An Introduction to the Physical Basis for Deriving River Information by Optical Remote Sensing - Fluvial Remote Sensing for Science and Management - page 47

Now, using the concept of a solid angle to account
for the directional structure of a three-dimensional light
field, we can define another radiometric quantity central
to remote sensing of rivers: radiance is the amount of
radiant energy per unit time per unit area perpendicular
to the direction of photon travel per unit solid angle
(Figure 3.2). Radiance thus has units of Wm − 2 sr − 1 and
is given the symbol L . Unlike irradiance, which varies
as the inverse square of the distance between the source
of radiant energy and the surface receiving the energy,
it can be shown that the radiance along a beam of light
is constant over distance (Schott, 1997). Radiance thus
has the important advantage of being independent of
both distance and geometric effects; illumination and
viewing angles are accounted for by normalising the
radiant energy to a unit solid angle. Because of these
unique properties, radiance measurements made under a
range of illumination conditions, at various heights above
the Earth's surface, and from different view angles can all
be directly compared to one another. For these reasons,
radiance is the quantity actually measured by remote
sensors. Also, note that each of the radiometric terms
defined above can be normalised to a unit wavelength
to provide additional information on the distribution
of radiant energy across the spectrum. Passive optical
remote sensing is thus based upon measurements of
a fundamental physical quantity: the spectral radiance
L (

First, the irradiance reflectance is defined as

E u ( λ )

R ( λ ) = (3.6)

E d ( λ )

Here, E u ( ) refers to the upwelling spectral irradiance,

which represents the radiant flux density integrated over
the entire upper hemisphere, with light arriving from
various directions weighted in proportion to the cosine
of the incidence angle (Equation 3.4) to account for
the projected area effect illustrated in Figure 3.1. The
downwelling spectral irradiance, denoted by E d (

λ

), is
defined similarly for the lower hemisphere. Simply put,
R (

λ

) is the ratio of the amount of energy traveling upward
away from a surface to the amount of energy impinging
upon the surface, irrespective of the directional structure
of the light field. In other words, R (

λ

) is the proportion of
the total energy incident upon a surface that is reflected
back away from that surface. A perfectly absorbing black
object would have R (

λ

λ ) = 0, and brighter objects would

have R ( ) values approaching 1. In an aquatic context,

we can define an irradiance reflectance for any location
above or within the water column. Within the water,
R ( λ ) is referred to as the volume reflectance and will
depend on the optical properties of pure water and other
suspended or dissolved materials. The volume reflectance
can also vary with depth, a dependence made explicit by
the notation R ( z ,

λ

λ ) = E u ( z , λ ) / E d ( z , λ ), where z denotes

), reported in units of Wm − 2 sr − 1 nm − 1 .
The spectral radiance measured by a remote detector
depends on the amount of radiant energy incident upon a
surface and the proportion of that energy that is reflected
into the sensor's field of view. Where and when the
surface of interest, a river channel in our case, receives
a smaller amount of irradiance, the amount of reflected
radiance will necessarily be smaller as well, even if the
properties of the surface remain the same. Amore general
description of the innate characteristics of a surface or
water body can be obtained by normalising the measured,
upwelling spectral radiance by the incident, downwelling
radiation. To provide such information, we can relate
the amount of energy reflected away from a surface (in
various directions) to the amount of energy received by
the surface (from various directions) using quantities
analogous to the radiometric terms in Table 3.1. These
quantities are generally referred to as reflectance, and two
particular forms of reflectance are especially relevant to
remote sensing of rivers.

λ depth below the water surface.

A second, somewhat more complex reflectance quan-
tity is the bidirectional reflectance , which, as the name
implies, does account for the directional structure of the
light field. Intuition tells us that such directional effects are
important, particularly in rivers, where the water surface
can act as a mirror-like reflector for certain illumination
and viewing angles, producing strong sun glint. Nearby
sandbars, in contrast, appear to have a similar brightness
regardless of sun angle and viewing direction. The smooth,
glassy water surface shown in Figure 3.3a is an example of
a specular reflector - all of the radiation reflected fromthe
surface is directed in the exact opposite direction from the
incident light ray. Sandbars behave more like Lambertian
or diffuse reflectors, with a similar amount of reflectance
in all directions (Figure 3.3a). Other objects might appear
brighter when viewed from the specular or backscatter
directions and darker from grazing angles. These varia-
tions in visual brightness can be quantified by considering
the magnitude of the radiance emanating from a surface