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Talk abstract
IMA Tutorial: Spatio-temporal Patterns in the Geosciences
September 24, 2001

Mathematics in the Geosciences, September 2001 - June 2002

Audio Recordings      Material from Talks

Michael Ghil (Professor of Atmospheric Sciences and Director, IGPP/UCLA)  http://www.atmos.ucla.edu/tcd/    ghil@atmos.ucla.edu  or   mghil@igpp.ucla.edu

Spatio-temporal pattern analysis in the atmosphere and oceans    Slides

The study of large-scale atmospheric and oceanic motions depends critically on a description of these motions that should be as reliable and as precise as possible. There exists thus a trade-off between the wealth of information that one wishes to extract from limited data sets, on the one hand, and the statistical confidence in that information, on the other.

This trade-off is optimized by applications of Karhunen- Loeve theory in the space and time domain, separately, as well as concurrently in the spatio-temporal domain. I shall illustrate this approach by applications to climate variability on the intraseasonal (10--100 days), interannual (1--10 years) and interdecadal (10--1000 years) time scales. The applications will involve phenomena that reside mainly in the atmosphere (so-called low-frequency variability), mainly in the ocean (its wind-driven and thermohaline circulation), and finally in the coupled atmosphere-ocean (the El Nino-Southern Oscillation).

Connections between this parsimonious, optimal description of climate variability and its explanation via dynamical systems theory will be outlined. I shall select one of the examples above to illustrate this connection in greater depth.

References

  • Ghil M., R. M. Allen, M. D. Dettinger, K. Ide, D. Kondrashov, M. E. Mann, A. Robertson, A. Saunders, Y. Tian, F. Varadi, and P. Yiou (2001) "Advanced spectral methods for climatic time series," Rev. Geophys., accepted. http://www.atmos.ucla.edu/tcd/MG/mg_ref_preprints.html

  • Pascal Yiou, Didier Sornette and Michael Ghil Data-adaptive wavelets and multi-scale singular-spectrum analysis
    Physica D: Nonlinear Phenomena, Volume 142, Issue 3-4 (2000), pp. 254-290
    [Abstract] [Full text] (PDF 1.1 Mb) (one of the 8 Hottest Papers in Physica D) http://www.elsevier.com/inca/publications/store/5/0/5/7/1/4/

Software (free)

  • SSA-MTM Toolkit for spectral analysis http://www.atmos.ucla.edu/tcd/ssa/ This software will be illustrated during Workshop #3 by one of the co-organizers, Ferenc Varadi.

Vladimir Keilis-Borok (Institute of Geophysics and Planetary Physics and Department of Earth and Space Science, University of California, Los Angeles and International Institute for Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow)  vkb@ess.ucla.edu

Complexity of Lithosphere and Earthquake Prediction

1. Earthquakes occur in the lithosphere - an upper shell of the solid Earth. Its thickness ranges from few km near oceanic ridges, to few hundred km in some continental regions. The lithosphere is set in motion by the large-scale currents in the underlying Earth's mantle and some internal processes like gravitational and chemical differentiation. In seismically active regions large part of this motion is realized through the earthquakes in a stick-slip fashion.

2. Two major factors cause complexity of the lithosphere: (i) Hierarchical structure, extending from about 10 tectonic plates to the about 10^25 grains of rocks. (ii) Instability, caused by a multitude of non-linear mechanisms, controlling the {strength - stress} field. In the time scale relevant to earthquake prediction, these factors turn the lithosphere into a hierarchical dissipative complex system. Strong earthquakes are regarded as the critical phenomena; an earthquake may be a critical phenomenon in certain volume of lithosphere, and a part of the background seismicity in a larger volume.

3. Development of the earthquake prediction algorithms brought together three methodologies: (i) phenomenological analysis of observations - I. Gelfand's type of pattern recognition and J. Tukey's kind of exploratory data analysis; (ii) "universal" lattice models of complex systems such as considered in statistical mechanics and non-linear dynamics; and (iii) Earth-specific models of tectonic faults' networks. In addition, (iv) theory of optimal control is used to link earthquake prediction with the earthquake preparedness.

4. Ongoing global test of the intermediate term prediction algorithms is discussed.


Donald L. Turcotte (Department of Geological Sciences Cornell University)  Turcotte@Geology.Cornell.edu

Spatio-temporal patterns in solid-earth geophysics    Slides

Examples of spatio-temporal patterns considered: Earthquakes, landslides, stratigraphy (sediment deposition), earth's magnetic field. Examples of spatio patterns considered: Oil fields, mineral deposits, faults, river networks. Examples of temporal patterns considered: Time series of river flows (floods), climate.


Eli Tziperman (Department of Environmental Sciences, Weizmann Institute of Science)  eli@beach.weizmann.ac.il  http://www.weizmann.ac.il/~eli

Glacial cycles: extreme natural climate change events, and still an unsolved puzzle...

The major glaciations that have occurred over the earth every 100,000 years during the past 1 million years are the largest natural climate variability signal in recent geological history. During each glaciation, global sea level dropped by 120 meters and this large amount of water has then accumulated as 2-3 km high glaciers over land; global temperature varied by many degrees, and the concentration of CO2 in the atmosphere changed by 30%. Understanding these large amplitude natural climatic events seems essential for us to be able to predict future climate change with reasonable certainty. In spite of the large amplitude and obvious significance of these events we still do not have an accepted theory for them. Some extremely varied theories were proposed for the mechanism of glacial cycles. Some of these theories suggested that the glacial cycles are a result of random noise processes, due to a stochastic resonance, external forcing by variations in solar radiation due to earth orbital changes, due to effects of the elasticity of the earth crust, self-sustained internal variability of the climate system, nonlinear chaotic dynamics, relaxation oscillations due to nonlinear sea ice-land ice interaction, and more. The phenomenology of glacial cycles will be briefly introduced and some of the main theoretical ideas regarding the possible mechanism of these cycles will be presented.


Audio Recordings     Material from Talks

Tutorial: Spatio-temporal Patterns in the Geosciences

Workshop: Spatio-temporal Patterns in the Geosciences

2001-2002 IMA Thematic Year on Mathematics in the Geosciences

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