# Statistical Analysis of Non-Gaussian Environmental Loads and Responses

## Ulla Eduarda Botelho Machado

### Centre for Mathematical Sciences Mathematical Statistics Lund Institute of Technology 2002

##### ISBN 91-628-5254-X LUTFMS--1019--2002

Abstract:
The thesis deals mainly with offshore engineering related problems where the dominant source of uncertainty is related to the loading. Loads arise from environmental random processes; e.g. waves, currents and winds. Complex as they are, such processes beg the consideration of randomness whence the need of associating probabilistic models to the engineering problems treated here.

Two different types of problems are investigated. Given a sea-state, or wind condition, we model: (i) the sea surface elevation at a fixed location, and (ii) the response of structures to environmental loads.

We start by assuming the sea surface elevation, at a fixed location, as a Gaussian process. For this case, exact integral forms of the joint long run distributions for the wave characteristics (wave periods, lengths, and heights) are derived. As the water depth decreases or the sea severity increases, the sea surface elevation departs from the Gaussian assumption and the wave profile becomes asymmetric. From a practical point of view this fact has several important consequences. Thus, the sea surface elevation is then considered to be a stationary non-Gaussian process: i.e. a sum of a Gaussian process and a second-order correction term. For such processes the problem of estimating the marginal probability density function is considered. The statistical analysis proceeds with the problem of calculating the mean upcrossing intensity function. Two different methods
to obtain numeric estimates of the latter function are presented: (i) a method based on the saddlepoint approximation, and (ii) a method based on numerical integration. The mean upcrossing intensity function as estimated by these methods is then used to estimate the distribution of wave characteristics through a transformed Gaussian model.

In engineering applications the process which represents the response of structures to environmental loads can often be written as a sum of a Gaussian process and a second-order correction term. The statistical analysis of such responses follows the same pattern as the one outlined above.

Keywords:
Stokes waves, random waves, responses, mean upcrossing intensity, Rice's formula, saddlepoint approximation,
distribution of wave characteristics