Simulation of climatic series with nonstationary trends

Journal of Hydrology 398 (2011) 33–43

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Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol

Simulation of climatic series with nonstationary trends and periodicities N.T. Kottegoda ⇑, L. Natale, E. Raiteri Dipartimento di Ingegneria Idraulica e Ambientale, Universita degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy

a r t i c l e

i n f o

Article history: Received 1 April 2010 Received in revised form 30 November 2010 Accepted 2 December 2010 Available online 5 December 2010 This manuscript was handled by G. Syme, Editor-in-Chief Keywords: Rainfall Rainfall simulation Gibbs sampling Trend Periodicity Fourier analysis

s u m m a r y A Gibbs sampling technique was recently proposed to model nonstationary trend and periodic components in monthly climatic series (Kottegoda et al., 2007). This paper describes the development of new techniques for simulating trends and periodicities estimated by the Gibbs procedure from observed rainfall data. The objective is to provide possible future climatic scenarios for planning, design and operation of water resource systems. In particular, trend is considered as an important indicator of climate change. Historical series of rainfall observations from Italy and Switzerland are used in this study. Support is found for the approach from the results of Mann–Kendall and associated tests for trend which do not provide evidence of long-term trends. Analysis of amplitudes and periodicities of waves in trend curves from Gibbs sampling highlights the diversity of the procedure. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Availability of water resources depends on future climatic scenarios. Some series of recorded or reconstructed data indicate that climatic processes have been changing from prehistoric times. How these changes are accounted for in time series analysis does not seem to follow a clearly defined approach. For example, the possibility of long-term trends is often investigated. Precipitation trends over past centuries have been analyzed by Diaz et al. (1989), Groisman and Easterling (1994) and Schönweise and Rapp (1997). Recently there has been increased use of the Mann–Kendall and Spearman’s rho tests in trend analysis: see, for example, Burn and Hag Elnur (2002), Cong et al. (2009), Hamed (2009), Libiseller and Grimvall (2002) and Yue et al. (2002). These nonparametric rank-based tests applied to detect monotonic trends are also suitable for nonnormal and censored data. If a trend is detected it is commonly treated as a deterministic component. There is the problem of differentiating between trends and natural variability. For instance, Diffenbaugh et al. (2008) showed that the number of tornados reported in the United States has been increasing steadily over the past half century by about 14 per year. However, because the collection of records began in the 1950s they could not determine whether there was a robust trend. ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (N.T. Kottegoda). 0022-1694/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2010.12.003

Koutsoyiannis (2006) deemed that a stochastic approach with the admission of scaling behaviour can reproduce climatic trends by considering them as large-scale fluctuations in a manner that is basically consistent. Budyko and Drozdov (1976) and Hulme (1995) had previously explored the relationship between the hydrological cycle and climatic change. Elsewhere, Underwood (2009) found by fitting smoothing functions to data that generalised additive models provide a framework for modelling nonlinear relationships in long-term rainfall series. On the practical side, Hurrell et al. (2003) extensively discuss climate changes induced by the atmospheric variability of the North Atlantic Oscillation. As in the common approach to trend analysis, the periodic component of a time series is related to natural cyclicity and in usual practice it is treated as deterministic. Variations caused by changes in solar intensity and other climatic factors are not taken into account. Adopting a different procedure, Kottegoda et al. (2007) combined a time series model and Bayesian statistics through the Markov chain Monte Carlo (MCMC) method of Gibbs sampling, treating trend and periodicity as nondeterministic components. In this way a stochastic dynamic system is applied to model climatic time series. Here the state functions that incorporate trend and periodicity which change in a process with respect to time are termed dynamic. Furthermore, the unknown variances of the trend and periodicity components are explicitly modelled. Thus a representative time series model can be adopted advantageously. This is

34

N.T. Kottegoda et al. / Journal of Hydrology 398 (2011) 33–43

equivalent to a random walk procedure in which the state variables change dynamically to capture the intrinsic structure of a historic series of climate. The Bayesian estimation procedure makes the method general and usable. The objective is to use trend and periodicity curves from Gibbs sampling derived from historical rainfall series as a basis for simulating possible climate scenarios. These simulations can have wideranging uses in design, planning and operation of water resources systems. Initially possible trends in rainfall series are investigated using Mann–Kendall tests and other methods. In addition, further insights are provided into the approach for modelling rainfall series by analysing amplitudes and periodicities of oscillations in trend curves from Gibbs sampling; these properties are compared with those observed in moving-averaged historical data. Rainfall series from Italy and Switzerland are used in this study.

