Abstract

We derive analytical equations for uncertainties in parameters extracted by nonlinear least-squares fitting of a Gaussian emission function with an unknown continuum background component in the presence of additive white Gaussian noise. The derivation is based on the inversion of the full curvature matrix (equivalent to Fisher information matrix) of the least-squares error, χ2, in a four-variable fitting parameter space. The derived uncertainty formulas (equivalent to Cramer–Rao error bounds) are found to be in good agreement with the numerically computed uncertainties from a large ensemble of simulated measurements. The derived formulas can be used for estimating minimum achievable errors for a given signal-to-noise ratio and for investigating some aspects of measurement setup trade-offs and optimization. While the intended application is Fabry–Perot spectroscopy for wind and temperature measurements in the upper atmosphere, the derivation is generic and applicable to other spectroscopy problems with a Gaussian line shape.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, and O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447-466 (2008).
[CrossRef]

2007

2006

E. J. Mierkiewicz, F. L. Roesler, S. M. Nossal, and R. J. Reynolds, “Geocoronal hydrogen studies using Fabry-Perot interferometers, part 1: instrumentation, observations, and analysis,” J. Atmos. Sol. Terr. Phys. 68, 1520-1552 (2006).
[CrossRef]

2005

D. Pallamraju and S. Chakrabarti, “First ground-based measurements of OI 6300 angstrom daytime aurora over Boston in response to the 30 October 2003 geomagnetic storm,” Geophys. Res. Lett. 32, L03S10 (2005).
[CrossRef]

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083 nm emission in the upper thermosphere,” J. Geophys. Res. Space Phys. 110, A08304 (2005).
[CrossRef]

J. Ireland, “Precision limits to emission-line profile measuring experiments,” Astrophys. J. 620, 1132-1139 (2005).
[CrossRef]

2002

D. Pallamraju, J. Baumgardner, and S. Chakrabarti, “HIRISE a ground-based high-resolution imaging spectrograph using echelle grating for measuring daytime air glow/auroral emissions,” J. Atmos. Sol. Terr. Phys. 64, 1581-1587 (2002).
[CrossRef]

M. Conde, “Deriving wavelength spectra from fringe images from a fixed-gap single-etalon Fabry-Perot spectrometer,” Appl. Opt. 41, 2672-2678 (2002).
[CrossRef] [PubMed]

2001

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

1998

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

S. Chakrabarti, “Ground based spectroscopic studies of sunlit air glow and aurora,” J. Atmos. Sol. Terr. Phys. 60, 1403-1423(1998).
[CrossRef]

1997

S. Nossal, F. L. Roesler, M. M. Coakley, and R. J. Reynolds, “Geocoronal hydrogen balmer-alpha line profiles obtained using Fabry-Perot annular summing spectroscopy: effective temperature results,” J. Geophys. Res. Space Phys. 102, 14541-14553 (1997).
[CrossRef]

1996

1995

1992

D. D. Lenz and T. R. Ayres, “Errors associated with fitting Gaussian profiles to noisy emission-line spectra,” Publ. Astron. Soc. Pac. 104, 1104-1106 (1992).
[CrossRef]

1991

S. C. Solomon, “Optical aeronomy,” Rev. Geophys. 29, 1089-1109 (1991).

1990

G. R. Swenson, S. B. Mende, and S. P. Geller, “Fabry-Perot imaging observations of OH(8-3)--rotational temperatures and gravity-waves,” J. Geophys. Res. Space Phys. 95, 12251-12263 (1990).
[CrossRef]

1984

D. Rees, A. H. Greenaway, R. Gordon, I. McWhirter, P. J. Charleton, and A. Steen, “The Doppler imaging-system--initial observations of the auroral thermosphere,” Planet. Space Sci. 32, 273 (1984).
[CrossRef]

C. A. Tepley, R. G. Burnside, J. W. Meriwether, P. B. Hays, and L. L. Cogger, “Spatial-mapping of the thermospheric neutral wind-field,” Planet. Space Sci. 32, 493-501 (1984).
[CrossRef]

T. L. Killeen and P. B. Hays, “Doppler line-profile analysis for a multichannel fabry-perot interferometer,” Appl. Opt. 23, 612-620 (1984).
[CrossRef] [PubMed]

1983

1982

D. A. Landman, R. Rousseldupre, and G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732-735 (1982).
[CrossRef]

1980

1978

1977

F. E. Barmore, “High-resolution observations of 6300-a oxygen line in day air glow,” Planet. Space Sci. 25, 185-191 (1977).
[CrossRef]

1971

Ayres, T. R.

