A new method of spectral analysis and its application to the Earth's free oscillations: The “Sompi” method
Authors: Hori Sadaki , Fukao Yoshio , Kumazawa Mineo , Furumoto Muneyoshi , Yamamoto Akihiko 
Published: February 2008

A new and powerful method of spectral analysis, which is named the “Sompi” method after a Japanese word, is introduced and applied to low‐frequency seismograms. The Sompi method is based on an autoregressive (AR) process model, which is different from an ordinary prediction type AR model. In the ordinary AR model the AR coefficients predict the present observation from the past observations so that they are determined by minimizing the prediction error. The model implicitly assumes that the past observations are noise‐free and that only the present observation consists of signal and noise. The method thus gives an unbiased estimate of the linear relationship among the past data and the present signal. In our AR model the AR coefficients extract signals from the time series so that they are determined by minimizing the extraction residual. The model assumes that each observation, either past or present, consists of signal and noise. The method thus gives an unbiased estimate of the linear relationship among the successive signals. Those who wish to make an unbiased spectral estimation for the signal must therefore minimize the extraction residual, rather than the prediction error. Minimization of the extraction residual leads to an eigenvalue problem of a non‐Toeplitz matrix of autocovariance. The minimum eigenvalue yields the extraction residual which is an estimate of the noise power. The corresponding eigenvector constitutes the AR coefficients whose characteristic equation gives the complex frequencies (frequencies and decay or growth rates) of the signals extracted. The complex amplitudes (amplitudes and phases) are then determined through a least squares procedure. The Sompi method thus first retrieves medium‐sensitive parameters, frequencies, and Q and then excitation‐sensitive parameters, amplitudes, and phases. From the practical point of view it is not feasible to apply the Sompi method directly to such data as the Earth's free oscillations in which spectral peaks are densely and nonuniformly distributed over the entire frequency domain. We propose an algorithm of aliased sampling which enables us to apply the Sompi method to such data. In this algorithm the time series is first narrowly and sharply band‐pass filtered. The resultant time series is then decimated at a rate corresponding to the bandwidth of the filter. The contributions of all the decimated time series, which may mutually lag in time, are stacked into the matrix elements of autocovariance for the eigenvalue problem. The Sompi method with this algorithm is tested against a synthetic seismogram to see the resolvability of the two modes closely spaced in frequency. The Sompi method is next applied to the seismograms of the 1977 Sumbawa Island earthquake recorded in the International Deployment of Accelerometers network. First, the two radial modes 0S0 and 1S0 are analyzed to examine the uniformity of their spectral parameters among different stations. Second, the fundamental multiplets 0Sl (l = 5–43) are analyzed to see how their initial amplitudes and phases oscillate with respect to l at one station. Third, the gravest mode 0S2 is analyzed in an attempt to resolve the five singlets with their Q values. Fourth, the analysis is made for the coupled multiplets 0Sl‐0Tl+1 (l = 10–12) to observe their mutual repulsion in frequency and their mutual attraction in decay rate. Fifth, the core mode 7S3, a mode with most of its energy confined to the inner core, is detected. All five experiments demonstrate remarkable accuracy and resolvability of the Sompi method.