the 19th century. Brunetti et al. (2006) give a comprehensive list of names and locations of rainfall stations in Italy. They find that high intensity precipitation has increased in recent decades and also during the early part of the previous century. In addition, Buffoni et al. (1999) discuss precipitation in Italy from 1883. Records in Switzerland used in this paper commenced in the late 19th century. All these data series have been sufficiently verified to confirm that the data used in this study, which pertains to monthly rainfall, are of good quality.

4. Initial investigations Results of two preliminary investigations are presented here. They support the particular stochastic approach for simulation that follows.

2. The trend and periodicity components 4.1. Mann–Kendall test For the analysis a time series is represented in the additive form

yi ¼ t i þ pi þ gi ;

ð1Þ

in which at time i, yi represents the observed value of the variable, ti and pi are the components of trend and periodicity, respectively, and gi is a random component of white noise. This can be applied to a time series of monthly rainfall. The trend component of Eq. (1) can be written as

t iþ1 ¼ ti þ si ;

ð2Þ

using an auxiliary variable si. The uncertainty in the trend component is modelled as

siþ1 ¼ si þ xi ;

ð3Þ

in which xi is a ‘latent’ variable (in Bayesian terminology) that describes the random fluctuations in this component. As regards the periodicity component pi in Eq. (1), one can model its deterministic part as a truncated Fourier series using two harmonics, representing the 6 months cycle and the annual cycle (see Kottegoda et al., 2004). To incorporate the nondeterministic elements in the periodicity component West and Harrison (1997) can be generally followed. In this way the trend component is treated as a simple linear process whereas the periodicity component is a linear sum of sine and cosine curves. Moreover, both components have important random elements. This leads to a generalised approach for modelling the state variables as random walks in which the variances are treated as parameters within a Bayesian estimation framework (in the manner of West and Harrison (1997)). Details of the particular Gibbs sampling procedure adopted to model trend and periodicity are given in Kottegoda et al. (2008). The properties defined here are illustrated in Fig. 1. This shows the power spectra from the trend, periodicity and random components resulting from Gibbs sampling applied to observed monthly rainfall at Lugano in Switzerland from 1864 to 2004. Fig. 1 is discussed further in Sections 5and 6. 3. Rainfall observations Fig. 2 shows the locations of rainfall observation stations in Italy and Switzerland referred to this paper. Meteorological observations have been made in Italy since the first network was established in 1654 by some of Galileo’s students. Consequently, stations at Bologna, Padova, Torino, Milan, Rome and Palermo have been in operation since the 18th century. At more than 15 other stations, such as Genova, observations started in the first half of

Initially the Mann–Kendall test for trend was applied to series of annual maxima of monthly rainfall in Italy and Switzerland recorded at sites shown in Fig. 2. See Appendix A for the statistic used in the test and its variance. The test should be applied to annual data (that is without seasonality). However, to take account of additional variability in the data monthly maximums from each year were tested rather than annual rainfalls. This results in a normally distributed nonperiodic series with insignificant lag1 serial correlation, such as 0.07 for Rome rainfalls for instance. Results of the tests are given in Table 1. Using a null hypothesis of no trend and a level of significance a = 0.05 with a two-tailed test, 7 out of 10 rainfall series in Italy show no trends. Also, 9 out of 12 series in Switzerland are trend-free. Please note that the probability of making a Type I Error in rejecting the null hypothesis (in one or more cases) when 22 series are tested is not 5% but 100(1–0.9522) = 68%. This will be higher for correlated series. The outcomes of the tests are confirmed by the results of other tests such as Sen’s slope method (Sen, 1968) and Spearman’s rho (Sneyers, 1990). These results do not indicate justification for incorporating trend as a deterministic component for the simulation of rainfall. The Mann–Kendall and other methods are applied to test for global or monotonic trends. Consequent to the findings reported here variable stochastic trends as specified in Eqs. (1) and (2) and in Gibbs sampling are simulated. 4.2. Amplitudes and periodicities of oscillations in trend curves Some important aspects of the procedure were investigated by analysing amplitudes and periodicities of oscillations in trend curves from Gibbs sampling. See, for example, Fig. 3a which comes from the application of the Gibbs sampler to obtain a trend curve of the monthly rainfall at Lugano in Switzerland for the period 1864– 2004. As in oceanography the term wave is used to represent an oscillation above or below a mean level. The height of a wave is referred to as its amplitude and its wave length, measured by the corresponding distance between two consecutive upcrossings of the mean level, as its periodicity. The amplitudes and periodicities show the diverse nature and extent of the oscillations in a trend curve from Gibbs sampling which are used for the simulation of the trend component. Likewise, the historical trend curve resulting from Gibbs sampling applied to monthly rainfall at Rome from 1782 to 2008 is shown in Fig. 3b. Also shown is a 5-year moving average of historical monthly rainfall over the same period.

