# A Stochastic Weather Model for Drought Derivatives in Arid Regions: A Case Study in Qatar

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data

#### 2.2. Reconnaissance Drought Index (RDI)

#### 2.3. Evapotranspiration

#### 2.4. Temperature Model

#### 2.5. Humidity Model

#### 2.6. Precipitation Model

#### 2.6.1. Precipitation Occurrence Model

#### 2.6.2. Distribution of Precipitation Amount

- Kolmogorov–Smirnov test (KS test [33])The Kolmogorov–Smirnov test is used to determine if a dataset comes from a specified distribution. It measures the differences between the empirical distribution of the sample and the cumulative distribution of the specified distribution, providing a test statistic, D, and p-values that can be used as criteria for hypothesis testing.
- Akaike information criteria (AIC [35])Firstly, the AIC developed by Hirotugu Akaike was used to evaluate the performance of the model in a simple linear regression. It was created to select the model that has the smallest loss of information from the given data. It measures the loss based on a likelihood function and is defined by:$$AIC=-2ln\left(L\right)+2K,$$
- Bayesian information criterion (BIC [36])Similar to AIC, BIC evaluates the model performance by using the likelihood, and the model with the smallest BIC is preferred. Compared to AIC, it has a larger penalty term for the number of parameters and observations. It is defined by$$BIC=-2ln\left(L\right)+Kln\left(n\right).$$

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Evapotranspiration Formulations

- Blaney–Criddle:The Blaney–Criddle equation can be written as follows:$$E{T}_{0}=P\left(0.46{T}_{mean}+8.13\right)$$
- Hargreaves:The Hargreaves equation can be written as follows:$$E{T}_{0}=0.0023\times {R}_{a}({T}_{mean}+17.80)\sqrt{{T}_{max}-{T}_{min}}$$
- Jensen–Haise:The method developed by Jensen–Haise for the arid and semiarid regions has the following equation:$$\begin{array}{cc}\hfill PET=& {\displaystyle \frac{1}{{\displaystyle 38-\left({\displaystyle 2\times \frac{Elevat}{305}}\right)+7.6\frac{50}{({e}_{s\left({T}_{max}\right)}-{e}_{s\left({T}_{min}\right)})}}}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \times \left[{T}_{mean}-\left({\displaystyle -2.5-0.14\left({e}_{s\left({T}_{max}\right)}-{e}_{s\left({T}_{min}\right)}\right)-\frac{Elevat}{550}}\right)\right]{R}_{a}\hfill \end{array}$$
- LinacreThe Linacre method can be written as follows:$$E{T}_{0}=\frac{{\displaystyle \frac{700\left({T}_{mean}+0.0006Z\right)}{100-L}+15\left({T}_{mean}-{T}_{d}\right)}}{80-{T}_{mean}}$$
- TurcFinally, the Turc method can be written as follows:$$\begin{array}{cc}\hfill PET& =0.013\left(\frac{{T}_{mean}}{15+{T}_{mean}}\right)\left({R}_{s}+50\right),\phantom{\rule{2.em}{0ex}}RH>50\%\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =0.013\left(\frac{{T}_{mean}}{15+{T}_{mean}}\right)\left({R}_{s}+50\right)\left(1+\frac{50-RH}{70}\right),\phantom{\rule{2.em}{0ex}}RH<50\%\hfill \end{array}$$

## Appendix B. Maximum Likelihood Estimation (MLE)

## References

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Blaney and Criddle | Hargreaves and Samani | Jensen and Haise | Lincare | Turc | |
---|---|---|---|---|---|

${R}^{2}$ | 0.5107 | 0.7283 | 0.5765 | 0.4367 | 0.7623 |

RMSE | 2.8507 | 1.6317 | 2.4806 | 10.3425 | 1.4817 |

MAE | 2.4389 | 1.3574 | 2.1244 | 8.9496 | 1.2232 |

MAXE | 5.636 | 3.8584 | 5.8988 | 24.0355 | 3.4935 |

Crop | Growing Period | Growth Stage | ${\mathit{k}}_{\mathit{c}}$ (Crop Coefficient) | ||||||
---|---|---|---|---|---|---|---|---|---|

