IEEE Geoscience and Remote Sensing Magazine - June 2019 - 12

to the target; S (m) is the atmospheric spherical scattering
albedo (i.e., the effective diffuse reflectivity of the sky to
upwelling radiation); and T u (m) is the upward atmospheric
transmittance of the target-to-sensor path. The denominator in the first term of (8) is due to the multiple reflections
between the adjacent material surface and the atmosphere
due to multiple scattering. This phenomenon, also known
as the trapping effect, has been modeled for a flat and homogeneous surface as the summation of infinite terms of
reflected contributions, leading to a geometric series that
converges to 61 - ra (m) S (m)@-1. This effect has been found
negligible for clear-sky conditions and low reflectivity of
adjacent materials, i.e., ra (m) S (m) 1 0.02 [16]. The second
term in (8) accounts for potential shadowing of the direct
solar unscattered term, with a sh = 1 for full shadowing and
a sh = 0 for full illumination [33].
For spectral remote sensing, the factor we care most
about in these equations is r (m), the reflectance. This is
what corresponds to the intrinsic properties of the material

TABLE 1. SUMMARY OF THE RADIOMETRIC TERMS INVOLVED
IN RADIATIVE TRANSFER MODELING.
TERM

SPECTRAL RADIANCE

L s ( m)

Sensor-reaching spectral radiance

Lgr (m)

Total ground-reflected spectral radiance

L sc (m)

Solar scattered spectral radiance

Ldir (m)

Direct solar spectral radiance

Ldif (m)

Diffuse (downwelling) solar-spectral radiance

Lobst (m)

Nearby obstacles/objects spectral radiance (secondary
illumination)

L p ( m)

Solar path-scattered radiance (path radiance)

Ladj (m)

Adjacency spectral radiance
SPECTRAL IRRADIANCE

Etoa (m)

Exo-atmospheric spectral irradiance at the top of the
atmosphere

Esu (m)

Solar unscattered (direct) spectral irradiance incident on
the surface

E d ( m)

Downwelling spectral irradiance from the sky incident on
the surface

Esu (m; x, y)

Solar unscattered spectral irradiance incident on a tilted
surface at (x, y)

Ed (m; x, y)

Downwelling spectral irradiance incident on a tilted
surface at (x, y)
SPECTRAL TRANSMITTANCE

Tu (m)

Upward atmospheric transmittance (target-to-sensor
path)

Td (m)

Downward atmospheric transmittance (sun-to-surface
path)
SPECTRAL REFLECTANCE

r(m)

Spectral reflectance of the target (assumed Lambertian)

r a ( m)

Spectral reflectance of the adjacent material

S(m)

Atmospheric spherical-scattering albedo

12

on the ground, that is, the material we are trying to find,
identify, or characterize. Extracting r (m) from the sensorreaching radiance Ls (m) is no small feat, because, in doing
so, we not only have to estimate all of the other terms and
factors in (5)-(9) but also have to deal with the variability
of each of those terms and factors.
The terms Lobst (m) and Ladj (m) in (6) and (7) are not expanded here, but details can be found in [16], [35], [37], and
[38]. Ladj (m) has basically the same form as (8), with ra (m)
in place of r (m) and with the diffuse upward transmittance instead of T u (m). As for Lobst (m), this has been modeled by accounting for the presence of a nearby object that
obstructs part of the sky dome viewable from the target;
thus, the average spectral radiance reflected by the object
itself is scaled by a factor (1 - F) [16], [35]. Depending on
the relative position of the sun, target, and obstacle, Lobst (m)
may also include the solar direct radiation reflected by the
nearby object toward the target.
TOPOGRAPHIC EFFECTS
So far, we have assumed a flat surface, but topographic effects and their possible incorporation into the model have
been thoroughly studied (see [34], [37], and [39]-[42]), and
(8) and (9) can be modified accordingly. Specifically, for a
local target surface at location (x, y) tilted by a slope angle
i sl (x, y) (subtended by the surface normal and the vertical
direction), (8) and (9) should be modified by accounting
for the local direct and diffuse irradiances at (x, y):
E su (m; x, y) = E toa (m) cos 6i ill (x, y)@ T d (m)
F (x, y)·E d (m; x, y) = E d (m)· ' A (m)

(10)

cos 6i ill (x, y)@
cos i

+ 61 - A (m)@ F (x, y)2

(11)

cos 6i ill (x, y)@ = cos i cos 6i sl (x, y)@
+ sin i sin 6i sl (x, y)@ cos 6z - z sl (x, y)@ .
(12)
Here, z sl (x, y) is the azimuth angle of the local sloped
terrain at (x, y), z is the sun azimuth angle, and i ill (x, y)
is the local (actual) target illumination angle. In (11), the
diffuse component is expressed as a linear combination
of two terms: one circumsolar diffuse irradiance from
the solid angle near the sun and one isotropic contribution from the remaining sky dome [37], [39], [40]. A (m)
is the anisotropy index, which gives the proportion of
diffuse radiation to be treated as circumsolar and isotropic and has been modeled as A (m) = (1 - a sh) T d (m) [37],
[39], [40]. The sky-view factor for the tilted target surface
in (11) should be evaluated from digital elevation models
[40], [43], although simplified expressions have also been
derived (e.g., for an infinitely long slope [40]). Of note
is that (10)-(12) can be modified or enriched to account
for topography at different levels, such as for mountainous terrain [37], or for the combined effect of topography
and structured forests [41], [42]. Additionally, some works
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2019
IEEE Geoscience and Remote Sensing Magazine - June 2019

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