Water resources in Australia are under pressure due to rapid population growth and anticipated climate change and variabilities. Total water consumption in Australia in 2004–2005 was 18 800 GL, of which 65% was used for agriculture, 11% for household purposes, 3% for manufacturing, 11% for distribution losses in water supply, sewerage and drainage services and the remainder was used for rural consumption (ABS, 2011). The country's population has doubled since 1955 and was 22 million in 2009. It is anticipated that an additional 4.5 million people will be added to the state capital cities in the next quarter of a century. This projected population growth will increase mains water demand. The IPCC (2007) has predicted that Australian average temperatures will increase by between 0.6 and 1.5 °C by 2030, while annual rainfall will decrease in the southern and north-eastern region and available moisture will also decrease all over Australia. Drought frequency will increase in the southern and south-eastern region, whereas heavy rainfall and tropical cyclone events will increase in the north-eastern regions. Therefore both population growth and climate change will impose significant pressures on the country's urban water security. Consequently several different measures have been considered for improving the resilience of urban water systems. Diversification of water sources by capturing and reusing rainwater, stormwater and treated wastewater is the key solution to reduce mains water demand and to increase water security (Beecham, 2003). A number of other studies have examined the potential for water sensitive urban design (WSUD) to reduce potable water demand in Australia (Beecham et al., 2005; Kandasamy et al., 2008).
Accurate information on spatial and temporal rainfall characteristics are essential in the design of different WSUD components such as bioretention basins, permeable pavements, constructed wetlands, infiltration trenches and roof gardens (Beecham et al., 2005). In particular, stormwater and rainwater harvesting schemes require rainfall information as a primary input variable. Analysis of rainfall characteristics is also becoming very important for rainfall forecasting in the context of climate change and variability. Australian rainfall exhibits a high degree of spatial and temporal variability, and the variability is influenced by natural climate phenomena such as the Southern Oscillation Index (SOI) (Beecham and Chowdhury, 2010; Chowdhury and Beecham, 2010). A significant number of studies have been conducted on Australian rainfall characteristics and modelling (Thyer and Kuczera, 2000; Harrold et al., 2003a, 2003b; Lambert et al., 2003; Srikanthan et al., 2005) and their relation to natural climate phenomena such as the El Niño-Southern Oscillation (ENSO), the Pacific and Indian Ocean sea surface temperature (SST) variabilities, the sub-tropical ridge, the Southern Annular Mode (SAM) and the Madden-Julian Oscillation (MJO) (Smith et al., 2000; Donald et al., 2006; Chowdhury and Beecham, 2010).
Although several studies have investigated rainfall properties, few have considered rainfall dry and wet spell properties. Dry and wet spells are defined as a series of consecutive dry and wet days, respectively. For reliable water supply, dry spell properties need to be considered in the design of storage components. Alternative water resources (rainwater and stromwater harvesting) require both wet and dry spell characteristics information for ensuring their supply reliability. In addition, the design of vegetative WSUD systems, such as bioretention basins and swales, should allow for the inter-event dry periods to ensure plant survival. The statistical behaviour of spells, their distribution and critical length are important parameters that need to be considered in the design of integrated urban water management systems. Several studies have been carried out on the distribution of dry and wet spells in various parts of the world (Gabriel and Neumann, 1957; Srinivasan, 1958; Green, 1970; Yap, 1973; Blairfish, 1975; Berger and Goosens, 1983; Theoharatos and Tselepidaki, 1990; Tolika and Maheras, 2005; Deni et al., 2008). A number a statistical distributions have been identified to fit the spell distributions in different regions. These distributions include geometric, compound geometric, logarithmic series, modified logarithmic series, Polya and truncated negative binomial distributions. To the best of our knowledge, no previous study has been conducted on characterizing dry and wet spells in Australia. This study is an attempt to fill this gap. Rainfall from two cities, Adelaide and Melbourne, is used. These two regions exhibit different rainfall patterns. The objectives of this study are to compare transitional probabilities of daily rainfall, to fit probability distributions to dry and wet spells and to compare the statistical properties of spells between the two regioins. An example analysis for WSUD has been provided in order to demonstrate how designs can be improved by considering spell characteristics.
