Abstract

High-resolution absorption spectroscopy is a promising method for non-invasive process monitoring, but the computational effort required to evaluate the data can be prohibitive in high-speed, real-time applications. This study presents a fast method to estimate absorbance spectra from transmitted intensity signals. We employ Bayesian statistics to combine a measurement model with prior information about the shape of the baseline intensity and absorbance spectrum. The resulting linear least-squares problem shifts most of the computational effort to a preparation step, thereby facilitating quick processing and low latency for any number of measurements. The method is demonstrated on simulated tunable diode laser absorption spectroscopy data with additive noise and a fluctuating fringe. Results were highly accurate and the method was computationally efficient, having a processing time of only 2 ms per spectrum.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (3)

J. Emmert, N. G. Blume, A. Dreizler, and S. Wagner, “Data analysis and uncertainty estimation in supercontinuum laser absorption spectroscopy,” Sci. Rep. 8(1), 10312 (2018).
[Crossref]

S. Bürkle, A. Dreizler, V. Ebert, and S. Wagner, “Experimental comparison of a 2d laminar diffusion flame under oxy-fuel and air atmosphere,” Fuel 212, 302–308 (2018).
[Crossref]

C. Wei, D. I. Pineda, C. S. Goldenstein, and R. M. Spearrin, “Tomographic laser absorption imaging of combustion species and temperature in the mid-wave infrared,” Opt. Express 26(16), 20944–20951 (2018).
[Crossref]

2017 (1)

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

2016 (1)

N. G. Blume, V. Ebert, A. Dreizler, and S. Wagner, “Broadband fitting approach for the application of supercontinuum broadband laser absorption spectroscopy to combustion environments,” Meas. Sci. Technol. 27(1), 015501 (2016).
[Crossref]

2015 (1)

2014 (2)

A. Hartmann, R. Strzoda, R. Schrobenhauser, and R. Weigel, “Ultra-compact tdlas humidity measurement cell with advanced signal processing,” Appl. Phys. B 115(2), 263–268 (2014).
[Crossref]

R. K. Hanson and D. F. Davidson, “Recent advances in laser absorption and shock tube methods for studies of combustion chemistry,” Prog. Energy Combust. Sci. 44, 103–114 (2014).
[Crossref]

2013 (1)

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

2011 (2)

P. Werle, “Accuracy and precision of laser spectrometers for trace gas sensing in the presence of optical fringes and atmospheric turbulence,” Appl. Phys. B 102(2), 313–329 (2011).
[Crossref]

U. von Toussaint, “Bayesian inference in physics,” Rev. Mod. Phys. 83(3), 943–999 (2011).
[Crossref]

2010 (1)

B. Lins, P. Zinn, R. Engelbrecht, and B. Schmauss, “Simulation-based comparison of noise effects in wavelength modulation spectroscopy and direct absorption tdlas,” Appl. Phys. B 100(2), 367–376 (2010).
[Crossref]

2009 (1)

D. Masiyano, J. Hodgkinson, S. Schilt, and R. P. Tatam, “Self-mixing interference effects in tunable diode laser absorption spectroscopy,” Appl. Phys. B 96(4), 863–874 (2009).
[Crossref]

2007 (2)

X. Liu, J. B. Jeffries, and R. K. Hanson, “Measurement of non-uniform temperature distributions using line-of-sight absorption spectroscopy,” AIAA J. 45(2), 411–419 (2007).
[Crossref]

L. A. Kranendonk, A. W. Caswell, and S. T. Sanders, “Robust method for calculating temperature, pressure, and absorber mole fraction from broadband spectra,” Appl. Opt. 46(19), 4117–4124 (2007).
[Crossref]

2005 (1)

2004 (1)

P. Paci, Y. Zvinevich, S. Tanimura, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson, and B. McManus, “Spatially resolved gas phase composition measurements in supersonic flows using tunable diode laser absorption spectroscopy,” J. Chem. Phys. 121(20), 9964–9970 (2004).
[Crossref]

1999 (1)

Chou Baer Hanson, “Spectral intensity and lineshape measurements in the first overtone band of hf using tunable diode lasers,” J. Mol. Spectrosc. 195(1), 123–131 (1999).
[Crossref]

An, X.

