# A Counting Process Approach for Trend Assessment of Drought Condition

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

#### 2.2. Data Preparation for Trend Analysis

#### 2.3. Statistical Tests for Trend Analysis

#### 2.3.1. Mann–Kendall Trend Test

#### 2.3.2. Non-Homogeneous Poisson Process (NHPP) with Power Law Trend Test

## 3. Results

#### 3.1. Differences between the Two Methods

- 1 for positive trend, i.e., the increasing of drought episodes
- 0 for non-significant trend
- $-1$ for negative trend, i.e., the decreasing of drought episodes

#### 3.2. Characterizing Trend Results by Drought Risk Class

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SPEI | Standardized Precipitation Evapotranspiration Index |

PDSI | Palmer Drought Severity Index |

NHPP | Non Homogeneous Poisson Process |

SPI | Standardized Precipitation Index |

GEV | Generalized Extreme Value Distribution |

CRU | Climatic Research Unit |

MK | Mann–Kendall |

EDI | Effective Drought Index |

CMI | Crop Moisture Index |

SWSI | Surface Water Supply Index |

## Appendix A

- /Dati (where to download CRU data)
- /Outputs
- /Plots

- (1)
**DroughtIndexGenerator.sh**(parent)- Look for the last CRU version among those in /Data then launch the childs
**DryMask.sh**and ***.R**to compute SPI and SPEI for each grid cell (the outputs is a NetCDF file) - The default SPEI time scales are set to 3, 4, 6, 12, 24, if they need to be changed in
**CRU_SPEI_calculation.R**file - The needed functions are in
**TrendFunctions.R** - DryMask.sh (child) set to NA the whole values of time series where yearly average precipitation is less than 73 mm (0.2 mm per day by 365 days) to guarantee the presence of missing values in the SPEI computation

- (2)
**DroughtTrendTest-Generator.sh**(parent) (ONLY FOR SPEI)- Check in /Outputs/../ if time series of SPEI have been generated
- Launch
**CRU_SPEI_TrendAnalysis.R**to compute the trend analysis - The outputs is a NetCDF file composed of four layers (Nhpp, MK, MK-classic, Difference Nhpp-MK), each one having −1, 0, or +1 values. Notice that an increasing trend of drought events is marked by −1 fo M-K whilst 1 for Nhpp.
- Another output is composed of the maps of the trend results (in /Plots)

- (1)
- Include ./ in the PATH
- (2)
- Add the bash command to call for the correspondent shell before the file name: e.g.,
**nohup bash DroughtIndexGenerator.sh &**

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**Figure 1.**Technical flow chart for the proposed method (the asterisk symbol refers to the different scale of aggregation used for the SPEI computation).

**Figure 2.**Time series of SPEI-6 March values from 1901 to 2018 at location 37.00° N, 2.25° W of Iberian peninsula: (

**a**) value for March of each year; and (

**b**,

**c**) only the values belonging to Drought and Severe+Extreme classes, respectively. The connecting lines in (

**b**,

**c**) identify two consecutive episodes occurred in the same class.

**Figure 3.**Cumulated function of episodes identified by SPEI-6 for the month of March in the period 1901–2018 at location 37.00° N, 2.25° W of Iberian peninsula and the correspondent fit of count data to the Poisson distribution: (

**a**,

**b**) Drought class; and (

**c**,

**d**) Severe+Extreme class.

**Figure 4.**Relative number of grid cells where NHPP and M-K test give significant (positive or negative) and non-significant trend for each month in the period 1901–2018: (

**a**) SPEI-3; (

**b**) SPEI-6; (

**c**) SPEI-12; and (

**d**) SPEI-24.

**Figure 5.**Spatial distribution of cases where the M-K gives a negative (

**top**) and positive trend (

**bottom**) and the response of the NHPP test on the same areas for SPEI-6 March: In the top panel, the M-K negative trend cases are blue and grey, while the NHPP non-significant trends are in grey. In the bottom panel, the M-K positive trend cases are yellow and grey, while the NHPP non-significant trends are in grey.

**Figure 6.**Areas of the globe interested by an increasing trend of drought episodes identified by SPEI-6 for the month of March. Significant positive trend: (1) only for NHPP (red areas); and (2) for NHPP and M-K (yellow areas).

**Figure 7.**Circular plots of the frequency of SPEI drought events when the NHPP test results in a positive significant trend of droughts for each class and month given that M-K test also results in a positive significant trend in the period 1901-2018: (

**a**) SPEI-3; and (

**b**) SPEI-12. (The y-axis is drawn as square root of frequency to highlight the differences between classes.)

**Table 1.**The units of deviation from the mean of a standardized Gaussian distribution and the correspondent measure of the severity of a dry event.

Class | SPEI Values |
---|---|

Moderately dry | −1 to −1.49 |

Severely dry | −1.50 to −1.99 |

Extremely dry | −2 and less |

Dry Risk Class | SPEI Values |
---|---|

Drought | −1 and less |

Moderate | −1 to −1.49 |

Severe | −1.50 to −1.99 |

Severe+Extreme | −1.50 and less |

**Table 3.**Variation in the percentage of NHPP positive trend cases to M-K positive trend, for each month and SPEI-time scale.

January | February | March | April | May | June | July | August | September | October | November | December | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

SPEI-3 | $18.0$ | $17.2$ | $14.0$ | $10.9$ | $9.3$ | $13.2$ | $15.5$ | $16.5$ | $18.0$ | $15.3$ | $12.9$ | $14.0$ |

SPEI-6 | $15.4$ | $13.4$ | $12.5$ | $12.1$ | $11.6$ | $11.4$ | $12.5$ | $12.1$ | $13.6$ | $13.1$ | $14.3$ | $15.2$ |

SPEI-12 | $13.5$ | $13.2$ | $12.7$ | $11.9$ | $11.6$ | $11.9$ | $12.4$ | $11.1$ | $11.5$ | $11.6$ | $11.3$ | $11.3$ |

SPEI-24 | $24.7$ | $24.8$ | $25.4$ | $24.9$ | $24.7$ | $24.7$ | $24.6$ | $24.1$ | $24.0$ | $24.0$ | $24.2$ | $23.8$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Di Giuseppe, E.; Pasqui, M.; Magno, R.; Quaresima, S.
A Counting Process Approach for Trend Assessment of Drought Condition. *Hydrology* **2019**, *6*, 84.
https://doi.org/10.3390/hydrology6040084

**AMA Style**

Di Giuseppe E, Pasqui M, Magno R, Quaresima S.
A Counting Process Approach for Trend Assessment of Drought Condition. *Hydrology*. 2019; 6(4):84.
https://doi.org/10.3390/hydrology6040084

**Chicago/Turabian Style**

Di Giuseppe, Edmondo, Massimiliano Pasqui, Ramona Magno, and Sara Quaresima.
2019. "A Counting Process Approach for Trend Assessment of Drought Condition" *Hydrology* 6, no. 4: 84.
https://doi.org/10.3390/hydrology6040084