Abstract

An algorithm is presented that corrects a multichannel fiber-coupled spectrograph for stray or scattered light within the system. The efficacy of the algorithm is evaluated based on a series of validation measurements of sources with different spectral distributions. This is the first application of a scattered-light correction algorithm to a multichannel hyperspectral spectrograph. The algorithm, based on characterization measurements using a tunable laser system, can be extended to correct for finite point-spread response in imaging systems.

© 2012 Optical Society of America

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References

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  1. Y. Zong, S. W. Brown, B. C. Johnson, K. R. Lykke, and Y. Ohno, “Simple spectral stray light correction method for array spectrometers,” Appl. Opt. 45, 1111–1119 (2006).
    [CrossRef]
  2. M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
    [CrossRef]
  3. Y. Zong, S. W. Brown, G. Meister, R. A. Barnes, and K. R. Lykke, “Characterization and correction of stray light in optical instruments,” Proc. SPIE 6744, 67441L (2007).
    [CrossRef]
  4. Y. Zong, S. W. Brown, K. R. Lykke, and Y. Ohno, “Correction of stray light in spectroradiometers and imaging instruments,” presented at CIE 26th Session, Beijing, China, 4–11 July 2007.
  5. P. A. Jansson and R. P. Breault, “Correction color-measurement error caused by stray light in image scanners,” in The Sixth Color Imaging Conference: Color Science, Systems, and Applications (Wiley, 1998), pp. 69–73.
  6. A. S. Bolton and D. J. Schlegel, “Spectro-perfectionism: an algorithmic framework for photon noise-limited extraction of optical fiber spectroscopy,” Publ. Astron. Soc. Pac. 122, 248–257 (2010).
    [CrossRef]
  7. C. Huang, J. R. G. Townshend, S. Liang, S. N. V. Kalluri, and R. S. DeFries, “Impact of a sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sensing Environ. 80, 203–212 (2002).
    [CrossRef]
  8. S. Qiu, G. Godden, X. Wang, and B. Guenther, “Satellite-Earth remote sensor scatter effects on Earth scene radiometric accuracy,” Metrologia 37, 411–414 (2000).
    [CrossRef]
  9. G. Meister and C. R. McClain, “Point-spread function of the ocean color bands of the moderate resolution imaging spectroradiometer on Aqua,” Appl. Opt. 49, 6276–6285 (2010).
    [CrossRef]
  10. D. Hong and K. J. Voss, “Effects of point-spread response on calibration and radiometric accuracy of CCD camera,” Appl. Opt. 43, 665–670 (2004).
    [CrossRef]
  11. B. Bitlis, P. A. Jansson, and J. P. Allebach, “Parametric point spread function modelling and reduction of stray light effects in digital still cameras,” Proc. SPIE 6498, 64980V (2007).
    [CrossRef]
  12. P. A. Jansson, ed., Deconvolution of Images and Spectra(Academic, 1997).
  13. S. W. Brown, G. P. Eppeldauer, and K. R. Lykke, “Facility for spectral irradiance and radiance responsivity using uniform sources,” Appl. Opt. 45, 8218–8237 (2006).
    [CrossRef]
  14. C. Huang, J. Townshend, S. Liang, S. Kalluri, and R. DeFries, “Impact of sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sens. Environ. 80, 203–212 (2002).
    [CrossRef]
  15. G. Meister, Y. Zong, and C. R. McClain, “Derivation of the MODIS Aqua point-spread function for ocean color bands,” Proc. SPIE 7081, 70811F (2008).
    [CrossRef]
  16. J. A. Esposito, A. Wu, and J. Sun, “MODIS reflective solar bands uncertainty analysis,” Proc. SPIE 5542, 448–458 (2004).
    [CrossRef]
  17. “MOBY @ MLML,” http://moby.mlml.calstate.edu/ .
  18. “BOUSSOLE,” http://www.obs-vlfr.fr/Boussole/html/home/home.php .

