Most agricultural, hydrological and ecological models require long sequences of daily rainfall as a major meteorological input. However, at many sites, such data series are often too short to allow a good estimation of the probability of extreme events or such data are simply unavailable. This has led to the development of mathematical models, known as stochastic weather generators, frequently used to produce long synthetic weather series that are statistically similar to historical records (e.g., Wilks and Wilby, 1999). Numerous approaches for the generation of daily rainfall data at single point are available in the hydrological and climatological literature (e.g., Richardson, 1981; Srikanthan and McMahon, 1985; Woolhiser, 1992; Sharma and Lall, 1999; Hayhoe, 2000; Wan et al., 2005; Srikanthan et al., 2005; Zheng and Katz, 2008; Liu et al., 2009). These models are widely used because they are easy to formulate and fast to implement (Wilks, 1999). Nevertheless, an important limitation of commonly used single site daily rainfall models is their inability to represent the monthly characteristics of historical rainfall. Therefore, the resultant daily series, when aggregated into monthly totals, will not adequately represent important statistical characteristics of monthly series.
Because rainfall data at the daily scale form the basic data set for the monthly precipitation series, a proper daily model should preserve monthly characteristics in addition to preserving the daily characteristics (Srikanthan and McMahon, 2001). Wang and Nathan (2007) pointed out the importance of preserving the statistical characteristics of rainfall at different time scales for many applications such as the assessment of water supply systems. In order to address this outstanding problem, in their investigation of rainfall generation Wang and Nathan (2002) developed a daily and monthly mixed model for the simulation of precipitation at a single site. The model first generates two rainfall series, reproducing daily and monthly statistics. Next, the monthly series are used to modify the generated daily rainfall values, after incorporating the serial correlation (Srikanthan and McMahon, 2001). Srikanthan and Chiew (2003) and Siriwardena et al. (2002) implemented a simplified approach for the Wang and Nathan model by generating only one sequence of daily rainfall amounts, but whereby the daily rainfall is adjusted to match the monthly characteristics.
This study attempts to extend our previous work on a single-site daily precipitation model (Mhanna and Bauwens, 2009) to a daily and monthly mixed model. The stochastic rainfall model is developed mainly for running a simulation model used to evaluate the performance of small-scale rainwater harvesting systems in arid and semi-arid areas in the Middle East. These areas are generally characterized by very high temporal variability of the rainfall. Therefore, a sufficiently long rainfall record is potentially important in order to ensure that the pattern of dry and wet periods is well represented within the rainfall time series. The paper is structured as follows. In Section 2, the study area and data variability are described. The daily model and the adjustment procedure are presented in Section 3. In Section 4, criteria used to evaluate the performance of the model are given. Section 5 compares the statistics of the generated series with the observations. Finally, conclusions drawn from the results end the paper in Section 6.
2. Data and study area
The study covers a wide geographical area and several climatic zones in the Middle East. The daily rainfall measurements for this study come from three meteorological stations whose locations are shown in Figure 1: Gaza, Al-Muwaqqar and Ras El-sudr. The available data series respectively cover the periods 1973–2006, 1986–2008 and 1976–1992.
In general, the Gaza Strip has a typical semi-arid climate and is located in the transitional zone between a temperate Mediterranean climate in the west and north, and the arid desert climate of the Sinai Peninsula in the east and south. As a result, despite the small area of the Strip (365 km2), the annual rainfall decreases from 450 mm in the north to about 200 mm in the south over a distance of 45 km. However, the average annual rainfall at the selected site (Gaza) is about 370 mm. Most rain falls in the period from mid-October until the end of March. The period May to September is dry with no rainfall. The climate in Al-Muwaqqar is typical Mediterranean arid with wet winters and dry summers. The annual rainfall, falling mostly during December, January and February, varies from 50 to 200 mm, with an annual mean of 125 mm. Ras El-sudr is situated at the low coastal area of the Gulf of Suez. The climate of the area is arid. Precipitation over the area is scarce and random. The spatial distribution of the precipitation appears to be controlled mainly by topographical features. The annual rainfall varies from less than 10–35 mm with an annual mean of about 15 mm and occurs mostly in December, January and March.
