213 American Academic Scientific Research Journal for Engineering, Technology, and Sciences ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 http://asrjetsjournal.org/ Generation of Synthetic 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 generate synthetic daily rainfall data for 39 meteorological stations in Jordan by estimating the distributional parameters of daily rainfall occurrence and amounts. Daily rainfall occurrence was modeled by the use of rainfall interarrival times, which were fitted to the one-parameter exponential distribution, except zero values which were represented using the ratio of the number of zero interarrival times to the total number of times, which was called zero ratio. Daily rainfall amounts were fitted to the two-parameter gamma distribution. Goodness-of-fit for one of the stations was tested using chi-square test. This test was performed using Microsoft Office Excel. Distributional parameters were calculated for both occurrence and amounts models, and 100 sequences of synthetic daily rainfall data were then generated, of which every sequence included 1000 non-zero daily rainfall data points (1000 wet days) and 1000 interarrival times (1000 dry spells of which some have a length of zero). These sequences were generated using the embedded random number generators in Python, for one-parameter exponential distribution, two-parameter gamma distribution, and uniform distribution. Percent errors were then calculated and found all to be less than 10%, which was considered acceptable. Keywords: rainfall; daily rainfall; synthetic data; occurrence model; amounts model; gamma distribution; exponential distribution; Jordan. 1. Introduction 1.1. Overview Rainfall maintains the hydrological cycle and plays a critical role for sustainable agro-economical, water management and sustenance of human livelihoods [1]. Also, rainfall precipitation is one of the most important weather variables in simulation models [2]. Since the availability of the weather data limits the applicability of the simulation method [3], stochastic rainfall models (SRMs) are used as tools for creating long unlimited rainfall time series data whose statistical properties are close to those of observational records [4]. They are used for augmenting rainfall time series, producing multiple climate realizations for vulnerability assessment, and generating synthetic rainfall records for ungauged stations through interpolating model parameters from adjacent gauged sites [5, 4]. ------------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 214 Precipitation is the most important variable in the rainfall–runoff models and extreme flood runoff models for analyzing dam safety risk, assessing flood and drought risk, designing infrastructure, managing water resources, and examining reservoir operations and performance [6, 4]. Stochastic rainfall data are also often used as inputs into hydrological models to quantify uncertainty in environmental systems associated with climatic variability, which facilitates making decisions about risk-based design and system operations [7]. Due to the scarcity of historical data, randomly generated synthetic data (stochastic replicates of the historical data) that are based on the statistical characteristics of the historical data are important to overcome the scarcity issue, since the synthetic data can be used for estimating the missing historical data due to their similarity in the statistical characteristics. Also, forecast accuracy of rainfall models depends in the first place on the reliability of the past rainfall data provided to the model [8]. Simulation of rainfall over a region for long time-sequences can be very useful for planning and policymaking, especially when the economy is heavily reliant on rainfall [9]. SRMs are useful in the fields of water resources management, hydrology, ecology, meteorology, and agricultural science and engineering, and help in water resources planning, reservoir and watershed management, flood risk assessment, drought risk assessment, rainfall–runoff models, hydraulic structure design, infrastructure design, erosion prediction, design of landfills, design of facilities for storage and disposal of hazardous wastes, agricultural production and planning, crop- yield