NATIONAL AND KAPODISTRIAN UNIVERSITY OF ATHENS SCHOOL OF SCIENCE – FACULTY OF PHYSICS DIVISION OF ENVIRONMENTAL PHYSICS DOCTORAL THESIS Statistical Modeling of Extreme Environmental Values Christos Tsalis M.Sc Advisory Committee Emeritus Professor George Kallos Professor Kostas A. Belibassakis Research Director Takvor H. Soukissian Athens, July 2021 Examination Committee George Kallos Emeritus Professor, National & Kapodistrian University of Athens School of Physics Division of Environment Physics - Meteorology Takvor H. Soukissian Research Director, Hellenic Centre for Marine Research - Institute of Oceanography, Anavissos Kostas A. Belibassakis Professor, School of Naval Architecture & Marine Eng. NTUA George N. Galanis Professor, Hellenic Naval Academy Helena Flocas Professor, National & Kapodistrian University of Athens School of Physics Division of Environment Physics - Meteorology Sofianos Sarantis Associate Professor, National & Kapodistrian University of Athens School of Physics Division of Environment Physics - Meteorology Nikolaos Themelis Assistant Professor, School of Naval Architecture & Marine Eng. NTUA i Acknowledgments This doctoral thesis would not have been possible without the guidance and the help of several people who in one way or another contributed in the preparation and completion of this study. First and foremost, I would like to fully express my thanks and gratitude to my supervisor Prof. George Kallos, and Dr. Takvor Soukissian and Prof. Konstantinos Belibassakis for their guidance. Their great thinking, intuition, patient guidance and encouragement has proven a valuable lesson for me. It has been a pleasure and honour to work under their supervision and both contributed many ideas, useful suggestions and constructive comments. I am greatly indebted to them for proof reading the final draft of this thesis. In addition, I would like to thank all my fellow PhD researchers from the department of Physics at the University of Athens and Naval Architecture & Marine Eng. department of NTUA and my fellow researchers of the Institute of Oceanography of HCMR for their help and moral support. I am greatly indebted for the help of several people who contributed in the preparation and completion of this study, especially Dr Flora Karathanasi for her help and moral support, Mrs Fani Anagnostou and Mr Miltos Stamos for their assistance in computational issues and finally Mr Panagiotis Triantafyllopoulos for his assistance in graphical issues. I would also like to thank Dr Ioannis Papastathopoulos from the department of Mathematics at the University of Edinburgh for his scientific advisory and his advice to extend the present research to other useful scientific areas. Finally, I would like to thank my family, my father Vasilis, my mother Paraskevi and my sisters Maria and Effie for their constant love and unconditional support over the years. Their encouragement has played a vital role throughout my studies. ii ii Dedicated to all my students ii Declaration I declare that this thesis entitled Statistical Modeling of Extreme Environmental Values is the result of my own research under the guidance of Prof. George Kallos, Prof. Kostas A. Belibassakis and Dr. Takvor Soukissian except as cited in the references. This thesis has not been submitted in any form for the award of a higher degree elsewhere. This doctoral thesis was partially supported by the Operational Program "Human Resources Development, Education and Lifelong Learning" and co-financed by the European Union (European Social Fund) and Greek national funds, (MIS 5007050), in (http://www.edulll.gr). Copyright © Tsalis Christos, 2021. All rights reserved. iii Abstract (in English) The accurate estimation of extreme values for metocean parameters (e.g., wind speed) plays a crucial role in the marine renewable energy industry and in coastal and offshore engineering applications. Typical challenges that arise in these fields of interest among others are the limited source of information in samples, commonly associated by the scarcity of long datasets, and the accurate estimation of the underlying dependence structure of the stochastic models that can be used for inference on applied problems with extremes. The present analysis, aims to assess the effect of the asymptotic distributional behavior of two types of extreme wind speed sampling data that form the basis of all subsequent predictions in the long term time scale. The first type of sampling data considered will be subsets of observations extracted from blocks of annual length and the second type are subset of observations exceeding a high enough threshold. The challenges closely related to the special attributes that form these types of sampling data motivate the present thesis which focuses on constructing and improving extreme value models to assess the risk associated to extreme wind speed episodes. In particular, in this thesis we focus on The identification of the combined effects of the samples of wind speed that influence the stability of the parameter estimates as well as the efficiency of the estimators to the modelling of extremes. Providing alternative methods of modelling extremes of wind speed that are less known to the relative fields of interest and infer to demonstrate better in comparison to the standard modelling approach. Extending the formulation of the stationary model of extremes to the parameterization of a nonstationary model in order to incorporate subject specific knowledge in the presence of trends under the assumption of climate wind changes. Extending the classical methods that identify the dependence structure in sample of observations in order to effectively model the extremes that are irregularly spaced in time. Specifically, the reconstruction of a dependent sample of extremes that are irregularly spaced in time is focused on relatively small samples of wind speed where the scarcity of long and complete time series is a common restriction in climatological studies. In this setting, the statistical analysis of the most used and less known estimators that model the extremes of wind speed is inferred from a twofold approach. A simulation study is performed first to assess the effect of the sample size to the estimators of the asymptotic distribution that model extremes. The evaluation of the simulation results is based on several statistical measures. Afterwards, the optimum methods from the simulation analysis are applied to wind speed datasets of different sample size and different direction step of sampling. The evaluation is based on datasets originated from databases of relatively moderate horizontan resolution to the regional locations at the North Sea, at the Pacific coast of central America and at the eastern Atlantic Ocean where these locations are exposed to a strong wind climate with evidence of extreme wind speeds. Inference of the sample size effect and the directional step of sampling to the demonstration of the model estimators is made on the obtained 50- and 100- year wind speed design values. From this assessment, the combined method of moments is advised as the suitable method when the sample size is limited. Other challenges that motivated this study is the modelling of extremes when the extremal characteristics are expected not constant over time. To this effect, seasonality and long-term trends are probably the main reasons that influence the stationary hypothesis of the wind speed processes. In this part of this study, an attempt is made to model the possible trends of extremes in the long-term behavior of the process. Since in practice the trend is unknown, various formulations of the trend as a function of time are assessed to represent the extremes of wind speed when the stationary assumption is not valid in order to alleviate the bias effect from iv the attempt of de-trending the process before the time series is used. Statistical tests challenged the modeling of the trend of rejection or not in favor of stationarity. For the extremes of non- stationary sequences and the application to wind speed design values, our analysis is based on coarse historical data of long datasets at regional locations at the North Sea where trends are notable to influence the wind speed variability. From this assessment, the simplest form of parameterization in the parameters of the extreme value distribution is advised in modeling extremes when stationarity is violated. Another common problem of design to assess risk associated to extremes of wind speed in met-ocean fields of interest, is the scarcity of long datasets. To this limitation, many applications utilize as many as possible extremes from the available dataset by re-sampling to a subset of extremes. However, the re-samples are often affected by dependency and the diagnostics related to the independence limitations is usually violated when the observations of these samples are irregularly spaced in time. To alleviate this effect, a resampling strategy is proposed that effectively models extremes irregularly in time when re-sampling of relatively small datasets of wind speed is advised. The proposed DeCA Uncorrelated (DeCAUn) model provides an improvement to the current physical De-Clustering Algorithm (DeCA) modelling the samples of DeCA irregularly in time. Specifically, the resampling strategy proposed analyzes the correlation effect in samples based on the extension of the standard correlation operator setting weight functions to observations irregularly spaced in time. To infer in terms of precision