Abstract

We propose and experimentally demonstrate the use of principal component analysis (PCA) based pattern recognition to extract temperature distribution from the measured Brillouin gain spectra (BGSs) along the fiber under test (FUT) obtained by Brillouin optical time domain analysis (BOTDA) system. The proposed scheme employs a reference database consisting of relevant ideal BGSs with known temperature attributes. PCA is then applied to the BGSs in the reference database as well as to the measured BGSs so as to reduce their size by extracting their most significant features. Now, for each feature vector of the measured BGS, we determine its best match in the reference database comprised of numerous reduced-size feature vectors of the ideal BGSs. The known temperature attribute corresponding to the best-matched BGS in the reference database is then taken as the extracted temperature of the measured BGS. We analyzed the performance of PCA-based pattern recognition algorithm in detail and compared it with that of curve fitting method. The experimental results validate that the proposed technique can provide better accuracy, faster processing speed and larger noise tolerance for the measured BGSs. Therefore, the proposed PCA-based pattern recognition algorithm can be considered as an attractive method for extracting temperature distributions along the fiber in BOTDA sensors.

© 2017 Optical Society of America

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References

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2016 (4)

2015 (6)

F. Wang, W. Zhan, Y. Lu, Z. Yan, and X. Zhang, “Determining the change of Brillouin frequency shift by using the similarity matching method,” J. Lightwave Technol. 33(19), 4101–4108 (2015).
[Crossref]

M. Alem, M. A. Soto, and L. Thévenaz, “Analytical model and experimental verification of the critical power for modulation instability in optical fibers,” Opt. Express 23(23), 29514–29532 (2015).
[Crossref] [PubMed]

F. N. Khan, Y. Yu, M. C. Tan, W. H. Al-Arashi, C. Yu, A. P. T. Lau, and C. Lu, “Experimental demonstration of joint OSNR monitoring and modulation format identification using asynchronous single channel sampling,” Opt. Express 23(23), 30337–30346 (2015).
[Crossref] [PubMed]

F. N. Khan, C. H. Teow, S. G. Kiu, M. C. Tan, Y. Zhou, W. H. Al-Arashi, A. P. T. Lau, and C. Lu, “Automatic modulation format/bit rate classification and signal-to-noise ratio estimation using asynchronous delay-tap sampling,” Elsevier J. Com. Elect. Eng. 47, 126–133 (2015).
[Crossref]

R. Ruiz-Lombera, J. Urricelqui, M. Sagues, J. Mirapeix, J. M. López-Higuera, and A. Loayssa, “Overcoming nonlocal effects and Brillouin threshold limitations in Brillouin optical time-domain sensors,” IEEE Photonics J. 7(6), 6803609 (2015).
[Crossref]

A. K. Azad, L. Wang, N. Guo, C. Lu, and H. Y. Tam, “Temperature sensing in BOTDA system by using artificial neural network,” Electron. Lett. 51(20), 1578–1580 (2015).
[Crossref]

2014 (3)

2013 (3)

X. Wang, A. Mueen, H. Ding, G. Trajcevski, P. Scheuermann, and E. Keogh, “Experimental comparison of representation methods and distance measures of time series data,” Data Min. Knowl. Discov. 26(2), 275–309 (2013).
[Crossref]

M. A. Farahani, E. Castillo-Guerra, and B. G. Colpitts, “A Detailed Evaluation of the Correlation-Based Method Used for Estimation of the Brillouin Frequency Shift in BOTDA Sensors,” IEEE Sens. J. 13(12), 4589–4598 (2013).
[Crossref]

M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
[Crossref] [PubMed]

2012 (4)

S. Xie, M. Pang, X. Bao, and L. Chen, “Polarization dependence of Brillouin linewidth and peak frequency due to fiber inhomogeneity in single mode fiber and its impact on distributed fiber Brillouin sensing,” Opt. Express 20(6), 6385–6399 (2012).
[Crossref] [PubMed]

X. Bao and L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
[Crossref] [PubMed]

C. A. Galindez-Jamioy and J. M. Lopez-Higuera, “Brillouin distributed fiber sensors: an overview and applications,” J. Sens. 2012, 204121 (2012).
[Crossref]

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-Spatial-Resolution Brillouin Optical Time-Domain Analysis Based on Bright Pulse Brillouin Gain and Complementary Code,” IEEE Photonics J. 4(6), 2242–2248 (2012).
[Crossref]

2010 (1)

2004 (1)

V. Perlibakas, “Distance measures for PCA-based face recognition,” Pattern Recognit. Lett. 25(6), 711–724 (2004).
[Crossref]

1999 (1)

1997 (1)

M. Niklès, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

1991 (1)

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3(1), 71–86 (1991).
[Crossref] [PubMed]

Airoldi, D.