35


80

(a) trend component

power

60 40 20 0 10

100

1000

period in months

power

4000

(b) periodicity component

3000 2000 1000 0 10

100

1000

period in months

power

150

(c) random component

100 50 0 10

100

1000

period in months Fig. 1. Power spectra applied to the trend, periodicity and random components resulting from Gibbs sampling applied to observed monthly rainfall at Lugano from 1864 to 2004.

Fig. 2. Locations of rainfall stations in Italy and Switzerland used in the study.

Fig. 4a gives normal plots of amplitudes of waves in trend curves from Gibbs sampling applied to monthly rainfall series at five diverse locations in Italy. This shows a mean of 21 mm and a range of 8 mm in the amplitudes around the mean value, i.e. from 17 mm to 25 mm. These are indicated on the horizontal normal deviate of zero. Similarities are seen in these plots from a regional

perspective, considering the five stations of Lugano, Roma, Perugia, Bologna and Torino. A comparison is made with corresponding properties after 5-year moving averages are applied to the historical monthly rainfall series at Rome as shown in Fig. 3b. Fig. 4b gives normal plots of periodicities of waves in trend curves from Gibbs sampling. It shows a mean of 60 months with

36


Table 1 Mann–Kendall test for trend applied to annual maximum monthly rainfalls in Italy and Switzerland. Rainfall station

n (years)

S

z

p

N.H.: no trend. Level of significance a = 0.05

Italy Asti Biella Milan Casale Torino Varallo Bologna Genova Perugia Rome

126 141 146 137 141 136 188 148 86 227

1137 561 232 379 26 18 727 427 560 2264

2.396 0.998 0.391 0.703 0.045 0.032 0.842 0.706 2.085 2.153

0.008 0.840 0.348 0.759 0.518 0.487 0.800 0.240 0.019 0.016

N.H. rejected No trend No trend No trend No trend No trend No trend No trend N.H. rejected N.H. rejected

Switzerland Basel 141 155 0.275 0.392 No trend Berne 141 736 1.310 0.905 No trend Chateau d’Oex 108 777 2.060 0.980 N.H. rejected Chaumont 141 184 0.326 0.628 No trend Davos 131 408 0.810 0.791 No trend Engelberg 141 380 0.676 0.250 No trend Geneva 141 986 1.756 0.960 No trend Lugano 141 345 0.613 0.730 No trend Santis 122 921 2.036 0.979 N.H. rejected Sils 141 663 1.180 0.881 No trend Sion 140 269 0.483 0.315 No trend Zurich 141 1276 2.273 0.988 N.H. rejected Rz p ¼ 1 ð2pÞ1=2 expðt2 =2Þdt; n is the number of years of data, S is the Mann–Kendall statistic (Eq. (A1)) and z is a N(0,1) variate normalized from S (Eq. (A3)) with probability of nonexceedance p. A two-tailed test is applied. The null hypothesis, N.H., is that there is no trend. See comments in text.