2. Data and Methods
Details of the rainfall stations used in this study are presented in Table I. Rainfall in Adelaide and Melbourne exhibits seasonal (where total rainfalls exhibit monthly or seasonal fluctuation) and non-seasonal (where total rainfalls do not exhibit monthly or seasonal variation) patterns respectively. Daily rainfall data collected by the Australian Bureau of Meteorology have been used in this study. Dry days were defined using a rainfall threshold value of 0.1 mm, which is consistent with several previous studies (Gabriel and Neumann, 1957; Green, 1970; Berger and Goosens, 1983; Tolika and Maheras, 2005; Deni et al., 2008). Consecutive days with less than 0.1 mm rainfall are defined as dry spells. Similarly, consecutive days with equal to or greater than 0.1 mm rainfall are defined as wet spells.
Table I. Details of rainfall stations used in this study
In this study the length of spells in days was considered. Let wi and di be wet and dry spells respectively. The corresponding length of such spells in days are Lwi and Ldi, where wi and di = 1, 2, …, i and Ldi and Lwi = 0, 1, 2, … For the case of drought management or water supply security, the highest value of Ldi is critical and for flood and water harvesting, the highest value of Lwi is critical. The rainfall amount in each wet spell was not analysed. The spell lengths were calculated on a monthly basis. In some cases a spell started in the previous month. For example, if a spell started on 31st January and ended on 3rd February, it was considered as a 4 d long spell in February. On the other hand, if a spell started on 29th January and ended on 1st February, it was a 4 d long spell in January. When a spell started on 31st January and ended on 1st February, it was considered a 2 d long spell in January. Figure 1 schematically shows the dry and wet spells and their relationships in the time horizon.
Information about transition probabilities from one rainfall state (wet, dry or a specific rainfall depth) to other states are important for wet and dry spells forescasting. In this study the first, second and third order two-state transitional probabilities (TP) of wet and dry days were estimated. Two-states indicate either a wet day (W) if daily rainfall was at least 0.1 mm or a dry day (D) if daily rainfall was less than 0.1 mm. Consider P(W) and P(D) as probabilities of a wet day and a dry day respectively, then the respective transitional probabilities can be expressed as P(W/W), P(W/D), P(D/D) and P(D/W), where P(W/W) and P(W/D) refer to probabilities of a wet day followed by a wet day and a dry day, respectively and P(D/D) and P(D/W) refer to probabilities of a dry day followed by a dry day and a wet day, respectively. Only the duration of consecutive wet and dry days are important in analysis of the critical wet and dry spells for first, second and third order transitional probabilities. The first order and second order transitional probability matrix can be written as:
Six statistical distributions were selected on the basis of their previous applications in wet and dry spell analyses (Yap, 1973; Tolika and Maheras, 2005; Deni et al., 2008). These distributions were geometric, compound geometric, generalized Pareto, logarithmic series, Polya and truncated negative binomial distributions. Estimation of distribution parameters is an important task in fitting theoretical distributions. The method of maximum likelihood was applied for the geometric and logarithmic series distributions. The method of factorial moments was used for the Polya and compound geometric distributions. The method of moments was used for the generalized Pareto and truncated negative binomial distributions. The probability density functions and associated parameters for these theoretical distributions are listed in Table II. Goodness of fit for theoretical to the observed distributions was estimated using the Chi-square (χ2) test statistic (Salas, 1993).
Table II. Theoretical distributions used in this study
Probability density function
(N is number of spells, N1 is number of spells having 1 day length and a, b, c, d, r, p, m are distribution parameters).
Generalized Pareto (GPD)
f(x) = p(1 − p)x − 1
Logarithmic series (LSD)
x̄ = [(1 − p−1)ln(1 − p)]−1
f(1) = (1 + d)−m/d
m − µ− 1
Compound geometric (CGD)
f(0) = b(a − 1)−1
a = (µ− 1)(b − 1)
Truncated negative binomial (TNBD)
Statistical moments (mean, variance, skewness and kurtosis) and different percentile values of dry and wet spells were calculated and compared between the Adelaide and Melbourne stations. For the spell length series xt, where t is the spell length in days and N is the total number of spells in any month, the various statistical moments used in this study are listed below:
where µi is the ith moment about its mean value.