F. Davò, S. Alessandrini, S. Sperati, L. D. Monache, D. Airoldi, and M. T. Vespucci, “Post-processing techniques and principal component analysis for regional wind power and solar irradiance forecasting,” Elsevier J. Solar. Energy 134, 327–338 (2016).
[Crossref]

Al-Arashi, W. H.

Alem, M.

Alessandrini, S.

F. Davò, S. Alessandrini, S. Sperati, L. D. Monache, D. Airoldi, and M. T. Vespucci, “Post-processing techniques and principal component analysis for regional wind power and solar irradiance forecasting,” Elsevier J. Solar. Energy 134, 327–338 (2016).
[Crossref]

Angulo-Vinuesa, X.

Azad, A. K.

A. K. Azad, L. Wang, N. Guo, H. Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016).
[Crossref] [PubMed]

A. K. Azad, L. Wang, N. Guo, C. Lu, and H. Y. Tam, “Temperature sensing in BOTDA system by using artificial neural network,” Electron. Lett. 51(20), 1578–1580 (2015).
[Crossref]

Bao, X.

Bolognini, G.

Brown, A.

Castillo-Guerra, E.

M. A. Farahani, E. Castillo-Guerra, and B. G. Colpitts, “A Detailed Evaluation of the Correlation-Based Method Used for Estimation of the Brillouin Frequency Shift in BOTDA Sensors,” IEEE Sens. J. 13(12), 4589–4598 (2013).
[Crossref]

Catalano, E.

Chen, L.

Colpitts, B. G.

M. A. Farahani, E. Castillo-Guerra, and B. G. Colpitts, “A Detailed Evaluation of the Correlation-Based Method Used for Estimation of the Brillouin Frequency Shift in BOTDA Sensors,” IEEE Sens. J. 13(12), 4589–4598 (2013).
[Crossref]

Davò, F.

F. Davò, S. Alessandrini, S. Sperati, L. D. Monache, D. Airoldi, and M. T. Vespucci, “Post-processing techniques and principal component analysis for regional wind power and solar irradiance forecasting,” Elsevier J. Solar. Energy 134, 327–338 (2016).
[Crossref]

Demerchant, M.

Di Pasquale, F.

Ding, H.

X. Wang, A. Mueen, H. Ding, G. Trajcevski, P. Scheuermann, and E. Keogh, “Experimental comparison of representation methods and distance measures of time series data,” Data Min. Knowl. Discov. 26(2), 275–309 (2013).
[Crossref]

Dominguez-Lopez, A.

Farahani, M. A.

M. A. Farahani, E. Castillo-Guerra, and B. G. Colpitts, “A Detailed Evaluation of the Correlation-Based Method Used for Estimation of the Brillouin Frequency Shift in BOTDA Sensors,” IEEE Sens. J. 13(12), 4589–4598 (2013).
[Crossref]

Galindez-Jamioy, C. A.

C. A. Galindez-Jamioy and J. M. Lopez-Higuera, “Brillouin distributed fiber sensors: an overview and applications,” J. Sens. 2012, 204121 (2012).
[Crossref]

Gonzalez-Herraez, M.

Guo, N.

A. K. Azad, L. Wang, N. Guo, H. Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016).
[Crossref] [PubMed]

A. K. Azad, L. Wang, N. Guo, C. Lu, and H. Y. Tam, “Temperature sensing in BOTDA system by using artificial neural network,” Electron. Lett. 51(20), 1578–1580 (2015).
[Crossref]

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-Spatial-Resolution Brillouin Optical Time-Domain Analysis Based on Bright Pulse Brillouin Gain and Complementary Code,” IEEE Photonics J. 4(6), 2242–2248 (2012).
[Crossref]

Hadar, R.

