Detecting harmonic signals in a noisy time-series: the z-domain Autoregressive (AR-z) spectrum

We develop a new method referred to as the AR-z spectrum for detecting harmonic signals with exponential decay/growth contained in a noisy time-series by extending the autoregressive (AR) method of Chao & Gilbert. The method consists of (i) ‘blindly’ forcing one 2nd-order AR fit to the signal content in the frequency domain for any chosen frequency whether or not there is truly a signal; (ii) finding the corresponding AR (complex-conjugate pair of) poles in the complex z-domain; (iii) converting the pole locations into the corresponding complex frequencies of the harmonic signals via the Prony's relation and (iv) constructing the Lorentzian power spectrum in the z-domain, conceptually constituting the analytical continuation of the spectrum from the (real) frequency domain to the complex z-domain, where a true harmonic signal is manifested as a Lorentzian peak. The AR-z spectrum can be further enhanced by forming the product spectrum from multiple records as available. We apply the AR-z spectral method to detect and to estimate the complex frequencies of the Earth's normal-modes of free oscillation using superconducting gravimeter records after recent large earthquakes. Specifically we show examples of detection and precise estimation of the frequencies and Q values of the split singlets of the spheroidal modes
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, and report the mode couplings manifested by the gravimeter recording of the toroidal modes
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. The AR-z spectrum proves to be highly sensitive for harmonic signal of decaying sinusoids in comparison to the conventional Fourier-based spectrum, particularly when the signal in question is weak and where high spectral resolution is desired.