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The synopsis presented above pertains to approaches for the design, analysis and risk assessments of water infrastructure that is central to this thesis and serves to set the scene and provide a contextual overview that frames the specific research objectives of the thesis.  These objectives have their origins in knowledge gaps identified in the very extensive body of scientific literature that relates to the significant issues and very real challenges associated with both understanding and accounting for the complexities of hydro-meteorological systems, particularly in the face of increasing climate related uncertainty.  It was through the identification of these gaps and the relationship of these to water infrastructure design practices in Ireland, that the thesis research questions were developed.  The relevant gaps and their linkages to the thesis objectives are detailed in the sections that follow.

1.4.1 Stationarity

The assumptions that underpin the statistical (frequency) modelling of design rainfalls require that historical observations are independent and, despite the inherent variability exhibited by data, that the long-term statistical characteristics (e.g. mean and variance) remain stationary in time i.e. have an identical distribution [Stedinger, 1993; Meehl et al., 2000; Khaliq et al., 2006]. The violation of these assumptions will result in unreliable design estimates and poor risk assessments.

Climate change is altering the patterns of the rainfall that we experience with the consequence that the intensity and frequency of extreme rainfalls is heavily modified.  This results in significant changes to the distributions of these rainfalls, ensuring they are not stationary in time [Mailhot and Duchesne, 2009; Wang et al., 2013]. The increase in intensity and frequency of rainfall is associated with the rise in global temperature due to elevated greenhouse gas concentrations (Figure '1.7). As mentioned, studies have demonstrated a warming of 0.85''C in the period from 1880 to 2012, with an average increase of 0.07 ''C per decade [IPCC, 2012]. This rise causes an increase in the water holding capacity of the atmosphere which in turn will intensify the hydrologic cycle [Trenberth, 1999]. Climate change studies that are based on simulations obtained from climate models suggest that this intensification and consequent flooding will continue.  This is expected to be most pronounced in the high latitudes and tropical regions, but the frequency of extreme precipitation events in the northern mid-latitudes is also predicted to increase in winter months with the likelihood that serious flooding will remain an issue across Europe [Fowler, 2003; IPCC, 2012]. However, it has been recognised that global or continental scale observations of historical climate, or projections of future climate, are not particularly useful for local or regional scale planning.  This, according to Martinez et al. [2012] provides strong motivation and reinforces the need to evaluate historical trends and future projections on a regional or local scale for improving climate impact research.

Observed regional climate change and variability reported in recent international climate studies (see for example Robson et al., 1998; Haylock and Nicholls, 2000; Meehl et al., 2000; Osborn and Hulme, 2000; Burn and Hag Elnur, 2002; Sen Roy and Balling, 2004; Alexander et al., 2006; Rodda et al., 2009; Costa and Soares, 2009; De Toffol et al., 2009; Jung et al., 2011; Dawadi and Ahmad, 2012; IPCC, 2012; Saidi et al., 2013) have shown that extreme rainfall exhibits statistically significant trends in means and/ or variance, and these could be induced by climate influences.  The assumption of stationarity is therefore, very questionable [Milly et al., 2008]. In the context of future climate change and the associated effects on extreme rainfall patterns in terms of changing frequencies, intensities, and durations, this merits serious scrutiny.  The violation of this assumption and failing to quantify and allow for trends in extreme rainfall can result in errors in water infrastructure design estimates with the consequence that levels of risk associated with infrastructure failure are increased or that the economic costs of the infrastructure are higher than necessary [Kundzewicz and Robson, 2000]. Thus, the long-term design and risk assessment of these water infrastructures relies on changes or trends in rainfall patterns being correctly identified and appropriately accounted for in design rainfall estimates.