Keogh, E.

X. Wang, A. Mueen, H. Ding, G. Trajcevski, P. Scheuermann, and E. Keogh, “Experimental comparison of representation methods and distance measures of time series data,” Data Min. Knowl. Discov. 26(2), 275–309 (2013).
[Crossref]

Khan, F. N.

Kiu, S. G.

F. N. Khan, C. H. Teow, S. G. Kiu, M. C. Tan, Y. Zhou, W. H. Al-Arashi, A. P. T. Lau, and C. Lu, “Automatic modulation format/bit rate classification and signal-to-noise ratio estimation using asynchronous delay-tap sampling,” Elsevier J. Com. Elect. Eng. 47, 126–133 (2015).
[Crossref]

Lau, A. P. T.

F. N. Khan, C. H. Teow, S. G. Kiu, M. C. Tan, Y. Zhou, W. H. Al-Arashi, A. P. T. Lau, and C. Lu, “Automatic modulation format/bit rate classification and signal-to-noise ratio estimation using asynchronous delay-tap sampling,” Elsevier J. Com. Elect. Eng. 47, 126–133 (2015).
[Crossref]

F. N. Khan, Y. Yu, M. C. Tan, W. H. Al-Arashi, C. Yu, A. P. T. Lau, and C. Lu, “Experimental demonstration of joint OSNR monitoring and modulation format identification using asynchronous single channel sampling,” Opt. Express 23(23), 30337–30346 (2015).
[Crossref] [PubMed]

Li, C.

C. Li and Y. Li, “Fitting of Brillouin spectrum based on LabVIEW,” in Proceedings of 5th International Conference on Wireless Communications, Networking and Mobile Computing (IEEE, 2009), pp. 3495–3498.
[Crossref]

Li, Y.

C. Li and Y. Li, “Fitting of Brillouin spectrum based on LabVIEW,” in Proceedings of 5th International Conference on Wireless Communications, Networking and Mobile Computing (IEEE, 2009), pp. 3495–3498.
[Crossref]

Loayssa, A.

R. Ruiz-Lombera, J. Urricelqui, M. Sagues, J. Mirapeix, J. M. López-Higuera, and A. Loayssa, “Overcoming nonlocal effects and Brillouin threshold limitations in Brillouin optical time-domain sensors,” IEEE Photonics J. 7(6), 6803609 (2015).
[Crossref]

Lopez-Gil, A.

Lopez-Higuera, J. M.

C. A. Galindez-Jamioy and J. M. Lopez-Higuera, “Brillouin distributed fiber sensors: an overview and applications,” J. Sens. 2012, 204121 (2012).
[Crossref]

López-Higuera, J. M.

R. Ruiz-Lombera, J. Urricelqui, M. Sagues, J. Mirapeix, J. M. López-Higuera, and A. Loayssa, “Overcoming nonlocal effects and Brillouin threshold limitations in Brillouin optical time-domain sensors,” IEEE Photonics J. 7(6), 6803609 (2015).
[Crossref]

Lu, C.

A. K. Azad, L. Wang, N. Guo, H. Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016).
[Crossref] [PubMed]

F. N. Khan, Y. Yu, M. C. Tan, W. H. Al-Arashi, C. Yu, A. P. T. Lau, and C. Lu, “Experimental demonstration of joint OSNR monitoring and modulation format identification using asynchronous single channel sampling,” Opt. Express 23(23), 30337–30346 (2015).
[Crossref] [PubMed]

A. K. Azad, L. Wang, N. Guo, C. Lu, and H. Y. Tam, “Temperature sensing in BOTDA system by using artificial neural network,” Electron. Lett. 51(20), 1578–1580 (2015).
[Crossref]

F. N. Khan, C. H. Teow, S. G. Kiu, M. C. Tan, Y. Zhou, W. H. Al-Arashi, A. P. T. Lau, and C. Lu, “Automatic modulation format/bit rate classification and signal-to-noise ratio estimation using asynchronous delay-tap sampling,” Elsevier J. Com. Elect. Eng. 47, 126–133 (2015).
[Crossref]

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-Spatial-Resolution Brillouin Optical Time-Domain Analysis Based on Bright Pulse Brillouin Gain and Complementary Code,” IEEE Photonics J. 4(6), 2242–2248 (2012).
[Crossref]

Lu, Y.

