Geography Reference
In-Depth Information
from a level water surface is essentially independent of
wavelength and is given by
Fresnel's formula
:
This concept is expressed via
Beer's law
, which relates the
radiant flux
) entering an absorbing medium to
the radiant flux
Φ
(
r
,
λ
) at a distance
r
into the medium:
Φ
(0,
λ
5
sin(
2
θ
i
− θ
r
)
ρ
(
θ
i
,
θ
r
)
=
0
.
)
e
−
a
(
λ
)
r
Φ
(
r
,
λ
)
= Φ
(0,
λ
(3.10)
sin(
θ
i
+ θ
r
)
5
tan(
2
θ
i
− θ
r
)
)isan
absorption coefficient
with units of m
−
1
.
The radiant flux thus decreases exponentially with dis-
tance traveled through the water column as photons are
absorbed.
The radiant flux is also reduced due to scattering, at
a rate quantified by the
scattering coefficient
b
(
where
a
(
λ
+
0
.
(3.9)
tan(
θ
i
+ θ
r
)
where
θ
r
are the angles of incidence and refrac-
tion, respectively. The magnitude of the surface-reflected
radiance is thus
L
ra
(
θ
i
and
). For incidence angles
up to 30
◦
, the Fresnel reflectance from a level water sur-
face is essentially a constant value of
λ
)
= ρ
L
da
(
λ
). A
closely related quantity, the
scattering phase function
,is
illustrated schematically in Figure 3.5 and describes how
scattering interactions redirect photons onto new paths
and thus provides directional information. For example,
by integrating the scattering phase function over the
λ
02-0.03. For
larger incidence angles (i.e., as the sun sinks in the sky),
ρ
ρ =
0
.
65
◦
,
and then rises rapidly to 1 at a grazing angle of incidence.
Although a level water surface provides a convenient
starting point and a useful first approximation for the
surface-reflected radiance, this simple geometry is clearly
not realistic.Water surfaces in rivers are oftenhighly irreg-
ular, with a temporally and spatially variable topography
dictated by flow hydraulics and surface turbulence. For
such non-uniform water surfaces, alternative approaches
must be used to characterise radiative transfer processes
at the air-water interface. This is not to say that the
physics pertaining to a level surface are no longer perti-
nent; instead, these principles are applied to a probability
distribution of water surface orientations. A stochastic
approach of this kind has been developed in marine set-
tings (Mobley, 1994), where probability distributions of
wave slopes can be related to wind speed (Cox andMunk,
1954). These models have also been applied to rivers
as a surrogate for flow-related surface turbulence. The
primary effect of water surface irregularity is to increase
surface-reflected radiance (Legleiter et al., 2004, 2009). In
essence, rougher water surfaces appear brighter because
more of the incident radiation is reflected, which reduces
the amount of energy transmitted across the surface and
into the water.
Continuing with our image chain analogy, the next
link is the water column itself. Here, the downwelling
light stream is attenuated as photons are absorbed and
scattered by pure water and other optically significant
materials. Both absorption and scattering interactions
act to reduce the intensity of the transmitted radiance,
denoted by
L
tw
(
increases gradually, does not exceed 0.1 until
θ
i
>
Backward Sca
t
ter
Fo
ward Scatte
r
90
°
45
°
135
°
180
0
°
°
225
°
315
°
270
°
AIR
WATER
Figure 3.5
Schematic, two-dimensional illustration of the
scattering phase function, which describes how scattering
interactions redirect photons onto new paths. For the scattering
center shown in the inset, the phase function is the probability
that a scattering event will cause the scattered photon to travel
in a specific direction, with an angle of 0
◦
indicating no change
in direction and an angle of 180
◦
implying that the photon was
scattered back into the direction from which it came. A forward
scattering probability can be defined by integrating the phase
function over angles from 0
◦
to 90
◦
and 270
◦
to 0
◦
(the right
half of the unit circle). The back-scattering probability is the
integral of the phase function over angles from 90
◦
to 270
◦
(the
left half of the unit circle). The scattering phase function
illustrated here is strongly biased in the forward direction.
Figure adapted from Schott, J.R.
Remote Sensing: The Image
Chain Approach
. New York, Oxford University Press, 1997.
λ
) (Figure 3.4a); scattering processes also
modify the directional structure of
L
tw
(
λ
). The farther
photons propagate through the water column, the more
likely they are to be absorbed or scattered, implying that
radiance varies as a function of path length (i.e., depth).
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