Digital Signal Processing Reference
In-Depth Information
being S.k/ the omni-directional spectrum
Z
S.k/
D
‰.k; / k d
(9.13)
The SWH is defined as SWH
D
4
p
z
.
The following spectra are a few popular ones. Those given as frequency spectrum
can be transformed into wavenumber spectrum using Eq.
9.10
.
Pierson-Moskowitz:
A simple wave spectrum is the Pierson-Moskowitz spec-
trum (
Pierson and Moskowitz 1964
), empirically obtained from measurements
in North Atlantic (1964)
˛g
2
!
5
g
!U
19:4
/
4
e
ˇ.
S
PM
.!/
D
(9.14)
where ˛ and ˇ are constant values ˛
D
0:0081, ˇ
D
0:74, and U
19:4
represents
the speed of the wind at 19.4 m altitude. It was revised in
Henrique et al.
(
2003
).
JONSWAP:
A well-accepted long-wave spectrum after experimental campaigns
in the North Sea (
Hasselmann et al. 1973
), and formulated as a modification of
the Pierson-Moskowitz spectrum, and it include limited fetch conditions. It is
expected to
be a re
asonable mo
del for
seas with peak period T
p
and SWH in the
range 3:6
p
SWH < T
p
<5
p
SWH
˛g
2
!
5
e
ˇ.
!
!
/
4
r
S
J
.!/
D
(9.15)
where r
D
exp
f
.!
!
p
/
2
=.2
2
!
p
/
g
and !
p
is the peak wave-frequency. The
empirical values are: !
p
D
22.g
2
=.U
10
F//
1=3
;
D
3:3;
D
0:07 (!
!
p
)or
D
0:09 (!>!
p
). U
10
is the wind speed at 10 m altitude, and F is the fetch
length.
Elfouhaily et al.:
The directional spectrum suggested in
Elfouhaily et al.
(
1997
)
is widely used in the GNSS-R community. It is a full wavenumber spectrum
to properly describe long and short waves. Both fetch and wave age (stage of
development of the sea) are included as input parameters.
1
2
k
4
ŒB
l
C
B
h
Œ1
C
.k/ cos.2/
‰.k; /
D
(9.16)
where B
l
, B
h
, and .k/ are given in (
Elfouhaily et al. 1997
, Eqs. 31, 40, 57)
respectively.
Besides waves generated by the blowing wind, the surface might present waves
from distant or past storms, what is called the
swell
. They typically present regular
waves of relatively long wavelength. The reason is that long waves travel faster than
the short ones,
escaping
from the storm. Typical swell wavelength are of the order of
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