Instantaneous Unit Hydrograph (IUH) — relationship with the S-curve For a catchment, which statement correctly describes the ordinate of the IUH at any time t?

Introduction / Context:
The Instantaneous Unit Hydrograph (IUH) idealizes the direct runoff response to an infinitesimally short burst (unit impulse) of effective rainfall. It is central to linear systems analysis in hydrology, linking rainfall excess to runoff via convolution.

Given Data / Assumptions:

• Linear time-invariant watershed response.
• Unit hydrographs and S-curves are defined for unit depth of effective rainfall.
• Effective rainfall intensity considered: 1 cm/hr for developing the S-curve.

Concept / Approach:
The S-curve (or S-hydrograph) is formed by continuously applying a unit hydrograph (of given duration) such that 1 cm of excess rainfall is supplied each hour. The IUH is the time derivative of the S-curve corresponding to a unit-depth impulse. Therefore, the IUH ordinate at time t equals the slope (time derivative) of the S-curve generated by 1 cm/hr effective rainfall input.

Step-by-Step Solution:
1) Construct the S-curve by superposing unit hydrographs at 1-hour intervals for a 1 cm/hr effective rainfall.2) Recognize that differentiating this cumulative response yields the impulse response (IUH).3) Conclude: IUH(t) = d/dt [S-curve(t)] for the 1 cm/hr input.4) Select the statement that matches this definition.

Verification / Alternative check:
In linear systems terms, the S-curve is the step response; IUH is the impulse response; derivative of step equals impulse, validating the relationship.

Why Other Options Are Wrong:
(a) and (b) refer to slopes of finite-duration unit hydrographs, not the cumulative S-curve; (c) has no standard meaning in UH theory.

Common Pitfalls:
Confusing IUH with a short-duration unit hydrograph; mixing cumulative (S) and incremental (UH/IUH) concepts.

Final Answer:
The slope of the S-curve with effective rainfall intensity of 1 cm/hr