Intensity-duration-frequency curve

There is a large number of empirical approaches that relate the intensity (I), the duration (t) and the return period (p), from fits to power laws such as:

• Sherman's formula,[4] with three parameters (a, c and n), which are a function of the return period, p:

${\displaystyle I(t)={\frac {a}{(t+c)^{n}}}}$

• Chow's formula,[5] also with three parameters (a, c and n), for a particular return period p:

${\displaystyle I(t)={\frac {a}{t^{n}+c}}}$

• Power law according to Aparicio (1997),[6] with four parameters (a, c, m and n), already adjusted for all return periods of interest:

${\displaystyle I(t,p)=a\cdot {\frac {p^{m}}{(t+c)^{n}}}}$

In hydrometeorology, the simple power law (taking ${\displaystyle \ c=0}$) is used according to Monjo (2016) as a measure of the time-structure of the rainfall:[2]

${\displaystyle I(t)={\frac {a}{t^{n}}}=I_{o}\left({\frac {t_{o}}{t}}\right)^{n}}$

where ${\displaystyle \ I_{o}}$ is defined as an intensity of reference for a fixed time ${\displaystyle \ t_{o}}$, i.e. ${\displaystyle \ a=I_{o}t_{o}^{n}}$, and ${\displaystyle \ n}$ is a non-dimensional parameter known as n-index. In a rainfall event, the equivalent to the IDF curve is called Maximum Averaged Intensity (MAI) curve.[7]

To get an IDF curves from a probability distribution, ${\displaystyle \ F(x)}$ it is necessary to mathematically isolate the precipitation${\displaystyle \ x}$, which is directly related to the average intensity ${\displaystyle \ I}$ and the duration ${\displaystyle \ t}$, by the equation ${\displaystyle \ x=It}$, and since the return period ${\displaystyle p}$ is defined as the inverse of ${\displaystyle \ 1-F(x)}$, the function ${\displaystyle \ f(p)}$ is found as the inverse of ${\displaystyle \ F(x)}$, according to:

${\displaystyle It=f(p)\quad \Leftarrow \quad p={\frac {1}{1-F(It)}}}$

• Power law with the return period, derived from the Pareto distribution, for a fixed duration ${\displaystyle \ t}$:

${\displaystyle \ I(p)=kp^{m}\quad \Leftarrow \quad F(It)=1-\left({\frac {kt}{It}}\right)^{1/m}=1-{\frac {1}{p}}}$

where the Pareto distribution constant has been redefined as${\displaystyle \ k'=kt}$, since it is a valid distribution for a specific duration of precipitation, ${\displaystyle \ x}$, it has been taken as${\displaystyle \ x=It}$.

• Function derived from the generalized Pareto distribution, for a given duration ${\displaystyle \ t}$:

${\displaystyle I(p)={\begin{cases}\mu +{\frac {\sigma }{m}}\cdot (p^{m}-1)\quad \Leftarrow \quad F(I)=1-\left(1+{\frac {m(I-\mu )}{\sigma }}\right)^{-1/m}=1-{\frac {1}{p}}&{\text{if }}m>0,\\\quad \mu +\sigma \ln(p)\quad \quad \Leftarrow \quad F(I)=1-\exp \left(-{\frac {I-\mu }{\sigma }}\right)=1-{\frac {1}{p}}&{\text{if }}m=0.\end{cases}}}$

Note that for ${\displaystyle \ m>0}$ y ${\displaystyle \ \mu ={\frac {\sigma }{m}}}$, the generalized Pareto distribution retrieves the simple form of the Pareto distribution, with ${\displaystyle \ k'={\frac {\sigma }{m}}}$. However, with ${\displaystyle \ m=0}$ the exponential distribution is retrieved.

• Function deduced from the Gumbel distribution and the opposite Gumbel distribution, for a given duration ${\displaystyle \ t}$:

${\displaystyle I(p)=\mu +\sigma \ln \left(\ln \left(1-{\frac {1}{p}}\right)\right)\quad \Leftarrow \quad \quad F(I)=\exp \left(-\exp \left(-{\frac {I-\mu }{\sigma }}\right)\right)=1-{\frac {1}{p}}}$

${\displaystyle I(p)=\mu +\sigma \ln(\ln p)\quad \quad \quad \quad \quad \Leftarrow \quad \quad F(I)=1-\exp \left(-\exp \left({\frac {I-\mu }{\sigma }}\right)\right)=1-{\frac {1}{p}}}$