models, and climate change impact studies [5, 4, 10, 11, 6, 7, 12, 1]. Thus, proper monitoring and forecasting of rainfall are inevitable [1]. Simulation of rainfall precipitation has two proposed basic approaches: physical and mathematical approaches [10]. Physical approaches are limited in their scope and applicability due to the complexity of the underlying rainfall generation mechanisms [10]. Mathematical approaches are more widely used and called stochastic approaches as they consider rainfall precipitation as a random process [10]. Rainfall precipitation is considered a stochastic process. The basic stochastic processes follow the point process theory [6]. A point process is the process of occurrence of an event continuously along a temporal or spatial dimension. For instance, the arrival of a vehicle to a certain point on a road is a time series event, since the position is fixed, and the events occur during a time interval. This is called a temporal point process. If the time is fixed and a picture is taken to the whole road, the positions of the vehicles on the road form a spatial point process. The point process of interest in this study is rainfall precipitation, which is a temporal point process. Point processes are Poisson processes, of which the time intervals among successive points follow the exponential probability distribution. When the time is divided into equal-width classes, e.g., 24-hour classes, frequency analysis can be done and histograms result. The shapes observed in the histograms usually follow the gamma probability distribution. In this study, daily rainfall precipitation is modeled. Thus, the width of classes into which the time is divided is equal to one day. The time interval between each two successive rainfall precipitation events is called interarrival time. Interarrival times describe the occurrence model, while the frequency analysis of the rainfall precipitation depths describes the amounts model. One of the challenges in the simulation of rainfall precipitation is achieving satisfactory representation of the American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 215 observed process with high comprehensibility, applicability, and computational soundness, without falling into over-parameterization [10]. Different timescales are used in rainfall modeling and range from a year to a few minutes, but the daily timescale has been gaining the highest attention [10]. The simulation scheme of the SRMs is based on two steps of random processes: occurrence model and amounts model [4, 10]. One of approaches for simulating rainfall occurrence is the alternating renewal process. In the alternating renewal process, lengths of consecutive wet and dry spells are considered a random variable in a truncated negative binomial or truncated geometric probability distribution, or a mixture of two geometric distributions [10]. Unfortunately, parameter estimation in the alternating renewal process is problematic. This problem can be overcome by using another approach for simulating rainfall occurrence that is the Markov chain (MC) process, which is the most widely used base of the occurrence models due to its simplicity and effectiveness [4, 13]. Transformation of MC transition probabilities are used to rewrite MC-based models as generalized linear models (GLMs), which can be incorporated in commonly available statistical packages [10]. In an MC, the state of a day is estimated based upon the state(s) of its preceding day(s) [10]. The number of the preceding days on which the state of the day of interest depends is called the order of the MC [11]. Despite the wide use of first-order MCs for daily rainfall simulation, higher-order MCs are getting more popular in sake of improving the dependence structure of wet and dry spells, especially for long dry spells [4]. Low-order MC- based models (i.e., first- and second-order) cannot satisfactorily reproduce observed low-frequency variability; cannot be generalized to represent spatial dependence across multiple point locations; underestimate the interannual variability of wet days, which is governed by the day-to-day and low-frequency variations in the rainfall; and provide poor distribution of number of wet days and rainfall totals at annual time scale [2]. These shortcomings