320
G. Zibordi and K.J. Voss
0 (z,λ,t 0 ) =
(z,λ,t)
E d (0 + ,λ,t)
E d (0
+ ,λ,t 0 )
(18.14)
where 0 (z,λ,t 0 ) indicates radiometric quantities as if they were taken at each depth
z at the same time t 0 , and E d (0 + ,λ,t 0 ) specifies the above-water downward irradiance
at time t 0 (with t 0 generally chosen to coincide with the beginning of the acquisition
sequence).
Omitting the variable t, the sub-surface quantities 0 (0 − ,λ) (i.e., L u (0 − ,λ),
E u (0 − ,λ) and E d (0 − ,λ) are then determined as the exponentials of the intercepts
resulting from the least-squares linear regressions of ln 0 (z,λ) versus z within the
extrapolation interval identified by z 0 < z < z 1 and chosen to satisfy the requirement of linear decay of ln 0 (z,λ) with depth. The negative values of the slopes of
the regression fits are the so-called diffuse attenuation coefficients K (λ) (i.e. K l (λ),
K u (λ) and K d (λ)) for the selected extrapolation interval.
Derived radiometric data products are then the dimensionless irradiance
reflectance at depth 0 − , R(0 − ,λ), defined as the ratio of E u (0 − ,λ) to E d (0 − ,λ), and
the Q-factor at nadir, Q n (0 − ,λ) in units of sr, defined as the ratio of E u (0 − ,λ) to
L u (0 − ,λ).
Additional data products are the remote sensing reflectance, R rs (λ), in units of
sr −1
R rs (λ) =
L w (λ)
E d (0 + ,λ)
(18.15)
and the normalized water-leaving radiance, L wn (λ), in units of W/m 2 /nm/sr
L wn (λ) = R rs (λ)E 0 (λ)
(18.16)
where E 0 (λ) is the average extra-atmospheric solar irradiance (Thuillier et al., 2003)
and L w (λ) the so called water-leaving radiance, i.e., the radiance leaving the sea and
quantified just above the surface as
L w (λ) = 0.543L u (0
− ,λ).
(18.17)
The factor 0.543 has been computed assuming n w is independent of wavelength (Austin, 1974), and accounts for the reduction in radiance from below to
above the water surface due to the change in the refractive index at the air-water
interface.
Both R rs (λ) and L wn (λ) are then quantities corrected for the illumination
effects dependent on the sun zenith angle, Sun-Earth distance and the atmospheric
transmittance (Mueller and Austin, 1995).
G. Zibordi and K.J. Voss
0 (z,λ,t 0 ) =
(z,λ,t)
E d (0 + ,λ,t)
E d (0
+ ,λ,t 0 )
(18.14)
where 0 (z,λ,t 0 ) indicates radiometric quantities as if they were taken at each depth
z at the same time t 0 , and E d (0 + ,λ,t 0 ) specifies the above-water downward irradiance
at time t 0 (with t 0 generally chosen to coincide with the beginning of the acquisition
sequence).
Omitting the variable t, the sub-surface quantities 0 (0 − ,λ) (i.e., L u (0 − ,λ),
E u (0 − ,λ) and E d (0 − ,λ) are then determined as the exponentials of the intercepts
resulting from the least-squares linear regressions of ln 0 (z,λ) versus z within the
extrapolation interval identified by z 0 < z < z 1 and chosen to satisfy the requirement of linear decay of ln 0 (z,λ) with depth. The negative values of the slopes of
the regression fits are the so-called diffuse attenuation coefficients K (λ) (i.e. K l (λ),
K u (λ) and K d (λ)) for the selected extrapolation interval.
Derived radiometric data products are then the dimensionless irradiance
reflectance at depth 0 − , R(0 − ,λ), defined as the ratio of E u (0 − ,λ) to E d (0 − ,λ), and
the Q-factor at nadir, Q n (0 − ,λ) in units of sr, defined as the ratio of E u (0 − ,λ) to
L u (0 − ,λ).
Additional data products are the remote sensing reflectance, R rs (λ), in units of
sr −1
R rs (λ) =
L w (λ)
E d (0 + ,λ)
(18.15)
and the normalized water-leaving radiance, L wn (λ), in units of W/m 2 /nm/sr
L wn (λ) = R rs (λ)E 0 (λ)
(18.16)
where E 0 (λ) is the average extra-atmospheric solar irradiance (Thuillier et al., 2003)
and L w (λ) the so called water-leaving radiance, i.e., the radiance leaving the sea and
quantified just above the surface as
L w (λ) = 0.543L u (0
− ,λ).
(18.17)
The factor 0.543 has been computed assuming n w is independent of wavelength (Austin, 1974), and accounts for the reduction in radiance from below to
above the water surface due to the change in the refractive index at the air-water
interface.
Both R rs (λ) and L wn (λ) are then quantities corrected for the illumination
effects dependent on the sun zenith angle, Sun-Earth distance and the atmospheric
transmittance (Mueller and Austin, 1995).