Mao, Y.

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-Spatial-Resolution Brillouin Optical Time-Domain Analysis Based on Bright Pulse Brillouin Gain and Complementary Code,” IEEE Photonics J. 4(6), 2242–2248 (2012).
[Crossref]

Martin-Lopez, S.

Minardo, A.

Mirapeix, J.

R. Ruiz-Lombera, J. Urricelqui, M. Sagues, J. Mirapeix, J. M. López-Higuera, and A. Loayssa, “Overcoming nonlocal effects and Brillouin threshold limitations in Brillouin optical time-domain sensors,” IEEE Photonics J. 7(6), 6803609 (2015).
[Crossref]

Monache, L. D.

F. Davò, S. Alessandrini, S. Sperati, L. D. Monache, D. Airoldi, and M. T. Vespucci, “Post-processing techniques and principal component analysis for regional wind power and solar irradiance forecasting,” Elsevier J. Solar. Energy 134, 327–338 (2016).
[Crossref]

Motil, A.

Muanenda, Y.

Mueen, A.

X. Wang, A. Mueen, H. Ding, G. Trajcevski, P. Scheuermann, and E. Keogh, “Experimental comparison of representation methods and distance measures of time series data,” Data Min. Knowl. Discov. 26(2), 275–309 (2013).
[Crossref]

Niklès, M.

M. Niklès, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Pang, M.

Pasquale, F. D.

Pentland, A.

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3(1), 71–86 (1991).
[Crossref] [PubMed]

Perlibakas, V.

V. Perlibakas, “Distance measures for PCA-based face recognition,” Pattern Recognit. Lett. 25(6), 711–724 (2004).
[Crossref]

Robert, P. A.

M. Niklès, L. Thévenaz, and P. A. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Ruiz-Lombera, R.

R. Ruiz-Lombera, J. Urricelqui, M. Sagues, J. Mirapeix, J. M. López-Higuera, and A. Loayssa, “Overcoming nonlocal effects and Brillouin threshold limitations in Brillouin optical time-domain sensors,” IEEE Photonics J. 7(6), 6803609 (2015).
[Crossref]

Sagues, M.

R. Ruiz-Lombera, J. Urricelqui, M. Sagues, J. Mirapeix, J. M. López-Higuera, and A. Loayssa, “Overcoming nonlocal effects and Brillouin threshold limitations in Brillouin optical time-domain sensors,” IEEE Photonics J. 7(6), 6803609 (2015).
[Crossref]

Scheuermann, P.

X. Wang, A. Mueen, H. Ding, G. Trajcevski, P. Scheuermann, and E. Keogh, “Experimental comparison of representation methods and distance measures of time series data,” Data Min. Knowl. Discov. 26(2), 275–309 (2013).
[Crossref]

Smith, J.

Soto, M. A.

Sovran, I.

Sperati, S.

F. Davò, S. Alessandrini, S. Sperati, L. D. Monache, D. Airoldi, and M. T. Vespucci, “Post-processing techniques and principal component analysis for regional wind power and solar irradiance forecasting,” Elsevier J. Solar. Energy 134, 327–338 (2016).
[Crossref]

Taki, M.

Tam, H. Y.

A. K. Azad, L. Wang, N. Guo, H. Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016).
[Crossref] [PubMed]

A. K. Azad, L. Wang, N. Guo, C. Lu, and H. Y. Tam, “Temperature sensing in BOTDA system by using artificial neural network,” Electron. Lett. 51(20), 1578–1580 (2015).
[Crossref]

Y. Mao, N. Guo, K. L. Yu, H. Y. Tam, and C. Lu, “1-cm-Spatial-Resolution Brillouin Optical Time-Domain Analysis Based on Bright Pulse Brillouin Gain and Complementary Code,” IEEE Photonics J. 4(6), 2242–2248 (2012).
[Crossref]

Tan, M. C.

Tao Lau, A. P.

Teow, C. H.