3. The rainfall generation model
In a previous study (Mhanna and Bauwens, 2009), a single-site rainfall model was developed to generate daily precipitation data for arid and semi-arid areas in the Middle East. The daily precipitation generator is a Markov chain—Exponential or a Markov chain-Gamma model, depending on the rainfall station. A first-order two-state Markov chain is used to determine the occurrence of rainfall. When a wet day is generated, a two-parameter Gamma distribution or a one-parameter Exponential distribution is used to generate the precipitation amount. In this study, the daily and monthly rainfall generator is a Markov-Gamma model that consists of three parts: a rainfall occurrence model, a rainfall amount model and an adjustment process. These parts are described in the following paragraphs.
3.1. The rainfall occurrence process
The occurrence process is simulated using a first-order Markov chain. This model was chosen, in preference to higher order Markov chains, based on its adequacy as expressed by the Bayesian Information Criterion (BIC) (Katz, 1981) and the Akaike Information Criterion (AIC) (Akaike, 1974).
The first-order Markov model involves the assumption that the probability of rain on a certain day is conditioned on the wet or dry status of the previous day. Let Xt represent the binary event of precipitation or no precipitation occurring on day t. A wet day is defined as occurring whenever a trace or larger amount of precipitation was recorded, while dry days are days which are not wet. In this study, a day with a total rainfall of 0.1 mm or more is considered a wet day. The process is determined by using the two conditional transition probabilities, which give the probabilities of change from one state to another, for the wet-day occurrence pattern: P01, the conditional probability of a wet day (Xt = 1) given that the previous day was dry (Xt−1 = 0) and P11, the conditional probability of a wet day given that the previous day was wet. The unconditional probability of a wet day, π, can be derived as (e.g., Katz and Parlange, 1998):
For each month separately, the transition probabilities P01 and P11 need to be determined to provide a transition from 1 month to the next. As discussed by Wilks (2006), the parameter estimation procedure consists simply of computing the conditional relative frequencies, which yield the maximum likelihood estimators (MLEs).
3.2. The rainfall amount process
A two-parameter Gamma distribution was used to generate the rainfall amounts on wet days, as it was shown that this distribution well preserves the important daily statistical characteristics, i.e., mean, standard deviation and skewness of observed rainfall amounts, as compared to the normally used distributions, such as the exponential distribution (Mhanna and Bauwens, 2009). The probability density function for the Gamma distribution is (e.g., Katz, 1977; Vlček and Huth, 2009):
where α and β denote the parameters of shape and scale, respectively, and Γ(α) is the Gamma function evaluated at α.
The parameters of the model are calculated separately for each site and for each month in the (rainy) winter half of the year, by using the method of maximum likelihood through the Thom approach (Thom, 1958). The parameters are estimated from the non-zero observed rainfall amounts after subtracting 0.1 mm, as the Gamma distribution may generate precipitation less than the resolution in the observed data (0.1 mm). To recover the original distribution of rainfall amounts on wet days, 0.1 mm is then added to all generated values on wet days (Srikanthan and Pegram, 2009).
3.3. The adjustment process
The basic structure of the adjustment process, adopted from Srikanthan and Chiew (2003) and Siriwardena et al. (2002) is based on the Thomas-Fiering monthly model. Once the daily precipitation data (Yi) are generated for a given month, the monthly precipitation total (Z̃i) is obtained by summing the daily precipitation amounts. Next, a new monthly total (Zi) is produced by using the Thomas-Fiering model (Equation (3) below). Finally, all the simulated daily precipitation data are multiplied by the Thomas-Fiering monthly factor (Zi/Z̃i), resulting in a new daily rainfall series for that month.
The Thomas-Fiering monthly model can be described mathematically as:
where µ(Zi) is the historical mean monthly rainfall for month i; µ′(Zi) is the theoretical mean monthly rainfall for month i; σ(Zi) is the historical standard deviation of monthly rainfall for month i; σ′(Zi) is the theoretical standard deviation of monthly rainfall for month i, and, ri, i−1 is the historical correlation coefficient of monthly rainfall amounts between months i and i − 1.
