# Crossings and maxima in Gaussian fields and seas

Abstract: 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 passage density. 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 waves.

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