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Fig. 1  Calculated time-area histogram.

3.  ROUTING PRINCIPLES

In surface-water hydrology, the term "routing" refers to the calculation of flows in time and space. The objective is to transform an inflow, either (a) an effective precipitation hyetograph in the case of watersheds/basins, or (b) a streamflow hydrograph in the case of reservoirs and channels, into an outflow hydrograph. In general, flow routing embodies two distinct physical processes:

1. convection, commonly referred to as translation or concentration, and

2. diffusion, commonly referred to as attenuation or storage.

Convection is interpreted as the movement of water in a direction parallel to the channel bottom. Diffusion may be interpreted as the movement of water in a direction perpendicular to the channel bottom. Mathematically, convection is a first-order process, while diffusion is a second-order process (Ponce, 1989).

Three types of physical features are recognized:

1. reservoirs,

2. stream channels, and

3. watersheds/basins.

The behavior of these features with respect to convection and diffusion varies. Significantly:

1. In reservoir routing, convection is zero while diffusion is finite; therefore, reservoir routing lacks convection and produces only diffusion.

2. In stream channel routing, convection is typically the dominant process, of first order, while diffusion is usually much smaller, of second order. For kinematic waves, diffusion is nonexistent; for diffusion waves, diffusion is relatively small; for mixed kinematic/dynamic waves, in practice referred to as dynamic waves, diffusion is large (Ponce and Simons, 1977).

3. In watershed/basin routing, convection and diffusion are usually about equal in size and, therefore, they may be accounted for separately. This is the basis of the Clark methodology.

The time-area method of watershed routing provides only convection; in contrast, the linear reservoir routing method provides only diffusion. The unit hydrograph is an elemental hydrograph for a given watershed/basin, subsuming both convection and diffusion in a hydrograph for a unit rainfall impulse. The Clark unit hydrograph accounts for these two processes separately, first using the time-area method to provide convection and second, using the linear reservoir method to provide diffusion.

4.  TIME-AREA METHOD

The time-area method of hydrologic watershed/basin routing transforms an effective storm hyetograph into a runoff hydrograph (Ponce, 1989). The method accounts for convection only and does not provide for diffusion. Therefore, hydrographs calculated with the time-area method show a characteristic lack of diffusion, resulting in higher peaks and shorter time bases than those that would have been obtained if diffusion had been taken into account.

The time-area method is based on the concept of time-area histogram. To develop a time-area histogram, the watershed'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 watershed into zones delimited by isochrone lines, i.e., the loci of points of equal travel time to the outlet. For any point inside the watershed, the travel time refers to the time that it would take a parcel of water to travel from that point to the outlet. The subareas delimited by the isochrones are measured and plotted in histogram form, as shown in Fig. 2.

Fig. 2  Time-area method:  (a) isochrone delineation; (b) time-area histogram.

For the method to work properly, the time interval of the time-area histogram (Fig. 2b) must be the same as that of the effective rainfall hyetograph (Fig. 3). The rationale of the time-area method is that, according to the runoff concentration principle, the partial flow at the end of each time interval is equal to the product of effective rainfall times the contributing watershed subarea (Ponce, 1989). The lagging and summation of the partial flows results in a flood hydrograph corresponding to the given time-area histogram and effective rainfall hyetograph.

Fig. 3  Effective rainfall hyetograph.

Table 2 shows the calculation of the time-area method using the data of Figs. 2 and 3. Note that the histogram subareas, highlighted in red, are defined for a time interval; for instance, the value 10 is applicable between time t = 0 to t = 1 hr; 30 between t = 1 to t = 2 hr, and so on. Also, note the six (6) effective rainfall increments, highlighted in blue.

In Table 2, the partial flows for the 0.5 cm/hr rainfall increment are obtained by multiplying each one of the subareas of Col. 2 times 0.5 cm/hr. Likewise, the partial flows for the 1.0 cm/hr rainfall increment are obtained by multiplying each one of the subareas of Col. 2 times 1.0 cm/hr, and lagged 1 hr, because the 1.0 cm/hr rainfall increment occurs 1 hr later; and so on. The sum across Cols. 3 to 8, shown in Col. 9, gives the ordinates of the outflow hydrograph in convenient (km²-cm/hr) units. Column 10 contains the values of Col. 9, converted to discharge units (m³/s).

