Annuality

Annuality , also called annuity , or frequency , is called the recurrence probability of natural events in the geosciences . It is measured in 1 / a ("per year"), or in time units , then one speaks of the return interval . The term is relevant for the estimation of extreme events.

calculation

Natural events are mostly provided with a statistical evaluation. The annuality is calculated “from the statistical and probabilistic analysis of the events observed in the past, which are regarded as random events ”. For this purpose, a probability distribution is adapted to the measurement data and this is extrapolated in order to also be able to estimate events in the unobserved period.

The basis are series of values measured over many years . The annual highs are selected from these. As part of a statistical analysis, a distribution function is adapted from which quantiles are then determined for certain probabilities , i.e. H. Groups of certain failure probabilities.

Since the output series contains maximum annual values, the probabilities of exceeding them are also referred to as annualities , with the unit ¹ ⁄ _a , i.e. a measure of the frequency .

In general, the following applies to the probability of exceeding a standard value ( quantile E _T as extreme event) per year, i.e. for the annuality:

{\ displaystyle {\ begin {aligned} P (E> E_ {T}) & = 1 / T \\ & = 1-F (E) \ end {aligned}}}

With

the statistical recurrence interval T (with the unit year) as the reciprocal of the annuality
the underlying probability function F ( E ).

The same applies to the probability of occurrence of a normal value per year:

{\ displaystyle {\ begin {aligned} P (E <E_ {T}) & = 1-P (E \ geq E_ {T}) \\ & = 1-1 / T \\ & = F (E) \ end {aligned}}}

The underlying probability function results, depending on the discipline, from empirical values of the estimation, but also from the performance of the modeling of the underlying physical processes ( climate models , hydrological precipitation and runoff modeling , etc.).
The underlying distributions are typically:

Normal distribution
Gaussian distribution
Gamma distributions like
- the Pearson III distribution
- the Weibull distribution
the Gumbel distribution
for interpolations to large T (extremely rare events): methods of adjustment calculation such as the plotting position method according to Gringorten or Nguyen.

Interpretation of the calculation results

An event with the annuality or probability of exceedance P _ü = 0.01 / a has a recurrence interval of 100 years, i.e. H. it is exceeded (statistically speaking) once every 100 years. In each of these years, however, the respective largest event can be exceeded (the probability of this being 0.01 in each individual year). An event with an annuality of 0.01 is therefore (statistically) exceeded about 10 times in 1000 years, without a period of 100 years having to be between these exceedances.

The longer the observation period (actually the elapsed share of the statistical return period), the greater the probability that an over underrange occurs ( stochastic risk) . The decisive factor here is the multiplication principle of the probability calculation for independent events :

The lower underrange probability for the elapsed share of the return period increases from year to year from : ${\ displaystyle P_ {u} = P (E <E_ {T})}$

{\ displaystyle P_ {u, \, x \, years} = (P_ {u, \, 1 \, year}) ^ {x}}

So is the bottom stays probability of an event with the return period ( About underrange Probability) 0.01 and the recurrence interval T = 100 a:

in one year 0.99
for the period of two years 0.99 * 0.99 = 0.99²
for three years 0.99³ etc.

the About underrange probability for the elapsed share of the return period increases from year to year to : ${\ displaystyle P_ {ue} = P (E \ geq E_ {T})}$

{\ displaystyle {\ begin {aligned} P_ {ue, \, x \, years} & = 1-P_ {u, \, x \, years} \\ & = 1- (P_ {u, \, 1 \ , Year}) ^ {x} \ end {aligned}}}

For example, there is a risk that a flood with a return interval of 100 years will be exceeded:

within a period of 25 years ${\ displaystyle 1-0 {,} 99 ^ {25} \ approx 0 {,} 22}$
for a period of 50 years . ${\ displaystyle 1-0 {,} 99 ^ {50} \ approx 0 {,} 40}$

Simply looking up when the comparable events were before and after is not sufficient, because around three "100-year" events can be close to one another. Whether something like this is a statistical coincidence (" outlier "), or whether the probabilities have really changed compared to the reference interval, or whether the forecast models are incorrect , is one of the difficult questions, such as those in the context of "climate change" ( climate change / global warming ) are the central study area.

Typical criteria for extreme events

Depending on the event, one uses for example:

Flow rate in hydrology and hydraulic engineering , indexed HQ ₁₀₀ u. Ä., Also water levels , high water warning levels , accordingly also for low water

Wind speed , Beaufort wind strength for storms , Saffir-Simpson hurricane wind scale for hurricanes , Fujita scale for tornadoes

Earthquake scales (e.g. Richter or Mercallis scale ) in earthquake research

Warning levels in disaster control or the occurrence of disaster alarms, for example

Turin scale for risky approaches to near-Earth asteroids and comets in astrophysics .

Typical annualities and return intervals

Annuality in 1 / a	T in a	annotation
1	0001	"Annual event", occurring regularly, assessment level for example for normal water to high water
0.1	0010	"10-year event", measurement of floods
0.05	0020th	"20-year event", measurement of floods
~ 0.033	0030th	"30-year event", standard averaging period in meteorology
0.02	0050	"50-year event", measurement of floods
0.01	0100	" Event of the century ", common practice, corresponds roughly to the expression "since living memory"
~ 0.0066	0150	Limit value in flood protection, such as danger zone planning in Austria
0.0033	0300	Exceptional event in climatology , the written measurement series go back about a maximum of 300 years : "since the beginning of the weather recording"
0.001	1000	"Millennium event", due to the sources of written event reports, mostly in the sense of "never been there"
0.0002	5000	Estimation based on geological findings for floods or mass movements for Central Europe, in the sense of "since the end of the last ice age "

literature

HP Nachtnebel, C. Gamperling, K. Leroch, J. Fürst, H. Holzmann (red. Rev.): Hydrology . Study sheets, winter semester 2003/04. Ed .: Institute for Water Management, Hydrology and Structural Hydraulic Engineering, University of Natural Resources and Life Sciences, Vienna. 1. Statistical basics , 2. Extreme value statistics , 3. Correlation and regression , 4. Time series analysis , 5. Regionalization and spatial interpolation .

Individual evidence

↑ Nachtnebel: hydrology . 2. Extreme value statistics , p. 2-2 (55) .
↑ Nachtnebel: hydrology . 2.2.1 Flood statistics , p. 2-4 (57) .
↑ Nachtnebel: hydrology . 1.8 Continuous distributions in hydrology , p. 1-30 (42) ff .
↑ Nachtnebel: hydrology . Tab 2.1 Alternative ploting positions and their application , p. 58 .

[Nachtnebel_55-1] Nachtnebel: hydrology . 2. Extreme value statistics , p. 2-2 (55) .

[Nachtnebel_57-2] Nachtnebel: hydrology . 2.2.1 Flood statistics , p. 2-4 (57) .

[Nachtnebel_42-3] Nachtnebel: hydrology . 1.8 Continuous distributions in hydrology , p. 1-30 (42) ff .

[Nachtnebel_58-4] Nachtnebel: hydrology . Tab 2.1 Alternative ploting positions and their application , p. 58 .