233 American Academic Scientific Research Journal for Engineering, Technology, and Sciences ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 http://asrjetsjournal.org/ Determination of Markov Chain Transition Probabilities for Daily Rainfall Data in Jordan Ahmad Osama Musleh a* , Fayez Ahmad Abdulla b a,b Jordan University of Science and Technology, Department of Civil Engineering, Irbid, Jordan a Email: aomusleh17@eng.just.edu.jo Abstract This study aims to determine Markov chain transition probabilities for daily rainfall data of 39 meteorological stations across Jordan. Two states were imposed to the chains, namely dry and wet, and first order was used as the dependence structure. This leads to four transition probabilities for each station in each month, namely dry- to-dry (pdd), dry-to-wet (pdw), wet-to-dry (pwd), and wet-to-wet (pww). In the end of the study, it is concluded that pdd > pdw for all stations in all months, and pww β‰₯ pwd in only 15.1% of the times, which are concentrated in the middle of the rainy season (i.e., December–March) at North of Jordan. Also, all months tend to be dry in the long term, especially October, November, April, and May. Most of the expected dry spell lengths range from 5 to 100 days, while the expected wet spell lengths range mostly from 1 to 2 days, which indicates the tendency of the Jordanian weather to be dry across the country. Keywords: rainfall; daily rainfall ;Markov chain; transition probabilities; equilibrium probabilities; spell lengths;Jordan. 1. Introduction 1.1. Overview Markov chain is widely used in the prediction of the occurrence of daily rainfall events on the basis of past observed data. It can be applied in different orders depending on the temporal extent of the effect of the state of a certain day. Zero-order Markov chain assumes no dependence in the states of the days along the rainfall sequence. First-order means that the state of a day is affected by the state of one preceding day. Second-order means that the state of a day is affected by the state of two preceding days, and so on. First order is a convenient choice to use in modeling rainfall [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] [11, 12, 13, 14, 15, 16, 17]. Different resolutions of rainfall can be simulated in a Markov chain, e.g., seasonal, monthly, daily, or hourly. Simulation of daily rainfall is commonly used and sufficiently useful for hydrological and agricultural applications [3, 18, 5, 6, 19, 10, 20, 11, 12, 21] [22, 23, 13, 24, 14, 15, 16, 17]. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 234 In a Markov chain, rainfall volumes are classified into states. Two states of dry and wet is a commonly selected choice in rainfall modeling, where dry refers to days of precipitation lower than or equal a small number (e.g., zero or 0.1 mm) and wet refers to days of precipitation amounts greater than that number [1, 2, 3, 18, 25, 4, 26, 10, 20, 11] [12, 21, 22, 23, 13, 24, 14, 15, 16, 17]. Depending on the observed data, the probability of each state to be followed by a certain state is calculated. These probabilities are called transition probabilities since they describe the probability of the transition from a state to a state. Transition probabilities are put in a matrix called transition probability matrix. For instance, for a two-state Markov chain, the probability matrix includes dry-to- dry, dry-to-wet, wet-to-dry, and wet-to-wet transition probabilities (i.e., pdd, pdw, pwd, and pww, respectively). From this matrix can the weather tendency be concluded whether it is more to the dry or to the wet state. However, Markov chain behaves poorly in long dry spells and when variations in seasonal trends of rainfall exist. Consequently, researchers have developed several improvements to Markov chain in order to handle these limitations [1, 2, 4, 7, 20, 11, 22, 13, 24]. These improvements are out of the scope of this paper. 1.2. Study area This study includes 39 rainfall stations across Jordan, of which the IDs, names and locations are shown in Figure 1. As a summary of their descriptive information, the years of record of the stations range from 22 to 78 years, except one station that has only 9 years of record. In more detail, 15% of the stations have 78 years of record, 33% more than 70, 40% more than 60, and 60% 50 years or more. All rainfall precipitation records are from October to May, except one record in June for Ras Muneif evaporation station. The number of rainy days in each month. These records are all in October and May, which are the beginning and the end of the rainy season, respectively. 1.3. The significance of the paper Research in Jordan lacks focus on rainfall stations. One previous research paper was found to study 13 meteorological stations in Jordan [27]. Another paper studied 6 stations [2]. Other research papers were found to study only 3 stations [28, 1]. This paper studies 39 stations across Jordan. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 235 # ID Name # ID Name 1 AD0019 Mafraq Airport 21 AN0003 Na'ur 2 AD0021 Turra 22 CC0001 Madaba 3 AD0032 Baqura 23 CC0004 Mushaqqar 4 AE0002 Irbid 24 CD0001 Sahab 5 AH0003 Ras Muneif 25 CD0002 Yaduda 6 AL0010 Deir Alla 26 CD0005 Jiza 7 AL0015 Zarqa 27 CD0010 Rabba 8 AL0016 Ruseifa 28 CF0006 Ghores-Safi 9 AL0018 Jubeiha 29 CF0007 Hasa 10 AL0019 Amman Airport 30 DA0002 Shaubak 11 AL0020 Ain Ghazal 31 ED0001 Aqaba 12 AL0035 Baq'a 32 ED0012 Ram 13 AL0048 Khaldiya 33 F 0002 H5 14 AL0053 King Talal Dam 34 F 0003 Azraq Police Post 15 AL0054 Hashimiya 35 F 0009 Azraq Evap. Station 16 AL0055 Wadi Dhuleil 17 AL0059 Um El-Jumal 36 G 0002 Jafr Police Post 18 AL0066 Khirebit Es Samra 37 G 0003 Ma'an 19 AM0001 Salt 38 G 0008 Jafr Evap. Station 20 AN0002 Wadi Es-Sir 39 H 0001 H4 Figure 1: The 39 rainfall stations included in this study across Jordan. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 236 Table 1: Number of rainy days in each month for each station Name Oct Nov Dec Jan Feb Mar Apr May Name Oct Nov Dec Jan Feb Mar Apr May Mafraq Airport 92 181 312 403 368 238 107 44 Na'ur 86 253 477 537 539 424 176 40 Turra 68 162 294 370 331 260 125 25 Madaba 77 264 455 544 511 405 141 39 Baqura 95 249 381 447 386 320 117 28 Mushaqqar 36 94 185 248 249 148 35 11 Irbid 149 318 513 609 569 483 224 69 Sahab 41 151 252 346 303 198 81 11 Ras Muneif 139 268 433 492 435 393 180 46 Yaduda 6 56 116 125 120 103 36 11 Deir Alla 122 279 458 559 488 411 160 53 Jiza 45 147 275 367 310 227 71 12 Zarqa 68 191 312 414 359 279 95 42 Rabba 58 209 380 473 442 335 121 18 Ruseifa 55 166 303 385 340 264 85 28 Ghores-Safi 25 51 96 145 131 89 38 5 Jubeiha 107 326 536 665 602 508 202 61 Hasa 22 61 68 104 81 82 24 6 Amman Airport 137 354 600 734 712 564 255 99 Shaubak 67 150 278 382 305 251 112 26 Ain Ghazal 4 18 51 57 65 46 22 10 Aqaba 27 47 105 119 87 77 45 9 Baq'a 86 196 350 413 413 316 119 39 Ram 8 14 21 43 24 23 13 2 Khaldiya 29 77 112 173 154 98 33 18 H5 64 162 251 304 310 227 111 49 King Talal Dam 46 146 242 300 310 218 68 21 Azraq Police Post 15 28 42 48 25 40 12 5 Hashimiya 30 94 141 174 182 115 45 15 Azraq Evap. Station 33 79 131 199 153 112 49 20 Wadi Dhuleil 52 134 236 312 301 208 71 23 Jafr Police Post 13 33 37 34 29 30 24 5 Um El-Jumal 49 160 250 320 288 210 85 26 Ma'an 52 86 138 218 171 152 57 26 Khirebit Es Samra 35 87 161 188 190 95 29 8 Jafr Evap. Station 21 19 28 35 32 27 18 4 Salt 105 302 526 618 610 503 196 64 H4 104 154 253 292 272 236 157 81 Wadi Es-Sir 100 277 520 587 545 431 190 41 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 237 2. Methodology The main concept of the Markov chain is the prediction of the state of a day based on the state of the previous day(s). Previous studies showed the validity of the first-order Markov chain for daily rainfall precipitation data [9], which means that the state of a day is dependent on the state of its previous day, not days. Two states are applied for Markov chain in this study: dry and wet. Dry state refers to the days that have zero rainfall precipitation depth. Wet state refers to the days that have rainfall precipitation depths greater than zero. Since two states are used, four transition probabilities (TPs) result for the model: dry-to-dry, dry-to-wet, wet-to-dry, and wet-to-wet. These probabilities are calculated using the following formulae: 𝑝𝑑𝑑 = 𝑛𝑑𝑑 𝑛𝑑 (11) 𝑝𝑑𝑀 = 𝑛𝑑𝑀 𝑛𝑑 (12) 𝑝𝑀𝑑 = 𝑛𝑀𝑑 𝑛𝑀 (13) 𝑝𝑀𝑀 = 𝑛𝑀𝑀 𝑛𝑀 (14) where 𝑝𝑑𝑑, 𝑝𝑑𝑀 , 𝑝𝑀𝑑 and 𝑝𝑀𝑀 are the dry-to-dry, dry-to-wet, wet-to-dry and wet-to-wet transition probabilities, respectively; 𝑛𝑑𝑑, 𝑛𝑑𝑀, 𝑛𝑀𝑑 and 𝑛𝑀𝑀 are the number of dry-to-dry, dry-to-wet, wet-to-dry and wet-to-wet days, respectively; and 𝑛𝑑 and 𝑛𝑀 the number of dry and wet days, respectively. By definitions, the following formulae can be concluded: 𝑛𝑑 = 𝑛𝑑𝑑 + 𝑛𝑑𝑀 (15) 𝑛𝑀 = 𝑛𝑀𝑑 + 𝑛𝑀𝑀 (16) 𝑝𝑑𝑑 + 𝑝𝑑𝑀 = 1 (17) 𝑝𝑀𝑑 + 𝑝𝑀𝑀 = 1 (18) These formulae were used as a final check for the model. Consequently, equilibrium probabilities (EPs) can be calculated using the following equations [6]: πœ‹π‘‘ = 1 βˆ’ 𝑝𝑀𝑀 (1 βˆ’ 𝑝𝑑𝑑) + (1 βˆ’ 𝑝𝑀𝑀) (19) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 238 πœ‹π‘€ = 1 βˆ’ 𝑝𝑑𝑑 (1 βˆ’ 𝑝𝑑𝑑) + (1 βˆ’ 𝑝𝑀𝑀) (20) where πœ‹π‘‘ and πœ‹π‘€ are the equilibrium probabilities of a dry and wet day, respectively. Since first order is assumed, the expected lengths of dry and wet spells can be calculated through the following equations, respectively [5]: 𝐸(𝑑) = 1 1 βˆ’ 𝑝𝑑𝑑 (21) 𝐸(𝑀) = 1 1 βˆ’ 𝑝𝑀𝑀 (22) Weather cycle (WC) can then be calculated as: π‘ŠπΆ = 𝐸(𝑑) + 𝐸(𝑀) (23) 3. Results and Discussion Half of the transition probabilities is shown in Table 2. The other half can be calculated through equations 17– 18. It can be noticed that pdd > 0.5 (i.e., pdd > pdw) for all stations in all months, and pwd > 0.5 (i.e., pwd > pww) for all stations in October, April, and May, except Ram station in May where pwd = 0.5 (i.e., pwd = pww). In November, pwd > 0.5 (i.e., pwd > pww) for all stations except Ras Muneif and Ain Ghazal. In December, pwd > 0.5 (i.e., pwd > pww) for all stations except Baqura, Irbid, Ras Muneif, Deir Alla, Amman Airport, Ain Ghazal, Baq'a, and Wadi Es-Sir. In January, pwd > 0.5 (i.e., pwd > pww) for all stations except Baqura, Irbid, Ras Muneif, Deir Alla, Jubeiha, Amman Airport, Ain Ghazal, Baq'a, King Talal Dam, Salt, Wadi Es-Sir, Mushaqqar, and Rabba. In February, pwd > 0.5 (i.e., pwd > pww) for all stations except Mafraq Airport, Baqura, Irbid, Ras Muneif, Deir Alla, Jubeiha, Amman Airport, Ain Ghazal, Baq'a, King Talal Dam, Salt, Na'ur, Mushaqqar, and Rabba. In March, pwd > 0.5 (i.e., pwd > pww) for all stations except Baqura, Irbid, Ras Muneif, Deir Alla, Jubeiha, Amman Airport, Ain Ghazal, Baq'a, and King Talal Dam. The equilibrium probability of a dry day for each month in each station (i.e., πœ‹π‘‘) is shown in Table 3. It can be concluded form the table that all values of πœ‹π‘‘ are greater than 0.65, and that in October, November, April, and May, they are all greater than 0.80. This indicates the tendency of those month to be dry in the long term. As noticed form the equations 19–20, the equilibrium probability of a wet day (i.e., πœ‹π‘€) can be calculated through the equation πœ‹π‘‘ + πœ‹π‘€ = 1 [6]. Expected dry and wet spell lengths (SLs) are shown in Table 4. All stations show higher dry spell lengths than wet spell lengths for all months. Dry spell lengths range from 5 to 1000 days, while wet spell lengths from 1 to 2 days, except Amman Airport station that has a wet spell length of 3 days in February. The wide variety in the dry spell lengths can be summarized as follows. Dry spells in October range from 15 to 167 days, with 76.9% American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 239 being less than 60 days; in