Note that the time-area method conserves mass exactly. For this example, the total rainfall is 6.5 cm and the watershed area is 100 km². Thus, the total rainfall volume is: 100 × 6.5 = 650 km²-cm. The sum of the ordinates of the runoff hydrograph (Col. 9) is 650 km²-cm/hr. These are hourly ordinates; thus, the hydrograph volume is: 650 km²-cm/hr × 1 hr = 650 km²-cm. Also, note that for this example, the time of concentration is T_c = 4 hr, the storm duration is t_r = 6 hr, and the hydrograph time base is T_b = 10 hr. The following relation holds:

Tb = Tc + tr (2)

While the time-area method accounts for convection only, it has the distinct advantage that the watershed/basin shape is reflected in the time-area histogram and, therefore, in the runoff hydrograph. When warranted, diffusion may be provided by routing the hydrograph calculated by the time-area method through a linear reservoir with an appropriate storage constant. Thus, a complete rainfall-runoff transform model using the time-area method and adding diffusion consists of two steps:

1. Use of the time-area method to provide pure convection, i.e., generating a translated-only hydrograph.

2. Routing of the translated-only hydrograph through a linear reservoir with an appropriate storage constant, to provide the desired diffusion effect.

5.  LINEAR RESERVOIR ROUTING

The linear reservoir method is a computational procedure that provides diffusion to an inflow hydrograph. In other words, a hydrograph routed through a linear reservoir effectively diffuses, that is, it lowers its peak flow while it increases its time base. The amount of diffusion depends on the value of the storage constant K relative to the time interval Δt. Values of Δt/K less than 2 provide diffusion; the lesser the Δt/K, the greater the diffusion. Values of Δt/K greater than 2 provide negative diffusion, i.e., hydrograph amplification. therefore, values of Δt/K greater than 2 are not used in reservoir routing (Ponce, 1989).

Given inflow I, outflow O, time interval Δt, and reservoir storage constant K, the general formula for the linear reservoir is (Fig. 4):

O2 = C0 I2 + C1 I1 + C2 O1 C0 = (Δt/K) / [ 2 + (Δt/K) ] C1 = C0 C2 = [ 2 - (Δt/K) ] / [ 2 + (Δt/K) ]  (3)

Fig. 4  Linear reservoir discretization.

Table 3 shows the routing of the discharge hydrograph obtained with the time-area method (Table 2, Col. 9) through a linear reservoir of storage constant K = 2 hr, with Δt = 1 hr. Following Eq. 3, the routing coefficients for this example are: C₀ = 0.2; C₁ = 0.2; C₂ = 0.6.

Note that the peak flow has decreased from 145 km²-cm/hr for time t = 6 hr at inflow (Col. 2), to 109.88 km²-cm/hr for t = 7 hr at outflow (Col. 6). Also, note that the time base of the hydrograph has increased from 10 hr at inflow, to 26 hr at outflow. The sum of hydrograph ordinates, 650 km²-cm/hr, is the same at inflow (Col. 2) and outflow (Col.6), indicating that routing through the linear reservoir has conserved mass exactly. Figure 5 shows the effect of routing the time-area hydrograph (Table 2, Col. 9) through a linear reservoir (Table 3, Column 6).

Fig. 5  Effect of routing the time-area hydrograph through a linear reservoir.

6.  UNIT HYDROGRAPH CONCEPTS

The unit hydrograph is used in flood hydrology as a means to develop the hydrograph for a given effective storm hyetograph. The word "unit" is normally taken to refer to a unit depth of effective rainfall. However, the word "unit" also refers to a unit depth of effective rainfall lasting a "unit" duration, or "unit" increment of time, i.e., an indivisible increment (Ponce, 1989). Typical unit hydrograph durations are 1 hr, 2 hr, 3 hr, 6 hr, 12 hr, and 24 hr. In flood hydrology, unit hydrograph durations from 1 hr to 6 hr are common.