2010 (2)

A. S. Bolton and D. J. Schlegel, “Spectro-perfectionism: an algorithmic framework for photon noise-limited extraction of optical fiber spectroscopy,” Publ. Astron. Soc. Pac. 122, 248–257 (2010).
[CrossRef]

G. Meister and C. R. McClain, “Point-spread function of the ocean color bands of the moderate resolution imaging spectroradiometer on Aqua,” Appl. Opt. 49, 6276–6285 (2010).
[CrossRef]

2009 (1)

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

2008 (1)

G. Meister, Y. Zong, and C. R. McClain, “Derivation of the MODIS Aqua point-spread function for ocean color bands,” Proc. SPIE 7081, 70811F (2008).
[CrossRef]

2007 (2)

Y. Zong, S. W. Brown, G. Meister, R. A. Barnes, and K. R. Lykke, “Characterization and correction of stray light in optical instruments,” Proc. SPIE 6744, 67441L (2007).
[CrossRef]

B. Bitlis, P. A. Jansson, and J. P. Allebach, “Parametric point spread function modelling and reduction of stray light effects in digital still cameras,” Proc. SPIE 6498, 64980V (2007).
[CrossRef]

2006 (2)

2004 (2)

J. A. Esposito, A. Wu, and J. Sun, “MODIS reflective solar bands uncertainty analysis,” Proc. SPIE 5542, 448–458 (2004).
[CrossRef]

D. Hong and K. J. Voss, “Effects of point-spread response on calibration and radiometric accuracy of CCD camera,” Appl. Opt. 43, 665–670 (2004).
[CrossRef]

2002 (2)

C. Huang, J. Townshend, S. Liang, S. Kalluri, and R. DeFries, “Impact of sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sens. Environ. 80, 203–212 (2002).
[CrossRef]

C. Huang, J. R. G. Townshend, S. Liang, S. N. V. Kalluri, and R. S. DeFries, “Impact of a sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sensing Environ. 80, 203–212 (2002).
[CrossRef]

2000 (1)

S. Qiu, G. Godden, X. Wang, and B. Guenther, “Satellite-Earth remote sensor scatter effects on Earth scene radiometric accuracy,” Metrologia 37, 411–414 (2000).
[CrossRef]

Allebach, J. P.

B. Bitlis, P. A. Jansson, and J. P. Allebach, “Parametric point spread function modelling and reduction of stray light effects in digital still cameras,” Proc. SPIE 6498, 64980V (2007).
[CrossRef]

Barnes, R. A.

Y. Zong, S. W. Brown, G. Meister, R. A. Barnes, and K. R. Lykke, “Characterization and correction of stray light in optical instruments,” Proc. SPIE 6744, 67441L (2007).
[CrossRef]

Bitlis, B.

B. Bitlis, P. A. Jansson, and J. P. Allebach, “Parametric point spread function modelling and reduction of stray light effects in digital still cameras,” Proc. SPIE 6498, 64980V (2007).
[CrossRef]

Bolton, A. S.

A. S. Bolton and D. J. Schlegel, “Spectro-perfectionism: an algorithmic framework for photon noise-limited extraction of optical fiber spectroscopy,” Publ. Astron. Soc. Pac. 122, 248–257 (2010).
[CrossRef]

Breault, R. P.

P. A. Jansson and R. P. Breault, “Correction color-measurement error caused by stray light in image scanners,” in The Sixth Color Imaging Conference: Color Science, Systems, and Applications (Wiley, 1998), pp. 69–73.

Brown, S. W.

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

Y. Zong, S. W. Brown, G. Meister, R. A. Barnes, and K. R. Lykke, “Characterization and correction of stray light in optical instruments,” Proc. SPIE 6744, 67441L (2007).
[CrossRef]

Y. Zong, S. W. Brown, B. C. Johnson, K. R. Lykke, and Y. Ohno, “Simple spectral stray light correction method for array spectrometers,” Appl. Opt. 45, 1111–1119 (2006).
[CrossRef]

S. W. Brown, G. P. Eppeldauer, and K. R. Lykke, “Facility for spectral irradiance and radiance responsivity using uniform sources,” Appl. Opt. 45, 8218–8237 (2006).
[CrossRef]

Y. Zong, S. W. Brown, K. R. Lykke, and Y. Ohno, “Correction of stray light in spectroradiometers and imaging instruments,” presented at CIE 26th Session, Beijing, China, 4–11 July 2007.

Clark, D. K.

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

DeFries, R.