D. D. Lenz and T. R. Ayres, “Errors associated with fitting Gaussian profiles to noisy emission-line spectra,” Publ. Astron. Soc. Pac. 104, 1104-1106 (1992).
[CrossRef]

Barmore, F. E.

F. E. Barmore, “High-resolution observations of 6300-a oxygen line in day air glow,” Planet. Space Sci. 25, 185-191 (1977).
[CrossRef]

Baumgardner, J.

D. Pallamraju, J. Baumgardner, and S. Chakrabarti, “HIRISE a ground-based high-resolution imaging spectrograph using echelle grating for measuring daytime air glow/auroral emissions,” J. Atmos. Sol. Terr. Phys. 64, 1581-1587 (2002).
[CrossRef]

Baumgardner, J. L.

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3rd ed. (McGraw-Hill, 2003).

Biondi, M. A.

Bishop, J.

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

Burnside, R. G.

C. A. Tepley, R. G. Burnside, J. W. Meriwether, P. B. Hays, and L. L. Cogger, “Spatial-mapping of the thermospheric neutral wind-field,” Planet. Space Sci. 32, 493-501 (1984).
[CrossRef]

Chakrabarti, S.

D. Pallamraju and S. Chakrabarti, “First ground-based measurements of OI 6300 angstrom daytime aurora over Boston in response to the 30 October 2003 geomagnetic storm,” Geophys. Res. Lett. 32, L03S10 (2005).
[CrossRef]

D. Pallamraju, J. Baumgardner, and S. Chakrabarti, “HIRISE a ground-based high-resolution imaging spectrograph using echelle grating for measuring daytime air glow/auroral emissions,” J. Atmos. Sol. Terr. Phys. 64, 1581-1587 (2002).
[CrossRef]

S. Chakrabarti, “Ground based spectroscopic studies of sunlit air glow and aurora,” J. Atmos. Sol. Terr. Phys. 60, 1403-1423(1998).
[CrossRef]

Charleton, P. J.

D. Rees, A. H. Greenaway, R. Gordon, I. McWhirter, P. J. Charleton, and A. Steen, “The Doppler imaging-system--initial observations of the auroral thermosphere,” Planet. Space Sci. 32, 273 (1984).
[CrossRef]

Coakley, M. M.

S. Nossal, F. L. Roesler, M. M. Coakley, and R. J. Reynolds, “Geocoronal hydrogen balmer-alpha line profiles obtained using Fabry-Perot annular summing spectroscopy: effective temperature results,” J. Geophys. Res. Space Phys. 102, 14541-14553 (1997).
[CrossRef]

M. M. Coakley, F. L. Roesler, R. J. Reynolds, and S. Nossal, “Fabry-Perot CCD annular-summing spectroscopy: study and implementation for aeronomy applications,” Appl. Opt. 35, 6479-6493 (1996).
[CrossRef] [PubMed]

Cogger, L. L.

C. A. Tepley, R. G. Burnside, J. W. Meriwether, P. B. Hays, and L. L. Cogger, “Spatial-mapping of the thermospheric neutral wind-field,” Planet. Space Sci. 32, 493-501 (1984).
[CrossRef]

Conde, M.

M. Conde, “Deriving wavelength spectra from fringe images from a fixed-gap single-etalon Fabry-Perot spectrometer,” Appl. Opt. 41, 2672-2678 (2002).
[CrossRef] [PubMed]

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Craven, J. D.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Dereniak, E. L.

Doe, R. A.

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

Faivre, M.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, and O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447-466 (2008).
[CrossRef]

Fesen, C.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, and O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447-466 (2008).
[CrossRef]

Fisher, G.

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in Fortran 77: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1996), Chap. 15.

Frank, L. A.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Friedman, J.

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

Garcia, R.

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Geller, S. P.

G. R. Swenson, S. B. Mende, and S. P. Geller, “Fabry-Perot imaging observations of OH(8-3)--rotational temperatures and gravity-waves,” J. Geophys. Res. Space Phys. 95, 12251-12263 (1990).
[CrossRef]

Gonzalez, S. A.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083 nm emission in the upper thermosphere,” J. Geophys. Res. Space Phys. 110, A08304 (2005).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Gordon, R.