can lead to improper evaluation of the hydrological or agricultural behavior of a region and suboptimal policies for system management [2]. The variance in the rainfall that is unexplained by low-order MC-based models (i.e., overdispersion) is said to be associated with climatic nonstationarity and/or longer time scale variations in the rainfall [2]. In order to incorporate these factors in an SRM, a covariate containing atmospheric signals can be imposed to allow variations in the SRM parameters, which can also be conditionally modified based solely on previous values of aggregated time scale predictors [2]. Amounts models are classified into parametric, semi-parametric and nonparametric models [5]. In parametric methods, the shape of the underlying probability density function (pdf) and correlation structure are presumed, while in nonparametric methods neither is presumed, and the pdf is characterized by the observed time series [10, 14]. In parametric methods, rainfall amounts can be simulated conditionally or unconditionally [10]. In conditional simulation, rainfall amounts have different probability distributions depending on the states of the corresponding wet days, whether they are similar to or different from the day of interest, while unconditional simulation assumes a generic distribution for the rainfall amounts [10]. Probability distributions incorporated in parametric approaches are highly positively skewed and include two-parameter gamma, shifted gamma, one- parameter exponential, three-parameter mixed exponential, kappa, and Weibull distributions [10, 11]. The advantage of the parametric techniques is their capability for extrapolation of non-observed extreme values [5, 4], but the nonparametric techniques are more widely used due to their simplicity and high capability of American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 216 reproducing observations, although the latter lacks the extrapolation ability for unobserved extremes [5, 14]. Recently, mixed and hybrid probability distributions have been used in parametric techniques, e.g., mixed exponential, double gamma, and GP-Type III distributions [5]. The advantage of these probability distributions is their ability to reproduce the whole range of rainfall amounts, including the most extreme rainfall [4]. Examples of nonparametric techniques include histograms, nearest-neighbor algorithm, and kernel density estimation approach [5, 10, 14]. WGEN, CLIGEN, ClimGen and WeaGETS are examples of recently generated MC-based SRMs with parametric probability distributions [4]. Seasonality (periodicity) in rainfall simulation can be expressed in two approaches; one is imposing a seasonal trend on the important model parameters, such as a polynomial or Fourier function, and the other is handling seasons discretely [10, 11]. Different seasons may have different probability distributions or different dependence characteristics [10]. In Fourier series, every day of the year has its own unique model parameters, while in other models, days are grouped into seasonal groups for which the parameters are estimated [14]. SRMs can represent one point location (single-site models) or multiple point locations (multi-site models), of which the latter is comparatively complex [5, 2]. A simple approach for developing a multi-site model is to extend single-site models by driving them with temporally independent but spatially correlated random numbers [5, 2]. Also, multi-site models sometimes use transformations of the multivariate normal distribution, but when nonparametric methods are adopted, nearest-neighbor resampling is an easy choice [14]. The spatial and temporal intermittence of daily rainfall makes it the most difficult weather variable to simulate [11]. SRMs suffer the limitation of not representing rainfall event characteristics, which include wet spell (rainfall duration), total rainfall amount in one rainfall event (rainfall depth), distribution and dependence structure of daily rainfall amounts in a wet spell (temporal rainfall patterns), and the correlation structure of these three characteristics [4]. Rainfall event characteristics have essential influences on runoff and flood modeling. In multi-day extreme rainfall events, flood volume and duration are highly influenced