Wavelet Spectral Analysis - University of Colorado Boulder：小波功率譜分析-科羅拉多大學(xué)博爾德分校

1、Wavelet Spectral AnalysisKen Nowak7 December 2010Need for spectral analysisMany geo-physical data have quasi-periodic tendencies or underlying variabilitySpectral methods aid in detection and attribution of signals in dataFourier Approach LimitationsResults are limited to globalScales are at specifi

2、c, discrete intervalsPer fourier theory, transformations at each scale are orthogonal Wavelet BasicsWf(a,b)= f(x) y(a,b) (x) dxMorlet wavelet with a=0.5Function to analyzeIntegrand of wavelet transform|W(a=0.5,b=6.5)|2=0|W(a=0.5,b=14.1)|2=.44b=2graphics courtesy of Matt DillinWavelets detect non-sta

3、tionary spectral componentsWavelet BasicsHere we explore the Continuous Wavelet Transform (CWT)No longer restricted to discrete scalesAbility to see “l(fā)ocal” features Mexican hat wavelet Morlet waveletGlobal Wavelet Spectrum|Wf (a,b)|2functionWavelet spectruma=2a=1/2Global wavelet spectrumSlide court

4、esy of Matt DillinWavelet DetailsConvolutions between wavelet and data can be computed simultaneously via convolution theorem.Wavelet transformWavelet functionAll convolutions at scale “a”Analysis of Lees Ferry DataLocal and global wavelet spectraCone of influence Significance levelsAnalysis of ENSO

5、 DataCharacteristic ENSO timescaleGlobal peakSignificance LevelsBackground Fourier spectrum for red noise process (normalized)Square of normal distribution is chi-square distribution, thus the 95% confidence level is given by:Where the 95th percentile of a chi-square distribution is normalized by th

6、e degrees of freedom. Scale-Averaged Wavelet PowerSAWP creates a time series that reflects variability strength over time for a single or band of scales Band ReconstructionsWe can also reconstruct all or part of the original data PACF indicates AR-1 modelStatistics capture observed values adequately

7、 Spectral range does not reflect observed spectrum Lees Ferry Flow Simulation Wavelet Auto Regressive Method (WARM) Kwon et al., 2007WARM and Non-stationary SpectraPower is smoothed across time domain instead of being concentrated in recent decadesWARM for Non-stationary SpectraResults for Improved

8、WARMWavelet Phase and CoherenceAnalysis of relationship between two data sets at range of scales and through timeCorrelation = .06Wavelet Phase and CoherenceCross Wavelet TransformFor some data X and some data Y, wavelet transforms are given as:Thus the cross wavelet transform is defined as:Phase An

9、gleCross wavelet transform (XWT) is complex. Phase angle between data X and data Y is simply the angle between the real and imaginary components of the XWT: Coherence and CorrelationCorrelation of X and Y is given as:Which is similar to the coherence equation:SummaryWavelets offer frequency-time localization of spectral powerSAWP visualizes how power changes for a given scale or band as a time series“Band pass” reconstructions can be performed from the wavelet transformWARM is an attractive simulation method that capture