A generalization for the $\delta^L$ operator

Although I have not yet explored the practical implications of this, the operator $\delta^L$, previously defined, can be described as a particular case of a more general operator ${\cal D}_L^{\alpha}$: 

$${\bf r} = {\cal D}_L^{\alpha}\{{\bf c}\} \;\;\; \longrightar... ...r {\displaystyle \sum_{j=i-n}^{i+n}} \mid c_j^{\alpha} \mid },$$ (10)

$$\mbox{\hspace{-4.1cm} where \hspace{1.0cm}} n = {L + 1 \ove... ...d \hspace{0.6cm} $\alpha$ \hspace{0.6cm} is any odd integer}.$$

The operator $\delta^L$ can be defined as  

$$\delta^L = \lim_{\alpha \rightarrow \infty} \: {\cal D}_L^{\alpha}.$$ (11)

Figure 7 shows that using this operator with finite values of $\alpha$ is equivalent to the introduction of the damping factor in equation (7). Also we can see that the results obtained with small values of $\alpha$ are similar to those obtained with the predictive deconvolution. An L^p norm deconvolution can be formulated as an iterative, reweighted least-squares inversion with weighting factors given by $W(i) = \mid \Delta t(i) \mid^{p-2}$, where $\Delta t(i)$ are the residuals of the previous iteration (Darche, 1989). Since the value of the exponent $\alpha$ will directly affect the relative weighting of the residuals, we can consider the ICP decon as somehow equivalent to an $L^\infty$ norm deconvolution.

 

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Figure 7 (a) Original reflectivity series. The other traces correspond to the results of the ICP algorithm using the generalized operator of equation (10) with different values of $\alpha$: (b) $\alpha = 3$,(c) $\alpha = 5$, (d) $\alpha = 11$,and (e) $\alpha = \infty$.