# Characteristics of Hydrological and Meteorological Drought Based on Intensity-Duration-Frequency (IDF) Curves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data Collection

#### 2.2. Standardized Drought Indices

#### 2.2.1. Standard Precipitation Index (SPI)

#### 2.2.2. Standard Precipitation Evapotranspiration Index (SPEI)

#### 2.2.3. Standardized Streamflow Index (SSI)

## 3. Methodology and Basic Concepts

#### 3.1. Basic Concepts

_{SD}, where SD is the specific drought duration. In a year without any CI, a zero value is assigned. The drought event for each duration is assigned to the month it ends with its year. This assumption applies regardless of the drought event’s starting and ending months. For example, a drought event with 9 months specific drought duration can start in a year and ends in another year. CI

_{9}starts from June 2022 to February 2023. IDF calculations should assign the CI value to the second year (2023).

#### 3.2. Methodology

- ∗
- The proposed methodology calculates the DI with any drought index. In this research, there are three ways, two of which are meteorological drought indices SPI and SPEI, and the other is the hydrological drought index SSI.
- ∗
- Determination of whether the month is dry or wet based on the drought definition using DI according to run and SPI theories. This point provides added value to this research because it is based on a comprehensive definition of drought characteristics.
- ∗
- Determination of the specific drought duration (SD).
- ∗
- Calculation of the maximum consecutive severity (MCS) and critical intensity (CI) for each specific duration (SD). CI
_{SD}is assigned to the last month and year of the drought duration. (See Figure 2).

**Figure 2.**The main concepts used in this research (DI: drought index, D: drought duration, SD: specific drought duration, MCS: maximum consecutive severity, CI: critical intensity).

- ∗
- Finding the years with CI value and those without drought events.
- ∗
- Frequency analysis includes the return period (T) determination, which can be 2-year, 10-year, or 500-year. The probability for each return period P(T) is calculated using Equation (4); the probability of years without any critical intensity P(0) by Equation (5); the probability of drought years P(d) or non-zero years by Equation (6).

_{D}will not be equaled or exceeded in any year of the return period:

- ∗
- The probability cumulative distribution function (CDF) of the drought years (non-zero values) at zero level is calculable in Equation (8). This probability will predict the critical intensity (CI) for each drought duration and return period.

_{SD_T}, where SD is the specific drought duration, and T is the return period. For each specific drought duration and return period, there is a critical intensity. Table 3 below summarizes how the CI values will be:

## 4. Results and Discussion

## 5. Conclusions

- To comprehensively understand and analyze drought characteristics, there is a need to divide a drought event into many specific drought duration intervals. Consequently, the severity and intensity of each specific drought duration should be evaluated.
- Drought characteristics are different based on drought definition. Both SPI and run theory must then be carried out to obtain the extreme drought characteristics.
- The normal probability distribution function was the dominant and suitable PDF for critical intensity values.
- Developed IDF drought curves should be used directly by designers and decision-makers for design and risk assessment purposes because of their simplicity and ease of interpretation, precise quantification, and practical application.
- Based on the dry years for both the run and SPI theories, the specific drought duration remained between 9 and 14 months. Data were insufficient for a specific drought duration longer than 14 months for deriving a suitable PDF.
- For 2 years return period, the IDF drought curve was calculated up to 4 months of specific drought duration. The CI for a longer specific drought duration cannot be calculated.
- Compared to the conventional risk analysis methods which depend on the drought events, the new concepts and framework provide us with a comprehensive understanding and range of choices regarding drought duration and return period.
- One of the major outcomes of our research is to provide water managers and suppliers with a new tool and approach to making decisions on appropriate management actions regarding drought frequency.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Drought duration based on run theory (red line) and SPI theory (blue line). D is the drought duration [40].

