IEEE Geoscience and Remote Sensing Magazine - June 2019 - 37

which is also similar, with the exception of adjacency, to the forward model
used in the remote-sensing ray-tracing code DIRSIG (i.e., digital imaging and
remote sensing image generation model) [85].
The spectral transmission factors x u (m) and x d (m) are computed from the
layered optical depths, d, as
z

x= e -d = e - b ext ,

(S5)

where b ext is the extinction coefficient (accounting for the sum of absorption
and scattering) and z is a particular path length, typically a small segment path
length in a layered atmosphere.
SENSOR-REACHING RADIANCE: LAMBERTIAN ASSUMPTIONS
AND SIMPLIFICATIONS
In the previous section, we developed a full at-sensor radiance model to
account for the various paths solar photons could take on their way to an airborne or space-based sensor. In this section, we recognize that some of these
paths may be inconsequential under certain conditions. Furthermore, we
examine the model when assuming Lambertian behavior, which is commonly
done in remote-sensing applications.
In (S4), the target BRDF (in units of sr -1) is used, i.e., a function of illumination angle, view angle, and wavelength. Under a Lambertian assumption, a simple integration over the hemisphere shows that the angular-dependent BRDF
reduces to
t (m)
t brdf(m) = r [sr -1],

(S6)

where t(m) can be thought of as the directional hemispheric reflectance.
Therefore, under this assumption, the direct solar term of (S1) reduces to

L Lamb
direct (m) =

E s (m) cos(i s) x d (i s, z s, m) x u (m) t t (m)
.
r(1- t b (m) S (m))

(S7)

Furthermore, if we assume a uniform Lambertian sky (i.e., radiance is independent of direction) such that L d (i, z, m) " L d (m), the integral in (S2) can be evaluated over the hemisphere and simplified as

#X L d (m) cos(i) dX = L d (m) r = E d (m),

to assume that the atmospheric effects are spatially uniform
and that the adjacency term is uniform and/or negligible.
This leads to a linear form of the radiance model, characterized by an offset term that corresponds to atmospheric
scattering and a gain factor that scales the reflectance. The
accuracy of the linear model depends on how well these
assumptions hold and, more importantly, on how the
gain and offset are calculated. In particular, the ELM uses
a simplified version of (2), where we assume minimal impacts from adjacency and trapping effects [i.e., t b(m) S(m) is
small]. Additional information on the trapping effect can
be found in "Forward Sensor-Reaching Radiance Model."
We also assume that we can combine the diffuse or
adjacency term with the upwelled term, a reasonable assumption when the scene materials are fairly uniformly
interspersed, when the image covers a small geographic
area, or when the visibility is high, making the diffuse
JUNE 2019

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

(S8)

where E d (m) is the irradiance (because E = Lr under Lambertian assumptions).
This then results in a simplified diffuse illumination component of the form
L Lamb
diffuse (m) =

E d (m) x u (m) t t (m)
.
r(1- t b (m) S (m))

(S9)

Using these Lambertian assumptions and approximations, we can simplify
our at-sensor radiance model in (S4) to
LLamb(m) =

E s (m) cos(i s) x d (i s, z s, m) x u (m) t t (m) E d (m) x u (m) t t (m)
+
r(1- t b (m) S (m))
r(1- t b (m) S (m))
L env (m) t b (m)
+
+ L up (m) .
(S10)
1- t b (m) S (m)

We can further simplify (S10) if we simply group together the direct and diffuse
solar fluxes that interact with the target as
A(m) =

E s (m) cos(i s) x d (i s, z s, m) x u (m) + E d (m) x u (m)
.
r

(S11)

Finally, we have
LLamb(m) =

A(m) t t (m)
B (m) t b (m)
+
+ L up (m),
1- t b (m) S (m) 1- t b (m) S (m)

(S12)

where we let B (m) = L env (m), a term that does not interact with the target. This
is our at-sensor radiance model, similar to that used throughout the literature
[23]-[25] when modeling the at-sensor radiance.
Strictly speaking, t b in both the numerator and denominator is not
exactly the same. As stated by Berk et al. [59], the t b in the numerator represents a
... weighted average reflectance over the surface region (typically
around 1 km in width when viewed from a high-altitude sensor) that
contributes to atmospherically scattered photons leaving the ground
and arriving at the sensor. The denominator variant represents an average reflectance over an even larger region, representative of photons
that left the surface once, but have then been rescattered via S, back to
the surface and then into the sensor. However, because S is small and
the differences in size of the two averaging regions are not critical, the
two averaged reflectances may be equated with little error.

transmittance small. With this assumption, (2) simplifies to
L Lamb
elm (m) = A(m) t t(m) + B(m) t b(m) + L up(m).

(3)

If we combine the adjacency term with the upwelled terms
as b(m) = B(m) t b(m) + L up(m) and let g (m) = A(m), we have
L Lamb
elm (m) = g (m)·t t (m) + b(m),

(4)

where we see a linear relationship between reflectance and
radiance through gain (i.e., irradiance and transmission,
in this case) and bias (i.e., upwelled and adjacent radiance)
terms. This can be written in terms of reflectance as
tt t (m) =

t
L Lamb
elm (m) - b(m)
,
gt (m)

(5)
37
IEEE Geoscience and Remote Sensing Magazine - June 2019

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