Options
Ergodicity of the recent geomagnetic field
Language
English
Obiettivo Specifico
3.4. Geomagnetismo
Status
Published
JCR Journal
JCR Journal
Peer review journal
Yes
Title of the book
Issue/vol(year)
3-4/186 (2011)
Publisher
Elsevier
Pages (printed)
103-110
Issued date
June 2011
Abstract
The geomagnetic field is a fundamental property of our planet: its study would allow us to understand
those processes of Earth’s interior, which act in its outer core and produce the main field. Knowledge
of whether the field is ergodic, i.e. whether time averages correspond to phase space averages, is an
important question since, if this were true, it would point out a strong spatio-temporal coupling amongst
the components of the dynamical system behind the present geomagnetic field generation. Another consequence
would be that many computations, usually undertaken with many difficulties in the phase
space, can be made in the conventional time domain. We analyse the temporal behaviour of the deviation
between predictive and definitive geomagnetic global models for successive intervals from 1965 to 2010,
finding a similar exponential growth with time. Also going back in time (at around 1600 and 1900 by
using the GUFM1 model) confirms the same findings. This result corroborates previous chaotic analyses
made in a reconstructed phase space from geomagnetic observatory time series, confirming the chaotic
character of the recent geomagnetic field with no reliable prediction after around 6 years from definitive
values, and disclosing the potentiality of estimating important entropic quantities of the field by time
averages. Although more tests will be necessary, some of our analyses confirm the efforts to improve
the representation of the geomagnetic field with more detailed secular variation and acceleration.
those processes of Earth’s interior, which act in its outer core and produce the main field. Knowledge
of whether the field is ergodic, i.e. whether time averages correspond to phase space averages, is an
important question since, if this were true, it would point out a strong spatio-temporal coupling amongst
the components of the dynamical system behind the present geomagnetic field generation. Another consequence
would be that many computations, usually undertaken with many difficulties in the phase
space, can be made in the conventional time domain. We analyse the temporal behaviour of the deviation
between predictive and definitive geomagnetic global models for successive intervals from 1965 to 2010,
finding a similar exponential growth with time. Also going back in time (at around 1600 and 1900 by
using the GUFM1 model) confirms the same findings. This result corroborates previous chaotic analyses
made in a reconstructed phase space from geomagnetic observatory time series, confirming the chaotic
character of the recent geomagnetic field with no reliable prediction after around 6 years from definitive
values, and disclosing the potentiality of estimating important entropic quantities of the field by time
averages. Although more tests will be necessary, some of our analyses confirm the efforts to improve
the representation of the geomagnetic field with more detailed secular variation and acceleration.
References
Anderson, J.G., Brune, J.N., 1999. Probabilistic hazard analysis without the ergodic
assumption. Seismol. Res. Lett 70, 19–23. Anosov, D.V., 2001. Ergodic theory. In: Hazewinkel, M. (Ed.), Encyclopaedia of
Mathematics. Springer, ISBN 978-1556080104.
Baranger, M., Latora, V., Rapisarda, A., 2002. Time evolution of thermodynamic
entropy for conservative and dissipative chaotic maps. Chaos Soliton. Fract. 13,
471–478.
Barraclough, D.R., De Santis, A., 1997. Some possible evidence for a chaotic
geomagnetic field from observational data. Phys. Earth Planet. Int. 99, 207–220.
Buonomano, V., Bartmann, F., 1986. Testing ergodic assumption in the low-intensity
interferences experiments. Il Nuovo Cimento 95 (2), 99–108.
Chen, Y.-H., Tsai, C.-C.P., 2002. A new method of the attenuation relationship with
variance components. Bull. Seismol. Soc. Am. 92 (5), 1984–1991.
Cross, M.C., Hohenberg, P.C., 1994. Spatiotemporal chaos. Science 263, 1569.
De Santis, A., 2007. How persistent is the present trend of the geomagnetic field to
decay and, possibly, to reverse? Phys. Earth Planet. Inter. 162, 217–226.
De Santis, A., 2008. Erratum to ‘‘How persistent is the present trend of the
geomagnetic field to decay and, possibly, to reverse?’’. Phys. Earth Planet. Inter.
170, 149.
De Santis, A., Barraclough, D.R., Tozzi, R., 2002. Nonlinear variability of the recent
geomagnetic field. Fractals 10, 297–303.
