range of catchment routing problems. Furthermore, the method can be applied to each runoff component (surface runoff, subsurface runoff, and baseflow) separately, and the catchment response can be taken as the sum of the responses of the individual components. For instance, assume that a certain basin has 10 cm of runoff, of which 7 cm are surface runoff, 2 cm are subsurface runoff, and 1 cm is baseflow. Since surface runoff is the less diffused process, it can be simulated with a high Courant number, say C = 1, and a small number of reservoirs, say N = 3.
Subsurface runoff is much more diffused than surface runoff; therefore, it can be simulated with Baseflow, being very diffused, can be simulated with C = 0.1 and N = 7. In practice, the parameters C and N are determined by extensive calibration. In this sense, the cascade of linear reservoirs remains essentially a conceptual model [17]. 10.4 KINEMATIC WAVES
Hydraulic catchment routing using kinematic waves was introduced by Wooding in 1965 [21, 22, 23]. Since then, the kinematic wave approach has been widely used in deterministic catchment modeling. The approach can be either lumped or distributed, depending on whether the parameters are kept constant or allowed to vary in space. Analytical solutions are suited to lumped modeling, whereas numerical solutions lend themselves to distributed modeling. Wooding used an openbook geometric configuration (Fig. 415) to represent the catchmentstream problem physically. As its name implies, an openbook configuration consists of two rectangular catchments separated by a stream and draining laterally into it; in turn the streamflow drains out of the catchment outlet. Wooding used analytical solutions of kinematic waves and the method of characteristics to formulate his method. Since diffusion is absent from these solutions, the method is strictly applicable only to kinematic waves. Criteria for the applicability of kinematic waves have been developed by Woolhiser and Liggett [24] (Eq. 455) for overland flow, and by Ponce et al. [12] for stream channel flow (Eq. 944). Kinematic catchment routing models can be approached in a variety of ways. Methods can be either (1) analytical or numerical, (2) lumped or distributed, (3) linear or nonlinear, or (4) single plane, twoplane, or a cascade of planes [7, 8]. Analytical models take advantage of the nondiffusive properties of kinematic waves, whereas numerical models are usually based on the method of finite differences. Linear models assume a constant wave celerity, but nonlinear models relax this restriction. The feature of variable wave celerity often renders the nonlinear models impractical because of wave steepening and associated kinematic shock development [8, 13]. Single and twoplane models are used in hydrologic engineering practice [7]. The application of kinematic wave modeling to catchment routing is illustrated here with an example of a twoplane finite difference numerical model. The model could be either lumped or distributed, depending on whether the inputs and parameters are allowed to vary in space or not. For simplicity, this example considers constant input (i.e., constant effective rainfall) and constant parameters (i.e., a linear mode of computation). In practice, a computeraided solution may relax this restriction. TwoPlane Linear Kinematic Catchment Routing Model Assume a catchment configured as two rectangular planes adjacent to each other, draining laterally into a stream channel located between them. Each of the planes is 100 m long by 200 m wide, and the channel is 200 m long (Fig. 105). The bottom friction in the planes and channel is such that the average velocity in the planes is 0.0417 m/s and the average velocity in the channel is 0.3 m/s. It is desired to obtain the hydrograph at the catchment outlet resulting from an effective rainfall of 9 cm/h lasting 20 min.
Calculation of flow parameters Since the model is linear, it is first necessary to calculate the flow parameters on which to base the calculation of the routing parameters and coefficients. The flow per unit width in the midlength of each plane is equal to the effective rainfall intensity times the contributing area (50 m × 1 m):
Since the average velocity in the planes is v_{p} = 0.0417 m/s, the average flow depth in the planes is: Laminar flow in the planes is assumed, with dischargedepth rating exponent β_{p} = 3. Therefore, the wave celerity in the planes is: c_{p} = β_{p} v_{p} = 3 × 0.0417 = 0.125 m/s The flow in the midlength of the channel is equal to the effective rainfall intensity times the contributing area (2 planes × 100 m × 100 m):
Assume a channel top width T_{c} = 5 m. Therefore, the flow per unit width in the channel is:
q_{c} = Since the average velocity in the channel is v_{c} = 0.3 m/s, the average flow depth in the channel (at midlength) is: d_{c} = A wide channel and turbulent Manning friction is assumed, with dischargearea rating exponent β_{c} = 1.67. Therefore, the wave celerity in the channel is: c_{c} = β_{c} v_{c} = 1.67 × 0.3 = 0.5 m/s The time of conncentration equal to the travel time in the planes plus the travel time in the channel. The travel time in the planes is: (100 m)/(0.125 m/s) = 800 s.
