Extreme Value Theory nds application in problems concerning low probability but high consequence events. In hydrology the study of heavy rainfall is important in regional ood risk assessment. In particular, the N-year return level is a key output of an extreme value analysis, hence care needs to be taken to ensure that the model is accurate and that the level of imprecision in the parameter estimates is made explicit. Rainfall is a process that evolves over time and space. Therefore, it is anticipated that at extreme levels the process would continue to show temporal and spatial correlation. In this study interest is in whether any trends in heavy rainfall can be detected for the Western Cape. The focus is on obtaining the 50-year daily winter rainfall return level and investigating whether this quantity is homogenous over the study area. The study is carried out in two stages. In the rst stage, the point process approach to extreme value theory is applied to arrive at the return level estimates at each of the fteen sites. Stationarity is assumed for the series at each station, thus an issue to deal with is that of short-range temporal correlation of threshold exceedances. The proportion of exceedances is found to be smaller (approximately 0.01) for stations towards the east such as Jonkersberg, Plettenbergbay and Tygerhoek. This can be attributed to rainfall values being mostly low, with few instances where large amounts of rainfall were observed. Looking at the parameters of the point process extreme value model, the location parameter estimate appears stable over the region in contrast to the scale parameter estimate which shows an increase towards in a south easterly direction. While the model is shown to t exceedances at each station adequately, the degree of uncertainty is large for stations such as Tygerhoek, where the maximum observed rainfall value is approximately twice as large as the high rainfall values. This situation was also observed at other stations and in such cases removal of these high rainfall values was avoided to minimize the risk of obtaining inaccurate return level estimates. The key result is an N-year rainfall return level estimate at each site. Interest is in mapping an estimate of the 50-year daily winter rainfall return level, however to evaluate the adequacy of the model at each site the 25-year return level is considered since a 25 year return period is well within the range of the observed data. The 25-year daily winter rainfall return level estimate for Ladismith is the smallest at 22:42 mm. This can be attributed to the station's generally low observed winter rainfall values. In contrast, the return level estimate for Tygerhoek is high, almost six times larger than that of Ladismith at 119:16 mm. Visually design values show di erences between sites, therefore it is of interest to investigate whether these di erences can be modelled. The second stage is the geostatistical analysis of the 50-year 24-hour rainfall return level The aim here is to quantify the degree of spatial variation in the 50-year 24-hour rainfall return level estimates and to use that association to predict values at unobserved sites within the study region. A tool for quantifying spatial variation is the variogram model. Estimation of the parameters of this model require a su ciently large sample, which is a challenge in this study since there is only fteen stations and therefore only fteen observations for the geostatistical analysis. To address this challenge, observations are expanded in space and time and then standardized and to create a larger pool of data from which the variogram is estimated. The obtained estimates are used in ordinary and universal kriging to derive the 50-year 24-hour winter rainfall return level maps. It is shown that 50-year daily winter design rainfall over most of the Western Cape lies between 40 mm and 80 mm, but rises sharply as one moves towards the east coast of the region. This is largely due to the in uence of large design values obtained for Tygerhoek. In ordinary kriging prediction uncertainty is lowest around observed values and is large if the distance from these points increases. Overall, prediction uncertainty maps show that ordinary kriging performs better than universal kriging where a linear regional trend in design values is included.

Floods is a serious concern across Europe due to the enormous material damage and death toll. Of alltypes of floods, flash floods and large-scale river floods have become major natural hydrological hazardsin most countries. The city of Halmstad was chosen due to its placement on the southern west coast ofSweden, a region for which climate projections have indicated more precipitation and potential forflooding. In recent years a number of floods have also been observed with associated damages. Usingextreme value analysis on observed data these events can be interpreted in terms of return level valuesand their frequency of occurrence. The seasonal variation of the precipitation and discharge of thecatchment were analyzed based on 43-year precipitation and 25-year discharge observation data and therelationship to NAO index was investigated to give a preliminary overview of the hydrologicalconditions in Halmstad and its causes. The results showed that Halmstad was seasonally characterizedby high discharge in winter and lower discharge in summer with the highest rainfall. The effect of stormtracks represented by the NAO index on the precipitation and discharge in winter months was evident.This study focused on the analysis of extreme data of precipitation and discharge. The return levels forup to 50-year return period were estimated by GEV fitting. The estimated return level of discharge fora 50-year return period is 250 m³/s, and the return levels of precipitation for a 50-year flood was foundto be 68 mm/day. Two cases were selected from a compiled annual maxima discharge data set foranalyzing and comparing their weather conditions based on ERA5 data. The results showed that differentweather conditions do have an impact on the total rainfall, and there were similar patterns but largedifferences between ERA5 reanalysis data and observed SMHI data was also shown emphasizing theneed for long-term observational data sets and further evaluation of reanalysis data.