November, they range from 8 to 100 days, with 79.5% being less than 25 days; in December, they range from 5 to 71 days, with 79.5% being less than 20 days; in January, they range from 5 to 62 days, with 89.7% being less than 25 days; in February, they range from 5 to 62 days, with 87.2% being less than 20 days; in March, they range from 6 to 77 days, with 87.2% being less than 25 days; in April, they range from 12 to 100 days, with 76.9% being less than 40 days; in May, they range from 26 to 1000 days, with 61.5% being less than 100 days. Weather cycle for each month–station crosscheck can be calculated using equation 23. Table 2 : Markov chain transition probabilities Station name TP Oct Nov Dec Jan Feb Mar Apr May Mafraq Airport pdd 0.957 0.914 0.865 0.807 0.831 0.893 0.95 0.977 pww 0.315 0.37 0.462 0.453 0.522 0.412 0.346 0.227 Turra pdd 0.965 0.921 0.863 0.842 0.844 0.896 0.949 0.985 pww 0.25 0.364 0.429 0.495 0.489 0.485 0.432 0.08 Baqura pdd 0.956 0.892 0.841 0.801 0.832 0.87 0.947 0.986 pww 0.379 0.494 0.553 0.55 0.588 0.534 0.41 0.286 Irbid pdd 0.947 0.89 0.823 0.797 0.797 0.846 0.924 0.969 pww 0.396 0.487 0.536 0.583 0.599 0.561 0.469 0.188 Ras Muneif pdd 0.935 0.881 0.801 0.787 0.796 0.828 0.914 0.976 pww 0.403 0.504 0.545 0.596 0.593 0.547 0.428 0.283 Deir Alla pdd 0.955 0.901 0.844 0.814 0.826 0.865 0.944 0.978 pww 0.352 0.452 0.514 0.556 0.559 0.513 0.412 0.226 Zarqa pdd 0.976 0.942 0.905 0.876 0.877 0.919 0.966 0.985 pww 0.191 0.356 0.372 0.408 0.376 0.387 0.211 0.19 Ruseifa pdd 0.981 0.942 0.902 0.88 0.874 0.92 0.972 0.99 pww 0.236 0.307 0.376 0.423 0.374 0.402 0.318 0.25 Jubeiha pdd 0.965 0.913 0.855 0.823 0.825 0.866 0.943 0.982 pww 0.271 0.472 0.499 0.54 0.543 0.504 0.406 0.328 Amman Airport pdd 0.958 0.903 0.846 0.813 0.806 0.862 0.927 0.968 pww 0.307 0.46 0.54 0.578 0.601 0.553 0.412 0.253 Ain Ghazal pdd 0.981 0.963 0.878 0.859 0.839 0.891 0.935 0.962 pww 0 0.529 0.529 0.536 0.594 0.522 0.364 0.1 Baq'a pdd 0.959 0.918 0.859 0.835 0.83 0.883 0.949 0.982 pww 0.306 0.444 0.501 0.535 0.576 0.532 0.395 0.308 Khaldiya pdd 0.971 0.933 0.905 0.846 0.857 0.913 0.972 0.983 pww 0.172 0.338 0.357 0.347 0.386 0.278 0.273 0.176 King Talal Dam pdd 0.975 0.934 0.89 0.874 0.857 0.911 0.964 0.99 pww 0.283 0.473 0.492 0.543 0.558 0.518 0.324 0.333 Hashimiya pdd 0.971 0.925 0.882 0.856 0.848 0.909 0.956 0.985 pww 0.2 0.404 0.39 0.408 0.47 0.377 0.178 0.133 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 240 Wadi Dhuleil pdd 0.976 0.941 0.895 0.864 0.866 0.913 0.968 0.989 pww 0.288 0.388 0.403 0.449 0.482 0.413 0.338 0.261 Um El-Jumal pdd 0.971 0.917 0.88 0.853 0.86 0.906 0.96 0.983 pww 0.163 0.35 0.42 0.478 0.497 0.438 0.365 0.077 Khirebit Es Samra pdd 0.965 0.923 0.87 0.856 0.839 0.928 0.969 0.992 pww 0.143 0.322 0.4 0.457 0.468 0.389 0.103 0.125 Salt pdd 0.971 0.91 0.852 0.832 0.821 0.864 0.94 0.981 pww 0.362 0.397 0.476 0.518 0.541 0.487 0.352 0.297 Wadi Es-Sir pdd 0.97 0.922 0.852 