Auwera, J. V.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Baer Hanson, Chou

Chou Baer Hanson, “Spectral intensity and lineshape measurements in the first overtone band of hf using tunable diode lasers,” J. Mol. Spectrosc. 195(1), 123–131 (1999).
[Crossref]

Barbe, A.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Bernath, P. F.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Bertsekas, D. P.

D. P. Bertsekas and J. N. Tsitsiklis, Introduction to probability (Athena Scientific, Belmont, Mass., 2002).

Birk, M.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Blume, N. G.

J. Emmert, N. G. Blume, A. Dreizler, and S. Wagner, “Data analysis and uncertainty estimation in supercontinuum laser absorption spectroscopy,” Sci. Rep. 8(1), 10312 (2018).
[Crossref]

N. G. Blume, V. Ebert, A. Dreizler, and S. Wagner, “Broadband fitting approach for the application of supercontinuum broadband laser absorption spectroscopy to combustion environments,” Meas. Sci. Technol. 27(1), 015501 (2016).
[Crossref]

Boudon, V.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Brittelle, M. S.

Bürkle, S.

S. Bürkle, A. Dreizler, V. Ebert, and S. Wagner, “Experimental comparison of a 2d laminar diffusion flame under oxy-fuel and air atmosphere,” Fuel 212, 302–308 (2018).
[Crossref]

Campargue, A.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Carter, C. D.

Caswell, A. W.

Chance, K. V.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Cole, R. K.

R. K. Cole, A. S. Makowiecki, N. Hoghooghi, and G. B. Rieker, Baseline-free quantitative absorption spectroscopy based on cepstral analysis, arXiv preprint arXiv:1906.11349 (2019).

Császár, A. G.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Daun, K. J.

S. J. Grauer, J. Emmert, A. M. Steinberg, S. Wagner, and K. J. Daun, Hyperspectral absorption tomography with a lineshape prior., in 11th US National Combustion Meeting, Pasadena, CA, Mar. 24-27, 2019 (2019).

Davidson, D. F.

R. K. Hanson and D. F. Davidson, “Recent advances in laser absorption and shock tube methods for studies of combustion chemistry,” Prog. Energy Combust. Sci. 44, 103–114 (2014).
[Crossref]

Demtröder, W.

W. Demtröder, Experimentalphysik 2 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2006).

Devi, V. M.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Dodge, Y.

Y. Dodge, Variance–covariance matrix, in The Concise Encyclopedia of Statistics, (Springer New York, New York, NY, 2008), pp. 559–561

Dreizler, A.

J. Emmert, N. G. Blume, A. Dreizler, and S. Wagner, “Data analysis and uncertainty estimation in supercontinuum laser absorption spectroscopy,” Sci. Rep. 8(1), 10312 (2018).
[Crossref]

S. Bürkle, A. Dreizler, V. Ebert, and S. Wagner, “Experimental comparison of a 2d laminar diffusion flame under oxy-fuel and air atmosphere,” Fuel 212, 302–308 (2018).
[Crossref]

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I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
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Spearrin, R. M.

Starikova, E.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Steinberg, A. M.

S. J. Grauer, J. Emmert, A. M. Steinberg, S. Wagner, and K. J. Daun, Hyperspectral absorption tomography with a lineshape prior., in 11th US National Combustion Meeting, Pasadena, CA, Mar. 24-27, 2019 (2019).

Strand, C.

C. Strand and R. Hanson, Thermometry and velocimetry in supersonic flows via scanned wavelength-modulation absorption spectroscopy, in 47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 493, (American Institute of Aeronautics and Astronautics, Reston, Virigina, 2011).

Strzoda, R.

A. Hartmann, R. Strzoda, R. Schrobenhauser, and R. Weigel, “Ultra-compact tdlas humidity measurement cell with advanced signal processing,” Appl. Phys. B 115(2), 263–268 (2014).
[Crossref]

Sung, K.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Tan, Y.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Tanimura, S.

