IEEE Geoscience and Remote Sensing Magazine - June 2019 - 11

In the LWIR, this could, for instance, be due to temperature
variation. An example in the VIS wavelengths arises when
a target in the shade exhibits a lower apparent reflectance
than that same target in the bright sun. More sophisticated
models (e.g., those that reduce the dependence on the temperature [30] or achieve temperature-emissivity separation
[31] or those that explicitly account for shadows [32]) may
ultimately be preferable to the scaled replacement model,
but they are also more complicated.
ENVIRONMENTALLY INDUCED TARGET VARIABILITY
Target spectral variability not only depends on the target itself but is also caused by the surrounding environment. In
this subsection, we examine some of the most important
environmental sources of spectral variability by considering the typical remote sensing scenario of a hyperspectral
sensor mounted onboard a platform (such as an aircraft
or a satellite) and imaging the targeted surface materials.
MAIN RADIATIVE TRANSFER MODEL EQUATIONS
The radiative transfer between a surface target and a sensor
across the VIS and NIR (VNIR) (~400-1,000 nm) and SWIR
(~1,000-2,500 nm) spectral ranges (i.e., VNIR-SWIR)
can be expressed by the following simplified physics-based
model for the sensor-reaching radiance Ls (m) [16], [33]-[36]:
Ls (m) = Lgr (m) + Lsc (m)
Lgr (m) = Ldir (m) + Ldif (m) + Lobst (m)
sc

p

adj

L (m) = L (m) + L (m).

(5)
(6)
(7)

potential nearby objects and obstacles in view of the target
(such as trees and buildings) [35]. The solar scattered radiance Lsc (m) includes the so-called path radiance Lp (m)-that
is, the solar spectral radiance scattered by the atmosphere
toward the sensor-and the adjacency term Ladj (m), accounting for the direct and diffuse solar spectral radiances
reflected from the materials adjacent to the target toward
the sensor (including the aforementioned multiple-reflections phenomenon).
By expanding the first two terms on the righthand side
of (6), we obtain the following [16], [33], [36]:
Ldir (m) + Ldif (m) =

E su (m) + F·E d (m) r (m) u
r T (m)
1 - ra (m) S (m)
r (m)
- a sh E su (m) r T u (m),

E su (m) = E toa (m) cos i T d (m).

(8)
(9)

In (8) and (9), E su (m) is the solar unscattered (direct) irradiance incident on the surface, E toa (m) is the solar exoatmospheric spectral irradiance at the top of the atmosphere incident on a surface orthogonal to the sun's rays, i
is the sun zenith angle (subtended by the sun's rays and the
surface normal), and T d (m) is the downward atmospheric
transmittance of the sun-to-surface path. The term E d (m) is
the downwelling spectral irradiance from the sky incident
on the surface, which undergoes a scaling by the sky view
factor (or shape factor) F ! 60, 1@ accounting for the fraction
of sky dome visible from the target [35], [36]; r (m) is the diffuse spectral reflectance (or albedo) of the targeted surface
[assumed to be Lambertian; i.e., r (m) = r BRDF (m)·r]; ra (m)
is the diffuse spectral reflectance of the material adjacent

JUNE 2019

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

Lobst(λ)

Ldif(λ)
Ldir(λ)

Lp(λ)
Ladj(λ)

A sketch of the various radiation contributions entering
the sensor is provided in Figure 3,
with Table 1 summarizing the major radiometric terms involved. In
(5)-(7), the total ground-reflected
Top of Atmosphere
spectral radiance Lgr (m) includes
the contributions reflected by the
target toward the sensor, and the solar scattered spectral radiance Lsc (m)
includes the contributions scattered
toward the sensor without having
interacted with the targeted surface.
Ldir (m) and Ldif (m) are the major contributors to Lgr (m) and represent,
respectively, the direct and diffuse
(downwelling) solar spectral radiances reflected by the target toward
the sensor-also accounting for the
multiple ref lections between the
adjacent material surface and the
atmosphere due to multiple scatterra (λ)
r (λ)
ing [16]. The ground-reflected radiance also includes the spectral radi- FIGURE 3. A sketch of the radiative-transfer model along the sun-to-target-to-sensor path, as
ance Lobst (m) arriving on the target described in the "Main Radiative-Transfer Model Equations" section. The individual terms are
due to secondary illumination from listed in Table 1.

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IEEE Geoscience and Remote Sensing Magazine - June 2019

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