0.825 0.815 0.879 0.944 0.984 pww 0.36 0.469 0.513 0.511 0.494 0.494 0.416 0.146 Na'ur pdd 0.972 0.926 0.853 0.838 0.828 0.882 0.945 0.986 pww 0.291 0.443 0.46 0.488 0.521 0.495 0.381 0.225 Madaba pdd 0.974 0.924 0.874 0.846 0.832 0.889 0.958 0.986 pww 0.224 0.413 0.466 0.48 0.454 0.457 0.355 0.179 Mushaqqar pdd 0.969 0.928 0.87 0.83 0.821 0.904 0.969 0.991 pww 0.25 0.404 0.476 0.532 0.566 0.493 0.2 0.273 Sahab pdd 0.981 0.942 0.898 0.865 0.87 0.921 0.967 0.995 pww 0.22 0.397 0.381 0.445 0.439 0.374 0.346 0.273 Yaduda pdd 0.993 0.946 0.884 0.876 0.864 0.898 0.965 0.986 pww 0.333 0.446 0.457 0.48 0.467 0.456 0.417 0.182 Jiza pdd 0.986 0.951 0.924 0.887 0.898 0.931 0.977 0.995 pww 0.267 0.286 0.425 0.381 0.384 0.339 0.268 0 Rabba pdd 0.98 0.93 0.877 0.844 0.846 0.889 0.959 0.993 pww 0.345 0.435 0.492 0.514 0.532 0.463 0.397 0.278 Ghores-Safi pdd 0.987 0.973 0.954 0.932 0.932 0.959 0.98 0.997 pww 0.24 0.255 0.312 0.345 0.344 0.337 0.237 0.2 Hasa pdd 0.989 0.956 0.953 0.933 0.942 0.95 0.981 0.995 pww 0.409 0.267 0.294 0.365 0.333 0.378 0.208 0.167 Shaubak pdd 0.969 0.93 0.885 0.842 0.847 0.899 0.954 0.987 pww 0.299 0.36 0.45 0.492 0.416 0.454 0.402 0.192 Aqaba pdd 0.99 0.982 0.969 0.955 0.966 0.972 0.982 0.997 pww 0.185 0.17 0.343 0.185 0.218 0.169 0.133 0.222 Ram pdd 0.993 0.988 0.981 0.97 0.978 0.982 0.989 0.999 pww 0.125 0.214 0.19 0.349 0.208 0.304 0.231 0.5 H5 pdd 0.976 0.946 0.923 0.905 0.889 0.924 0.96 0.983 pww 0.188 0.34 0.39 0.395 0.381 0.33 0.27 0.224 Azraq Police Post pdd 0.988 0.978 0.969 0.964 0.979 0.976 0.989 0.995 pww 0.2 0.214 0.286 0.271 0.24 0.4 0.083 0 Azraq Evap. Station pdd 0.979 0.952 0.935 0.899 0.905 0.938 0.973 0.989 pww 0.091 0.203 0.359 0.377 0.288 0.268 0.265 0.25 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 241 Jafr Police Post pdd 0.994 0.985 0.983 0.983 0.984 0.985 0.989 0.997 pww 0.154 0.212 0.162 0.088 0.069 0.1 0.167 0 Ma'an pdd 0.983 0.97 0.954 0.932 0.935 0.95 0.979 0.992 pww 0.231 0.233 0.246 0.321 0.24 0.27 0.175 0.231 Jafr Evap. Station pdd 0.99 0.99 0.986 0.984 0.982 0.987 0.99 0.998 pww 0.238 0.211 0.25 0.314 0.188 0.259 0.167 0.25 H4 pdd 0.964 0.943 0.908 0.902 0.89 0.92 0.946 0.976 pww 0.26 0.253 0.281 0.349 0.29 0.322 0.312 0.358 Table 3: Markov chain equilibrium probabilities Station EP Oct Nov Dec Jan Feb Mar Apr May Mafraq Airport Ο€d 0.941 0.88 0.799 0.739 0.739 0.846 0.929 0.971 Turra Ο€d 0.955 0.89 0.806 0.762 0.766 0.832 0.918 0.984 Baqura Ο€d 0.934 0.824 0.738 0.693 0.71 0.782 0.918 0.981 Irbid Ο€d 0.919 0.823 0.724 0.673 0.664 0.74 0.875 0.963 Ras Muneif Ο€d 0.902 0.807 0.696 0.655 0.666 0.725 0.869 0.968 Deir Alla Ο€d 0.935 0.847 0.757 0.705 0.717 0.783 0.913 0.972 Zarqa Ο€d 0.971 0.917 0.869 0.827 0.835 0.883 0.959 0.982 Ruseifa Ο€d 0.976 0.923 0.864 0.828 0.832 0.882 0.961 0.987 Jubeiha Ο€d 0.954 0.859 0.776 0.722 0.723 0.787 0.912 0.974 Amman Airport Ο€d 0.943 0.848 0.749 0.693 0.673 0.764 0.89 0.959 Ain Ghazal Ο€d 0.981 0.927 