**Figure 4.**Resulting IDF curves using SPI for both run and SPI theories for 3- and 12-month timescales: (

**a**) IDF using SPI3 based on run theory. (

**b**) IDF using SPI12 based on run theory. (

**c**) IDF using SPI3 based on SPI theory. (

**d**) IDF using SPI12 based on SPI theory.

**Figure 5.**Resulting IDF curves using SPEI for both run and SPI theories for 3- and 12-month timescales: (

**a**) IDF using SPEI3 based on run theory. (

**b**) IDF using SPEI12 based on run theory. (

**c**) IDF using SPEI3 based on SPI theory. (

**d**) IDF using SPEI12 based on SPI theory.

**Figure 6.**Resulting IDF curves using SSI for both run and SPI theories for 3- and 12-month timescales: (

**a**) IDF using SSI3 based on run theory. (

**b**) IDF using SSI12 based on run theory. (

**c**) IDF using SSI3 based on SPI theory. (

**d**) IDF using SSI12 based on SPI theory.

Station’s Name | Lat (N) | Long (W) | Monthly Precipitation (P)—mm | Standard Deviation (mm) | Monthly Temperature (T)—°C | Standard Deviation °C |
---|---|---|---|---|---|---|

Durham University Station | 54.77 | 1.59 | 54.37 | 31.74 | 8.6 | 4.46 |

Station’s Name | Lat (N) | Long (E) | Monthly Stream flow (Q)—m^{3}/s | Standard Deviation (m ^{3}/s) | - | - |

Lüleburgaz station | 41.35 | 27.35 | 9.57 | 13.2 | - | - |

Specific drought duration | SD |

Total number of years | N |

Number of drought years | N_{D} |

Number of years without any drought | N_{0} = N – N_{D} |

The probability of years without any drought event | $\mathrm{P}\left(0\right)=\frac{{\mathrm{N}}_{0}}{\mathrm{N}}$ |

The probability of drought years | $\mathrm{P}\left(\mathrm{d}\right)=\frac{{\mathrm{N}}_{\mathrm{D}}}{\mathrm{N}}$ |

Return period | T |

The probability of each return period | $\mathrm{P}\left(\mathrm{T}\right)=\frac{1}{\mathrm{T}}$ |

The cumulative probability distribution function of the drought years | ${\mathrm{P}}^{*}\left(\mathrm{d}\right)=\frac{1-\frac{1}{\mathrm{T}}-\mathrm{P}\left(0\right)}{1-\mathrm{P}\left(0\right)}$ |

SD (Month) | |||||
---|---|---|---|---|---|

T (Year) | $\frac{1}{\mathbf{T}}$ | 1 | 2 | 3 | 4 |

2 | 0.5 | ${\mathrm{CI}}_{1\_2}$ | ${\mathrm{CI}}_{2\_2}$ | ${\mathrm{CI}}_{3\_2}$ | ${\mathrm{CI}}_{4\_2}$ |

5 | 0.2 | ${\mathrm{CI}}_{1\_5}$ | ${\mathrm{CI}}_{2\_5}$ | ${\mathrm{CI}}_{3\_5}$ | ${\mathrm{CI}}_{4\_5}$ |

10 | 0.1 | ${\mathrm{CI}}_{1\_10}$ | ${\mathrm{CI}}_{2\_10}$ | ${\mathrm{CI}}_{3\_10}$ | ${\mathrm{CI}}_{4\_10}$ |

25 | 0.04 | ${\mathrm{CI}}_{1\_25}$ | ${\mathrm{CI}}_{2\_25}$ | ${\mathrm{CI}}_{3\_25}$ | ${\mathrm{CI}}_{4\_25}$ |

50 | 0.02 | ${\mathrm{CI}}_{1\_50}$ | ${\mathrm{CI}}_{2\_50}$ | ${\mathrm{CI}}_{3\_50}$ | ${\mathrm{CI}}_{4\_50}$ |

100 | 0.01 | ${\mathrm{CI}}_{1\_100}$ | ${\mathrm{CI}}_{2\_100}$ | ${\mathrm{CI}}_{3\_100}$ | ${\mathrm{CI}}_{4\_100}$ |

200 | 0.005 | ${\mathrm{CI}}_{1\_200}$ | ${\mathrm{CI}}_{2\_200}$ | ${\mathrm{CI}}_{3\_200}$ | ${\mathrm{CI}}_{4\_200}$ |

500 | 0.002 | ${\mathrm{CI}}_{1\_500}$ | ${\mathrm{CI}}_{2\_500}$ | ${\mathrm{CI}}_{3\_500}$ | ${\mathrm{CI}}_{4\_500}$ |

_{SD_T}: SD is the specific drought duration and T is the return period.