1.4.2 Trend Analysis

The detection of two types of trend in hydro-meteorological data is of particular concern for assessments of non-stationarity. The first of these are monotonic linear trends whereby there exists a gradual linear increase/ decrease in the test variable with time, and the second reflects a step change, where an abrupt shift in the test variable is encountered in the time series. To evaluate the significance of trends, both parametric and rank-based non-parametric tests are commonly applied.  The testing method chosen is influenced by whether the analysed data meets certain assumptions such as independence and being identically normally distributed [Costa and Soares, 2009].  The violation of test assumptions can result in Type-I errors (false detection of trends). Certain parametric tests require data to be independent and normally distributed [Hirsch et al., 1991; Fathian et al., 2014]. Non-parametric tests offer advantages in the sense that they do not make any assumption about the distribution of the data and are more suitable for non-normally distributed and censored data, which are frequently encountered in hydro-meteorological time series [Ehsanzadeh et al., 2011; Westra et al., 2013].  Moreover, non-parametric procedures can have significantly higher statistical power (or efficiency) than parametric procedures in cases where the sample size is large [Hirsch et al., 1991], as is the case for the data records analysed in this thesis.  To test the significance of monotonic trends in the context of dependence between two variables, the non-parametric Mann-Kendall (MK) test, Spearman's rank correlation coefficient (rho), together with parametric linear regression tests are commonly applied [Kundzewicz and Robson, 2004]. Of more relevance to the current study is the frequent application of the MK and rho tests in hydro-meteorological studies which have been shown by Yue et al. [2002] to have equal power in detecting monotonic trends in these data. The significance of step trends in hydro-meteorological data is also evaluated using either parametric or non-parametric techniques and again, particular assumptions are inherent in each. Parametric tests include the two sample t-test [Iman and Conover, 1983] and the Standard Normal Homogeneity [Alexandersson, 1986] test. The most common non-parametric tests however, for studies of hydro-meteorological data are the tests of Pettitt [Pettitt, 1979] and Mann-Whitney-Wilcoxon [Wilcoxon, 1945]. As is the case of monotonic trend detection, non-parametric tests for identifying step changes are considered more powerful than parametric techniques because no prior conditions are assumed about the analysed data such as it having to be normally and/ or identically distributed [Reeves et al., 2007; Madsen et al., 2014].

Common practice in trend analysis focuses on changes in the value of the analysed data set.  However, some studies (see for example Meehl et al., 2000; Katz and Brown, 1992; Al Saji et al., 2015) have suggested that considering only mean values and ignoring other (higher order) statistical characteristics such as the variance can mask significant non-stationarities.  Non-stationarities in either the mean or variance can significantly change the frequency of the test variable (extreme temperature, rainfall etc.). Figure '1.8 illustrates the impact of a shift in the mean (associated with the location parameter) and variance (associated with the scale parameter) on a given probability density function.

The proper identification of non-stationarity in rainfall records may therefore be more fully captured by exploring trends in both the mean and/ or variance of the data.  Considering only a single characteristic may give a false sense that the analysed series is stationary, when in fact it is not and by extension, may result in erroneous design rainfall estimates.  This issue is explored in Chapter 2 of the thesis by applying the Mann-Kendall linear and Mann-Whitney-Wilcoxon step change trend detection techniques to investigate the changes in the mean and variance of annual maximum daily rainfalls at seven stations in Dublin, Ireland, where long and high quality rainfall records were available.  Results from these detection techniques were assessed against available station metadata in addition to reported extreme weather events that caused flooding episodes over Dublin in the past decades to distinguish between climatological and non-climatological infkuences. The seven stations are located in a geographically similar and spatially compact region (< 950 km2) and from Pearson's r coefficient; their rainfalls were shown to be well correlated.  Design estimates were obtained from frequency analysis of annual maximum daily rainfalls using the GEV distribution, identified through application of the MAD Goodness of Fit criterion. To evaluate the impact of the observed non-stationarity in variance on rainfall design estimates, two sets of depth-frequency relationships at each station for return periods from 5 to 100-years were constructed. These relationships were constructed with bootstrapped confidence intervals based on the full rainfall record assuming stationarity and the second was based on a partial record commencing in the year that followed the observed shift in variance.

1.4.3 Climate Influences

The analysis of identifying statistically significant trends in design rainfall remains somewhat incomplete in the absence of investigating any temporal variability and climate associations. As has been already mentioned, observed trends in extreme rainfall are predominantly influenced by the temporal variability of the climate and these influences need to be considered in the proper modelling of extreme rainfall for increased reliability in engineering design.