P. Paci, Y. Zvinevich, S. Tanimura, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson, and B. McManus, “Spatially resolved gas phase composition measurements in supersonic flows using tunable diode laser absorption spectroscopy,” J. Chem. Phys. 121(20), 9964–9970 (2004).
[Crossref]

Tashkun, S. A.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Tatam, R. P.

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

D. Masiyano, J. Hodgkinson, S. Schilt, and R. P. Tatam, “Self-mixing interference effects in tunable diode laser absorption spectroscopy,” Appl. Phys. B 96(4), 863–874 (2009).
[Crossref]

Tennyson, J.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Toon, G. C.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Tran, H.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Tsitsiklis, J. N.

D. P. Bertsekas and J. N. Tsitsiklis, Introduction to probability (Athena Scientific, Belmont, Mass., 2002).

Tyuterev, V. G.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

von Toussaint, U.

U. von Toussaint, “Bayesian inference in physics,” Rev. Mod. Phys. 83(3), 943–999 (2011).
[Crossref]

Wagner, G.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Wagner, S.

J. Emmert, N. G. Blume, A. Dreizler, and S. Wagner, “Data analysis and uncertainty estimation in supercontinuum laser absorption spectroscopy,” Sci. Rep. 8(1), 10312 (2018).
[Crossref]

S. Bürkle, A. Dreizler, V. Ebert, and S. Wagner, “Experimental comparison of a 2d laminar diffusion flame under oxy-fuel and air atmosphere,” Fuel 212, 302–308 (2018).
[Crossref]

N. G. Blume, V. Ebert, A. Dreizler, and S. Wagner, “Broadband fitting approach for the application of supercontinuum broadband laser absorption spectroscopy to combustion environments,” Meas. Sci. Technol. 27(1), 015501 (2016).
[Crossref]

S. J. Grauer, J. Emmert, A. M. Steinberg, S. Wagner, and K. J. Daun, Hyperspectral absorption tomography with a lineshape prior., in 11th US National Combustion Meeting, Pasadena, CA, Mar. 24-27, 2019 (2019).

Wcislo, P.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Wei, C.

Weigel, R.

A. Hartmann, R. Strzoda, R. Schrobenhauser, and R. Weigel, “Ultra-compact tdlas humidity measurement cell with advanced signal processing,” Appl. Phys. B 115(2), 263–268 (2014).
[Crossref]

Werle, P.

P. Werle, “Accuracy and precision of laser spectrometers for trace gas sensing in the presence of optical fringes and atmospheric turbulence,” Appl. Phys. B 102(2), 313–329 (2011).
[Crossref]

Wilzewski, J.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Wyslouzil, B. E.

P. Paci, Y. Zvinevich, S. Tanimura, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson, and B. McManus, “Spatially resolved gas phase composition measurements in supersonic flows using tunable diode laser absorption spectroscopy,” J. Chem. Phys. 121(20), 9964–9970 (2004).
[Crossref]

Yu, S.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Zahniser, M.

P. Paci, Y. Zvinevich, S. Tanimura, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson, and B. McManus, “Spatially resolved gas phase composition measurements in supersonic flows using tunable diode laser absorption spectroscopy,” J. Chem. Phys. 121(20), 9964–9970 (2004).
[Crossref]

Zak, E. J.

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Zinn, P.

B. Lins, P. Zinn, R. Engelbrecht, and B. Schmauss, “Simulation-based comparison of noise effects in wavelength modulation spectroscopy and direct absorption tdlas,” Appl. Phys. B 100(2), 367–376 (2010).
[Crossref]

Zvinevich, Y.

P. Paci, Y. Zvinevich, S. Tanimura, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson, and B. McManus, “Spatially resolved gas phase composition measurements in supersonic flows using tunable diode laser absorption spectroscopy,” J. Chem. Phys. 121(20), 9964–9970 (2004).
[Crossref]

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X. Liu, J. B. Jeffries, and R. K. Hanson, “Measurement of non-uniform temperature distributions using line-of-sight absorption spectroscopy,” AIAA J. 45(2), 411–419 (2007).
[Crossref]

Appl. Opt. (3)

Appl. Phys. B (4)

