Geography Reference
In-Depth Information
hemisphere ahead of the incident flux (the right side of
Figure 3.5 inset), we can determine a forward scatter-
ing probability for the photons in the incident beam.
Similarly, integrating over the hemisphere trailing the
incident beam (left side of Figure 3.5 inset) yields a back-
scattering probability. Scattering in natural waters tends
to be strongly biased toward the forward direction - that
is, most scattered photons continue to propagate in a sim-
ilar direction - but back-scattering is largely responsible
for imparting a volume reflectance, defined by Equation
(3.6), to the water column.
Attenuation
of the downwelling light stream thus
results from both absorption and scattering. The total
attenuation coefficient
c
(
the radiance distribution comprising
E
(
,
z
) and is thus
commonly referred to as the diffuse attenuation coeffi-
cient. We must also consider the angle at which light
is propagating through the aquatic medium in order to
calculate irradiance at a particular depth using a depth-
averaged attenuation coefficient
K
(
λ
λ
). In this case, Beer's
law can be re-written as
E
(
λ
,
z
)
=
E
(
λ
,0
−
)
e
−
K
(
λ
)
z
(3.13)
,0
−
) is the spectral irradiance just beneath the
water surface.
In shallow stream channels, however, the attenuation
of light is more complex than Equation (3.13) would
appear to imply. Strictly speaking, Beer's law is not
appropriate for these environments because the atten-
uation of the downwelling flux differs from that of the
upwelling flux due to the presence of a reflective bottom
that acts as a source of radiant energy (Dierssen et al.,
2003). Because irradiance can be readily partitioned into
downwelling and upwelling components, separate atten-
uation coefficients for the downwelling and upwelling
irradiance -
K
d
(
where
E
(
λ
) is defined as the fraction of
radiant energy removed from an incident beam per unit
distance due to the combined effects of both types of
processes:
λ
c
(
λ
)
=
a
(
λ
)
+
b
(
λ
)
(3.11)
These three coefficients are
inherent optical properties
of
the water column in the sense that they do not depend on
how they are measured nor how the water is illuminated.
Because inherent optical properties are independent of
the spatial distribution and intensity of the incident radi-
ation, these quantities could be directly related to certain
attributes of interest, such as the suspended sediment con-
centration. Remote sensing, however, involves observing
rivers under a range of illumination conditions, so we
must instead rely on
apparent optical properties
,which
are used to normalise for the effects of external envi-
ronmental conditions, such as changes in the incident
illumination. Inferring water column characteristics from
remotely sensed data thus relies heavily upon relation-
ships between apparent and inherent optical properties
(Bukata et al., 1995).
Attenuation, in many ways, forms the basis for inter-
preting measurements of radiance that has interacted
with the water in a stream channel. As a result, one
apparent optical property of particular interest is the
irra-
diance attenuation coefficient
. This quantity, denoted by
K
(
,
z
), respectively - can be
defined by analogy with Equation (3.12). By making such
a distinction, Legleiter et al. (2004) demonstrated that
whereas
K
d
(
λ
,
z
)and
K
u
(
λ
λ
,
z
) was essentially independent of depth,
K
u
(
,
z
) varied considerably within the water column.
Although the downwelling flux decreases monotonically
with depth, the upwelling flux can actually increase
with depth as more and more photons reflected from
the bottom contribute to
E
u
(
λ
,
z
). These effects clearly
complicate the definition of a single, representative atten-
uation coefficient. In practice, however, the two-way trip
photons must make to return from the streambed to the
water surface is accounted for by using 2
K
d
(
λ
)asan
effective attenuation coefficient
. This approach has been
shown to provide reasonable results for a range of plausi-
ble stream conditions (Legleiter et al., 2009; Legleiter and
Roberts, 2009).
One of the most critical links along the image chain
is the manner in which those photons that do propa-
gate through the full depth of the water column interact
with the streambed. Some proportion of the downwelling
irradiance impinging upon the bed, at depth
z
b
,willbe
reflected upward and begin a second traverse of the water
column. Both of the reflectance concepts described in
Section 3.2.1 apply here, and we can define an irradiance
reflectance for the bottom,
R
b
(
λ
λ
,
z
), can be derived from Beer's law and is defined as
the change in the spectral irradiance, denoted by
Δ
E
(
λ
,
z
)
corresponding to a unit change in depth
z
at an initial
depth
z
:
Δ
1
E
(
λ
,
z
)
K
(
λ
,
z
)
=−
(3.12)
), using Equation (3.6). A
more complete description of the reflectance properties
of the streambed would account for directional struc-
ture by considering the BRDF of the substrate. To fully
λ
E
(
λ
,
z
)
Δ
z
This is an apparent optical property because, unlike
c
(
λ
λ
),
K
(
,
z
) depends on the directional structure of
Search WWH ::
Custom Search