Planting | Harvest | Initial | Dev | Mid | Late | Initial | Mid | Late | |

Alfalfa | 1-Jan | 2-Mar | 10 | 20 | 20 | 10 | 0.40 | 0.95 | 0.40 |

Bean | 15-Sep | 29-Nov | 15 | 25 | 25 | 10 | 0.50 | 1.15 | 0.90 |

Carrot | 15-Oct | 12-Feb | 20 | 30 | 50 | 20 | 0.70 | 1.05 | 0.95 |

Cucumber | 15-Nov | 25-Mar | 25 | 35 | 50 | 20 | 0.60 | 1.00 | 0.75 |

Maize | 1-Jan | 21-May | 25 | 40 | 45 | 30 | 0.30 | 1.20 | 0.60 |

Onion | 15-Oct | 31-May | 20 | 35 | 110 | 45 | 0.70 | 1.05 | 0.75 |

Potato | 1-Dec | 20-Apr | 30 | 35 | 50 | 25 | 0.50 | 1.15 | 0.75 |

Rice | 1-Dec | 30-Apr | 30 | 30 | 60 | 30 | 1.05 | 1.20 | 0.70 |

Tomato | 1-Jan | 16-May | 30 | 40 | 40 | 25 | 0.60 | 1.15 | 0.80 |

Wheat | 15-Dec | 24-May | 20 | 50 | 60 | 30 | 0.70 | 1.15 | 0.30 |

Crop | Observation | Simulation | ||||||
---|---|---|---|---|---|---|---|---|

Mean | SD | Min | Max | Mean | SD | Min | Max | |

Alfalfa | 1.6790 | 0.6127 | 0.3651 | 3.4470 | 1.1920 | 0.4732 | 0.2531 | 2.6430 |

Bean | 2.6640 | 0.6936 | 0.7510 | 4.6270 | 2.0430 | 0.5325 | 0.8086 | 3.7270 |

Carrot | 2.1840 | 0.3488 | 0.9584 | 3.8760 | 1.5560 | 0.3148 | 0.5188 | 2.9350 |

Maize | 2.7510 | 1.2636 | 0.2738 | 6.4760 | 2.4020 | 1.2528 | 0.1638 | 6.3320 |

Tomato | 2.8830 | 1.0954 | 0.5477 | 6.2060 | 2.5150 | 1.2096 | 0.3277 | 6.0680 |

Wheat | 2.5730 | 1.1111 | 0.6206 | 6.2060 | 2.1510 | 1.1037 | 0.4033 | 6.0680 |

Month | $\mathit{\mu}$ | $\mathit{\lambda}$ | $\mathit{\gamma}$ |
---|---|---|---|

January | 17.4780 | 0.4011 | 1.6294 |

February | 18.8917 | 0.4824 | 1.7278 |

March | 22.3762 | 0.4331 | 1.7986 |

April | 28.2374 | 0.2821 | 1.5822 |

May | 33.1215 | 0.3567 | 1.6389 |

June | 35.0124 | 0.6396 | 1.9190 |

July | 35.7646 | 0.6965 | 1.8501 |

August | 35.1229 | 0.9300 | 1.5149 |

September | 32.8898 | 0.5524 | 1.2789 |

October | 28.6960 | 0.2412 | 0.8825 |

November | 23.7035 | 0.1493 | 1.1662 |

December | 18.8716 | 0.3034 | 1.4249 |

Month | $\mathit{\mu}$ | $\mathit{\lambda}$ | $\mathit{\gamma}$ |
---|---|---|---|

January | 67.4488 | 0.6768 | 11.4293 |

February | 64.0371 | 0.9049 | 14.8098 |

March | 54.9218 | 0.7767 | 15.1913 |

April | 46.4332 | 0.5920 | 12.7813 |

May | 36.3553 | 0.7387 | 12.2807 |

June | 35.3932 | 0.7081 | 14.4253 |

July | 45.2631 | 0.5689 | 15.4916 |

August | 55.2619 | 0.7355 | 14.2856 |

September | 56.2345 | 1.1614 | 15.9156 |

October | 58.7774 | 0.9829 | 11.9194 |

November | 62.1151 | 1.1193 | 11.0523 |

December | 67.9820 | 0.8075 | 10.6235 |

Month | Probability Vector | Month | Probability Vector |
---|---|---|---|

January | $\left[0.965\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.035\right]$ | July | $\left[1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\right]$ |

February | $\left[0.971\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.029\right]$ | August | $\left[0.999\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.001\right]$ |

March | $\left[0.97\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.030\right]$ | September | $\left[0.999\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.001\right]$ |

April | $\left[0.976\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.024\right]$ | October | $\left[0.998\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.002\right]$ |

May | $\left[0.993\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.007\right]$ | November | $\left[0.976\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.024\right]$ |

June | $\left[1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\right]$ | December | $\left[0.960\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0.040\right]$ |

Distribution | |
---|---|

Exponential (density) | $f(x;\lambda )=\lambda exp(-\lambda x),\phantom{\rule{2.em}{0ex}}x\ge 0;\phantom{\rule{4pt}{0ex}}\lambda >0$ |

Log-normal (density) | $f(x;\mu ,\sigma )=\frac{1}{x\sigma \sqrt{2\pi}}exp\left(\frac{-{(lnx-\mu )}^{2}}{2{\sigma}^{2}}\right),\phantom{\rule{2.em}{0ex}}x>0;\phantom{\rule{4pt}{0ex}}\sigma >0$ |

Gamma (density) | $f(x;\alpha ,\beta )=\frac{{\beta}^{-\alpha}{x}^{\alpha -1}{e}^{-x/\beta}}{\Gamma \left(\alpha \right)},\phantom{\rule{2.em}{0ex}}x>0;\phantom{\rule{0.166667em}{0ex}}\alpha ,\phantom{\rule{4pt}{0ex}}\beta >0$ |

