Unit Hydrograph Method

Open Design Flood Estimation

Generate the full flood hydrograph — rising limb, peak, and recession — from a rainfall excess hyetograph using the Unit Hydrograph method. This guide covers the classical Sherman definition, synthetic unit hydrographs (Snyder, SCS, SANRAL HRU 1/72), duration conversion via the S-curve, convolution, and a worked South African example.

Overview

The Unit Hydrograph (UH) method is the standard linear technique for converting a storm’s rainfall excess (direct runoff depth) into a discharge hydrograph at the catchment outlet. Introduced by L.K. Sherman in 1932, it is still the workhorse of design hydrology for medium to large catchments where the full shape of the flood — not merely the peak — is required.

A unit hydrograph is the direct runoff hydrograph resulting from one unit depth (typically 1 mm or 1 inch) of uniformly distributed rainfall excess falling over a specified duration on the catchment. Because the method assumes linearity and time-invariance of the catchment response, the hydrograph for any storm can be built from the UH by scaling (for volume) and convolving (for temporal distribution).

Unit hydrographs are widely used in South Africa through the HRU 1/72 synthetic method (Midgley & Pitman, 1972), which classifies the country into nine veld-type zones with regional lag and peak-rate coefficients. Internationally, the SCS dimensionless unit hydrograph and Snyder’s synthetic UH remain standard.

Core concept

A catchment is treated as a linear, time-invariant system. For a unit rainfall excess of duration $D$:

• the direct runoff hydrograph (DRH) for any storm composed of multiple D-hour rainfall-excess blocks is obtained by scaling the UH by each block’s depth, lagging each scaled UH by the block’s start time, and summing — this is convolution.
• the peak discharge scales linearly with the total rainfall-excess depth.
• the time base of the DRH is invariant to the storm magnitude (for a fixed $D$).

The three foundational assumptions of linearity (Sherman, 1932) are:

1. Linearity of response — runoff ordinates are proportional to rainfall-excess depth.
2. Time-invariance — the UH shape does not change from storm to storm.
3. Spatial uniformity — rainfall excess is distributed uniformly over the catchment.

What the UH does not include

The unit hydrograph transforms rainfall excess (already net of losses) into direct runoff. It does not include baseflow, and it does not include the loss process itself. You must first compute rainfall excess — via the SCS method, a φ-index, or Green-Ampt — before applying the UH. See also DRH Method.

The unit hydrograph formula

For a unit hydrograph of duration $D$ with ordinates $u(t)$, the direct runoff hydrograph $Q(t)$ for a rainfall-excess hyetograph $P_e$ with $N$ D-hour blocks of depth $P_j$ is the discrete convolution:

$$Q(t) \;=\; \sum_{j=1}^{\min(t, N)} P_j \cdot u(t - j + 1)$$

Convolution — discrete form

Where $Q(t)$ is the discharge at time-step $t$, $P_j$ is the effective rainfall depth in the $j$-th time block (mm), and $u(k)$ is the $k$-th ordinate of the D-hour unit hydrograph (m³/s per mm).

In continuous form this is the convolution integral:

$$Q(t) \;=\; \int_{0}^{t} P_e(\tau) \cdot u(t - \tau) \,\mathrm{d}\tau$$

Convolution — continuous form

Synthetic unit hydrographs

When observed streamflow data are not available to derive a UH from observed storms, a synthetic unit hydrograph is constructed from catchment characteristics. Three families are in common use.

Snyder’s synthetic UH

Developed by F.F. Snyder (1938) for the Appalachian catchments of the US, Snyder’s method characterises the UH with a lag time and a peak discharge:

$$t_p \;=\; C_t \cdot (L \cdot L_c)^{0.3}$$

Snyder lag time

$$Q_p \;=\; \frac{C_p \cdot A}{t_p}$$

Snyder peak flow

Where $t_p$ = basin lag (hr), $L$ = mainstream length (km), $L_c$ = length to centroid (km), $A$ = area (km²), and $C_t$ and $C_p$ are regional coefficients calibrated from observed hydrographs ($C_t \approx 1.35 – 1.65$, $C_p \approx 0.56 – 0.69$ for Appalachian catchments). Different values apply to other regions and must be calibrated locally.

SCS dimensionless UH

The SCS (NRCS) dimensionless UH defines the unit hydrograph as a dimensionless curve of $Q/Q_p$ vs $t/t_p$, with a single shape applicable everywhere. Peak discharge and time to peak are:

$$t_p \;=\; \frac{D}{2} + T_L \qquad\qquad Q_p \;=\; \frac{C \cdot A}{t_p}$$

SCS time to peak and peak discharge

Where $C$ = 2.08 (SI: m³/s per mm per km² per hr) for standard catchments, and $T_L$ is the lag time (from rainfall centroid to peak), typically $T_L \approx 0.6\,T_c$.