D. Masiyano, J. Hodgkinson, S. Schilt, and R. P. Tatam, “Self-mixing interference effects in tunable diode laser absorption spectroscopy,” Appl. Phys. B 96(4), 863–874 (2009).
[Crossref]

P. Werle, “Accuracy and precision of laser spectrometers for trace gas sensing in the presence of optical fringes and atmospheric turbulence,” Appl. Phys. B 102(2), 313–329 (2011).
[Crossref]

A. Hartmann, R. Strzoda, R. Schrobenhauser, and R. Weigel, “Ultra-compact tdlas humidity measurement cell with advanced signal processing,” Appl. Phys. B 115(2), 263–268 (2014).
[Crossref]

B. Lins, P. Zinn, R. Engelbrecht, and B. Schmauss, “Simulation-based comparison of noise effects in wavelength modulation spectroscopy and direct absorption tdlas,” Appl. Phys. B 100(2), 367–376 (2010).
[Crossref]

Fuel (1)

S. Bürkle, A. Dreizler, V. Ebert, and S. Wagner, “Experimental comparison of a 2d laminar diffusion flame under oxy-fuel and air atmosphere,” Fuel 212, 302–308 (2018).
[Crossref]

J. Chem. Phys. (1)

P. Paci, Y. Zvinevich, S. Tanimura, B. E. Wyslouzil, M. Zahniser, J. Shorter, D. Nelson, and B. McManus, “Spatially resolved gas phase composition measurements in supersonic flows using tunable diode laser absorption spectroscopy,” J. Chem. Phys. 121(20), 9964–9970 (2004).
[Crossref]

J. Mol. Spectrosc. (1)

Chou Baer Hanson, “Spectral intensity and lineshape measurements in the first overtone band of hf using tunable diode lasers,” J. Mol. Spectrosc. 195(1), 123–131 (1999).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

I. E. Gordon, L. S. Rothman, C. Hill, R. V. Kochanov, Y. Tan, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, B. J. Drouin, J.-M. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, V. I. Perevalov, A. Perrin, K. P. Shine, M.-A. Smith, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, A. Barbe, A. G. Császár, V. M. Devi, T. Furtenbacher, J. J. Harrison, J.-M. Hartmann, A. Jolly, T. J. Johnson, T. Karman, I. Kleiner, A. A. Kyuberis, J. Loos, O. M. Lyulin, S. T. Massie, S. N. Mikhailenko, N. Moazzen-Ahmadi, H. Muller, O. V. Naumenko, A. V. Nikitin, O. L. Polyansky, M. Rey, M. Rotger, S. W. Sharpe, K. Sung, E. Starikova, S. A. Tashkun, J. V. Auwera, G. Wagner, J. Wilzewski, P. Wcisło, S. Yu, and E. J. Zak, “The hitran2016 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer 203, 3–69 (2017).
[Crossref]

Meas. Sci. Technol. (2)

N. G. Blume, V. Ebert, A. Dreizler, and S. Wagner, “Broadband fitting approach for the application of supercontinuum broadband laser absorption spectroscopy to combustion environments,” Meas. Sci. Technol. 27(1), 015501 (2016).
[Crossref]

J. Hodgkinson and R. P. Tatam, “Optical gas sensing: a review,” Meas. Sci. Technol. 24(1), 012004 (2013).
[Crossref]

Opt. Express (1)

Prog. Energy Combust. Sci. (1)

R. K. Hanson and D. F. Davidson, “Recent advances in laser absorption and shock tube methods for studies of combustion chemistry,” Prog. Energy Combust. Sci. 44, 103–114 (2014).
[Crossref]

Rev. Mod. Phys. (1)

U. von Toussaint, “Bayesian inference in physics,” Rev. Mod. Phys. 83(3), 943–999 (2011).
[Crossref]

Sci. Rep. (1)

J. Emmert, N. G. Blume, A. Dreizler, and S. Wagner, “Data analysis and uncertainty estimation in supercontinuum laser absorption spectroscopy,” Sci. Rep. 8(1), 10312 (2018).
[Crossref]

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F. Mayinger, Optical measurements: techniques and applications (Springer Science & Business Media, 2013).

