Abstract

Results of optical frequency transfer over a carrier-grade dense-wavelength-division-multiplexing (DWDM) optical fiber network are presented. The relation between soil temperature changes on a buried optical fiber and frequency changes of an optical carrier through the fiber is modeled. Soil temperatures, measured at various depths by the Royal Netherlands Meteorology Institute (KNMI) are compared with observed frequency variations through this model. A comparison of a nine-day record of optical frequency measurements through the 2×298km fiber link with soil temperature data shows qualitative agreement. A soil temperature model is used to predict the link stability over longer periods (days–months–years). We show that optical frequency dissemination is sufficiently stable to distribute and compare, e.g., rubidium frequency standards over standard DWDM optical fiber networks using unidirectional fibers.

© 2015 Optical Society of America

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References

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2014 (2)

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

T. P. Heavner, E. A. Donley, F. Levi, G. Costanzo, T. E. Parker, J. H. Shirley, N. Ashby, S. Barlow, and S. R. Jefferts, “First accuracy evaluation of NIST-F2,” Metrologia 51, 174–182 (2014).
[Crossref]

2013 (2)

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10−18 instability,” Science 341, 1215–1218 (2013).
[Crossref]

2012 (3)

S. Weyers, V. Gerginov, N. Nemitz, R. Li, and K. Gibble, “Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2,” Metrologia 49, 82–87 (2012).
[Crossref]

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

O. Lopez, A. Haboucha, B. Chanteau, C. Chardonnet, A. Amy-Klein, and G. Santarelli, “Ultra-stable long distance optical frequency distribution using the internet fiber network,” Opt. Express 20, 23518–23526 (2012).
[Crossref]

2011 (1)

2010 (1)

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

2009 (1)

2008 (5)

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

D. Piester, A. Bauch, L. Breakiron, D. Matsakis, B. Blanzano, and O. Koudelka, “Time transfer with nanosecond accuracy for the realization of international atomic time,” Metrologia 45, 185–198 (2008).
[Crossref]

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D 48, 35–41 (2008).
[Crossref]

R. Emardson, P. O. Hedekvist, M. Nilsson, S.-C. Ebenhag, K. Jaldehag, P. Jarlemark, C. Rieck, J. Johansson, L. R. Pendrill, P. Löthberg, and H. Nilsson, “Time transfer by passive listening over a 10-Gb/s optical fiber,” IEEE Trans. Instrum. Meas. 57, 2495–2501 (2008).
[Crossref]

P. A. Williams, W. C. Swann, and N. R. Newbury, “High-stability transfer of an optical frequency over long fiber-optic links,” J. Opt. Soc. Am. B 25, 1284–1293 (2008).
[Crossref]

2004 (2)

E. A. Elias, R. Cichota, H. H. Torriani, and Q. de Jong van Lier, “Analytical soil-temperature model: correction for temporal variation of daily amplitude,” Soil Science Society of America Journal 68, 784–788 (2004).

K.-C. Lin, C.-J. Lin, and W.-Y. Lee, “Effects of gamma radiation on optical fibre sensors,” IEE Proc. Optoelectron. 151, 12–15 (2004).
[Crossref]

2002 (1)

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74, 1–6 (2002).
[Crossref]

2001 (1)

M. A. Lombardi, L. M. Nelson, A. N. Novick, and V. S. Zhang, “Time and frequency measurements using the global positioning system,” Cal. Lab. Int. J. Metrol. 8, 26–33 (2001).

1999 (1)

K. M. Larson and J. Levine, “Carrier-phase time transfer,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 1001–1012 (1999).

1998 (1)

1994 (1)

Abgrall, M.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

Achkar, J.

V. Zhang, T. E. Parker, J. Achkar, A. Bauch, L. Lorini, D. Matsakis, D. Piester, and D. G. Rovera, “Two-way satellite time and frequency transfer using 1  Mchip/s codes,” in Proceedings of the 41st Annual Precise Time and Time Interval (PTTI) Meeting (2009), pp. 371–382.

Allan, D. W.

D. W. Allan and M. A. Weiss, “Accurate time and frequency transfer during common-view of a GPS satellite,” in Proceedings of the 34th Annual Frequency Control Symposium (1980), pp. 334–346.

