*2.5. Signal Processing Using Wavelet Analysis*

Fourier analysis does not provide any information about when the frequencies are present during the time-span covered by the time series. Conversely, the wavelet transform is especially suited to identify localized intermittent periodicities from low signal-to-noise ratio time-series [21,22]. The Morlet wavelet [23] is adapted to geophysical time series as described by [24] who developed the software provided at http://paos.colorado.edu/ research/wavelets (accessed on 23 September 2021). We used the Matlab wavelet coherence toolbox as adapted by [25] and provided at http://www.glaciology.net/wavelet-coherence

(accessed on 23 September 2021). The Continuous Wavelet Power Spectrum (CWT) expands time-series records into time/frequency space. The time-series input data must be equally spaced in time. Although Copernicus satellites have a regular revisit interval (6 days for Sentinel-1A/B and 3 days for SMOS), some acquisitions may be missing or excluded from processing. These missing values are linearly interpolated using a constant time interval. The other time-series data of extensometers and soil moistures sensors are down-sampled using the same time interval. Two individual CWTs can be combined by using the Cross Wavelet Transform (XWT) tool which is computed by multiplying the CWT of one timeseries by the complex conjugate of the CWT of the second time-series. XWT image is the 2-D representation of the absolute value and the phase of the complex number in the time-frequency space. For many geophysical phenomena, an appropriate background spectrum is the red/Brownian noise (increasing power with decreasing frequency) [26]. This background spectrum was recently used in a wavelet analysis for land subsidence [27] and clay expansion [7]. ANGLEMEAN function calculates the mean angle found by XWT during a time period, and the associated sigma which corresponds to a standard deviation.
