IEEE Geoscience and Remote Sensing Magazine - June 2019 - 78

advanced to the point that these challenging requirements
can be met.
GENERAL DESCRIPTION
The critical attributes that, in part, govern imaging spectrometer performance are succinctly captured by the measurement equation. This equation, which describes the signal measured by a detector element of the FPA, is given by
S (m c) = t int

#0 3 U6L (m) x o (m) h (m)@P (m c - m) dm,

(5)

where U is the étendue, i.e., optical throughput; L (m) is
the at-aperture spectral radiance assumed to be uniform
at the entrance pupil; x o (m) is the optical efficiency; h (m)
is the detector response function, which can be the quantum efficiency with the signal in terms of photoelectrons
generated; P^m c - mh is the instrumental profile with center wavelength m c; and t int is the integration time for one
measurement cycle. The signal described by (5) is only that
portion that has undergone spectral discrimination and
does not include the thermal background signal because
of self-emission and the FPA's dark current. Additionally,
the detector output is assumed to be linear. The spatial
dependence has been suppressed in the notation. Each
quantity critically impacts the performance of an imaging spectrometer.
The étendue is the geometric factor that determines the
maximum amount of light flux that can be transmitted by

Scanning

Staring

Spectrometers

Interferometers

Grating Based

Michelson

Fabry-Pérot
Prism Based
Q

∼
vo
(a)

∼
∼
vo + Q vo + 2Q
(b)

FIGURE 5. Imaging spectrometer categories. (a) Dispersive designs
require a slit and a scanning mechanism, and (b) interferometer
designs stare at an area as the spectral information is recorded. The
Michelson encodes the spectral information in an interferogram,
and the Fabry-Pérot transmission function displays a channel
spectrum as it is scanned under monochromatic illumination at
wavenumber ou o, with the free spectral range given by Q.

78

a lossless optical system. It is often simplified to the " AX"
product, which is inversely proportional to the square of
the system F-number; therefore, optical designers strive to
produce optically fast systems with low F-numbers. Similarly, the optical efficiency is maximized through the selection of optical materials, and, for dispersive systems, the
grating verses prism efficiency is weighed.
The instrumental profile P ^m c -m h is the function that
describes the spectral discrimination. This is typically
called the spectral response function (SRF) in the remote
sensing literature; it ignores the long-standing terminology of spectroscopists, for which the instrumental line
shape or instrumental profile is common practice. The full
width at half maximum of P is known as the limit of resolution and describes the energy spread of a spectral channel.
The instrumental profile determines the extent to which
the true radiance spectrum can be recovered. The scene
radiance is comprised of the surface-leaving radiance, as
determined by the surface properties and the atmospheric transmission and emission. The spectral information
is tremendously detailed, and the instrumental profile
smooths this detail through the convolution in (5). In the
context of remote sensing, imaging spectrometers are high
spectral resolution systems with the ability to recover relatively subtle spectral features in comparison to multispectral imagers.
OPTICAL FORMS
The development of LWIR imaging spectrometers for terrestrial remote sensing, as opposed to atmospheric science,
has been dominated by dispersive forms. These designs require a slit that forms a spatial mask at the telescope focus
at the input of the spectrometer, which limits the range of
angles that pass through to the dispersing element. The slit
is projected to the surface by the telescope, and the data are
acquired by advancing the slit image along the surface at
an appropriate sampling rate using either the platform motion (push broom) or a scan mirror whose axis of rotation
is aligned with the direction of travel (whisk broom). For
high-performing designs, the instrumental profile (which
results from a convolution of the slit width, the line spread
function, and the pixel pitch) is uniform (or nearly so)
across the full field in a relative sense, with only the center
wavelength changing. This implies that point aberrations
are well corrected across the field so that the point spread
function is contained within a pixel, and field aberrations,
such as distortion, vary a small fraction of a detector element over the full field. The effects of distortion are referred to as the smile, the curvature of a monochromatic
image of the slit as a function of spatial position, and the
keystone, the variation in the magnification of the slit as a
function of wavelength. Keystone is the more deleterious
of the two effects, causing wavelength-dependent mixing
between adjacent spatial samples. Often, the instrumental
profile can be well modeled using a simple function such
as a Gaussian.
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

JUNE 2019
IEEE Geoscience and Remote Sensing Magazine - June 2019

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