by rainfall depth and duration, and the resulting surface runoff is significantly affected by their dependence structure [4]. Also, rainfall peak delay in rainfall events increases the severity of the resulting runoff and floods, which is an example of the importance of the effect of temporal rainfall patterns [4]. Consequently, the importance of including rainfall event characteristics in SRMs can be concluded. Thus, event-based rainfall models (rainfall event models), which reproduce rainfall event characteristics, have been developed to overcome the limitations of SRMs [4]. However, the output of rainfall event models can only be used as input to event-based hydrological models, since it is in the image of a sequence of rainfall events [4]. For this reason, a recently published research paper has suggested an MC-based SRM named as SDRM-MCREM (stochastic daily rainfall model with Markov chain rainfall event model) that generates rainfall time series while maintaining rainfall event characteristics [4]. Rainfall event models are classified into profile-based and pulse-based models, with representative Barlett- Lewis and Neyman-Scott models [5]. Rainfall depth and duration can be simulated jointly by copula functions or stochastically using Monte-Carlo American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 217 method which is preferred to be used for simulating temporal rainfall patterns as well [4]. 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 [15]. Another paper studied 6 stations [16]. Other research papers were found to study only 3 stations [17, 18]. This paper studies 39 stations across Jordan. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 218 # 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 213-232 219 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 213-232 220 2. Methodology 2.1. Goodness-of-fit Chi-square test is a well-known statistical goodness-of-fit test for testing whether a dataset follows a certain probability distribution. In this study, chi-square test was applied for Mafraq Airport station dataset for both occurrence and amounts model. That is, it was used for testing whether the interarrival times followed the exponential distribution and whether non-zero rainfall precipitation depth data followed the gamma distribution. It was shown that both sets of data followed their corresponding probability distributions except for a simple modification that was needed for the interarrival times’ test. This modification was made to exclude zero interarrival times from the model and represent them by a new parameter called zero ratio. This parameter is explained in section 3.3.1. 2.2. Model calibration 2.2.1. Occurrence model Occurrence model was constructed based on the concept of interarrival time. An interarrival time is the period between two successive rainfall events. In other words, it is the number of dry days between every two successive wet days. According to previous studies, interarrival times follow a statistical probability distribution called exponential distribution. The exponential distribution has one parameter called lambda. Lambda is calculated using the following equations: πœ† = 1 π‘šπ‘’π‘Žπ‘› (6) πœ† = 1 π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› (7) Therefore, lambda in this study was calculated using the following equation: πœ† = 1 2 ( 1 π‘šπ‘’π‘Žπ‘› + 1 π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› ) (8) In this study, however, this parameter was found not to be enough for modeling interarrival times. A huge inclination was detected in the zero values of interarrival times. Therefore, a new parameter was created to overcome this error. Lambda was hence used for modeling only interarrival times greater than zero. This means that interarrival times with a minimum of one day were found to follow the exponential distribution. Zero interarrival times were represented by taking a normalized ratio of their number to the total number of interarrival times, which was called zero ratio. Both of those parameters (i.e., lambda and zero ratio) were determined for the real rainfall datasets and the randomly generated synthetic datasets. Percent errors were then calculated for the two parameters. This was done by programming using Python. The code is shown in Appendix A. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 221 2.2.2. Amounts model After knowing whether a day is dry or wet using the occurrence model, the amounts of rainfall precipitation (i.e., rainfall precipitation depths) are estimated using the amounts model. Thus, the amounts model is only concerned with non-zero rainfall precipitation days (i.e., wet days). Previous studies show that it is acceptable to assume that rainfall precipitation depths for wet days follow the gamma distribution. Gamma distribution has two parameters: alpha and beta, which can be calculated as follows. 𝛼 = ( π‘šπ‘’π‘Žπ‘› π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘› ) 2 (9) 𝛽 = (π‘ π‘‘π‘Žπ‘›π‘‘π‘Žπ‘Ÿπ‘‘ π‘‘π‘’π‘£π‘–π‘Žπ‘‘π‘–π‘œπ‘›)2 π‘šπ‘’π‘Žπ‘› (10) These two parameters were calculated for both the original data and the randomly generated synthetic data, and percent errors were then calculated. This was done by programming using Python. The code is shown in Appendix A. 2.3. Model validation 2.3.1. Occurrence model 2.3.1.1. Graphical method For visual comparison, a graphical method was used for the validation of the occurrence model for Mafraq Airport station. This was done by imposing the histogram of the interarrival times of the synthetic data on the histogram of those of the observed data. The histogram bars of the observed data are in red, while those of the synthetic data are in blue; thus, the areas where the two histograms overlap are in violet. 2.3.1.2. Percent errors After calculating the exponential distribution parameters (i.e., lambda and zero ratio) of the interarrival times of the observed data, these parameters were calculated also for the synthetic data. However, the synthetic data parameters are not shown in the results. Instead, the percent errors of the synthetic data parameters based on the observed data parameters are calculated and shown in the results table. 2.3.2. Amounts model 2.3.2.1. Graphical method For visual comparison, a graphical method was used for the validation of the amounts model for Mafraq Airport station. This was done by imposing the histogram of the non-zero daily rainfall amounts of the synthetic data on the histogram of those of the observed data. The histogram bars of the observed data are in red, while those of American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 222 the synthetic data are in blue; thus, the areas where the two histograms overlap are in violet. 2.3.2.2. Percent errors After calculating the gamma distribution parameters (i.e., alpha and beta) of the non-zero daily rainfall amounts of the observed data, these parameters were calculated also for the synthetic data. However, the synthetic data parameters are not shown in the results. Instead, the percent errors of the synthetic data parameters based on the observed data parameters are calculated and shown in the results table. 3. Results and Discussion 3.1. Goodness-of-fit Goodness-of-fit was tested using the well-known chi-square test for Mafraq Airport station. Interarrival times were successfully fitted to the exponential distribution, and non-zero daily rainfall amounts to the gamma distribution. Results of chi-square tests for interarrival times and non-zero depths for Mafraq Airport station – October are shown in Tables 2–3. Table 2: Chi-square test for the occurrence model of Mafraq Airport station – October (fitting interarrival times to one-parameter exponential distribution) (1/mean = 0.046062407; 1/standard deviation = 0.043082865; πœ† = 0.044572636; 𝑋2 = 12.99295; πœ’2 = 23.68; 𝑋2 ≀ πœ’2 β‡’ H0 accepted; Zero frequency = 29; Zero ratio = 0.318681319) π‘˜ 𝑓 𝑝 𝐹 (𝑓 βˆ’ 𝐹)2 𝐹⁄ 2 8 0.085287 5.287814 1.391114 4 6 0.078013 4.83683 0.279721 6 2 0.07136 4.42431 1.328406 8 5 0.065274 4.046972 0.22443 10 5 0.059707 3.701817 0.455257 12 2 0.054615 3.386099 0.567399 14 3 0.049957 3.097308 0.003057 16 6 0.045696 2.833147 3.539866 18 2 0.041799 2.591515 0.135014 20 3 0.038234 2.370492 0.167172 22 1 0.034973 2.168319 0.629506 24 2 0.03199 1.983389 0.000139 26 3 0.029262 1.814231 0.775011 28 0 0.026766 1.6595 1.6595 30 0 0.024483 1.517966 1.517966 More 14 0.262585 16.28029 0.319388 Sum 62 1 62 12.99295 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 223 Table 3: Chi-square test for the amounts model of