Theor Appl Climatol DOI 10.1007/s00704-015-1667-8
ORIGINAL PAPER
Changing patterns in rainfall extremes in South Australia Mohammad Kamruzzaman 1 & Simon Beecham 1 & Andrew V. Metcalfe 2
Received: 22 April 2015 / Accepted: 25 October 2015 # Springer-Verlag Wien 2015
Abstract Daily rainfall records from seven stations in South Australia, with record lengths from 50 to 137 years and a common period of 36 years, are investigated for evidence of changes in the statistical distribution of annual total and annual average of monthly daily maxima. In addition, the monthly time series of monthly totals and monthly daily maxima are analysed for three stations for which records exceed 100 years. The monthly series show seasonality and provide evidence of a reduction in rainfall when the Southern Oscillation Index (SOI) is negative, which is modulated by the Pacific Decadal Oscillation (PDO). However, the monthly series do not provide any evidence of a consistent trend or of any changes in the seasonal pattern. Multivariate analyses, typically used in statistical quality control (SQC), are applied to time series of yearly totals and of averages of the 12 monthly daily maxima, during the common 36-year period. Although there are some outlying points in the charts, there is no evidence of any trend or step changes. However, some supplementary permutation tests do provide weak evidence of an increase of variability of rainfall measures. Furthermore, a factor analysis does provide some evidence of a change in the spatial structure of extremes. The variability of a factor which represents the difference
* Mohammad Kamruzzaman
[email protected] Simon Beecham
[email protected] Andrew V. Metcalfe
[email protected] 1
Centre for Water Management and Reuse, School of Natural and Built Environments, University of South Australia, Mawson Lakes, SA 5095, Australia
2
School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
between extremes in the Adelaide Hills and the plains increases in the second 18 years relative to the first 18 years. There is also some evidence that the mean of this factor has increased in absolute magnitude.
1 Introduction The intensity, duration and spatial distribution of extreme rainfall are crucial for urban water management applications such as water sensitive urban design (WSUD), which need to consider rainfall over very short durations (typically 6 min in Australia) (e.g. Kamruzzaman et al., 2015). It is important to monitor any changes in the statistical distribution of these characteristics as these will have a major impact on the performance of WSUD systems (e.g. Willems et al., 2012; Harremoës and Mikkelsen, 1995; Beecham and Chowdhury, 2012). Furthermore, research into possible changes in extreme rainfall is particularly important for many ecological systems. For example, the unique flora and fauna in Australia are sensitive to even slight variations in climate (e.g. Alexander and Arblaster 2009). In addition, Nowreen et al. (2015) showed that changes in rainfall extremes will add to the environmental stress in the Haor basin area of Bangladesh. Several studies provide evidence of changes in rainfall patterns over periods of around 50 years, based on data from multiple meteorological stations in the study area. For example, Salinger and Griffiths (2001) found that trends in rainfall indices showed a zonal pattern of response, with the frequency of 1-day 95th percentile extremes decreasing in the north and east, and increasing in the west over the period 1951–1996. van den Besselaar et al. (2013) found evidence of a decrease of around 20 % in the times between extreme rainfall events in Europe between 1951 and 2010. Willems (2013) analysed long precipitation records at 724 stations across Europe and
