How do you calculate an SCS unit hydrograph?

Find the time to peak, Tp = D/2 + 0.6 Tc, where D is the rainfall burst. The unit peak is qp = 0.208 A / Tp in m³/s per mm of runoff (A in km², Tp in hours), or 484 A / Tp in cfs per inch (A in square miles). Multiply the time column of the dimensionless table by Tp and the flow column by qp. A 12 km² catchment with Tc = 1.5 h gives Tp = 0.9 h and qp = 2.77 m³/s per mm.

What does the peak rate factor 484 mean?

It is the constant in qp = 484 A Q / Tp in US units, and it fixes the share of runoff volume under the rising limb at 3/8 of the total. Converted to SI it becomes 0.208 for m³/s, km², mm and hours. NRCS guidance uses lower factors, down to about 100 to 300, for flat or swampy catchments with a lot of storage, and higher ones, up to about 575, for steep terrain. The table on this page is the standard 484 shape.

What is the difference between a unit hydrograph and a design hydrograph?

A unit hydrograph is the catchment's response to one unit of effective rainfall. A design hydrograph is its response to the effective rainfall of the design storm. Convolution builds one from the other: each burst of effective rainfall produces a copy of the unit hydrograph scaled by its depth and lagged to its start time, and the copies are summed. Studio's generator instead draws the design hydrograph shape around a peak computed by another method.

When should I use Snyder or Clark instead of the SCS unit hydrograph?

Snyder's method was developed on Appalachian catchments and suits medium to large catchments with a defined main channel, where its coefficients Ct and Cp can be calibrated against gauged floods. Clark's method routes a time-area curve through a linear reservoir, so it represents catchment shape and storage explicitly; HEC-HMS offers it as a standard transform for basins with marked attenuation. For small, ungauged catchments the SCS shape is the common default.

What time step should I use for a unit hydrograph?

Use a time step no larger than one fifth of the time to peak, which is the usual guidance from Ponce and the HEC-HMS documentation. A coarser step describes the rising and falling limbs with too few ordinates, and when the hydrograph is convolved with a rainfall series the peak tends to fall between ordinates and is understated. For a catchment with Tp = 54 minutes, a 6 to 10-minute step is suitable.

Unit Hydrograph Calculator: SCS, Snyder and Clark Shapes

Hydrograph summary
Peak flow, Qp0.208 A Q / Tp, Q = 1 mm	2.77m³/s
Time to peak, Tp0.6 × Tc	0.90h
Base time, Tb5 Tp	4.50h
Volume, V	12 005m³
Runoff depth carried, V / ACompare with the storm's runoff depth	1.00mm
Ordinates	46 at 6 min

Hydrograph ordinates
Time (min)	Time (h)	Q (m³/s)
0	0.00	0.00
6	0.10	0.10
12	0.20	0.33
18	0.30	0.64
24	0.40	1.06
30	0.50	1.60
36	0.60	2.13
42	0.70	2.51
48	0.80	2.73
54	0.90	2.77
60	1.00	2.73
66	1.10	2.54
72	1.20	2.31
78	1.30	2.04
84	1.40	1.70
90	1.50	1.37
96	1.60	1.12
102	1.70	0.93
108	1.80	0.78
114	1.90	0.66
120	2.00	0.56
126	2.10	0.46
132	2.20	0.38
138	2.30	0.32
144	2.40	0.27

Using the calculator above

The calculator opens on the worked example below: a 12 km² catchment, Tc = 1.5 h, the SCS dimensionless shape, 1 mm of runoff and a 6-minute step. The 1 mm unit hydrograph peaks at 2.77 m³/s at 0.90 h and holds 12 005 m³.

Shape. SCS dimensionless, SCS triangular, Snyder, Clark, gamma, or a triangle from its rise time.
Peak from. For the SCS shapes and Snyder, a runoff depth Q sets the peak: 0.208 A Q / Tp for SCS (484 in US units), or Snyder’s unit peak times Q. Enter 1 mm or 1 inch for the unit hydrograph. Design peak scales the shape to another method’s peak; 45 m³/s gives the example’s second half. Clark, gamma and the triangle take a design peak only.
Catchment. Area and Tc (SCS, Snyder and Clark).
Shape parameters. Snyder: Ct, Cp, the channel lengths L and Lca, and an optional unit duration. Clark: R (blank uses 0.7 Tc). Gamma: n and k. Triangle: rise time and recession ratio.
Time step. In minutes, Tp/5 or finer.

