New Research on Cone of Influence and Computing Scheme of Mexican Hat Wavelet Transform

Using the effective domain of wavelet function, the problems of cone of influence (COI) and high-frequency distortion of Mexican Hat wavelet transform in the paper of Torrence and Compo (1998) are analyzed and the solution are explored. The analytic expressions of wavelet coefficient and global wavelet power spectrum for time series of sinusoid and the relationship between period and wavelet scale are deduced. The effective domain of Mexican Hat wavelet transform is b-2.12a, b+2.12a for localized time b and wavelet scale a. The authentic cone of influence for Mexican Hat wavelet transform is 2.12a. The maximum value of wavelet scale is N/4.24 (N is the length of time series). When wavelet scale approaches zero, wavelet function is singular. So the simple sum scheme replacing the integration for wavelet coefficient does not include the singularity of wavelet function, therefore the pseudo high-frequency oscillation is deduced. A new computing scheme for wavelet coefficient is designed. The major elements include improved-resolution and located time b in the new grids. The difference of effective domain of wavelet function for small different wavelet scales can be distinguished by the improved-resolution. Located time b has great contribution to its wavelet coefficient because the wavelet function is the biggest at the located point b. The pseudo significant high-frequency oscillation that is produced by the computing scheme proposed by Torrence and Compo (1998) can be eliminated by the new computing scheme of wavelet coefficient, well using sharply descending characteristics of wavelet function and cubic spline interpolation. The rationality of new computing scheme is tested by the analytic solution of wavelet transform of sinusoid time sequence. The global wavelet power spectrum from the new computing scheme indicates that the variability of winter Niño3.4 index exhibits interdecadal oscillation of about 12 years and interannual oscillation of quasi-four years, and no significant quasi-biennial oscillation. The questions of cone of influence and pseudo high-frequency oscillation have two adverse aftermaths. First is that the correct judgment of interdecadal variation can be affected and second is that pseudo quasi-biennial variation can be deduced. Therefore, the results about COI and the new computing scheme for wavelet coefficient make the conclusions induced from wavelet analysis more credible.