M. Kamruzzaman et al.
found evidence of multidecadal oscillatory behaviour, which he attributed to persistence in atmospheric circulation patterns over the North Atlantic. Goul et al. (2012) analysed data from 44 rainfall stations in the Ivory Coast for the period 1942– 2002 and after subdividing the area into climatically homogeneous regions found downward trends in the mean or variance of extreme rainfall in two of them. In contrast, Shahid (2011) found evidence of an increase in magnitude and frequency of extreme events in Northwest Bangladesh. Some studies have considered longer periods. For example, Gallant and Karoly (2010) analysed gridded data for the continent of Australia between 1911 and 2008 and concluded that there is evidence of an increase of between 1 and 2 % per decade. Westra and Sisson (2011) found that trends in extreme rainfall in Australia depend on the duration over which the intensity is calculated and the location within Australia. For example, they found evidence of an increase in 6-min duration rainfall on the East Coast and a decline in daily rainfall in Western Australia. It is generally accepted that the East Coast of Australia is affected by the Southern Oscillation Index (SOI), and Kamruzzaman et al. (2013) demonstrated statistically significant influence of SOI in the Murray Darling River basin. Martín-Vide and López-Bustins (2006) determined a relationship between extreme rainfall patterns in Andalusia in Spain and principal teleconnection indices such as the North Atlantic Oscillation (NAO) and the Western Mediterranean Oscillation (WeMO). Kuo et al. (2011) recommended a dynamic time series model for long duration extreme rainfalls to observe trends in annual maxima in Taiwan. Overall, the literature demonstrates that trends in extremes not only differ around the world but also within continental areas. Willems (2013) argues that relatively short-term trends may be part of longer cycles, but IPCC argues that recent trends should not necessarily be accounted for in this way (CSIRO, 2001). Analysis of changing patterns in rainfall, and rainfall extremes, is based on an assumed underlying stochastic process. A stochastic process is second-order stationary if the mean, standard deviation and autocorrelation do not change over time. However, stationarity is a property of a mathematical model and it is unrealistic to suppose that environmental, or any other physical variables, follow a model. In some cases, environmental time series may be well modelled by random errors distributed about a deterministic trend, such as a linear increase or decrease, over some particular time period. However, over past records at least, Willems (2013) claims that such trends have been transient. An alternative approach to investigating trends is to apply the methods of statistical quality control (SQC) which aims to identify out-of-control signals. In the industrial context, out of control signals lead to additional monitoring followed by corrective action. When necessary, in the environmental context, the system may itself return to a normal level due to complex feedback mechanisms but corrective action implies changes to anthropogenic
practice. We apply some techniques of SQC to time series derived from rainfall records at seven meteorological stations in South Australia. These time series are annual total rainfall and annual average of the 12-month extreme 24 h rainfall. SQC techniques, at least in their standard form, assume the time series are sequences of independent normally distributed varieties. Taking annual values removes any seasonality and is expected to make the assumption of independence reasonable. Annual rainfall will be close to being normally distributed, and averaging 12 monthly extremes will bring the variable closer to normality. However, an assumption of normality is not critical as permutation tests, which are distribution free, can be applied. Alternatively, the variable might be transformed to near normality before analysis. We do not expect environmental time series to be in statistical control (SC), so one aim of the SQC techniques is to identify years, or periods, during which rainfall has been atypically high or low. Also important is to investigate whether these years are associated with any known climate indicators such as SOI. The second aim of the SQC techniques is to identify any systematic trend or step change in the underlying process, if such sustained changes have occurred. The first stage of the analysis was to investigate the long time series of monthly maximum rainfall and of total rainfall for the three stations that had record lengths exceeding 100 years. The objectives were to investigate whether or not there is evidence of: trends in the mean, trends in the variability, changes in the seasonal pattern, and association with SOI, PDO, and their interactions. The second stage of the analysis is to apply multivariate SQC techniques to annual data from the 36-year common period dating back from 2012. The third stage is a factor analysis for the annual average monthly maximum rainfall.
2 Data The analysis is based on data from seven rainfall stations around Adelaide in South Australia (Fig. 1): Adelaide Airport (AA), Glen Osmond (GO), Kent Town (KT), Morphett Vale (MV), Mount Bold Reservoir (MBR), Pooraka (Po) and Salisbury Bowling Club (SBC). These stations were selected because they had the longest records of rainfall at sub-daily scales and also had the highest Australian Bureau of Meteorology quality designations for rainfall records for the period 1st January 1977 to 31st December 20012 (BoM 2011, Haylock and Nicholls 2000). A 24 hourly interval is considered dry if recorded rainfall is less than 0.001 mm, and in these cases, the record is then set to 0. For example, the Pooraka rainfall station has 16,985 non-zero records, so
Changing patterns in rainfall extremes in South Australia
Fig. 1 Location of daily rainfall stations: i. Adelaide Airport (AA), ii. Glen Osmond (GO), iii. Kent Town (KT), iv. Morphett Vale (MV), v. Mount Bold Reservoir (MBR),vi. Pooraka (Po), vii. Salisbury Bowling Club (SBC)
the proportion of wet to dry 24 h intervals is 34 % over 135 years. The proportion of wet intervals and other details for the seven stations are shown in Table 1. The rainfall data quality is shown graphically in Appendix 1, where the height of the record indicates the number of days of the year with complete data.