**Table 4.**The main variables including total years, P(0), P(d), PDF with its parameters, and T with CI for IDF calculation using SPI12 based on SPI theory at Durham meteorological station.

D (month) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Total Years | 154 | 154 | 154 | 154 | 154 | 154 | 154 | 154 | 154 | 154 | 154 | 154 |

Years with drought | 81 | 79 | 75 | 65 | 63 | 61 | 55 | 51 | 47 | 43 | 37 | 34 |

Years without drought | 73 | 75 | 79 | 89 | 91 | 93 | 99 | 103 | 107 | 111 | 117 | 120 |

P(0) | 0.47 | 0.49 | 0.51 | 0.58 | 0.59 | 0.60 | 0.64 | 0.67 | 0.69 | 0.72 | 0.76 | 0.78 |

P(d) | 0.53 | 0.51 | 0.49 | 0.42 | 0.41 | 0.40 | 0.36 | 0.33 | 0.31 | 0.28 | 0.24 | 0.22 |

Nor. | Nor. | Nor. | Nor. | Nor. | Nor. | Nor. | Nor. | Nor. | Nor. | Nor. | Nor. | |

${\mu}^{*}$ | −1.5 | −1.39 | −1.34 | −1.35 | −1.35 | −1.33 | −1.35 | −1.34 | −1.28 | −1.29 | −1.14 | −1 |

$\sigma {}^{*}$ | 0.57 | 0.59 | 0.58 | 0.59 | 0.59 | 0.59 | 0.58 | 0.57 | 0.55 | 0.54 | 0.49 | 0.44 |

T (Year) | Critical Intensity (CI) for 12 months | |||||||||||

5 | −1.67 | −1.55 | −1.47 | −1.39 | −1.36 | −1.32 | −1.26 | −1.15 | −1.06 | −0.98 | −0.67 | −0.42 |

10 | −2.00 | −1.90 | −1.82 | −1.77 | −1.76 | −1.72 | −1.69 | −1.60 | −1.52 | −1.49 | −1.24 | −1.05 |

25 | −2.31 | −2.23 | −2.15 | −2.12 | −2.11 | −2.08 | −2.06 | −1.96 | −1.90 | −1.87 | −1.62 | −1.40 |

50 | −2.51 | −2.43 | −2.35 | −2.33 | −2.32 | −2.30 | −2.27 | −2.19 | −2.11 | −2.08 | −1.82 | −1.59 |

100 | −2.68 | −2.61 | −2.52 | −2.51 | −2.51 | −2.48 | −2.46 | −2.37 | −2.29 | −2.26 | −2.00 | −1.75 |

200 | −2.83 | −2.76 | −2.69 | −2.68 | −2.68 | −2.64 | −2.62 | −2.54 | −2.46 | −2.42 | −2.14 | −1.88 |

500 | −3.02 | −2.95 | −2.88 | −2.87 | −2.87 | −2.85 | −2.81 | −2.73 | −2.63 | −2.61 | −2.32 | −2.04 |

**Table 5.**Maximum drought duration, maximum intensity, and slope of IDF curves for each return period using SPI and SPEI.

