Geology 6600/7600 Signal Analysis 12 Oct 2015 © A.R. Lowry 2015 Last time: 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! Wavelet methods are used to estimate power spectral properties local to a particular time/location in a nonstationary random variable sequence The wavelet transform convolves a wavelet of a particular scale with the signal (by multiplying the spectral signal amplitudes by the FT of the wavelet, and IFT’ing) Can form a scalogram (localized power spectrum) at a point using the modulus-squared of the resulting WT’s for multiple wavelet scales at that point

Geology 6600/7600 Signal Analysis 08 Oct 2013 © A.R. Lowry 2013 Last time: PSE (Localization with Wavelets) Wavelet methods are used to estimate power spectral properties local to a particular time/location in a nonstationary random variable sequence The wavelet transform convolves a wavelet of a particular scale with the signal (by multiplying the spectral signal amplitudes by the FT of the wavelet, and IFT’ing) Can form a scalogram (localized power spectrum) at a point using the modulus-squared of the resulting WT’s for multiple wavelet scales at that point Wavelet types include Derivative of Gaussian (strictly real), Morlet (complex), Fan (superposition of Morlets to get complex power for |k| instead of particular (k x, k y ) …) Wavelet coherence (like all coherence) is real, but complex coherency can identify load correlations!

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

Wavelet Coherency Wavelet coherence is real: Wavelet coherency is complex: eastings northings wavelength

“Noise” Detection (1/2) Consider two surfaces, u & v, which can be correlated by varying the correlation coefficient, R, between them, to give a new surface v : Form cross-spectrum between U and V : Imaginary part: (Note: “Noise” is Jon’s terminology based on McKenzie’s; really, “noise” as used here means correlated loading!)

Use complex coherency instead of coherence: Define normalised coherency-squared: “Noise” Detection (2/2)

North American T e (Bouguer) T e from the Bouguer coherence; load deconvolution method.

“Noise” Locations Coherence noiseAdmittance noise Imaginary parts show location of correlation between surface and internal loads.

Masking T e Bouguer coherenceFree-air admittance “Noise” mainly in the Superior Province, southern Interior Platform, and southern Appalachians. Includes a number of intracratonic basins (e.g., Illinois and Michigan Basins) and the Mid-continental Rift.