Alnis, J.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

Altschul, B.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

Amy-Klein, A.

O. Lopez, A. Haboucha, B. Chanteau, C. Chardonnet, A. Amy-Klein, and G. Santarelli, “Ultra-stable long distance optical frequency distribution using the internet fiber network,” Opt. Express 20, 23518–23526 (2012).
[Crossref]

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D 48, 35–41 (2008).
[Crossref]

Ashby, N.

T. P. Heavner, E. A. Donley, F. Levi, G. Costanzo, T. E. Parker, J. H. Shirley, N. Ashby, S. Barlow, and S. R. Jefferts, “First accuracy evaluation of NIST-F2,” Metrologia 51, 174–182 (2014).
[Crossref]

Barlow, S.

T. P. Heavner, E. A. Donley, F. Levi, G. Costanzo, T. E. Parker, J. H. Shirley, N. Ashby, S. Barlow, and S. R. Jefferts, “First accuracy evaluation of NIST-F2,” Metrologia 51, 174–182 (2014).
[Crossref]

Bauch, A.

D. Piester, A. Bauch, L. Breakiron, D. Matsakis, B. Blanzano, and O. Koudelka, “Time transfer with nanosecond accuracy for the realization of international atomic time,” Metrologia 45, 185–198 (2008).
[Crossref]

V. Zhang, T. E. Parker, J. Achkar, A. Bauch, L. Lorini, D. Matsakis, D. Piester, and D. G. Rovera, “Two-way satellite time and frequency transfer using 1  Mchip/s codes,” in Proceedings of the 41st Annual Precise Time and Time Interval (PTTI) Meeting (2009), pp. 371–382.

Beloy, K.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10−18 instability,” Science 341, 1215–1218 (2013).
[Crossref]

Bergquist, J. C.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Beyer, A.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

Bishof, M.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

Blanzano, B.

D. Piester, A. Bauch, L. Breakiron, D. Matsakis, B. Blanzano, and O. Koudelka, “Time transfer with nanosecond accuracy for the realization of international atomic time,” Metrologia 45, 185–198 (2008).
[Crossref]

Bloom, B. J.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

Breakiron, L.

D. Piester, A. Bauch, L. Breakiron, D. Matsakis, B. Blanzano, and O. Koudelka, “Time transfer with nanosecond accuracy for the realization of international atomic time,” Metrologia 45, 185–198 (2008).
[Crossref]

Bromley, S. L.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

Brown, A.

A. Brown, R. Silva, and E. Powers, “A GPS receiver designed for carrier-phase time transfer,” in Proceedings of ION National Technical Meeting, Anaheim, California, January, 2000, pp. 32–41.

Brusch, A.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Campbell, S. L.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

Chanteau, B.

Chardonnet, C.

O. Lopez, A. Haboucha, B. Chanteau, C. Chardonnet, A. Amy-Klein, and G. Santarelli, “Ultra-stable long distance optical frequency distribution using the internet fiber network,” Opt. Express 20, 23518–23526 (2012).
[Crossref]

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D 48, 35–41 (2008).
[Crossref]

Chou, C. W.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Cichota, R.

E. A. Elias, R. Cichota, H. H. Torriani, and Q. de Jong van Lier, “Analytical soil-temperature model: correction for temporal variation of daily amplitude,” Soil Science Society of America Journal 68, 784–788 (2004).

Costanzo, G.

T. P. Heavner, E. A. Donley, F. Levi, G. Costanzo, T. E. Parker, J. H. Shirley, N. Ashby, S. Barlow, and S. R. Jefferts, “First accuracy evaluation of NIST-F2,” Metrologia 51, 174–182 (2014).
[Crossref]

Daussy, C.

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D 48, 35–41 (2008).
[Crossref]

de Jong van Lier, Q.

E. A. Elias, R. Cichota, H. H. Torriani, and Q. de Jong van Lier, “Analytical soil-temperature model: correction for temporal variation of daily amplitude,” Soil Science Society of America Journal 68, 784–788 (2004).

Diddams, S. A.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Donley, E. A.