The travel time in the channel is:
Therefore, the time of concentration is: This assures concentrated flow at the catchment outlet. The maximum possible (i.e., equilibrium) peak flow is equal to the product of rainfall intensity and catchment area:
The total volume of runoff is:
Selection of discrete intervals For simplicity, a space interval Δx = 100 m is chosen for the planes. This amounts to one spatial increment in the planes. In an actual application using a computer, a smaller value of Δx would be indicated. The time interval is chosen as Δt = 10 min (the rainfall duration is t_{r} = 20 min). This leads to a Courant number in the planes: C_{p} = c_{p} (Δt /Δx) = 0.75
In the case of the channel, a space interval This leads to a Courant number in the channel: C_{c} = c_{c} (Δt /Δx) = 1.5 Selection of routing scheme There are many possible choices for routing scheme. Either first or secondorder schemes may be used (Section 9.2). In practice, firstorder schemes are preferred because they are more stable than secondorder schemes (compare the results of a first order scheme, Table 95, with those of a second order scheme, Table 94). Two firstorder schemes are chosen here: (1) Scheme I, forwardintime, backwardinspace, stable for Courant numbers C ≤ 1 (similar to the convex method, see Example 95), and (2) Scheme II, forwardinspace, backwardintime, stable for Courant numbers C ≥ 1 (exact opposite of the convex method). The use of these two schemes guarantees that the solution will remain stable because scheme I is used for Courant numbers C ≤ 1, whereas scheme II is used for Courant numbers C > 1 [7]. In the present application, scheme I is used for routing in the planes (C_{p} = 0.75), and scheme II is used for routing in the channel (C_{c} = 1.5). Lateral inflows are an integral part of catchment routing. For routing in the planes, lateral inflow is the effective rainfall; for channel routing, lateral inflow is the lateral contribution from the planes. Therefore, it is necessary to discretize the kinematic wave equation with lateral inflow, Eq. 943. The discretization of Eq. 943 in a forwardintime, backwardinspace linear scheme, as shown in Fig. 106 (a), leads to:
in which
with the Courant number C = βv Δt /Δs, with Δs either Δx (planes) or Δy (channel). The term Q_{L} is the lateral inflow in cubic meters per second. For routing in the planes, the lateral inflow is equal to the effective rainfall (centimeters per hour) times the applicable area (square meters). For channel routing, the lateral inflow is the average distributed lateral inflow (cubic meters per second per meter) multiplied by the channel length (meters).
The discretization of Eq. 943 in a forwardinspace, backwardintime linear scheme, as shown in Fig. 106 (b), leads to:
in which
The catchment routing is shown in Table 104.
Assessment of kinematic wave method The calculated outflow hydrograph peak is 0.8437 m^{3}/s, and it occurs at 20 min. This value is less than the maximum peak flow, Eq. 1014: 1 m^{3}/s. Since the rainfall duration is equal to the time of concentration this implies that the hydrograph has undergone a certain amount of runoff diffusion.
This diffusion is really numerical diffusion, due primarily to the coarse grid size and secondarily to the Courant numbers
(of planes and channel) being different To prove this assertion, it is necessary to reduce the grid size and test the convergence of the kinematic wave schemes, Eqs. 1016 and 1017. Convergence refers to the ability of the numerical scheme to approach the analytical solution as the grid is refined. Due to the large number of calculations involved, the procedure is better accomplished with the aid of a computer program. Table 105 shows the results obtained by successive grid refinement, using program EH1000B.
It is seen that the results are a function of grid size and that the peak flow value converges to the maximum possible value (1 m^{3}/s) as the grid is refined. It is concluded that a kinematic wave numerical solution for catchment routing is grid dependent. If necessary, numerical diffusion can be eliminated by successive grid refinement (while keeping the Courant number as close to 1 as possible). However, in this case the calculated hydrograph would be translated only, with no diffusion. This may be adequate for catchments with negligible runoff diffusion (e.g., small catchments with slopes on the order of 1% or more), but is generally not adequate for catchments showing substantial amounts of runoff diffusion (e.g., midsize catchments of mild relief). For the latter, the diffusion wave technique may be used as a viable alternative to the kinematic wave. 10.5 DIFFUSION WAVES
Catchment routing with diffusion waves is applicable to cases where both translation and diffusion are important, that is, for routing in midsize catchments where catchment slope is such that the kinematic wave criterion is not satisfied. Although the concept of diffusion waves and catchment routing dates back to the work of Dooge [1], actual numerical applications are relatively more recent [14, 16]. Diffusion wave routing can provide gridindependent results, and is therefore regarded as an improvement over griddependent techniques. The diffusion wave catchment routing approach is illustrated here by using the same example as in the previous section. The MuskingumCunge method (Chapter 9) is used as the routing scheme of the diffusion wave method [14]. TwoPlane Linear Diffusion Catchment Routing Model This example is similar to that of the previous section. Assume a catchment configured as two rectangular planes adjacent to each other, draining laterally into a stream channel located between them. Each of the planes is 100 m long by 200 m wide, and the channel is 200 m long (Fig. 105). The slopes of planes and channel (in the direction of the flow) are S_{op} = 0.01; and S_{oc} = 0.01, respectively. The bottom friction in the planes and channel is such that the average velocity in the planes is v_{p} = 0.0417 m/s, and the average velocity in the channel is v_{c} = 0.3 m/s. It is desired to calculate the runoff hydrograph at the catchment outlet resulting from an effective rainfall of 9 cm/h lasting 20 min. Calculation of flow parameters Since the model is linear, it is first necessary to calculate the flow parameters on which to base the calculation of the routing parameters and coefficients. As described in the preceding section, the flow per unit width in the mid length of each plane is equal to q_{p} = 0.00125 m^{2}/s. Since the average velocity in the planes is v_{p} = 0.0417 m/s, the average flow depth in the planes is d_{p} = q_{p}/v_{p} = 0.03 m. Laminar flow in the planes is assumed, with a dischargedepth rating exponent β_{p} = 3. Therefore, the wave celerity in the planes is c_{p} = β_{p}v_{p} = 0.125 m/s. The flow in the midlength of the channel is equal to 0.5 m^{3}/s.
