Time-frequency analysis of seismic data using synchrosqueezing wavelet transform

Synchrosqueezing wavelet transform

The well-known continuous wavelet transform is defined by:

$$W_x(a,t)=\int_{-\infty}^{\infty}x(\tau)a^{-1/2}\psi^*\left(\frac{\tau-t}{a}\right)d\tau,$$ (1)

where $\psi(t)$ is the chosen mother wavelet, $\psi^*$ denotes the complex conjugate of $\psi$ . The instantaneous frequency of $w_x(a,t)$ for signal $x(t)$ can be got by:

$$w_x(a,t) = -i(W_x(a,t))^{-1}\frac{\partial}{\partial t}W_x(a,t).$$ (2)

The synchrosqueezed transform $T_x(w,t)$ can be determined only at the centers $w_l$ of the successive bins $[w_l-\frac{1}{2}\Delta w,w_l+\frac{1}{2}\Delta w]$ , with $w_l-w_{l-1}=\Delta w$ , by summing different contributions (Daubechies et al., 2011):

$$T_x(w_l,t)=(\Delta w)^{-1} \sum_{a_k:\vert w(a_k,t)-w_l\vert\le \Delta w/2}^{} W_x(a_k,t)a_k^{-3/2}(\Delta a)_k.$$ (3)

The SSWT is invertible and the original signal can be obtained by:

$$\begin{split}x(t) &= \mathcal{Re}\left[C_{\psi}^{-1}\int_{0}^... ...t[C_{\psi}^{-1}\sum_{l}T_x(w_l,t)(\Delta w)\right]. \end{split}$$ (4)

Time-frequency analysis of seismic data using synchrosqueezing wavelet transform