Ocean Remote Sensing Using GNSS-R - GNSS Remote Sensing: Theory, Methods and Applications - page 228

points in Eqs. 8.30 - 8.33 , 8.42 ,or 8.43 ). In Ocean applications, both statistical
characterization and numerical surface realizations can be obtained from the Ocean
waves' spectrum.
This section reports about a few popular spectra; then it describes how to use
them to model the statistics of the slopes; and finally a brief summary on how to
generate numerical realizations of surfaces making use of the spectra.

9.2.1.1 Ocean Wave Spectra

Ocean wave spectra inform about the distribution of the Ocean wave energy
density with respect to the waves' temporal or spatial frequencies, as well as their
directions. The waves' spatial frequency (wavenumber k D 2=) and the time-
frequency spectra can be converted into each other making use of a particular
dispersion relation between k and !. For dee p waters (depth larger than half wave's
wavelength) the dispersion reads: ! D p gk,so

S.k/ D S.!/ @!

g
2! @k D S.!/

(9.10)

g being the acceleration due to gravity.
Spectra can be measured for a particular site and epoch, or it can also be modelled
as functions of a given set of parameters. Typical parameters that relate to the
sea state conditions are the wind and the significant wave height (SWH). Other
parameters that can affect the spectrum are the fetch (length of water over which the
wind has blown), and stage of development of the sea. The waves grow higher in
altitude and longer in period as the wind blows longer or over larger areas. As the
wind keeps blowing and the waves keep growing the sea is under developing stage,
the so called developing sea . However, after some time of steady wind conditions the
phase velocity of the waves' crests matches the wind speed, waves and wind reach a
point of equilibrium. From this point on, the waves neither grow higher nor increase
their period, regardless of the wind duration. The sea is then fully developed . When
the wind drops it will enter in the decaying phase . Many spectra consider fully
developed seas solely, but some of them account for the stage of development.
‰. k / represents the distribution of the waves' energy as a function of their 2-
dimensional wavenumber k D 2= u . In particular, it is the Fourier transform of
the auto-covariance function of the surface displacements

‰. k / D FT f < z . r 0 / z . r 0 C r /> g (9.11)

where z . r / is the elevation of the sea surface at coordinates r D .x; y/. The spectrum
is normalized so that

Z 1 Z 1 Z 1

z D < z 2 > D

‰.k x ;k y /dk x dk y D S.k/dk (9.12)

1 1 0