Multiresolution Analysis of Shell Growth Increments to Detect Variations in Natural Cycles

Summary

Conventional spectral analysis is mainly based on Fourier transform. This kind of transform provides excellent information in terms of frequencies (with their associated amplitudes) constituting the original signal, but does not keep the spatial information: it is possible to determine the elementary bricks that compose the signal, but not the way they are ordered along the signal. This limitation of the method has been noticed by Gabor (1946) who proposed a sliding window along the signal, in which the Fourier transform could be performed. In this way, part of the local (spatial) information is not lost. Nevertheless, this time-frequency tiling is still rigid and not really appropriate for natural complex signals. In the eighties, mathematicians introduced the concept of wavelet transform. The wavelet is a localized function, sort of a probe, capable of dilation (spreading out of the wavelet along the Oy axis) and translation (along the Ox axis). The transformation of the original signal by the wavelet results in coefficients, which are another expression of the signal. In addition, the wavelet transform acts as a mathematical microscope. In the discrete wavelet transform, two wavelets are used: the mother wavelet (the probe) and the scaling function. Therefore, it is possible to observe the signal at various scales, which is equivalent to the extent of the smoothing effect on the signal. This results in approximation coefficients computed by the scaling function. However, the mother wavelet will provide the detail coefficients. In conclusion, the signal is decomposed in two series of coefficients for each scale of observation. This extremely powerful tool has been used to detect cycles in the growth of lacustrine shells. By removing detail and/or approximation coefficients at different scales, and using image reconstruction (the wavelet transform has an inverse wavelet transform), annual, seasonal, tidal (monthly), and fortnight cycles in shell growth increments can easily be detected. Because of the very low amplitude of some of these cycles, it would not be possible to detect them without using the scaling effect and the detail coefficients associated with the lower scales. This method is much more powerful than the conventional Fourier transform when the aim of the study is to look for specific local periods and scale-sensitive information.