Unit Hydrograph S Curve Calculator
Calculate unit hydrograph curve with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
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Where Tp is time to peak, D is unit duration, tL is lag time, qp is peak flow, A is watershed area, and the S-curve approaches equilibrium discharge equal to area times unit rainfall rate.
Last reviewed: December 2025
Worked Examples
Example 1: SCS Synthetic UH Derivation
Example 2: Duration Conversion via S-Curve
Background & Theory
The Unit Hydrograph S Curve Calculator applies the following established principles and formulas. Earth science calculators draw on a wide range of measurement scales and physical principles that quantify natural phenomena across geological, atmospheric, and hydrological systems. Earthquake magnitude is most precisely described by the Moment Magnitude Scale (Mw), which replaced the original Richter scale for larger events. Mw is calculated as Mw = (2/3) log10(M0) − 10.7, where M0 is the seismic moment in dyne-centimeters. The Richter scale, while still referenced colloquially, is a local magnitude (ML) measurement derived from peak seismograph amplitude at a standard 100 km distance. Wind intensity is classified using the Beaufort Scale, a 13-point empirical scale (0–12) relating wind speed in knots to observable sea and land effects, with Beaufort 12 corresponding to hurricane-force winds above 64 knots. Tropical cyclone intensity is further categorized by the Saffir-Simpson Hurricane Wind Scale, which assigns Categories 1 through 5 based on sustained wind speed, correlating with expected structural damage. Mineral hardness is quantified on the Mohs scale (1–10), comparing scratch resistance relative to reference minerals from talc (1) to diamond (10). Soil composition analysis measures the proportions of sand, silt, and clay by particle size, alongside organic matter content, bulk density, and porosity, which together determine engineering and agricultural suitability. Seismic wave velocity in rock varies by material: P-waves travel at approximately 5–7 km/s in granite and 1.5 km/s in water, while S-waves travel at roughly 60% of P-wave speeds. Atmospheric pressure decreases with altitude according to the barometric formula: P = P0 × exp(−Mgh / RT), where M is molar mass of air, g is gravitational acceleration, h is altitude, R is the universal gas constant, and T is temperature in Kelvin. Standard sea-level pressure is 101,325 Pa. Tidal calculations use harmonic analysis of gravitational forcing by the Moon and Sun, with the principal lunar semidiurnal tidal constituent (M2) having a period of approximately 12.42 hours.
History
The history behind the Unit Hydrograph S Curve Calculator traces back through the following developments. The systematic study of Earth's structure and processes spans millennia, but the scientific foundations were laid in the seventeenth century. In 1669, Danish naturalist Nicolas Steno published his principles of stratigraphy, establishing the laws of superposition, original horizontality, and lateral continuity — foundational rules for reading rock layers that remain in use today. Scottish geologist James Hutton introduced the concept of uniformitarianism in 1788, proposing that geological processes observable in the present have operated throughout Earth's history at broadly consistent rates. This idea of deep time challenged prevailing biblical chronologies and set the stage for modern geology. Charles Lyell systematized these ideas in his landmark three-volume work Principles of Geology, published beginning in 1830, which directly influenced Charles Darwin's thinking on biological evolution during the voyage of the Beagle. The nineteenth century saw growing curiosity about continental shapes, but a coherent theory awaited Alfred Wegener, a German meteorologist who proposed continental drift in 1912, arguing that the continents had once formed a supercontinent he called Pangaea. His evidence included matching fossil records and geological formations across the Atlantic, but his mechanism was disputed for decades. The theory gained acceptance in the 1960s when seafloor spreading was confirmed through paleomagnetic studies, and plate tectonics emerged as the unifying framework of modern geoscience. The United States Geological Survey was established by Congress in 1879 to classify public lands and examine the geological structure, mineral resources, and products of the national domain. The twentieth century brought instrumental advances, including the global seismograph network deployed after World War II, initially to monitor nuclear tests, which dramatically improved earthquake detection and characterization. Satellite Earth observation began in earnest with the Landsat program launched in 1972, enabling continuous global monitoring of land use, glacier retreat, and vegetation patterns. Today, GPS networks, LIDAR scanning, and ocean-floor mapping provide centimeter-scale precision for tracking tectonic motion, sea level rise, and volcanic deformation in near real time.
