(1) Department of Geology and
Geophysics and Minnesota Supercomputing
Institute, University of Minnesota, MN USA
(2) Dept.
Physique, University of Montreal, Montreal, Quebec, Canada
(3) Department of Geophysics,
Faculty of Mathematics and Physics, Charles University, Prague, Czech
Republic
(4) JAMSTEC,
Yokosuka, Japan
Wavelets are linear mathematical transformations (e.g. Resnikoff and Wells, 1998) which can analyze both temporal signals and spatial images at different scales. The wavelet transform is sometimes called a mathematical microscope. Large wavelets give an approximate image of the signal, while smaller and smaller wavelets zoom in on small details. Until recently most of the applications of wavelets in geophysics have been focused on the use of one-dimensional wavelets to analyse time-series of the Chandler wobble or one-dimensional spatial tracks, such as topography and gravity anomalies. Recently fast multi-dimensional wavelet transforms(Bergeron et al., 1999, 2000 a and b, Yuen et al., 2000), based on second-derivatives of the Gaussian function, have been developed and they have allowed us to construct rapidly two- and three-dimensional wavelet-transforms of geophysically relevant fields, such as geoid anomalies, temperature-fields in high Rayleigh number convection, and mixing of passive heterogeneities. With wavelet transform one can view the character of a field at a particular length-scale. This continuous transform is different from orthogocal wavelet functions, which locks on to a particular hierarchy of length-scales, like a lattice. We have constructed scalograms of the two-dimensional wavelet-transformed quantities of the geoid anomalies, temperature fields in thermal convection and the scalar field depicting the hetereogeneities in mixing driven by thermal convection. With these scalograms one can discern the detail changes for the different length scales ranging from long wavelength to short wavelength, ranging from about one-tenth to one-hundredth of the entire domain. We have also constructed two proxy field variables, which are associated with the maximum of the wavelet transformed quantity, E-max, and the associated local wavenumber of the wavelet, k-max. Both E-max and k-max are functions of the spatial variables and their peak functional values indicate places with sharp variations in the field values, be it the geoid, seismic velocity anomalies or the thermal fields in convection. Both mixing dynamics and thermal convection in high Rayleigh numbers display different behavior at different length-scales, as revealed by the wavelet-transformation. Long-wavelength anomalies are mixed much faster than intermediate wavelength anomalies. The thickness of the thermal boundary layer depends also on the length-scale of the wavelet interrogation.
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References
[1] Resnikoff, H.L. and R.O. Welss, Jr., Wavelet Analysis: The Scalable
Structure of Information, 435 pp., Springer Verlag, New York, 1998.
[2] Bergeron, S.Y., Vincent, A.P. , Yuen, D.A., Tranchant, B.J.S. and C.
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[5] Yuen, D.A., Vincent, A.P., Bergeron, S.Y., Dubuffet, F., Ten, A.A.,
Steinbach, V.C. and L. Starin, Crossing of scales and non-linearities in
geophysical processes, in Problems in Geophysics for the New Millenium, Ed. by
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