T. P. Heavner, E. A. Donley, F. Levi, G. Costanzo, T. E. Parker, J. H. Shirley, N. Ashby, S. Barlow, and S. R. Jefferts, “First accuracy evaluation of NIST-F2,” Metrologia 51, 174–182 (2014).
[Crossref]

Droste, S.

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

Drullinger, R. E.

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K. Jaldehag, S.-C. Ebenhag, P. O. Hedekvist, and C. Rieck, “Time and frequency transfer using asynchronous fiber-optical networks: progress report,” in Proceedings of the 41st Annual Precise Time and Time Interval (PTTI) Meeting (2009), pp. 383–396.

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R. Emardson, P. O. Hedekvist, M. Nilsson, S.-C. Ebenhag, K. Jaldehag, P. Jarlemark, C. Rieck, J. Johansson, L. R. Pendrill, P. Löthberg, and H. Nilsson, “Time transfer by passive listening over a 10-Gb/s optical fiber,” IEEE Trans. Instrum. Meas. 57, 2495–2501 (2008).
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Matsakis, D.

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N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10−18 instability,” Science 341, 1215–1218 (2013).
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A. Brown, R. Silva, and E. Powers, “A GPS receiver designed for carrier-phase time transfer,” in Proceedings of ION National Technical Meeting, Anaheim, California, January, 2000, pp. 32–41.

Predehl, K.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, “Optical frequency transfer via 146  km fiber link with 10−19 relative accuracy,” Opt. Lett. 34, 2270–2272 (2009).
[Crossref]

Raupach, S. M. F.

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

Rieck, C.

R. Emardson, P. O. Hedekvist, M. Nilsson, S.-C. Ebenhag, K. Jaldehag, P. Jarlemark, C. Rieck, J. Johansson, L. R. Pendrill, P. Löthberg, and H. Nilsson, “Time transfer by passive listening over a 10-Gb/s optical fiber,” IEEE Trans. Instrum. Meas. 57, 2495–2501 (2008).
[Crossref]

K. Jaldehag, S.-C. Ebenhag, P. O. Hedekvist, and C. Rieck, “Time and frequency transfer using asynchronous fiber-optical networks: progress report,” in Proceedings of the 41st Annual Precise Time and Time Interval (PTTI) Meeting (2009), pp. 383–396.

Rosenband, T.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Rovera, D.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

Rovera, D. G.

V. Zhang, T. E. Parker, J. Achkar, A. Bauch, L. Lorini, D. Matsakis, D. Piester, and D. G. Rovera, “Two-way satellite time and frequency transfer using 1  Mchip/s codes,” in Proceedings of the 41st Annual Precise Time and Time Interval (PTTI) Meeting (2009), pp. 371–382.

Salomon, C.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

Santarelli, G.

O. Lopez, A. Haboucha, B. Chanteau, C. Chardonnet, A. Amy-Klein, and G. Santarelli, “Ultra-stable long distance optical frequency distribution using the internet fiber network,” Opt. Express 20, 23518–23526 (2012).
[Crossref]

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D 48, 35–41 (2008).
[Crossref]

Schioppo, M.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10−18 instability,” Science 341, 1215–1218 (2013).
[Crossref]

Schmidt, P. O.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Schnatz, H.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, “Optical frequency transfer via 146  km fiber link with 10−19 relative accuracy,” Opt. Lett. 34, 2270–2272 (2009).
[Crossref]

Sherman, J. A.

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10−18 instability,” Science 341, 1215–1218 (2013).
[Crossref]

Shirley, J. H.

T. P. Heavner, E. A. Donley, F. Levi, G. Costanzo, T. E. Parker, J. H. Shirley, N. Ashby, S. Barlow, and S. R. Jefferts, “First accuracy evaluation of NIST-F2,” Metrologia 51, 174–182 (2014).
[Crossref]

Silva, R.

A. Brown, R. Silva, and E. Powers, “A GPS receiver designed for carrier-phase time transfer,” in Proceedings of ION National Technical Meeting, Anaheim, California, January, 2000, pp. 32–41.

Stalnaker, J. E.

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Stenger, J.

