IEEE Geoscience and Remote Sensing Magazine - June 2019 - 34

in published books by Bellman et al. [15], Lenoble [16],
Goody and Yung [17], Thomas and Stamnes [18], and, more
recently, Liou [19]. For this article, rather than start from
an integro-differential (RT) equation that includes a source
function to account for scattering and absorptions gains
and losses based on atmospheric slab geometry, we instead describe a single at-sensor radiance model that itemizes dominant sources of energy in the reflective spectral
region. This particular model has been used by many, including Manolakis et al. (called the at-aperture model) [20],
Schott (called the sensor-reaching-radiance model) [21], and
Eismann (called the pupil-plane sensor radiance model) [22]. A
high-level description of solar sensor-reaching or measured
radiance, omitting self-emission and the sensor response,
can be expressed, typically in units of [W/m -2 sr -1 nm -1],
as a summation of solar energy from a variety of paths, outlined in Figure 4. That is,
L = L direct + L diffuse + L adj + L up ,

atmosphere is called the trapping effect [23] and is explained in more detail in "Forward Sensor-Reaching
Radiance Model." L diffuse is the scattered skylight (i.e.,
diffuse component) incident onto the target and up to
the sensor and incident onto the background and scattered off the atmosphere, onto the target, and up to the
sensor. L adj is the radiance that is scattered up to the
sensor from adjacent illuminated surfaces, sometimes
called the environmental effect. Finally, L up is the radiance backscattered by the atmosphere itself on its way
up to the sensor. This radiance path never interacts
with target or background surfaces and is thought of
as an interference term.
If we assume that the surface reflectance is Lambertian, then we can write the at-sensor radiance of (1) as
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)

(2)

(1)

where L direct is the radiance directly from the sun, off the
target, and up to the sensor.
This solar radiance can also interact with the background and scatter off the atmosphere (i.e., S), onto
the target, and up to the sensor. This reflecting off the

which is a more detailed version of (1) and is used throughout the literature [23]-[25] for modeling at-sensor radiance. This expression applies to the ultraviolet (UV)VNIR-SWIR wavelength region and would also have to
be convolved with the sensor response to obtain the effective radiance (i.e., what the detector actually sees). In

TABLE 1. APPROXIMATE VNIR-SWIR ABSORPTION
WAVELENGTHS FOR VARIOUS ATMOSPHERIC CONSTITUENTS
(FIGURE 2).
WAVELENGTH
( µm)

COMMENT

Oxygen (O2)

0.69

Weak

0.762

Strong

1.279

Weak

Ozone (O3)

0.602

Weak Chappuis band

Carbon monoxide (CO)

2.334

Very weak

2.362

Very weak

1.68

Weak

1.6-1.8

Weak band

2.2-2.4

Strong

1.43

Weak

1.576

Weak

1.606

Weak

1.93-2.14

Strong band

0.59

Very weak

0.65

Very weak

0.72

Weak

0.82

Weak

0.91

Weak

0.94

Strong

1.13

Strong

1.4

Very strong band

1.9

Very strong band

2.7

Very strong band

Methane (CH4)

Carbon dioxide (CO2)

Water vapor (H2O)

(a)
900

800

700

(cm)

ATMOSPHERIC
CONSTITUENT

600
(b)

FIGURE 3. (a) A color image of an AVIRIS-collected image over

Woodside, California. (b) A FLAASH-derived water-vapor map [13].
34

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2019
IEEE Geoscience and Remote Sensing Magazine - June 2019

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2019