Table 1
3 Method of analysis 3.1 Analysis of long-term time series by univariate analysis In the case of long-term monthly rainfall records, a univariate time series regression model can be used to
Rainfall station information
Station name
Adelaide Airport Glen Osmond Kent Town Morphett Vale Mount Bold Reservoir Pooraka Salisbury Bowling Club
ID
23034 23005 23090 23098 23734 23026 23023
Rainfall record
Location
Start
End
Latitude
Longitude
1/01/1955 1/01/1884 1/01/1977 1/01/1947 1/01/1951 1/01/1878 1/01/1874
31/12/2012 31/12/2012 31/12/2012 31/12/2012 31/12/2012 31/12/2012 31/12/2012
−34.95 −34.95 −34.92 −34.97 −35.12 −34.83 −34.77
138.52 138.65 138.62 138.54 138.68 138.61 138.64
% of wet days
% of missing data
32.98 39.94 33.76 26.22 44.33 34.28 33.48
0.4 14.63 1.193 8.84 1.18 7.29 11.93
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investigate any association with climatic indices and to determine whether or not there is evidence of trends or
changes in seasonal patterns. A general model has the form:
2 Y t ¼ β0 þ β1 t−t þ β2 t−t þ β3 Febt þ ::::::: þ β13 Dect þ β14 t−t Febt þ :::::::þ β 24 t−t Dect þ β25 SOIt þ β26 PDOt þ β 27 SOI PDO þ εt
where Yt is a measure of rainfall, such as monthly daily maximum, and t is the time from the beginning of the record measured in months, and t is the mean value of t. Then, β0 is the expected rainfall for January when t equals t, and both SOI and PDO are 0. The terms 2 β1 t−t þ β2 t−t allow for a quadratic trend, with β0 equal 0 giving the particular use of a linear trend. A quadratic trend can also approximate one or two step changes. The indicator variables: 1 if month t is February Febt ¼ 0 otherwise 1 if month t is December Dect ¼ 0 otherwise together with their coefficients β3 up to β13 allow for additive monthly seasonal effects relative to January. The set of interaction terms: β 14 t−t Febt þ þβ24 t−t Dect allow for a trend in seasonal effects. The possible effects of SOI, PDO and their interaction (Kamruzzaman et al., 2011) are allowed for by the terms: β25 SOIt þ β26 PDOt þ β 27 SOIt PDOt εt represents the error about the deterministic model. In a standard regression model, εt are assumed to be a sequence of independent random variables with a mean of 0, and a constant variance Gaussian (normal) distribution. When the model is fitted the residuals (rt), the differences between observed and fitted values are estimates of the errors. The residuals are defined by: ^0 þ β ^ 1 t−t þ þβ ^ 27 SOI PDO r t ¼ yt − β ^ i are least square estimates of the βi. It is a consewhere β quence of the least square algorithm that: n X t¼1
rt ¼ 0
where n is the record length. However, the residuals can be used to check whether the standard assumptions of serial independence, constant variance and a Gaussian distribution are ^ðk Þ is an estimate of the serial reasonable. The correlogram ρ correlation at lag k. n−k X
^ ðk Þ ¼ ρ
rt rtþk
t¼1 n X
r2t
t¼1
The assumption of equal variance and a Gaussian distribution can be checked by a plot of residuals against t and a normal quantile-quantile plot, respectively. A regression of |rt| against t is a more formal test for a possible increase or decrease in variability. The variance of the errors is estimated by: n X
S2 ¼
r2t
t¼1
n−p
where p is the number of parameters in the model, which is 28 in the case of the model introduced in this section. A GARCH model can be used to investigate volatility in the model error. Typically, the GARCH (1,1) model has the form: σ2t ¼ α0 þ α1 ε2t−1 þ γ 1 σ2t−1 where σ2t is the variance of the errors at time t. The standard assumption is that the coefficients α1 and γ1 are both equal to 0. A least squares analysis provides an estimate of the stan ^ i , under the standard dard errors of the coefficients, sd^ β assumption about errors. There is evidence of a non-zero coefficient at the 5 % level of significance with a two sided alternative hypothesis, if the absolute value of the ratio of the estimated coefficient to its standard error exceeds about 2.0. In particular, there would be evidence of a linear trend if: ^1 β > 2:0 ^ ^ sd β 1
Changing patterns in rainfall extremes in South Australia
The quadratic term ðt−t Þ2 is uncorrelated with t−t, provided the time step is constant (and approximately uncorrelated if there are a few missing values), so there would be evidence of a quadratic trend if: ^2 β > 2:0 ^ ^ sd β 2 If an investigation of the residuals suggests a trend in variability or serial correlation of the errors, a generalised linear regression can be used. The assumption of normality of the ^ i will tend errors is not too critical because the distribution of β to normality as the time series becomes long (provided the errors have a distribution with a finite variance) as it is a consequence of the Central Limit Theorem. A trend in seasonal effects is modelled with the 11 coefficients of the interaction term between t−t and monthly indicator variables. It is more appropriate to consider a null hypothesis that they are all 0, corresponding to such a trend, than it is to consider the coefficients individually. If S2NI and S2I are the estimates of the variance of the errors in the model without these interaction terms (NI) and with these interaction terms, respectively, then . ðn−pÞ S 2N I −ðn−p−11Þ S 2I 11 e F 11; n−p−11 S2