The dimensionless shape is approximated by a triangle with $t_b = 2.67\,t_p$ (time base) for hand calculation, or by the smoother curvilinear form with $t_b = 5\,t_p$ for more detailed work. The constant $C$ can be adjusted between roughly 1.3 (flat) and 2.6 (steep) to reflect catchment responsiveness — this is the peak rate factor (PRF).

SANRAL HRU 1/72 synthetic UH (South Africa)

The South African synthetic unit hydrograph method — developed by Midgley & Pitman (HRU Report 1/72, 1972), with subsequent revisions in Report 1/85 — classifies South Africa into nine veld-type zones, each with calibrated regional coefficients $C_t$ and $K_u$:

$$T_L \;=\; C_t \cdot \left(\frac{L \cdot L_c}{\sqrt{S}}\right)^{m}$$

HRU 1/72 lag time

$$Q_p \;=\; \frac{K_u \cdot A}{T_L}$$

HRU 1/72 unit peak flow

Where $L$ is the main watercourse length (km), $L_c$ is the distance to the centroid (km), $S$ is the 10-85 slope (m/m), and $A$ is the catchment area (km²). $m \approx 0.36$ (regional exponent). The catchment index $I_c = (L \cdot L_c)/\sqrt{S}$ captures shape and steepness in a single parameter.

Representative HRU 1/72 coefficients by veld zone:

Zone	Description	$C_t$	$K_u$
1	Coastal tropical forest	0.99	0.261
2	Schlerophyllous bush	0.62	0.306
3	Mountain sourveld	0.35	0.277
4	Grasslands of the interior plateau (tall)	0.32	0.386
5	Highland sourveld and Dohne sourveld	0.21	0.351
5a	As Zone 5 but more responsive	0.53	0.488
6	Karoo	0.19	0.265
7	False karoo	0.19	0.315
8	Bushveld	0.19	0.367
9	Tall grassveld	0.13	0.321

The storm loss factor $k$ (the proportion of design rainfall lost to infiltration, interception and depression storage) is tabulated by veld zone and return period in HRU 1/72 Tables 4-5 through 4-13. Effective rainfall is $P_e = P_d \cdot (1 - k)$.

Source: Midgley & Pitman (1972), HRU Report 1/72, Table 4-1 (values are representative — consult SANRAL Drainage Manual Table 3.16 for full regional parameters).

Which synthetic method to use

For design flood estimation in South Africa, prefer the HRU 1/72 method as it is calibrated to SA veld zones and is the SANRAL-endorsed approach. The SCS dimensionless UH with local lag equations is useful for cross-checking and in regions where HRU 1/72 coefficients are not well-constrained. Snyder is primarily of historical interest in South African practice.

Duration conversion — the S-curve

A UH is defined for a specific unit duration $D$. If the design storm is discretised into blocks of a different duration $D'$, the UH must be converted. The S-curve (S-hydrograph) technique accomplishes this:

1. Sum an infinite series of lagged D-hour UHs, each lagged by $D$ hours — the result is the S-curve, the discharge from continuous 1 mm/D rainfall excess.
2. Lag the S-curve by $D'$ hours and subtract from the original S-curve to get a hydrograph of continuous rainfall for $D'$ hours.
3. Scale by $D/D'$ to convert from 1 mm/D intensity back to a 1 mm/D’ unit UH.

$$u_{D'}(t) \;=\; \frac{D}{D'} \left[\, S(t) \;-\; S(t - D') \,\right]$$

S-curve duration conversion

A practical rule: choose the computational time step $\Delta t \le 0.25\,T_c$ per HEC guidance, and use it as both the UH duration and the hyetograph block width to avoid S-curve conversion.

Convolution — building the DRH

Once a UH of matching duration is available, the direct runoff hydrograph is built by convolving the rainfall-excess hyetograph with the UH. For a storm with $N$ blocks of effective rainfall:

$$Q(n\,\Delta t) \;=\; \sum_{j=1}^{\min(n, N)} P_j \cdot u\bigl((n - j + 1)\Delta t\bigr)$$

Discrete convolution

Baseflow — typically estimated by simple straight-line separation on an observed hydrograph or as a recession constant for synthetic applications — is then added to the DRH to obtain the total streamflow hydrograph. For DRH Method computations the same convolution is used but with additional Muskingum routing through the channel reach.

Worked example

A 45 km² veld-zone-4 catchment (interior tall grasslands) in the Free State is analysed for a 1:100-year design flood using HRU 1/72. The SCS method has already been used to derive effective rainfall from a 165 mm, 6-hour design storm.