C. Huang, J. Townshend, S. Liang, S. Kalluri, and R. DeFries, “Impact of sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sens. Environ. 80, 203–212 (2002).
[CrossRef]

DeFries, R. S.

C. Huang, J. R. G. Townshend, S. Liang, S. N. V. Kalluri, and R. S. DeFries, “Impact of a sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sensing Environ. 80, 203–212 (2002).
[CrossRef]

Eppeldauer, G. P.

Esposito, J. A.

J. A. Esposito, A. Wu, and J. Sun, “MODIS reflective solar bands uncertainty analysis,” Proc. SPIE 5542, 448–458 (2004).
[CrossRef]

Feinholz, M. E.

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

Flora, S. J.

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

Godden, G.

S. Qiu, G. Godden, X. Wang, and B. Guenther, “Satellite-Earth remote sensor scatter effects on Earth scene radiometric accuracy,” Metrologia 37, 411–414 (2000).
[CrossRef]

Guenther, B.

S. Qiu, G. Godden, X. Wang, and B. Guenther, “Satellite-Earth remote sensor scatter effects on Earth scene radiometric accuracy,” Metrologia 37, 411–414 (2000).
[CrossRef]

Hong, D.

Huang, C.

C. Huang, J. Townshend, S. Liang, S. Kalluri, and R. DeFries, “Impact of sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sens. Environ. 80, 203–212 (2002).
[CrossRef]

C. Huang, J. R. G. Townshend, S. Liang, S. N. V. Kalluri, and R. S. DeFries, “Impact of a sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sensing Environ. 80, 203–212 (2002).
[CrossRef]

Jansson, P. A.

B. Bitlis, P. A. Jansson, and J. P. Allebach, “Parametric point spread function modelling and reduction of stray light effects in digital still cameras,” Proc. SPIE 6498, 64980V (2007).
[CrossRef]

P. A. Jansson and R. P. Breault, “Correction color-measurement error caused by stray light in image scanners,” in The Sixth Color Imaging Conference: Color Science, Systems, and Applications (Wiley, 1998), pp. 69–73.

Johnson, B. C.

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

Y. Zong, S. W. Brown, B. C. Johnson, K. R. Lykke, and Y. Ohno, “Simple spectral stray light correction method for array spectrometers,” Appl. Opt. 45, 1111–1119 (2006).
[CrossRef]

Kalluri, S.

C. Huang, J. Townshend, S. Liang, S. Kalluri, and R. DeFries, “Impact of sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sens. Environ. 80, 203–212 (2002).
[CrossRef]

Kalluri, S. N. V.

C. Huang, J. R. G. Townshend, S. Liang, S. N. V. Kalluri, and R. S. DeFries, “Impact of a sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sensing Environ. 80, 203–212 (2002).
[CrossRef]

Liang, S.

C. Huang, J. R. G. Townshend, S. Liang, S. N. V. Kalluri, and R. S. DeFries, “Impact of a sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sensing Environ. 80, 203–212 (2002).
[CrossRef]

C. Huang, J. Townshend, S. Liang, S. Kalluri, and R. DeFries, “Impact of sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sens. Environ. 80, 203–212 (2002).
[CrossRef]

Lykke, K. R.

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

Y. Zong, S. W. Brown, G. Meister, R. A. Barnes, and K. R. Lykke, “Characterization and correction of stray light in optical instruments,” Proc. SPIE 6744, 67441L (2007).
[CrossRef]

Y. Zong, S. W. Brown, B. C. Johnson, K. R. Lykke, and Y. Ohno, “Simple spectral stray light correction method for array spectrometers,” Appl. Opt. 45, 1111–1119 (2006).
[CrossRef]

S. W. Brown, G. P. Eppeldauer, and K. R. Lykke, “Facility for spectral irradiance and radiance responsivity using uniform sources,” Appl. Opt. 45, 8218–8237 (2006).
[CrossRef]

Y. Zong, S. W. Brown, K. R. Lykke, and Y. Ohno, “Correction of stray light in spectroradiometers and imaging instruments,” presented at CIE 26th Session, Beijing, China, 4–11 July 2007.

McClain, C. R.