R. K. Hanson, R. M. Spearrin, and C. S. Goldenstein, Spectroscopy and optical diagnostics for gases (Springer, 2016).

S. J. Grauer, J. Emmert, A. M. Steinberg, S. Wagner, and K. J. Daun, Hyperspectral absorption tomography with a lineshape prior., in 11th US National Combustion Meeting, Pasadena, CA, Mar. 24-27, 2019 (2019).

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Supplementary Material (1)

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» Code 1       Sample code for Bayesian absorbance inference

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Figures (6)

Fig. 1.
Fig. 1. Fringe absorbance of two resonators with a very slight difference in the FSR: $\epsilon_\mathrm{FSR}=10^{-4}~\mathrm{cm}^{-1}$. The fringes nearly coincide at small wavenumbers but are visibly pseudo-shifted at larger wavenumbers.
Fig. 2.
Fig. 2. Flowchart of Bayesian absorbance inference. Transmitted intensity signals are used to estimate key parameters and construct the system matrix. Subsequent signals are evaluated in real-time to obtain MAP estimates, which contain the absorbance spectrum.
Fig. 3.
Fig. 3. Five of the 1000 simulated intensity signals. Points in the red area were evaluated with our Bayesian technique. The intensity trace indicated in red was used to estimate $\Delta_\mathrm{FSR}$.
Fig. 4.
Fig. 4. Estimating the FSR from a reference trace: a) the TDLAS signal $I_\nu$, third-order polynomial fit $I_{\nu,0}^*$, and resulting fringe; and b) the amplitude of the Fourier transform of the absorbance. The peak in the amplitude spectrum yields an estimate of $\Delta_\mathrm{FSR}$.
Fig. 5.
Fig. 5. Correlation (a) and covariance (b) matrices for the absorbance prior, determined from simulated spectra (c). Lighter colors represent higher covariance/correlation values. The covariance matrix is displayed on a logarithmic color scale.
Fig. 6.
Fig. 6. Bayesian, ideal, and ground truth absorbance spectra plus residua for the Bayesian estimates. Ideal absorbances were calculated from the noisy transmitted intensity and the exact simulated baseline intensity. That is, the absorbance that could be computed given perfect knowledge of the baseline and fringe.

Tables (2)

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Table 1. Parameters Used in Bayesian Absorbance Inference.

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Table 2. Computation Time of Each Step in the Inference, Evaluations Were Performed in MATLAB R2017b on an Intel Core i7 6500U CPU with 16 GB RAM.

Equations (40)

