Edge preserving regularization with the Cauchy norm

Imagine that the model residuals in equations (3) and (4) consisted of spikes separated by relatively large distances. Then the interval velocity $\bf u$ would be piecewise smooth with jumps at the spike locations, which is what we desire. However in solving (2)-(4) we use the least-squares criterion - minimization of the $\ell_2$ norm of the residual. Any spikes in the residual will be attenuated. To do this, the solver smooths the velocity across the spike location.

It is known that the $\ell_1$ norm is less sensitive to spikes in the residual Claerbout and Muir (1973); Darche (1989); Nichols (1994). $\ell_1$ norm minimization makes the assumption that the residuals have an exponential distribution, a ``long-tailed'' distribution relative to the Gaussian. Here, we compute nonlinear model residual weights which force a Cauchy distribution, another long-tailed distribution which approximates an exponential distribution Youzwishen (2001).

We perform the following non linear iterations: starting with ${\bf Q_{\tau}}^{0}= {\bf Q_{x}}^{0} = \bf I$, at the k^th iteration the algorithm solves

$$\begin{eqnarray} {\bf W ( Cu}^k - {\bf d ) \approx 0} \nonumber \\ \epsilon_{\t... ...\ \epsilon_{x} {\bf Q_{x}}^{k-1} { \bf D_x u}^k { \bf \approx 0 }\end{eqnarray}$$ (5)

where

$$\begin{eqnarray} {\bf Q_{\tau}}^{k-1} = \frac{1}{\left[1+\left( \frac{{ \bf D_{\... ...ac{{ \bf D_x u}^{k-1}}{\alpha_x} \right)^2 \right]^{\frac{1}{2}}},\end{eqnarray}$$ (6) (7)

and ${\bf u}^{k}$ is the result of the k^th nonlinear iteration, ${\bf Q_{\tau}}^{k-1}$ and ${\bf Q_{x}}^{k-1}$ are the (k-1)^th diagonal weighting operators, ${ \bf D_{\tau} }$ and ${ \bf D_x }$ are the first order derivatives in time and midpoint, $\bf I$ is the identity matrix, the scalars $\alpha_{\tau}$ and $\alpha_x$ are the trade-off parameters controlling the discontinuities in the solution, and the scalars $\epsilon_{\tau}$ and $\epsilon_{x}$ balance the relative importance of the two model residuals.