Envelope crossing distributions for Gaussian fields

Krzysztof Podgorski and Igor Rychlik

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
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. Vanmarcke (1975) improved this approximation by accounting for the proportion of empty excursions of the envelope. Ditlevsen and Lindgren (1988) proposed an accurate approximation of this proportion by using the Slepian model method. This approximation was further studied in Ditlevson (1994). In the first part of the paper, we review the approach as well as give a brief account of the results.
In the second main part of the paper, the method of sampling distribution is applied to the envelope field that is a generalization of the envelope process. The need of considering a field rather than a process is particularly important in these applications for which both spatial and temporal variability has to be taken into account. Here we notice that the envelope field is not uniquely defined and that its statistical properties depend on a chosen version. We utilize convienient envelope sampling distributions to decide for 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. It is particularly important in marine applications as the velocity of the envelope is related to the rate at which energy is transported by propagating waves. 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 waves groups.
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