750 700 650 600 550

monthly raainfall, mm

500 450 400 350 300 250 200 150 100 50

1864 1867 1870 1873 1876 1879 1882 1885 1888 1891 1894 1897 1900 1903 1906 1909 1912 1915 1918 1921 1924 1927 1930 1933 1936 1939 1942 1945 1948 1951 1954 1957 1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005

0

year Fig. 3a. Historical trend curve resulting from Gibbs sampling applied to monthly rainfall at Lugano from 1864 to 2004.

a range of 36 months in the periodicities around the mean value, i.e. from 42 to 78 months. As in Fig. 4a a comparison is possible with the 5-year moving-averaged series from historical data at Rome of Fig. 3b. It follows from Fig. 4a that the mean amplitude is 17 mm from the trend curve for Rome rainfalls from Gibbs sampling compared with 6 mm in the 5-year moving-averaged historical series. Also,

the mean periodicity in Fig. 4b for Rome rainfalls is 66 months in the trend curve from Gibbs sampling compared with 22 months in the moving-averaged historical series. That is, the mean amplitude in the trend curve from Gibbs sampling is much higher than that of a moving-averaged historical series. Likewise the mean periodicity is much higher in the trend curve from Gibbs sampling than from a moving-averaged historical series. Results from data at

37


350 monthly total rainfall trend component, t 5-year moving average applied to historical monthly rainfalls trend curve from Gibbs sampling

300

monthly rainfall, mm

250

200

150

100

50

1782 1787 1792 1797 1802 1807 1812 1817 1822 1827 1832 1837 1842 1847 1852 1857 1862 1867 1872 1877 1882 1887 1892 1897 1902 1907 1912 1917 1922 1927 1932 1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 2007

0

year Fig. 3b. Historical trend curve resulting from Gibbs sampling applied to monthly rainfall at Rome from 1782 to 2008. Also shown is a 5-year moving average applied to historical monthly rainfall over the same period.

2

0.95

1

0.85

0

0.79 0.74 0.69 0.64 0.59 0.54 0.49 0.44 0.38 0.33 0.28 0.23 Rome Lugano Perugia Bologna Torino From 5 year moving-averaged monthly rainfall at Rome

-1

probability of non-exceedance

normal deviate

0.90

0.18 0.13 0.08

0.03

-2 0

10

20 amplitude in mm

30

40

Fig. 4a. Normal plot of amplitudes of waves in trend curves from Gibbs sampling applied to some monthly rainfall series in Italy and amplitudes from moving-averaged historical data at Rome.

other sites and the use of moving averages also show similar disparities. Furthermore, it is seen that for all stations the distributions in the outcomes from Gibbs sampling tend to the normal distribution,

an essential consequence of the Gibbs procedure, whereas those of the moving-averaged series are not normally distributed. Thus it is apparent that a simulation model based on moving averages rather than the trend curve from Gibbs sampling cannot

38


2

0.95

1

0.85

0

0.79 0.74 0.69 0.64 0.59 0.54 0.49 0.44 0.38 0.33 0.28 0.23 Rome Lugano Perugia Bologna Torino From 5 year moving-averaged monthly rainfall at Rome

-1

probability of non-exceedance

normal deviate

0.90

0.18 0.13 0.08

0.03

-2 0

20

40

60

80

100

periodicity in months Fig. 4b. Normal plot of periodicities of waves in trend curves from Gibbs sampling applied to some monthly rainfall series in Italy and periodicities from moving-averaged historical data at Rome.

5. Simulation using trend, periodicity and random components Initially attempts were made to simulate a trend curve from Gibbs sampling such as Figs. 3a and 3b for Lugano monthly rainfall using different types of time series models, for example, ARIMA and the like. These trials were not successful. This is because it is not possible to capture the extraordinary features of such a trend curve using conventional time series models. The main reason is that the trend, and likewise the periodicity component, in the Gibbs sampling procedure incorporates a random variable (see Eq. (3)) with a variance, treated as a random parameter in the Bayesian framework, that has a posterior distribution conditioned on sample data and previous values of the random variables (Kottegoda et al., 2007, pp. 56–57 and Fig. 16). The approach followed in this paper is to use a historical trend curve from Gibbs sampling as a basis for simulation of supplementary trend curves. This will be discussed shortly. The historical periodicity curve is similarly treated. These can be summed together with a random component, as shown in Fig. 1 and specified by Eq. (1), to obtain a synthetic series of monthly rainfall. Application is made to monthly series of rainfall. It is seen that the power spectra for the trend component in Fig. 1a and random component in Fig. 1c have relatively low power. It means that there are no hidden periodicities apart from the obvious ones of Fig. 1b. The oscillations represented in Fig. 1a for periods greater than, say, 50 months are discussed in Section 6. It is proposed to use harmonic analysis of the historical trend curve and hence a Fourier representation of the curve. The main reason is because in this way one can introduce random elements, as follows, that provide a means of simulation based on a long ser-

ies of observed rainfalls and hence obtain an unlimited set of trend curves, as possible climatic scenarios. In a system so devised one can substantially differentiate one trend curve from another. Thus the trend curve from Gibbs sampling can be represented by the sum of N harmonics. The numbers of parameters are reduced to a bare minimum through methods of regression as shown here. Each harmonic i is associated with coefficients, say, ai and bi

