An Introduction to the Physical Basis for Deriving River Information by Optical Remote Sensing - Fluvial Remote Sensing for Science and Management - page 48

Specular
Reflector
( θ i = θ r )

Z

Ε (θ i , φ i ) L (θ v , φ v )

θ i

Diffuse
Reflector
(Lambertian)

θ V

θ i θ r

φ i

φ V

x

SAND

WATER

y

(a) (b)

Figure 3.3 Bidirectional reflectance concepts. (a) A level water surface can act as a specular reflector, with all of the incident radiation
reflected in the exact opposite direction. (b) A sandbar acts as a diffuse or Lambertian reflector, with a similar amount of radiance in
all directions. (c) The bidirectional reflectance distribution function (BRDF) describes reflectance values for all combinations of
illumination and viewing geometry. These geometries are specified by an incidence angle

θ i and azimuth φ i for the downwelling

φ v for the direction in which the upwelling radiance is measured. The BRDF is defined
by Equation (3.7) in the text. Figure adapted from Schott, J.R. Remote Sensing: The Image Chain Approach . New York, Oxford
University Press, 1997.

θ v and azimuth

irradiance and the view angle

in each direction as a probability distribution - what is
the likelihood that a photon incident upon the surface
will be scattered in a particular direction? Guided by this
concept, we can define a bidirectional reflectance distri-
bution function (BRDF) , illustrated in Figure 3.3b. This
distribution describes reflectance values for all combina-
tions of illumination and viewing geometry. The BRDF
is defined as the ratio of the radiance scattered into
the direction described by the view angle

Lambertian, 100% reflectant surface, L ( λ ) = E d ( λ ) / π
and the BRDF for such a surface thus reduces to 1/ .

A bidirectional reflectance can thus be converted to the
more common and intuitive irradiance reflectance by
assuming Lambertian behaviour and multiplying by

π

π .

3.2.2 Radiative transferprocessesalong
the imagechain

θ v and view
azimuth φ v to the downwelling irradiance arriving from
an incidence angle of

For passive optical remote sensing, the first link in the
image chain is the sun, the source of radiant energy.
The top of Earth's atmosphere receives an irradiance
of 1367Wm − 2 , an average, spectrally-integrated value
known as the solar constant. Upon entering the atmo-
sphere, this radiation is modified as photons interact with
air molecules and aerosols through various absorption
and scattering mechanisms. Importantly, these mecha-
nisms depend on wavelength, the type and concentration
of certain atmospheric constituents, and the path length
radiation must traverse to reach the surface, which is
primarily a function of the solar incidence angle. A full
discussion of atmospheric effects is beyond the scope of
this chapter, and more complete treatments are avail-
able elsewhere (Bukata et al., 1995; Schott, 1997). For

θ i and an azimuth of φ i :

r BRDF ( θ i , φ i , θ v , φ v , λ )
=

L ( θ v , φ v , λ )
E d ( θ i , φ i , λ ) =

dL ( θ v , φ v , λ )
L ( θ v , φ v , λ )cos( θ i ) d Ω ( θ i , φ i )
(3.7)

The last expression on the right explicitly indicates that
the BRDF uses the projected horizontal area of the
incident radiance, as seen in the cos(

θ i ) factor in the
denominator. The BRDF thus has units of sr − 1 . Although
these directional effects can be important (Mobley
et al., 2003), in practice, surfaces are typically assumed
to behave as Lambertian reflectors. For a perfectly