3. Result and Discussion
3.1. Transitional probability
It is assumed that the probability of rainfall on any day depends only on whether the previous day was dry or wet. Given the event on the previous day, the probability of rainfall is assumed independent of events on further preceding days. Such a probability model is often referred to as a first order (depends on the previous day) two-state (wet or dry) Markov chain model (Gabriel and Neumann, 1957, 1962). The first order Markov chain process is flexible enough to give a variety of useful statistical information such as transitional and stationary probabilities, the recurrence interval and the lengths of expected wet and dry spells (Lana and Burgueňo, 1998; Ochola and Kerkides, 2003). The transitional probabilities of Markov chains have also been used to generate synthetic sequences of wet and dry days (Jimoh and Webster, 1996). However the generation needs to be repeated at least 20 times for the average value (McMahon and Mein, 1986). The estimated first, second and third order two-state transitional probabilities of dry and wet days in Adelaide and Melbourne are shown in Figures 2a and b. For the reliability of water supply, the consecutive dry and wet day probabilities [P(D/D), P(DD/D), P(DDD/D), P(W/W), P(WW/W) and P(WWW/W)] are important. Persistent behaviour of dry and wet spells was previously modelled by some researchers using a first order Markov process (Sharma, 1996).
The first, second and second order TPs of wet day followed by a wet day/s [P(W/W), P(WW/W) and P(WWW/W)] in Adelaide exhibit seasonal patterns, which are mainly due to Adelaide's seasonal rainfall distribution. Unlike the first order TPs, the second and third order [P(DD/D) and P(DDD/D)] in Adelaide also exhibits seasonality. From Figure 2b, it is clear that Adelaide has a greater probability of dry periods than Melbourne. On average, during the Australian summer season (December–February) Adelaide has a 7.7% higher probability of dry periods than Melbourne. In autumn (March–May) and spring (September–November) seasons Adelaide has a 6% and 6.8% higher probability of dry periods, respectively. In the winter season (June–August) there is a 2% higher probability of a dry period in Adelaide. Individually in different months, Adelaide has a 10.4 and 8.6% higher probability of dry periods in November and February respectively. In the case of wet day transition probabilities, Adelaide has less probability of wet periods than Melbourne during the summer, autumn and spring seasons. In summer, spring and autumn, a 13.4, 11.4 and 1.3% lower probability of wet periods was observed in Adelaide, respectively. Winter shows a different scenario, with Adelaide having a 4.6 higher probability of wet periods than Melbourne. Except for May–August, Adelaide has less transitional probability of wet days. The maximum difference of 18% was observed in February, followed by a 16% in October, 14.7% in December and 13.2% in November. From the higher order TP analysis it can be postulated that the probability of longer dry spells is higher in Adelaide than in Melbourne.
3.2. Statistical properties of spells
Barron et al. (2003) defined dry and wet spells in two ways namely as meteorological and agricultural spells. The meteorological spell considers only the rainfall characteristics while the agricultural spell considers rainfall, evapo-transpiration and soil characteristics in a water balance approach. The meteorological spells are considered in this study and have been estimated on the basis of historical wet and dry day sequences. Different statistical moments estimated from the observed spells are listed in Tables III and IV. The mean dry spell length was found to be longest in summer and the maximum lengths were observed in February for both stations. On average, during November to March, the mean dry spell length in Adelaide is about 1.5 days longer than in Melbourne. In addition, the mean wet spell length was observed to be longer in Adelaide than in Melbourne during the autumn and winter seasons (March–September). The maximum mean wet spell lengths were 2.44 days in July in Adelaide and 2 days in September in Melbourne. All dry and wet spell lengths exhibited positively skewed distributions. In terms of reliability of supply, dry spell characteristics are of most importance. The design of water infrastructure such as storage components of WSUD systems needs to consider dry spell variability between cities in order to maintain the reliability of water supply. Whilst analysed for Adelaide and Melbourne in this study, a similar methodology can be applied to other cities in order to understand dry spell properties for water infrastructure design.