De Santis, A., Barraclough, D.R., Tozzi, R., 2003. Spatial and temporal spectra of the
geomagnetic field and their scaling properties. Phys. Earth Planet. Inter. 135,
125–134.
De Santis, A., Tozzi, R., Gaya-Piqué, L.R., 2004. Information content and K-Entropy of
the present geomagnetic field. Earth Planet. Science Lett. 218, 269–275.
De Santis, A., Qamili, E., 2010. Shannon information of the geomagnetic field for the
past 7000 years. Nonlinear Proc. Geophys. 17, 77–84.
De Santis, A., Cianchini, G., Qamili, E., Frepoli, A., 2010. The 2009 L’Aquila (Central
Italy) seismic sequence as a chaotic process. Tectonophysics 496, 44–52.
Eckmann, J.P., Ruelle, D., 1985. Ergodic theory of chaos and strange attractors. Part I.
Rev. Mod. Phys. 57 (3), 617–656.
Egolf, D., 2000. Equilibrium regained: from nonequilibrium chaos to statistical
mechanics. Science 287, 101–104.
Finlay, C.C., Maus, S., Beggan, C.D., Hamoudi, M., Lowes, F.J., Olsen, N., Thebault, E.,
2010. Evaluation of candidate geomagnetic field models for IGRF-11. Earth
Planets Space 62, 787–804.
Hongre, L., Sailhac, P., Alexandrescu, M., Dubois, J., 1999. Non linear and multifractal
approaches of the geomagnetic field. Phys. Earth Planet. Inter. 110, 157–190.
Jackson, A., Jonkers, A.R.T., Walker, M.R., 2000. Four centuries of geomagnetic
secular variation from historical records. Philos. Trans. R. Soc. Lond. A 358, 957–
990.
Kantz, H., Schreiber, T., 1997. Nonlinear Time Series Analysis. Cambridge University
Press.
Mandea, M., Holme, R., Pais, A., Pinheiro, K., Jackson, A., Verbanac, G., 2010.
Geomagnetic jerks: rapid core field variations and core dynamics. Space Sci.
Rev. 155, 147–175.
Maus, S., Macmillan, S., Chernova, T., Choi, S., Dater, D., Golovkov, V., Lesur, V.,
Lowes, F., Luhr, H., Mai, W., McLean, S., Olsen, N., Rother, M., Sabaka, T.,
Thomson, A., Zvereva, T., 2005. The 10th-generation international geomagnetic
reference field. Geophys. J. Int. 161, 561–565.
Maus, S., Rother, M., Stolle, C., Mai, W., Choi, S., Lühr, H., Cooke, D., Roth, C., 2006.
Third generation of the Potsdam Magnetic Model of the Earth (POMME).
Geochem. Geophys. Geosyst. 7, Q07008, doi:10.1029/2006GC001269.
Maus, S., Silva, L., Hulot, G., 2008. Can core-surface flow models be used to improve
the forecast of the Earth’s main magnetic field? J. Geophys. Res. 113, B08102,
doi:10.1029/2007JB005199.
Maus, S., Manoj, C., Rauberg, J., Michaelis, I., Lühr, H., 2010. NOAA/NGDC candidate
models for the 11th generation International Geomagnetic Reference Field and
the concurrent release of the 6th generation POMME magnetic model. Earth
Planets Space 62, 729–735.
McLean, S., Macmillan, S., Maus, S., Lesur, V., Thomson, A., Dater, D., 2004. The US/
UK World Magnetic Model for 2005–2010. NOAA Technical Report NESDID/
NGDC-1.
Olsen, N., Luhr, H., Sabaka, T.J., Mandea, M., 2006. CHAOS – a model of Earth’s
magnetic field derived from CHAMP, Oersted, and SAC-C magnetic satellite
data. Geophys. J. Int. 166, 67–75.
Olsen, N., Mandea, M., Sabaka, T.J., Tøffner-Clausen, L., 2009. CHAOS-2 – a
geomagnetic field model derived from one decade of continuous satellite
data. Geophys. J. Int. 179, 1477–1487.
Paine, A.D.M., 1985. Ergodic reasoning in geomorphology: time for a review of the
term? Prog. Phys. Geogr. 9 (1), 1–15.
Sabaka, T.J., Olsen, N., Purucker, M.E., 2004. Extending comprehensive models of the
Earth’s magnetic field with Ørsted and CHAMP data. Geophys. J. Int. 159, 521–
547.