Assume a channel top width
Therefore, the flow per unit width in the channel is q_{c} = Since the average velocity in the channel is v_{c} = 0.3 m/s, the average flow depth in the channel (at midlength) is d_{c} = q_{c} / v_{c} = 0.333 m. A wide channel with turbulent Manning friction is assumed, with a dischargearea rating exponent β_{c} = 1.67. Therefore, the wave celerity in the channel is c_{c} = β_{c} v_{c} = 0.5 m/s. The time of concentration is equal to 1200 s, which is equal to the rainfall duration. The maximum possible peak flow is 1.0 m^{3}/s. The total volume of runoff is 1200 m^{3}. For simplicity, a space interval of Δx = 100 m is chosen for the planes, and Δy = 200 m for the channel. This amounts to one spatial increment in planes and channel. The time interval is chosen as Δt = 10 minutes. In an actual computer application, a finer grid size would be indicated. Selection of routing scheme
The chosen routing scheme is the MuskingumCunge method, For this purpose, Eq. 943 is discretized in the same way as Eq. 961, leading to:
which has the same meaning as Eq. 962 with Eqs. 974 to 976, except for the addition of the lateral inflow term, with routing coefficient [14]:
The term Q_{L} is the lateral inflow in cubic meters per second. For overland flow routing, the lateral inflow is equal to the effective rainfall (centimeters per hour) times the applicable area (square meters). For channel routing, the lateral inflow is the average distributed lateral inflow (cubic meters per second per meter) multiplied by the channel length (meters). The grid size and physical parameters allow the calculation of the routing parameters. The Courant numbers in the planes and channel (Eq. 969) are: C_{p} = 0.75 and C_{c} = 1.5, respectively. The cell Reynolds numbers in the planes and channel (Eq. 971) are: D_{p} = 0.01 and D_{c} = 0.1, respectively.
With Eqs. 974 to 976 and 1020, the routing coefficients in the planes are: C_{0} =
Likewise, the routing coefficients in the channel are: C_{0} = 0.231, C_{1} = 0.923,
C_{2} = The catchment routing is shown in Table 106.
Assessment of Diffusion Wave Method The calculated outflow hydrograph peak is 0.9802 m^{3}/s, and it occurs at 20 min. This value is very close to the maximum possible peak flow (1 m^{3}/s), revealing that the amount of physical diffusion for this particular example is relatively small. To study the effect of grid size on the hydrograph calculated by the diffusion wave method, a test similar to that of the previous section is performed with the aid of a computer program. Table 107 shows the results obtained by successive grid refinement, using program EH1000C. It is shown that the results are essentially independent of grid size. A coarse grid (one space increment in both planes and channel) results in a peak flow of 0.9802, whereas a fine grid (16 increments in both planes and channel) results in a value of 0.9845. However, it should be noted that the coarsegrid solution (one space increment in planes and channel) exhibits a small but perceptible amount of numerical dispersion, as demonstrated by its peak (0.9802) being somewhat greater than the peak obtained with two increments (0.9716).