Timescale = 3 Months | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

SPI_Run Theory | SPI_SPI Theory | SPEI_Run Theory | SPEI_SPI Theory | |||||||||

T (Year) | Max. D | Max. I | Slope | Max. D | Max. I | Slope | Max. D | Max. I | Slope | Max. D | Max. I | Slope |

2 | 6.5 | 1.79 | 0.274 | 4.4 | 2.06 | 0.495 | 9.1 | 1.55 | 0.171 | 6.3 | 1.73 | 0.276 |

5 | 11.0 | 2.17 | 0.197 | 9.0 | 2.31 | 0.257 | 14.4 | 1.86 | 0.129 | 11.4 | 1.97 | 0.173 |

10 | 14.1 | 2.33 | 0.165 | 10.5 | 2.53 | 0.24 | 20.4 | 1.94 | 0.095 | 13.3 | 2.13 | 0.16 |

25 | 15.9 | 2.59 | 0.163 | 14.6 | 2.63 | 0.18 | 23.5 | 2.14 | 0.091 | 18.1 | 2.21 | 0.122 |

50 | 16.5 | 2.77 | 0.168 | 16.3 | 2.77 | 0.17 | 24.1 | 2.27 | 0.094 | 20.1 | 2.31 | 0.115 |

100 | 16.8 | 2.94 | 0.175 | 16.8 | 2.92 | 0.174 | 24.6 | 2.41 | 0.098 | 21.6 | 2.4 | 0.111 |

200 | 16.9 | 3.1 | 0.183 | 17.3 | 3.05 | 0.176 | 25.0 | 2.52 | 0.101 | 22.1 | 2.48 | 0.112 |

500 | 17.1 | 3.31 | 0.1942 | 17.9 | 3.2 | 0.179 | 25.0 | 2.67 | 0.107 | 24.1 | 2.58 | 0.107 |

Timescale = 12 Months | ||||||||||||

SPI_Run Theory | SPI_SPI Theory | SPEI_Run Theory | SPEI_SPI Theory | |||||||||

T (Year) | Max. D | Max. I | Slope | Max. D | Max. I | Slope | Max. D | Max. I | Slope | Max. D | Max. I | Slope |

2 | 9.6 | 1.28 | 0.134 | - | - | - | 11.2 | 1.1 | 0.098 | - | - | - |

5 | 27.9 | 1.7 | 0.061 | 19.0 | 1.82 | 0.096 | 34.3 | 1.61 | 0.047 | 13.8 | 1.8 | 0.13 |

10 | 33.5 | 2.01 | 0.06 | 29.2 | 2.1 | 0.072 | 43.0 | 1.85 | 0.043 | 20.6 | 1.98 | 0.096 |

25 | 36.7 | 2.35 | 0.064 | 36.1 | 2.42 | 0.067 | 48.2 | 2.12 | 0.044 | 30.0 | 2.16 | 0.072 |

50 | 38.4 | 2.57 | 0.067 | 39.1 | 2.62 | 0.067 | 50.0 | 2.3 | 0.046 | 34.8 | 2.3 | 0.066 |

100 | 39.0 | 2.77 | 0.071 | 41.8 | 2.8 | 0.067 | 51.0 | 2.45 | 0.048 | 38.1 | 2.44 | 0.064 |

200 | 39.3 | 2.95 | 0.075 | 43.7 | 2.97 | 0.068 | 52.0 | 2.6 | 0.05 | 40.6 | 2.56 | 0.063 |

500 | 39.6 | 3.17 | 0.08 | 45.3 | 3.17 | 0.07 | 51.5 | 2.78 | 0.054 | 45.0 | 2.7 | 0.06 |

**Table 6.**Maximum drought duration, maximum intensity, and slope of IDF curves for each return period using SSI.

Timescale = 3 Months | Timescale = 12 Months | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

SSI_Run Theory | SSI_SPI Theory | SSI_Run Theory | SSI_SPI Theory | |||||||||

T (Year) | Max. D | Max. I | Slope | Max. D | Max. I | Slope | Max. D | Max. I | Slope | Max. D | Max. I | Slope |

2 | 9.30 | 1.06 | 0.114 | - | - | - | 9.1 | 0.48 | 0.053 | - | - | - |

5 | 35.00 | 1.4 | 0.04 | 32.4 | 1.46 | 0.045 | 46.2 | 1.34 | 0.029 | 46.5 | 1.44 | 0.031 |

