A theoretical investigation of the wave height and level-crossing distributions

In the general theory off stochastic processes, among the most challenging and still mainly unsolved problems are those concerning the zero- and level-crossing behaviour and extreme distribution of sample functions of stochastic processes. In this work, we investigate these properties for a class of stationary Gaussian ergodic random processes, with special reference to the ocean wave process. The motivation for this research was provided by the desire to find a useful distribution of wave heights for the broad-band ocean wave process. It was found that the most meaningful definition of wave heights is the so called zero-up-crossing wave heights. We show that the distribution of zero-up-crossing wave heights is closely related to the distributions of absolute maxima (or minima), of the first passage time, and of the adjacent zero-crossings. Thus, these distributions are the main concern of this thesis. A dynamical model for the ocean wave process is first constructed. With a Gaussian white noise input, the output process of this dynamic system is a vector Markov process. Although many statistical properties of one dimensional Markov processes are known and reviewed in Chapter 2, there has been little similar progress for vector Markov processes. In the case of random initial velocities for the one-dimensional Markov processes, an approximate formula for the probability density function of absolute maxima between adjacent zero crossings is derived. We use a simple example to show how this formula can be used. We then study second order (two-dimensional) vector Markov processes in order to find a more accurate expression and better results for the absolute maxima distribution. The only existing analytical formula for two dimensional vector Markov process is the joint probability density function of adjacent zero crossing intervals and crossing velocities for a second order oscillator given McKean (1963). Wong (1966, 1970) extended McKean's result to a wider class of random processes (Wong's processes) which includes the ocean wave process as a special case. In this thesis, we establish formal relationships between the level-crossing and absolute maxima distribution of McKean's process and those of Wong's process. In fact, we show that these relationships are applicable to a class of random processes (i.e. generalised Wong's processes), slightly wider than Wong's process. We then proceed to find the joint probability density function of the first passage time and crossing velocities for the McKean's process. From this joint probability density, we can find the probability density function of absolute maxima for both McKean's process and generalised Wong's processes. A simulation study via digital filtering is performed to confirm the analytical results. Finally, an optimal weighting function is proposed for the estimation of directional ocean wave spectrum.