IEEE Geoscience and Remote Sensing Magazine - June 2019 - 36

FORWARD SENSOR-REACHING RADIANCE MODEL
In the "Sensor-Reaching Radiance" section, we stated a Lambertian model for
the sensor-reaching radiance in the form of (2). In this sidebar, starting with (1),
we derive and illustrate the assumptions and approximations required to obtain
(2), including discussions of the trapping and adjacency effect, bidirectional
reflectance distribution function (BRDF ), and radiance due to background objects.

same scatter light, which may exhibit the trapping effect with the background,
as mentioned in the previous section. We can write the diffuse term as

DIRECT ILLUMINATION
Examining (1) and Figure 4(a), we see that the downward direct solar flux
(L direct) is the reflected direct solar radiation originating from the sun, through
the atmosphere, and up to the sensor [i.e., within the sensor's field of view
(FOV)]. A second illustrated path shows the direct solar flux interacting with the
background and being backscattered by the atmosphere back down to the surface, also within the sensor's FOV.
Thus, we can write the per-pixel spectral direct solar radiance as

This is the integration of a directional radiance, L d (i, z, m), over the hemisphere, dX. This diffuse energy reflects off the target of reflectance, t brdf
t , on
its way to the sensor through transmission, x u (m). We again notice that this
downwelling radiance can exhibit the trapping effect, as indicated by the multiplicative quantity previously mentioned.

L direct (x, y, m) =6E s (m) cos(i s) $ x d (i s, z s, m) t brdf
t (m) x u (m)@ $

1
,
1- t b (m) S (m)
(S1)

where E s (m) is the exoatmospheric spectral irradiance from the sun in units of
Wm -2 nm -1 . at a zenith angle of i s, x d (i s, z s, m) is the spectral transmission
through the atmosphere along the sun-target path at (zenith, azimuth) position
-1
(i s, z s), t brdf
t (m) is the angularly dependent target BRDF in units of sr , and
x u (m) is the spectral transmission along the target-sensor path.
In Figure 4(a), we also see a path that interacts with the background. For
more dense atmospheres, we consider photons that originate from the sun,
reflect off the background albedo toward the sensor, but then get effectively
reflected back down toward Earth's surface by a quantity called the spherical
(scattering) albedo, S, which can be thought of as the effective reflectance of
the atmosphere to upwelling radiance. This iterative bouncing back and forth,
also called trapping, due to multiply scattered photons, can continue indefinitely where scattered background photons can interact with the target itself
when making their way to the sensor. This results in the multiplicative quantity
1/(1- t b (m) S (m)), where t b (m) is the background albedo (i.e., the directional
hemispheric reflectance). We can see that the effects of multiple scattering are
minimized if either the background reflectance or the spherical albedo is zero.
DIFFUSE ILLUMINATION
In Figure 4(b), we see the diffuse illumination component. Here, the radiance,
L diffuse , is due to various scattered paths that illuminate the ground. Some flux
scatters onto the target inside the sensor's FOV, while other paths show the

minimum values at each wavelength. The DOS method has
been incorporated in some more recent atmospheric compensation methods, such as QUAC, as we describe later.
INTERNAL AVERAGE REFLECTANCE
AND FLAT FIELD CORRECTION
In the late 1980s, ratio approaches, used with multi- and
hyperspectral data, were developed to normalize radiance
data to relative reflectance. These included the IAR [35] and
the flat field correction (FFC) approaches [36]. The IAR
simply divides every radiance pixel by the image-wide averaged radiance spectrum to produce an estimate of relative
reflectance. The assumption is that the averaged spectrum
is representative of the direct solar term, L direct . This has the
effect of normalizing the data to a scene-average spectrum
and works best with arid scenes lacking vegetation. The FFC
approach also takes a ratio. It assumes that there is a bright,
36

L diffuse (x, y, m) =

1
$ x (m) # t tbrdf(m) L d (i, z, m) cos(i) dX. (S2)
1- t b (m) S (m) u
X

UPWELLED ILLUMINATION
In Figure 4(c), we see upwelled radiance, L up , backscattered toward space and
into the sensor's instantaneous FOV. This energy path is due to scattered solar
radiation in the atmosphere that makes its way to the sensor but never interacts
with the target or background. Sometimes, this interference term is referred to
as atmospheric haze, and it has a tendency to reduce overall contrast in
sensed imagery.
ADJACENT ILLUMINATION
Finally, in Figure 4(d), we consider those photons that are scattered in the
atmosphere, interact with the background, and make their way into the sensor's FOV. This phenomenon is called the adjacency effect [84], and it can be
described as
L adj (x, y, m) =[L env (m) t b (m)]

1
,
1- t b (m) S (m)

(S3)

where L env (m) is the radiance from the immediate environment that interacts
with the background and spherical albedo, tb (m) and S, respectively. This
effect is usually combined with the upwelled radiance.
TOTAL AT-SENSOR RADIANCE
Finally, by combining (S1), (S2), (S3), and L up , we can re-express (1) as
L(x, y, m) =6E s (m) cos(i s) $ x d (i s, z s, m) t brdf
t (m) x u (m)@

1
1- t b (m) S (m)
1
+ :x u (m) # t brdf
t (m) L d (i, z, m) cos (i) dXD $
1- t b (m) S (m)
X
1
,
+ L up (m) +6L env (m) t b (m)@
1- t b (m) S (m)

(S4)

spectrally flat (except for the absorption features) object in
the scene that is user defined via an ROI. Ideally, this would
be a white calibration panel, for example. Every image radiance pixel is then divided by this flat radiance spectrum to
produce an estimate of relative reflectance. Both of these
methods are quick and do not require any field or laboratory measurements. However, the results represent only
crude estimates of the actual reflectance spectra.
EMPIRICAL LINE METHOD
A much more accurate scene-based technique is the ELM
[37]. It requires reflectance field measurements of at least
one dark and one bright object (e.g., calibration panels)
in the scene, ideally Lambertian, homogenous, large, and
measured at the time of overflight. The approach creates
per-band coefficients that linearly map scene radiance (or
digital counts) to reflectance. In many scenes, it is adequate
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2019
IEEE Geoscience and Remote Sensing Magazine - June 2019

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