Geography Reference
In-Depth Information
overhead (Figure 3.1a), whereas larger values of
like that
shown in Figure 3.1b imply more glancing, oblique angles
of approach. The irradiance incident upon a surface is
given by
Lambert's cosine law
:
θ
θ
= I/r
E
θ
=
E
0
cos(
θ
)
(3.4)
I
where
E
0
is the irradiance onto a surface oriented per-
pendicular to the direction in which the light beam
is propagating. For larger incidence angles, the energy
encompassed by the beam is projected, or spread, over a
lager surface area, resulting in smaller amount of energy
per unit area and thus a smaller irradiance (Figure 3.1b).
The important implication of Equation (3.4) is that the
amount of radiant energy incident upon a river, some frac-
tion of which will be reflected and ultimately recorded by
a remote sensor, depends on the incidence angle, which
in turn depends on the time of day, time of year, geo-
graphic location, degree of cloud cover, and surface slope
and aspect. For example, images acquired late in the day,
when
= A/r
2
Ω
L
A
r
θ
θ
r
r
AIR
WATER
Figure 3.2
Geometric principles involved in the concept of
radiance. On the left is an angle
is large, will have a weaker energy signal than
data acquired when the sun is higher in the sky. Similarly,
local variations in streambed orientation relative to the
sun dictate that some areas of the bottom will receive less
energy than others. If the instrumental noise associated
with an imaging system is assumed constant, those times
and places for which irradiance is reduced will have lower
signal-to-noise ratios, and information derived therefrom
will be less reliable.
Irradiance provides spatial information about the flux
of radiant energy but not angular or directional informa-
tion. This radiometric quantity is an integrated measure
in that the irradiance includes energy incident upon a
surface from all directions encompassed by a hemisphere
above that surface: incidence angles from 0 to 90
◦
and
all
azimuths
(i.e., compass directions from 0 to 360
◦
).
For remote sensing, however, we are interested in the
amount of energy arriving at a detector from a specific
direction. To obtain this type of directional information,
the irradiance is normalised to a unit
solid angle
,which
is a three-dimensional angular measure. To understand
the concept of a solid angle, first consider an angle
θ
on
a two-dimensional plane like that shown on the left side
of Figure 3.2. We can define such an angle as the ratio
of the length
l
of an arc bracketed by two lines origi-
nating from the centre of a circle to the radius
r
of the
circle:
θ
on a two-dimensional plane,
defined as the ratio of the length
l
of an arc bracketed by two
radii of a circle of radius
r
:
θ
θ =
/
r
. By analogy with this
two-dimensional case, the three-dimensional solid angle shown
in the center of the figure is defined as the ratio of the surface
area
A
encompassed by a range of incidence angles and
azimuths to the square of the radius
r
of the sphere:
l
r
2
.
On the right side of the figure is an illustration of radiance,
L
,
defined as the amount of radiant energy per unit time per unit
area perpendicular to the direction of photon travel per unit
solid angle. Figure adapted from Schott, J.R.
Remote Sensing:
The Image Chain Approach
. New York, Oxford University Press,
1997.
Ω =
A
/
entire circle has an arc length equivalent to the circum-
ference of the circle,
l
r
; dividing this arc length by
the circle's radius yields the angle in radians - there are
2
=
2
π
radians in a circle. For the three-dimensional
case depicted in the centre of Figure 3.2, we can define a
solid angle analogously by dividing the area of a spher-
ical surface encompassed by a specific set of incidence
angles and azimuths by the square of the radius of
the sphere
π
r
/
r
=
2
π
A
r
2
Ω =
(3.5)
where
A
is the area of the surface and
denotes the solid
angle. The units of a solid angle are called steradians (sr),
and there are 4
Ω
θ =
l
/
r
. For example, an angle encompassing the
r
2
r
2
π
/
=
4
π
steradians in a sphere.
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