Applications of Rice's Formula in Oceanographic and Environmental Problems
Sofia Åberg
Centre for Mathematical Sciences
Mathematical Statistics
Lund University
2007
ISBN 9789162871246
LUTFMS10302007

Abstract:


This thesis is based on five papers (AE), 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 halfwavelength, 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 halfwavelength 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 PiersonMoskowitz 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








