Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer

R. Glenn Sellar; Glenn D. Boreman

doi:10.1364/AO.44.001614

Applied Optics
Vol. 44,
Issue 9,
pp. 1614-1624
(2005)
•https://doi.org/10.1364/AO.44.001614

Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer

R. Glenn Sellar and Glenn D. Boreman

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R. Glenn Sellar and Glenn D. Boreman, "Comparison of relative signal-to-noise ratios of different classes of imaging spectrometer," Appl. Opt. 44, 1614-1624 (2005)

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Abstract

The continued development of new and fundamentally different classes of imaging spectrometer has increased both the scope and the complexity of comparisons of their relative signal-to-noise ratios. Although the throughput and multiplex advantages of Fourier-transform spectrometers were established in the early 1950s, the application of this terminology to imaging spectrometers is often ambiguous and has led to some confusion and debate. For comparisons of signal-collection abilities to be useful to a system designer, they must be based on identical requirements and constraints. We present unambiguous definitions of terminology for application to imaging spectrometers and comparisons of signal-collection abilities and signal-to-noise-ratios on a basis that is useful to a systems designer and inclusive of six fundamentally different classes (both traditional and novel) of imaging spectrometers.

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	Along-Track Scanning

Type of Spectral Scanner
Filtering	No known examples	Filter array	Band-sequential
		Wedge filter	Filter wheel
		Linear variable filter	Tunable filter^a
Dispersive	Grating or prism	(No known examples)	Image slicer
			Tomographic
Interferometric	Static FTS (Sagnac)	Static FTS (Mach-Zehnder, Sagnac)	Traditional FTS (Michelson)

Dimension	Range	Sample Extent	Number of Samples
Cross track	X_o (m)	Δx_o (m)	M_x
Along track	Y_o (m)	Δy_o (m)	M_y
Spectral	Λ₀ (μm)	Δλ_o (μm)	M_λ
Temporal	τ (s)	τ (s)	l

Constraint	Array	Pixel	Number of Elements
Width	X_i (m)	Δx_i (m)	M_i
Height	Y_i (m)	Δy_i (m)	M_j
Detector Noise		N_d

Object spectral radiance	L_λ (photons s⁻¹ m⁻² sr⁻¹ μm⁻¹)
Range to object	d(m)
Aperture	Constrained by mass limitation
Effective focal length	f (mm, derived from other constraints)

Spectral Class (Scan Class)	Signal Factor	Relative Signal for Example: M_y = 4000, M_λ = 200, M_j = 2000
Filtering (Framing)	$(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y}})$	2.5
Filtering (Windowing)	$(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y} + M_{j}})$	1
Dispersive (Pushbroom)	$(1) (\frac{1}{M_{y}})$	2
Interferometric (Pushbroom)	$(\frac{1}{2}) (\frac{1}{M_{y}})$	0.5
Interferometric (Framing)	$(\frac{1}{2}) (\frac{M_{j}}{M_{y}})$	500
Interferometric (Windowing)	$(\frac{1}{2}) (\frac{M_{j}}{M_{y} + M_{j}})$	400
Panchromatic (Framing)	$(1) (\frac{M_{j}}{M_{y}})$	2000

Spectral Class (Scan Class)	Δλ_i	M_k	Δτ_i
Filtering (Framing)	$\frac{Λ}{M_{λ}}$	$M_{λ} (\frac{M_{y}}{M_{j}})$	$\frac{τ M_{j}}{M_{λ} M_{y}}$
Filtering (Windowing)	$\frac{Λ}{M_{λ}}$	M_y + M_j	$\frac{τ}{M_{y} + M_{j}}$
Dispersive (Pushbroom)	$\frac{Λ}{M_{λ}}$	M_y	$\frac{τ}{M_{y}}$
Interferometric (Pushbroom)	$\frac{Λ}{M_{λ}}$	M_y	$\frac{τ}{M_{y}}$
Interferometric (Framing)	$\frac{Λ}{2}$	$2 M_{λ} (\frac{M_{y}}{M_{j}})$	$\frac{τ M_{j}}{2 M_{λ} M_{y}}$
Interferometric (Windowing)	$\frac{Λ}{2}$	M_y + M_j	$\frac{τ}{M_{y} + M_{j}}$
Panchromatic (Framing)	Λ	$\frac{M_{y}}{M_{j}}$	$\frac{τ M_{j}}{M_{y}}$

