Problem 1: Baseflow Separation & Direct Runoff Hydrograph (DRH) A stream gauge records total hydrograph ordinates at 1-hour intervals: 15, 20, 50, 90, 60, 30, and $15 \text{ m}^3/\text{s}$. Using a constant baseflow separation method of $15 \text{ m}^3/\text{s}$, calculate the ordinates of the Direct Runoff Hydrograph (DRH). If the catchment area is $32.4 \text{ km}^2$, calculate the total depth of effective rainfall (direct runoff) in mm.

Problem 2: Unit Hydrograph Application A 2-hour Unit Hydrograph (UH) for a catchment has ordinates at 1-hour intervals of: $0, 10, 25, 15, 5, 0 \text{ m}^3/\text{s/cm}$. A storm produces $2 \text{ cm}$ of effective rainfall in the first 2 hours, and $3 \text{ cm}$ of effective rainfall in the next 2 hours. Calculate the resulting Direct Runoff Hydrograph (DRH).

Problem 3: S-Curve Method Derivation You are given a 2-hour Unit Hydrograph. You need to derive a 4-hour Unit Hydrograph for the same catchment. The S-Curve ordinates (derived from summing infinite 2-hr UHs lagged by 2 hours) are as follows at 1-hour intervals (t=0 to 6): $0, 10, 35, 60, 80, 95, 100$. Calculate the peak ordinate of the new 4-hour Unit Hydrograph.

Problem 4: Snyder's Synthetic Unit Hydrograph An ungauged catchment has an area ($A$) of $400 \text{ km}^2$. From a topographic map, the length of the main stream ($L$) is $30 \text{ km}$, and the distance from the outlet to a point on the stream opposite the basin centroid ($L_c$) is $15 \text{ km}$. Regional studies indicate that the coefficient $C_t = 1.5$ and $C_p = 0.6$. Calculate the basin lag ($t_p$) and the peak discharge ($Q_p$) of the standard Unit Hydrograph using Snyder's equations.

Case Study 1: The Principle of Superposition in Linear Systems The foundation of Unit Hydrograph theory relies on the catchment acting as a "linear system." Discuss why the Principle of Superposition is critical for predicting complex storm events and under what physical conditions this linearity assumption fails.

Case Study 2: Synthesizing Hydrographs for Ungauged Basins A civil engineer is designing a culvert for a highway crossing in a remote, mountainous region where no historical streamflow data (no stream gauges) exists. Discuss how the engineer can estimate the design flood hydrograph without recorded data.