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De-bursting

Most signals are smooth, but running medians assume they have no curvature. An alternate expression of this assumption is that the signal has minimal curvature $ 0 \approx h_{i+1} -2 h_{i} + h_{i-1} $;in other words, $ \bold 0 \approx \nabla^2 \bold h$.Thus we propose to create the cleaned-up data $\bold h$from the observed data $\bold d$ with the fitting problem
\begin{displaymath}
\begin{array}
{lll}
 0 &\approx & \bold W (\bold h - \bold d) \\  0 &\approx & \epsilon\ \nabla^2 \bold h
 \end{array}\end{displaymath} (18)
where $\bold W$ is a diagonal matrix with weights sprinkled along the diagonal, and where $\nabla^2$ is a matrix with a roughener like (1,-2,1) distributed along the diagonal. This is shown in Figure [*] with $\epsilon = 1$.Experience showed similar performances for $0 \approx \nabla \bold h$ and $0 \approx \nabla^2 \bold h$.Better results, however, will be found later in Figure [*], where the $\nabla^2$ operator is replaced by an operator designed to predict this very predictable signal.
next up previous print clean
Next: MEDIAN BINNING Up: NOISE BURSTS Previous: De-spiking with median smoothing
Stanford Exploration Project
4/27/2004