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Fig. 1  Calculated time-area histogram.

3.  MODCLARK IN HEC-HMS

The HEC-HMS model continues the U.S. Army Corps of Engineers tradition of using the Clark unit hydrograph, incorporating it as part of their suite of rainfall-runoff transform procedures (Hydrologic Engineering Center 2010). When HEC-HMS was introduced, Kull and Feldman (1998) reported that about 40% of Corps offices were using the methodology. The original Clark model is a lumped model, i.e., it provides only one hydrograph at the watershed/basin outlet. In line with modern methods of watershed analysis, in HEC-HMS, the Clark model is extended to accomodate spatially distributed rainfall data. This practice effectively extends the Clark model to the realm of quasi-distributed surface runoff modeling. The revised version of the Clark model is referred to as the ModClark method (Kull and Feldman 1998).

4.  ROUTING PRINCIPLES

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

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

2. diffusion, 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 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 hydrologic/hydraulic features are recognized:

1. reservoirs,

2. stream channels, and

3. watersheds/basins.

In reservoir routing, convection is zero and diffusion is finite; therefore, reservoir routing incorporates only diffusion. In stream channel routing, convection is the dominant process (first order) and diffusion is typically small (second order). For kinematic waves, diffusion is zero; for diffusion waves, diffusion is finite but relatively small; and for dynamic waves, diffusion is large to very large (Ponce and Simons 1977) In watershed/basin routing, convection and diffusion are about equally important and, therefore, they are often accounted for separately, as in the Clark methodology.

The time-area method of watershed routing provides only convection (translation, or concentration) (Section 5). The linear reservoir method provides only diffusion (attenuation, or storage) (Section 6). The unit hydrograph is an elemental hydrograph for a given basin, subsuming convection and diffusion in a hydrograph for a unit rainfall impulse (Section 7). The Clark unit hydrograph uses: (a) the time-area method to provide convection, and (b) the linear reservoir method to provide diffusion (Section 8).

It is worth noting that the linear reservoir method can be applied in series, to provide varying amounts of diffusion. This method, referred to as the cascade of linear reservoirs (CLR), theoretically can provide only diffusion (Ponce 2009). In practice, however, the use of several reservoirs in series results in an outflow hydrograph that appears to have both convection and diffusion properties (Ponce 1989; p. 307). In essence, a lot of diffusion is being used as a way to simulate convection.

5.  TIME-AREA METHOD

The time-area method transforms an effective storm hyetograph into a runoff hydrograph (Ponce 1989). The method accounts for convection only and does not include 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 (storage) 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 effective storm hyetograph must be the same as the time interval of the time-area histogram (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; p. 308). The lagging and summation of the partial flows results in a runoff [flood] hydrograph for the given time-area histogram and effective storm hyetograph.

Fig. 3  Effective rainfall hyetograph.

An example of the calculation of the time-area method with the data of Figs. 1 and 2 is shown in Table 2. Note that the histogram subareas (highlighted in red in Table 2) are defined for a time interval; for instance, the value 10 is applicable from time t = 0 to t = 1 hr; 30 from t = 1 to t = 2 hr, and so on.

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. In general, the following relation holds:

Tb = Tc + tr

(2)

While the time-area method accounts for convection only (i.e, translation, or concentration), it has the distinct advantage that the watershed shape is reflected in the time-area histogram and, therefore, in the runoff hydrograph. Diffusion can 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 model using the time-area method, and adding diffusion, would consist of two steps:

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

2. Routing of the translated-only hydrograph through a linear reservoir to provide the desired diffusion effect.

6.  LINEAR RESERVOIR

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 would diffuse, that is, lower its peak flow and increase 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 or equal to 2 provide hydrograph diffusion; the lesser the Δt/K, the greater the diffusion (Ponce 1980). Values of Δt/K greater than 2 provide hydrograph amplification, that is, negative diffusion; therefore, values of Δt/K greater than 2 are not used in linear reservoir routing (Ponce 1989; p.259).

Given inflow I, outflow O, time interval Δt, reservoir storage constant K, the general formula for the linear reservoir is:

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

(3)

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 25 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 the routing through the linear reservoir has conserved volume exactly. Figure 4 shows the effect of routing the time-area hydrograph (Table 2, Col. 9) through a linear reservoir (Table 3, Column 6).

