Environmental Engineering Reference
In-Depth Information
characterizes the angular distribution of single-event scattering). Sometimes the
volume scattering function is referred to as the scattering phase function of the
aquatic medium (Preisendorfer
1961
; Kirk
1991
). Optical properties of water at
any given point in the medium are dependent on the irradiance distribution at
that point (Preisendorfer
1961
). The apparent optical properties of water bodies
(e.g. vertical attenuation coefficient, irradiance and reflectance) are significantly
dependent on the inherent optical properties (e.g. absorption, scattering and back-
scattering coefficients), which depend on the shape of the volume scattering func-
tion (Kirk
1984
,
1991
,
1994
; Belzile et al.
2002
). Kirk (
1984
,
1991
,
1994
) derived
an empirical relationship between the apparent and inherent optical properties of
natural waters, depending on the angle of incidence of the photons on the surface,
based on field observation data followed by Monte Carlo modeling of the under-
water light field. He stated that the vertical attenuation coefficient for downward
irradiance at the midpoint of the euphotic zone,
K
d
, can be expressed as a func-
tion of the absorption coefficient, the scattering coefficient, and the solar altitude
in accordance with (Eq.
2.3
):
1
/
2
1
µ
0
A
2
+
G
(µ
0
)
AB
K
D
=
(2.3)
where
a
is the absorption coefficient,
b
is the scattering coefficient,
μ
0
is the
cosine of the zenith angle of refracted solar photons (direct beam) just beneath the
surface (calculated from the incident zenith angle using Snell's Law).
G
(
μ
0
) is a
coefficient function that specifies the relative contribution of scattering to the ver-
tical attenuation of irradiance. Its value is determined by the shape of the volume
scattering function
β
(
θ
) and by
μ
0
.
G
(
μ
0
) is a linear function of
μ
0
that can be
expressed as (Eq.
2.4
):
G
(µ
0
) =
G
1
µ
0
−
G
2
(2.4)
where g
1
and g
2
are numerical constants that depend on the volume scattering
function used in the calculations.
In case of irradiance reflectance, Gordon et al. (
1975
) fit their data to another
power series that has been simplified by Jerlov (
1976
) to (Eq.
2.5
):
b
b
a
+
b
b
R
(
0
)
=
Constant
(2.5)
where
R(0)
is the irradiance reflectance just below the surface,
b
b
is the backscat-
tering coefficient, and the constant has the value 0.32 for zenith sun and 0.37 for
an overcast sky. The irradiance reflectance can be expressed as (Eq.
2.6
) (Kirk
1981
)
R
(
0
)
=
0. 328
b
b
/
a
(2.6)
This equation is in good agreement with the result obtained by Prieur (
1976
)
using a different calculation procedure (Eq.
2.7
):
(2.7)
R
(
0
) =
0. 33
b
b
/
a