Mafraq Airport station – October (fitting non-zero data points to two-parameter gamma distribution) (mean = 3.031521739; standard deviation = 3.458352503; 𝛼 = 0.768392041; 𝛽 = 3.945279982; 𝑋2 = 3.811961161; πœ’2 = 15.51; 𝑋2 ≀ πœ’2 β‡’ H0 accepted) π‘˜ 𝑓 𝑝 𝐹 (𝑓 βˆ’ 𝐹)2 𝐹⁄ 0.5 22 0.209742 19.29624 0.378847 1 9 0.129102 11.87742 0.697084 1.5 10 0.100616 9.256651 0.059694 2 8 0.081893 7.534124 0.028808 2.5 7 0.068028 6.258608 0.087825 3 7 0.057192 5.26169 0.574288 3.5 4 0.048464 4.458652 0.047181 4 2 0.041298 3.799422 0.852214 4.5 2 0.03534 3.251255 0.481549 5 4 0.030339 2.791233 0.523467 More 17 0.197986 18.2147 0.081006 Sum 92 1 92 3.811961 3.2. Model calibration 3.2.1. Occurrence model Values of lambda and zero ratio for the observed daily rainfall data are shown in Table 4. 3.2.2. Amounts model Values of alpha and beta for the observed daily rainfall data are shown in Table 4. 3.3. Model validation 3.3.1. Occurrence model 3.3.1.1. Graphical method A graphical representation for original and synthetic data of Mafraq Airport station is provided in the histograms shown in Figure 2. Visual comparison shows a good similarity between observed and synthetic daily rainfall data except for May due to the lack of the observed records. 3.3.1.2. Percent errors Percent errors for the exponential distribution parameters (i.e., lambda and zero ratio) were calculated for the interarrival times of the synthetic daily rainfall data. All the percent errors for lambda and zero ratio are less than 10%, which is considered acceptable. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 224 3.3.2. Amounts model 3.3.2.1. Graphical method A graphical illustration for both non-zero observed data and non-zero synthetic data for Mafraq Airport station is shown as histograms in Figure 3. Visual comparison shows a good similarity between observed and synthetic daily rainfall data. (a) (b) (c) (d) (e) (f) (g) (h) Figure 2: Histogram illustration of observed and synthetic interarrival times for Mafraq Airport station, in (a) October (b) November (c) December (d) January (e) February (f) March (g) April (h) May (it: observed interarrival times; itr: synthetic (random) interarrival times) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 225 (a) (b) (c) (d) (e) (f) (g) (h) Figure 3 : Histogram illustration of observed and synthetic non-zero rainfall depths for Mafraq Airport station (a) October (b) November (c) December (d) January (e) February (f) March (g) April (h) May (nz: observed non-zero rainfall depths; nzr: synthetic (random) non-zero rainfall depths) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 226 3.3.2.2. Percent errors Percent errors for the exponential distribution parameters (i.e., lambda and zero ratio) were calculated for the interarrival times of the synthetic daily rainfall data. All the percent errors for lambda and zero ratio are less than 10%, which is considered acceptable. Table 4: Distributional parameters for the observed daily rainfall data Station Name Parameter Oct Nov Dec Jan Feb Mar Apr May Mafraq Airport Ξ» 0.044 0.083 0.135 0.193 0.176 0.112 0.05 0.022 z 0.319 0.372 0.463 0.452 0.522 0.412 0.346 0.222 Ξ± 0.76 0.648 0.465 0.526 0.647 0.73 0.689 0.339 Ξ² 0.251 0.127 0.102 0.11 0.15 0.146 0.216 0.168 Turra Ξ» 0.034 0.076 0.121 0.137 0.14 0.093 0.051 0.015 z 0.25 0.364 0.429 0.496 0.489 0.485 0.435 0.077 Ξ± 0.763 0.922 0.839 0.766 0.724 0.766 0.794 1.109 Ξ² 0.155 0.119 0.099 0.094 0.09 0.09 0.128 0.366 Baqura Ξ» 0.048 0.11 0.146 0.19 0.153 0.115 0.052 0.015 z 0.379 0.496 0.551 0.55 0.59 0.536 0.41 0.276 Ξ± 0.288 0.655 0.636 0.519 0.615 0.778 0.727 0.5 Ξ² 0.052 0.08 0.068 0.057 0.075 0.11 0.134 0.075 Irbid Ξ» 0.057 0.113 0.185 0.213 0.198 0.149 0.083 0.027 z 0.399 0.487 0.536 0.583 0.599 0.561 0.469 0.186 Ξ± 0.615 0.647 0.555 0.62 0.57 0.719 0.54 0.388 Ξ² 0.134 0.076 0.056 0.064 0.056 0.075 0.089 0.093 Ras Muneif Ξ» 0.064 0.113 0.17 0.196 0.188 0.157 0.086 0.023 z 0.406 0.506 0.545 0.596 0.594 0.547 0.428 0.277 Ξ± 0.53 0.611 0.647 0.628 0.608 0.611 0.506 0.49 Ξ² 0.098 0.068 0.06 0.057 0.053 0.057 0.071 0.079 Deir Alla Ξ» 0.045 0.102 0.155 0.214 0.178 0.14 0.061 0.022 z 0.352 0.453 0.513 0.556 0.559 0.513 0.415 0.222 Ξ± 0.606 0.471 0.565 0.677 0.631 0.631 0.536 0.557 Ξ² 0.163 0.064 0.079 0.096 0.096 0.102 0.094 0.167 Zarqa Ξ» 0.02 