G. Meister and C. R. McClain, “Point-spread function of the ocean color bands of the moderate resolution imaging spectroradiometer on Aqua,” Appl. Opt. 49, 6276–6285 (2010).
[CrossRef]

G. Meister, Y. Zong, and C. R. McClain, “Derivation of the MODIS Aqua point-spread function for ocean color bands,” Proc. SPIE 7081, 70811F (2008).
[CrossRef]

Meister, G.

G. Meister and C. R. McClain, “Point-spread function of the ocean color bands of the moderate resolution imaging spectroradiometer on Aqua,” Appl. Opt. 49, 6276–6285 (2010).
[CrossRef]

G. Meister, Y. Zong, and C. R. McClain, “Derivation of the MODIS Aqua point-spread function for ocean color bands,” Proc. SPIE 7081, 70811F (2008).
[CrossRef]

Y. Zong, S. W. Brown, G. Meister, R. A. Barnes, and K. R. Lykke, “Characterization and correction of stray light in optical instruments,” Proc. SPIE 6744, 67441L (2007).
[CrossRef]

Ohno, Y.

Y. Zong, S. W. Brown, B. C. Johnson, K. R. Lykke, and Y. Ohno, “Simple spectral stray light correction method for array spectrometers,” Appl. Opt. 45, 1111–1119 (2006).
[CrossRef]

Y. Zong, S. W. Brown, K. R. Lykke, and Y. Ohno, “Correction of stray light in spectroradiometers and imaging instruments,” presented at CIE 26th Session, Beijing, China, 4–11 July 2007.

Qiu, S.

S. Qiu, G. Godden, X. Wang, and B. Guenther, “Satellite-Earth remote sensor scatter effects on Earth scene radiometric accuracy,” Metrologia 37, 411–414 (2000).
[CrossRef]

Schlegel, D. J.

A. S. Bolton and D. J. Schlegel, “Spectro-perfectionism: an algorithmic framework for photon noise-limited extraction of optical fiber spectroscopy,” Publ. Astron. Soc. Pac. 122, 248–257 (2010).
[CrossRef]

Sun, J.

J. A. Esposito, A. Wu, and J. Sun, “MODIS reflective solar bands uncertainty analysis,” Proc. SPIE 5542, 448–458 (2004).
[CrossRef]

Townshend, J.

C. Huang, J. Townshend, S. Liang, S. Kalluri, and R. DeFries, “Impact of sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sens. Environ. 80, 203–212 (2002).
[CrossRef]

Townshend, J. R. G.

C. Huang, J. R. G. Townshend, S. Liang, S. N. V. Kalluri, and R. S. DeFries, “Impact of a sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sensing Environ. 80, 203–212 (2002).
[CrossRef]

Voss, K. J.

Wang, X.

S. Qiu, G. Godden, X. Wang, and B. Guenther, “Satellite-Earth remote sensor scatter effects on Earth scene radiometric accuracy,” Metrologia 37, 411–414 (2000).
[CrossRef]

Wu, A.

J. A. Esposito, A. Wu, and J. Sun, “MODIS reflective solar bands uncertainty analysis,” Proc. SPIE 5542, 448–458 (2004).
[CrossRef]

Yarbrough, M. A.

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

Zong, Y.

G. Meister, Y. Zong, and C. R. McClain, “Derivation of the MODIS Aqua point-spread function for ocean color bands,” Proc. SPIE 7081, 70811F (2008).
[CrossRef]

Y. Zong, S. W. Brown, G. Meister, R. A. Barnes, and K. R. Lykke, “Characterization and correction of stray light in optical instruments,” Proc. SPIE 6744, 67441L (2007).
[CrossRef]

Y. Zong, S. W. Brown, B. C. Johnson, K. R. Lykke, and Y. Ohno, “Simple spectral stray light correction method for array spectrometers,” Appl. Opt. 45, 1111–1119 (2006).
[CrossRef]

Y. Zong, S. W. Brown, K. R. Lykke, and Y. Ohno, “Correction of stray light in spectroradiometers and imaging instruments,” presented at CIE 26th Session, Beijing, China, 4–11 July 2007.