D. Rees, A. H. Greenaway, R. Gordon, I. McWhirter, P. J. Charleton, and A. Steen, “The Doppler imaging-system--initial observations of the auroral thermosphere,” Planet. Space Sci. 32, 273 (1984).
[CrossRef]

Greenaway, A. H.

D. Rees, A. H. Greenaway, R. Gordon, I. McWhirter, P. J. Charleton, and A. Steen, “The Doppler imaging-system--initial observations of the auroral thermosphere,” Planet. Space Sci. 32, 273 (1984).
[CrossRef]

D. Rees and A. H. Greenaway, “Doppler imaging -system--an optical device for measuring vector winds: 1. general-principles,” Appl. Opt. 22, 1078-1083 (1983).
[CrossRef] [PubMed]

Haffner, M.

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

Hagen, N.

Hallinan, T.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Hallinan, T. J.

Hays, P. B.

He, X.

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

Hecht, J. H.

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Hernandez, G.

Hoch, E.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Immel, T.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Ireland, J.

J. Ireland, “Precision limits to emission-line profile measuring experiments,” Astrophys. J. 620, 1132-1139 (2005).
[CrossRef]

Jensen, A. S.

Kamalabadi, F.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083 nm emission in the upper thermosphere,” J. Geophys. Res. Space Phys. 110, A08304 (2005).
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Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

Kerr, R. B.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083 nm emission in the upper thermosphere,” J. Geophys. Res. Space Phys. 110, A08304 (2005).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Killeen, T. L.

Kupinski, M.

Lading, L.

Lancaster, R. S.

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

Landman, D. A.

D. A. Landman, R. Rousseldupre, and G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732-735 (1982).
[CrossRef]

Lappen, M.

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
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D. D. Lenz and T. R. Ayres, “Errors associated with fitting Gaussian profiles to noisy emission-line spectra,” Publ. Astron. Soc. Pac. 104, 1104-1106 (1992).
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McCormack, B.

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

McWhirter, I.

D. Rees, A. H. Greenaway, R. Gordon, I. McWhirter, P. J. Charleton, and A. Steen, “The Doppler imaging-system--initial observations of the auroral thermosphere,” Planet. Space Sci. 32, 273 (1984).
[CrossRef]

Mende, S. B.

G. R. Swenson, S. B. Mende, and S. P. Geller, “Fabry-Perot imaging observations of OH(8-3)--rotational temperatures and gravity-waves,” J. Geophys. Res. Space Phys. 95, 12251-12263 (1990).
[CrossRef]

Meriwether, J.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, and O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447-466 (2008).
[CrossRef]

Meriwether, J. W.

C. A. Tepley, R. G. Burnside, J. W. Meriwether, P. B. Hays, and L. L. Cogger, “Spatial-mapping of the thermospheric neutral wind-field,” Planet. Space Sci. 32, 493-501 (1984).
[CrossRef]

Mierkiewicz, E. J.

E. J. Mierkiewicz, F. L. Roesler, S. M. Nossal, and R. J. Reynolds, “Geocoronal hydrogen studies using Fabry-Perot interferometers, part 1: instrumentation, observations, and analysis,” J. Atmos. Sol. Terr. Phys. 68, 1520-1552 (2006).
[CrossRef]

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

Nossal, S.

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

S. Nossal, F. L. Roesler, M. M. Coakley, and R. J. Reynolds, “Geocoronal hydrogen balmer-alpha line profiles obtained using Fabry-Perot annular summing spectroscopy: effective temperature results,” J. Geophys. Res. Space Phys. 102, 14541-14553 (1997).
[CrossRef]

M. M. Coakley, F. L. Roesler, R. J. Reynolds, and S. Nossal, “Fabry-Perot CCD annular-summing spectroscopy: study and implementation for aeronomy applications,” Appl. Opt. 35, 6479-6493 (1996).
[CrossRef] [PubMed]

Nossal, S. M.

E. J. Mierkiewicz, F. L. Roesler, S. M. Nossal, and R. J. Reynolds, “Geocoronal hydrogen studies using Fabry-Perot interferometers, part 1: instrumentation, observations, and analysis,” J. Atmos. Sol. Terr. Phys. 68, 1520-1552 (2006).
[CrossRef]

Noto, J.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083 nm emission in the upper thermosphere,” J. Geophys. Res. Space Phys. 110, A08304 (2005).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Olson, J.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Pallamraju, D.