The characteristics of oscillatory patterns of extreme rainfall indices can influence the presence of these statistically significant trends and have been used to describe the natural climate variability in scientific literature and for establishing drivers of both monotonic and step trends. Climate induced non-stationarity in extreme hydro-meteorological variables can be caused by anthropogenic climate change but it can also be triggered by the influence of the inherent variability in the climate [Ntegeka and Willems, 2008; Willems, 2013b]. Regardless of their origin, trends should be accounted for in the frequency modelling of design rainfall.   It is recommended in some literature that step changes be dealt with by truncating the record from the year of the break point and using the shorter but more recent data that better reflects the current climate to establish more reliable estimates of design rainfall quantiles [McCabe, 2002; Leahy and Kiely, 2010].  However, this approach is arguable biased when the changes are an artefact of an oscillatory pattern and the analysed time series is of a limited duration in the sense that it does not span a full oscillation cycle [Willems, 2013a].  Incorporating linear trends in the frequency analyses of design rainfall on the other hand, remains considerably more complex.  An approach by Mailhot et al. [2010] incorporated time as a co-variate when evaluating the parameters of the extreme rainfall distributions.  Time however, is arguably an unsuitable parameter for such an application because firstly, it is not a physically based covariate and secondly, because it is defined in only a single, increasing dimension.  While these approaches would be appropriate for irreversible climate signals, it is clearly less suitable for addressing trends that are triggered by natural climate variability. Therefore, distinguishing between the trends from anthropogenic climate change and the inherent variability of the natural climate is important for engineering design purposes given that trend analysis alone without consideration of inherent temporal variability of the data may not capture fully the evolution of design rainfall.  Including for trends by evaluating truncated data series assumes that the direction and magnitude of any detected changes will continue into the future.  Such an approach does not allow for the possibility of a return to a previous climate regime which may occur from climate variability that can follow (multi) decadal natural cycles and cause oscillatory behaviour in extreme rainfall [Burroughs, 2003; Willems, 2013b].

Testing for the presence of these climate influenced periodicities or oscillatory patterns in hydro-meteorological variables is commonly done with approaches that include the use of the Hurst exponent [Hurst, 1951], spectral analysis (Fourier and wavelet analysis) [G''miz-Fortis et al., 2002], low-pass filters [McCuen, 2002], and more recently, the Quantile Perturbation Method [Ntegeka and Willems, 2008]. To better understand the large scale climatic influences on oscillatory patterns in regional rainfall, climate studies have tended to focus on investigating the presence of links between these oscillations and those of large scale climate teleconnections (e.g. El-Ni''o Southern Oscillation (ENSO); North Atlantic Oscillation (NAO); Indian Ocean Dipole (IOD); Pacific Decadal Oscillation (PDO); Interdecadal Pacific Oscillation (IPO), etc.) [Cai and Van Rensch, 2012].

Ireland is located along the Atlantic freeboard in the north-west of Europe.  The influence of the NAO on local temperature and rainfall in this region is well reported [Sweeney, 1985; Hurrell and Van Loon, 1997; Arnell, 1999; Sweeney et al., 2006; Scaife et al., 2008].  Furthermore, and specific to Ireland, changes in the seasonal patterns of the NOA have been strongly associated with previously reported changes to wetter conditions (see for example Arnell, 1999; Kiely, 2007).  Trends in annual accumulations of rainfall and streamflow at selected monitoring stations have also been well correlated with winter NAO variability [Kiely, 1999].  The significance of the NAO as the most significant large scale mode of climate variability relevant to Ireland and the primary driver of extreme rainfall variability is of particular significance to this research and is used in the analyses of the thesis and in developing the insights that are presented.  The NAO Index is usually defined as the normalised pressure difference between a station in the Azores and one in Iceland [Jones et al., 1997; Visbeck and Hurrell, 2001].  NAO indices, as is the case for most climatic variables, follow a cyclical pattern as shown in Figure '1.9 for the annual NAO index.  Seasonal and monthly NAO indices can be shown to follow similar patterns.

