Chapter TwoHydrograph Convolution and Deconvolution

UH Convolution

• Suppose we had an excess (net) rainfall hyetograph and a 1-hour UH for an area, and

• we’d like to know what the actual storm hydrograph might look like.

The Procedure

• The procedure is a little like the time-area method we dealt with earlier.

• If you have the excess rainfall intensity in one hour increments, and the 1-hour UH, you can multiply the 1-hour UH by the rainfall intensity for each time interval, lag each resulting UH by the appropriate amount, and sum.

The Convolution Equation

• The UH Convolution formula is

Where n is the time, Pi is the rainfall excess at time increment i and Ui is the unithydrograph ordinate at time increment i.

An Example

• I’ll go through the example in your text, first the way your author does it, then by a much easier method.

• Next slide.

You can do this with a spreadsheet—you write the UH ordinates down columns, each multiplied by the rainfall intensity for that hour, then lag each successive UH until the rain stops, then sum.

And for homework, we will do a unit hydrograph convolution to derive a storm hydrograph for area 1

The Intent

• So if we can make a UH, we can construct hydrographs based on hyetographs. While these aren’t perfect, they can be calibrated for watersheds, with a goal of being able to adequately predict flooding.

Deconvolution

• If we had an actual storm hydrograph, and an excess rainfall hyetograph, we could produce the UH from them.

• We can solve each line of the convolution equation we used above IN ORDER for the U ordinates because each successive line adds only one unknown.

• The example in the text, which we will do in detail, illustrates this.

The deconvolution method• Suppose we had a 4-hour rainfall that resulted in 7 hours of runoff,

and we have a hyetograph and hydrograph for each. To get a UH, write out the convolution equations for this situation:

You can solve these equations for the U ordinates one at a time, starting at the first.

Note that you wouldn’t have to solve the last three, because you’d already have all seven of the U ordinates.