Wavelet Spectral Analysis - University of Colorado Boulder：小波功率谱分析-科罗拉多大学博尔德分校.ppt

Summary Wavelets offer frequency-time localization of spectral power SAWP visualizes how power changes for a given scale or band as a time series “Band pass” reconstructions can be performed from the wavelet transform WARM is an attractive simulation method that captures spectral features Summary Cross wavelet transform can offer phase and coherence between data sets Additional Resources: /research/wavelets/ /~nowakkc/wave J is range of scales over which to reconstruct C and psi are empirically derived wavelet values * * Similar spectral results for K-NN * * * Denotes complex conjugate * Instead of top term being “average” across 2 time series, values from square of XWT are “point values” thus we need to smooth over some range in time and frequency to see how 2 data sets relate “locally” also, wavelet transforms must be normalized by scale. * Wavelet Spectral Analysis Ken Nowak 7 December 2010 Need for spectral analysis Many geo-physical data have quasi-periodic tendencies or underlying variability Spectral methods aid in detection and attribution of signals in data Fourier Approach Limitations Results are limited to global Scales are at specific, discrete intervals Per fourier theory, transformations at each scale are orthogonal Wavelet Basics Wf(a,b)= f(x) y(a,b) (x) dx ? Morlet wavelet with a=0.5 Function to analyze Integrand of wavelet transform |W(a=0.5,b=6.5)|2=0 |W(a=0.5,b=14.1)|2=.44 b=2 b=6.5 b=14.1 graphics courtesy of Matt Dillin ∫ Wavelets detect non-stationary spectral components Wavelet Basics Here we explore the Continuous Wavelet Transform (CWT) No longer restricted to discrete scales Ability to see “local” features Mexican hat wavelet Morlet wavelet Global Wavelet Spectrum |Wf (a,b)|2 function Wavelet spectrum a=2 a=1/2 Global wavelet spectrum Slide courtesy of Matt Dillin Wavelet Details Convolutions between wavelet and data can be computed simultaneously via convolution theorem. Wavelet transfor