F. N. Khan, C. H. Teow, S. G. Kiu, M. C. Tan, Y. Zhou, W. H. Al-Arashi, A. P. T. Lau, and C. Lu, “Automatic modulation format/bit rate classification and signal-to-noise ratio estimation using asynchronous delay-tap sampling,” Elsevier J. Com. Elect. Eng. 47, 126–133 (2015).
[Crossref]

Thévenaz, L.

Trajcevski, G.

X. Wang, A. Mueen, H. Ding, G. Trajcevski, P. Scheuermann, and E. Keogh, “Experimental comparison of representation methods and distance measures of time series data,” Data Min. Knowl. Discov. 26(2), 275–309 (2013).
[Crossref]

Tur, M.

Turk, M.

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3(1), 71–86 (1991).
[Crossref] [PubMed]

Urricelqui, J.

R. Ruiz-Lombera, J. Urricelqui, M. Sagues, J. Mirapeix, J. M. López-Higuera, and A. Loayssa, “Overcoming nonlocal effects and Brillouin threshold limitations in Brillouin optical time-domain sensors,” IEEE Photonics J. 7(6), 6803609 (2015).
[Crossref]

Vespucci, M. T.

F. Davò, S. Alessandrini, S. Sperati, L. D. Monache, D. Airoldi, and M. T. Vespucci, “Post-processing techniques and principal component analysis for regional wind power and solar irradiance forecasting,” Elsevier J. Solar. Energy 134, 327–338 (2016).
[Crossref]

Wang, F.

Wang, L.

A. K. Azad, L. Wang, N. Guo, H. Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016).
[Crossref] [PubMed]

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M. C. Tan, F. N. Khan, W. H. Al-Arashi, Y. Zhou, and A. P. Tao Lau, “Simultaneous optical performance monitoring and modulation format/bit-rate identification using principal component analysis,” J. Opt. Commun. Netw. 6(5), 441–448 (2014).
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X. Wang, A. Mueen, H. Ding, G. Trajcevski, P. Scheuermann, and E. Keogh, “Experimental comparison of representation methods and distance measures of time series data,” Data Min. Knowl. Discov. 26(2), 275–309 (2013).
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Electron. Lett. (1)

A. K. Azad, L. Wang, N. Guo, C. Lu, and H. Y. Tam, “Temperature sensing in BOTDA system by using artificial neural network,” Electron. Lett. 51(20), 1578–1580 (2015).
[Crossref]

Elsevier J. Com. Elect. Eng. (1)

F. N. Khan, C. H. Teow, S. G. Kiu, M. C. Tan, Y. Zhou, W. H. Al-Arashi, A. P. T. Lau, and C. Lu, “Automatic modulation format/bit rate classification and signal-to-noise ratio estimation using asynchronous delay-tap sampling,” Elsevier J. Com. Elect. Eng. 47, 126–133 (2015).
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Figures (14)

