Given The Following 2-Hr Unit Hydrograph Calculate The 1-Hr Hydrograph

When designing stormwater management systems, hydrologists often need to convert unit hydrographs between different time intervals. This guide explains how to derive a 1-hour unit hydrograph from a given 2-hour unit hydrograph using mathematical transformations.

Introduction

Unit hydrographs are fundamental tools in hydrology that represent the runoff response of a watershed to a unit of effective rainfall. When you have a 2-hour unit hydrograph and need a 1-hour version, you're essentially changing the time base of the hydrograph.

This transformation is necessary when:

Your rainfall data is available in 1-hour increments
You need to model stormwater systems with different time steps
You're comparing results with other studies using different time bases

Note: This method assumes the rainfall intensity is constant over each time interval. For more complex rainfall patterns, additional adjustments may be needed.

Methodology

The process involves three main steps:

Divide the 2-hour hydrograph into two 1-hour hydrographs
Apply the S-curve transformation to each 1-hour hydrograph
Combine the transformed hydrographs

Step 1: Divide the 2-hour hydrograph

First, split the 2-hour hydrograph into two equal parts. If the original hydrograph has values Q₁ and Q₂ for the first and second hours, the divided hydrographs will be:

First 1-hour hydrograph: Q₁/2 and Q₂/2

Second 1-hour hydrograph: Q₁/2 and Q₂/2

Step 2: Apply S-curve transformation

The S-curve transformation accounts for the fact that rainfall intensity changes when converting between different time intervals. The transformation formula is:

Transformed hydrograph = Original hydrograph × (1 + Δt/2)

Where Δt is the time step change (in this case, from 2 hours to 1 hour)

Step 3: Combine the hydrographs

After transforming both 1-hour hydrographs, you can combine them to get the final 1-hour unit hydrograph. The peak flow will be the sum of the peaks from both transformed hydrographs.

Worked Example

Let's work through an example with a simple 2-hour unit hydrograph:

Time (hours)	Runoff (cfs)
0-1	10
1-2	20

Step 1: Divide the hydrograph

First 1-hour hydrograph: 5 and 10

Second 1-hour hydrograph: 5 and 10

Step 2: Apply S-curve transformation

Using Δt = 1 hour (since we're converting from 2-hour to 1-hour):

First hydrograph transformed: 5 × (1 + 1/2) = 7.5 and 10 × (1 + 1/2) = 15

Second hydrograph transformed: 5 × (1 + 1/2) = 7.5 and 10 × (1 + 1/2) = 15

Step 3: Combine the hydrographs

The final 1-hour unit hydrograph will have:

First hour: 7.5 + 7.5 = 15 cfs
Second hour: 15 + 15 = 30 cfs

FAQ

Why do I need to convert between different time intervals?: Different time intervals are used for different purposes. Rainfall data might be available in hourly increments, while runoff models might require daily values. Converting between these time bases allows for consistent analysis across different datasets.
What if my original hydrograph has more than two time periods?: For hydrographs with more than two time periods, you would need to apply the same transformation to each pair of time periods. The process remains the same, but you'll need to perform the calculations for each segment.
How accurate is this method?: This method provides a reasonable approximation for many practical applications. However, for highly accurate results, especially in complex watersheds, more sophisticated methods like the Clark or Nash cascade models may be needed.