The results list the peak, time to peak, base time, volume, runoff depth and the first 25 ordinates.

What a unit hydrograph is

Sherman (1932) introduced the unit hydrograph: the direct-runoff hydrograph at the outlet from one unit depth of effective rainfall, spread evenly over the catchment and falling for a fixed duration D. It rests on four assumptions: the catchment responds the same way in every storm, runoff scales linearly with effective rainfall, the responses to successive bursts add up, and the rain is spatially uniform.

With those assumptions, the runoff from any storm follows by discrete convolution of the effective rainfall $P_i$ with the unit hydrograph ordinates $U_j$ :

Q_k = \sum_{i=1}^{\min(k,\,N_P)} P_i \, U_{k-i+1}

Here $N_P$ is the number of rainfall pulses. A synthetic unit hydrograph gives you $U_j$ for an ungauged catchment from its area and response time instead of from recorded floods.

The SCS unit hydrograph

The NRCS (formerly SCS) dimensionless unit hydrograph in the National Engineering Handbook Part 630, Chapter 16 is the most widely used synthetic shape. It needs two numbers, the time to peak and the peak:

T_p = \frac{D}{2} + 0.6\,T_c

q_p = \frac{0.208\,A\,Q}{T_p} \quad \text{(SI)} \qquad\qquad q_p = \frac{484\,A\,Q}{T_p} \quad \text{(US customary)}

Symbol	Meaning	SI	US customary
$T_p$	Time from start of effective rainfall to peak	h	h
$D$	Duration of the unit rainfall burst	h	h
$T_c$	Time of concentration (lag $= 0.6\,T_c$ )	h	h
$q_p$	Peak discharge	m³/s	cfs
$A$	Catchment area	km²	mi²
$Q$	Runoff depth (1 for a unit hydrograph)	mm	in

The 0.208 is 484 converted: 484 × 0.02832 / (2.590 × 25.4) = 0.208. The curvilinear hydrograph ends at 5 Tp. Its triangular equivalent has the same volume, a recession of 1.67 Tp and a base of 2.67 Tp.

SCS dimensionless unit hydrograph table

Multiply $t/T_p$ by $T_p$ and $q/q_p$ by $q_p$ to get the dimensional unit hydrograph (peak rate factor 484):

t/Tp	q/qp	t/Tp	q/qp	t/Tp	q/qp
0.0	0.000	1.1	0.990	2.4	0.147
0.1	0.030	1.2	0.930	2.6	0.107
0.2	0.100	1.3	0.860	2.8	0.077
0.3	0.190	1.4	0.780	3.0	0.055
0.4	0.310	1.5	0.680	3.2	0.040
0.5	0.470	1.6	0.560	3.4	0.029
0.6	0.660	1.7	0.460	3.6	0.021
0.7	0.820	1.8	0.390	3.8	0.015
0.8	0.930	1.9	0.330	4.0	0.011
0.9	0.990	2.0	0.280	4.5	0.005
1.0	1.000	2.2	0.207	5.0	0.000

Worked example: 12 km² catchment

A 12 km² rural catchment has Tc = 1.5 h from the time of concentration calculator. For a short burst the D/2 term is small (3 minutes for a 6-minute burst), so the generator takes $T_p = 0.6 \times 1.5 = 0.90$ h.

q_p = \frac{0.208 \times 12 \times 1}{0.90} = 2.77\ \text{m}^3/\text{s per mm}

Running the SCS dimensionless shape with that peak and a 6-minute step gives the 1 mm unit hydrograph (selected ordinates):

Time (min)	0	18	36	54	72	90	108	126	144	180	216	270
Q (m³/s)	0	0.64	2.13	2.77	2.31	1.37	0.78	0.46	0.27	0.09	0.03	0

The volume under the curve is 12 005 m³, which is 1.00 mm over 12 km², so the shape and the peak are consistent. The SCS triangle with the same peak ends at 2.40 h and holds 11 996 m³.

Now scale the same shape to a design peak. If a peak-flow method gives 45 m³/s at this outlet, the generator returns a hydrograph holding 194 800 m³, which is 16.2 mm of runoff over the catchment. Compare that implied depth with the runoff depth of the design storm, from the curve number method for example. If the storm produces more runoff than the hydrograph carries, the hydrograph understates the volume a pond downstream would have to store.