6

4

2

ln (power)

reproduce adequately the range of oscillations seen in historical data. Besides, the moving average procedure involves the prior specification of a window (e.g. 5-year moving average) whereas this is not a pre-requisite in the Gibbs sampling procedure.

0

-2

-4

-6

-8 0

5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

harmonic number Fig. 5. Relationship between ln(power) and harmonic number used in simulating trend curves resulting from Gibbs sampling applied to monthly rainfall at Lugano for the period 1864–2004 after Fourier analysis is applied to the trend curve.

39


3500 3000 2500 2000 1500 1000

power

500 10 9 8 7 6 5 4 3 2 1 0 1

10

period in months Fig. 6. Power spectrum with enlargement before and after two points representing the significant periods of 6 and 12 months in the periodicity component of Fig. 1.

-30

2

6 months period

-35

1

12 months period

beta

normal plot

-40

0

-45

-50

-1 -55

-60

-2

1000 0

1000

2000

3000

4000

5000

2000

3000

4000

5000

power

power Fig. 7. Normal plot of power for the periods of 12 and 6 months in the periodicity component of Fig. 1 resulting from Gibbs sampling applied to 5-year sub-samples of monthly rainfall at Lugano from 1864 to 2004.

that scale the inherent sin and cos terms. These coefficients are estimated from the data. If one denotes ða2i þ b2i Þ1=2 by Pi which represents power then harmonic analysis leads to a relationship between power and harmonic number. For the Lugano monthly rainfall this is shown in Fig. 5. The residuals (measured by the vertical distances between the points and the linear plot of Fig. 5) are found to be normally distributed. It was found that N = 80 is sufficient to represent the main features in the trend curve. To put it differently, by using N = 80 the standard error of regression in the plot of Fig. 5 is reduced compared, say, to the case N = 40.

Fig. 8a. Relationship of the beta coefficient with power for the period of 12 months in the periodicity component resulting from Gibbs sampling applied to 5-year subsamples of monthly rainfall at Lugano from 1864 to 2004. In the case of the 12 months period, the beta coefficient provides a better fit to power than the alpha coefficient.

The application to monthly rainfall at Lugano is used as an example for further elaboration. The procedure for simulation of trend is as follows: (1) The linear relationship of Fig. 5 in which the power tends to decrease with the harmonic number is used to simulate a set of powers of harmonics that represent a trend curve, commencing with the first harmonic. In Fig. 5 the errors have constant variance and are normally distributed. Addition-

40


(3) Thus one obtains a set of N randomized harmonics and the resulting Fourier curves. These are summed to obtain a simulated trend curve. (4) By repeating the procedure an unlimited set of simulated trend curves can be obtained.

-14 -16 -18

As regards the power spectrum represent periods significant, with shortly. The suggested follows:

-20

alfa

-22 -24 -26 -28 -30 -32 0

500

1000

1500

power Fig. 8b. Relationship of the alpha coefficient with power for the period of 6 months in the periodicity component resulting from Gibbs sampling applied to 5-year subsamples of monthly rainfall at Lugano from 1864 to 2004. In the case of the 6 months period, the alpha coefficient provides a better fit to power than the beta coefficient.

ally, the random errors around the mean power of the harmonic are also incorporated. For this purpose one can generate a set of N random standard normal numbers, scale them with the standard error associated with the linear regression of Fig. 5 and add to the corresponding mean powers. (2) The a coefficients are set from the relationship between a and power in the historical series. That is, after the power for a particular harmonic is estimated using Fig. 5 the a coefficient from the a and power relationship is then estimated. The b coefficient is determined from the power. (See the procedure for simulating the periodicity component that follows.)

periodicity component of the rainfall data, the is shown in Fig. 1b. Two harmonics, that of 12 months and 6 months, are found to be adaptations for randomization as discussed procedure for simulation of periodicity is as