0.794 0.767 0.716 0.814 0.907 0.959 Baq'a Ο€d 0.944 0.871 0.78 0.738 0.714 0.8 0.922 0.975 Khaldiya Ο€d 0.966 0.908 0.871 0.809 0.811 0.892 0.963 0.98 King Talal Dam Ο€d 0.966 0.889 0.822 0.784 0.756 0.844 0.949 0.985 Hashimiya Ο€d 0.965 0.888 0.838 0.804 0.777 0.873 0.949 0.983 Wadi Dhuleil Ο€d 0.967 0.912 0.85 0.802 0.794 0.871 0.954 0.985 Um El-Jumal Ο€d 0.967 0.887 0.829 0.78 0.782 0.857 0.941 0.982 Khirebit Es Samra Ο€d 0.961 0.898 0.822 0.79 0.768 0.895 0.967 0.991 Salt Ο€d 0.957 0.87 0.78 0.742 0.719 0.79 0.915 0.974 Wadi Es-Sir Ο€d 0.955 0.872 0.767 0.736 0.732 0.807 0.913 0.982 Na'ur Ο€d 0.962 0.883 0.786 0.76 0.736 0.811 0.918 0.982 Madaba Ο€d 0.968 0.885 0.809 0.772 0.765 0.83 0.939 0.983 Mushaqqar Ο€d 0.96 0.892 0.801 0.734 0.708 0.841 0.963 0.988 Sahab Ο€d 0.976 0.912 0.859 0.804 0.812 0.888 0.952 0.993 Yaduda Ο€d 0.99 0.911 0.824 0.807 0.797 0.842 0.943 0.983 Jiza Ο€d 0.981 0.936 0.883 0.846 0.858 0.905 0.97 0.995 Rabba Ο€d 0.97 0.89 0.805 0.757 0.752 0.829 0.936 0.99 Ghores-Safi Ο€d 0.983 0.965 0.937 0.906 0.906 0.942 0.974 0.996 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 242 Hasa Ο€d 0.982 0.943 0.938 0.905 0.92 0.926 0.977 0.994 Shaubak Ο€d 0.958 0.901 0.827 0.763 0.792 0.844 0.929 0.984 Aqaba Ο€d 0.988 0.979 0.955 0.948 0.958 0.967 0.98 0.996 Ram Ο€d 0.992 0.985 0.977 0.956 0.973 0.975 0.986 0.998 H5 Ο€d 0.971 0.924 0.888 0.864 0.848 0.898 0.948 0.979 Azraq Police Post Ο€d 0.985 0.973 0.958 0.953 0.973 0.962 0.988 0.995 Azraq Evap. Station Ο€d 0.977 0.943 0.908 0.86 0.882 0.922 0.965 0.986 Jafr Police Post Ο€d 0.993 0.981 0.98 0.982 0.983 0.984 0.987 0.997 Ma'an Ο€d 0.978 0.962 0.943 0.909 0.921 0.936 0.975 0.99 Jafr Evap. Station Ο€d 0.987 0.987 0.982 0.977 0.978 0.983 0.988 0.997 H4 Ο€d 0.954 0.929 0.887 0.869 0.866 0.894 0.927 0.964 Table 4: Expected dry and wet spell lengths (numbers are rounded to be integers to be expressive for numbers of days) Station SL Oct Nov Dec Jan Feb Mar Apr May Mafraq Airport E(d) 23 12 7 5 6 9 20 43 E(w) 1 2 2 2 2 2 2 1 Turra E(d) 29 13 7 6 6 10 20 67 E(w) 1 2 2 2 2 2 2 1 Baqura E(d) 23 9 6 5 6 8 19 71 E(w) 2 2 2 2 2 2 2 1 Irbid E(d) 19 9 6 5 5 6 13 32 E(w) 2 2 2 2 2 2 2 1 Ras Muneif E(d) 15 8 5 5 5 6 12 42 E(w) 2 2 2 2 2 2 2 1 Deir Alla E(d) 22 10 6 5 6 7 18 45 E(w) 2 2 2 2 2 2 2 1 Zarqa E(d) 42 17 11 8 8 12 29 67 E(w) 1 2 2 2 2 2 1 1 Ruseifa E(d) 53 17 10 8 8 13 36 100 E(w) 1 1 2 2 2 2 1 1 Jubeiha E(d) 29 11 7 6 6 7 18 56 E(w) 1 2 2 2 2 2 2 1 Amman Airport E(d) 24 10 6 5 5 7 14 31 E(w) 1 2 2 2 3 2 2 1 Ain Ghazal E(d) 53 27 8 7 6 9 15 26 E(w) 1 2 2 2 2 2 2 1 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 243 Baq'a E(d) 24 12 7 6 6 9 20 56 E(w) 1 2 2 2 2 2 2 1 Khaldiya E(d) 34 15 11 6 7 11 36 59 E(w) 1 2 2 2 2 1 1 1 King Talal Dam E(d) 40 15 9 8 7 11 28 100 E(w) 1 2 2 2 2 2 1 1 Hashimiya E(d) 34 13 8 7 7 11 23 67 E(w) 1 2 2 2 2 2 1 1 Wadi Dhuleil E(d) 42 17 10 7 7 11 31 91 E(w) 1 2 2 2 2 2 2 1 Um El-Jumal E(d) 34 12 8 