Increased extreme precipitation has been documented in many regions around the world, in- cluding central and northern Europe. Global warming increases average temperature, which in turn enhances atmospheric water holding capacity. These changes are believed to increase the frequency and/or intensity of extreme precipitation events. In determining the design storm, or a worst probable storm, for infrastructure design and failure risk assessment, experts commonly assume that statistics of extreme precipitation do not change significantly over time. This so- called notion of stationarity assumes that the statistics of future extreme precipitation events will be similar to those of historical observations. This study investigates the consequences of using a stationary assumption as well as the alternative: a non-stationary framework that con- siders temporal changes in statistics of extremes. Here we evaluate stationary and non-stationary return levels for 10-year to 50-year extreme precipitation events for different durations (1-day, 2-day, ..., 7-day precipitation events), based on the observed daily precipitation from Sweden. Non-stationary frequency analysis is only considered for stations with statistically significant trends over the past 50 years at 95% confidence (i.e., 15 to 39 % out of 139 stations, depend- ing on duration, 1-day, 2-day, ..., 7-day). We estimate non-stationary return levels using the General Extreme Value distribution with time-dependent parameters, inferred using a Bayesian approach. The estimated return levels are then compared in terms of duration, recurrence in- terval and location. The results indicate that a stationary assumption might, when a significant trend exists, underestimate extreme precipitation return levels by up to 40 % in Sweden. This report highlights the importance of considering better methods for estimating the recurrence in- terval of extreme events in a changing climate. This is particularly important for infrastructure design and risk reduction. / Ökad extrem nederbörd har dokumenterats globalt, däribland centrala och norra Europa. Den globala uppvärmningen medför en förhöjd medeltemperatur vilket i sin tur ökar avdunstning av vatten från ytor samt atmosfärens förmåga att hålla vatten. Dessa förändringar tros kunna öka och intensifiera nederbörd. Vid bestämning av dimensionerande nederbördsintensiteter för byggnationsprojekt antas idag att frekvensen och storleken av extrem nederbörd inte kommer att förändras i framtiden (stationäritet), vilket i praktiken innebär ingen förändring i klimatet. Den här studien syftar till att undersöka effekten av en icke-stationärt antagande vid skattning av dimensionerande nederbördsintensitet. Icke-stationära och stationära nerderbördsintensiteter föråterkomsttider mellan 10 och 100år bestämdes utifrån daglig och flerdaglig svensk nederbörds- data. Nederbördintensiteterna bestämdes med extremvärdesanalys i mjukvaran NEVA, där den generella extremvärdesfördelningen anpassades till årlig maximum nederbörd på platser i Sverige som påvisade en ökande trend under de senaste 50åren (15% till 39 % utav 139 stationer, beroende på varaktighet). De dimensionerande nederbördsintensiteterna jämfördes sedan med avseende på varaktighet, återkomsttid och plats. Resultaten indikerade på att ett stationärt antagande riskerar att underskatta dimensionerande nederbördsintensiteter för en viss återkomsttid med upp till 40 %. Detta indikerar att antagandet om icke-stationäritet har större betydelse för olika platser i Sverige, vilket skulle kunna ge viktig information vid bestämning av dimensionerande regnintensiteter.

