Hydrographs are usually based on measured data but it can be useful to be able to express a hydrograph as a mathematical function. A colleague, John Fenton, sent me this equation.
The plot below shows a hydrograph where:
- Qmin = 1
- Qmax = 10
- tp = 1 (time of peak)
- = 5
The effect of the Qmin, Qmax and tp parameters is self explanatory; spreads out, or closes up, the hydrograph.
For some techniques, such as the unit hydrograph, it is important to be able to find the “points of contraflexure” or inflection points of the curve. These are the points, on either side of the peak, where the gradient is at a maximum. Their location can be determined by setting the second derivative of equation 1 to zero.
The two solutions are:
The inflection points can also be calculated numerically as shown in this gist which includes the code to produce the graphs.
Other hydrograph functions are provided in Yevdjevich (1959). The most useful is:
and are constants.
The hydrograph will peak when at a maximum flow of:
Yevdjevich, V. M. (1959) Analytical integration of the differential equation for water storage. Journal of Research of the National Bureau of Standards – B. Mathematics and Mathematical Physics. 63(1):43-52.