Isochrones for the
Appomattox River at Petersburg, Virginia 
 


ABSTRACT
Clark's original unit hydrograph (Clark 1945) and Ponce's somewhat improved
version (Ponce 1989; p. 311) are explained and compared.
Clark's procedure routes, through a linear reservoir, a discrete unitrunoff hyetograph,
while Ponce's procedure routes a continuous unit hydrograph.
Since the unit hydrograph has a longer time base than the unitrunoff hyetograph,
Ponce's procedure provides a somewhat smaller peak discharge and a 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.

1. INTRODUCTION
The Clark unit hydrograph (Clark 1945)
was included in the original HEC1 model of the
U.S. Army Corps of Engineers (Hydrologic Engineering Center 1990).
A variation of the Clark method, referred to as ModClark, forms part of Hydrologic Engineering Center's HECHMS, the second generation hydrologic model
of the U.S. Army Corps of Engineers, which superseded HEC1 in 1998. Despite its
apparent ubiquitousness in Corps literature, details about the Clark model are not
widely available, with a notable exception in Ponce's textbook (1989).
Herein, we endeavor to clarify the origins of the
Clark methodology, to explain its theoretical basis,
and to compare the original Clark model with Ponce's (1989) version.
In a nutshell, the Clark unit hydrograph is derived by routing the discrete, timeareaderived,
unitrunoff hyetograph
through a linear reservoir. The storage coefficient of the linear reservoir is chosen empirically in such a way as
to provide the hydrograph diffusion that is necessary to simulate a realistic unit hydrograph.
2. HEC1 CLARK UNIT HYDROGRAPH
The HEC1 Clark unit hydrograph requires the specification of the watershed/basin drainage area A,
the time of concentration T_{c} , the linear reservoir's storage constant K, and the timearea histogram.
If no basinspecific, timearea histogram is available, HEC1 provides a default timearea curve [a two endtoend parabolashaped basin] from which
a timearea histogram can be obtained (Kull and Feldman 1998). In cases where the actual watershed/basin shape deviates substantially from this standard shape, it
is advisable to construct a basinspecific timearea histogram.
HEC's default timearea curve is:
T_{*} = T_{i} / T_{c}
A_{*} = A_{i} / A
A_{*} = 1.414 T_{*}^{1.5} [0 ≤ T_{*} ≤ 0.5]
A_{*} = 1  1.414 (1  T_{*})^{1.5} [0.5 ≤ T_{*} ≤ 1]

 (1) 
where T_{i} and A_{i} are cumulative time and cumulative area, respectively.
For example, given A = 1000 km^{2} and T_{c} = 6 hr, the calculated default timearea histogram
for this watershed data is shown in Table 1. The time of concentration is conveniently divided into six (6) 1hr intervals.
Using the default timearea curve (Eq. 1), the watershed area is divided into six (6) corresponding subareas, shown in Col. 4 of Table 1.
The calculated timearea histogram is shown in Fig. 1.
Table 1. HEC's default timearea histogram for A
= 1,000 km^{2} and T_{c} = 6 hr.

[1] 
[2] 
[3] 
[4] 
Histogram increment i 
Cumulative time T_{i} at the end of increment i (hr) 
Cumulative area A_{i} at the end of cumulative time T_{i} (km^{2}) 
Incremental area ΔA_{i} (km^{2}) 
1 
1 
96.2 
96.2 
2 
2 
272.1 
175.9 
3 
3 
500 
227.9 
4 
4 
727.9 
227.9 
5 
5 
903.8 
175.9 
6 
6 
1,000 
96.2 
Fig. 1 Calculated timearea histogram.
3. MODCLARK IN HECHMS
The HECHMS model continues the U.S. Army Corps of Engineers tradition of using the Clark unit hydrograph, incorporating it as part of their suite
of rainfallrunoff transform procedures (Hydrologic Engineering Center 2010).
When HECHMS 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 HECHMS,
the Clark model is extended to accomodate spatially distributed rainfall data. This practice
effectively extends the Clark model to the realm of quasidistributed 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 surfacewater 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:  convection, commonly referred to as translation or concentration, and
 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 firstorder process, while diffusion is a secondorder process (Ponce 1989).
Three types of hydrologic/hydraulic features are recognized:  reservoirs,
 stream channels, and
 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 timearea 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 timearea 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. TIMEAREA METHOD
The timearea 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 timearea 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 timearea method is based on the concept of timearea histogram.
To develop a timearea 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 Timearea method: (a) isochrone delineation; (b) timearea 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 timearea histogram (Fig. 3). The rationale of the timearea 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 timearea histogram and effective storm hyetograph.
Fig. 3 Effective rainfall hyetograph.
An example of the calculation of the timearea 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.
Table 2. Timearea method of watershed routing.

