Deconvolution, Bandwidth, and the Trispectrum

Abstract

In the largest application area of time series analysis—geophysical exploration—the underlying innovations sequence is of primary interest and must be estimated. This sequence is estimated by deconvolving the non-Gaussian, noninvertible time series. This involves estimation of a phase-shift correction from the time series, which can be carried out by maximizing the kurtosis of the series. Unfortunately, the method is hampered by the fact that the time series is typically deficient in power in certain bands of frequencies (“band-limited”). The consequences of this can be analyzed by studying the trispectrum—the third of the polyspectra—of the series. This reveals two important results. First, we are able to easily appreciate why for certain types of band-limitation, kurtosis cannot be used to determine a phase correction. Second, by looking at the inner and outer subvolumes of the support volume for the discrete-parameter trispectrum, we see that for the standard linear model the trispectrum is non-0 in both the inner and outer volumes, whereas the trispectrum of a series sampled finely enough to avoid aliasing from a continuous fourth-order stationary process is equal to the same continuous-parameter trispectrum value in the inner volumes, but always 0 in the outer volumes. Hence, when looking at higher-order structure, the standard linear model need not give results that accord with physical reality.

Key Words:

• Kurtosis
• Non-Gaussian time series
• Phase
• Polyspectra