Given information

• Area: $A = 45$ km²
• Main watercourse length: $L = 14$ km
• Length to centroid: $L_c = 6.5$ km
• 10-85 slope: $S = 0.0085$ m/m
• Veld zone: 4 → $C_t = 0.32$, $K_u = 0.386$, $m = 0.36$
• Effective rainfall blocks (1-hour, mm): 4, 10, 18, 12, 6, 3

Step 1 — Catchment index

$$I_c \;=\; \frac{L \cdot L_c}{\sqrt{S}} \;=\; \frac{14 \times 6.5}{\sqrt{0.0085}} \;=\; \frac{91}{0.0922} \;=\; 987$$

Step 2 — Lag time

$$T_L \;=\; 0.32 \times (987)^{0.36} \;=\; 0.32 \times 11.86 \;=\; 3.80 \text{ hr}$$

Step 3 — Unit peak flow (per mm of rainfall excess)

$$Q_p \;=\; \frac{K_u \cdot A}{T_L} \;=\; \frac{0.386 \times 45}{3.80} \;=\; 4.57 \text{ m}^3/\text{s per mm}$$

Step 4 — UH shape

Using the SCS-style triangular approximation with $t_p = D/2 + T_L = 0.5 + 3.80 = 4.30$ hr and $t_b = 2.67\,t_p = 11.48$ hr, the UH ordinates at 1-hour intervals (from $Q = 0$ at $t=0$, linear rise to $Q_p$ at $t_p$, linear fall to 0 at $t_b$) are:

$t$ (hr)	0	1	2	3	4	4.3	5	6	7	8	9	10	11	11.5
$u$ (m³/s/mm)	0	1.06	2.13	3.19	4.25	4.57	4.00	3.36	2.72	2.07	1.43	0.79	0.15	0

Step 5 — Convolution

Using the discrete convolution $Q(n) = \sum_{j=1}^{6} P_j \cdot u(n - j + 1)$ with the 6 rainfall blocks and the 1-hour UH, the peak discharge works out to approximately:

$$Q_{\text{peak}} \;\approx\; P_3 \cdot u(1) + P_2 \cdot u(2) + P_1 \cdot u(3) + \ldots$$

For the combined storm, the peak is approximately 185 m³/s at $t \approx 6$ hr after the start of effective rainfall. Full convolution tables are produced automatically by HydroDesign’s Design Flood Estimation tool.

Cross-check with SDF

Always cross-check a UH-derived flood estimate against a regional method such as the SDF or empirical methods. For this example, the SDF 1:100 year estimate for a Region 1 veld-4 catchment of 45 km² is typically within ±30% of the UH result; larger divergences warrant investigation.

Limitations

• Linearity assumption. Real catchments are non-linear: peak discharge per unit rainfall excess tends to increase with storm magnitude, so UH-derived peaks for very large storms are usually conservative (low). Conversely, for very small storms the method may overestimate.
• Time-invariance. Seasonal changes in vegetation, channel geometry and antecedent moisture all change the UH shape. Using a fixed UH ignores these effects.
• Spatial uniformity of rainfall excess. For catchments larger than ~500 km², storm cells rarely cover the whole area uniformly, and spatial routing is better handled by a distributed or semi-distributed model.
• Dependence on loss estimation. The UH is applied to rainfall excess — any error in the loss model (e.g. CN, $\lambda$) propagates directly to the hydrograph.
• Synthetic UH coefficients. Regional coefficients ($C_t$, $K_u$, PRF) are calibrated against a limited set of catchments; extrapolation to atypical catchments (very small, very urbanised, highly karstic, etc.) should be done cautiously.
• Duration sensitivity. The unit duration $D$ must match the computational time step. Poor choice of $D$ (too coarse) smooths the peak; too fine a $\Delta t$ risks numerical noise.

Related tools and guides

• Rational Method — peak-only alternative for small catchments
• SCS Method — derive rainfall excess from storm rainfall
• DRH Method — UH convolution with Muskingum channel routing
• SDF Method — regional method for SA SANRAL projects
• Tc Calculator — time of concentration and lag-time methods
• Design Rainfall — storm depths for the chosen return period
• Design Storm — temporal distribution of the design hyetograph

References

• Sherman, L.K. (1932). Streamflow from rainfall by the unit-graph method. Engineering News-Record, 108, 501 – 505. (The foundational paper.)
• Snyder, F.F. (1938). Synthetic unit-graphs. Transactions of the American Geophysical Union, 19, 447 – 454.
• Midgley, D.C. & Pitman, W.V. (1972). Surface Water Resources of South Africa — Report No. 1/72. Hydrological Research Unit, University of the Witwatersrand, Johannesburg. (The SA synthetic UH method and veld-zone coefficients.)
• Hydrological Research Unit (HRU). (1985). Report 1/85 — Revision of the synthetic unit hydrograph method. University of the Witwatersrand.
• SANRAL. (2013). Drainage Manual (6th ed.). South African National Roads Agency, Pretoria. Chapter 3 — Hydrology; Section 3.6: Synthetic Unit Hydrograph Method.
• Chow, V.T., Maidment, D.R. & Mays, L.W. (1988). Applied Hydrology. McGraw-Hill, New York. Chapter 7 — Unit Hydrograph.
• USACE. (1994). Engineering and Design — Flood-Runoff Analysis, Engineer Manual EM 1110-2-1417. US Army Corps of Engineers, Washington DC. Chapter 6 — Unit Hydrograph.
• USDA-NRCS. (2007). National Engineering Handbook, Part 630, Chapter 16 — Hydrographs.

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Open Design Flood Estimation