0.048 0.074 0.098 0.1 0.066 0.032 0.012 z 0.194 0.356 0.372 0.408 0.376 0.387 0.211 0.186 Ξ± 0.777 0.718 0.484 0.668 0.649 0.812 0.465 0.607 Ξ² 0.21 0.133 0.086 0.126 0.117 0.165 0.131 0.166 Ruseifa Ξ» 0.017 0.048 0.079 0.096 0.098 0.066 0.025 0.009 z 0.241 0.307 0.376 0.423 0.374 0.402 0.318 0.241 Ξ± 0.92 0.689 0.637 0.719 0.744 0.733 1.053 0.669 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 227 Ξ² 0.188 0.119 0.099 0.12 0.117 0.113 0.241 0.202 Jubeiha Ξ» 0.031 0.088 0.143 0.184 0.173 0.133 0.061 0.016 z 0.271 0.474 0.498 0.541 0.543 0.505 0.406 0.323 Ξ± 0.402 0.512 0.598 0.648 0.578 0.736 0.503 0.729 Ξ² 0.06 0.051 0.047 0.049 0.043 0.06 0.067 0.115 Amman Airport Ξ» 0.041 0.102 0.148 0.193 0.192 0.143 0.072 0.028 z 0.309 0.462 0.541 0.578 0.601 0.554 0.412 0.25 Ξ± 0.366 0.404 0.432 0.515 0.484 0.509 0.378 0.378 Ξ² 0.111 0.071 0.069 0.078 0.075 0.084 0.107 0.139 Ain Ghazal Ξ» 0.024 0.037 0.12 0.158 0.165 0.113 0.068 0.044 z 0 0.529 0.529 0.526 0.594 0.522 0.364 0.091 Ξ± 1.01 1.481 0.462 0.457 0.542 0.583 0.834 1.349 Ξ² 0.43 0.34 0.053 0.052 0.066 0.114 0.353 0.349 Baq'a Ξ» 0.037 0.088 0.142 0.169 0.167 0.115 0.053 0.019 z 0.302 0.446 0.501 0.535 0.576 0.532 0.395 0.3 Ξ± 0.866 0.446 0.49 0.624 0.597 0.735 0.684 0.885 Ξ² 0.182 0.053 0.054 0.064 0.066 0.073 0.149 0.246 Khaldiya Ξ» 0.029 0.06 0.097 0.142 0.142 0.088 0.026 0.017 z 0.172 0.338 0.357 0.349 0.383 0.278 0.281 0.167 Ξ± 1.107 0.763 0.667 0.514 0.639 0.786 0.624 0.382 Ξ² 0.38 0.166 0.11 0.09 0.124 0.165 0.206 0.092 King Talal Dam Ξ» 0.026 0.066 0.099 0.103 0.12 0.077 0.039 0.01 z 0.283 0.473 0.492 0.545 0.558 0.518 0.324 0.318 Ξ± 0.563 0.733 0.713 1.018 0.792 0.924 1.308 2.519 Ξ² 0.107 0.098 0.085 0.112 0.1 0.106 0.311 0.514 Hashimiya Ξ» 0.029 0.074 0.123 0.129 0.147 0.089 0.042 0.016 z 0.2 0.404 0.39 0.41 0.467 0.377 0.182 0.125 Ξ± 1.305 0.813 0.712 0.694 0.832 0.72 0.635 1.231 Ξ² 0.481 0.19 0.162 0.119 0.173 0.184 0.186 0.424 Wadi Dhuleil Ξ» 0.024 0.052 0.077 0.104 0.095 0.07 0.027 0.01 z 0.288 0.388 0.403 0.449 0.483 0.413 0.338 0.25 Ξ± 0.81 0.72 0.583 0.723 0.527 0.608 0.924 0.907 Ξ² 0.22 0.153 0.128 0.154 0.125 0.142 0.329 0.548 Um El-Jumal Ξ» 0.03 0.077 0.125 0.154 0.141 0.098 0.035 0.016 z 0.167 0.35 0.42 0.478 0.497 0.44 0.369 0.074 Ξ± 0.45 0.799 0.79 0.715 0.83 0.747 0.742 0.603 Ξ² 0.125 0.179 0.205 0.175 0.213 0.185 0.241 0.238 Khirebit Es Samra Ξ» 0.042 0.074 0.134 0.164 0.174 0.072 0.03 0.009 z 0.147 0.326 0.4 0.457 0.468 0.389 0.103 0.111 Ξ± 1.191 0.876 0.754 1.005 0.828 0.943 1.549 2.072 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 228 Ξ² 0.472 0.181 0.182 0.199 0.172 0.249 0.601 0.786 Salt Ξ» 0.026 0.094 0.153 0.185 0.186 0.143 0.066 0.019 z 0.365 0.399 0.475 0.519 0.541 0.488 0.352 0.292 Ξ± 0.55 0.638 0.698 0.838 0.76 0.797 0.669 0.94 Ξ² 0.069 0.044 0.041 0.044 0.047 0.051 0.068 0.124 Wadi Es-Sir Ξ» 0.032 0.079 0.151 0.182 0.194 0.124 0.062 0.016 z 0.364 0.469 0.512 0.511 0.494 0.494 0.416 0.143 Ξ± 0.567 0.552 0.582 0.732 0.64 0.776 0.479 0.88 Ξ² 0.086 0.046 0.041 0.045 0.044 0.054 0.052 0.118 Na'ur Ξ» 0.027 0.078 0.148 0.174 0.178 0.116 0.053 0.015 z 0.294 0.443 0.459 0.488 0.522 0.495 0.381 0.22 Ξ± 0.712 0.565 0.588 0.626 0.689 0.822 0.513 0.657 Ξ² 0.08 0.046 0.045 0.043 0.053 0.065 0.061 0.138 Madaba Ξ» 0.022 0.076 0.124 0.164 0.175 0.117 0.044 0.014 z 0.224 0.413 0.466 0.48 0.454 0.458 0.355 0.175 Ξ± 1.12 0.49 0.559 0.786 0.629 0.794 0.781 0.703 Ξ² 0.228 0.052 0.053 0.075 0.055 0.076 0.094 0.098 Mushaqqar Ξ» 0.04 0.073 0.132 0.194 0.188 0.093 0.03 0.011 z 0.25 0.404 0.478 0.532 0.569 0.493 0.206 0.25 Ξ± 1.146 0.654 0.484 0.724 0.725 0.546 0.545 0.946 Ξ² 0.19 0.066 0.045 0.069 0.067 0.055 0.065 0.177 Sahab Ξ» 0.022 0.056 0.088 0.122 0.119 0.075 0.032 0.005 z 0.22 0.4 0.382 0.445 0.439 0.374 0.346 0.25 Ξ± 0.792 0.588 0.593 0.735 0.702 0.695 0.547 0.648 Ξ² 0.143 0.083 0.058 0.076 0.07 0.06 0.069 0.065 Yaduda Ξ» 0.005 0.05 0.108 0.115 0.133 0.093 0.047 0.015 z 0.333 0.455 0.461 0.48 0.467 0.456 0.417 0.167 Ξ± 5.765 0.472 0.671 0.709 1.103 1.069 0.942 3.185 Ξ² 0.997 0.048 0.055 0.067 0.099 0.084 0.09 0.622 Jiza Ξ» 0.014 0.042 0.066 0.103 0.084 0.062 0.021 0.005 z 0.267 0.286 0.425 0.383 0.384 0.339 0.268 0 Ξ± 0.87 0.649 0.776 0.831 0.731 0.932 0.596 0.862 Ξ² 0.254 0.087 0.103 0.115 0.095 0.109 0.083 0.15 Rabba Ξ» 0.021 0.075 0.118 0.163 0.154 0.109 