7 7 11 25 59 E(w) 1 2 2 2 2 2 2 1 Khirebit Es Samra E(d) 29 13 8 7 6 14 32 125 E(w) 1 1 2 2 2 2 1 1 Salt E(d) 34 11 7 6 6 7 17 53 E(w) 2 2 2 2 2 2 2 1 Wadi Es-Sir E(d) 33 13 7 6 5 8 18 62 E(w) 2 2 2 2 2 2 2 1 Na'ur E(d) 36 14 7 6 6 8 18 71 E(w) 1 2 2 2 2 2 2 1 Madaba E(d) 38 13 8 6 6 9 24 71 E(w) 1 2 2 2 2 2 2 1 Mushaqqar E(d) 32 14 8 6 6 10 32 111 E(w) 1 2 2 2 2 2 1 1 Sahab E(d) 53 17 10 7 8 13 30 200 E(w) 1 2 2 2 2 2 2 1 Yaduda E(d) 143 19 9 8 7 10 29 71 E(w) 1 2 2 2 2 2 2 1 Jiza E(d) 71 20 13 9 10 14 43 200 E(w) 1 1 2 2 2 2 1 1 Rabba E(d) 50 14 8 6 6 9 24 143 E(w) 2 2 2 2 2 2 2 1 Ghores-Safi E(d) 77 37 22 15 15 24 50 333 E(w) 1 1 1 2 2 2 1 1 Hasa E(d) 91 23 21 15 17 20 53 200 E(w) 2 1 1 2 1 2 1 1 Shaubak E(d) 32 14 9 6 7 10 22 77 E(w) 1 2 2 2 2 2 2 1 Aqaba E(d) 100 56 32 22 29 36 56 333 E(w) 1 1 2 1 1 1 1 1 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 244 Ram E(d) 143 83 53 33 45 56 91 1000 E(w) 1 1 1 2 1 1 1 2 H5 E(d) 42 19 13 11 9 13 25 59 E(w) 1 2 2 2 2 1 1 1 Azraq Police Post E(d) 83 45 32 28 48 42 91 200 E(w) 1 1 1 1 1 2 1 1 Azraq Evap. Station E(d) 48 21 15 10 11 16 37 91 E(w) 1 1 2 2 1 1 1 1 Jafr Police Post E(d) 167 67 59 59 62 67 91 333 E(w) 1 1 1 1 1 1 1 1 Ma'an E(d) 59 33 22 15 15 20 48 125 E(w) 1 1 1 1 1 1 1 1 Jafr Evap. Station E(d) 100 100 71 62 56 77 100 500 E(w) 1 1 1 1 1 1 1 1 H4 E(d) 28 18 11 10 9 13 19 42 E(w) 1 1 1 2 1 1 1 2 4. Conclusions and Recommendations 4.1. Conclusions pdd > pdw for all stations in all months. pww β‰₯ pwd in only 15.1% of the times, which are concentrated in the middle of the rainy season (i.e., December–March) at North of Jordan. In the long term, all months tend to be dry, especially October, November, April, and May. The expected dry spell lengths range from 5 to 100 days, except 13 stations that have dry spell lengths greater than 100 days in May, while the expected wet spell lengths range from 1 to 2 days, except one station that has a wet spell length of 3 days in February. 4.2. Recommendations for Future Studies ο‚· Nonparametric methods for data resampling (e.g., kernel and nearest-neighbor estimators) are recommended to use before studying the data. ο‚· Spatial correlations of daily rainfall data are recommended to consider among the meteorological stations. 5. Ethical Statement We will conduct ourselves with integrity, fidelity, and honesty. We will openly take responsibility for my actions, and only make agreements, which we intend to keep. We will not intentionally engage in or participate in any form of malicious harm to another person or animal. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 233-247 245 6. Conflict of Interests We declare that we have NO conflict of interests in the subject matter or materials discussed in this paper. 7. Data Availability Statement The data associated with this paper are available with the authors and can be accessed if needed. 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