GEVD (CDF) | $F(x;\xi ,\mu ,\alpha )=exp\left\{-{\left[1+\xi \left(\frac{x-\mu}{\alpha}\right)\right]}^{-1/\xi}\right\},$ $\mathrm{where}x\in \left\{z\right|1+\xi (z-\mu )/\alpha >0\};\phantom{\rule{4pt}{0ex}}\sigma >0$ |

K4D (CDF) | $F(x;\xi ,\alpha ,h,k)={\left[1-h{\left[1-\frac{k(x-\xi )}{\alpha}\right]}^{\frac{1}{k}}\right]}^{\frac{1}{h}}$, $k\ne 0,\phantom{\rule{4pt}{0ex}}h\ne 0$ (See [34] for a description of the support of K4D.) |

Distribution | Estimation of Parameters | |||
---|---|---|---|---|

Exponential | $\widehat{\mu}=0.108$ | |||

Log-normal | $\widehat{\mu}=0.834$ | $\widehat{\sigma}=1.510$ | ||

Gamma | $\widehat{\alpha}=0.463$ | $\widehat{\beta}=0.050$ | ||

GEVD | $\widehat{\mu}=1.060$ | $\widehat{\alpha}=1.322$ | $\widehat{\xi}=1.364$ | |

K4D | $\widehat{\xi}=1.225$ | $\widehat{\alpha}=1.424$ | $\widehat{h}=-0.375$ | $\widehat{k}=-1.329$ |

Distribution | KS Test (p-Value) | AIC | BIC |
---|---|---|---|

Exponential | $2.2\times {10}^{-16}$ | 1174.589 | 1177.793 |

Log-normal | $9.937\times {10}^{-4}$ | 973.921 | 980.329 |

Gamma | $1.467\times {10}^{-6}$ | 1075.677 | 1082.085 |

GEVD | $1.467\times {10}^{-6}$ | 953.895 | 963.507 |

K4D | 0.0929 | 999.214 | 1008.03 |

Statistics | Observation | Simulation |
---|---|---|

Mean | 27.67 | 27.57 |

SD | 6.89756 | 6.903637 |

Min | 9.80 | 11.48 |

Max | 42.20 | 41.78 |

Percentile | Observation | Simulation |

1 | 14.22 | 14.99241 |

10 | 17.90 | 17.95062 |

25 | 21.40 | 20.99015 |

50 | 28.70 | 28.54294 |

75 | 33.90 | 34.00301 |

90 | 35.90 | 35.83885 |

99 | 38.60 | 38.14206 |

Statistics | Observation | Simulation |
---|---|---|

Mean | 54.01 | 54.210 |

SD | 16.2691 | 14.92881 |

Min | 8.00 | 2.252 |

Max | 95.00 | 98.900 |

Percentile | Observation | Simulation |

1 | 18.00 | 17.39385 |

10 | 30.00 | 33.39348 |

25 | 42.00 | 44.32667 |

50 | 56.00 | 55.76611 |

75 | 66.00 | 64.95842 |

90 | 74.00 | 72.4125 |

99 | 85.00 | 83.9390 |

Observation | Simulation 2 (K4D) | |
---|---|---|

Minimum | 0.25 | 0.19 |

Q1 | 0.76 | 0.67 |

Q2 (Median) | 1.78 | 1.53 |

Mean | 9.22 | 8.68 |

Q3 | 6.10 | 4.71 |

Maximum | 182.88 | 186.04 |

Crop | Observation | Simulation | ||||
---|---|---|---|---|---|---|

Mean | SD | Max | Mean | SD | Max | |

Carrot | 0.1268 | 0.1827 | 0.7161 | 0.1460 | 0.2133 | 0.7788 |

Maize | 0.0700 | 0.1104 | 0.4678 | 0.1141 | 0.1827 | 0.7602 |

Tomato | 0.0692 | 0.1093 | 0.4631 | 0.1130 | 0.1813 | 0.7567 |

Wheat | 0.0799 | 0.1047 | 0.4303 | 0.1153 | 0.1823 | 0.7341 |

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## Share and Cite

**MDPI and ACS Style**

Paek, J.; Pollanen, M.; Abdella, K. A Stochastic Weather Model for Drought Derivatives in Arid Regions: A Case Study in Qatar. *Mathematics* **2023**, *11*, 1628.
https://doi.org/10.3390/math11071628

**AMA Style**

Paek J, Pollanen M, Abdella K. A Stochastic Weather Model for Drought Derivatives in Arid Regions: A Case Study in Qatar. *Mathematics*. 2023; 11(7):1628.
https://doi.org/10.3390/math11071628

**Chicago/Turabian Style**

Paek, Jayeong, Marco Pollanen, and Kenzu Abdella. 2023. "A Stochastic Weather Model for Drought Derivatives in Arid Regions: A Case Study in Qatar" *Mathematics* 11, no. 7: 1628.
https://doi.org/10.3390/math11071628