D. Pallamraju and S. Chakrabarti, “First ground-based measurements of OI 6300 angstrom daytime aurora over Boston in response to the 30 October 2003 geomagnetic storm,” Geophys. Res. Lett. 32, L03S10 (2005).
[CrossRef]

D. Pallamraju, J. Baumgardner, and S. Chakrabarti, “HIRISE a ground-based high-resolution imaging spectrograph using echelle grating for measuring daytime air glow/auroral emissions,” J. Atmos. Sol. Terr. Phys. 64, 1581-1587 (2002).
[CrossRef]

Percival, J.

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in Fortran 77: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1996), Chap. 15.

Rees, D.

D. Rees, A. H. Greenaway, R. Gordon, I. McWhirter, P. J. Charleton, and A. Steen, “The Doppler imaging-system--initial observations of the auroral thermosphere,” Planet. Space Sci. 32, 273 (1984).
[CrossRef]

D. Rees and A. H. Greenaway, “Doppler imaging -system--an optical device for measuring vector winds: 1. general-principles,” Appl. Opt. 22, 1078-1083 (1983).
[CrossRef] [PubMed]

Reynolds, R. J.

E. J. Mierkiewicz, F. L. Roesler, S. M. Nossal, and R. J. Reynolds, “Geocoronal hydrogen studies using Fabry-Perot interferometers, part 1: instrumentation, observations, and analysis,” J. Atmos. Sol. Terr. Phys. 68, 1520-1552 (2006).
[CrossRef]

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

S. Nossal, F. L. Roesler, M. M. Coakley, and R. J. Reynolds, “Geocoronal hydrogen balmer-alpha line profiles obtained using Fabry-Perot annular summing spectroscopy: effective temperature results,” J. Geophys. Res. Space Phys. 102, 14541-14553 (1997).
[CrossRef]

M. M. Coakley, F. L. Roesler, R. J. Reynolds, and S. Nossal, “Fabry-Perot CCD annular-summing spectroscopy: study and implementation for aeronomy applications,” Appl. Opt. 35, 6479-6493 (1996).
[CrossRef] [PubMed]

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3rd ed. (McGraw-Hill, 2003).

Roble, R. G.

Roesler, F. L.

E. J. Mierkiewicz, F. L. Roesler, S. M. Nossal, and R. J. Reynolds, “Geocoronal hydrogen studies using Fabry-Perot interferometers, part 1: instrumentation, observations, and analysis,” J. Atmos. Sol. Terr. Phys. 68, 1520-1552 (2006).
[CrossRef]

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

S. Nossal, F. L. Roesler, M. M. Coakley, and R. J. Reynolds, “Geocoronal hydrogen balmer-alpha line profiles obtained using Fabry-Perot annular summing spectroscopy: effective temperature results,” J. Geophys. Res. Space Phys. 102, 14541-14553 (1997).
[CrossRef]

M. M. Coakley, F. L. Roesler, R. J. Reynolds, and S. Nossal, “Fabry-Perot CCD annular-summing spectroscopy: study and implementation for aeronomy applications,” Appl. Opt. 35, 6479-6493 (1996).
[CrossRef] [PubMed]

Rousseldupre, R.

D. A. Landman, R. Rousseldupre, and G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732-735 (1982).
[CrossRef]

Rudy, J.

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Shea, E. M.

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Sherwood, P.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, and O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447-466 (2008).
[CrossRef]

Sigwarth, J.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Sipler, D. P.

Sivjee, G. G.

Smith, R. W.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
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S. C. Solomon, “Optical aeronomy,” Rev. Geophys. 29, 1089-1109 (1991).

Steen, A.

D. Rees, A. H. Greenaway, R. Gordon, I. McWhirter, P. J. Charleton, and A. Steen, “The Doppler imaging-system--initial observations of the auroral thermosphere,” Planet. Space Sci. 32, 273 (1984).
[CrossRef]

Stenbaek-Nielsen, H.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Sulzer, M. P.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083 nm emission in the upper thermosphere,” J. Geophys. Res. Space Phys. 110, A08304 (2005).
[CrossRef]

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

Sun, W.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

Swenson, G. R.