Table III. Dry spell length statistics for Adelaide (A) and Melbourne (M)
Table IV. Wet spell length statistics for Adelaide (A) and Melbourne (M)
3.3. Statistical distribution of spells
Six theoretical distributions have been fitted to the observed spells. Details of these distributions are presented in Table II. Dry and wet spell distributions are usually location dependent. Several authors have observed that wet and dry spells follow geometric or related distributions (Gabriel and Neumann, 1957, 1962). Some previous studies assumed a Poisson distribution for the occurrence of wet and dry spells (Lana and Burgueňo, 1998; Ochola and Kerkides, 2003) and a geometric distribution for the lengths of spells (Chapman, 1998; Bedient and Huber, 2002). The dry and wet spells observed at the two rainfall stations in Adelaide and Melbourne are statistically best fitted (at a 95% confidence level using the Chi-square test statistic) by geometric (GD) and compound geometric distributions (CGD). The Polya distribution (PLD) was also found to fit at a 95% confidence level to the Melbourne dry spell lengths and the Adelaide wet spell lengths. Figures 3 and 4 show observed and fitted distributions for Adelaide and Melbourne for selected months (February, June and October). The estimated parameters for the fitted distributions are presented in Table V.
Table V. Estimated parameters for fitted distributions
The results are consistent with previous findings. In earlier studies, the GD has been found to fit the dry and wet spells at Tel Aviv (Gabriel and Neumann, 1957). Similarly, the PLD was found to successfully fit the dry and wet spells in Belgium (Berger and Goosens, 1983) and in the Aegean area (Theoharatos and Tselepidaki, 1990). Yap (1973) successfully fitted the CGD to wet spells in Malaysia. The TNBD was found to successfully fit the sequence of dry and wet days in Greece (Anagnostopoulou et al., 2003; Tolika and Maheras, 2005). Deni et al. (2008) successfully described the characteristics of dry and wet spells in Peninsular Malaysia using the TNBD and the GD respectively.
3.4. Implications for water sensitive urban design
The guiding principles of WSUD are to provide source control of wastewater and stormwater and where possible to use stormwater in the urban landscape for maximizing recreational amenity. WSUD also promotes the use of alternative water sources for potable water demand management (Beecham, 2003; Kandasamy et al., 2008). Widely used WSUD technologies in Australia include rainwater tanks, swales, bioretention basins, constructed wetlands and permeable pavements. Green roofs and rainwater gardens are also becoming popular. In addition, flood control can be achieved using on-site stormwater retention technologies. Stormwater storage within porous paving systems has potential for later reuse (Myers et al., 2007). Appropriate design of storage capacity of WSUD components is also of great concern not only for ensuring supply reliability but also for sustaining vegetation growth in bioretential basins. Large rainwater storages providing supply to domestic users are often impractical at the residential allotment scale, so it is important to optimize storage volumes to minimize the tank size while maximizing supply during long dry periods, particularly where there are no other supplies available (Argue, 2004).
Rainfall spell properties are not directly considered as a design parameter in current WSUD practices. For example, Argue (2004) and ARQ (2005) provide indicative supply yields from rainwater harvesting systems for all State capitals in Australia. The average annual supply (kL) curves are expressed as a function of effective roof area (m2), tank storage volume (kL) and daily water demand (L). Using a continuous simulation method, curves for percentage of average annual runoff retained or harvested have been developed as a function of demand (L/s/m2) and storage (% of annual runoff volume). Incorporation of dry spell properties in this design procedures will increase security of water supply for end uses. Using the fitted distribution parameters to dry spell lengths (given in Table V), cumulative distribution curves for each month can be generated. Figure 5 shows cumulative distributions of dry spell lengths for both cities for February using the fitted compound geometric distribution. From these curves different percentile values of dry spell lengths can be obtained. From Figure 5, 50th and 75th percentile values of dry spell lengths for Adelaide and Melbourne are 5 and 3.75 d, and 9.5 and 7.25 d, respectively.