[1] 
[2] 
[3] 
[4] 
[5] 
[6] 
[7] 
[8] 
[9] 
[10] 
Time (hr) 
Timearea histogram subareas (km^{2}) 
Partial flows (km^{2}cm/hr) for indicated rainfall increments 
Outflow (km^{2}cm/hr) 
Outflow (m^{3}/s) 
0.5 cm/hr 
1.0 cm/hr 
2.0 cm/hr 
1.5 cm/hr 
1.0 cm/hr 
0.5 cm/hr 
0 
 
0 
 
 
 
 
 
0 
0 
1 
10 
5 
0 
 
 
 
 
5 
13.9 
2 
30 
15 
10 
0 
 
 
 
25 
69.4 
3 
20 
10 
30 
20 
0 
 
 
60 
166.7 
4 
40 
20 
20 
60 
15 
0 
 
115 
319.4 
5 
 
0 
40 
40 
45 
10 
0 
135 
375 
6 
 
 
0 
80 
30 
30 
5 
145 
402.8 
7 
 
 
 
0 
60 
20 
15 
95 
263.9 
8 
 
 
 
 
0 
40 
10 
50 
138.9 
9 
 
 
 
 
 
0 
20 
20 
55.6 
10 
 
 
 
 
 
 
0 
0 
0 
Sum 
100 
 
 
 
 
 
 
650 
 
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 km^{2}cm/hr units.
Column 10 contains the values of Col. 9, converted to discharge units (m^{3}/s).
Note that the timearea method conserves mass exactly. For this example, the total rainfall is 6.5 cm and the
watershed area is 100 km^{2}. Thus, the total rainfall volume is: 100 × 6.5 = 650 km^{2}cm.
The sum of the ordinates of the runoff hydrograph (Col. 9) is 650 km^{2}cm/hr.
These are hourly ordinates; thus, the hydrograph volume is: 650 km^{2}cm/hr × 1 hr = 650 km^{2}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:
While the timearea method accounts for convection only (i.e, translation, or concentration), it has the distinct advantage that the watershed shape is reflected
in the timearea histogram and, therefore, in the runoff hydrograph.
Diffusion can be provided by routing the hydrograph calculated by the timearea method through a linear reservoir
with an appropriate storage constant.
Thus, a complete rainfallrunoff model using the timearea method, and adding diffusion,
would consist of two steps:
Use of the timearea method to provide pure convection, i.e., a translatedonly hydrograph.
Routing of the translatedonly 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:
O_{2} = C_{0} I_{2} + C_{1} I_{1} + C_{2} O_{1}
C_{0} = (Δt/K) / [2 + (Δt/K)]
C_{1} = C_{0}
C_{2} = [2  (Δt/K)] / [2 + (Δt/K)]

 (3) 
Table 3 shows the routing of the discharge hydrograph obtained with the timearea 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:
C_{0} = 0.2; C_{1} = 0.2;
C_{2} = 0.6.
Table 3.
Routing of a timearea hydrograph through a linear reservoir.

[1] 
[2] 
[3] 
[4] 
[5] 
[6] 
[7] 
Time (hr) 
Timearea hydrograph ordinates (km^{2}cm/hr) 
Partial flows (Eq. 3) (km^{2}cm/hr) 
Outflow from the linear reservoir (km^{2}cm/hr) 
Outflow (m^{3}/s) 
C_{0} I_{2} 
C_{1} I_{1} 
C_{2} O_{1} 
0 
0 
 
 
 
