300
Richard C. Zimmerman and Arnold G. Dekker
D. Apparent Optical Properties
Apparent optical properties (AOPs) are derived from
the IOPs defining the medium, combined with the
angular distribution of the ambient light field. As
such, AOPs will be affected by the time of day, degree
of cloud cover, sea surface state and depth within the
water column. The most commonly used AOPs are
the diffuse attenuation coefficients or “K ” functions.
K functions, also referred to as quasi-inherent optical properties, because the effects of changes in the
incident radiation field are generally small, although
that is not always the case (Baker and Smith, 1979,
but see Miller and McPherson, 1995). Because the
irradiance within a sunlit + skylit water body decreases approximately exponentially with depth, it
is conventional to describe that relationship using
the Lambert-Beer equation. Thus, the downwelling
plane irradiance anywhere within the water column
[i.e. E d (z)] is a function of the in-water irradiance
at the surface [E d (0)], the diffuse attenuation coefficient for downwelling irradiance (K d ), and the depth
(z) of the overlying water column:
E d (z) = E d (0) exp(−K d z)
(12)
Solving for K d yields:
K d =
− ln
E d (z)
E d (0)
z
(13)
Thus, the diffuse attenuation coefficient is easily approximated from vertical profiles of spectral irradiance. The diffuse attenuation coefficients for upwelling plane irradiance and scalar irradiance are
defined by similar equations. K functions of broadband irradiance (e.g. PAR) provide convenient, albeit much less perfect, descriptors of light attenuation because they do not properly account for the
strong wavelength-dependency of absorption. This
can be an especially significant problem for estimating light availability for seagrass photosynthesis
in green coastal waters (see Chapter 13). The K -
functions are dimensionalized with the same units
(inverse length) as the IOPs, but it is important to
remember that the IOPs refer to the loss of radiant
power from a collimated beam. K functions represent the attenuation of naturally diffuse light comprised of photons traveling in all directions.
The average cosine ¯
µ, another AOP, provides a
useful way to approximate the angular structure of
the submarine light field in terms of a single parameter. For a collimated beam oriented at angle (θ)
relative to the zenith, ¯
µ d = cos θ. Thus, the average
pathlength of photons traveling downward through
the water, and therefore the magnitude of the diffuse
attenuation coefficient, is proportional to
1
¯
µ d
. The
downwelling average cosine can also be calculated
from the ratio of the downwelling plane irradiance
normalized to the downwelling scalar irradiance:
¯
µ d (z) ≡
E d (z)
E od (z)
(14)
The upwelling average cosine is defined similarly as:
¯
µ u (z) ≡
E u (z)
E ou (z)
(15)
Values of ¯
µ d typically range from 0.9 to 0.75 in natural waters. A typical value for ¯
µ u is about 0.39. If
the light field is isotropic (i.e. equal intensity in all
directions), ¯
µ d = ¯
µ u = 0.5.
III. Radiative Transfer in Natural Waters
Radiative transfer theory provides a macroscopic,
linear approximation for calculating the loss of radiant energy of a beam due to absorption and scattering along a defined path, and the gain of energy
by scattering into the path. As such, it creates a robust framework to develop mechanistic models of
the submarine light environment, system-level productivity and remotely sensed reflectance of seagrass
meadows in optically shallow waters. Exact solutions to the radiance transfer equations have been
developed for natural waters in which the optical
medium is a continuous material composed of randomly arranged scattering elements separated by
large distances relative to the wavelength of light
(Mobley, 1994). Plant leaves, however, represent a
dense packaging of optically active material, which
violates the single-scattering assumptions of these
exact solutions. Consequently, models of irradiance
distribution in submerged plant canopies must rely
on more empirical relationships between leaf optical
properties and light attenuation by the bulk canopy
as well as the water column (Goudriaan, 1988;
Shultis and Myneni, 1988; Ganapol and Myneni,
1992; Zimmerman, 2003).
