Abstract
1 Department of Geophysics, Graduate School of Science, Kyoto University, Sakyo, Kyoto, Japan * Corresponding author address: Prof. Taishi Okamoto, Department of Geophysics, Graduate School of Science, Kyoto University, Sakyo, Kyoto, Japan; E-mail: dollar@kugi.kyoto-u.ac.jp doi: 10.3319/TAO.2007.18.4.681(T) Yamada and Yomogida (1997) applied the discrete wavelet transform (DWT) to group velocity measurements for the first time. Although their study is one of the pioneering works in application of DWT to seismological analysis, their method gives an inaccurate value as a group velocity in some cases and requires modification. In this report, we point out the problem and propose a modified DWT method for overcoming the problem. In our method, DWT is carried out not for an analysed signal itself but for its complex envelope (Farnbach 1975). A computation algorithm for DWT coefficients for our method is given and shown to be almost the same as that by Yamada and Ohkitani (1991). The influence of the difference between the conventional method and our method on identification of group arrival times of a wave is also shown by a numerical experiment. If analysts want to identify group arrival times using DWT, our method must be adopted instead of the conventional method. (
Highlights
The discrete wavelet transform (DWT) and the continuous wavelet transform (CWT) are methods for time-frequency analysis of time history data, and a squared DWT coefficient of the data is supposed to represent energy in corresponding time and frequency range (e.g., Yamada and Ohkitani 1991), because some types of wavelet functions can form an orthonormal basis for the function space L2(R), the set of square-integrable functions
Note that DWT coefficients are not computed for an original signal with a complex-valued wavelet, and that the wavelet still has the orthonormality
DWT coefficients are computed for a complex envelope of an original signal with a conventional real-valued orthonormal wavelet
Summary
The discrete wavelet transform (DWT) and the continuous wavelet transform (CWT) are methods for time-frequency analysis of time history data, and a squared DWT coefficient of the data is supposed to represent energy in corresponding time and frequency range (e.g., Yamada and Ohkitani 1991), because some types of wavelet functions can form an orthonormal basis for the function space L2(R), the set of square-integrable functions. Yamada and Yomogida (1997) applied the DWT to group velocity measurement of dispersed surface waves based on this idea They measure group velocities, defining a time for which an absolute value of a DWT coefficient for an observed or synthetic record has a relative maximum as a group arrival time of a wave having the same frequency as each wavelet. Except for the difference, the methods used in the two studies seem to be the same, because in both studies a time for which an absolute value of a wavelet coefficient for a dispersed wave has a relative maximum is interpreted as a group arrival time of a wave having the same frequency as each wavelet They are by no means the same, it is not important whether CWT is used or DWT is used. The influence of this difference on wavelet coefficients is described below
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