0 
0 
1 
5 
1 
0 
0 
1 
2.78 
2 
25 
5 
1 
0.6 
6.6 
18.33 
3 
60 
12 
5 
3.96 
20.96 
58.22 
4 
115 
23 
12 
12.58 
47.58 
132.17 
5 
135 
27 
23 
28.55 
78.55 
218.19 
6 
145 
29 
27 
47.13 
103.13 
286.47 
7 
95 
19 
29 
61.88 
109.88 
305.22 
8 
50 
10 
19 
65.93 
94.93 
263.69 
9 
20 
4 
10 
56.96 
70.96 
197.11 
10 
0 
0 
4 
42.58 
46.58 
129.37 
11 
0 
0 
0 
27.95 
27.95 
77.64 
12 
0 
0 
0 
16.77 
16.77 
46.58 
13 
0 
0 
0 
10.06 
10.06 
27.94 
14 
0 
0 
0 
6.04 
6.04 
16.77 
15 
0 
0 
0 
3.62 
3.62 
10.07 
16 
0 
0 
0 
2.17 
2.17 
6.03 
17 
0 
0 
0 
1.30 
1.30 
3.62 
18 
0 
0 
0 
0.78 
0.78 
2.17 
19 
0 
0 
0 
0.47 
0.47 
1.30 
20 
0 
0 
0 
0.28 
0.28 
0.78 
21 
0 
0 
0 
0.17 
0.17 
0.47 
22 
0 
0 
0 
0.10 
0.10 
0.28 
23 
0 
0 
0 
0.06 
0.06 
0.17 
24 
0 
0 
0 
0.04 
0.04 
0.10 
25 
0 
0 
0 
0.02 
0.02 
0.07 
Sum 
650 
 
 
 
650.00 
 
Note that the peak flow has decreased from 145 km^{2}cm/hr for time t = 6 hr at inflow (Col. 2),
to 109.88 km^{2}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 km^{2}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 timearea hydrograph (Table 2, Col. 9) through a linear reservoir (Table 3, Column 6).
Fig. 4 Effect of routing the timearea 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 1hr unit hydrograph corresponds to
1 cm [or in] or effective precipitation lasting 1 hr; the 2hr 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 timeareaderived unitrunoff
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):
 Drainage area A = 1,335 square miles;
 Time of concentration T_{c} = 6 days;
 Time interval Δt = 0.5 days = 12 hr;
 Hydrograph duration t_{r} = 12 hr;
and
 Linear reservoir storage constant K = 15.428 hr ≅ 15 hr.
Using Eq. 3: C_{0} = 0.28;
C_{1} = 0.28;
C_{2} = 0.44.
Fig. 5 Isochrones for the Appomattox River at Petersburg, Virginia (Clark 1945).
By definition, C_{0} = C_{1}. Furthermore, within one time step,
Clark used the unitrunoff hyetograph, wherein I_{2} = I_{1}. Thus, the
linear reservoir routing equation reduces to:
O_{2} = 2 C_{0} I_{2} + + C_{2} O_{1}

 (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] timearea 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 (mi^{2}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 mi^{2}in/hr, which, when integrated over the time interval (12 hr)
and distributed over the basin area (1,335 mi^{2}) 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 timearea histogram through a linear reservoir;
indeed one cannot route an area; only a discharge.
Effectively, Clark routed the inflow [unitrunoff 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).
Table 4.
Calculation of Clark's Appomattox River example.

