# Joint Modeling of Severe Dust Storm Events in Arid and Hyper Arid Regions Based on Copula Theory: A Case Study in the Yazd Province, Iran

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Copula Theory

_{X1}(x1), F

_{X2}(x2), …, F

_{XN}(x

_{N}))) is a connection function for the association of random variables X

_{1}, X

_{2}, …, X

_{N}with marginal functions ${F}_{{X}_{1}}\left({x}_{1}\right),{F}_{{X}_{2}}\left({x}_{2}\right),\dots .,{F}_{{X}_{N}}\left({x}_{N}\right)$ that and the parameter of $\theta $ is defined in Equation (1) [44].

#### 2.1.1. Estimation of the Parameters of the Copula Functions

_{1k}, x

_{2k}, …, x

_{pk}(k = 1, …, n) are random variables.

#### 2.1.2. Selecting the Copula Function

_{ei}is the empirical copula value; P

_{i}is the theoretical copula value; k is the number of model parameters; n is the number of observations; and L is the maximum log- likelihood function.

#### 2.1.3. Analysis of Bivariate Dust Storm Return Period

_{X}(x) is the marginal function of the one of the variables of maximum wind speed, geopotential height, and vertical velocity.

## 3. Results

#### 3.1. Determine the Marginal Functions Of’ Dust Storm Variables

#### 3.2. Choosing the Best Copula Function for Bivariate Modeling of Dust Storms

#### 3.3. Joint and Conditional Probability of Dust Storm

#### 3.4. Bivariate Return Period of Dust Storm

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The joint probability distribution function (JPDF) and corresponding contour for each pair of dust storm variables (

**a**) Student t copula, (

**b**) Gaussian copula.

**Figure 4.**The joint probability P(U ≥ u, V ≥ v) for (

**a**) geopotential height and wind speed, (

**b**): vertical velocity and wind speed, based on copula functions (left: Student t copula and right: Gaussian copula).

**Figure 5.**The conditional probability of geopotential height (

**a**) and vertical velocity (

**b**) given wind speed exceeding a certain value, ${u}^{\prime}$.

**Figure 6.**Univariate return period of dust storm events based on maximum wind speed (

**a**), geopotential height (

**b**), and vertical velocity (

**c**).

**Figure 7.**Bivariate return period (in years) and corresponding contour lines based on two pairs of dust storm variables ((

**a**,

**b**): ${T}_{UV}^{AND}$, (

**c**,

**d**): ${T}_{UV}^{OR}$).

Archimedean Family | Frank | Joint CDF | Generator Function |

$C\left(u,v;\theta \right)=\frac{1}{\theta}ln\left[1+\frac{\left({e}^{-\theta u}-1\right)\left({e}^{-\theta v}-1\right)}{{e}^{-\theta}-1}\right],\theta \ne 0$ | $-ln\frac{{e}^{-\theta \upsilon}-1}{{e}^{-\theta}-1}$ | ||

Gumbel | $C\left(u,v;\theta \right)=exp\left\{-{\left[-{(Lnu)}^{\theta}+{(-Lnv)}^{\theta}\right]}^{\frac{1}{\theta}}\right\},\theta \ge 1$ | ${(-ln\upsilon )}^{\theta}$ | |

Clayton | $C\left(u,v;\theta \right)={({u}^{-\theta}+{v}^{-\theta}-1)}^{-1/\theta},\theta >0$ | $\frac{{\upsilon}^{-\theta}-1}{\theta}$ | |

Rotated Joe | $1-[1-{\displaystyle {\displaystyle \prod}_{i=1}^{m}}(1-{(1-{u}_{i})}^{\theta}){]}^{1/\theta}$ | $-ln[1-\left(1-t{)}^{\theta}\right]$ | |

Rotated Gumbel | $C\left(u,v;\theta \right)=u+v-1+C\left(1-u,1-v\right)$ | ${(-ln\upsilon )}^{\theta}$ | |

Rotated Clayton | $C\left(u,v;\theta \right)=u+v-1+C\left(1-u,1-v\right)$ | $\frac{{\upsilon}^{-\theta}-1}{\theta}$ | |

Elliptical Family | Student-t | ${{\displaystyle \int}}_{-\infty}^{{t}_{\upsilon}{}^{-1}\left(u\right)}{{\displaystyle \int}}_{-\infty}^{{t}_{\upsilon}{}^{-1}\left(v\right)}\frac{1}{2\pi \sqrt{\left(1-{r}^{2}\right)}}\left\{1+\frac{{x}^{2}-2rxy+{y}^{2}}{\upsilon \left(1-{r}^{2}\right)}\right\}{d}_{x}{d}_{y}$ ${t}_{\upsilon}\left(x\right)={{\displaystyle \int}}_{-\infty}^{x}\frac{\Gamma \left(\left(\upsilon +1\right)/2\right)}{\sqrt{\pi \upsilon \Gamma \left(\upsilon /2\right)}}{(1+{y}^{2})}^{-\left(\upsilon +1\right)/2}{d}_{y}\begin{array}{cc}& \upsilon \ne 0\end{array}$ | - |

Gaussian | $C\left(u,v\right)={{\displaystyle \int}}_{0}^{u}\Phi \left(\frac{{\Phi}^{-1}\left(v\right)-\rho xy{\Phi}^{-1}\left(t\right)}{\sqrt{1-{\rho}^{2}xy}}\right)dt$ | - |

**Table 2.**Kendall correlation coefficient between two pairs of maximum wind speed–geopotential height and maximum wind speed–vertical velocity at different levels.

