The three equations are formally identical. A derivation is given in Mobley (1994). The
first term on the right-hand side of each equation describes reflection in the water, the
second at the surface. Frequently, the first term alone is called remote sensing
reflectance (e.g. Mobley, 1994). In WASI, the reflection at the surface is also included
in the R rs definition. It is calculated using eq. (13a) or (13b) and can easily be excluded
by setting the reflection factor ı L equal to zero.
R rs
− (Ȝ) is calculated using eq. (17) or (19), R(Ȝ) using eq. (14) or (16). The factors
ı, ı L
− , and ı
− are the reflection factors for E d , L u
− , and E u
− , respectively. ı depends on
the radiance distribution and on surface waves. Typical values are 0.02 to 0.03 for clear
sky conditions and solar zenith angles below 45°, and 0.05 to 0.07 for overcast skies
(Jerlov, 1976; Preisendorfer and Mobley, 1985, 1986). The default value is ı = 0.03.
ı L
− can either be calculated as a function of ș v using eq. (12), or a constant value can be
inserted. ı
− is in the range of 0.50 to 0.57 with a value of 0.54 considered typical
(Jerome et al., 1990; Mobley, 1999). The defaults of the other constants are set to Q = 5
sr and n W = 1.33.
Selection of the equation to use depends on the application:
• Eq. (20a) links remote sensing reflectance in water to that in air. Since the
same spectrum type is used above and below the water surface, it is the most
convenient parameterisation. This equation is used by default.
• Eq. (20b) is useful when R rs (Ȝ) is linked to R(Ȝ), for example if in situ
measurements of R(Ȝ) were performed as “ground truth” for a remote sensing
instrument.
• Eq. (20c) avoids the use of the factor Q, which is difficult to assess. The
equation is useful, for example, for optical closure experiments which
investigate the consistency of measurements above and below the water
surface by measuring simultaneously the spectra R rs (Ȝ), R(Ȝ), and R rs
− (Ȝ).
2.7 BOTTOM REFLECTANCE
The irradiance reflectance of a surface is called albedo. When N different surfaces
of albedo a n (Ȝ) are viewed simultaneously, the measured albedo is the following sum:
¦
−
=
λ
⋅
=
λ
1
N
0
n
n
n
b
,
)
(
a
f
)
(
R
(21)
where f n is the areal fraction of surface number n within the sensor’s field of view; it is
Ȉ f n = 1. This equation is implemented in WASI for N = 6 bottom types. Three of the
spectra a n (Ȝ) provided with WASI represent bare bottom, the other green makrophytes:
0 = a constant reflectance of 10%, 1 = sand, 2 = silt, 3 = Chara aspera, 4 =
Potamogeton perfoliatus, 5 = Potamogeton pectinatus. All spectra were measured by
Pinnel (2005). The sand spectrum is from a coastal shallow area in South Australia
(Bolivar), the other spectra were measured at German lakes (Lake Constance and
Starnberger See).
When the upwelling radiation is measured by a radiance sensor, the corresponding
remote sensing reflectance can be expressed as follows:
90
Gege and Albert
first term on the right-hand side of each equation describes reflection in the water, the
second at the surface. Frequently, the first term alone is called remote sensing
reflectance (e.g. Mobley, 1994). In WASI, the reflection at the surface is also included
in the R rs definition. It is calculated using eq. (13a) or (13b) and can easily be excluded
by setting the reflection factor ı L equal to zero.
R rs
− (Ȝ) is calculated using eq. (17) or (19), R(Ȝ) using eq. (14) or (16). The factors
ı, ı L
− , and ı
− are the reflection factors for E d , L u
− , and E u
− , respectively. ı depends on
the radiance distribution and on surface waves. Typical values are 0.02 to 0.03 for clear
sky conditions and solar zenith angles below 45°, and 0.05 to 0.07 for overcast skies
(Jerlov, 1976; Preisendorfer and Mobley, 1985, 1986). The default value is ı = 0.03.
ı L
− can either be calculated as a function of ș v using eq. (12), or a constant value can be
inserted. ı
− is in the range of 0.50 to 0.57 with a value of 0.54 considered typical
(Jerome et al., 1990; Mobley, 1999). The defaults of the other constants are set to Q = 5
sr and n W = 1.33.
Selection of the equation to use depends on the application:
• Eq. (20a) links remote sensing reflectance in water to that in air. Since the
same spectrum type is used above and below the water surface, it is the most
convenient parameterisation. This equation is used by default.
• Eq. (20b) is useful when R rs (Ȝ) is linked to R(Ȝ), for example if in situ
measurements of R(Ȝ) were performed as “ground truth” for a remote sensing
instrument.
• Eq. (20c) avoids the use of the factor Q, which is difficult to assess. The
equation is useful, for example, for optical closure experiments which
investigate the consistency of measurements above and below the water
surface by measuring simultaneously the spectra R rs (Ȝ), R(Ȝ), and R rs
− (Ȝ).
2.7 BOTTOM REFLECTANCE
The irradiance reflectance of a surface is called albedo. When N different surfaces
of albedo a n (Ȝ) are viewed simultaneously, the measured albedo is the following sum:
¦
−
=
λ
⋅
=
λ
1
N
0
n
n
n
b
,
)
(
a
f
)
(
R
(21)
where f n is the areal fraction of surface number n within the sensor’s field of view; it is
Ȉ f n = 1. This equation is implemented in WASI for N = 6 bottom types. Three of the
spectra a n (Ȝ) provided with WASI represent bare bottom, the other green makrophytes:
0 = a constant reflectance of 10%, 1 = sand, 2 = silt, 3 = Chara aspera, 4 =
Potamogeton perfoliatus, 5 = Potamogeton pectinatus. All spectra were measured by
Pinnel (2005). The sand spectrum is from a coastal shallow area in South Australia
(Bolivar), the other spectra were measured at German lakes (Lake Constance and
Starnberger See).
When the upwelling radiation is measured by a radiance sensor, the corresponding
remote sensing reflectance can be expressed as follows:
90
Gege and Albert