Geology 6600/7600Signal Analysis

14 Oct 2015

© A.R. Lowry 2015

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 (kx, ky)…)• Wavelet coherence (like all coherence) is real, but complex coherency can identify load correlations!

Thought Exercise (I):• The complex-coherency method of Kirby & Swain (2009) uses the imaginary component of complex coherency to identify the presence of correlated loading at flexural wavelengths in gravity and Bouguer topography data. This works because flexural response is zero-phase, implying that non-zero-phase contributions to coherence are not a result of flexure but rather of some other process…

Loads Mass Flux Processes:Surface Loads Erosion Deposition Dip-slip Fault Offsets Volcanic Construction

Subsurface Loads Thermal Variations Lithologic Variations Crustal Thickness (Strain; Magmatism)

What happens if a process that correlates surface and subsurface loading is zero-phase?What can we do about it?

Thought Exercise (II):• Your assignment for next Friday (to cross-correlate GPS time series) ideally could be done as either a convolution in the time-domain or a multiplication in the frequency domain…

What fundamental hurdle(s) would need to be overcome in order to do the multiplication in the spectral domain?

What else would change about the approach if you did this in the spectral domain? What might be one way to overcome this problem?