# Hydrograph as a function

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.

$Q = Q_{min} + (Q_{max} - Q_{min}) ( (\frac{t}{t_p}) ( \exp({1-\frac {t}{t_p}}) )) ^{\beta }$         (1)

The plot below shows a hydrograph where:

• Qmin = 1
• Qmax = 10
• tp = 1 (time of peak)
• $\beta$ = 5

Hydrograph defined by a mathematical function

The effect of the Qmin, Qmax and tp parameters is self explanatory; $\beta$ spreads out, or closes up, the hydrograph.

The influence of the beta parameter on 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:

$\frac{t}{t_p} = 1 \pm \frac{1}{\sqrt{\beta}}$

Hydrograph inflection points

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:

$Q = Q_0 t^a \exp(-bt)$         (2)

$a$ and $b$ are constants.

The hydrograph will peak when $t = \frac{a}{b}$ at a maximum flow of:

$Q_0 \left(\frac{a}{b} \right)^a exp(-a)$

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.

## 4 thoughts on “Hydrograph as a function”

1. AIi

Hi Tony,
Thanks for sharing your information. I have a question I appreciate if you could answer to me.
What is the best approach for producing a hydrograph with 100 and 500 year return period for a highly regulated catchment? I have a gauging station with 50 years data at the downstream of the catchment . The max flood in this period has just 40 year return period. I don’t want to use a rainfall run off model because I do not have enough information about the reservoirs within the catchment. Is it possible to build 100 and 500 years hydrographs from available time series? . The only method which I have found in text books is unit hydrograph method which is very simple and I am sure it is not enough accurate for this study. Do u have any idea?

Cheers,