Envelope crossing distributions for Gaussian fields
Krzysztof Podgorski and Igor Rychlik
Centre for Mathematical Sciences
Lund Institute of Technology,
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.