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l n ( I ν I ν , 0 ) = α ν ,
α ν = 0 l κ ν [ T ( x ) , p ( x ) , q ( x ) ) ] d x ,
I ν , f r i n g e I ν , 0 = 1 1 + F sin 2 ( π ν / Δ F S R ) ,
Δ F S R = 1 2 n g d .
α ν , f r i n g e = l n ( I ν , f r i n g e I ν , 0 ) = l n [ 1 + F sin 2 ( π ν Δ F S R ) ] = l n [ 1 + F 2 F 2 cos ( 2 π ν Δ F S R ) ] .
α ν , f r i n g e F 2 F 2 cos ( 2 π ν Δ F S R ) .
α ν , f r i n g e F 2 + β 0 cos ( 2 π ν Δ F S R ) + β 1 sin ( 2 π ν Δ F S R ) , w h e r e F = β 0 2 + β 1 2 .
l n ( I ν ) = l n ( I ν , 0 ) α ν α ν , f r i n g e = l n ( I ν , 0 ) α ν F 2 β 0 cos ( 2 π ν Δ F S R ) β 1 sin ( 2 π ν Δ F S R ) .
l n ( I ν ) = l n ( I ν , 0 ) α ν β 0 cos ( 2 π ν Δ F S R ) β 1 sin ( 2 π ν Δ F S R ) ,
π ( x | b ) = π ( b | x ) π p r ( x ) π ( b ) ,
x M A P = a r g m a x x π ( x | b ) ,
x = [ l n ( I ν 1 , 0 ) , l n ( I ν 2 , 0 ) , , l n ( I ν N , 0 ) , α ν 1 , α ν 2 , , α ν N , β 0 , β 1 ] T ,
b = [ l n ( I ν 1 ) , l n ( I ν 2 ) , , l n ( I ν N ) ] T .
b ^ i = l n ( I ν i , 0 ) α ν i β 0 cos ( 2 π ν i Δ F S R ) β 1 sin ( 2 π ν i Δ F S R ) + ϵ i .
b ^ ( x ) = A b x + ϵ ,
A b = [ 1 1 1 1 1 0 0 0 1 0 0 0 1 N × N  identity mat. 1 0 0 0 1 0 0 0 1 Negative  N × N  identity mat. cos ( 2 π Δ F S R ν 1 ) sin ( 2 π Δ F S R ν 1 ) cos ( 2 π Δ F S R ν 2 ) sin ( 2 π Δ F S R ν 2 ) cos ( 2 π Δ F S R ν N ) sin ( 2 π Δ F S R ν N ) N × 2  fringe model 1 1 1 1 ] .
π ( b | x ) = 1 ( 2 π ) N det ( Σ b ) exp [ 1 2 ( b ^ ( x ) b ) T Σ b 1 ( b ^ ( x ) b ) ] .
( Σ b ) i , j = { 0 , f o r i j ( σ I / I ν i ) 2 , f o r i = j .
Σ b = ( σ I / I ¯ ) 2 E N × N ,
π α ( α ) = α { χ | g ( χ ) α } π χ ( χ ) d χ , w h e r e   α = g ( χ ) .
π α ( α ) = 1 ( 2 π ) N det ( Σ α ) exp [ 1 2 ( α μ α ) T Σ α 1 ( α μ α ) ] ,
μ α = 1 M k = 1 M α k ,
Σ α = 1 M 1 k = 1 M ( α k μ α ) ( α k μ α ) T .
A α = [ 0 N × N , E N × N , 0 N × 2 ] ,
π p r , α ( x ) = 1 ( 2 π ) N det ( Σ α ) exp [ 1 2 ( A α x μ α ) T Σ α 1 ( A α x μ α ) ] .
( A d d I ) i , j = { 1 2 , f o r i = j 1 , f o r i = j 1 1 2 , f o r i = j 2 0 , o t h e r w i s e ,
π p r , d d I ( x ) = 1 ( 2 π ) N 2 det ( Σ d d I ) exp [ 1 2 ( A d d I x ) T Σ d d I 1 ( A d d I x ) ] .
Σ d d I = σ d d I 2 E N 2 × N 2 .
σ d d I = max j | ( A d d I [ l n ( x ) 0 N + 1 × 1 ] ) j |
π ( x | b ) = π ( b | x ) π p r , α ( x ) π p r , d d I ( x ) π ( b ) π ( b | x ) π p r , α ( x ) π p r , d d I ( x ) .
x M A P = a r g m a x x exp { 1 2 [ Σ T ( A b x b ) T Σ b 1 ( A b x b ) ( A α x μ α ) T Σ α 1 ( A α x μ α ) ( A d d I x ) T Σ d d I 1 ( A d d I x ) Σ T ] 1 2 } .
L b T L b = Σ b 1 , L α T L α = Σ α 1 , a n d L d d I T L d d I = Σ d d I 1 ,
x M A P = a r g m a x x exp { 1 2 [ Σ T ( L b A b x L b b ) T ( L b A b x L b b ) + ( L α A α x L α μ α ) T ( L α A α x L α μ α ) + ( L d d I A d d I x ) T ( L d d I A d d I x ) Σ T ] 1 2 } .
x M A P = a r g m i n x [ L b A b L α A α L d d I A d d I ] x [ L b b L α μ α 0 ( N 2 ) x 1 ] 2 2 = a r g m i n x A x γ 2 2 .
x M A P = ( A T A ) 1 A T γ = A # γ ,
δ p h a s e = 2 π δ F S R | ν N ν 1 | Δ F S R 2 .
Γ = [ γ 1 , γ 2 , , γ 1000 ] ,
X M A P = A # Γ .
μ κ = l 1 μ α a n d Σ κ = l 2 Σ α .
μ α = 2 l 2 μ κ = μ α a n d Σ α = 2 l 2 4 Σ κ = 1 2 Σ α .

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