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74, 1–6 (2002).
[Crossref]

Sterr, U.

Swann, W. C.

P. A. Williams, W. C. Swann, and N. R. Newbury, “High-stability transfer of an optical frequency over long fiber-optic links,” J. Opt. Soc. Am. B 25, 1284–1293 (2008).
[Crossref]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Telle, H. R.

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74, 1–6 (2002).
[Crossref]

Terra, O.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

G. Grosche, O. Terra, K. Predehl, R. Holzwarth, B. Lipphardt, F. Vogt, U. Sterr, and H. Schnatz, “Optical frequency transfer via 146  km fiber link with 10−19 relative accuracy,” Opt. Lett. 34, 2270–2272 (2009).
[Crossref]

Tetsuya, I.

Torriani, H. H.

E. A. Elias, R. Cichota, H. H. Torriani, and Q. de Jong van Lier, “Analytical soil-temperature model: correction for temporal variation of daily amplitude,” Soil Science Society of America Journal 68, 784–788 (2004).

Udem, T.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

van der Hoeven, P. C. T.

P. C. T. van der Hoeven and W. N. Lablans, “Grondtemperaturen,” (Koninklijk Nederlands Meteorologisch Instituut, 1992).

Vogt, F.

Weiss, M. A.

V. S. Zhang, T. E. Parker, and M. A. Weiss, “Multi-channel GPS/GLONASS common-view between NIST and USNO,” in Proceedings of the 2000 IEEE/EIA International Frequency Control Symposium and Exhibition(2000), pp. 598–606.

D. W. Allan and M. A. Weiss, “Accurate time and frequency transfer during common-view of a GPS satellite,” in Proceedings of the 34th Annual Frequency Control Symposium (1980), pp. 334–346.

Weyers, S.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

S. Weyers, V. Gerginov, N. Nemitz, R. Li, and K. Gibble, “Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2,” Metrologia 49, 82–87 (2012).
[Crossref]

Wilken, T.

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

Williams, J. R.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

Williams, P. A.

Wineland, D. J.

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

Woudenberg, D. J. P. W.

D. J. P. W. Woudenberg, “Vergelijkende metingen van de grondtemperatuur te De Bilt in 1961,” (KNMI, 1966).

Yajima, H.

Yamaguchi, A.

Ye, J.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

L.-S. Ma, P. Jungner, J. Ye, and J. L. Hall, “Delivering the same optical frequency at two places: accurate cancellation of phase noise introduced by an optical fiber or other time-varying path,” Opt. Lett. 19, 1777–1779 (1994).
[Crossref]

Zhang, V.

V. Zhang, T. E. Parker, J. Achkar, A. Bauch, L. Lorini, D. Matsakis, D. Piester, and D. G. Rovera, “Two-way satellite time and frequency transfer using 1  Mchip/s codes,” in Proceedings of the 41st Annual Precise Time and Time Interval (PTTI) Meeting (2009), pp. 371–382.

Zhang, V. S.

M. A. Lombardi, L. M. Nelson, A. N. Novick, and V. S. Zhang, “Time and frequency measurements using the global positioning system,” Cal. Lab. Int. J. Metrol. 8, 26–33 (2001).

V. S. Zhang, T. E. Parker, and M. A. Weiss, “Multi-channel GPS/GLONASS common-view between NIST and USNO,” in Proceedings of the 2000 IEEE/EIA International Frequency Control Symposium and Exhibition(2000), pp. 598–606.

Zhang, W.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

Zhang, X.

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

Appl. Phys. B (1)

H. R. Telle, B. Lipphardt, and J. Stenger, “Kerr-lens, mode-locked lasers as transfer oscillators for optical frequency measurements,” Appl. Phys. B 74, 1–6 (2002).
[Crossref]

Cal. Lab. Int. J. Metrol. (1)

M. A. Lombardi, L. M. Nelson, A. N. Novick, and V. S. Zhang, “Time and frequency measurements using the global positioning system,” Cal. Lab. Int. J. Metrol. 8, 26–33 (2001).

