Envelope crossing distributions for Gaussian fields

Journal article, 2008

The envelope process is an analytical tool often used to study extremes and wave groups. In an approach to approximate the first passage probability for the underlying response, the average number of envelope crossings is used to obtain an upper bound. In the first part of the paper, we review the approach as well as give a brief account of the previous results with some focus on the contribution of Ove Ditlevsen. In the main part of the paper, the method of sampling distribution is applied to the envelope field that is a generalization of the envelope process. Here we notice that the envelope field is not uniquely defined and that its statistical properties depend on a chosen version. We utilize convenient envelope sampling distributions to decide on a version that has desired smoothing properties. The spatial-temporal Gaussian sea-surface model is used to illustrate this approach. One intrinsically multivariate problem is studying velocities of moving spatial records. Under the Gaussian model we derive sampling properties of the envelope velocity measured at the level contours. By associating the properties of envelope with the properties of group waves we present differences between statistical distributions of individual waves and wave groups.