This question involves understanding the concept of a unit hydrograph (UH) and how hydrographs combine when a rainfall event is split into sequential parts.

Understanding Unit Hydrographs and Staggering

A unit hydrograph is a hydrograph resulting from a direct runoff of a unit amount (typically 1 cm or 1 inch) of rainfall excess occurring uniformly over the drainage basin at a uniform rate for a specified duration.

In this scenario, we have a 2-hour unit hydrograph. This means it represents the basin's response to 2 hours of effective rainfall producing 1 cm of runoff.

When this 2-hour unit hydrograph is staggered by 2 hours, it implies that the rainfall event that generates this hydrograph is essentially treated as two consecutive 2-hour periods of rainfall excess, each producing 1 cm of runoff. The second pulse of rainfall excess starts 2 hours after the first.

Calculating the Resulting Hydrograph Characteristics

Let's denote the first 2-hour unit hydrograph as UH₁ and the second, staggered 2-hour unit hydrograph as UH₂.

• UH₁ starts at time $t=0$ and lasts for 2 hours. It represents 1 cm of runoff.
• UH₂ starts at time $t=2$ hours (staggered by 2 hours) and also lasts for 2 hours. It represents another 1 cm of runoff.

When these two hydrographs are added graphically, the resulting hydrograph's characteristics are determined as follows:

• Total Runoff Depth: The total runoff depth generated is the sum of the runoff from each pulse: $1 \text{ cm} + 1 \text{ cm} = 2 \text{ cm}$.
• Duration (Time Base): The time base of the resulting hydrograph is the duration of the first unit hydrograph plus the staggering interval. This is because the second hydrograph starts after the first one has begun, and the total time span covered is from the start of the first to the end of the second.

  Time base = (Duration of UH) + (Staggering Interval)

  Time base = $2 \text{ hours} + 2 \text{ hours} = 4 \text{ hours}$

Therefore, the resulting hydrograph represents the combined response to 2 cm of runoff, spread over a duration of 4 hours.

Conclusion

The combination of two 2-hour unit hydrographs, staggered by 2 hours, results in a hydrograph that reflects a total runoff depth of 2 cm and has a time base of 4 hours. This is described as a 4-hour unit hydrograph with 2 cm runoff, indicating the duration characteristics derived from the process and the total volume of runoff represented.