The theoretical mean and variance of the rainfall total Zi, for the Gamma distribution model used above, over a month of n days are given by Katz (1985):
4. Model performance evaluation
The performance of the model is evaluated using statistical indicators at the daily and monthly scales. These statistics are calculated for every month in the (rainy) winter half of the year (October to March) and they include:
the mean, standard deviation and skewness of the monthly rainfall;
the mean daily rainfall in a month;
the mean daily rainfall, considering only wet days, in a month, and,
the standard deviation and skewness of the daily rainfall in a month.
Furthermore, the distributions of the 1 day precipitation as well as the distributions of precipitation totals over several consecutive days of the historical and generated rainfall are compared. These include the precipitation totals on 2, 3 and 5 consecutive days.
5. Results and discussion
This section presents a comparison of the historical and the synthetic series. For more details concerning the parameters employed to generate daily precipitation and the statistics of original series, readers are referred to Mhanna and Bauwens (2009).
The Thomas-Fiering monthly factors (Zi/Z̃i) are given in Table I. Generally, the factor is typically within the range 1 ± 20%. The monthly statistics derived from the generated and the historical rainfalls are compared in Figure 2. The mean monthly precipitation is successfully reproduced by the model and the quality of data is satisfactory for the three stations. The model also preserves the monthly standard deviation very well. The monthly skew coefficient is generally considered to be satisfactorily preserved by the model. All points, except one, lie within the 95% confidence level, which confirms that the synthetic values do not differ much from the observed ones.
Table I. Monthly values of the Thomas-Fiering factor
The various daily statistics derived from the generated and the historical rainfalls are compared in Figure 3. The results show that the generation model was successful in producing the mean daily precipitation at the three stations. The mean daily generated precipitation shows no significant differences with the observed ones. In addition, the model preserves the mean daily precipitation on wet days adequately. The daily standard deviation and daily skew coefficient are also well reproduced by the model.
Concerning the distributions of the 1 day of rainfall, the results show that the model can reproduce with high reliability the properties of the distributions of the precipitation amounts. The modelling of the distributions of the precipitation totals over 2, 3 or 5 days is somewhat less successful. As an example, Figure 4 shows the distributions of the exceedance probabilities of the accumulated precipitation amounts in 5 consecutive days at the selected stations. Around 67% of the two-sample Kolmogorov–Smirnov tests (K–S) indicate that the measured and the generated distributions of the rainfall totals over consecutive days are not significantly different at the 0.05 probability level. A similar result was observed in a previous investigation (Mhanna and Bauwens, 2009), where just 63% of all combinations (month-location and consecutive days) passed the K–S test. Consequently, preserving the monthly characteristics in addition to the daily ones seems to have no significant impact on the modelling of the distributions of consecutive days of rainfall, as it was assumed.
A single site daily and monthly precipitation generator is developed for the simulation of rainfall occurrences and amounts in arid and semi-arid areas in the Middle East. A first-order two-state Markov chain is used to determine the occurrence of rainfall. The rainfall amounts on wet days are generated by using the two-parameter Gamma distribution. The basic structure of the generation process consists of the generation of a sequence of daily rainfall amounts and the subsequent adjustment of the daily rainfall by using the Thomas-Fiering monthly model. This procedure ensures that the daily and monthly characteristics of the rainfall are reproduced.
Statistical analyzes of the historical and synthetic rainfall series show that the model generally performs well as all the important characteristics of rainfall at the daily and monthly scales are preserved. Only the simulation of the distributions of the precipitation totals over several consecutive days was less successful, although satisfactory for the majority of the stations. It seems that preserving the monthly characteristics in addition to the daily ones does not enhance the modelling of the distributions of the rainfall totals on several consecutive days, as it was assumed. Therefore, the model has to be further improved when the distributions of the rainfall totals over consecutive days is considered to be important. This could be done by, for example, fitting a probability distribution model to predict the lengths of wet and dry spells (e.g., Ochola and Kerkides, 2003).