Unit Hydrograph Transfer to an Ungauged Tributary
A 240 km² gauged catchment has 12 years of record and a derived 1-hour unit hydrograph with peak 38 m³/s and time-to-peak 6 h. A proposed 95 km² tributary within the same physiographic region has no streamflow record. You must estimate its 1-hour unit hydrograph for a dam spillway design. The tributary is steeper (mean slope 0.041 vs 0.018) and more impervious (urban fraction 0.22 vs 0.06). The gauged catchment's baseflow index is 0.48; regional regression suggests the tributary's baseflow index is 0.31. A colleague proposes simple area-ratio scaling of the gauged unit hydrograph. A reviewer questions whether the transfer is defensible.
1. Evaluate the area-ratio transfer and state what adjustments the physical differences require.
Read the worked answer
Area-ratio scaling alone assumes hydrologic similarity, which the data contradict. Area ratio gives peak = 38 × (95/240) = 15.0 m³/s and time-to-peak unchanged at 6 h. But the tributary is steeper and 3.7× more urbanized, so response should be faster and peakier. Steeper slope and higher imperviousness shorten time-to-peak and raise peak; lower baseflow index (0.31 vs 0.48) means a larger direct-runoff fraction, further raising the unit-hydrograph peak. The area-ratio estimate is therefore biased low on peak and long on time-to-peak. Defensible transfer requires: (1) scaling time-to-peak by a slope/urbanization adjustment (e.g., time-to-peak ∝ L/√S, plus an imperviousness factor), (2) rescaling peak to preserve unit volume, and (3) documenting that the tributary lies within the calibrated range of the regional method. The transfer is defensible only as a screening estimate, not as a design hydrograph without uncertainty bounds.
2. The tributary unit hydrograph must conserve 1 mm of direct runoff. Show the volume check and explain why it constrains the peak/time-to-peak pair.
Read the worked answer
One mm over 95 km² = 0.001 m × 95×10⁶ m² = 95,000 m³. A triangular unit hydrograph has volume = ½ × peak × base. If time-to-peak is 4 h and base is taken as 2.67× time-to-peak ≈ 10.7 h, then peak = 2 × 95,000 / (10.7 × 3600) ≈ 4.9 m³/s. Any peak chosen must satisfy this volume identity given the base length. This is why peak and time-to-peak cannot be adjusted independently: steepening the response (shorter time-to-peak) raises the peak for the same volume. The area-ratio peak of 15.0 m³/s with 6 h time-to-peak implies a much larger base than the tributary's flashy response supports, confirming inconsistency. The volume check is the guardrail against an over-peaked or under-peaked transferred hydrograph.
3. What would you report as the defensible design statement, and what residual uncertainty remains?
Read the worked answer
Report a range, not a single hydrograph: a best estimate with lower/upper bounds reflecting transfer uncertainty. State the assumptions explicitly (hydrologic similarity within the region, adjusted for slope and imperviousness), the volume-conserving derivation, and the baseflow-index correction. Residual uncertainty includes: ungauged peak-flow bias in regional methods (often ±30–50%), the unverified time-to-peak adjustment, and the absence of local calibration. Recommend a sensitivity run and, if the spillway is safety-critical, a short gauging program or rainfall-runoff model calibration before final design. The honest answer is a bounded estimate with stated limitations, not a false-precision single number.