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In the previous report (Zhang and Claerbout, 2010), we introduced the spiking deconvolution problem using the hybrid norm solver (Claerbout, 2009a). Synthetic examples (Zhang and Claerbout, 2010) showed that given a minimum-phase wavelet, it retrieved the sparse reflectivity model almost perfectly even with a reflection series that is far from white, while conventional L2 deconvolution did a poor job. However, if the assumption of a minimum-phase wavelet was removed, the hybrid norm spiking deconvolution failed quickly and gave a poor result similar to the conventional L2 deconvolution.

In this paper, we still rely on the hybrid norm solver to retrieve the sparse model, but we use a slightly more complex formulation that avoids the minimum-phase wavelet constraint.

We start by realizing that any (mixed-phase) wavelet $ C(Z)$ can be decomposed into a minimum-phase part $ A(Z)$ and a maximum-phase part $ B(1/Z)$ plus a certain time shift:

$\displaystyle C(Z)=A(Z)B(1/Z)Z^{k},$ (1)

where $ B(Z)$ is also a minimum-phase wavelet (therefore $ B(1/Z)Z^k$ is a maximum-phase wavelet) and the exponent $ k$ is the order of $ B(Z)$ . This $ Z^{k}$ term makes the wavelet $ C(Z)$ causal. In the time domain, (1) can be written as

$\displaystyle c = a*b^r*\delta(n-k),$ (2)

where $ b^r$ stands for the time reverse of series $ b$ .

Our original spiking deconvolution can find only a minimum-phase wavelet which has the same spectrum of real wavelet $ c$ . It can be defined as an inverse problem as follows:

$\displaystyle [d]f_c = r,$ (3)

where $ [d]$ is the data convolution operator, and $ f_c$ is the unknown filter. In this formulation, the filter is the only unknown, the hybrid norm is applied on the residual term $ r$ to enforce the sparseness constraint. In theory, the residual $ r$ itself is the reflectivity model. Such a method requires the wavelet in the data to be minimum-phase because only a minimum-phase wavelet has a causal stable inverse.

The following bidirectional deconvolution formulation utilizes a pair of conventional deconvolutions, trying to invert components $ a$ and $ b$ separately:

\begin{displaymath}\begin{array}{l} \left[(d*f_b^r) \right] f_a = r_{a}, \\ \left[(d*f_a)^r \right] f_b = r_{b}, \end{array}\end{displaymath} (4)

in which $ f_a$ and $ f_b$ are the corresponding filters that corresponds to the inverses of $ a$ and $ b$ denoted above, the superscript $ r$ means time-reverse. The operator in each equation is the convolution operator. Again the hybrid norm is applied to $ r_a$ and $ r_b$ , and the reflectivity model is simply $ r_{a}$ plus a time shift. Notice that this is a non-linear inversion, since the operator itself depends on the unknown $ f_a$ and $ f_b$ . In practice we have to solve these two inversions alternately and therefore iteratively.

To understand the meaning of (4), let

$\displaystyle d = m * c = m*a*b^r*\delta(n-k),$ (5)

where m is the reflectivity model and the $ \delta$ term is just a time shift. Assume $ f_a$ and $ f_b$ are perfectly known in the operators (which is not true in reality), i.e.

$\displaystyle f_a*a = \delta(n), f_b*b=\delta(n)

Substituting (5) into (4), since

$\displaystyle d*f_b^r$ $\displaystyle =$ $\displaystyle m*\delta(n-k)*a,$ (6)
$\displaystyle (d*f_a)^r$ $\displaystyle =$ $\displaystyle (m*b^r*\delta(n-k))^r = m^r*\delta(n+k)*b,$ (7)

we have

\begin{displaymath}\begin{array}{l} \left[(m*\delta(n-k))*a \right] f_a = r_{a}, \\ \left[(m^r*\delta(n+k))*b \right] f_b = r_{b}. \end{array}\end{displaymath} (8)

From (8) it is easier to see what is behind the bidirectional deconvolution formulation (4): It tries to separate the two parts of the wavelet, turning each one into a traditional deconvolution problem in which the wavelet ($ a,b$ ) is always minimum-phase.

As with all non-linear estimation, iteration is required. Convergence is assured if the starting solution is close enough. We expect the traditional PEF for $ a$ and an impulse function for $ b$ to be a pretty good first guess. The following section shows several examples (complexity varies from low to high) illustrating the effectiveness and limitations of the method.

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Next: Data Examples Up: Zhang and Claerbout: Hybrid Previous: Zhang and Claerbout: Hybrid