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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
. Suppose a complex time series is
. In Z-transform notation, the two-point prediction-error filter can be expressed as:
![$\displaystyle F(Z)=1-Z/Z_0.$](img50.png) |
(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
:
![$\displaystyle d(t)=e^{i\omega(t)\Delta t}d(t-\Delta t).$](img52.png) |
(14) |
In order to estimate
using equation 14, we need to first minimize the least-squares misfit of the true and predicted time series with a local-smoothness constraint:
![$\displaystyle \min_{\mathbf{a}} \parallel \mathbf{d} - \mathbf{D} \mathbf{a} \parallel_2^2 + \mathbf{R}(\mathbf{a}),$](img53.png) |
(15) |
where
and
are vectors composed of the entries
and
, respectively, and
.
is a diagonal matrix composed of the entries
.
denotes the local-smoothness constraint. Equation 15 can be solved using shaping regularization:
![$\displaystyle \hat{\mathbf{a}} = [\lambda^2\mathbf{I}+\mathcal{T}(\mathbf{D}^T\mathbf{D}-\lambda^2\mathbf{I})]^{-1}\mathcal{T}\mathbf{D}^T\mathbf{d},$](img61.png) |
(16) |
where
is a triangle smoothing operator and
is a scaling parameter that controls the physical dimensionality and enables fast convergence.
can be chosen as the least-squares norm of
. After the filter coefficient
is obtained, we can straightforwardly calculate the local angular frequency
by
![$\displaystyle \omega(t) = Re\left[\frac{\mbox{arg}(a(t))}{\Delta t}\right].$](img64.png) |
(17) |
Next: Preparing smoothly variable frequency
Up: Method
Previous: 1D non-stationary seislet transform
2019-02-12