It is desirable to reduce the potential threats that result from the variability of nature, such as droughts or heat waves that lead to food shortage, or the other extreme, floods that lead to severe damage. To prevent such catastrophic events, it is necessary to understand, and to be capable of characterising, nature's variability. Typically one aims to describe the underlying dynamics of geophysical records with differential equations. There are, however, situations where this does not support the objectives, or is not feasible, e.g., when little is known about the system, or it is too complex for the model parameters to be identified. In such situations it is beneficial to regard certain influences as random, and describe them with stochastic processes. In this thesis I focus on such a description with linear stochastic processes of the FARIMA type and concentrate on the detection of long-range dependence. Long-range dependent processes show an algebraic (i.e. slow) decay of the autocorrelation function. Detection of the latter is important with respect to, e.g. trend tests and uncertainty analysis. Aiming to provide a reliable and powerful strategy for the detection of long-range dependence, I suggest a way of addressing the problem which is somewhat different from standard approaches. Commonly used methods are based either on investigating the asymptotic behaviour (e.g., log-periodogram regression), or on finding a suitable potentially long-range dependent model (e.g., FARIMA[p,d,q]) and test the fractional difference parameter d for compatibility with zero. Here, I suggest to rephrase the problem as a model selection task, i.e.comparing the most suitable long-range dependent and the most suitable short-range dependent model. Approaching the task this way requires a) a suitable class of long-range and short-range dependent models along with suitable means for parameter estimation and b) a reliable model selection strategy, capable of discriminating also non-nested models. With the flexible FARIMA model class together with the Whittle estimator the first requirement is fulfilled. Standard model selection strategies, e.g., the likelihood-ratio test, is for a comparison of non-nested models frequently not powerful enough. Thus, I suggest to extend this strategy with a simulation based model selection approach suitable for such a direct comparison. The approach follows the procedure of a statistical test, with the likelihood-ratio as the test statistic. Its distribution is obtained via simulations using the two models under consideration. For two simple models and different parameter values, I investigate the reliability of p-value and power estimates obtained from the simulated distributions. The result turned out to be dependent on the model parameters. However, in many cases the estimates allow an adequate model selection to be established. An important feature of this approach is that it immediately reveals the ability or inability to discriminate between the two models under consideration. Two applications, a trend detection problem in temperature records and an uncertainty analysis for flood return level estimation, accentuate the importance of having reliable methods at hand for the detection of long-range dependence. In the case of trend detection, falsely concluding long-range dependence implies an underestimation of a trend and possibly leads to a delay of measures needed to take in order to counteract the trend. Ignoring long-range dependence, although present, leads to an underestimation of confidence intervals and thus to an unjustified belief in safety, as it is the case for the return level uncertainty analysis. A reliable detection of long-range dependence is thus highly relevant in practical applications. Examples related to extreme value analysis are not limited to hydrological applications. The increased uncertainty of return level estimates is a potentially problem for all records from autocorrelated processes, an interesting examples in this respect is the assessment of the maximum strength of wind gusts, which is important for designing wind turbines. The detection of long-range dependence is also a relevant problem in the exploration of financial market volatility. With rephrasing the detection problem as a model selection task and suggesting refined methods for model comparison, this thesis contributes to the discussion on and development of methods for the detection of long-range dependence. / Die potentiellen Gefahren und Auswirkungen der natürlicher Klimavariabilitäten zu reduzieren ist ein wünschenswertes Ziel. Solche Gefahren sind etwa Dürren und Hitzewellen, die zu Wasserknappheit führen oder, das andere Extrem, Überflutungen, die einen erheblichen Schaden an der Infrastruktur nach sich ziehen können. Um solche katastrophalen Ereignisse zu vermeiden, ist es notwendig die Dynamik der Natur zu verstehen und beschreiben zu können. Typischerweise wird versucht die Dynamik geophysikalischer Datenreihen mit Differentialgleichungssystemen zu beschreiben. Es gibt allerdings Situationen in denen dieses Vorgehen nicht zielführend oder technisch nicht möglich ist. Dieses sind Situationen in denen wenig Wissen über das System vorliegt oder es zu komplex ist um die Modellparameter zu identifizieren. Hier ist es sinnvoll einige Einflüsse als zufällig zu betrachten und mit Hilfe stochastischer Prozesse zu modellieren. In dieser Arbeit wird eine solche Beschreibung mit linearen stochastischen Prozessen der FARIMA-Klasse angestrebt. Besonderer Fokus liegt auf der Detektion von langreichweitigen Korrelationen. Langreichweitig korrelierte Prozesse sind solche mit einer algebraisch, d.h. langsam, abfallenden Autokorrelationsfunktion. Eine verläßliche Erkennung dieser Prozesse ist relevant für Trenddetektion