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Semi-Empirical Probability Distributions and Their SEMI-EMPIRICAL PROBABILITY DISTRIBUTIONS AND THEIR APPLICATION IN WAVE-STRUCTURE INTERACTION PROBLEMS A Dissertation by AMIR HOSSEIN IZADPARAST Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY December 2010 Major Subject: Ocean Engineering SEMI-EMPIRICAL PROBABILITY DISTRIBUTIONS AND THEIR APPLICATION IN WAVE-STRUCTURE INTERACTION PROBLEMS A Dissertation by AMIR HOSSEIN IZADPARAST Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Approved by: Chair of Committee, John M. Niedzwecki Committee Members, Kenneth F. Reinschmidt James M. Kaihatu Steven F. DiMarco Head of Department, John M. Niedzwecki December 2010 Major Subject: Ocean Engineering iii I.ABSTRACT Semi-empirical Probability Distributions and Their Application in Wave-Structure Interaction Problems. (December 2010) Amir Hossein Izadparast, B.S., Shiraz University; M.S., University of Tehran Chair of Advisory Committee: Dr. John Michael Niedzwecki In this study, the semi-empirical approach is introduced to accurately estimate the probability distribution of complex non-linear random variables in the field of wave- structure interaction. The structural form of the semi-empirical distribution is developed based on a mathematical representation of the process and the model parameters are estimated directly from utilization of the sample data. Here, three probability distributions are developed based on the quadratic transformation of the linear random variable. Assuming that the linear process follows a standard Gaussian distribution, the three-parameter Gaussian-Stokes model is derived for the second-order variables. Similarly, the three-parameter Rayleigh-Stokes model and the four-parameter Weibull- Stokes model are derived for the crests, troughs, and heights of non-linear process assuming that the linear variable has a Rayleigh distribution or a Weibull distribution. The model parameters are empirically estimated with the application of the conventional method of moments and the newer method of L-moments. Furthermore, the application of semi-empirical models in extreme analysis and estimation of extreme statistics is iv discussed. As a main part of this research study, the sensitivity of the model statistics to the variability of the model parameters as well as the variability in the samples is evaluated. In addition, the sample size effects on the performance of parameter estimation methods are studied. Utilizing illustrative examples, the application of semi-empirical probability distributions in the estimation of probability distribution of non-linear random variables is studied. The examples focused on the probability distribution of: wave elevations and wave crests of ocean waves and waves in the area close to an offshore structure, wave run-up over the vertical columns of an offshore structure, and ocean wave power resources. In each example, the performance of the semi-empirical model is compared with appropriate theoretical and empirical distribution models. It is observed that the semi-empirical models are successful in capturing the probability distribution of complex non-linear variables. The semi-empirical models are more flexible than the theoretical models in capturing the probability distribution of data and the models are generally more robust than the commonly used empirical models. v II.DEDICATION To my beloved family and friends vi III.ACKNOWLEDGMENTS I would like to gratefully acknowledge the partial financial support of the Texas Engineering Experiment Station and the R.P. Gregory ’32 Chair endowment during my Ph.D. studies. In addition, I would like to recognize Dr. Per S. Teigen of StatoilHydro and the Offshore Technology Research Center for permission to utilize the experimental data presented in this study. I would like to express my sincere gratitude and thankfulness to my academic advisor, Prof. John Niedzwecki, for his encouragement, supervision, and support. His great vision and guidance were crucial in the completion of this dissertation. I am also grateful to my advisory committee members: Dr. Kenneth Reinschmidt, Dr. James Kaihatu, and Dr. Steven DiMarco for their input and guidance throughout this research study. I am deeply thankful to my parents for their immense support, unconditional love, and inspiration throughout my life. I would also like to thank my sister for her constant encouragement. Finally, I am greatly thankful to my Noushin for her patience, support, and help. She has given me emotional solace and has been my greatest inspiration. vii IV.TABLE OF CONTENTS Page ABSTRACT ..................................................................................................................... iii DEDICATION ................................................................................................................... v ACKNOWLEDGMENTS ................................................................................................ vi TABLE OF CONTENTS ................................................................................................ vii LIST OF FIGURES ............................................................................................................ x LIST OF TABLES ........................................................................................................... xv CHAPTER I INTRODUCTION ............................................................................................. 1 I.1 Problem Definition .................................................................................... 4 I.2 Research Methodology .............................................................................. 5 I.2.1 Model development ..................................................................... 5 I.2.2 Model uncertainty analysis .......................................................... 8 I.2.3 Model application and evaluation .............................................. 10 II SEMI-EMPIRICAL MODEL DEVELOPMENT ........................................... 12 II.1 Mathematical Background ..................................................................... 12 II.2 Empirical Parameter Estimation ............................................................ 14 II.2.1 Method of moments .................................................................. 14 II.2.2 Method of L-moments .............................................................. 16 II.3 Gaussian-Stokes Model .......................................................................... 19 II.3.1 Model development .................................................................. 19 II.3.2 Parameter estimation ................................................................ 20 II.4 Rayleigh-Stokes Model .......................................................................... 23 II.4.1 Model development .................................................................. 23 II.4.2 Parameter estimation ................................................................ 25 viii CHAPTER Page II.5 Weibull-Stokes Model ........................................................................... 28 II.5.1 Model development .................................................................. 28 II.5.2 Parameter estimation ................................................................ 30 II.6 Extreme Analysis ................................................................................... 33 III MODEL UNCERTAINTY ANALYSIS ......................................................... 36 III.1 Introduction ........................................................................................... 36 III.2 Model Parameter Uncertainty ............................................................... 38 III.2.1 Gaussian-Stokes model ........................................................... 38 III.2.2 Rayleigh-Stokes model ........................................................... 42 III.2.3 Weibull-Stokes model ............................................................. 46 III.3 Sample Variability and Parameter Estimation Uncertainty .................. 51 III.3.1 Moments vs. L-moments ......................................................... 51 III.3.2 Parameter estimates ................................................................. 55 III.3.3 Quantile estimates and extreme statistics ................................ 59 III.4 Effects of Faulty Extreme Measurements ............................................. 68 IV SEMI-EMPIRICAL MODEL APPLICATIONS ............................................ 73 IV.1 Introduction .......................................................................................... 73 IV.2 Mini-TLP Model Test Data .................................................................. 75 IV.3 Probability Distribution of Wave Elevations ........................................ 79 IV.3.1 Introduction ............................................................................. 79 IV.3.2 Mathematical background ....................................................... 83 IV.3.3 Theoretical probability distributions ....................................... 86 IV.3.4 Empirical probability distributions ......................................... 89 IV.3.5 Sample data sets ...................................................................... 92 IV.3.6 Analysis and results ................................................................ 93 IV.4 Probability
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