Although Chapter 2 identified trends in two statistical characteristics, or indices, of extreme daily rainfall (mean and variance), the relationship of these trends to any large scale mode of climate variability was not explored. This linkage is however, investigated in Chapter 3.  The chapter broadens the scope of work in Chapter 2, which focused only on Dublin rainfall stations, to a national scale where high quality datasets from 12 stations across the country are used to underpin the analysis.  The studied stations were limited to those where record lengths were sufficient for climate signal detection.  Furthermore, Chapter 3 focusses on sub-daily short-duration (15-, 30-minute and 1-hour) rainfall extremes at these stations that are important critical inputs for the design of urban water infrastructure.  The presence of non-stationary signals in records at these 12 stations was identified through extensive trend analyses using the non-parametric Mann-Kendall and Mann-Whitney-Wilcoxon linear and step-change tests respectively.  The analysis was undertaken on a suite of extreme rainfall indices that are known to impact the parameters of the statistical distribution of extreme rainfall and which were extracted from the historical records.  These included annual maxima (AM), variance of AM and the extreme frequency series.  Given the importance of understanding and if necessary, accounting for non-stationary rainfall signals in engineering design processes, the impact of detected trends across the 12 stations on design rainfall estimates was evaluated by comparing two sets of depth-duration-frequency curves, the first using the full data record available and the second based on the truncated dataset that followed from the year of the identified step change.  The MAD goodness-of-fit criterion confirmed the suitability of the Generalised Extreme Value distribution for this analysis.

Furthermore, and as is common with studies of this type, linkages between detected non-stationarities and large scale modes of climate variability, the NAO in this case, was explored by examining the monthly and seasonal rainfall extremes and relating this to appropriate NAO indices using the Kendall's Tau measure of correlation.

The climatic association of extreme rainfall and attribution of trends in its extreme indices from Chapter 3 is further explored in Chapter 4.  This further exploration involved an assessment of the temporal variability of sub-daily extreme rainfall and its correlation with NAO indices using the Quantile Perturbation Method (QPM) [Ntegeka and Willems, 2008]. Applications of the QPM that have been reported in literature (see for example Taye and Willems, 2013; Mora et al., 2014; Mora and Willems, 2011) commonly utilise Peak-Over-Threshold (POT) series to combine frequency and perturbations of extremes. The method uses empirical statistical analyses to define a region of natural variability that can be used to identify any climate change trends in the oscillations of extreme rainfall. However, in establishing this region of natural variability, the use of the POT record can be computationally expensive given that the full series needs to be resampled many thousands of times.  Furthermore, the QPM using a POT approach requires a continuous data record to properly establish a region of natural climate variability.  However, a continuous record of sub-daily rainfall data was only available for hourly durations and data provided for the 15-minute and 30-minute durations by Met ''ireann, the Irish meteorological service, was not continuous (i.e. the available data related to that above a certain threshold).  As an alternative, the use of the Annual Maximum (AM) series in the QPM was explored in Chapter 4, where the analysis and the resulting identification of climate change signals was shown to be reasonably consistent for the corresponding analysis undertaken with the hourly POT series.  The AM series for the 15- and 30 minute rainfall durations was subsequently used in the QPM across the 12 study stations.  Furthermore, relationships between the temporal variability of monthly/seasonal extreme rainfall with that of corresponding NAO indices was determined to distinguish between natural climate variability and climate change, in being the primary drivers of statistically significant trends in these rainfalls.

Investigating these temporal variations and their statistical significance is central to enhancing our understanding of the climatic drivers of the observed trends in extreme rainfall and improves knowledge when interpreting and contextualising historical trends to ensure that they are properly accounted for in design rainfall estimation.

1.4.4 Climate Conditioned Modelling

Given that significant climatic influences on extreme rainfall were evident in Chapters 3 and 4 of the thesis, Chapters 5 and 6 of the thesis focus on accounting for these in the statistical modelling of extreme rainfalls.  Properly doing so will facilitate a more robust estimation of rainfall for engineering design and risk assessment purposes. Such work is particularly pertinent given the accepted engineering consensus, reported in Woodward et al. [2013], that future uncertainties of climate change need to be properly accounted for in the development of long-term design and risk assessment strategies to ensure economic efficiency.