Fig. 4  Effect of routing the time-area hydrograph through a linear reservoir.

7.  UNIT HYDROGRAPH

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 used in hydrology are 1 hr, 2 hr, 3 hr, 6 hr, 12 hr, and 24 hr. 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 in] or effective precipitation lasting 1 hr; the 2-hr unit hydrograph corresponds tp 1 cm [or 1 in] or 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 [or in/hr]. For a certain basin, once a unit hydrograph for a given duration has been determined, this unit hydrograph may be used to calculate unit hydrographs for other durations.

The unit hydrograph can be developed from streamgaging measurements or by synthetic means. Once developed, the unit hydrograph contains information on the runoff concentration and diffusion properties of the watershed/basin. It 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; p.185).

8.  CLARK UNIT HYDROGRAPH

Clark (1945) was able to provide hydrograph diffusion by routing the discrete time-area-derived unit-runoff hyetograph through a linear reservoir (Section 6). To demonstrate his methodology, Clark used the example of the Appomattox River at Petersburg, Virginia, with the following data (Fig. 5):

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. Hydrograph duration t_r = 12 hr; and

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

Fig. 5  Isochrones for the Appomattox River at Petersburg, Virginia (Clark 1945).

By definition, C₀ = C₁. 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; Col. 2 shows the time in hours; Col. 3 shows the stated [measured] time-area histogram percentages; Col. 4 shows the incremental areas. The unit hydrograph rainfall intensity is 1/12 in/hr. Column 5 shows the inflows, calculated as the areas of Col. 4 multiplied by 1 in/hr. Since the [unit hydrograph] rainfall intensity is 1/12 in/hr, each value of Col. 5 amounts to 12 times the inflow. Column 6 is the actual inflow, that is 1/12 of Col. 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 units. The sum of the hydrograph ordinates in Col. 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 Col. 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 Col. 5, 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 Table 5 Example).

9.  PONCE'S CLARK UNIT HYDROGRAPH

Ponce's version of the 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; p. 311 [2]). Ponce's Clark unit hydrograph retains the essence of the original methodology, while providing an improved unit hydrograph, consistent with established time-area routing principles.

Table 6 illustrates the computation of Ponce's version of the 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 Col. 5, is the translated-only unit hydrograph, with flow units (km²-cm/hr) and, following Eq. 2, a time base T_b = 6 hr.

Column 9 shows that the method conserves mass. Note that unlike the discrete unit-runoff hyetograph shown in Col. 5 of Table 4, Col. 5 of Table 5 shows a continuous unit hydrograph. That is the difference between the two methods and that is why the results are slightly different. Ponce's unit hydrograph has a longer time base (1 hr longer); consequently, it has a somewhat smaller peak discharge and a longer time lag.

Table 7 shows a side-by-side comparison of Clark's original unit hydrograph with Ponce's version. It is seen that the original unit hydrograph has a somewhat higher peak. The difference is balanced by the somewhat higher tail of Ponce's version. In addition, Ponce's version 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 6 shows a graphical portrayal of the differences between the two unit hydrographs.

Fig. 6  Comparison of Clark's original and Ponce's Clark unit hydrographs.

The program Online Routing Clark makes possible the calculation and side-by-side comparison of Clark's original and Ponce's Clark unit hydrographs.

10.  SUMMARY

Clark's original unit hydrograph (Clark 1945) and Ponce's somewhat improved version (Ponce 1989; p. 311 [3]) are explained and compared. Clark's procedure routes, through a linear reservoir, the discrete, time-area-derived, unit-runoff hyetograph, while Ponce's procedure routes the continuous, time-area-derived, unit hydrograph. Since the unit hydrograph has a longer time base than the unit-runoff hyetograph, Ponce's procedure provides a somewhat smaller peak discharge and longer time lag than Clark's. The difference, however, does not appear to be substantial. As expected, both methods are shown to be 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.

Kull, D. W., and A. D. Feldman. 1998. Evolution of Clark's unit graph method to spatially distributed runoff. Journal of Hydrologic Engineering, ASCE, Vol. 3, No. 1, January, 9-19.

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.