Duration Distribution and Up-crossings Rate of Generalized Hyperbolic Processes

Moh’D T Alodat,Khalid M Aludaat

doi:10.17713/ajs.v36i3.332

Abstract

A Gaussian process is usually used to model the sea surface elevation in the oceanography. As the depth of the water decreases or the sea severity increases, the sea surface elevation departs from symmetry and Gaussianity. In this paper, a stationary non-Gaussian random process called the generalized hyperbolic process is used as an alternative model. The process generates a family of processes. We derive the rate of up-crossings for this process and the distribution of the height of the process. We also derive the duration distribution of an excursion for the generalized hyperbolic process.

Highlights

In oceanography, the sea surface elevation, at a fixed location, is modelled by a stationary Gaussian random process
The Gaussian model will not capture the asymmetry in the data
The excursion set of the process X(t) above a level u is defined to be the set of points t ∈ [0, A], where X(t) exceeds u, i.e., the set {t ∈ [0, A] : X(t) ≥ u}

Summary

Introduction

The sea surface elevation, at a fixed location, is modelled by a stationary Gaussian random process. The statistical properties of the sea surface elevation are called the sea state These properties are very important for the reliability analysis in the ocean engineering (Baxevani et al, 2005). As the depth of the water decreases or the sea severity increases, the sea surface elevation departs from both symmetry and Gaussianity (Baxevani et al, 2005; Rychlik, 1993; Rychlik and Leadbetter, 2002; Cherneva et al, 2005) Under these sea states, the Gaussian model will not capture the asymmetry in the data. We should not ignore the asymmetry when modelling the sea (Baxevani et al, 2005) In this case, we need an alternative non-Gaussian stationary random process to model the sea surface elevation. The following basic result is from Adler (1981) and will be used later

Crossings and Duration of a Process

Generalized Hyperbolic Distribution

Generalized Hyperbolic Process

Special Cases

Approximating the Duration Distribution

Conclusions