IEEE Geoscience and Remote Sensing Magazine - June 2019 - 82

where S is (m c) is due to self-emission and includes the dark
signal from the detector element. The imaging spectrometer
views the hot and cold blackbodies, specified as B (m c, TH)
and B (m c, TC), enabling the determination of R and S is as
S H (m c) - S C (m c)
,
B (m c, TH) - B (m c, TC)

(8)

S C (m c) B (m c, TH) - S H (m c) B (m c, TC)
.
B (m c, TH) - B (m c, TC)

(9)

R (m c) =
and
S is (m c) =

The blackbody radiance is also band averaged. Once R and
Sis are known, then (7) yields the desired at-aperture scene
radiance. Typically, the blackbodies are measured frequently, e.g., before and after the collection of a scene or flight,
to ensure radiometric accuracy. This is a conservative approach because the detectors
THE IDEAL BLACKBODY
and optical temperature are
FOLLOWS THE PLANCK
often stable. Moreover, unity
RADIATION LAW AND
blackbody emissivity has been
SHOULD DEPEND ONLY
assumed [1]. The calibration
of a Michelson interferometerUPON THE TEMPERATURE
based imaging spectrometer is
AND NOT UPON THE
more challenging and utilizes
SURFACE OR MATERIAL
the complete complex specPROPERTIES OF THE
tra resulting from the Fourier
SOURCE.
transform (FT) of the interferograms. The resulting process
is similar to the one outlined
here except that it is modified to account for any nonlinearities in the phase [24].
A great deal of research has been devoted to the development of high-quality blackbodies for calibration [25].
The ideal blackbody follows the Planck radiation law and
should depend only upon the temperature and not upon
the surface or material properties of the source. A cavity
with a small opening compared to the internal surface
area is a close approximation to the ideal blackbody, motivating a broad range of cavity-based designs; however,
cavity blackbodies used for remote sensing tend to have
larger aperture openings and rely upon high-emissivity
surfaces and multiple reflections so that the exiting light
is unlikely to originate from outside. The coatings are
often specular reflectors in this case, and the effective
emissivity of these designs typically exceeds 0.99. The
internal geometry often includes cones, either pointed
toward or away from the opening, and V-shaped grooves
to increase the number of reflections. Great care is taken
to ensure that the cavities are also isothermal, which
is particularly challenging for large apertures. Carbon
nanotube coatings are a promising high-emissivity surface with excellent thermal conductivity, which ensures
that the temperature drop across the coating is small,
82

thereby facilitating isothermality [26]. The designs typically use thermistors whose characterization is traceable
to a national metrology laboratory that measures the
cavity temperature.
The characterization process also addresses other attributes such as nonlinear behavior, underperforming
detector elements, and system performance metrics.
Nonlinear behavior is characterized by using a variable
blackbody at several different temperatures. The signal
from each detector element is fit using a linear least
squares model so that any deviations from linearity are
quantified and the data calibration process is modified
if required. Nonlinearity is particularly deleterious for
Michelson interferometers, producing a signal at the second harmonic of the spectrum, an out-of-band signal,
a reduced in-band signal for all wavenumbers, and a dc
error [27]. The linearity data can also be used to calculate
noise metrics such as the noise-equivalent spectral radiance (NESR) through
NESR (i, j, T) =

v (i, j, T)
,
R (i, j)

(10)

where v (i, j, T) is the standard deviation calculated from
an average of a number of acquisitions for a particular
blackbody at temperature T. This is also done as a function of frame rate [28]. Pixels that are excessively noisy in
comparison to the average performance or have anomalous responsivity are cataloged based on the radiometric
characterization.
VICARIOUS CALIBRATION
Vicarious calibration techniques have long been used for
the on-orbit calibration maintenance of VNIR/SWIR spectral imagers, and similar techniques have been developed
for LWIR imaging spectrometers [29]. Vicarious calibration requires accurately known surface-leaving radiance,
which can be measured directly or modeled if the surface
emissivity and temperature as well as the emission and
transmission properties of the atmosphere are known.
Combining (3) and (4), the band averaged at-aperture radiance is given by

L (m c) =

# " x a 6eB (Tg) + (1 - e) L d@ + L u ,P (m c - m) dm
,
# P (m c - m) dm

(11)

where the spectral dependence of all the quantities except
the instrumental profile has been suppressed. Each term is
determined through a combination of measurements and
radiative transfer modeling.
The application of (11) requires the modeling of upwelling and downwelling radiance using a radiative
transfer code that is based upon in situ measurements
of the atmospheric conditions and surface properties.
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2019
IEEE Geoscience and Remote Sensing Magazine - June 2019

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