# Regional Analysis of Dust Day Duration in Central Iran
## 2 Materials and Methods
### Homogeneity Testing of DDD  
Discordancy Test  
[PERSON] [31] presented two statistical measures for regional homogeneity, which include the discordancy statistic D and the homogeneity statistic H.  
The discordancy test identifies inconsistent locations within the whole group. It is not easy to choose a value for D\({}_{\text{i}}\) (discordancy or mismatch criterion at station (i)) as a decision criterion for a mismatched station. Ref. [PERSON] [32] suggested that a station is awkward when the D\({}_{\text{i}}\) level is above a critical level. In large areas, stations with high D\({}_{\text{i}}\) are more likely to exist. However, it is still suggested that if a station has \(D_{i}>3\), that is considered a mismatch [31]. The discordancy statistic is defined as follows:  
\[\overline{U}=N^{-1}\sum_{i=1}^{N}u_{i} \tag{1}\]  
\[S=(N-1)^{-1}\sum_{i=1}^{N}(u_{i}-\overline{u})(u_{i}-\overline{u})^{T} \tag{2}\]  
\[D_{i}=\frac{1}{3}(u_{i}-\overline{u})^{T}S^{-1}(u_{i}-\overline{u}) \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Latitude** (\({}^{\circ}\)**N**) & **Longitude** (\({}^{\circ}\)**E**) & **Station Names** \\ \hline
30.88 & 55.25 & Anar \\
29.10 & 58.35 & Bam \\
28.73 & 57.67 & Jiroft \\
27.97 & 57.70 & Kahnooj \\
30.25 & 56.97 & Kerman \\
30.42 & 57.70 & Shahdad \\
29.55 & 55.68 & Sirjan \\
30.10 & 55.13 & Shahrbabak \\
33.38 & 52.38 & Ardestan \\
32.52 & 51.71 & Isfahan \\
33.78 & 55.08 & Khoor.Biabank \\
32.85 & 53.08 & Naeein \\
33.53 & 51.90 & Natanz \\
31.98 & 51.83 & Shahreza \\
32.52 & 51.85 & Kabotaabad \\
34.13 & 49.83 & Arak \\
33.88 & 50.48 & Mahalat \\
35.05 & 50.33 & Saveh \\
34.70 & 50.85 & Ghom \\
34.78 & 51.18 & Salafchegan \\
35.59 & 53.42 & Semnan \\
27.13 & 60.92 & Zabol \\
29.47 & 60.47 & Zahdan \\
32.43 & 53.62 & Aghda \\
31.90 & 54.28 & Yazd \\
30.50 & 54.25 & Marvast \\
36.18 & 57.65 & Sabzevar \\
32.89 & 52.28 & Birjand \\
34.03 & 58.18 & Ferdos \\
33.67 & 56.90 & Tabas \\
35.92 & 50.90 & Karaj \\
36.25 & 50.00 & Ghazvin \\
36.00 & 50.75 & Hashtgerd \\
35.75 & 51.88 & Abali \\
36.20 & 49.95 & Takestan \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Synoptic stations used in this study. Here \(u_{i}=\left[\tau_{2}^{i},\tau_{3}^{i},\tau_{4}^{i}\right]^{T}\) is the linear moment matrix at station i, N is the number of stations, and S is the sample covariance matrix [31; 32]. The homogeneity test statistics include three criteria: H\({}_{1}\), H\({}_{2}\), and H\({}_{3}\). If the value of this statistic is less than 1, the region is homogeneous; if it is between 1 and 2, the region is somewhat homogeneous; and if it is greater than 3, the region is completely discordant. In practice, it is stated that the H\({}_{1}\) criterion is more appropriate for this test [33]. The value of this statistic is obtained from the following equation:  
\[H_{i}=\left(V_{obs}-\mu_{V}\right)/\sigma_{V} \tag{4}\]  
with  
\[V_{obs}=\left\{\frac{\sum_{i=1}^{N}n_{i}\big{(}\tau_{2}^{i}-\tau_{2}^{R}\big{)} ^{2}}{\sum_{i=1}^{N}n_{i}}\right\}^{1/2} \tag{5}\]  
Here, \(n_{i}\) is the sample size at station i, \(\tau_{2}^{i}\) is the sample linear torque (L-CV), \(\tau_{2}^{R}\) is the mean sample linear torque (L-CV), \(\mu_{v}\) is the mean of V\({}_{\rm obs}\) values and \(\sigma_{v}\) is the standard deviation of the V\({}_{\rm obs}\) values [31; 32]  
To select the best distribution in an area, we can choose the one that gives the best value of a fit criterion. Tests that are used to select a distribution to fit data include the Chi-square test \(\chi^{2}\), Kolmogorov-Smirnov test, PPPC (Probability Plot Correlation Coefficient) test, correlation coefficient test (for example, normal distribution and normal log and Pearson type 3, Anderson-Darling test and linear moment curves) [31; 33; 34]. In this research, the statistics of Hosking and Wallis are used.