This is caused by the negative C_{0} in the planes (C_{0} =
Unlike in the kinematic wave method, in the diffusion wave technique the numerical diffusion is matched to the physical problem. As in the case of stream channel routing (Section 10.4), this procedure works best when numerical dispersion is minimized, that is, when Courant numbers are kept reasonably close to 1. In practice, substantial deviations from this condition may lead to increased numerical dispersion and associated numerical instability. As with other finite difference schemes, a test of grid independence similar to that shown in Table 107 is necessary to verify the convergence of the numerical scheme. 10.6 ASSESSMENT
Catchment routing techniques have evolved from the simple timearea methods to the more elaborate physically based kinematic and diffusion wave techniques. The variety of existing methods and techniques reflects the fact that no one method is applicable to all cases. Surface runoff in catchments is a complex phenomenon, and research continues to unveil improved ways of solving the problem. In nature, catchments can be either small, midsize, or large. Runoff processes are nonlinear and distributed; nonlinear in the sense that the parameters do not increase in the same proportion as the flow, and distributed in the sense that the parameters vary within the catchment. Hydrographs originating in catchment runoff are generally concentrated, diffused, and dispersed. From the mathematical standpoint, concentration is a firstorder process, diffusion a secondorder process, and dispersion a thirdorder process [5]. Being a firstorder process, runoff concentration is the primary mechanism; it is also referred to as translation or wave travel. For certain applications, diffusion also plays an important rolefor instance, in the modeling of runoff response in catchments with mild slopes. Early description of the diffusion mechanism referred to it as storage. The usage is so widespread that it has been preserved in this book. In a routing context, diffusion and (longitudinal channel) storage have essentially the same meaning. Dispersion is a third order process; therefore, it is usually much smaller than runoff concentration and diffusion. Physical dispersion translates into wave steepening, a concept similar to that of skewness, or the third moment of a statistical distribution (Chapter 6). The timearea method can calculate concentration but it cannot account for diffusion or dispersion. Therefore, the timearea method should be limited to small and midsize catchments where translation is by far the predominant mechanism. When used indiscriminately, the timearea method always overestimates the peak of the outflow hydrograph. If necessary, diffusion can be added by routing the timearea hydrograph through a linear reservoir. However, the storage constant would have to be determined either from measured data or by synthetic means. While the timearea method is a lumped method, it can be used as a component of larger network models which have a distributed structure [19]. In principle, the method of cascade of linear reservoirs accounts for runoff diffusion only. However, the connection of several linear reservoirs in series provides enough diffusion so that translation is actually being simulated by means of diffusion. The method is linear insofar as the routing parameters are determined by calibration. Nevertheless, by successive calibrations at different flow levels, from low to high, the method can be made to reflect the nonlinearity actually existing in nature. The method is lumped, but it can be used as a component of larger network models that have a distributed structure. The method has been successfully applied to very large basins (in excess of 10,000 km^{2}), which exhibit substantial amounts of diffusion. For such large basins, the distributed models of the kinematic and diffusion type may be impractical due to the prohibitive amount of data required to properly specify the spatial diversity. The kinematic wave model provides translation and diffusion, the latter, however, due only to the finite grid size. The method can be linear or nonlinear, and lumped or distributed, depending on the numerical scheme and input data. The method is applicable to small catchments with steep slopes where diffusion is small and can be controlled by grid refinement. Theoretically, the method could also be applicable to midsize catchments, as long as physical diffusion remains small. In practice, the larger the catchment, the more unlikely it is that physical diffusion is negligible. The distributed nature of kinematic wave models results in substantial data needs; the use of average parameters would render the model lumped, with the consequent loss of detail. Another important consideration in kinematic wave models is the validity of the geometric configuration. For instance, twoplane descriptions are adequate as long as the catchment geometry fits the twoplane model configuration. Otherwise, a certain amount of lumping would be introduced by the model's inability to properly account for the physical detail. In practice, the larger the catchment, the more difficult it is to fit catchments within twoplane descriptions. Multipleplane descriptions are possible but invariably lead to additional complexity [8]. The diffusion wave technique (MuskingumCunge scheme) provides translation and diffusion and, unlike the kinematic wave model, its solution is generally independent of grid size. Therefore, it is applicable to catchments with substantial amounts of physical diffusion, either small catchments of mild slope or midsize catchments with average slopes. The method is linear or nonlinear, and lumped or distributed, depending on the numerical scheme and input data. As with kinematic models, the validity of the geometric configuration is important in diffusion wave catchment models. Unless the model's geometric abstraction is a reasonable representation of the catchment's actual geometry, the degree of lumping introduced may tend to mask the distributed nature of the model. The preceding comments have referred specifically to catchment routing, i.e., the conversion of effective rainfall into runoff. In practice, comparative evaluations of the performance of catchment routing methods/models are hampered by the fact that it is seldom possible to determine effective rainfall with any degree of certainty. Hydrologic abstractions are timevariant, distributed, and nonlinear. Therefore, a proper estimation of hydrologic abstractions is crucial to the performance evaluation of catchment routing models. From this discussion, it can be concluded that no one method or model is suitable for all applications. All have strengths and weaknesses; they are either simple or complex, and suffer from lack of detail or require a substantial amount of data for their successful operation. In practice, the choice of catchment routing method remains one of individual preference and experience. QUESTIONS
PROBLEMS
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