# Applications of Rice's Formula in Oceanographic and Environmental Problems

## Sofia Åberg

### Centre for Mathematical Sciences Mathematical Statistics Lund University 2007

##### ISBN 978-91-628-7124-6 LUTFMS-1030-2007

Abstract:
This thesis is based on five papers (A-E), all treating various applications of Rice's formula. The areas of application are oceanographic and environmental problems.
The first three papers treat properties of encountering waves, that is, waves that a ship encounters while sailing on the ocean. Paper A addresses the problem of computing the intensity of such waves and, in particular, the intensity of encountering waves that overtake the ship. A general formula for this intensity is given and it is shown that for a Gaussian sea it takes an explicit form. The aim of Paper B is to compute distributions of properties of the overtaking, encountering waves in a Gaussian sea. Besides giving integral formulas for the distribution of wave slope and the joint distribution of waveheight and half-wavelength, we also give interpretation of the results in terms of wave velocity. The results are illustrated by numerical examples. Next, in Paper C, the distribution of waveheight and half-wavelength is further explored. More precisely, we propose an approximation of the distribution, which is inspired by the velocity interpretation in Paper B. This approximation is evaluated for a Gaussian sea having a Pierson-Moskowitz spectrum.
Paper D is devoted to the study of wave intensity and the distribution of slope in a Lagrangian sea model. Two different cases are considered, namely the sea surface observed at a fixed time point and at a fixed location, respectively. Formulas for the wave intensity and slope are given for these two cases, and the results are compared to the Gaussian sea model.
In the last paper, Paper E, we consider some aspects of the statistical quality of air quality standards. More precisely, we want to illuminate the danger of not taking the spatial variability of the air pollution concentrations into account in the design of the standards. This is illustrated by computing upper and lower bounds for the distribution of the maximum of the pollutant concentration in a region around the site where the measurement is taken. The maximum is computed conditional on a measurement being exactly at the level prescribed by the standard.
Key words:
Encountering waves; Gaussian random fields; level crossings; Palm distributions; Rice's formula; wave statistics