und Unsicherheitsanalysen. Um eine verläßliche Strategie für die Detektion langreichweitig korrelierter Prozesse zur Verfügung zu stellen, wird in der Arbeit ein anderer als der Standardweg vorgeschlagen. Gewöhnlich werden Methoden eingesetzt, die das asymptotische Verhalten untersuchen, z.B. Regression im Periodogramm. Oder aber es wird versucht ein passendes potentiell langreichweitig korreliertes Modell zu finden, z.B. aus der FARIMA Klasse, und den geschätzten fraktionalen Differenzierungsparameter d auf Verträglichkeit mit dem trivialen Wert Null zu testen. In der Arbeit wird vorgeschlagen das Problem der Detektion langreichweitiger Korrelationen als Modellselektionsproblem umzuformulieren, d.h. das beste kurzreichweitig und das beste langreichweitig korrelierte Modell zu vergleichen. Diese Herangehensweise erfordert a) eine geeignete Klasse von lang- und kurzreichweitig korrelierten Prozessen und b) eine verläßliche Modellselektionsstrategie, auch für nichtgenestete Modelle. Mit der flexiblen FARIMA-Klasse und dem Whittleschen Ansatz zur Parameterschätzung ist die erste Voraussetzung erfüllt. Hingegen sind standard Ansätze zur Modellselektion, wie z.B. der Likelihood-Ratio-Test, für nichtgenestete Modelle oft nicht trennscharf genug. Es wird daher vorgeschlagen diese Strategie mit einem simulationsbasierten Ansatz zu ergänzen, der insbesondere für die direkte Diskriminierung nichtgenesteter Modelle geeignet ist. Der Ansatz folgt einem statistischen Test mit dem Quotienten der Likelihood als Teststatistik. Ihre Verteilung wird über Simulationen mit den beiden zu unterscheidenden Modellen ermittelt. Für zwei einfache Modelle und verschiedene Parameterwerte wird die Verläßlichkeit der Schätzungen für p-Wert und Power untersucht. Das Ergebnis hängt von den Modellparametern ab. Es konnte jedoch in vielen Fällen eine adäquate Modellselektion etabliert werden. Ein wichtige Eigenschaft dieser Strategie ist, dass unmittelbar offengelegt wird, wie gut sich die betrachteten Modelle unterscheiden lassen. Zwei Anwendungen, die Trenddetektion in Temperaturzeitreihen und die Unsicherheitsanalyse für Bemessungshochwasser, betonen den Bedarf an verläßlichen Methoden für die Detektion langreichweitiger Korrelationen. Im Falle der Trenddetektion führt ein fälschlicherweise gezogener Schluß auf langreichweitige Korrelationen zu einer Unterschätzung eines Trends, was wiederum zu einer möglicherweise verzögerten Einleitung von Maßnahmen führt, die diesem entgegenwirken sollen. Im Fall von Abflußzeitreihen führt die Nichtbeachtung von vorliegenden langreichweitigen Korrelationen zu einer Unterschätzung der Unsicherheit von Bemessungsgrößen. Eine verläßliche Detektion von langreichweitig Korrelierten Prozesse ist somit von hoher Bedeutung in der praktischen Zeitreihenanalyse. Beispiele mit Bezug zu extremem Ereignissen beschränken sich nicht nur auf die Hochwasseranalyse. Eine erhöhte Unsicherheit in der Bestimmung von extremen Ereignissen ist ein potentielles Problem von allen autokorrelierten Prozessen. Ein weiteres interessantes Beispiel ist hier die Abschätzung von maximalen Windstärken in Böen, welche bei der Konstruktion von Windrädern eine Rolle spielt. Mit der Umformulierung des Detektionsproblems als Modellselektionsfrage und mit der Bereitstellung geeigneter Modellselektionsstrategie trägt diese Arbeit zur Diskussion und Entwicklung von Methoden im Bereich der Detektion von langreichweitigen Korrelationen bei.

The main objective of wave energy converters (WECs) is to ensure reliable electricity production at a competitive cost. Two challenges to achieving this are ensuring an efficient energy conversion and offshore survivability.         This thesis work is structured in three different sections: Control and maximum power optimization, forces and dynamics analysis in extreme wave conditions, and statistical modeling of extreme loads in reliability analysis.        The need for prediction and future knowledge of waves and wave forces is essential due to the non-causality of the optimal velocity relation for wave energy converters. Using generic concepts and modes of motion, the sensitivity of the prediction horizon to various parameters encountered in a real system is elaborated. The results show that through a realistic assumption of the dissipative losses, only a few seconds to about half a wave cycle is sufficient to predict the required future knowledge for the aim of maximizing the power absorption.          The results of a 1:30 scaled wave tank experiment are used to assess the line force and dynamic behaviour of a WEC during extreme wave events. Within the comparison of different wave type representations, i.e. irregular, regular and focused waves, of the same sea state, the results show that not all the wave types deliver the same maximum line forces. As a strategy of mitigating the line forces during extreme wave events, changing the power take-off (PTO) damping may be employed. With consideration of the whole PTO range, the results indicate an optimum damping value for each sea state in which the smallest maximum line force is obtained. Although wave breaking slamming and end-stop spring compression lead to high peak line forces, it is possible that they level out due to the overtopping effect. Waves with a long wavelength result in large surge motion and consequently higher and more damaging forces.         On the investigation of reliability assessment of the wave energy converter systems, computing the return period of the extreme forces is crucial. Using force measurement force data gathered at the west coast of Sweden, the extreme forces are statistically modelled with the peak-over-threshold method. Then, the return level of the extreme forces over 20 years for the calm season of the year is computed.