Eur. Phys. J. D (1)

O. Lopez, A. Amy-Klein, C. Daussy, C. Chardonnet, F. Narbonneau, M. Lours, and G. Santarelli, “86-km optical link with a resolution of 2 × 10−18 for RF frequency transfer,” Eur. Phys. J. D 48, 35–41 (2008).
[Crossref]

IEE Proc. Optoelectron. (1)

K.-C. Lin, C.-J. Lin, and W.-Y. Lee, “Effects of gamma radiation on optical fibre sensors,” IEE Proc. Optoelectron. 151, 12–15 (2004).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

R. Emardson, P. O. Hedekvist, M. Nilsson, S.-C. Ebenhag, K. Jaldehag, P. Jarlemark, C. Rieck, J. Johansson, L. R. Pendrill, P. Löthberg, and H. Nilsson, “Time transfer by passive listening over a 10-Gb/s optical fiber,” IEEE Trans. Instrum. Meas. 57, 2495–2501 (2008).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

K. M. Larson and J. Levine, “Carrier-phase time transfer,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 1001–1012 (1999).

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Metrologia (3)

D. Piester, A. Bauch, L. Breakiron, D. Matsakis, B. Blanzano, and O. Koudelka, “Time transfer with nanosecond accuracy for the realization of international atomic time,” Metrologia 45, 185–198 (2008).
[Crossref]

S. Weyers, V. Gerginov, N. Nemitz, R. Li, and K. Gibble, “Distributed cavity phase frequency shifts of the caesium fountain PTB-CSF2,” Metrologia 49, 82–87 (2012).
[Crossref]

T. P. Heavner, E. A. Donley, F. Levi, G. Costanzo, T. E. Parker, J. H. Shirley, N. Ashby, S. Barlow, and S. R. Jefferts, “First accuracy evaluation of NIST-F2,” Metrologia 51, 174–182 (2014).
[Crossref]

Nature (1)

B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (2)

A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. Hänsch, “Precision measurement of the hydrogen 1S–2S frequency via a 920-km fiber link,” Phys. Rev. Lett. 110, 230801 (2013).
[Crossref]

C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, “Frequency comparison of two high-accuracy Al+ optical clocks,” Phys. Rev. Lett. 104, 070802 (2010).
[Crossref]

Science (3)

N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10−18 instability,” Science 341, 1215–1218 (2013).
[Crossref]

T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Bergquist, “Frequency ratio of Al+ and Hg+ single-ion optical clocks; metrology at the 17th decimal place,” Science 319, 1808–1812 (2008).
[Crossref]

K. Predehl, G. Grosche, S. M. F. Raupach, S. Droste, O. Terra, J. Alnis, T. Legero, T. W. Hänsch, T. Udem, R. Holzwarth, and H. Schnatz, “A 920-kilometer optical fiber link for frequency metrology at the 19th decimal place,” Science 336, 441–444 (2012).
[Crossref]

Soil Science Society of America Journal (1)

E. A. Elias, R. Cichota, H. H. Torriani, and Q. de Jong van Lier, “Analytical soil-temperature model: correction for temporal variation of daily amplitude,” Soil Science Society of America Journal 68, 784–788 (2004).

Other (11)

P. C. T. van der Hoeven and W. N. Lablans, “Grondtemperaturen,” (Koninklijk Nederlands Meteorologisch Instituut, 1992).

A. Brown, R. Silva, and E. Powers, “A GPS receiver designed for carrier-phase time transfer,” in Proceedings of ION National Technical Meeting, Anaheim, California, January, 2000, pp. 32–41.

Made available by Koninklijk Nederlands Meteorologisch Instituut.

Koninklijk Nederlands Meteorologisch Instituut (KNMI), “Cesar, Cabauw experimental site for atmospheric research,” http://www.cesar-observatory.nl/ .

Koninklijk Nederlands Meteorologisch Instituut (KNMI), “KNMI DataCentrum,” https://data.knmi.nl/ .

D. J. P. W. Woudenberg, “Vergelijkende metingen van de grondtemperatuur te De Bilt in 1961,” (KNMI, 1966).

K. Jaldehag, S.-C. Ebenhag, P. O. Hedekvist, and C. Rieck, “Time and frequency transfer using asynchronous fiber-optical networks: progress report,” in Proceedings of the 41st Annual Precise Time and Time Interval (PTTI) Meeting (2009), pp. 383–396.

