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9.3 Characteristics
of Multipath Fading Due to Sea Surface Reflection
9.3.1 Fundamental Concepts
We use here the developments of Karasawa
and Shiokawa [1988] in describing fundamentals of fading due to sea
surface reflections. Their model adopted concepts described by Sandrin
and Fang [1986] who proposed a method for determining the fading depth
using a multipath power diagram as a function of antenna gain, elevation
angle and Nakagami-Rice statistics [ITU-R-1994,
pp. 59-61]. Although the model of Karasawa and Shiokawa was confirmed
for circular polarization, it is believed also applicable for horizontal
and vertical polarizations with appropriate caveats.
Multipath reflections from the sea are comprised of "specular" and "diffuse"
contributions, also referred to as "coherent" and "incoherent" components,
respectively. For calm seas, the specular component dominates but decreases
rapidly for increasingly rough seas. The total power received is obtained
by combining the power contributions of coherently scattered and incoherently
scattered waves, and the direct wave component. Under calm sea conditions,
the forward-scattered wave is composed entirely of the coherent component
that can be estimated by
 ,
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where 
is
the incident wave vector and where i denotes either the horizontal
or vertical components (H,V) and * denotes the complex conjugate.
That is,
 .
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Also, R is the Fresnel reflection coefficient matrix of the sea
given by
 ,
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where for the calm sea case, RHV = RVH
= 0. DR is the directivity of the antenna in the direction
of the specular reflection point and given by
 .
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The intensity of the coherent and incoherent components depends on the
roughness of the sea expressed by the roughness parameter u given
by
 ,
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where 
is
the wavelength in m, 
is the elevation angle to the satellite, h0 is the RMS
wave height in m related to the significant wave height H by
 .
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The significant wave height is the average height of waves corresponding
to the highest one-third of the actual waves that pass a fixed point. When
the sea is rough, the incoherent and coherent components of the scattered
powers, PI, PC may be expressed in
terms of the coefficients 
and 
defined
such that
and
 ,
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where PCO is given by (9-1). The
mean total power of the reflected waves is thus given by
 .
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The coherent component 
is related to the roughness parameter u by the relation [Beckmann
and Spizzichino, 1963],
 .
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For u > 1, Miller
et al. [1984] showed the results to be consistent with the measurements
of Beard [1961] (for
u
up to 3) where
is defined by
and where I0 is the zero-order modified Bessel function.
Karasawa and Shiokawa demonstrated from their measurements at 1.5 GHz using
right-handed circular polarization that
 .
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Combining (9-12) with (9-10) or (9-11) enables 
to be explicitly expressed by
or
 .
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The measurements of Karasawa and Shiokawa were sufficiently scattered such
that they fell within the bounds of both (9-13) and (9-14).
The level of the maximum coherent component PCO of
the reflected wave may be determined by varying the antenna height on board
the ship during calm sea conditions. By determining the level difference
between the minimum and maximum signals (for a calm sea), the intensity
of the coherent component of the scattered wave relative to the direct
wave component can be evaluated. The intensity of the incoherent component
of the scattered wave may be obtained through the assumption that the incoherent
signal follows a Nakagami-Rice distribution. Figure
9-1 shows the relation between the fading depth and the mean total
multipath power relative to the direct signal power assuming this distribution.
Karasawa
and Shiokawa [1984] define the "fading depth" as the dB difference
between the signal level of the direct incident wave and a threshold level
that the resultant signal level exceeds with a probability of P
percent of the time. The individual curves denote the indicated probability
of exceeding the ordinate fading depth. Shipborne experiments carried out
by Karasawa
and Shiokawa [1988] have corroborated use of this distribution for
values of
u
2.
As an illustration of the dependence of u on wave-height, Figure
9-2 is a plot of u given by (9-5) versus
the RMS wave height
h (upper abscissa) and significant wave height
H
(lower abscissa). We note that at =
5°, u2
implies H
> 1.5 m and h > 0.4 m.
Figure 9-1: Fading depth versus total multipath power relative
to the direct signal level for different exceedance probabilities (of the
incoherent level) based on the Nakagami-Rice distribution.
Figure 9-2: Roughness parameter versus significant wave height
(lower abscissa) and RMS wave height (upper abscissa).
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