[1] 
[2] 
[3] 
[4] 
[5] 
[6] 
[7] 
[8] 
[9] 
[10] 
Time (d) 
Time (hr) 
Timearea histogram
(%) 
Increment of area (mi^{2}) 
12 Inflow (mi^{2}in/hr) 
Inflow (mi^{2}in/hr) 
Partial flows (mi^{2}in/hr) 
Outflow (mi^{2}in/hr) 
Outflow (UH ordinates) (cfs) 
2 C_{0} I_{2} 
C_{2} O_{1} 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0.5 
12 
1.8 
24.030 
24.030 
2.003 
1.121 
0.000 
1.121 
723.673 
1 
24 
3.8 
50.730 
50.730 
4.228 
2.367 
0.493 
2.861 
1846.170 
1.5 
36 
6.9 
92.115 
92.115 
7.676 
4.299 
1.259 
5.557 
3586.395 
2 
48 
10.8 
144.180 
144.180 
12.015 
6.728 
2.445 
9.174 
5920.052 
2.5 
60 
19.1 
254.985 
254.985 
21.429 
11.899 
4.036 
15.936 
10283.798 
3 
72 
7.6 
101.460 
101.460 
8.455 
4.735 
7.012 
11.747 
7580.380 
3.5 
84 
6.5 
86.775 
86.775 
7.231 
4.050 
5.168 
9.218 
5948.631 
4 
96 
5.5 
73.425 
73.425 
6.119 
3.247 
4.056 
7.482 
4828.621 
4.5 
108 
9.0 
120.150 
120.150 
10.013 
5.607 
3.292 
8.899 
5742.958 
5 
120 
14.0 
186.900 
186.900 
15.575 
8.722 
3.916 
12.638 
8155.470 
5.5 
132 
9,5 
126.825 
126.825 
10.569 
5.919 
5.561 
11.479 
7407.792 
6 
144 
5.5 
73.425 
73.4250 
6.119 
3.427 
5.051 
8.477 
5470.652 
6.5 
156 
0 
0 
0 
0 
0 
3.730 
3.730 
2407.087 
7 
168 
0 
0 
0 
0 
0 
1.641 
1.641 
1059.118 
7.5 
180 
0 
0 
0 
0 
0 
0.722 
0.722 
466.012 
8 
192 
0 
0 
0 
0 
0 
0.318 
0.318 
205.045 
8.5 
204 
0 
0 
0 
0 
0 
0.140 
0.140 
90.220 
9 
216 
0 
0 
0 
0 
0 
0.062 
0.062 
039.697 
9.5 
228 
0 
0 
0 
0 
0 
0.027 
0.027 
17.467 
10 
240 
0 
0 
0 
0 
0 
0.012 
0.012 
7.685 
10.5 
252 
0 
0 
0 
0 
0 
0.005 
0.005 
3.382 
11 
264 
0 
0 
0 
0 
0 
0.002 
0.002 
1.488 
11.5 
276 
0 
0 
0 
0 
0 
0.001 
0.001 
0.655 
12 
288 
0 
0 
0 
0 
0 
0.00044 
0.00044 
0.288 
Sum 
 
100.00 
3315.000 
 
 
 
 
111.249 
71786.924 
Table 5 illustrates a more general computation of the Clark procedure,
one where the time interval Δt is not equal to the unit hydrograph duration t_{r}.
For this example, 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:
C_{0} = 0.2;
C_{1} = 0.2;
C_{2} = 0.6. To develop the flows,
the histogram subareas are multiplied by two (2) 1hr rainfall increments of 0.5 cm/hr each, and lagged appropriately.
The sum across Cols. 3 and 4, shown in Col. 5, is the discrete unitrunoff hyetograph,
with flow units (km^{2}cm/hr). The duration of the hyetograph is 5 hr, that is, the sum
of the time of concentration (4 hr) plus the unit hydrograph duration (2 hr) minus 1.
Table 5.
Calculation of the Clark unit hydrograph.

[1] 
[2] 
[3] 
[4] 
[5] 
[6] 
[7] 
[8] 
[9] 
Time (hr) 
Timearea histogram subareas
(km^{2}) 
Partial flows (km^{2}cm/hr) for indicated rainfall increments 
Sum (discrete unitrunoff hyetograph)
(km^{2}cm/hr) 
Partial flows (Eq. 4) (km^{2}cm/hr) 
Outflow from the linear reservoir (km^{2}cm/hr) 
Outflow (m^{3}/s) 
0.5 cm/hr 
0.5 cm/hr 
2 C_{0} I_{2} 
C_{2} O_{1} 
0 
 
 
 
 
 
 
0 
0 
1 
10 
5 
 
5 
2 
0 
2 
5.56 
2 
30 
15 
5 
20 
8 
1.2 
9.2 
25.56 
3 
20 
10 
15 
25 
10 
5.52 
15.52 
43.11 
4 
40 
20 
10 
30 
12 
9.31 
21.31 
59.19 
5 
 
 
20 
20 
8 
12.79 
20.79 
57.75 
6 
 
 
 
 
0 
12.47 
12.47 
34.65 
7 
 
 
 
 
0 
7.48 
7.48 
20.78 
8 
 
 
 
 
0 
4.49 
4.49 
12.47 
9 
 
 
 
