Least-Squares Wavelet Analysis and Its Applications in Geodesy and Geophysics

The Least-Squares Spectral Analysis (LSSA) is a robust method of analyzing unequally spaced and non-stationary data/time series. Although this method takes into account the correlation among the sinusoidal basis functions of irregularly spaced series, its spectrum still shows spectral leakage: power/energy leaks from one spectral peak into another. An iterative method called AntiLeakage Least-Squares Spectral Analysis (ALLSSA) is developed to attenuate the spectral leakages in the spectrum and consequently is used to regularize data series. In this study, the ALLSSA is applied to regularize and attenuate random noise in seismic data down to a certain desired level. The ALLSSA is subsequently extended to multichannel, heterogeneous and coarsely sampled seismic and related gradient measurements intended for geophysical exploration applications that require regularized (equally spaced) data free from aliasing effects.

A new and robust method of analyzing unequally spaced and non-stationary time/data series is rigorously developed. This method, namely, the Least-Squares Wavelet Analysis (LSWA), is a natural extension of the LSSA that decomposes a time series into the time-frequency domain and obtains its spectrogram. It is shown through many synthetic and experimental time/data series that the LSWA supersedes all state-of-the-art spectral analyses methods currently available, without making any assumptions about or preprocessing (editing) the time series, or even applying any empirical methods that aim to adapt a time series to the analysis method. The LSWA can analyze any non-stationary and unequally spaced time series with components of low or high amplitude and frequency variability over time, including datum shifts (offsets), trends, and constituents of known forms, and by taking into account the covariance matrix associated with the time series. The stochastic confidence level surface for the spectrogram is rigorously derived that identifies statistically significant peaks in the spectrogram at a certain confidence level; this supersedes the empirical cone of influence used in the most popular continuous wavelet transform.

All current state-of-the-art cross-wavelet transforms and wavelet coherence analyses methods impose many stringent constraints on the properties of the time series under investigation, requiring, more often than not, preprocessing of the raw measurements that may distort their content. These methods cannot generally be used to analyze unequally spaced and non-stationary time series or even two equally spaced time series of different sampling rates, with trends and/or datum shifts, and with associated covariance matrices. To overcome the stringent requirements of these methods, a new method is developed, namely, the Least-Squares Cross-Wavelet Analysis (LSCWA), along with its statistical distribution that requires no assumptions on the series under investigation. Numerous synthetic and geoscience examples establish the LSCWA as the method of methods for rigorous coherence analysis of any experimental series.