10 | 42.37 | 1.61 | 0.038 | 38.9 | 1.71 | 0.044 | 43.9 | 1.67 | 0.038 | 42.9 | 1.76 | 0.041 |

25 | 52.57 | 1.84 | 0.035 | 37.9 | 1.97 | 0.052 | 42.9 | 2 | 0.0466 | 39.8 | 2.07 | 0.052 |

50 | 43.91 | 2.02 | 0.046 | 36.6 | 2.12 | 0.058 | 42.3 | 2.2 | 0.052 | 37.5 | 2.25 | 0.06 |

100 | 43.20 | 2.16 | 0.05 | 35.3 | 2.26 | 0.064 | 41.8 | 2.38 | 0.057 | 37.1 | 2.41 | 0.065 |

200 | 42.59 | 2.3 | 0.054 | 34.5 | 2.38 | 0.069 | 41.0 | 2.54 | 0.062 | 36.1 | 2.56 | 0.071 |

500 | 41.17 | 2.47 | 0.06 | 33.4 | 2.54 | 0.076 | 39.7 | 2.74 | 0.069 | 35.7 | 2.71 | 0.076 |

**Table 7.**Comparison between conventional and IDF drought curves methods regarding drought intensity using SPI12 for Durham station.

SPI12 Using Conventional Method | SPI12 Using IDF Drought Curves | ||||||||
---|---|---|---|---|---|---|---|---|---|

T (Years) | |||||||||

Duration (m) | Average | Maximum | 5 | 10 | 25 | 50 | 100 | 200 | 500 |

2 | −0.82 | −1.28 | −1.55 | −1.9 | −2.23 | −2.43 | −2.61 | −2.76 | −2.95 |

3 | −1.36 | −0.94 | −1.47 | −1.82 | −2.15 | −2.35 | −2.52 | −2.69 | −2.88 |

4 | - (*) | - | −1.39 | −1.77 | −2.12 | −2.33 | −2.51 | −2.68 | −2.87 |

5 | - | - | −1.36 | −1.76 | −2.11 | −2.32 | −2.51 | −2.68 | −2.87 |

6 | −0.87 | −1.11 | −1.32 | −1.72 | −2.08 | −2.3 | −2.48 | −2.64 | −2.85 |

7 | −0.75 | −1.25 | −1.26 | −1.69 | −2.06 | −2.27 | −2.46 | −2.62 | −2.81 |

8 | −1.19 | −2.02 | −1.15 | −1.6 | −1.96 | −2.19 | −2.37 | −2.54 | −2.73 |

9 | −0.65 | −1.01 | −1.06 | −1.52 | −1.9 | −2.11 | −2.29 | −2.46 | −2.63 |

10 | −1.1 | −1.6 | −0.98 | −1.49 | −1.87 | −2.08 | −2.26 | −2.42 | −2.61 |

11 | −0.53 | −1.22 | −0.67 | −1.24 | −1.62 | −1.82 | −2 | −2.14 | −2.32 |

12 | - | - | −0.42 | −1.05 | −1.4 | −1.59 | −1.75 | −1.88 | −2.04 |

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## Share and Cite

**MDPI and ACS Style**

Abu Arra, A.; Şişman, E.
Characteristics of Hydrological and Meteorological Drought Based on Intensity-Duration-Frequency (IDF) Curves. *Water* **2023**, *15*, 3142.
https://doi.org/10.3390/w15173142

**AMA Style**

Abu Arra A, Şişman E.
Characteristics of Hydrological and Meteorological Drought Based on Intensity-Duration-Frequency (IDF) Curves. *Water*. 2023; 15(17):3142.
https://doi.org/10.3390/w15173142

**Chicago/Turabian Style**

Abu Arra, Ahmad, and Eyüp Şişman.
2023. "Characteristics of Hydrological and Meteorological Drought Based on Intensity-Duration-Frequency (IDF) Curves" *Water* 15, no. 17: 3142.
https://doi.org/10.3390/w15173142