Common approaches for this involve linking the parameters of the statistical distribution of extreme rainfall to some covariates such as time or time dependent variables (see for example Fowler et al., 2010; Khaliq et al., 2006; Katz et al., 2002; Smith, 2004).  It is often the case however, that climate controlled models for some geographic regions utilise influential indices of large atmospheric teleconnections as covariates.  Non-stationary models of this type have been shown to offer statistically significant improvements over traditional stationary approaches in modelling extreme daily rainfall in several regions of the world by addressing the primary shortcoming of stationary models, namely their omission of any effects of dynamic climatic influences [Villarini et al., 2011a; L''pez and Franc''s, 2013; Osman et al., 2013; Tramblay et al., 2013; Sun et al., 2014; Yin et al., 2014; Mondal and Mujumdar, 2015].  It is the approach of utilising large scale climatic variables, the NAO and global temperature anomalies in this case given their significance as drivers of Irish climate, which is adopted in this thesis, focussing particularly on finer temporal resolutions.  It is these durations that are most critical in climate based modelling for the design and risk assessment of urban water systems in regions such as Ireland, where the extreme rainfall is highly influenced by climate change and variability.

This work is particularly novel in that short-duration (sub-hourly) extremes have not been included in climate conditioned modelling approaches and the use of NAO indices as suitable covariates in climate conditioned modelling of extreme rainfalls of any duration has not been assessed for Irish rainfall.  A previous study by Osman et al. [2013], which incorporated climatic influences in statistical rainfall models, was limited to rainfall of daily duration and the climatic covariates considered were downscaled regional variables from General Circulation Models (GCMs), rather than large climate teleconnections, such as the NAO.  While downscaled climatic data are sometimes used as covariates in climate conditioned models to describe extreme rainfalls, they come with added layers of uncertainty [Fowler et al., 2007].  These arise from sparse data for the downscaling process, their representation of sub-daily precipitation and full precipitation fields on fine scales; their poor ability in capturing changes in small-scale processes and their subsequent feedback to larger scales, together with errors inherited from the driving GCMs [Maraun et al., 2010; Chen et al., 2011].  Consequently, climate models offer better skill levels for predicting large scale weather variables over regional variables, the prediction of which can be considerably more erratic [Katz et al., 2002; Gregersen et al., 2013].

In Chapter 5, a climate conditioned non-stationary statistical model for the estimation of design rainfall quantiles for these sub daily critical durations is developed and tested for Ireland.  For this, the three-parameter GEV distribution is evaluated with NAO indices introduced as covariates in its location and scale parameters. The GEV distribution is one of the most frequently used distributions in hydrology and climatology and its widespread adoption is associated with the added flexibility that it provides through the inclusion of a shape parameter [Katz et al., 2002; El Adlouni et al., 2007].  Adding a covariate to its parameters allows for climatic influences on the statistical characteristics of the distribution of extreme rainfall i.e. the mean and/or variance of the extreme rainfall time series, to be incorporated in the statistical model.

A similar analysis is presented in Chapter 6, but is limited in this case to the Dublin area to evaluate the efficacy of climate conditioned models to describe extreme daily rainfall in the region.  Chapter 2 illustrated that extreme daily rainfalls records in the region are non-stationary (largely in variance but also in the extreme values themselves in some stations).  The Dublin area is unique in the sense that a cluster of stations with high quality data at a daily resolution is available.  This presented an additional opportunity to evaluate the suitability of climate conditioned models to model extreme rainfall extracted from averaged regional data as opposed to at-site analysis on individual station data, as was done in Chapter 5 for stations across the country.

1.5 Summary of Thesis Research Questions and Objectives

The above narrative identifies critical knowledge gaps in our understanding and modelling of short-duration extreme rainfalls in Ireland.  From this, a well-defined set of research questions that underpin the thesis emerged.  These are summarised in Table '1.1 where they are also mapped to various chapters of the thesis.

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