Appl. Opt. (4)

J. Atmos. Ocean. Technol. (1)

M. E. Feinholz, S. J. Flora, M. A. Yarbrough, K. R. Lykke, S. W. Brown, B. C. Johnson, and D. K. Clark, “Stray light correction of the Marine Optical System,” J. Atmos. Ocean. Technol. 26, 57–73 (2009).
[CrossRef]

Metrologia (1)

S. Qiu, G. Godden, X. Wang, and B. Guenther, “Satellite-Earth remote sensor scatter effects on Earth scene radiometric accuracy,” Metrologia 37, 411–414 (2000).
[CrossRef]

Proc. SPIE (4)

Y. Zong, S. W. Brown, G. Meister, R. A. Barnes, and K. R. Lykke, “Characterization and correction of stray light in optical instruments,” Proc. SPIE 6744, 67441L (2007).
[CrossRef]

B. Bitlis, P. A. Jansson, and J. P. Allebach, “Parametric point spread function modelling and reduction of stray light effects in digital still cameras,” Proc. SPIE 6498, 64980V (2007).
[CrossRef]

G. Meister, Y. Zong, and C. R. McClain, “Derivation of the MODIS Aqua point-spread function for ocean color bands,” Proc. SPIE 7081, 70811F (2008).
[CrossRef]

J. A. Esposito, A. Wu, and J. Sun, “MODIS reflective solar bands uncertainty analysis,” Proc. SPIE 5542, 448–458 (2004).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

A. S. Bolton and D. J. Schlegel, “Spectro-perfectionism: an algorithmic framework for photon noise-limited extraction of optical fiber spectroscopy,” Publ. Astron. Soc. Pac. 122, 248–257 (2010).
[CrossRef]

Remote Sens. Environ. (1)

C. Huang, J. Townshend, S. Liang, S. Kalluri, and R. DeFries, “Impact of sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sens. Environ. 80, 203–212 (2002).
[CrossRef]

Remote Sensing Environ. (1)

C. Huang, J. R. G. Townshend, S. Liang, S. N. V. Kalluri, and R. S. DeFries, “Impact of a sensor’s point spread function on land cover characterization: assessment and deconvolution,” Remote Sensing Environ. 80, 203–212 (2002).
[CrossRef]

Other (5)

“MOBY @ MLML,” http://moby.mlml.calstate.edu/ .

“BOUSSOLE,” http://www.obs-vlfr.fr/Boussole/html/home/home.php .

P. A. Jansson, ed., Deconvolution of Images and Spectra(Academic, 1997).

Y. Zong, S. W. Brown, K. R. Lykke, and Y. Ohno, “Correction of stray light in spectroradiometers and imaging instruments,” presented at CIE 26th Session, Beijing, China, 4–11 July 2007.

P. A. Jansson and R. P. Breault, “Correction color-measurement error caused by stray light in image scanners,” in The Sixth Color Imaging Conference: Color Science, Systems, and Applications (Wiley, 1998), pp. 69–73.

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Figures (12)

Fig. 1.
Fig. 1.

Andor CCD image showing the four input channels to the spectrograph illuminated by an IR-filtered Xe arc source. The edges of each channel are denoted with solid black lines. The spectral, or dispersion, dimension and the spatial dimension are labeled in the figure.

Fig. 2.
Fig. 2.

(Top) Signal from channel 2 when illuminated by a Xe arc source. (Bottom) The signal from channels 1, 3, and 4, with only channel 2 illuminated by a Xe arc source.

Fig. 3.
Fig. 3.

Ratio of signals from channels 1, 3, and 4 to the signal from channel 2, with channel 2 illuminated by a Xe arc source.

Fig. 4.
Fig. 4.

Subset of the imaged laser lines used to characterize the along-track scattering from channel 2. The dark black horizontal line denotes the transition point between IB, or properly imaged, radiation and OOB, or scattered, radiation. It is set at 1% of the peak value. Note that the vertical axis is scaled in arbitrary units (A.U.).

Fig. 5.
Fig. 5.

Expanded image on the CCD for laser input between 450 nm and 480 nm. The peak for the laser images on the right-hand side of the graph was set to a value of 1. Note the light from second-order diffracted light falling on the CCD between 880 nm and 980 nm.

Fig. 6.
Fig. 6.