Fig. 1
Fig. 1 BOTDA experimental setup, PC: Polarization Controller, EOM: Electro-Optic Modulator, RF: Radio Frequency, VOA: Variable Optical Attenuator, PPG: Pump Pattern Generator, EDFA: Erbium-doped Fiber Amplifier, BPF: Band Pass Filter, PS: Polarization Scrambler, FBG: Fiber Bragg Grating, PD: Photodetector, FUT: Fiber Under Test.
Fig. 2
Fig. 2 Functional block diagram depicting temperature extraction from BOTDA measured BGS using PCA-based pattern recognition.
Fig. 3
Fig. 3 BFS-temperature characteristics of the FUT. The whole span of a ~200 m FUT is heated inside the oven each time at different temperatures. Several local BGSs along the FUT for a given temperature are first normalized and then averaged to obtain one measured BGS for that specific temperature. The BFSs of the measured BGSs are determined by using CFM.
Fig. 4
Fig. 4 (a) A total of P = 25 most significant eigenvalues λj plotted in descending order and (b) Parameter Ω versus number of PCs selected, P.
Fig. 5
Fig. 5 Distribution of BGSs along the 38.2 km long FUT obtained from BOTDA measurement using pump pulse of duration 20 ns, trace averaging of 1000, sampling interval of 0.4 m and frequency step of 1 MHz with last ~600 m section of the FUT heated inside the oven at 70 °C.
Fig. 6
Fig. 6 A typical normalized noisy BGS near the end of 38.2 km long FUT and its fitted curve obtained after employing CFM. The noise is computed by subtracting the noisy BGS from the fitted one. The noisy BGS is obtained from BOTDA experiment using pump pulse of duration 20 ns, frequency step of 1 MHz and trace averaging of 1000. The fitted curve gives gB ≈0.867, υB ≈10.8832 GHz and ΔυB ≈54.46 MHz.
Fig. 7
Fig. 7 (a) SNR distributions for the BGSs along the 38.2 km FUT with last ~600 m section of the FUT heated inside the oven at 40 °C, and (b) Average SNR calculated for the BGSs along the last 500 m (i.e., 37.7 km to 38.2 km) section of the FUT.
Fig. 8
Fig. 8 RMSE as a function of number of PCs selected to synthesize the feature vectors of BGSs obtained using two different frequency steps of (a) 1 MHz and (b) 2 MHz.
Fig. 9
Fig. 9 Temperature distributions along 38.2 km FUT determined using PCA-based pattern recognition and CFM for the BGSs obtained using 1 MHz frequency step, 1000 trace averaging and the last ~600 m section of the FUT heated at (i) 40 °C, (ii) 50 °C, (iii) 60 °C and (iv) 70 °C; inset: temperature distributions for 1.2 km section from 37 km to 38.2 km.
Fig. 10
Fig. 10 Temperature distributions along 38.2 km FUT determined using PCA-based pattern recognition and CFM for the BGSs obtained using 2 MHz frequency step, 1000 trace averaging and the last ~600 m section of the FUT heated at (i) 40 °C, (ii) 50 °C, (iii) 60 °C and (iv) 70 °C; inset: temperature distributions for 1.2 km section from 37 km to 38.2 km..
Fig. 11
Fig. 11 (a) RMSE, (b) SD computed for the extracted temperatures along the last 500 m (i.e., 37.7 km to 38.2 km) section of the FUT and (c) SD computed for the extracted temperatures along the 500 m (i.e., 37 km to 37.5 km) section of the FUT. The measured BGSs are obtained using 1 MHz frequency step.
Fig. 12
Fig. 12 (a) RMSE, (b) SD computed for the extracted temperatures along the last 500 m (i.e., 37.7 km to 38.2 km) section of the FUT and (c) SD computed for the extracted temperatures along the 500 m (i.e., 37 km to 37.5 km) section of the FUT. The measured BGSs are obtained using 2 MHz frequency step.
Fig. 13
Fig. 13 (a) Relative running time, (b) RMSE, (c) SD for the extracted temperatures along the last 500 m (i.e., 37.7 km to 38.2 km) section and (d) SD for the extracted temperatures along the 37 km to 37.5 km section of the FUT using PCA-based pattern recognition with P = 9. The BGSs are obtained using Δυ = 1 MHz frequency step.
Fig. 14
Fig. 14 (a) Relative running time, (b) RMSE, (c) SD for the extracted temperatures along the last 500 m (i.e., 37.7 km to 38.2 km) section and (d) SD for the extracted temperatures along the 37 km to 37.5 km section of the FUT using PCA-based pattern recognition with P = 9. The BGSs are obtained using Δυ = 2 MHz frequency step.

Equations (12)

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

g ¯ = 1 M i = 1 M g i
C = 1 M i = 1 M ψ i ψ i T = Ψ Ψ T
C μ j = λ j μ j for j = 1 , 2 , . . . , N
Ω = j = 1 P λ j / j = 1 N λ j > δ
k = 1 P w k μ k ψ for k = 1 , 2 , . . . , P
μ l μ m = { 1 if l = m 0 if l m
w k = μ k T ψ for k = 1 , 2 , . . . , P
r = [ w 1 w 2 . . . w P ] T
D ( s , r ) = s r = k = 1 P ( s k r k ) 2
g ( υ ) = g B 1 + 4 [ ( υ υ B ) / ( Δ υ B ) ] 2
υ B ( T ) = C T Δ T + υ B ( T o )
SNR = g B 2 σ n 2

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