Frequently Asked Questions
Formula
Tp = D/2 + tL; qp = 2.08 * A / Tp; S-curve equilibrium = A / (D * 3600)
Where Tp is time to peak, D is unit duration, tL is lag time, qp is peak flow, A is watershed area, and the S-curve approaches equilibrium discharge equal to area times unit rainfall rate.
Worked Examples
Example 1: SCS Synthetic UH Derivation
Problem: Watershed area 50 km2, unit duration 2 hr, lag time 3 hr. Derive the SCS triangular UH.
Solution: Tp = 2/2 + 3 = 4 hr\nTr = 1.67 x 4 = 6.68 hr\nTb = 4 + 6.68 = 10.68 hr\nqp = 2.08 x 50 / 4 = 26.0 m3/s
Result: Tp = 4.00 hr | Qp = 26.0 m3/s | Tb = 10.68 hr
Example 2: Duration Conversion via S-Curve
Problem: Convert a 2-hr UH to a 4-hr UH. Original peak 25 m3/s, lag 3 hr, area 50 km2.
Solution: Peak ratio = D/Dt = 2/4 = 0.5\nAdjusted peak = 25 x 0.5 = 12.5 m3/s (approx)\nNew rise time = 4/2 + 3 = 5 hr
Result: Adjusted Peak: 12.5 m3/s | Rise Time: 5.0 hr | Ratio: 0.500
Frequently Asked Questions
What is a unit hydrograph in hydrology?
A unit hydrograph (UH) is the direct runoff hydrograph resulting from one unit of effective rainfall (typically 1 cm or 1 inch) applied uniformly over a watershed for a specified duration. It is a fundamental tool in flood hydrology that allows engineers to predict runoff hydrographs for any rainfall event by superposition and scaling. The concept assumes linearity (runoff is proportional to rainfall excess) and time invariance (the response shape remains constant). Sherman introduced the concept in 1932.
What is an S-curve in unit hydrograph theory?
An S-curve is the cumulative runoff response to a continuous uniform rainfall of unit intensity, constructed by lagging and summing successive unit hydrographs at intervals equal to the unit duration D. The S-curve rises from zero and asymptotically approaches an equilibrium discharge equal to the watershed area times the unit rainfall rate. The S-curve method is essential for converting a unit hydrograph from one duration to another, which is needed when the available UH duration does not match the desired computational time step.
How do you use S-curves to change unit hydrograph duration?
To derive a new-duration UH from an existing one, first construct the S-curve by successively lagging and summing the original UH at intervals of its duration D. Then create a second S-curve lagged by the desired new duration Dt. The difference between the two S-curves, divided by the ratio Dt/D, gives the new-duration unit hydrograph. This method preserves the volume of the unit hydrograph while adjusting its temporal distribution. It works for both increasing and decreasing the duration.
What is the SCS triangular unit hydrograph?
The SCS triangular unit hydrograph is a simplified synthetic UH approximated by a triangle with time to peak Tp = D/2 + tL (where D is duration and tL is lag time), and recession time Tr = 1.67 * Tp. The peak flow rate is qp = 2.08 * A / Tp where A is area in km2 and Tp in hours. This gives a base time Tb = 2.67 * Tp. The triangle has the same volume as 1 cm of runoff over the watershed area. It is the most widely used synthetic UH for ungauged watersheds.
What is lag time and how does it relate to the unit hydrograph?
Lag time (tL) is the time interval between the centroid of effective rainfall and the peak of the direct runoff hydrograph. In the SCS method, lag time is estimated as tL = L^0.8 * (S+1)^0.7 / (1140 * Y^0.5) where L is hydraulic length, S is potential retention from CN, and Y is average slope percent. The time to peak of the UH is Tp = D/2 + tL. Lag time is approximately 0.6 times the time of concentration for most watersheds.
What assumptions underlie unit hydrograph theory?
Unit hydrograph theory assumes linearity (the principle of proportionality and superposition applies), time invariance (the watershed response shape does not change with time or antecedent conditions), and uniform spatial distribution of rainfall excess. In reality, these assumptions are only approximately met since infiltration is nonlinear, watershed response changes with soil moisture, and rainfall is rarely uniform. Despite these limitations, UH methods remain practical and widely used for engineering design.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy