An intensity-duration-frequency curve (IDF curve) is a mathematical function that relates the
rainfall
Rain is water droplets that have condensed from atmospheric water vapor and then fall under gravity. Rain is a major component of the water cycle and is responsible for depositing most of the fresh water on the Earth. It provides water f ...
intensity with its
duration
Duration may refer to:
* The amount of time elapsed between two events
* Duration (music) – an amount of time or a particular time interval, often cited as one of the fundamental aspects of music
* Duration (philosophy) – a theory of time and ...
and
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of occurrence.
These curves are commonly used in
hydrology
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...
for
flood forecasting and
civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewage ...
for
urban drainage design. However, the ''IDF curves'' are also analysed in
hydrometeorology
Hydrometeorology is a branch of meteorology and hydrology that studies the transfer of water and energy between the land surface and the lower atmosphere. Hydrologists often use data provided by meteorologists. As an example, a meteorologist might ...
because of the interest in the ''time concentration'' or ''time-structure'' of the
rainfall
Rain is water droplets that have condensed from atmospheric water vapor and then fall under gravity. Rain is a major component of the water cycle and is responsible for depositing most of the fresh water on the Earth. It provides water f ...
.
[(pdf)]
/ref> Additionally, Heidari et al., (2020) recently developed IDF curves for drought events
Mathematical approaches
The IDF curves can take different mathematical expressions, theoretical or empirically fitted to observed rainfall data. For each duration (e.g. 5, 10, 60, 120, 180 ... minutes), the empirical cumulative distribution function (ECDF), and a determined frequency or return period A return period, also known as a recurrence interval or repeat interval, is an average time or an estimated average time between events such as earthquakes, floods, landslides, or river discharge flows to occur.
It is a statistical measurement typ ...
is set. Therefore, the empirical IDF curve is given by the union of the points of equal frequency of occurrence and different duration and intensity Likewise, a theoretical or semi-empirical IDF curve is one whose mathematical expression is physically justified, but presents parameters that must be estimated by empirical fits.
Empirical approaches
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, with three parameters (''a'', ''c'' and ''n''), which are a function of the return period, ''p'':
::
* Chow's formula, also with three parameters (''a'', ''c'' and ''n''), for a particular return period ''p'':
::
* Power law according to Aparicio (1997),[Aparicio, F. (1997): Fundamentos de Hidrología de Superficie.
Balderas, México, Limusa. 303 p.] with four parameters (''a'', ''c'', ''m'' and ''n''), already adjusted for all return periods of interest:
::
In hydrometeorology
Hydrometeorology is a branch of meteorology and hydrology that studies the transfer of water and energy between the land surface and the lower atmosphere. Hydrologists often use data provided by meteorologists. As an example, a meteorologist might ...
, the simple power law (taking ) is used according to Monjo (2016) as a measure of the time-structure of the rainfall:
::
where is defined as an intensity of reference for a fixed time , i.e. , and 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.[Moncho, R.; Belda. F; Caselles, V. (2010): ''Climatic study of the exponent “n” in IDF curves: application for the Iberian Peninsula''. Tethys, nº6: 3–14. DOI: 10.3369/tethys.2009.6.0]
(pdf)
Theoretical approaches
To get an IDF curves from a probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
, it is necessary to mathematically isolate the precipitation, which is directly related to the average intensity and the duration , by the equation , and since the return period A return period, also known as a recurrence interval or repeat interval, is an average time or an estimated average time between events such as earthquakes, floods, landslides, or river discharge flows to occur.
It is a statistical measurement typ ...
is defined as the inverse of , the function is found as the inverse of , according to:
::
* Power law with the return period, derived from the Pareto distribution
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actua ...
, for a fixed duration :
::
::where the Pareto distribution constant has been redefined as, since it is a valid distribution for a specific duration of precipitation, , it has been taken as.
* Function derived from the generalized Pareto distribution
In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location \mu, scale \sigma, and shap ...
, for a given duration :
::
:: Note that for y , the generalized Pareto distribution retrieves the simple form of the Pareto distribution, with . However, with the exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
is retrieved.
*Function deduced from the Gumbel distribution
In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.
Thi ...
and the opposite Gumbel distribution, for a given duration :
::
::
References
{{reflist
Empirical process
Mathematical analysis
Hydrology and urban planning