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 originaIl 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, unlike in reservoir and channel routing. 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 should 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 at the end of each time interval is equal to the product of effective rainfall times contributing subarea. 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 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. 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 t_{r} = effective rainfall duration. 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 Eq. 101 in lieu of time base. Therefore time of concentration can be defined as the difference between the time to point of inflection 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 tailá 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.
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 t ime 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  l)th reservoir, is taken as inflow to the Nth reservoir. The outflow from the Nth reservoir is taken as the outflow from the cascade of linear reservoirs. 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 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 and stream channel routing modules [18]. 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 t!.tIK. This ratio is properly a Courant number (C = Δt / K). 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 105 gives
or, alternatively:
which is the routing equation used in the SSARR model [19]. Equations 1010 and lO11 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 greater than 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 (or Eq. 1011) results in negative outflow values (numerical instability). For this reason, Eq. 1010 is restricted in practice to C ≤ 2.
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 C = 0.4 and N = 5. 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 [16]. 10.4 CATCHMENT ROUTING WITH KINEMATIC WAVES
Hydraulic catchment routing using kinematic waves was introduced by Wooding in 1965 [ (20, 21, 22)]. 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 are more appropriate for 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 [23] (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 or the method of characteristics. 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 áa 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 mls 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 X 1 m):
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 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 the effective rainfall intensity times the contributing area (2 planes X 100 m X 100 m):
Assume a channel top width T_{c} = 5 m. Therefore, the flow per unit width in the channel is q_{c} = Q_{c} / T_{c} = 0.1 m^{2}/s. 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 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} = 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.l25 m/s) = 800 s. The travel time in the channel is (200 m)/(0.5 m/s) = 400 s. Therefore, the time of concentration is 800 + 400 = 1200 s, which is equal to the effective rainfall duration. 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. This leads to a Courant number in the planes C_{c} = c_{p}(Δt / Δx) = 0.75. In the case of the channel, a space interval Δy = 200 m is chosen, that is, one spatial increment in the channel. 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 less than or equal to 1 (similar to the convex method, see Example 95), and (2) Scheme II, forwardinspace, backwardintime, stable for Courant numbers greater than or equal to 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 less than or equal to 1, whereas scheme II is used for Courant numbers greater than 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 (Fig. 106(a)) leads to:
in which C_{1} = C; C_{2} = 1  C; and C_{3} = C, with C being the Courant number (C = βv Δt / Δs), with Δs either Δx (planes) or Δy (channel). The term Q_{r} 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 (Fig. 106(b)) leads to:
in which C_{0} = (C  l) / C; C_{1} = l/C; and C_{3} = 1. The catchment routing is shown in Table 104. Column 1 shows time in minutes. Columns 24 show the plane routing, and Cols. 57 show the channel routing. Column 2 shows the lateral inflow to each plane:
Notice that the lateral inflow is an average value for a time interval, and it lasts 20 min (i.e., the effective rainfall duration). Column 3 shows the upstream inflow to the plane, that is, zero (this example does not consider upstream inflow to the planes). Column 4 is obtained by routing with Eq. 1016, with Courant number in the planes C_{p} = 0.75. Column 4 is the outflow hydrograph from each plane. The sum of Col. 2 is 1.0; likewise, the sum of Col. 4 is 0.9999, which confirms that the volume under the
10.5 CATCHMENT ROUTING WITH 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 have only recently been attempted [6,14,15]. 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 ofthe flow) are Sop = 0.01; and Soc = 0.01 , respectively. The bottom friction in the planes and channel is such that the average velocity in the planes is 0.0417 mis, and the average velocity in the channel is 0.3 m/ s. It is desired to calculate the runoff hydrograph at the catchment outlet resulting from an effective rainfall of 9 cmlh 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 T_{c} = 5 m. Therefore, the flow per unit width in the channel is q_{c} = Q_{c} / T_{c} = 0.1 m^{2}/s. 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, Eq. 962, with the routing coefficients calculated by Eqs. 974 to 976 but modified with the addition of lateral inflow. For this purpose, Eq. 9 43 is discretized in the same way as Eq. 961, leading to
10.6 ASSESSMENT OF CATCHMENT ROUTING TECHNIQUES
Catchmentrouting 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 distributednonlinear 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 [18]. 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 substantiat amounts of runoff diffusion. For such large basins, the distributed models of the kinematic and diffusion type would 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 would 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 catchmentrouting methods 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 catchmentrouting models. From this discussion, it can be concluded that no one r::ethod or model is suitable for all applications. All have strengths and weaknesses; they are either siinple or complex and suffer from lack of detail or require a substantial amount of data for their successful operation. In practice, the choice of catchmentrouting method remains one of individual preference and experience. QUESTIONS
PROBLEMS
REFERENCES
Alley, W. M., T. E. Reilly, and. O. E. Franke. 1999. Sustainability of groundwater resources. U.S. Geological Survey Circular 1186, Denver, Colorado, 79 p.

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