The following simplified analysis demonstrates how dry spell lengths can be useful in the design of lot-scale roof water harvesting systems. The analysis does not consider the overall security of supply, but simply the minimum storage volume required. For small residential sites in Australia, Argue (2004) suggests small rainwater storages of 1–2 kL capacity. Depending on allotment size, the number of inhabitants and individual preferences, storage capacities of 5–10 kL are often suggested. Storage capacity needs to be sufficient to meet the water demand during dry spells, particularly when other sources are absent or limited. For the average domestic water demand per standard allotment in Adelaide (870 L/d) and Melbourne (1049 L/d) (Argue, 2004), the corresponding minimum storage capacities at 50th percentile dry spell length (or alternatively at 50% supply security) are (870 L/d × 5 d) = 4350 L and (1049 L/d × 3.75 d) = 3934 L, respectively. Using 75th percentile values of dry spell length (at 75% supply security) these values correspond to (870 L/d × 9.5 days) = 8265 L and (1049 L/d × 7.25 d) = 7606 L, respectively. Therefore, for the same size property, increase of supply security from 50 to 75% nearly doubles the storage volume requirements. Also for the same size property, storage capacity in Adelaide needs to be larger than in Melbourne, even though the average domestic water consumption is higher in Melbourne. Consideration of dry spell lengths, using the methods outlined in this study, provides a cost-effective solution to storage capacity design, particularly where water resources are limited.
Dry and wet spell characteristics were analysed and compared for the two State capital cities of Adelaide and Melbourne in Australia. This is perhaps the first study to characterize dry and wet spell behaviour in Australia. In the context of climate change and variability and rapid population growth in all the Australian major capital cities, sustainable urban water management is considered a primary challenge. Water sensitive urban design (WSUD) principles are now practised in most parts of Australia in order to facilitate source control of stormwater, to manage mains water demand, and to diversify water supplies. Three basic objectives of urban water supply are to maintain adequate water quantity, quality and reliability of supply. Dry spell characteristics are important in maintaining the reliability of alternative water supplies. The findings of this study are summarized below:
•Transitional probability analysis indicated that Adelaide has a greater probability of dry periods than Melbourne. During the Australian summer (December-February) Adelaide has an average 7.7% higher probability of dry periods than Melbourne. Monthly statistics showed that Adelaide has a 10.4 and 8.6% higher probability of dry periods in November and February, respectively. On the other hand, during summer, spring and autumn, Adelaide has a 13.4, 11.4 and 1.3% less probability, respectively, of wet periods than Melbourne. Winter shows a different scenario, when Adelaide has a 4.6% higher probability of wet periods than Melbourne. The highest value of 18% less probability of wet spells in Adelaide was observed in February, followed by 16% in October, 14.7% in December and 13.2% in November. In comparison to Melbourne, higher order transitional probabilities of dry days in Adelaide were found to be higher than the first order transitional probabilities. Therefore, the probability of longer dry spells is higher in Adelaide than in Melbourne.
•The study revealed that dry and wet spells in Adelaide and Melbourne are best fitted by geometric and compound geometric distributions. The Polya distribution has also been found to fit well to the Melbourne dry spell lengths and Adelaide wet spell lengths.
•Current design procedures generally do not explicitly consider dry spell lengths. This may be because of the perception that continuous simulation analysis implicitly includes interevent periods. However, in reality the influence of the dry spells is not generally incorporated into the design of WSUD systems. A simplified analysis of storage volume requirements considering 50th and 75th percentile dry spell lengths indicated that, storage capacities for WSUD systems for household water supply in Adelaide generally need to be larger than Melbourne for the same lot configuration, even though domestic water consumption is higher in Melbourne. Increase of supply reliability from 50 to 75% increases required storage volume twofold. Therefore, consideration of spell properties could assist in developing sustainable WSUD solutions.
The concepts for this paper were presented at the 8th International Workshop on Precipitation in Urban Areas, St Moritz, Switzerland, December 2009. The authors are grateful to the workshop participants whose feedback and commentary have led to the development of this paper.