Other synthetic shapes in the generator

SCS triangular. Rise Tp, recession 1.67 Tp, base 2.67 Tp. Use it for hand checks.
Snyder (1938). In SI the lag is $t_L = 0.75\,C_t (L\,L_{ca})^{0.3}$ hours with $L$ and $L_{ca}$ in km, the unit peak $q_p = 0.275\,C_p A / t_L$ m³/s per mm, and the widths at 50 % and 75 % of the peak $W_{50} = 0.18\,(q_p/A)^{-1.075}$ and $W_{75} = 0.10\,(q_p/A)^{-1.075}$ hours, one third before the peak and two thirds after it (FHWA HDS-2, 2002). The generator joins those seven points, closes 1 mm of runoff on the recession, and scales the shape to your design peak. Without $L$ or $L_{ca}$ the lag falls back to $0.6\,T_c$ . For HDS-2’s Example 6.9 (2 266 km², $L$ = 133.6 km, $L_{ca}$ = 65 km, $C_t$ = 1.32, $C_p$ = 0.63, 3-hour unit) it gives $t_L$ = 15.0 h, $q_p$ = 26.0 m³/s per mm, $W_{50}$ = 21.9 h and $W_{75}$ = 12.2 h, the published values.
Clark (1945). Rainfall reaches the outlet through a time-area curve (a standard curve or your own), then passes through a single linear reservoir with storage coefficient R, which attenuates and delays the peak.
Gamma. A two-parameter gamma function, $q \propto (t/k)^{n-1} e^{-t/k}$ , with its peak at $k(n-1)$ . With the defaults and a Tc, the generator sets $n = 3$ and $k = T_c/3$ , which puts the peak at $\tfrac{2}{3}T_c$ .
Simple triangle. Peak and rise time only, with a recession of 1.67 times the rise by default. It needs neither area nor Tc.

Choosing a shape

Start with the SCS dimensionless shape for small, ungauged catchments; it is the one most reviewers expect and the shape behind NRCS TR-20 and the HEC-HMS SCS unit hydrograph transform. Use Clark where the catchment is long and narrow, or holds storage that delays the peak, and you can estimate R. Use Snyder where regional Ct and Cp values have been fitted to gauged floods. Draw at least two shapes on the same peak before you choose: they carry different volumes, and for a pond or a reservoir the volume governs the design more than the peak does.

Inputs and where they come from

Area: from watershed delineation.
Time of concentration: from a Tc method suited to the catchment.
Design peak: from a peak-flow method such as the TR-55 calculator, the Rational Method or a flood frequency analysis at a gauged site.
Time step: Tp/5 or finer.
Snyder: Ct, Cp, the main-channel length L (the longest watercourse) and the distance from the outlet to the point opposite the catchment centroid, Lca; optionally the unit duration. Clark: the storage coefficient R in hours.

Limitations and when to use something else

Linearity. Doubling the effective rainfall doubles the runoff. Observed peaks in large floods can be 10 to 30 % higher than linear theory predicts, because flow spreads overbank and normally dry areas start to contribute.
Uniform rainfall. The method assumes rain falls evenly over the whole catchment. Once a catchment is larger than a typical storm cell, use a distributed rainfall-runoff model.
Fixed response. Urbanisation and land-use change shift the unit hydrograph, so a synthetic shape fitted to old conditions goes out of date.
Inputs dominate. Design rainfall depth, curve number and Tc carry more uncertainty than the choice between SCS, Snyder and Clark. Spend effort there first.
Design floods in South Africa. The HRU (1972) regional unit hydrograph, SDF and the regional statistical methods are run together in design flood estimation.

What a free Studio account adds

Results on this page are free. A free account opens the same calculation in Studio’s Hydrograph Generator, which:

takes its peak from a saved Rational, TR-55, flood frequency, RMF or QRT result, its Tc from a saved Tc calculation and its area from a saved watershed delineation, and records the link to each;
draws several shapes on one chart, with a table of their peak, time to peak, base time and volume;
lists every ordinate and copies the table for a spreadsheet;
sends the hydrograph to flood routing or a dam safety evaluation as the inflow;
saves to a project and produces Word and Excel reports;
in the workbench layout, now in preview, exports the ordinates to CSV, EPA SWMM (.dat), HEC-HMS (.gage), HEC-RAS unsteady flow (.csv) and InfoWorks ICM (.csv).

Unit hydrograph calculator