Multiscale regularization

Another approach to combat the slow convergence of least squares sparse data interpolation is to design a regularization operator that works at multiple scales simultaneously. Starting with equation (3), we replace the regularization operator, $\bf A$, with a composite regularization operator (for the two-scale case):  

$$\left[\begin{array} {c} \bf A \ {\bf AD}_k \end{array}\right]$$ (8)

${\bf D}_k$, mnemonic for downsampling, is a normalized binning operator which subsamples a vector of size n to a vector of size n/k, implicitly smoothing it in the process. Replacing $\bf A$ in equation (3) with this new regularization operator gives  

$$\bf \left[ \begin{array} {c} \bf B \ \epsilon_1 \bf A \... ...rray} {c} \bf d \ \bf 0 \ \bf 0 \end{array} \right].$$ (9)

$\epsilon_1$ and $\epsilon_2$ are scaling factors. In the fashion of equation (4), we can write the least squares inverse corresponding to the system of equation (9):  

$$\bold B^{\dagger} = (\bold B^T \bold B + \epsilon_1^2 \bol... ..._2^2 \bold D_k^T \bold A^T \bold A \bold D_k)^{-1} \bold B^T$$ (10)

Applying the downsampling operator ${\bf D}_k$ to the model vector attenuates high-frequency components while boosting low-frequency components, thus we infer that the eigenvalue spectrum of $\bold A^T \bold A + \bold D_k^T \bold A^T \bold A \bold D_k$ is better balanced than that of $\bold A^T \bold A$ alone, which speeds convergence to a smooth model.

Claerbout (1999) presents a very similar multiscale methodology with one important difference: the filters, not the data, are upscaled from one scale to the next. Crawley (2000) applies this methodology to interpolating seismic data with nonstationary prediction error filters (PEF). The PEF is more readily upscaled, since it is normally conceptualized as a dip annhilator, and it annhilates the same dips at all scales. Unfortunately, other filters, like the Laplacian finite difference filter used in this paper, do not have the self-similarity property of the PEF, so explicitly expanding the filter is a dangerous proposition.