G. R. Swenson, S. B. Mende, and S. P. Geller, “Fabry-Perot imaging observations of OH(8-3)--rotational temperatures and gravity-waves,” J. Geophys. Res. Space Phys. 95, 12251-12263 (1990).
[CrossRef]

G. G. Sivjee, T. J. Hallinan, and G. R. Swenson, “Fabry-Perot interferometer imaging system for thermospheric temperature and wind measurements,” Appl. Opt. 19, 2206-2209 (1980).
[CrossRef] [PubMed]

Tanigawa, G.

D. A. Landman, R. Rousseldupre, and G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732-735 (1982).
[CrossRef]

Tepley, C. A.

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Periodic variations of geocoronal balmer-alpha brightness due to solar-driven exospheric abundance variations,” J. Geophys. Res. Space Phys. 106, 28797-28817 (2001).
[CrossRef]

R. B. Kerr, R. Garcia, X. He, J. Noto, R. S. Lancaster, C. A. Tepley, S. A. Gonzalez, J. Friedman, R. A. Doe, M. Lappen, and B. McCormack, “Secular variability of the geocoronal balmer-alpha brightness: magnetic activity and possible human influences,” J. Geophys. Res. Space Phys. 106, 28819-28829 (2001).
[CrossRef]

C. A. Tepley, R. G. Burnside, J. W. Meriwether, P. B. Hays, and L. L. Cogger, “Spatial-mapping of the thermospheric neutral wind-field,” Planet. Space Sci. 32, 493-501 (1984).
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Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in Fortran 77: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1996), Chap. 15.

Tufte, S.

S. Nossal, F. L. Roesler, J. Bishop, R. J. Reynolds, M. Haffner, S. Tufte, J. Percival, and E. J. Mierkiewicz, “Geocoronal h alpha intensity measurements using the wisconsin h alpha mapper Fabry-Perot facility,” J. Geophys. Res. Space Physics 106, 5605-5615 (2001).
[CrossRef]

Veliz, O.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, and O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447-466 (2008).
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Vitterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vitterling, and B. P. Flannery, Numerical Recipes in Fortran 77: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1996), Chap. 15.

Waldrop, L. S.

L. S. Waldrop, R. B. Kerr, S. A. Gonzalez, M. P. Sulzer, J. Noto, and F. Kamalabadi, “Generation of metastable helium and the 1083 nm emission in the upper thermosphere,” J. Geophys. Res. Space Phys. 110, A08304 (2005).
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J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
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Zipf, M. E.

Ann. Geophys.

J. Meriwether, M. Faivre, C. Fesen, P. Sherwood, and O. Veliz, “New results on equatorial thermospheric winds and the midnight temperature maximum,” Ann. Geophys. 26, 447-466 (2008).
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G. G. Sivjee, T. J. Hallinan, and G. R. Swenson, “Fabry-Perot interferometer imaging system for thermospheric temperature and wind measurements,” Appl. Opt. 19, 2206-2209 (1980).
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L. Lading and A. S. Jensen, “Estimating the spectral width of a narrowband optical signal,” Appl. Opt. 19, 2750-2756 (1980).
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D. Rees and A. H. Greenaway, “Doppler imaging -system--an optical device for measuring vector winds: 1. general-principles,” Appl. Opt. 22, 1078-1083 (1983).
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T. L. Killeen and P. B. Hays, “Doppler line-profile analysis for a multichannel fabry-perot interferometer,” Appl. Opt. 23, 612-620 (1984).
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M. A. Biondi, D. P. Sipler, M. E. Zipf, and J. L. Baumgardner, “All-sky doppler interferometer for thermospheric dynamics studies,” Appl. Opt. 34, 1646-1654 (1995).
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M. M. Coakley, F. L. Roesler, R. J. Reynolds, and S. Nossal, “Fabry-Perot CCD annular-summing spectroscopy: study and implementation for aeronomy applications,” Appl. Opt. 35, 6479-6493 (1996).
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M. Conde, “Deriving wavelength spectra from fringe images from a fixed-gap single-etalon Fabry-Perot spectrometer,” Appl. Opt. 41, 2672-2678 (2002).
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Astrophys. J.

J. Ireland, “Precision limits to emission-line profile measuring experiments,” Astrophys. J. 620, 1132-1139 (2005).
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D. A. Landman, R. Rousseldupre, and G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732-735 (1982).
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Geophys. Res. Lett.