A watersheds/basin can have many unit hydrographs, each for a different duration. The 1-hr unit hydrograph corresponds to 1 cm [or, alternatively, 1 in] or effective precipitation lasting 1 hr; the 2-hr unit hydrograph corresponds to 1 cm of effective precipitation lasting 2 hr; and so on. Thus, given a t_r -hr unit hydrograph, the effective rainfall intensity is equal to (1/t_r) cm/hr. For a certain basin, once a unit hydrograph for a given duration has been determined, it may be used to calculate the unit hydrographs for other durations.

A unit hydrograph may be developed from streamgage measurements or by synthetic means. Once developed, the unit hydrograph contains information on the convection (runoff concentration) and diffusion (runoff diffusion) properties of the watershed/basin. The unit hydrograph is used as a building block in order to develop the flood hydrograph. To develop the flood hydrograph, the unit hydrograph is convoluted with the effective storm hyetograph (Ponce, 1989).

7.  CLARK'S ORIGINAL UNIT HYDROGRAPH

Clark (1945) pioneered the use of the time-area method in conjunction with a linear reservoir as a means to develop a unit hydrograph. To demonstrate his methodology, Clark used the example of the Appomattox River at Petersburg, Virginia, with the following data (Fig. 6):

1. Drainage area A = 1,335 square miles;

2. Time of concentration T_c = 6 days;

3. Time interval Δt = 0.5 days = 12 hr;

4. Unit hydrograph duration t_r = 12 hr; and

5. Linear reservoir storage constant K = 15.428 hr ≅ 15 hr.

Fig. 6  Isochrones for the Appomattox River at Petersburg, Virginia (Clark 1945).

By definition, C₁ = C₀. Furthermore, within one time step, Clark used the unit-runoff hyetograph, wherein I₂ = I₁. Thus, the linear reservoir routing equation reduces to:

O2 = 2 C0 I2 + C2 O1  (4)

Table 4 shows the computation of the Appomattox River example.

• Column 1 shows the time in days.

• Column 2 shows the time in hours.

• Column 3 shows the time-area histogram percentages.

• Column 4 shows the incremental areas.

• Column 5 shows the inflows, calculated as the areas of Column 4 multiplied by 1 in/hr. Since the unit hydrograph duration t_r = 12 hr, the rainfall intensity is 1/12 in/hr, each value of Column 5 amounts to 12 times the inflow.

• Column 6 is the actual inflow, that is 1/12 of Column 5.

• Columns 7 and 8 shows the partial flows, calculated using Eq. 4.

• Column 9 is the outflow from the linear reservoir, in the routed units (mi²-in/hr).

• Column 10 is the outflow (unit hydrograph ordinates), converted to cfs.

The sum of the hydrograph ordinates in Column 9 is 111.249 mi²-in/hr, which, when integrated over the time interval (12 hr) and distributed over the basin area (1,335 mi²) results in exactly 1 in of runoff (111.249 × 12 / 1,335 = 1). Likewise, the sum of the unit hydrograph ordinates in Column 10 is 71,786 cfs, which, when integrated over the time interval and distributed over the basin area results in exactly 1 in of runoff.

It is important to note that Clark did not route the time-area histogram through a linear reservoir; indeed one cannot route an area; only a discharge. Effectively, Clark routed the inflow [unit-runoff hyetograph] shown in Column 6, a value which is linearly related to the area. To avoid unnecessary complexity, Clark set the time interval Δt equal to the unit hydrograph duration t_r, thus avoiding the lagging and addition which would have been necessary had the time interval Δt and unit hydrograph duration t_r been different (See the example of Table 5).

8.   PONCE'S CLARK UNIT HYDROGRAPH

Ponce's Clark unit hydrograph differs slightly from the original Clark procedure. Ponce applied the time-area method to a given unit rainfall, of intensity 1/t_r, to obtain a continuous unit hydrograph based on the time-area histogram. He then routed this unit hydrograph through a linear reservoir to obtain the Clark unit hydrograph (Ponce, 1989). Ponce's Clark unit hydrograph retains the essence of the original methodology, while providing an improved unit hydrograph, consistent with established routing principles.

