Initial Publication Date: May 26, 2015

Application of Unit Hydrograph to Derive Runoff Hydrograph


The objective of this step is to learn how to use a unit hydrograph to derive a runoff hydrograph. To use this step, the user need to have observed excess rainfall data and the unit hydrograph for the duration of the excess rainfall. The output from this step is a direct runoff hydrograph.

Conceptual Student Learning Outcomes

  • Students demonstrate the understanding of unit hydrograph
  • Students can generate runoff hydrograph from excess rainfall by using unit hydrograph
  • Application of discrete convolution equation

Practical Student Learning Outcomes

  • Data analysis using MS Excel
  • Use of discrete form of convolution in MS Excel

Student Time

  • One hour

Reference Documents and Files


Data Inputs

  • Numerical Values: Excess rainfall hyetograph for a storm event
  • Numerical Values: Unit hydrograph for the watershed

The rainfall data and a 2.5 hour unit hydrograph for Hall Creek watershed in Indiana are provided in this Excel file: Unit Hydrograph Data (Excel 2007 (.xlsx) 10kB Nov10 14)

Data Outputs

  • Numerical Values: Direct Runoff Hydrograph

Required Hardware and Software

  • MS Excel

Related Steps


A graphical explanation of how a unit hydrograph is used to derive a direct runoff hydrograph is first provided. A unit hydrograph is a hydrograph resulting from one inch or one mm of rainfall falling uniformly over the total watershed area. For example, if a rainfall of P1 inches occurs during a time interval of Δt, the total runoff hydrograph is P1 multiplied by the total unit hydrograph, which is the blue curve (PUH_1) in the figure below. If the rainfall stops after Δt, this is the direct runoff hydrograph. If the rainfall continues with P2 inches (P2 may or may not be equal to P1) during the next time interval, the total unit hydrograph is again multiplied by the unit hydrograph to get the direct runoff hydrograph from P2 (PUH_2, shown in red). Because P2 occurs after Δt, the red curve is delayed or lagged by Δt in the figure. If the rainfall stops after P2, the total runoff hydrograph (green) is the addition of blue curve and red curve in the figure below. If the rainfall continues with P3, the same procedure is repeated with the P3 component of direct runoff hydrograph delayed by 2*Δt, and added to the red and blue curve to get the total runoff hydrograph.


Once the concept of how a unit hydrograph is used is clear from the graphical example, it is now easy to apply this concept in Excel. The data provided with this step has the following format (shown in figure below) for a direct runoff and unit hydrograph:


A direct runoff hydrograph can be obtained by using the discrete form of convolution integral as shown below.


Where Q is the runoff ordinate, P is the rainfall pulse and U is the unit hydrograph ordinate.

Application of this equation in Excel involves the following steps:

  1. Multiply the entire unit hydrograph (UH) by the first rainfall pulse (P1). Lets call this the PUH_1 hydrograph.
  2. Obtain the PUH_2 hydrograph by multiplying the unit hydrograph by the second rainfall pulse. Delay or lag Q2 by Δt.
  3. Repeat step 2 to obtain PUH_3, PUH_4,..., PUH_M, where M is the total number of rainfall pulses. In this dataset, M = 3 so we can stop after 3. Remember PUH_3 is delayed by additional Δt compared to PUH_2.
  4. Sum all the PUH columns (PUH_1 + PUH_2 + ....+ PUH_M) to get the direct runoff hydrograph. The total number of direct runoff ordinates (N) must be equal to U + M – 1. Where U is the total number of unit hydrograph ordinates.

Computation of PUH_1, PUH_2,..., PUH_M and the total runoff hydrograph is shown below for the given data. The yellow cells represent the unit hydrograph, the green cells represent the precipitation, purple cells represent PUH, and the blue cells represent the total runoff hydrograph. The runoff hydrograph is obtained by summing all the PUH columns. Note how the delay in each PUH is accomplished by just inserting one additional empty cell for each PUH. Because the last PUH (PUH_3) is lagged or delayed by 2 cells or 6Δt, the total runoff hydrograph will have 2 additional ordinates compared to the unit hydrograph ordinates.


OK, now you know how to apply unit hydrograph to derive a direct runoff hydrograph from rainfall data!

Additional Activities and Variants


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