What This Solves
Generates a runoff hydrograph by convolving (superimposing) a rainfall excess time series with a unit hydrograph using the principle of superposition.
Best Used When
- You have a unit hydrograph and a rainfall distribution and need the resulting runoff hydrograph
- You are performing detailed hydrologic modeling with time-varying rainfall
- You want to see the full flow-over-time response of a watershed to a design storm
Do NOT Use When
- You only need the peak flow rate and not the full hydrograph shape — Use Rational Method Calculator
- You need to generate the unit hydrograph itself before convolving — Use SCS Unit Hydrograph Calculator
Key Assumptions
- The watershed response is linear (principle of superposition applies)
- Rainfall excess is uniform over the watershed for each time step
- The unit hydrograph is time-invariant (does not change during the storm)
- Base flow is not included in the convolution (added separately if needed)
- Rainfall excess has already been separated from total rainfall (losses removed)
Input Quality Notes
The quality of the result depends entirely on the unit hydrograph and the rainfall excess hyetograph. Verify that the unit hydrograph time step matches the rainfall increment duration.
Convolve a unit hydrograph with a series of excess rainfall increments to build a complete direct runoff hydrograph. The calculator applies discrete convolution and the principle of superposition per NRCS NEH Part 630, and can also generate an SCS dimensionless unit hydrograph from your time of concentration and drainage area.
Hydrograph Convolution
For educational purposes only. Not a substitute for professional engineering judgment.
How hydrograph convolution works
A unit hydrograph (UH) is the direct runoff response of a watershed to one unit of excess rainfall (1 inch in US units, 1 mm in SI) applied uniformly over a fixed duration. Convolution scales and lags a copy of the UH for each rainfall increment, then sums those responses. This works because a linear, time-invariant watershed obeys superposition: the total runoff equals the sum of the responses to the individual rainfall pulses.
The governing discrete convolution equation is:
Q(n) = Σm=1n Pe(m) · U(n − m + 1)
- Q(n) — direct runoff ordinate at time step n (cfs or m³/s)
- Pe(m) — excess rainfall depth during time step m (in or mm)
- U(n − m + 1) — unit hydrograph ordinate at the lagged index
The number of ordinates in the output hydrograph is fixed by the input lengths:
nout = M + N − 1
where M is the number of excess rainfall pulses and N is the number of unit hydrograph ordinates. Optional constant base flow is added to every ordinate after convolution, and total volume is found by trapezoidal integration of the discharge ordinates over the time step.
To generate a synthetic UH, the calculator uses the SCS method: lag time TL = 0.6 × Tc, time to peak tp = D/2 + TL (D is the unit duration / time step), and peak discharge qp = K · A / tp, with K = 484 (US) or 2.08 (SI). The shape comes from the dimensionless ratios in the table below.
SCS dimensionless unit hydrograph ratios
Standard NRCS dimensionless unit hydrograph ordinates from NEH Part 630, Chapter 16 (Exhibit 16-1). Multiply the time ratio by tp to get time, and the discharge ratio by qp to get the unit hydrograph ordinate. Selected points are shown; the calculator interpolates linearly between them.
| Time ratio (t / tp) | Discharge ratio (q / qp) |
|---|---|
| 0.0 | 0.000 |
| 0.2 | 0.100 |
| 0.4 | 0.310 |
| 0.6 | 0.660 |
| 0.8 | 0.930 |
| 1.0 | 1.000 |
| 1.2 | 0.930 |
| 1.4 | 0.780 |
| 1.6 | 0.560 |
| 1.8 | 0.390 |
| 2.0 | 0.280 |
| 2.4 | 0.147 |
| 3.0 | 0.055 |
| 4.0 | 0.011 |
| 5.0 | 0.000 |
The curve peaks at t / tp = 1.0 (q / qp = 1.0) and returns to zero at t / tp = 5.0. Roughly 37.5% of the runoff volume occurs on the rising limb, which corresponds to the standard peak rate factor of 484 in US customary units.
Worked example
Convolve the unit hydrograph U = [0, 100, 200, 150, 75, 25, 0] cfs (N = 7 ordinates, 30-min time step) with excess rainfall Pe = [0.5, 1.0, 0.5] in (M = 3 pulses):
- Output ordinates = M + N − 1 = 3 + 7 − 1 = 9
- Peak discharge occurs where the lagged responses overlap most. At t = 90 min (n = 3): 0.5×150 + 1.0×200 + 0.5×100 = 75 + 200 + 50 = 325 cfs
- Time to peak = 3 × 30 min = 90 minutes
This matches the hand-calculated convolution peak for the McCuen (2005) example used to verify the calculator.
Frequently asked questions
What is hydrograph convolution?
Convolution is the mathematical process of combining a series of excess (effective) rainfall increments with a unit hydrograph to build the complete direct runoff hydrograph. Each rainfall increment produces a scaled, lagged copy of the unit hydrograph, and these responses are summed (superposed) time step by time step. It is the standard way to turn a design storm into a flow-versus-time hydrograph.
What is the convolution equation?
The discrete convolution equation is Q(n) = Σ P(m) × U(n − m + 1), summed over m = 1 to n. Here Q(n) is the direct runoff ordinate at time step n, P(m) is the excess rainfall depth in time step m, and U(n − m + 1) is the unit hydrograph ordinate. The resulting hydrograph has M + N − 1 ordinates, where M is the number of rainfall pulses and N is the number of unit hydrograph ordinates.
What assumptions does unit hydrograph convolution rely on?
Convolution treats the watershed as a linear, time-invariant system. The three core assumptions are linearity (runoff is proportional to excess rainfall), superposition (responses from separate rainfall increments can be added), and time invariance (the unit hydrograph shape does not change from storm to storm). It also assumes rainfall is uniform over the watershed and constant within each time step. It does not capture nonlinear response, spatially variable rainfall, or changing infiltration during a storm.
How do I get the excess rainfall to convolve?
Convolution uses excess (effective) rainfall — the depth left after losses such as interception, depression storage and infiltration are removed. Compute it first with a loss method such as the NRCS curve number (CN) method, then enter the per-time-step excess depths. The unit hydrograph already represents the watershed transform, so the inputs you supply here must be the runoff-producing rainfall, not the gross rainfall.
What peak rate factor should I use to generate a unit hydrograph?
The standard NRCS peak rate factor is 484 in US customary units (cfs per inch, area in square miles, tp in hours) and 2.08 in SI units (m³/s per mm, area in km², tp in hours). The factor of 484 reflects a hydrograph where about 37.5% of runoff occurs before the peak. Flat, marshy or storage-dominated watersheds may justify a lower factor (down to roughly 300), while steep, rapidly responding watersheds may use a higher factor (up to about 600).
Standards & related tools
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Last verified: February 2026