(1) The power spectrum in Fig. 6 shows the powers around the peak powers for periods of 6 months and 12 months shown in Fig. 1b for the periodicity component. From the periodicity curve obtained after Gibbs sampling applied to the Lugano historical rainfalls (from 1864 to 2004) 25 sub-samples each of 5 years length are initially obtained. The powers of the periods of 12 months and 6 months are calculated for each subsample. This gives the variability for simulating the peak powers. Fig. 7 shows normal plots of power so obtained. Figs. 8a and 8b give the beta and alpha coefficients, respectively, for the periods of 12 months and 6 months which have useful relationships with power. Our decision to use the alpha and beta coefficients in this particular way was on the basis of which plot gives the least standard error of regression. (1) To simulate a nonstationary periodicity curve for Lugano rainfall a random standard normal number is initially generated. Then one enters Fig. 7 and obtains the power for each of the periods of 12 months and 6 months. (2) From the powers obtained from Fig. 7 one estimates the alpha and beta coefficients using the relationships given in Figs. 8a and 8b for the periods of 12 months and 6 months, respectively. Randomness is also introduced to these coefficients through generated standard normal numbers scaled by the standard errors of regression in Figs. 8a and 8b. The

6 5 4

ln (power)

3 2 1 0 -1 -2 -3 125

130

135

140

145

150

155

160

harmonic number Fig. 9a. Plot of ln(power) vs. harmonic number around harmonic number 141 (the peak power of which is excluded here) corresponding to the period of 12 months shown in Fig. 7 and the 141 year rainfall record at Lugano.

41


powers, i.e., the amplitudes of these periodicities, are randomized.

missing coefficients are found from the power in each application. This gives initially the two main Fourier curves to sum for the periodicity component. (3) In conventional time series models one would use, in the case of applications shown here, deterministic periods of 12 and 6 months. In this approach, as already stated, the

In addition, the randomization procedure is extended further by using also secondary powers around those of the two main harmonics pertaining to the deterministic periods of 12 and 6 months.

6 5 4 3 2 1

ln(power)

0 -1 -2 -3 -4 -5 -6 -7 -8 -9 250

255

260

265

270

275

280

285

290

295

300

305

310

315

harmonic number Fig. 9b. Plot of ln(power) vs. harmonic number around harmonic number 282 (the peak power of which is excluded here) corresponding to the period of 6 months shown in Fig. 7 and the 141 year rainfall record at Lugano.

300

historical trend curve resulting from Gibbs sampling a typical simulated trend curve 1% and 99% credibility limits applied to simulated trend curves mean curve from 100 simulations

200

150

100

year Fig. 10. Simulated trend curves based on the historical trend curve from Gibbs sampling applied to monthly rainfall at Lugano.

2004

1999

1994

1989

1984

1979

1974

1969

1964

1959

1954

1949

1944

1939

1934

1929

1924

1919

1914

1909

1904

1899

1894

1889

1884

1879

1874

1869

50 1864

monthly rainfall, mm

250

42


For this purpose exponential fits are made to the secondary powers of the 15 harmonics preceding and 15 harmonics following harmonic number 141 which corresponds to the annual cycle of the Lugano monthly rainfalls spanning 141 years. These are as shown by the logarithmic plots of Fig. 9a. Similarly for the 6 months period corresponding to harmonic number 282 secondary powers represented by 30 harmonics preceding and 30 harmonics following number 282 are used. These are shown by the logarithmic plots of Fig. 9b. The periods of 12 and 6 months are shown in Figs. 1 and 6. Figs. 9a and 9b are used to simulate secondary powers around the two main harmonics. One can then use ai and bi relationships with power in a manner similar to that adopted before in simulating the peak harmonics in Figs. 8a and 8b. This gives additional non-peak Fourier curves to add to the main peak curves of step 3. Their sum gives the periodicity curve. By these procedures randomization is incorporated in the periodicity component. As in the case of the trend component, the number of parameters are minimized by using methods of regression. Consequently, one can use only six parameters for the trend component and 18 parameters for the periodicity component. As part of the procedure of simulating a time series, the random component of Eq. (1) has no serial correlation and deterministic properties. Its simulation does not present any problems.