 
0 
2.69 
2.69 
7.48 
10 
 
 
 
 
0 
1.61 
1.61 
4.488 
11 
 
 
 
 
0 
0.97 
0.97 
2.688 
12 
 
 
 
 
0 
0.58 
0.58 
1.62 
13 
 
 
 
 
0 
0.35 
0.35 
0.978 
14 
 
 
 
 
0 
0.21 
0.21 
0.58 
15 
 
 
 
 
0 
0.13 
0.13 
0.358 
16 
 
 
 
 
0 
0.08 
0.08 
0.22 
17 
 
 
 
 
0 
0.05 
0.05 
0.13 
18 
 
 
 
 
0 
0.03 
0.03 
0.08 
19 
 
 
 
 
0 
0.02 
0.02 
0.05 
20 
 
 
 
 
0 
0.01 
0.01 
0.03 
21 
 
 
 
 
0 
0.006 
0.006 
0.016 
22 
 
 
 
 
0 
0.004 
0.004 
0.011 
Sum 
100 
 
 
 
 
0 
100.00 
 
Note that the linear reservoir routing conserves mass exactly. In effect,
the sum of Col. 2 is 100 km^{2} and the effective rainfall is 1 cm; therefore, the
total rainfall volume is 100 km^{2}cm. The sum of the hydrograph ordinates in Col. 8 is
100 km^{2}cm/hr, which, when integrated over the time interval Δt = 1 hr,
gives 100 km^{2}cm.
9. PONCE'S CLARK UNIT HYDROGRAPH
Ponce's version of the Clark unit hydrograph differs slightly from the original Clark procedure.
Ponce applied the timearea method to a given unit rainfall, of intensity 1/t_{r},
to obtain a continuous
unit hydrograph based on the timearea 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 timearea 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: C_{0} = 0.2;
C_{1} = 0.2;
C_{2} = 0.6. To develop the flows,
the histogram subareas are multiplied by two (2) 1hr rainfall increments of 0.5 cm/hr each, and lagged appropriately.
The sum across Cols. 3 and 4, shown in Col. 5, is the translatedonly unit hydrograph, with flow units (km^{2}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 unitrunoff 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 6.
Calculation of the Clark unit hydrograph according to Ponce (1989).

[1] 
[2] 
[3] 
[4] 
[5] 
[6] 
[7] 
[8] 
[9] 
[10] 
Time (hr) 
Histogram subareas
(km^{2}) 
Partial flows (km^{2}cm/hr) for indicated rainfall increments 
Sum
(continuous unit hydrograph) (km^{2}cm/hr) 
Partial flows (Eq. 3) (km^{2}cm/hr) 
Outflow from the linear reservoir (km^{2}cm/hr) 
Outflow (m^{3}/s) 
0.5 cm/hr 
0.5 cm/hr 
C_{0} I_{2} 
C_{1} I_{1} 
C_{2} O_{1} 
0 
 
 
 
0 
 
 
 
0 
0 
1 
10 
5 
 
5 
1 
0 
0 
1 
2.78 
2 
30 
15 
5 
20 
4 
1 
0.6 
5.6 
15.55 
3 
20 
10 
15 
25 
5 
4 
3.36 
12.36 
34.33 
4 
40 
20 
10 
30 
6 
5 
7.42 
18.42 
51.17 
5 
 
 
20 
20 
4 
6 
11.05 
21.05 
58.47 
6 
 
 
 
0 
0 
4 
12.63 
16.63 
46.19 
7 
 
 
 
 
0 
0 
9.98 
9.98 
27.72 
8 
 
 
 
 
0 
0 
5.99 
5.99 
16.64 
9 
 
 
 
 
0 
0 
3.59 
3.59 
9.98 
10 
 
 
 
 
0 
0 
2.15 
2.15 
5.98 
11 
 
 
 
 
0 
0 
1.29 
1.29 
3.58 
12 
 
 
 
 
0 
0 
0.78 
0.78 
2.17 
13 
 
 
 
 
0 
0 
0.47 
0.47 
1.30 
14 
 
 
 
 
0 
0 
0.28 
0.28 
0.78 
15 
 
 
 
 
0 
0 
0.17 
0.17 
0.47 
16 
 
 
 