D. Pallamraju and S. Chakrabarti, “First ground-based measurements of OI 6300 angstrom daytime aurora over Boston in response to the 30 October 2003 geomagnetic storm,” Geophys. Res. Lett. 32, L03S10 (2005).
[CrossRef]

J. Atmos. Sol. Terr. Phys.

E. J. Mierkiewicz, F. L. Roesler, S. M. Nossal, and R. J. Reynolds, “Geocoronal hydrogen studies using Fabry-Perot interferometers, part 1: instrumentation, observations, and analysis,” J. Atmos. Sol. Terr. Phys. 68, 1520-1552 (2006).
[CrossRef]

D. Pallamraju, J. Baumgardner, and S. Chakrabarti, “HIRISE a ground-based high-resolution imaging spectrograph using echelle grating for measuring daytime air glow/auroral emissions,” J. Atmos. Sol. Terr. Phys. 64, 1581-1587 (2002).
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S. Chakrabarti, “Ground based spectroscopic studies of sunlit air glow and aurora,” J. Atmos. Sol. Terr. Phys. 60, 1403-1423(1998).
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J. Geophys. Res. Space Phys.

M. Conde, J. D. Craven, T. Immel, E. Hoch, H. Stenbaek-Nielsen, T. Hallinan, R. W. Smith, J. Olson, W. Sun, L. A. Frank, and J. Sigwarth, “Assimilated observations of thermospheric winds, the aurora, and ionospheric currents over Alaska,” J. Geophys. Res. Space Phys. 106, 10493-10508 (2001).
[CrossRef]

J. Noto, R. B. Kerr, E. M. Shea, L. S. Waldrop, G. Fisher, J. Rudy, J. H. Hecht, S. A. Gonzalez, M. P. Sulzer, and R. Garcia, “Evidence for recombination as a significant source of metastable helium,” J. Geophys. Res. Space Phys. 103, 11595-11603 (1998).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Least-squares error minimization requires that the sample errors weighted by the functions in the figure sum up to zero. The solid curve minimizes the error due to p 1 ; dashed, p 2 ; dash-dotted, p 3 ; dotted, p 4 .

Fig. 2
Fig. 2

Errors for each of the fitting parameters as a function of p 4 , the width of the line. The open circles represent the mean “expected errors” as calculated by the fitting algorithm. The crosses represent the “true error” calculated by taking the standard deviation of the results obtained from 400 Monte Carlo trials. The solid line represents the theoretical errors derived based on the incorrect assumption that the parameter errors are independent. Each of the four sets of curves represent a different noise level. Only the formula for p 3 accurately predicts the error.

Fig. 3
Fig. 3

Graphic representation of the line-to-range quotient, Q. The vertical lines show the spectral range at which the derived formulation of error breaks down completely. The spectral range of the measurement should be greater.

Fig. 4
Fig. 4

Errors for each of the fitting parameters as a function of p 4 , the width of the line. The open circles represent the mean “expected errors” as calculated by the fitting algorithm. The crosses represent the “true error” calculated by taking the standard deviation of the results obtained from 400 Monte Carlo trials. The solid line represents the theoretical errors derived based on the full curvature matrix. Each of the four sets of curves represent a different noise level. The results agree very well.

Equations (44)

