Matstat seminarioum fredagen 26 oktober kl 13:15 i MH:227

Anastassia Baxevani, Lund
Velocities of Moving Surfaces

Abstract

In this talk 
we discuss various concepts of velocity for the case of stationary
two-dimensional random fields evolving in time. This work is a
continuation of the ideas introduced by
Longuet-Higgins (1957) and an extension of the concepts presented for
one-dimensional waves in  Podgorski,
Rychlik and Sjö (2000).

Although there are a lot of different ways of
defining velocities, we focus on two cases: 
velocity defined in the
direction of the gradient, which was  introduced by
Longuet-Higgins and on the new notion of velocity, which is the
composition of the velocity in the direction of the gradient and the one in
the direction perpendicular to it. This composite velocity appears to
capture better some of the dynamical aspects of the two-dimensional waves, such
as the drift motion  as well as the motions due to
rising and lowering of a random sea.

Once the velocities are well defined we proceed with the
study of their statistical
properties. They can   be studied either for
arbitrary chosen points or
when observed 
at points on the sea-surface satisfying certain conditions, for
example, points at a high sea level.
This introduces the so-called
{\em sampling bias} which changes the distribution of the quantity at
hand. The form of the  sampling bias distributions can be obtained
by utilizing a generalized Rice formula. 


We derive explicit formulas for the distributions of these
velocities in terms of
spectral moments of the underlying process. This allows us to compare
properties of
different velocities as well as distributions collected at different
points in the sea. 


All that is needed for practical computations is some
of the spectral moments. We illustrate such  computations
for the case of a  Gaussian sea modeled by 
the JONSWAP spectrum.