2.4 SPECULAR REFLECTANCE
An above-water radiance sensor looking down to the water surface measures the
sum of two radiance components: one from the water body and one from the surface.
The first comprises the desired information about the water constituents, the second is
an unwanted add-on which requires correction. However, correction is difficult. For
example, the method from the SeaWiFS protocols (Mueller and Austin, 1995), which is
widely used in optical oceanography, leads to rms errors of the corrected water leaving
radiance as large as 90% under typical field conditions (Toole et al., 2000). Thus,
WASI offers different methods.
The radiance reflected from the surface, L r (Ȝ), is a fraction σ L of sky radiance
L s (Ȝ):
L r (Ȝ) = σ L · L s (Ȝ).
(11)
L s (Ȝ) is the average radiance of that area of the sky that is specularly reflected into the
sensor. It can be imported from file or calculated using eq. (25). σ L is the Fresnel
reflectance and depends on the angle of reflection. The value can either be specified by
the user or it can be calculated from the viewing angle ș v using the Fresnel equation for
unpolarized light (Jerlov 1976):
2
2
2
2
sin (
' ) tan (
' )
1
.
2 sin (
' ) tan (
' )
v
v
v
v
L
v
v
v
v
σ
−
−
=
+
+
+
(12)
ș’ v is the angle of refraction, which is related to ș v by Snell’s law n W sinș’ v = sinș v ,
where n W ≈ 1.33 is the refractive index of water. For viewing angles near nadir, ı L ≈
0.02.
The ratio of the radiance reflected from the water surface to the downwelling
irradiance,
,
)
(
E
)
(
L
)
(
E
)
(
L
)
(
R
d
s
L
d
r
surf
rs
λ
λ
⋅
σ
=
λ
λ
=
λ
(13a)
is called specular reflectance. E d (Ȝ) and L s (Ȝ) can either be imported from file, or one or
both can be calculated using eq. (23) or (25). If the wavelength-independent model of
surface reflection is chosen, it is
.
R
L
surf
rs
π
σ
=
(13b)
Toole et al. (2000) showed that R rs
surf (Ȝ) is nearly spectrally flat for an overcast day, but
is not flat for clear-sky conditions. Thus, eq. (13a) should be used in general, and eq.
(13b) only for overcast days.
ș ș
ș ș
ș ș
ș ș
87
Inverse Modeling of Spectral Measurements
An above-water radiance sensor looking down to the water surface measures the
sum of two radiance components: one from the water body and one from the surface.
The first comprises the desired information about the water constituents, the second is
an unwanted add-on which requires correction. However, correction is difficult. For
example, the method from the SeaWiFS protocols (Mueller and Austin, 1995), which is
widely used in optical oceanography, leads to rms errors of the corrected water leaving
radiance as large as 90% under typical field conditions (Toole et al., 2000). Thus,
WASI offers different methods.
The radiance reflected from the surface, L r (Ȝ), is a fraction σ L of sky radiance
L s (Ȝ):
L r (Ȝ) = σ L · L s (Ȝ).
(11)
L s (Ȝ) is the average radiance of that area of the sky that is specularly reflected into the
sensor. It can be imported from file or calculated using eq. (25). σ L is the Fresnel
reflectance and depends on the angle of reflection. The value can either be specified by
the user or it can be calculated from the viewing angle ș v using the Fresnel equation for
unpolarized light (Jerlov 1976):
2
2
2
2
sin (
' ) tan (
' )
1
.
2 sin (
' ) tan (
' )
v
v
v
v
L
v
v
v
v
σ
−
−
=
+
+
+
(12)
ș’ v is the angle of refraction, which is related to ș v by Snell’s law n W sinș’ v = sinș v ,
where n W ≈ 1.33 is the refractive index of water. For viewing angles near nadir, ı L ≈
0.02.
The ratio of the radiance reflected from the water surface to the downwelling
irradiance,
,
)
(
E
)
(
L
)
(
E
)
(
L
)
(
R
d
s
L
d
r
surf
rs
λ
λ
⋅
σ
=
λ
λ
=
λ
(13a)
is called specular reflectance. E d (Ȝ) and L s (Ȝ) can either be imported from file, or one or
both can be calculated using eq. (23) or (25). If the wavelength-independent model of
surface reflection is chosen, it is
.
R
L
surf
rs
π
σ
=
(13b)
Toole et al. (2000) showed that R rs
surf (Ȝ) is nearly spectrally flat for an overcast day, but
is not flat for clear-sky conditions. Thus, eq. (13a) should be used in general, and eq.
(13b) only for overcast days.
ș ș
ș ș
ș ș
ș ș
87
Inverse Modeling of Spectral Measurements