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γ = γ ( p ¯ , λ ) = p 1 + p 2 e - ( λ - p 3 ) 2 p 4 2 ,
γ i = γ ( p ¯ , λ i ) + n i ,
χ 2 ( p ¯ ) i χ i 2 ( p ¯ ) i R i 2 / σ i 2 i ( γ ( p ¯ , λ i ) - γ i ) 2 / σ i 2 ,
p ¯ ML = arg min p ¯ χ 2 ( p ¯ ) .
δ δ p k χ 2 ( p ¯ ) = δ δ p k i R i 2 σ i 2 = i δ γ δ p k R i σ i 2 = 0.
δ γ ( p ¯ , λ i ) δ p 1 = 1 ,
δ γ ( p ¯ , λ i ) δ p 2 = e - ( λ i - p 3 ) 2 p 4 2 ,
δ γ ( p ¯ , λ i ) δ p 3 = 2 p 2 ( λ i - p 3 ) p 4 2 e - ( λ i - p 3 ) 2 p 4 2 ,
δ γ ( p ¯ , λ i ) δ p 4 = 2 p 2 ( λ i - p 3 ) 2 p 4 3 e - ( λ i - p 3 ) 2 p 4 2 .
i R i σ i 2 = 0 ,
i e - ( λ i - p 3 ) 2 p 4 2 R i σ i 2 = 0 ,
i ( λ i - p 3 ) e - ( λ i - p 3 ) 2 p 4 2 R i σ i 2 = 0 ,
i ( λ i - p 3 ) 2 e - ( λ i - p 3 ) 2 p 4 2 R i σ i 2 = 0.
C ¯ = C - 1 = ( 1 2 H ) - 1 ,
C i j = 1 2 H i j = 1 2 δ 2 χ 2 δ p i δ p j = i ( δ γ δ p i δ γ δ p j + δ 2 γ δ p i δ p j R i ) σ i - 2 .
σ p k = ( 1 2 δ 2 χ 2 δ p k 2 ) - 1 / 2 = ( 1 2 i δ 2 δ p k 2 R i 2 σ i 2 ) - 1 / 2 ( i ( δ γ δ p k ) 2 σ i - 2 ) - 1 / 2 .
σ p 1 = σ ( i ( δ γ δ p 1 ) 2 ) - 1 / 2 = σ ( i 1 ) - 1 / 2 = σ N .
0 x m e - a x 2 d x = ( 2 a m + 1 2 ) - 1 Γ ( m + 1 2 ) ,
Γ ( n / 2 ) = π ( n - 2 ) !! 2 ( n - 1 ) / 2 = π , π / 2 , 3 π / 2 , for     n = 1 , 3 , 5.
σ p 2 = σ ( i ( δ γ δ p 2 ) 2 ) - 1 / 2 σ ( e - 2 ( λ - p 3 ) 2 / p 4 2 ( Δ λ ) - 1 d λ ) - 1 / 2 = σ ( ( Δ λ ) - 1 ( 2 / p 4 2 ) - 1 / 2 Γ ( 1 / 2 ) ) - 1 / 2 = σ ( p 4 Δ λ π 2 ) - 1 / 2 .
σ p 3 = σ ( i ( δ γ δ p 3 ) 2 ) - 1 / 2 = σ ( i ( 2 p 2 ( λ i - p 3 ) p 4 2 e - ( λ i - p 3 ) 2 p 4 2 ) 2 ) - 1 / 2 σ p 4 2 ( 2 p 2 ) - 1 ( ( Δ λ ) - 1 ( 2 p 4 - 2 ) - 3 / 2 Γ ( 3 / 2 ) ) - 1 / 2 = σ ( p 2 2 π p 4 Δ λ 2 ) - 1 / 2 .
σ p 4 = σ ( i ( δ γ δ p 3 ) 2 ) - 1 / 2 = σ ( 1 2 i 8 p 2 2 ( λ i - p 3 ) 4 p 4 6 e - 2 ( λ i - p 3 ) 2 p 4 2 ) - 1 / 2 σ p 4 ( 2 p 2 ) - 1 ( p 4 ( Δ λ ) - 1 ( 2 ) - 5 / 2 Γ ( 5 / 2 ) ) - 1 / 2 = σ ( 3 π p 2 2 4 2 p 4 Δ λ ) - 1 / 2 .
C 12 = σ - 2 i δ γ δ p 1 δ γ δ p 2 = σ - 2 i e - ( λ i - p 3 ) 2 p 4 2 2 p 4 π σ 2 Δ λ ,
C 13 = σ - 2 i δ γ δ p 1 δ γ δ p 3 = σ - 2 i 2 p 2 ( λ i - p 3 ) p 4 2 e - 2 ( λ i - p 3 ) p 4 2 0 ,