Variables | 500 hPa | 850 hPa | 1000 hPa |
---|---|---|---|

Maximum wind speed–geopotential height | −0.29 | −0.26 | −0.19 |

Maximum wind speed–vertical velocity | 0.14 | 0.27 | 0.3 |

Variables | CDF | Parameters | |
---|---|---|---|

Maximum wind speed | Wakeby | $F\left(x\right)=\xi +\frac{\alpha}{\beta}(1-\left(1-{u}^{\beta}\right)\frac{\gamma}{\delta}(1-\left(1-u{)}^{-\delta}\right)$ | α = 80.89, β = 12.59, γ = 4.56, δ = 0.057, ξ = 0.46 |

Geopotential height 500 hPa | GEV | $F\left(x\right)=exp(-\left(1+kz{)}^{-1/k}\right)$ $z=\frac{x-\mu}{\sigma}$ | κ = −0.54 σ = 98.19 μ = 5761.7 |

Vertical velocity | GEV | $F\left(x\right)=exp(-\left(1+kz{)}^{-1/k}\right)$ $z=\frac{x-\mu}{\sigma}$ | κ = −0.23 σ = 0.082 μ = −0.038 |

**Table 4.**Selection criteria of the best fit of copula functions for maximum wind speed–geopotential height.

Copula CDF | Parametric Estimation | Nonparametric Estimation | ||||||
---|---|---|---|---|---|---|---|---|

AIC | BIC | S_{OLS} | Parameter | AIC | BIC | S_{OLS} | Parameter | |

Frank | −5.38 | −3.85 | 0.3557 | −2.46 | −5.34 | −2.1 | 0.3557 | −2.3 |

Gaussian | −2.76 | −0.53 | 0.2533 | −0.46 | −2.31 | −0.88 | 0.3533 | −0.28 |

Rotated Clayton | −2.66 | −0.34 | 0.256 | −0.77 | −2.02 | −0.201 | 0.3590 | −0.48 |

Rotated Gumbel | −5.5 | −4.04 | 0.354 | −2.44 | −5.13 | −2.81 | 0.3543 | −1.14 |

Student t | −7.4 | −4.45 | 0.2312 | −0.57 2 | - | - | - | - |

Rotated Joe | −5.15 | −3.7 | 0.3451 | −2.68 | −4.2 | −2.6 | 0.3493 | −1.47 |

**Table 5.**Selection criteria of the best fit of copula functions for Maximum wind speed–vertical velocity.

Copula CDF | Parametric Estimation | Nonparametric Estimation | ||||||
---|---|---|---|---|---|---|---|---|

AIC | BIC | S_{OLS} | Parameter | AIC | BIC | S_{OLS} | Parameter | |

Frank | −2.82 | −0.489 | 0.036 | 1.85 | −2.24 | −0.0212 | 0.046 | 2.46 |

Clayton | −2.89 | −0.536 | 0.035 | 0.64 | −2.84 | −0.0157 | 0.047 | 0.643 |

Gumbel | −3.1 | −0.637 | 0.031 | 1.38 | −1.92 | −0.373 | 0.045 | 1.87 |

Gaussian | −3.81 | −2.31 | 0.029 | 0.289 | −2.28 | −1.86 | 0.042 | 0.571 |

Student t | −2.01 | −1.98 | 0.041 | 0.87, 1.58 | - | - | - | - |

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**MDPI and ACS Style**

Mesbahzadeh, T.; Mirakbari, M.; Mohseni Saravi, M.; Soleimani Sardoo, F.; Krakauer, N.Y. Joint Modeling of Severe Dust Storm Events in Arid and Hyper Arid Regions Based on Copula Theory: A Case Study in the Yazd Province, Iran. *Climate* **2020**, *8*, 64.
https://doi.org/10.3390/cli8050064

**AMA Style**

Mesbahzadeh T, Mirakbari M, Mohseni Saravi M, Soleimani Sardoo F, Krakauer NY. Joint Modeling of Severe Dust Storm Events in Arid and Hyper Arid Regions Based on Copula Theory: A Case Study in the Yazd Province, Iran. *Climate*. 2020; 8(5):64.
https://doi.org/10.3390/cli8050064

**Chicago/Turabian Style**

Mesbahzadeh, Tayyebeh, Maryam Mirakbari, Mohsen Mohseni Saravi, Farshad Soleimani Sardoo, and Nir Y. Krakauer. 2020. "Joint Modeling of Severe Dust Storm Events in Arid and Hyper Arid Regions Based on Copula Theory: A Case Study in the Yazd Province, Iran" *Climate* 8, no. 5: 64.
https://doi.org/10.3390/cli8050064