IEEE Geoscience and Remote Sensing Magazine - June 2019 - 10

Here, the bidirectional reflectance r BRDF is the ratio of
the radiance L (i O, z O) to the irradiance E (i i, z i), where L
is scattered into the direction described by the orientation
angles i O and z O, and E is irradiance from the i i, z i direction (see Figure 2 for an illustration of BRDF geometry).
The function in (1) describes the bidirectional reflectance
values for every combination of input/output angles and,
although not explicitly captured in the equation here, also
changes as a function of wavelength. The simplest BRDF is
constant with respect to the input and output angles; such a
surface is termed Lambertian.
Although the assumption of an ideal diffusely reflecting surface may not truly be achieved in practice, it is often a useful approximation and a natural starting point for
more sophisticated models. Some of these BRDF models
are theoretical physics-based models that use simulations
of light scattering from multifaceted and/or multilayered reflecting rough surfaces, while others are empirical models that provide simple formulations capable of
reproducing specific kinds of reflective behaviors. Often,
these models involve free parameters that are fit to comprehensive experimental measurements made by using sophisticated goniometers and gonioreflectometer systems;
the challenge there is that those systems are only recently
being designed for practical deployment in challenging
environments [17], [18]. The more complex models are
often used in computer graphic design to render realistic
surface illustrations [19], and the simpler models are more
commonly adopted in remote sensing. Schowengerdt [20]
argues that most natural surfaces are approximately Lambertian for viewing angles (i.e., i O in Figure 2) within 20°
to 40° from the zenith. But BRDF also depends on the illumination angle, with most surfaces brighter in the specular
and backscatter directions and darker elsewhere. Trees and
canopies, for instance, have a strong return from the backscatter direction [16]. Choosing the Lambertian model certainly simplifies hyperspectral data analysis, but it just as
clearly underestimates the spectral variability that will be
observed in both target and background materials.

E(θi , φi )

θo

θi

L(θO , φO )

y
φi
φo

FIGURE 2. An illustration of the BRDF geometry described in (1).

Intimate mixtures present their own set of challenges,
because their spectra result from complicated and varying
underlying phenomenology (e.g., reflections and scatterings between grains of different materials). A common
model for these nonlinear spectral interactions is Hapke's
radiative transfer model for intimate mineral mixtures
[21], [22]. This model considers several contributions to
the spectra of intimate mixtures: single volume scattering,
multiple volume scattering, coherent backscattering, the
shadow hiding opposition effect, macroscopic roughness,
and compactness. All of these contributions can lead to
highly variable resulting spectra. A number of approaches
have been explored for applications to hyperspectral unmixing, with particular focus on kernel-based methods
[23]-[27].
EXTRINSIC TARGET VARIABILITY
The signal observed at the sensor depends not only on the
nature of the material but also on how much material there
is. For solid subpixel targets, the target size relative to the
pixel size matters; for gas-phase plumes or thin layers of
powder, concentration and thickness matter. In the LWIR,
the temperature matters as well (i.e., warmer materials emit
more radiation and appear brighter at the sensor).
Perhaps the simplest, and probably the most common,
expression for how much material is in a pixel is given by
the additive model [1]:
x = z + et ,

(2)

where x is the observed spectrum at a pixel, z is the spectrum of the background material in that pixel, t is the spectral signature of the target material, and e is a scalar quantity that corresponds to how much of the target is present.
For opaque subpixel targets, the larger the target, the
more of the background it obscures, and a more appropriate expression is the replacement model [1]:
x = (1 - a) z + at,

(3)

where 0 # a # 1 corresponds to the fraction of the pixel
that is taken up by the target material.
Although the additive and replacement models are
the most popular choices, the actual interaction of target
and background can be more complicated. For chemical
plumes, for instance, the additive model is popular, but it is
only an approximation-accurate in the weak-plume limit
[28]-of the actual Beer's law absorption rule [29]. This approximation is often valid, but it is also motivated by the
fact that target detection algorithms are more easily derived
under the assumptions of the additive model.
The scaled replacement model, which is a kind of hybrid of
the additive and replacement models, considers a target that
can both occlude the background and have variable strength:
x = (1 - a) z + aet.
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IEEE Geoscience and Remote Sensing Magazine - June 2019

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