0.041 0.007 z 0.345 0.438 0.492 0.514 0.533 0.464 0.397 0.263 Ξ± 0.886 0.491 0.49 0.594 0.544 0.597 0.405 0.449 Ξ² 0.184 0.06 0.05 0.057 0.056 0.058 0.044 0.079 Ghores-Safi Ξ» 0.011 0.02 0.031 0.044 0.043 0.027 0.016 0.004 z 0.24 0.255 0.316 0.347 0.346 0.337 0.237 0.167 Ξ± 0.27 0.602 0.539 0.667 0.706 0.518 0.405 0.351 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 229 Ξ² 0.057 0.161 0.132 0.21 0.234 0.148 0.118 0.047 Hasa Ξ» 0.014 0.036 0.04 0.057 0.05 0.051 0.014 0.005 z 0.429 0.267 0.294 0.365 0.338 0.383 0.208 0.143 Ξ± 0.46 0.657 0.353 0.698 0.533 0.837 0.942 1.098 Ξ² 0.128 0.17 0.109 0.218 0.201 0.299 0.322 0.325 Shaubak Ξ» 0.035 0.065 0.113 0.162 0.155 0.098 0.044 0.014 z 0.299 0.36 0.451 0.492 0.416 0.454 0.402 0.185 Ξ± 0.562 0.375 0.458 0.506 0.592 0.444 0.453 0.465 Ξ² 0.16 0.059 0.049 0.054 0.063 0.046 0.06 0.076 Aqaba Ξ» 0.008 0.018 0.03 0.038 0.031 0.026 0.018 0.003 z 0.185 0.17 0.346 0.185 0.218 0.171 0.133 0.2 Ξ± 0.452 0.384 0.21 0.286 0.223 0.605 0.292 0.324 Ξ² 0.148 0.11 0.048 0.092 0.056 0.205 0.069 0.067 Ram Ξ» 0.006 0.009 0.014 0.022 0.017 0.014 0.009 0.002 z 0.143 0.214 0.19 0.357 0.217 0.304 0.231 0.333 Ξ± 0.673 0.945 1.056 1.085 0.858 0.588 2.394 0.681 Ξ² 0.379 0.204 0.322 0.268 0.207 0.076 0.663 0.194 H5 Ξ» 0.023 0.047 0.077 0.094 0.117 0.071 0.041 0.016 z 0.19 0.34 0.39 0.395 0.381 0.33 0.27 0.22 Ξ± 0.515 0.392 0.65 0.553 0.459 0.433 0.482 1.281 Ξ² 0.135 0.094 0.181 0.193 0.149 0.149 0.15 0.387 Azraq Police Post Ξ» 0.01 0.016 0.022 0.024 0.015 0.017 0.009 0.005 z 0.2 0.222 0.286 0.271 0.24 0.4 0.083 0 Ξ± 0.711 1.018 1.159 0.678 1.759 1.344 0.482 1.756 Ξ² 0.187 0.197 0.208 0.165 0.532 0.365 0.095 0.213 Azraq Evap. Station Ξ» 0.022 0.042 0.066 0.106 0.096 0.061 0.026 0.012 z 0.094 0.203 0.359 0.377 0.288 0.268 0.265 0.238 Ξ± 0.227 0.55 0.482 0.618 0.4 0.464 0.502 0.551 Ξ² 0.051 0.159 0.14 0.237 0.156 0.136 0.152 0.151 Jafr Police Post Ξ» 0.008 0.015 0.015 0.014 0.023 0.014 0.013 0.003 z 0.167 0.212 0.162 0.088 0.069 0.1 0.174 0 Ξ± 0.633 0.828 0.97 1.169 0.507 1.285 0.647 0.752 Ξ² 0.158 0.122 0.197 0.3 0.088 0.257 0.119 0.198 Ma'an Ξ» 0.015 0.027 0.045 0.067 0.065 0.05 0.022 0.008 z 0.235 0.233 0.246 0.321 0.24 0.27 0.175 0.222 Ξ± 0.34 0.447 0.423 0.663 0.663 0.524 1 1.398 Ξ² 0.073 0.115 0.126 0.253 0.191 0.138 0.265 0.393 Jafr Evap. Station Ξ» 0.007 0.007 0.01 0.011 0.013 0.009 0.041 0.002 z 0.25 0.211 0.25 0.314 0.188 0.259 0.167 0.2 Ξ± 0.573 0.408 0.929 0.353 0.198 0.559 0.722 2.014 American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 230 Ξ² 0.131 0.088 0.717 0.077 0.085 0.156 0.382 0.666 H4 Ξ» 0.029 0.057 0.083 0.098 0.11 0.073 0.055 0.021 z 0.262 0.253 0.281 0.349 0.29 0.322 0.312 0.354 Ξ± 0.418 0.454 0.577 0.628 0.475 0.593 0.416 0.607 Ξ² 0.121 0.109 0.176 0.221 0.144 0.186 0.101 0.208 4. Conclusions and Recommendations 4.1. Conclusions For all the stations included in this study, interarrival times of daily rainfall data can be represented using the one-parameter exponential distribution, except for zero values that can be represented using a ratio between the number of zeros and the total number of values called zero ratio. For all the stations included in this study, non- zero daily rainfall amounts can be represented using the two-parameter gamma distribution. Goodness-of-fit for one of the stations was tested using chi-square test. Interarrival times were successfully fitted to the exponential distribution, and non-zero daily rainfall amounts to the gamma distribution. 4.2. Recommendations for Future Studies ο‚· Markov chain is recommended to use in the calibration stage of the occurrence model. ο‚· Nonparametric methods (e.g., kernel and nearest-neighbor estimators) are recommended to use instead of the parametric method of assuming probability distributions beforehand. ο‚· Nonparametric methods for data resampling 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. 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. American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 213-232 231 References [1] A. M. Nyongesa, G. Zeng and V. Ongoma, "Non-homogeneous hidden Markov model for downscaling of short rains occurrence in Kenya," Theoretical and Applied Climatology, vol. 139, no. 3, pp. 1333-1347, 2020. [2] R. Mehrotra and A. 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