Table 6 illustrates the computation of Ponce's Clark unit hydrograph. To enable comparison, the data for this example is the same as that of Table 5. The unit hydrograph duration is t_r = 2 hr, the time interval is Δt = 1 hr, and the linear reservoir storage constant is K = 2 hr. Following Eq. 3, the routing coefficients are: C₀ = 0.2; C₁ = 0.2; C₂ = 0.6. To develop the flows, the histogram subareas are multiplied by two (2) 1-hr rainfall increments of 0.5 cm/hr each, and lagged appropriately. The sum across Cols. 3 and 4, shown in Column 5, is the translated-only unit hydrograph, with flow units (km²-cm/hr) and, following Eq. 2, a time base T_b = 6 hr. Columns 6, 7, and 8 are the partial flows of the linear reservoir routing. Column 9 is the sum of Cols. 6, 7, and 8, i.e., the outflow from the linear reservoir. Column 9 shows that the method conserves mass; the sum of Column 9 is 100.

Note that unlike the discrete unit-runoff hyetograph shown in Column 5 of Table 5, Column 5 of Table 6 shows a continuous unit hydrograph. That is the difference between the two methods and that is why the results are slightly different. Following Eq. 2, Ponce's Clark unit hydrograph has a 6-hr time base, i.e., 1 hr longer than Clark's; consequently, it has a smaller peak discharge.

In Table 6, the continuous unit hydrograph is highlighted in magenta color. Note that Ponce's Clark unit hydrograph conserves mass exactly, as shown by the last lines of Columns 2 and 8.

Table 7 shows a side-by-side comparison of the original Clark and Ponce's Clark unit hydrographs. It is seen that Clark's original unit hydrograph has a somewhat higher peak. The difference is balanced by the longer tail of Ponce's Clark unit hydrograph. In addition, Ponce's Clark is lagged about 1 hr to the right, which reflects the time base of the inflow unit hydrograph (6 hr), as opposed to that of the inflow unit-runoff hyetograph (5 hr). Both unit hydrographs conserve mass exactly (see bottom line of Cols. 5 and 9). Figure 7 shows a graphical portrayal of the differences between the two unit hydrographs.

Fig. 7  Comparison of Clark's original and Ponce's Clark unit hydrographs.

9.  ONLINE CALCULATION

The program ONLINE ROUTING CLARK enables the calculation and side-by-side comparison of Clark´s original and Ponce´s Clark's unit hydrographs. Figure 8 shows the comparison for the same example shown in Table 7.

Fig. 8  Online calculation of Clark's original and Ponce's Clark unit hydrographs.

10.  SUMMARY

Clark's original unit hydrograph (Clark, 1945) and Ponce's Clark unit hydrograph (Ponce, 1989) are explained and compared. Clark's procedure routes, through a linear reservoir, discrete time-area-derived unit-runoff hyetograph (highlighted in cyan in Table 5), while Ponce's procedure routes the continuous time-area-derived unit hydrograph (highlighted in magenta in Table 6). Since the unit hydrograph has a longer time base than the unit-runoff hyetograph, Ponce's procedure provides a longer time lag and a correspondingly smaller peak discharge than Clark's original methodology. The difference, however, does not appear to be substantial. As expected, both methods are shown to be 100% mass conservative.

REFERENCES

Clark, C. O., 1945. Storage and the unit hydrograph. Transactions, ASCE, Vol. 110, Paper No. 2261, 1419-1446.

Hydrologic Engineering Center, 1990. HEC-1, Flood Hydrograph Package. Davis, California.

Hydrologic Engineering Center, 2010. HEC-HMS, Hydrologic Modeling System. User's Manual, Version 3.5, August 2010, CDP-74A, Davis, California.

Ponce, V. M., and D. B. Simons. 1977. Shallow wave propagation in open channel flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, December, 1461-1476.

Ponce, V. M. 1980. Linear reservoirs and numerical diffusion. Journal of the Hydraulics Division, ASCE, Vol. 106, No. HY5, May, 691-699.

Ponce, V. M. 1989. Engineering Hydrology: Principles and Practices. Prentice Hall, Englewood Cliffs, New Jersey.

Ponce, V. M. 2009. Cascade and convolution: One and the same. Online article.