A general theory on frequency and time-frequency analysis of irregularly sampled time series based on projection methods. II. Extension to time-frequency analysis

Geophysical time series are sometimes sampled irregularly along the time axis. The situation is particularly frequent in palaeoclimatology. Yet, there is so far no general framework for handling continuous wavelet transform when the time sampling is irregular. Here we provide such a framework. To this end, we define the scalogram as the continuous-wavelet-transform-equivalent of the extended Lomb-Scargle periodogram defined in part I of this study (Lenoir and Crucifix, 2017). The signal being analyzed is modeled as the sum of a locally periodic component in the time-frequency plane, a polynomial trend, and a background noise. The mother wavelet adopted here is the Morlet wavelet classically used in geophysical applications. The background noise model is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, which is more general than the traditional Gaussian white and red noise processes. The scalogram is smoothed by averaging over neighboring times in order to reduce its variance. The Shannon-Nyquist exclusion zone is on the other hand defined as the area corrupted by local aliasing issues. The local amplitude in the time-frequency plane is then estimated with least-squares methods. We show that the squared amplitude and the scalogram are approximately proportional. Based on this property, we define a new analysis tool: the weighted smoothed scalogram, which we recommend for most analyses. The estimated signal amplitude also gives access to band and ridge filtering. Finally, we design a test of significance for the weighted smoothed scalogram against the stationary Gaussian CARMA background noise, and provide algorithms for computing confidence levels, either analytically or with Monte Carlo Markov Chain methods. All the analysis tools presented in this article are available to the reader in the Python package WAVEPAL.