Unit Hydrograph S Curve Calculator

A unit hydrograph is the outlet response of a catchment to one unit of effective rainfall, conventionally 1 cm or 1 inch, spread uniformly over the basin during a fixed duration D. Its usefulness rests on two assumptions: proportionality, so 3 cm of excess produces ordinates three times as large, and superposition, so a multi-period storm response is the sum of lagged and scaled unit responses. Every unit hydrograph is tied to the duration for which it was derived, and that is the problem the S-curve solves. Lagging copies of a D-hour unit hydrograph at intervals of D and summing them yields the response to continuous unit-intensity rain, a curve that climbs and flattens at an equilibrium discharge equal to catchment area times the rainfall rate. Offsetting that S-curve by a new duration Dt, differencing, and rescaling by D/Dt produces a unit hydrograph of the new duration with the volume preserved. The synthetic side of this calculation uses the SCS triangular approximation, where time to peak is Tp = D/2 + tL, recession runs 1.67 times Tp, and peak discharge follows qp = 2.08 A / Tp for area in square kilometers and Tp in hours. The 2.08 coefficient is fixed by requiring the triangle area to equal 1 cm of runoff, and it embeds the assumption that about 37.5 percent of runoff volume falls under the rising limb. Limitations are real: response is not truly linear at extreme discharges when floodplain storage engages, uniform rainfall is unrealistic over large basins, and S-curves built from noisy hydrographs commonly oscillate and need smoothing.

The unit hydrograph was introduced by LeRoy K. Sherman, an American engineer, in a 1932 article in Engineering News-Record titled around the derivation of streamflow from rainfall by the unit graph method. Sherman's insight was that a catchment could be treated as a fixed linear operator, so that a single characteristic response curve derived from one well-documented storm could be scaled and superposed to forecast others. It was an extraordinarily economical idea for an era with limited streamgauge records and no computation beyond arithmetic, and it remains the conceptual backbone of engineering flood hydrology. The method's obvious constraint, that a unit graph is valid only for its own rainfall duration, drove the development of the S-curve or summation curve technique in the following decade as a way to transfer a unit hydrograph between durations. Attention then turned to ungauged basins, where no observed hydrograph exists to derive from. Franklin Snyder published a synthetic unit hydrograph based on regional coefficients for Appalachian catchments in 1938, C. O. Clark introduced a routing approach using a time-area curve and a linear reservoir in 1945, and the Soil Conservation Service developed the dimensionless and triangular unit hydrographs used here, documented in the National Engineering Handbook. James Nash added the cascade of linear reservoirs in 1957, giving the instantaneous unit hydrograph an analytical form. These procedures are now embedded in modeling packages such as HEC-HMS.