Four D3,j submatrices that describe (a) scattering from channel 1 into channel 3 (D3,1), (b) scattering from channel 2 into channel 3 (D3,2), (c) scattering within channel 3 (D3,3) and (d) scattering from channel 4 into channel 3 (D3,4). Each submatrix has 1024×1024 elements (the number of elements in the detector array).

Fig. 7.
Fig. 7.

Full SL distribution function matrix, D. Note that the pixels along the diagonal (within the IB area) are 0 by definition.

Fig. 8.
Fig. 8.

Uncorrected and SL-corrected signals from the spectrograph for a single channel, channel 2, illuminated by radiation from a Xe arc lamp. Solid colored lines are uncorrected signals from all four channels; dashed colored lines are SL-corrected signals from channels 1, 3, and 4.

Fig. 9.
Fig. 9.

Transmittance filter types used with the Xe source, a BG-39 filter, a BG-28 filter, and a PER photopic filter.

Fig. 10.
Fig. 10.

Validation measurements from (a) channel 1 with an unfiltered Xe lamp; (b) channel 2 with a Xe lamp with a photopic PER filter; (c) channel 3 with a Xe lamp with a BG39 filter; and (d) channel 4 with a Xe lamp with a BG28 filter. Data are shown on a logarithmic scale. The black lines are for all four tracks illuminated, with no SL correction; the blue lines are the signals from the channels when only the single channel is illuminated; the red lines are the SL-corrected signal when the single channel is illuminated; and the green lines are the SL-corrected signals when all channels are illuminated.

Fig. 11.
Fig. 11.

Expanded view of validation measurements from (a) channel 1 with an unfiltered Xe lamp; (b) channel 2 with a photopic PER filter; (c) channel 3 with a Xe lamp with a BG39 filter; and (d) channel 4 with a Xe lamp with a BG28 filter shown on a linear scale. The black lines are for all 4 tracks illuminated, with no SL correction; the blue lines are the signals from the channels when only the single channel is illuminated; the red lines are the SL-corrected signal when the single channel is illuminated; and the green lines are the SL-corrected signals when all channels are illuminated.

Fig. 12.
Fig. 12.

Ratio of SL-corrected to uncorrected signals from the spectrograph looking at (a) channel with an unfiltered Xe lamp; (b) channel 2 with a photopic PER filter; (c) channel 3 with a Xe lamp with a BG39 filter; and (d) channel 4 with a Xe lamp with a BG28 filter.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

Ymeas=YIB+YSL.
Y⃗measi=[ymeasi,1ymeasi,2ymeasi,n],Y⃗IBi=[yIBi,1yIBi,2yIBi,n],andY⃗SLi=[ySLi,1ySLi,2ySLi,n].
Y⃗meas=[Y⃗meas1Y⃗meas2Y⃗measm],Y⃗IB=[Y⃗IB1Y⃗IB2Y⃗IBm],andY⃗SL=[Y⃗SL1Y⃗SL2Y⃗SLm].
Y⃗SL=D·Y⃗IB+δ⃗.
YSLi=jDi,jYIBj,
Y⃗SL=[Y⃗SL1Y⃗SL2Y⃗SLm]=[jD1,jY⃗IBjjD2,jY⃗IBjjDm,jY⃗IBj]=[D1,1D1,2D1,3D1,mD2,1D2,2D2,3D2,mDm,1Dm,2Dm,3Dm,m]·[Y⃗IB1Y⃗IB2Y⃗IBm].
D1,1=[d1,11,1d1,21,1··d1,j1,1··d1,n11,1d1,n1,1d2,11,1d2,21,1··d2,j1,1··d2,n11,1d2,n1,1············dx,11,1dx,21,1··dx,j1,1··dx,n11,1dx,n1,1··················dn1,11,1dn1,21,1··dn1,j1,1··dn1,n11,1dn1,n1,1dn,11,1dn,21,1··dn,j1,1··dn,n11,1dn,n1,1].
Y⃗meas=Y⃗IB+Y⃗SL=Y⃗IB+D·Y⃗IB.
Y⃗IB=I·Y⃗IB,
Y⃗meas=[I+D]·Y⃗IB.
Y⃗IB=[I+D]1·Y⃗meas=A1·Ymeas,A=[I+D].
Y⃗IB=A1·Y⃗meas=C·Y⃗meas.

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