the case of sufficiently transparent and shallow waters this assumption becomes
invalid. Indeed, in such waters a certain amount of photons traveling downwards is
reflected at the bottom back to the water-air interface instead of being absorbed by
water molecules at depth. It is conducive to the probability that a certain fraction of
reflected photons eventually reach the water surface. Thus, the resulting diffuse
reflectance just beneath the water surface R tot ðÀ0; lÞ can be decomposed:
R tot ðÀ0; lÞ ¼ R w ðÀ0; lÞ þ R bot ðÀ0; lÞ;
(16.7)
where R w ðÀ0; lÞ is the upwelling spectral reflectance due to water (by definition it
equals the ratio E u ðÀ0; lÞ=E d ðÀ0; lÞ; E u ; E d , being upwelling and downwelling
irradiances at z ¼ À0) and R bot ðÀ0; lÞ is the upwelling spectral reflectance due to
the bottom optical influence, which can be defined as:
R bot À0; l
ð
Þ¼ A À R e:s: À0; l
ð
Þ
ð
Þ exp À2K d l
ð Þh
ð
Þ ;
(16.8)
A being the bottom spectral albedo.
Under cloudless conditions, the coefficient of downwelling irradiance attenuation can be parameterized as follows:
K sun À0; l; y 0
ð
Þ¼ 1 m 0
=
ð
Þ a
2 l
ð Þ þ 0:473m 0 À 0:218
ð
Þ a l
ð Þb l
ð Þ
Â
à 1=2 ;
(16.9)
where m 0 , as above, is equal to cosðy
0
0 Þ; y
0
0 being the in-water refracted sun zenith
angle.
For overcast conditions:
K sky À0; l
ð
Þ¼1:168 a
2 l
ð Þ þ 0:168a l
ð Þb l
ð Þ
Â
à 1=2 :
(16.10)
Hence, the attenuation coefficient of downwelling global radiation, K d ðÀ0; l; y 0 Þ,
can be expressed as
K d À0; l; y 0
ð
Þ¼F w K sky l
ð Þ þ 1 À F w
ð
ÞK sun l; y 0
ð
Þ;
(16.11)
where F w ¼ F 1 À r sky
= F 1 À r sky
þ 1 À F
ð
Þ 1 À r sun y 0
ð Þ
ð
Þ
h
i
, r sky ¼ 0:066
and r sun ðy 0 Þ are the Fresnel reflectivities of sky and solar irradiance (directly
propagating from the zenith angle y 0 ) respectively; F ¼ E sky = E sky þ E sun
À
Á
is the
fraction of the incident irradiance that is diffuse. It is valid r sky ¼ 0.066, and the
value r sun ðy 0 Þ can be numerically assessed from the Fresnel formulas:
r sun ¼
1
2
sin
2 y i À y r
ð
Þ
sin
2 y i þ y r
ð
Þ
þ
tg
2 y i À y r
ð
Þ
tg 2 y i þ y r
ð
Þ
(16.12)
16.2 Mechanisms of Interactions of Solar Light
165
invalid. Indeed, in such waters a certain amount of photons traveling downwards is
reflected at the bottom back to the water-air interface instead of being absorbed by
water molecules at depth. It is conducive to the probability that a certain fraction of
reflected photons eventually reach the water surface. Thus, the resulting diffuse
reflectance just beneath the water surface R tot ðÀ0; lÞ can be decomposed:
R tot ðÀ0; lÞ ¼ R w ðÀ0; lÞ þ R bot ðÀ0; lÞ;
(16.7)
where R w ðÀ0; lÞ is the upwelling spectral reflectance due to water (by definition it
equals the ratio E u ðÀ0; lÞ=E d ðÀ0; lÞ; E u ; E d , being upwelling and downwelling
irradiances at z ¼ À0) and R bot ðÀ0; lÞ is the upwelling spectral reflectance due to
the bottom optical influence, which can be defined as:
R bot À0; l
ð
Þ¼ A À R e:s: À0; l
ð
Þ
ð
Þ exp À2K d l
ð Þh
ð
Þ ;
(16.8)
A being the bottom spectral albedo.
Under cloudless conditions, the coefficient of downwelling irradiance attenuation can be parameterized as follows:
K sun À0; l; y 0
ð
Þ¼ 1 m 0
=
ð
Þ a
2 l
ð Þ þ 0:473m 0 À 0:218
ð
Þ a l
ð Þb l
ð Þ
Â
à 1=2 ;
(16.9)
where m 0 , as above, is equal to cosðy
0
0 Þ; y
0
0 being the in-water refracted sun zenith
angle.
For overcast conditions:
K sky À0; l
ð
Þ¼1:168 a
2 l
ð Þ þ 0:168a l
ð Þb l
ð Þ
Â
à 1=2 :
(16.10)
Hence, the attenuation coefficient of downwelling global radiation, K d ðÀ0; l; y 0 Þ,
can be expressed as
K d À0; l; y 0
ð
Þ¼F w K sky l
ð Þ þ 1 À F w
ð
ÞK sun l; y 0
ð
Þ;
(16.11)
where F w ¼ F 1 À r sky
= F 1 À r sky
þ 1 À F
ð
Þ 1 À r sun y 0
ð Þ
ð
Þ
h
i
, r sky ¼ 0:066
and r sun ðy 0 Þ are the Fresnel reflectivities of sky and solar irradiance (directly
propagating from the zenith angle y 0 ) respectively; F ¼ E sky = E sky þ E sun
À
Á
is the
fraction of the incident irradiance that is diffuse. It is valid r sky ¼ 0.066, and the
value r sun ðy 0 Þ can be numerically assessed from the Fresnel formulas:
r sun ¼
1
2
sin
2 y i À y r
ð
Þ
sin
2 y i þ y r
ð
Þ
þ
tg
2 y i À y r
ð
Þ
tg 2 y i þ y r
ð
Þ
(16.12)
16.2 Mechanisms of Interactions of Solar Light
165