3.2.1 Multivariate control chart Suppose there are annual rainfall records, such as annual total or annual average monthly maximum from m stations over n years in a region. Let xt be an m×1 vector of the annual measurements at time t. The Hotelling T2 multivariate control chart is a plot of T2 =(xt −μ)TV−1(xt −μ) against t, where μ and V are the target value and the variance–covariance matrix, respectively. In our application, we set μ as the vector of ^ , and V as means for the first half of the record, n/2 years, μ the variance-covariance matrix for the first n/2 years V^ . ^ i jσ ^i σ ^ j ; f or 1 ≤ i; j≤ m V^ i j ¼ C where Ĉij is the correlation matrix, σ ^ i , and σ ^ j are the standard deviations at sites i and j, respectively, with all calculated from the first 18 years of data. The additional subscripts e or a denote average monthly maximum and annual total of rainfall series, respectively. The means and covariance matrix are given for the second half period for comparison, but are not used in the construction of the chart. A reasonable approximation to the upper α quantile of T2, 2 ^ and V^ are based on around 20 or more data, is χm,α , if the if μ data are from a multivariate normal distribution. More precise values of the upper α quantile for T2 when sampling from other multivariate distributions can be obtained by simulation.
I
where p is the number of parameters to be estimated in the NI model and F11, n-p-11 is the F-distribution with 11 and n-p-11 degrees of freedom. If there is a substantial trend in seasonal effects, the numerator will be large with respect to the denominator. There is evidence to reject the null hypothesis of no interaction at the 5 % level, if the ratio exceeds the upper 0.05 quantile of the F-distribution.
3.2.2 Cumulative Sum Method for Rainfall Analysis The cumulative sum (cusum) chart aims to detect a shift in the process mean from the target mean value (μ0). The cusum, Ci, is calculated from a time series {yt} as: Ci ¼
i X
ðyt −μ0 Þ
ð2Þ
t¼1
3.2 Statistical quality control for multivariate data In some regions, rainfall records may be available at several sites, but over shorter periods. The records cannot be treated as independent, but the methods of statistical quality control (SQC) for multivariate processes provide a suite of analyses that use all the available data while allowing for the spatial correlation. The analyses assume that data are temporally independent, and assume that there is no serial correlation. The null hypothesis is that the data are independent draws from a distribution with a fixed mean and variance, in which case the process is described as stable, or in control. The analyses are designed to detect individual points that are out of control, or some change in the process such as a shift in the mean or a trend or an increase in variability. Analysis of annual data removes the need to allow for seasonality and makes the assumption of no serial correlation reasonable.
where i=1,2,……n and μ0 is process mean when the process is in control. A statistical test for determining whether there is evidence of a change in the mean when yt are independent Gaussian varieties is based on the test statistics are C+ and C− defined by: þ Cþ i ¼ max 0; xi −ðμ0 þ k Þ þ C i−1 C −i ¼ max 0; ðμ0 −k Þ−xi þ C −i−1 ð3Þ C− ¼ 0 Cþ ¼. k¼σ 2 If either C+i or C−i exceed 5σ, then the process is considered to be out of statistical control (SC) at the 0.2 % level. Exceeding by C+i is evidence of an increase in the mean level, and exceeding by C−i is evidence of a decrease in the mean level. In the case of annual
M. Kamruzzaman et al.
rainfall statistics xt at m stations, yt can be defined as the mean rainfall over the stations: yt ¼
m X
xti
i¼1
where x is m×1, f is c×1, B is m×c and e is m×1 The underlying factors, the c elements of f, are assumed to independent with mean 0 and variance 1. The variance covariance matrix of x is: V ¼ BB= þ G
This averaging over stations justifies an assumption that the yt are approximately Gaussian. The values of μ and σ can be estimated from the first n/2 values. If a precise p value is required for the greatest values of C+i , C−i in the period n/2
