EMD-seislet transform

Estimating local frequency by complex non-stationary autoregression

According to the autoregressive spectral analysis theory (Marple, 1987), a complex time series that has a constant frequency component is predictable by a two-point prediction-error filter $(1,-e^{i\omega_0\Delta t})$ . Suppose a complex time series is $d(t)$ . In Z-transform notation, the two-point prediction-error filter can be expressed as:

$$F(Z)=1-Z/Z_0.$$ (13)

We assume that a 1D time series has a smooth frequency component, then the 1D time series can be locally predicted using different local two-point prediction-error filters $(1,-e^{i\omega(t)\Delta t})$ :

$$d(t)=e^{i\omega(t)\Delta t}d(t-\Delta t).$$ (14)

In order to estimate $\omega(t)$ using equation 14, we need to first minimize the least-squares misfit of the true and predicted time series with a local-smoothness constraint:

$$\min_{\mathbf{a}} \parallel \mathbf{d} - \mathbf{D} \mathbf{a} \parallel_2^2 + \mathbf{R}(\mathbf{a}),$$ (15)

where $\mathbf{d}$ and $\mathbf{a}$ are vectors composed of the entries $d(t)$ and $a(t)$ , respectively, and $a(t)=e^{i\omega(t)\Delta t}$ . $\mathbf{D}$ is a diagonal matrix composed of the entries $d(t-\Delta t)$ . $\mathbf{R}$ denotes the local-smoothness constraint. Equation 15 can be solved using shaping regularization:

$$\hat{\mathbf{a}} = [\lambda^2\mathbf{I}+\mathcal{T}(\mathbf{D}^T\mathbf{D}-\lambda^2\mathbf{I})]^{-1}\mathcal{T}\mathbf{D}^T\mathbf{d},$$ (16)

where $\mathcal{T}$ is a triangle smoothing operator and $\lambda$ is a scaling parameter that controls the physical dimensionality and enables fast convergence. $\lambda$ can be chosen as the least-squares norm of $\mathbf{D}$ . After the filter coefficient $a(t)$ is obtained, we can straightforwardly calculate the local angular frequency $\omega(t)$ by

$$\omega(t) = Re\left[\frac{\mbox{arg}(a(t))}{\Delta t}\right].$$ (17)

EMD-seislet transform