 
0 
0 
0.10 
0.10 
0.28 
17 
 
 
 
 
0 
0 
0.06 
0.06 
0.17 
18 
 
 
 
 
0 
0 
0.04 
0.04 
0.11 
19 
 
 
 
 
0 
0 
0.02 
0.02 
0.06 
20 
 
 
 
 
0 
0 
0.01 
0.01 
0.03 
21 
 
 
 
 
0 
0 
0.006 
0.006 
0.016 
22 
 
 
 
 
0 
0 
0.004 
0.004 
0.011 
Sum 
100 
 
 
 
 
 
0 
100.00 
 
Table 7 shows a sidebyside 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 unitrunoff 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.
Table 7.
Comparison of Clark's original unit hydrograph with Ponce's version.

[1] 
[2] 
[3] 
[4] 
[5] 
[6] 
[7] 
[8] 
[9] 
Time (hr) 
Partial flows (Eq. 3) (km^{2}cm/hr) 
Clark's original unit hydrograph (km^{2}cm/hr) 
Partial flows (Eq. 3) (km^{2}cm/hr) 
Ponce's Clark unit hydrograph (km^{2}cm/hr) 
C_{0} I_{2} 
C_{1} I_{1} 
C_{2} O_{1} 
C_{0} I_{2} 
C_{1} I_{1} 
C_{2} O_{1} 
0 
 
 
 
 
 
 
 
0 
1 
1 
1 
0 
2 
1 
0 
0 
1 
2 
4 
4 
1.2 
9.2 
4 
1 
0.6 
5.6 
3 
5 
5 
5.52 
15.52 
5 
4 
3.36 
12.36 
4 
6 
6 
9.31 
21.31 
6 
5 
7.42 
18.42 
5 
4 
4 
12.79 
20.79 
4 
6 
11.05 
21.05 
6 
0 
0 
12.47 
12.47 
0 
4 
12.63 
16.63 
7 
0 
0 
7.48 
7.48 
0 
0 
9.98 
9.98 
8 
0 
0 
4.49 
4.49 
0 
0 
5.99 
5.99 
9 
0 
0 
2.69 
2.69 
0 
0 
3.59 
3.59 
10 
0 
0 
1.61 
1.61 
0 
0 
2.15 
2.15 
11 
0 
0 
0.97 
0.97 
0 
0 
1.29 
1.29 
12 
0 
0 
0.58 
0.58 
0 
0 
0.78 
0.78 
13 
0 
0 
0.35 
0.35 
0 
0 
0.47 
0.47 
14 
0 
0 
0.21 
0.21 
0 
0 
0.28 
0.28 
15 
0 
0 
0.13 
0.13 
0 
0 
0.17 
0.17 
16 
0 
0 
0.08 
0.08 
0 
0 
0.10 
0.10 
17 
0 
0 
0.05 
0.05 
0 
0 
0.06 
0.06 
18 
0 
0 
0.03 
0.03 
0 
0 
0.04 
0.04 
19 
0 
0 
0.02 
0.02 
0 
0 
0.02 
0.02 
20 
0 
0 
0.01 
0.01 
0 
0 
0.01 
0.01 
21 
0 
0 
0.006 
0.006 
0 
0 
0.006 
0.006 
22 
0 
0 
0.004 
0.004 
0 
0 
0.004 
0.004 
Sum 
 
 
 
100.00 
 
 
0 
100.00 
Fig. 6 Comparison of Clark's original and Ponce's Clark unit hydrographs.
The program
Online Routing Clark
makes possible the calculation and sidebyside 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, timeareaderived, unitrunoff hyetograph,
while Ponce's procedure routes the continuous, timeareaderived, unit hydrograph.
Since the unit hydrograph has a longer time base than the unitrunoff 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, 14191446.
Hydrologic Engineering Center, 1990. HEC1, Flood Hydrograph Package. Davis, California.
Hydrologic Engineering Center, 2010. HECHMS, Hydrologic Modeling System. User's Manual, Version 3.5, August 2010, CDP74A, 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, 919.
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, 14611476.
Ponce, V. M. 1980. Linear reservoirs and numerical diffusion. Journal of the Hydraulics Division, ASCE, Vol. 106, No. HY5, May, 691699.
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.