Spectral Class (Scan Class)	Signal Factor	Relative Signal for Example: M_y = 4000, M_λ = 200, M_j = 1000
Filtering (Framing)	$(\frac{1}{2}) (\frac{1}{M_{λ}}) (\frac{M_{j}}{M_{λ} M_{y}})$	12.5
Filtering (Windowing)	$(\frac{1}{2}) (\frac{1}{M_{λ}}) (\frac{M_{j}}{M_{y} + M_{j}})$	2
Dispersive (Pushbroom)	$(1) (\frac{1}{M_{j}}) (\frac{1}{M_{y}})$	1
Interferometric (Pushbroom)	$(\frac{1}{2}) (\frac{1}{M_{j}}) (\frac{1}{M_{y}})$	0.5
Interferometric (Framing)	$(\frac{1}{2}) (1) (\frac{M_{j}}{2 M_{λ} M_{y}})$	5,000
Interferometric (Windowing)	$(\frac{1}{2}) (1) (\frac{1}{M_{y} + M_{j}})$	400
Panchromatic (Framing)	$(1) (1) (\frac{M_{j}}{M_{y}})$	1,000,000

Spectral Class (Scan Class)	M	Signal Factor
Filtering (Framing)	M_iM_λM_y	$(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y}})$
Filtering (Windowing)	M_iM_jM_y	$(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y} + M_{j}})$
Dispersive (Pushbroom)	M_iM_jM_y	$(1) (\frac{1}{M_{y}})$
Interferometric (Pushbroom)	M_iM_jM_y	$(\frac{1}{2}) (\frac{1}{M_{y}})$
Interferometric (Framing)	M_i2M_λM_y	$(\frac{1}{2}) (\frac{M_{j}}{M_{y}})$
Interferometric (Windowing)	M_iM_jM_y	$(\frac{1}{2}) (\frac{M_{j}}{M_{y} + M_{j}})$
Panchromatic (Framing)	M_iM_y	$(1) (\frac{M_{j}}{M_{y}})$

Spectral Class (Scan Class)	SNR Factor for Q = 0 (No Detector Noise)	SNR Factor for General Case
Filtering (Framing)	${[(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y}})]}^{1 / 2}$	$\frac{(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y}})}{Q + {[(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y}})]}^{1 / 2}}$
Filtering (Windowing)	${[(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y} + M_{j}})]}^{1 / 2}$	$\frac{(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y} + M_{j}})}{Q + {[(\frac{1}{2}) (\frac{1}{M_{λ}} \frac{M_{j}}{M_{y} + M_{j}})]}^{1 / 2}}$
Dispersive (Pushbroom)	${[(1) (\frac{1}{M_{y}})]}^{1 / 2}$	$\frac{(1) (\frac{1}{M_{y}})}{Q + {[(1) (\frac{1}{M_{y}})]}^{1 / 2}}$
Interferometric (Pushbroom)	${[\frac{S (λ)}{\bar{S}}] \sqrt{\frac{1}{M_{λ}}}} {[(\frac{1}{2}) (\frac{1}{M_{y}})]}^{1 / 2}$	${[\frac{S (λ)}{\bar{S}}] \sqrt{\frac{1}{M_{λ}}}} \frac{(\frac{1}{2}) (\frac{1}{M_{y}})}{Q + {[(\frac{1}{2}) (\frac{1}{M_{y}})]}^{1 / 2}}$
Interferometric (Framing)	${[\frac{S (λ)}{\bar{S}}] \sqrt{\frac{1}{M_{λ}}}} {[(\frac{1}{2}) (\frac{M_{j}}{M_{y}})]}^{1 / 2}$	${[\frac{S (λ)}{\bar{S}}] \sqrt{\frac{1}{M_{λ}}}} \frac{(\frac{1}{2}) (\frac{M_{j}}{M_{y}})}{Q + {[(\frac{1}{2}) (\frac{M_{j}}{M_{y}})]}^{1 / 2}}$
Interferometric (Windowing)	${[\frac{S (λ)}{\bar{S}}] \sqrt{\frac{1}{M_{λ}}}} {[(\frac{1}{2}) (\frac{M_{j}}{M_{y} + M_{j}})]}^{1 / 2}$	${[\frac{S (λ)}{\bar{S}}] \sqrt{\frac{1}{M_{λ}}}} \frac{(\frac{1}{2}) (\frac{M_{j}}{M_{y} + M_{j}})}{Q + {[(\frac{1}{2}) (\frac{M_{j}}{M_{y} + M_{j}})]}^{1 / 2}}$
Panchromatic (Framing)	${[(1) (\frac{M_{j}}{M_{y}})]}^{1 / 2}$	$\frac{(1) (\frac{M_{j}}{M_{y}})}{Q + {[(1) (\frac{M_{j}}{M_{y}})]}^{1 / 2}}$

	Relative SNR for Example: M_y = 4000, M_λ = 200, M_j = 2000, S(λ)/ $\bar{S}$ = 1

Spectral Class (Scan Class)	Q = 0	Q = 0.1
Filtering (Framing)	1.6	2.3
Filtering (Windowing)	1.4	1.9
Dispersive (Pushbroom)	1	1
Interferometric (Pushbroom)	0.05	0.04
Interferometric (Framing)	1.6	9.0
Interferometric (Windowing)	1.4	7.9
Panchromatic (Framing)	32	193

Abstract

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Applied Optics