Schuster, H.G., 1995. Deterministic Chaos: An Introduction, third ed. Wiley-VCH,
Weinheim, p. 291.
Silva, L., Maus, S., Hulot, G., Thebault, E., 2010. On the possibility of extending the
IGRF predictive secular variation model to a higher SH degree. Earth Planets
Space 62, 815–820.
Sugihara, G., May, R.M., 1990. Nonlinear forecasting as a way of distinguish chaos
from measurement error in time series. Nature 344, 734–741.
Takens, F., 1981. Detecting strange attractors in turbulence. In: Rand, D.A., Young,
L.S. (Eds.), Lecture Notes in Mathematics, vol. 898. Springer, Berlin, p. 366.
Wales, D.J., 1991. Calculating the rate of loss information from chaotic time series
by forecasting. Nature 350, 485–488.
Walters, P., 1982. An Introduction to Ergodic Theory. Springer, New York.
Woo, G., Wang, G.-Q., Tang, G., 2006. Insights from Parkfield array data for
probabilistic risk modelling. In: Proceedings of the 8th US National Conference
on Earthquake Engineering, San Francisco, CA, USA, Paper No. 219, p. 8.
assumption. Seismol. Res. Lett 70, 19–23. Anosov, D.V., 2001. Ergodic theory. In: Hazewinkel, M. (Ed.), Encyclopaedia of
Mathematics. Springer, ISBN 978-1556080104.
Baranger, M., Latora, V., Rapisarda, A., 2002. Time evolution of thermodynamic
entropy for conservative and dissipative chaotic maps. Chaos Soliton. Fract. 13,
471–478.
Barraclough, D.R., De Santis, A., 1997. Some possible evidence for a chaotic
geomagnetic field from observational data. Phys. Earth Planet. Int. 99, 207–220.
Buonomano, V., Bartmann, F., 1986. Testing ergodic assumption in the low-intensity
interferences experiments. Il Nuovo Cimento 95 (2), 99–108.
Chen, Y.-H., Tsai, C.-C.P., 2002. A new method of the attenuation relationship with
variance components. Bull. Seismol. Soc. Am. 92 (5), 1984–1991.
Cross, M.C., Hohenberg, P.C., 1994. Spatiotemporal chaos. Science 263, 1569.
De Santis, A., 2007. How persistent is the present trend of the geomagnetic field to
decay and, possibly, to reverse? Phys. Earth Planet. Inter. 162, 217–226.
De Santis, A., 2008. Erratum to ‘‘How persistent is the present trend of the
geomagnetic field to decay and, possibly, to reverse?’’. Phys. Earth Planet. Inter.
170, 149.
De Santis, A., Barraclough, D.R., Tozzi, R., 2002. Nonlinear variability of the recent
geomagnetic field. Fractals 10, 297–303.
De Santis, A., Barraclough, D.R., Tozzi, R., 2003. Spatial and temporal spectra of the
geomagnetic field and their scaling properties. Phys. Earth Planet. Inter. 135,
125–134.
De Santis, A., Tozzi, R., Gaya-Piqué, L.R., 2004. Information content and K-Entropy of
the present geomagnetic field. Earth Planet. Science Lett. 218, 269–275.
De Santis, A., Qamili, E., 2010. Shannon information of the geomagnetic field for the
past 7000 years. Nonlinear Proc. Geophys. 17, 77–84.
De Santis, A., Cianchini, G., Qamili, E., Frepoli, A., 2010. The 2009 L’Aquila (Central
Italy) seismic sequence as a chaotic process. Tectonophysics 496, 44–52.
Eckmann, J.P., Ruelle, D., 1985. Ergodic theory of chaos and strange attractors. Part I.
Rev. Mod. Phys. 57 (3), 617–656.
Egolf, D., 2000. Equilibrium regained: from nonequilibrium chaos to statistical
mechanics. Science 287, 101–104.
Finlay, C.C., Maus, S., Beggan, C.D., Hamoudi, M., Lowes, F.J., Olsen, N., Thebault, E.,
2010. Evaluation of candidate geomagnetic field models for IGRF-11. Earth
Planets Space 62, 787–804.
Hongre, L., Sailhac, P., Alexandrescu, M., Dubois, J., 1999. Non linear and multifractal
approaches of the geomagnetic field. Phys. Earth Planet. Inter. 110, 157–190.