D. W. Allan and M. A. Weiss, “Accurate time and frequency transfer during common-view of a GPS satellite,” in Proceedings of the 34th Annual Frequency Control Symposium (1980), pp. 334–346.

V. S. Zhang, T. E. Parker, and M. A. Weiss, “Multi-channel GPS/GLONASS common-view between NIST and USNO,” in Proceedings of the 2000 IEEE/EIA International Frequency Control Symposium and Exhibition(2000), pp. 598–606.

D. W. Hanson, “Fundamentals of two-way time transfer by satellite,” in Proceedings of the 43th Annual Symposium on Frequency Control (1989), pp. 174–178.

V. Zhang, T. E. Parker, J. Achkar, A. Bauch, L. Lorini, D. Matsakis, D. Piester, and D. G. Rovera, “Two-way satellite time and frequency transfer using 1  Mchip/s codes,” in Proceedings of the 41st Annual Precise Time and Time Interval (PTTI) Meeting (2009), pp. 371–382.

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Figures (12)

Fig. 1.
Fig. 1. Top: schematic map of the SURFnet fiber-optic network. The fiber link between VU University Amsterdam and Van Swinderen Institute Groningen is shown by the green line. The green squares are amplifier sites. the red circle is the KNMI measurement site at Cabauw. Bottom: schematic representation of the duplex fiber link between Amsterdam and Groningen in the SURFnet network (length: 2 × 295 km ) . The unidirectional Er 3 + amplifiers are used by all active DWDM channels. In Amsterdam, 500 m of intra-office fiber bridges the distance between the SURFnet node and the laboratory. In Groningen, 2 km of additional underground fiber is needed to bridge the distance from the SURFnet node at the computing center of the University of Groningen to the laboratory, adding to a total link length of 2 × 298 km .
Fig. 2.
Fig. 2. Amsterdam–Groningen fiber link, overview of the experimental setup. This arrangement allows for optical-versus-GPS comparisons (essentially a fiber-optic frequency comparison of the GPS-linked Rb clocks in Amsterdam and Groningen), and for measurements of the round-trip stability of the fiber-link in Amsterdam. Details of the CW laser lock setup are given in Fig. 3. The setup for the round-trip analysis and for the optical versus Rb/GPS comparison are shown in detail in Fig. 4 and Fig. 5, respectively.
Fig. 3.
Fig. 3. CW-laser stabilization setup (Amsterdam). The 1559.79 nm, 3 kHz (Lorentzian linewidth) diode laser is frequency stabilized by a phase-locked loop to a mode of the Er 3 + -fiber frequency comb laser. The photodiode signal of the fiber-coupled beat unit is amplified, filtered, and split by a 3 dB power splitter (S) for input to the phase detector and the counter. The stabilization setup is fully referenced to the GPS-disciplined Rb frequency standard. The monitor ports are used to observe optical power variations of the Planex laser before and after the variable optical attenuator (VOA), which regulates the laser power to a constant level before injection into the telecommunication network.
Fig. 4.
Fig. 4. Experimental setup for the characterization of the passive frequency stability of the fiber link (Amsterdam). For the long round-trip measurements, the free-space AOM unit (300 MHz) was replaced by a fiber-coupled AOM ( 42 MHz ). In both cases the AOM was driven by a Rb-referenced DDS unit with a set accuracy of 3.55 μHz . Frequency deviations of the link are recorded with a Rb-referenced counter.
Fig. 5.
Fig. 5. Experimental setup for the remote optical frequency measurement (Groningen). Of the received optical power, 90% is sent back to Amsterdam. To improve the signal of the free-space beat unit, the link light is amplified with a BOA-6434 semiconductor optical amplifier (SOA). The amplified light is then combined with light from the fiber frequency comb laser in a free-space beat unit to obtain an rf beat between the nearest frequency comb mode and the CW link laser.
Fig. 6.