C 14 = σ - 2 i δ γ δ p 1 δ γ δ p 4 = σ - 2 i p 2 2 ( λ i - p 3 ) 2 p 4 3 e - ( λ i - p 3 ) 2 p 4 2 2 p 2 π σ 2 Δ λ ,
C 23 = σ - 2 i δ γ δ p 2 δ γ δ p 3 = σ - 2 i p 2 2 ( λ i - p 3 ) p 4 2 e - ( λ i - p 3 ) 2 p 4 2 0 ,
C 24 = σ - 2 i δ γ δ p 2 δ γ δ p 4 = σ - 2 i 2 p 2 ( λ i - p 3 ) 2 p 4 3 e - 2 ( λ i - p 3 ) 2 p 4 3 p 2 π 2 σ 2 Δ λ ,
C 34 = σ - 2 i δ γ δ p 3 δ γ δ p 4 = σ - 2 i p 2 2 4 ( λ i - p 3 ) 3 p 4 5 e - 2 ( λ i - p 3 ) 2 p 4 2 0.
C = 1 σ 2 ( N p 4 π Δ λ 0 p 2 π Δ λ p 4 π Δ λ p 4 π 2 Δ λ 0 p 2 π 2 2 Δ λ 0 0 p 2 2 π p 4 2 Δ λ 0 p 2 π Δ λ p 2 π 2 2 Δ λ 0 3 p 2 2 π 4 2 p 4 Δ λ ) .
C ¯ = σ 2 ( 2 Δ λ - 3 p 4 2 π + 2 N Δ λ Δ λ 3 p 4 π - 2 N Δ λ 0 2 p 4 Δ λ 3 p 2 p 4 π - 2 N p 2 Δ λ Δ λ 3 p 4 π - 2 N Δ λ Δ λ ( 4 p 4 2 π - 3 N Δ λ ) p 4 ( 3 p 4 π - N 2 π Δ λ ) 0 2 Δ λ ( - 2 p 4 2 π + N Δ λ ) p 2 ( 3 p 4 π - N 2 π Δ λ ) 0 0 p 4 2 π Δ λ p 2 2 0 2 p 4 Δ λ 3 p 2 p 4 π - 2 N p 2 Δ λ 2 Δ λ ( - 2 p 4 2 π + N Δ λ ) p 2 ( 3 p 4 π - N 2 π Δ λ ) 0 4 p 4 Δ λ ( p 4 2 π - N Δ λ ) p 2 2 ( 3 p 4 π - N 2 π Δ λ ) ) .
σ p 1 = σ [ ( 1 N ) 1 1 - Q ] 1 2 ,
σ p 2 = σ [ 3 2 ( Δ λ 2 p 4 π ) 1 - 8 9 Q 1 - Q ] 1 2 ,
σ p 3 = σ [ p 4 Δ λ 2 p 2 2 π ] 1 2 ,
σ p 4 = σ [ 3 2 ( 4 p 4 Δ λ 2 3 p 2 2 π ) 1 - 2 3 Q 1 - Q ] 1 2 .
Q 3 p 4 π N Δ λ 2
δ 2 γ δ p 1 2 = δ 2 γ δ p 1 δ p 2 = δ 2 γ δ p 1 δ p 3 = δ 2 γ δ p 1 δ p 4 = δ 2 γ δ p 2 2 = 0 ,
δ 2 γ δ p 2 δ p 3 = 2 ( λ i - p 3 ) p 4 2 e - ( λ i - p 3 ) 2 p 4 2 ,
δ 2 γ δ p 2 δ p 4 = 2 ( λ i - p 3 ) 2 p 4 3 e - ( λ i - p 3 ) 2 p 4 2 ,
δ 2 γ δ p 3 2 = 4 p 2 ( λ i - p 3 ) 2 p 4 4 e - ( λ i - p 3 ) 2 p 4 2 - 2 p 2 p 4 2 e - ( λ i - p 3 ) 2 p 4 2 ,
δ 2 γ δ p 3 δ p 4 = 4 p 2 ( λ i - p 3 ) 3 p 4 5 e - ( λ i - p 3 ) 2 p 4 2 - 4 p 2 ( λ i - p 3 ) 2 p 4 3 e - ( λ i - p 3 ) 2 p 4 2 ,
δ 2 γ δ p 4 2 = 4 p 2 ( λ i - p 3 ) 4 p 4 6 e - ( λ i - p 3 ) 2 p 4 2 - 6 p 2 ( λ i - p 3 ) 2 p 4 4 e - ( λ i - p 3 ) 2 p 4 2 .
i f ( p ¯ ) ( λ i - p 3 ) n e - ( λ i - p 3 ) 2 p 4 2 R i σ i - 2 = 0 , for     n = 0 , 1 , 2.
C 34 = C 34 + i 4 p 2 ( λ i - p 3 ) 3 p 4 5 e - ( λ i - p 3 ) 2 p 4 2 R i σ i - 2 ,
C 44 = C 44 + i 4 p 2 ( λ i - p 3 ) 4 p 4 6 e - ( λ i - p 3 ) 2 p 4 2 R i σ i - 2 .

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