Jackson, A., Jonkers, A.R.T., Walker, M.R., 2000. Four centuries of geomagnetic
secular variation from historical records. Philos. Trans. R. Soc. Lond. A 358, 957–
990.
Kantz, H., Schreiber, T., 1997. Nonlinear Time Series Analysis. Cambridge University
Press.
Mandea, M., Holme, R., Pais, A., Pinheiro, K., Jackson, A., Verbanac, G., 2010.
Geomagnetic jerks: rapid core field variations and core dynamics. Space Sci.
Rev. 155, 147–175.
Maus, S., Macmillan, S., Chernova, T., Choi, S., Dater, D., Golovkov, V., Lesur, V.,
Lowes, F., Luhr, H., Mai, W., McLean, S., Olsen, N., Rother, M., Sabaka, T.,
Thomson, A., Zvereva, T., 2005. The 10th-generation international geomagnetic
reference field. Geophys. J. Int. 161, 561–565.
Maus, S., Rother, M., Stolle, C., Mai, W., Choi, S., Lühr, H., Cooke, D., Roth, C., 2006.
Third generation of the Potsdam Magnetic Model of the Earth (POMME).
Geochem. Geophys. Geosyst. 7, Q07008, doi:10.1029/2006GC001269.
Maus, S., Silva, L., Hulot, G., 2008. Can core-surface flow models be used to improve
the forecast of the Earth’s main magnetic field? J. Geophys. Res. 113, B08102,
doi:10.1029/2007JB005199.
Maus, S., Manoj, C., Rauberg, J., Michaelis, I., Lühr, H., 2010. NOAA/NGDC candidate
models for the 11th generation International Geomagnetic Reference Field and
the concurrent release of the 6th generation POMME magnetic model. Earth
Planets Space 62, 729–735.
McLean, S., Macmillan, S., Maus, S., Lesur, V., Thomson, A., Dater, D., 2004. The US/
UK World Magnetic Model for 2005–2010. NOAA Technical Report NESDID/
NGDC-1.
Olsen, N., Luhr, H., Sabaka, T.J., Mandea, M., 2006. CHAOS – a model of Earth’s
magnetic field derived from CHAMP, Oersted, and SAC-C magnetic satellite
data. Geophys. J. Int. 166, 67–75.
Olsen, N., Mandea, M., Sabaka, T.J., Tøffner-Clausen, L., 2009. CHAOS-2 – a
geomagnetic field model derived from one decade of continuous satellite
data. Geophys. J. Int. 179, 1477–1487.
Paine, A.D.M., 1985. Ergodic reasoning in geomorphology: time for a review of the
term? Prog. Phys. Geogr. 9 (1), 1–15.
Sabaka, T.J., Olsen, N., Purucker, M.E., 2004. Extending comprehensive models of the
Earth’s magnetic field with Ørsted and CHAMP data. Geophys. J. Int. 159, 521–
547.
Schuster, H.G., 1995. Deterministic Chaos: An Introduction, third ed. Wiley-VCH,
Weinheim, p. 291.
Silva, L., Maus, S., Hulot, G., Thebault, E., 2010. On the possibility of extending the
IGRF predictive secular variation model to a higher SH degree. Earth Planets
Space 62, 815–820.
Sugihara, G., May, R.M., 1990. Nonlinear forecasting as a way of distinguish chaos
from measurement error in time series. Nature 344, 734–741.
Takens, F., 1981. Detecting strange attractors in turbulence. In: Rand, D.A., Young,
L.S. (Eds.), Lecture Notes in Mathematics, vol. 898. Springer, Berlin, p. 366.
Wales, D.J., 1991. Calculating the rate of loss information from chaotic time series
by forecasting. Nature 350, 485–488.
Walters, P., 1982. An Introduction to Ergodic Theory. Springer, New York.
Woo, G., Wang, G.-Q., Tang, G., 2006. Insights from Parkfield array data for
probabilistic risk modelling. In: Proceedings of the 8th US National Conference
on Earthquake Engineering, San Francisco, CA, USA, Paper No. 219, p. 8.
Type
article
File(s)
No Thumbnail Available
Name
Ergodicity_PEPI2011.pdf
Size
424.61 KB
Format
Adobe PDF
Checksum (MD5)
e5cb1f1ac2aee3a78316bcd0f8a75d69