6. Conclusions and further discussion There is no evidence of global trend in the rainfall series from the results of Mann–Kendall of Table 1 and other tests. By considering amplitudes and periodicities of waves in trend curves in Figs. 4a and 4b respectively the versatility of the trend curves from Gibbs sampling are highlighted and the limitations of moving average methods are shown. The results of these preliminary investigations provide justification for the stochastic approach adopted here. The objective is firstly to provide a method of simulating a trend component using a trend curve from Gibbs sampling applied to a historical rainfall series. Each trend curve is represented by the sum of N harmonics. A harmonic number has its associated power and the a and b coefficients that scale the intrinsic sin and cos terms. Randomization procedures described here provide an unlimited set of simulated curves as possible future climatic scenarios. The trend component is of particular interest in the context of climatic change. In Fig. 10 the historical trend curve from Gibbs sampling and a typical simulated curve are shown. Fig. 10 also shows a mean simulated curve consequent to 100 repeated simulations. After ranking the ordinates 99% and 1% credibility limits are provided. Previously, Kottegoda et al. (2008) proposed the fitting of trend curves to sub-samples of a historical series of rainfall (such as five sub-samples of 25 years from the Lugano series) for similar use as possible future climatic scenarios. However, this has limitations imposed by the number and extent of sub-samples so obtained. The periodicity component is also simulated as the sum of N harmonics but it is concentrated around the harmonics for periods of 12 months and 6 months. This follows an initial division of the periodicity curve from Gibbs sampling obtained from a rainfall series into sub-samples of 5 years as a means of representing variability in harmonic power. In simulating both components the number of parameters are minimized by methods of regression. Thirdly, a random component is added as in Fig. 1c. Thus one can simulate a monthly rainfall series by summing the three components as specified in Eq. (1). It was found that conventional time series models are ineffective for simulating the curves from Gibbs sampling. For example,

ARIMA and related class of models fail to capture the properties of the trend curves shown in Figs. 3a and 3b mainly because of the conditioned distribution of the variances which is used in the derivation of the trend curve. Simulated trend curves account for shifts in the commencement and end of a wet or dry period, as encountered in practice. They also show differences in rainfall amounts between groups of years. Furthermore, there are differences in trend between one century and another. For example, these can be associated with oscillations that have periods greater than, say, 50 months seen in Fig. 1a for the trend component from Gibbs sampling from Lugano monthly rainfalls. Such outcomes are not possible by using conventional time series models. These aspects are important when one simulates a hydrological phenomenon that can be used for the planning and design of a water resources system. The simulation summaries of Fig. 10, for instance, can be adopted for a variety of possibilities. If one verifies the efficiency of a water resources system over its designed life span a wide range of different situations are encountered. Simulation can be helpful for investigating the variability of a phenomenon such as rainfall. The irrigation systems in Italy, for instance, have been in operation for periods longer than 100 years, as discussed by Kottegoda and Natale (1994). Their impact on the environment should be overhauled. For this purpose, the methods suggested here will provide an engineer possible climate scenarios that essentially incorporate oscillations in trend and periodicity. From the Gibbs sampling procedure one can simulate long spells of low flows over a number of years such as those observed during of the past 10 years in northern Italy; other models usually account for seasonal variability within an annual cycle and the variance is incorporated in the random component. Acknowledgement Financial support provided by Fondazione Lombardia per l’Ambiente is gratefully acknowledged. Appendix A A.1. Mann Kendall analysis for trend The Mann–Kendall test compares the relative magnitudes of sample data. Thus it is used as a nonparametric or distribution-free test for identifying trends in time series. Let x1, x2, . . . , xn represent a time series of, say, n annual rainfalls. The Mann–Kendall statistic is given by

S¼

n1 X n X

½xi xk

ðA1Þ

k¼1 i¼kþ1

where [xi xk] = 1 for (xi xk)>0, [xi xk] = 1 for (xi xk) < 0. The variance of S is defined by

r2S ¼

[xi xk] = 0

1 ½nðn 1Þð2n þ 5Þ 18

for

xi = xk,

ðA2Þ

if xi – xk in Eq. (A1) (that is, there are no ties).

For S > 0; n > 10 Z ¼ ðS 1Þ=rS Nð0; 1Þ S < 0; n > 10

Z ¼ ðS þ 1Þ=rS Nð0; 1Þ

ðA3Þ

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