Crossings and Maxima in Gaussian Fields and Seas
Centre for Mathematical Sciences
Lund Institute of Technology
In this thesis the focus is on crossing points in random fields and the
probability distributions of various crossing variables in different
applications.The crossing points are generalisations to random fields of
points of level crossings by stochastic processes. Only crossing events that
occur at distinct points in the parameter space are considered, for example
stationary points (zero-crossings by the gradient). Crossing variables are
variables that are defined attached to a crossing point, for example the
height of a local maximum.
The intensity of crossing points is given by a generalisation of Rice's formula
for the expected number of level crossings, and the formulation of Zähle
(1984) [Stoch. Proc. Appl. 17: 255-283], which is valid under very mild
conditions, is used.
The distributions of the crossing variables are derived in the ergodic, or
Palm, sense conditioned on the related crossing points. An ergodic distribution
is given by a ratio of intensities, for which the generalised Rice's formula
is used. Further, the density function of the ergodic distribution is for
some types of crossing variables derived using Durbin's formula for the first
The thesis consists of an introductory survey of the subject and related
theory, followed by five included papers (A-E) where different applications
are studied. In addition to the derivation of formulae for the ergodic
distributions and their densities, emphasis is on computation of the formulae
under the assumption that the random field is Gaussian. All presented results
are numerically exemplified, and when numerical approximations are necessary,
the method used and its accuracy is discussed. Some of the results are also
verified by simulations.
In Paper A, the intensity of local maxima of nonhomogeneous random fields
is the basis for evaluation of approximative confidence regions for the local
maxima of reconstructed surfaces.
In Paper B, the global maximum is studied. The density of the height and
position of the global maximum is
derived for absolutely continuous stochastic processes. The result can be
applied to both stationary and non-stationary processes.
A Gaussian homogeneous spatio-temporal random field is often used to model
the water free-surface of a sea. In Papers C, D, and E, this application
is studied, and examples of analysed crossing variables are spatio-temporal
wave characteristics, wave velocities, and wave characteristics of extremal
Gaussian random fields, level-crossings, generalised Rice's formula, extremes,