Geology 6600/7600 Signal Analysis 09 Oct 2015 © A.R. Lowry 2015 Last time: A Periodogram is the squared modulus of the signal FFT! Blackman-Tukey estimates autocorrelation from signal, then FFT’s: If triangle-tapered ( 1/N instead of 1/N–l ), this gives a spectrum identical to the periodogram!

Geology 5600/6600 Signal Analysis 09 Oct 2015 Last time Cont’d: PSE (Tapers and Multitapers) Two significant problems with Periodogram or Blackman- Tukey power spectral estimates include: (1) Large variance in large lags of estimated ACF (& hence at ALL wavelengths of PSE) (2) Bias, or leakage, of power from a given frequency to nearby frequencies in the main lobe or to side lobes of the sinc 2 function convolved with the “true” PSE Can reduce (1) with stacking if there are redundant data Can reduce (2) with longer data set, if stationary & available Tapering multiplies a finite data window by a different function (i.e., not a box) with different FT that reduces some aspect of leakage (usually, smaller side lobes). The Multitaper Method multiplies by several minimum-bias orthogonal tapers and sums resulting PSEs! © A.R. Lowry 2015

The Multitaper Method, or MTM (Thomson), partially overcomes both the variance problem and the bias problem by (1) generating multiple power spectra from the same signal using orthogonal, minimum bias (Slepian) tapers, and (2) averaging these independently tapered spectra to form the final spectral estimate. The Slepian tapers (also called discrete prolate spheroidal sequences) look like this: Note that each successively higher-order taper has one additional zero-crossing. In practice, spectra are most commonly generated by averaging over the first three, five, or seven tapers…

An example comparison of a nine-taper MTM (red) with a periodogram (black) power spectral estimate:

Wavelet Methods: Suppose we have a nonstationary random process (e.g., an output sequence for which the linear system is changing in time or space… For example, lithospheric response with changing flexural rigidity). We could use small, moving windows to examine the changes, but these will suffer from variance/bias effects as previously noted (& particularly at lags/wavelengths ~ the window dimension). We could use different window sizes, centered at the same point, to represent different frequencies (but then bias is changing in a complicated way with frequency, and other problems may arise). Alternatively, one could get really clever and design localization approaches that capture localized properties at multiple scales. These are called wavelet methods …

These notes adapted from notes by Jon Kirby. Jon uses fan- wavelets to get flexural response of the lithosphere, e.g., Kirby & Swain, JGR, But there are many types… Morlet wavelets are (complex-valued) sinusoids multiplied by a Gaussian exponential: References include: Morlet et al. (1982a) Geophysics, 47(2): Morlet et al. (1982b) Geophysics, 47(2): Grossmann & Morlet (1984) SIAM J. Math. Anal., 15(4): Real-valued MorletComplex Morlet

CWT in spatial domain * = * = small scalelarge scale Derivative of Gaussian The wavelet transform is a convolution in the time/space domain = multiplication in the frequency/wavenumber domain: Continuous Wavelet Transform: Real

CWT in wavenumber domain small scale X F -1 X large scale Derivative of Gaussian The wavelet transform is a convolution in the time/space domain = multiplication in the frequency/wavenumber domain: Continuous Wavelet Transform: Real

CWT in spatial domain * small scale * large scale == Morlet The wavelet transform is a convolution in the time/space domain = multiplication in the frequency/wavenumber domain: Continuous WT: Complex

CWT in wavenumber domain X small scale large scale X F -1 Morlet The wavelet transform is a convolution in the time/space domain = multiplication in the frequency/wavenumber domain: Continuous WT: Complex

CWT in wavenumber domain X  F -1 X small scale large scale Morlet  F -1 The wavelet transform is a convolution in the time/space domain = multiplication in the frequency/wavenumber domain: Continuous wavelet transform: Fan

Power Spectral Profiles Derivative of Gaussian