Fig. 6. Comparison of two round-trip stability measurements (ODEV, here denoted as Overlapping σ y ) for averaging time τ . Measurement of almost nine days from 30 September 2013 to 9 October 2013 (solid, outliers due to accidental low beat signal in this period where taken out and replaced with the median of the dataset, see text). The peak at 0.5 days and dip at 1 day are typical for frequency deviations with a one-day period. Measurement of more than 13 h performed on 6 July 2012, all data were included (dashed).
Fig. 7.
Fig. 7. Comparison of ODEVs of the in-loop link laser stability relative to the frequency comb (short dashed), the round-trip stability (long dashed), and the remote link laser frequency stability measured in Groningen (solid) of the 13 h 2012 measurement series, divided by 2 , giving the Rb clock stabilities (dotted). The (red) straight dashed lines schematically indicate the Rb clock limit (SRS PRS10 datasheet), and the TWSTFT limit reported in [14].
Fig. 8.
Fig. 8. Fiber-link round-trip stability (dashed–dotted) compared with round-trip stabilities as calculated from the KNMI soil temperature measurements for different fiber depths: on the surface (solid curve), at 20 cm depth (long-dashed curve), at 30 cm depth (short-dashed curve), and at 50 cm depth (dotted curve). At shorter averaging times, the model curves display a 1 / τ slope, which indicates that, on shorter time scales, temperature noise is significantly more prominent in the KNMI measurements than in the temperature-dependent link stability.
Fig. 9.
Fig. 9. Frequency deviations after a roundtrip through the fiber link (solid) and frequency deviations calculated from the soil temperature data at 20 cm (dashed) and 30 cm depth (dotted).
Fig. 10.
Fig. 10. Fit parameters obtained by least-squares fitting to (partly overlapping) 24 h subsets of round-trip frequency data, with a spacing of 6 h between each subset. Day of year represents the center of the data range. Top, frequency offset for fit. Bottom three panels, values of the c n for the most important depths; the c n found for the other depths are negligibly small. Averages and standard deviations over this dataset are f offset = 8.1 ( 20.3 ) c 20 cm = 0.13 ( 0.17 ) , c 30 cm = 1.01 ( 0.72 ) , and c 50 cm = 0.10 ( 0.24 ) .
Fig. 11.
Fig. 11. Annual variation of soil temperature at various depths. Modeled temperature at the surface (solid, appearing as a wide band due to diurnal variations, which are not resolved at the time scale of the plot), and at 50 cm depth (dashed). Measured temperature at the surface (dotted) and at 50 cm depth (dashed–dotted). Temperatures at 50 cm depth are offset by 10 deg centigrade for visibility; arrows indicate true position. The inset shows a 10-day subset of the data to visualize the diurnal variations of the surface temperature.
Fig. 12.
Fig. 12. Frequency stability comparison between soil temperature measurements and the model [Eq. (11)] at several depths: surface measurement (solid), model (solid gray); 50 cm depth measurement (long-dashed), model (long-dashed gray); 200 cm depth model (short-dashed gray). The straight lines indicate the 1 / τ behavior with a maximum instability of 2.6 × 10 12 at half a day for the diurnal variation (dashed) and 2.5 × 10 14 at half a year for the annual variation (dotted) of the sinusoidal model.

Tables (1)

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Table 1. Parameters of the Soil Temperature Model of Eq. (11) Retrieved by a Least-Squares Fit to Data Obtained from [27] and [33]

Equations (11)

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φ = ω 0 c n L ,
d φ d t = ω 0 c ( L n T d T d t + n L T d T d t ) .
1 L L T = α Λ ,
n T = α n ,
T ( z , t ) = T 0 + A T 0 e z C φ × sin ( 2 π P T 0 ( t t 0 ) z C φ ) ,
C φ = 1 C s π P T 0
C s = λ / ρ C m .
Δ f = 2 π f 0 L c ( α n + n α Λ ) Δ T Δ t .
y = Δ f n = 0 N c n Δ f n ( T KNMI , n ) + f offset .
A T d , year ( t ) = T d , year + A d , year sin ( 2 π P year ( t t 0 , year ) ) ,
T annual ( z , t ) = T 0 + T day ( z , t , A T d , year ( t ) ) + T year ( z , t ) ,

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