OpenHydroLab · The catchment as a linear system: identify its
response (derive), use it (apply), re-express it (S-curve)
📋 Given: observed storm hydrograph at the gauge
(A = 423 km²) and, from the rain gauges, 3.0 cm of effective rainfall in 6 h.
Step through the derivation
Your judgment call
Aha demo: baseflow separation is a judgment call —
set it wrong (try 5 m³/s) and step 3's runoff depth no longer reconciles with the
3.0 cm the rain gauges measured. The ✗ tells you your separation, not the rain,
is at fault.
Storm pattern
Changing Dᵣ automatically switches to the
Dᵣ-hour UH (built by the S-curve method of mode 3) — a UH may only be applied to
pulses of its own duration.
Show
Aha demo: the total flood is the colored bands stacked —
each band is one rain pulse pushed through the same UH, lagged by when it fell. Drag R₂
to zero and back: its whole band vanishes while the others stand still (linearity +
superposition). Then shorten Dᵣ: the same rain depths arrive faster, the UH in use
gets peakier, and the flood peak climbs.
Target duration
Show
Aha demo: drag D′ from 1 h to 24 h and watch the UH morph:
short-duration UHs are tall and peaky, long ones low and flat, volume pinned at 1 cm.
At D′ = 12, 18, 24 h the superposition method (sum n lagged 6-h UHs, ÷ n) draws
its own white dashed curve — it lands exactly on the S-curve result, because
superposition IS the S-curve algebra restricted to integer multiples. Now set
D′ = 9 h: superposition has no answer, and that is precisely why the S-curve
method exists.
The catchment
Area A
423 km²
UH duration D
6 h
1 cm over A
4.23 ×10⁶ m³
Derivation ledger
Chosen baseflow
DRH volume V
Runoff depth d = V/A
Rain-gauge ER (given)
3.00 cm
Reconciles?
Derived UH peak
Derived UH volume
Predicted flood
UH in use
UH_Dᵣ peak
Total ER
Peak discharge
Time of peak
DRH volume ∫Q_d dt
= ΣRᵢ × A
S-curve & new UH
Equilibrium Q_eq
= A·(1 cm)/6 h
D′-hour UH peak
Time of peak
(6-h UH peak, for comparison)
D′-UH volume ÷ A
Superposition method
Underlying response: a smooth gamma IUH (Nash cascade,
n = 4, k = 3 h) scaled to 423 km². All three modes are computed from this one function,
so deriving, applying and re-expressing are exactly consistent — as linear-systems
theory says they must be.