# The 50- and 100-year Exceedance Probabilities as New and Convenient Statistics for a Frequency Analysis of Extreme Events: An Example of Extreme Precipitation in Israel

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

^{N}

^{N}

^{N}

- N is the length of the period, in years;
- Q[N]y is the non-exceedance probability of the event over the N-year period;
- q is the annual non-exceedance probability of the event;
- p is the annual exceedance probability of the event;
- T is the return period of the event, in years.

^{N}

^{N}

^{1/N}

^{1/N})

## 3. Results

#### 3.1. Examples of Calculating P[N]y as a Function of T and Vice Versa

**Example A**. The exceedance probability over a 100-year period (N = 100 years) for a 500-year event (T = 500 years) is (with help of Formula (4c)):

^{100}≈ 0.18, or 18%

**Example B**. A return period T of an event with a required exceedance probability over 100 years, P100y = 1%, or 0.01, is an inverse of an annual exceedance probability p, which is (with help of Formula (5a)):

^{1/100}≈ 0.0001005, or about 0.01%

**Example C.**Let us require an exceedance probability of an extreme event to be 2% over a 50-year period: P50y = 2%, or 0.02. The corresponding return period T would be the following (with help of Formula (5b)):

^{1/50}) ≈ 2475 years, or about 2500 years

**Example D**. To provide a 10% exceedance probability of an extreme event over a 100-year period (P100y = 10%, or 0.1), a corresponding return period T should be the following (with Formula (5b)):

^{1/100}) ≈ 950 years, or about 1000 years

#### 3.2. Graphical Representation of Exceedance Probabilities over a Period of Years P[N]y

#### 3.2.1. Exceedance Probabilities P[N]y as Functions of a Length of the Period of N Years

#### 3.2.2. The 50- and 100-year Exceedance Probabilities as Functions of a Return Period

#### 3.3. Application

#### 3.3.1. Study Area

#### 3.3.2. Long-Term Series of Annual Maxima of Rainfall Intensities

- Historical data (HD): The long-term digitized records from the rainfall strip-charts of the 90+ historical stations (each record started in 1938 at the earliest and ended in 2005 at the latest) [13];
- Automated 1-min data (A1): The 1-min rainfall data (started sometimes in 2006 and mostly in 2009) calibrated for high intensities from the 80+ automated stations, up to 2021 [14];
- Automated 10-min data (A10): The 10-min rainfall data [15], mostly for the short period from around 2006 to around 2008, to fill as much as possible the gap between HD and A1 (despite some underestimation of A10 due to coarser time resolution as compared to A1 and lack of calibration for high intensities).

#### 3.3.3. Estimating the Return Periods

- F is the cumulative distribution function (CDF) of the GEV distribution;
- x is the return value (return level). In our application, it is the rainfall intensity RI;
- μ is a location parameter;
- σ is a scale parameter;
- k is a shape parameter.

#### 3.3.4. The Exceedance Probabilities over 50- and 100-year Periods

## 4. Summary, Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The study area: Israel. Left panel: its locations in the SE Mediterranean; central panel: its rainfall climatology (based on data for 1991–2020 provided by the Israel Meteorological Service) and numbers of subregions referred to in other figures; right panel: topography [16].

**Figure 4.**Annual maxima of 10-, 20-, 30-, 60-, and 120-min rainfall intensities (in units of rainfall amounts, mm per interval).

**Figure 6.**Return periods (RP). The RP for each annual maximum value was estimated with the GEV parameters fitted separately for each region and each short-time interval (from 10 to 120 min). All RP were estimated using Formulas (6) and (7).

**Figure 7.**The 100-year exceedance probabilities. Red lines denote P100y = 63% below which all points correspond to the short-interval rainfall intensities with return periods longer than 100 years. All the 100-year probabilities were calculated using Formula (4c) after substitutions: P100y = 1 − (1 − 1/T)100.

**Figure 8.**The 50-year exceedance probabilities. Blue lines denote P50y = 50% corresponding to return periods longer than 72 years, while red lines—P50y = 20% corresponding to return periods longer than 225 years. All the 50-year probabilities were calculated using Formula (4c) after substitutions: P50y = 1 − (1 − 1/T)50.

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**MDPI and ACS Style**

Osetinsky-Tzidaki, I.; Fredj, E. The 50- and 100-year Exceedance Probabilities as New and Convenient Statistics for a Frequency Analysis of Extreme Events: An Example of Extreme Precipitation in Israel. *Water* **2023**, *15*, 44.
https://doi.org/10.3390/w15010044

**AMA Style**

Osetinsky-Tzidaki I, Fredj E. The 50- and 100-year Exceedance Probabilities as New and Convenient Statistics for a Frequency Analysis of Extreme Events: An Example of Extreme Precipitation in Israel. *Water*. 2023; 15(1):44.
https://doi.org/10.3390/w15010044

**Chicago/Turabian Style**

Osetinsky-Tzidaki, Isabella, and Erick Fredj. 2023. "The 50- and 100-year Exceedance Probabilities as New and Convenient Statistics for a Frequency Analysis of Extreme Events: An Example of Extreme Precipitation in Israel" *Water* 15, no. 1: 44.
https://doi.org/10.3390/w15010044