Richard C. Zimmerman and Arnold G. Dekker
D. Apparent Optical Properties
Apparent optical properties (AOPs) are derived from
the IOPs defining the medium, combined with the
angular distribution of the ambient light field. As
such, AOPs will be affected by the time of day, degree
of cloud cover, sea surface state and depth within the
water column. The most commonly used AOPs are
the diffuse attenuation coefficients or “K ” functions.
K functions, also referred to as quasi-inherent optical properties, because the effects of changes in the
incident radiation field are generally small, although
that is not always the case (Baker and Smith, 1979,
but see Miller and McPherson, 1995). Because the
irradiance within a sunlit + skylit water body decreases approximately exponentially with depth, it
is conventional to describe that relationship using
the Lambert-Beer equation. Thus, the downwelling
plane irradiance anywhere within the water column
[i.e. E d (z)] is a function of the in-water irradiance
at the surface [E d (0)], the diffuse attenuation coefficient for downwelling irradiance (K d ), and the depth
(z) of the overlying water column:
E d (z) = E d (0) exp(−K d z)
(12)
Solving for K d yields:
K d =
− ln
E d (z)
E d (0)
z
(13)
Thus, the diffuse attenuation coefficient is easily approximated from vertical profiles of spectral irradiance. The diffuse attenuation coefficients for upwelling plane irradiance and scalar irradiance are
defined by similar equations. K functions of broadband irradiance (e.g. PAR) provide convenient, albeit much less perfect, descriptors of light attenuation because they do not properly account for the
strong wavelength-dependency of absorption. This
can be an especially significant problem for estimating light availability for seagrass photosynthesis
in green coastal waters (see Chapter 13). The K -
functions are dimensionalized with the same units
(inverse length) as the IOPs, but it is important to
remember that the IOPs refer to the loss of radiant
power from a collimated beam. K functions represent the attenuation of naturally diffuse light comprised of photons traveling in all directions.
The average cosine ¯
µ, another AOP, provides a
useful way to approximate the angular structure of
the submarine light field in terms of a single parameter. For a collimated beam oriented at angle (θ)
relative to the zenith, ¯
µ d = cos θ. Thus, the average
pathlength of photons traveling downward through
the water, and therefore the magnitude of the diffuse
attenuation coefficient, is proportional to
1
¯
µ d
. The
downwelling average cosine can also be calculated
from the ratio of the downwelling plane irradiance
normalized to the downwelling scalar irradiance:
¯
µ d (z) ≡
E d (z)
E od (z)
(14)
The upwelling average cosine is defined similarly as:
¯
µ u (z) ≡
E u (z)
E ou (z)
(15)
Values of ¯
µ d typically range from 0.9 to 0.75 in natural waters. A typical value for ¯
µ u is about 0.39. If
the light field is isotropic (i.e. equal intensity in all
directions), ¯
µ d = ¯
µ u = 0.5.
III. Radiative Transfer in Natural Waters
Radiative transfer theory provides a macroscopic,
linear approximation for calculating the loss of radiant energy of a beam due to absorption and scattering along a defined path, and the gain of energy
by scattering into the path. As such, it creates a robust framework to develop mechanistic models of
the submarine light environment, system-level productivity and remotely sensed reflectance of seagrass
meadows in optically shallow waters. Exact solutions to the radiance transfer equations have been
developed for natural waters in which the optical
medium is a continuous material composed of randomly arranged scattering elements separated by
large distances relative to the wavelength of light
(Mobley, 1994). Plant leaves, however, represent a
dense packaging of optically active material, which
violates the single-scattering assumptions of these
exact solutions. Consequently, models of irradiance
distribution in submerged plant canopies must rely
on more empirical relationships between leaf optical
properties and light attenuation by the bulk canopy
as well as the water column (Goudriaan, 1988;
Shultis and Myneni, 1988; Ganapol and Myneni,
1992; Zimmerman, 2003).