# Canonical representation

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 7 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

What is the canonical representation of a causal wavelet? Let ${\displaystyle w_{M}}$ be the minimum-phase wavelet with the same amplitude spectrum as that of the causal wavelet w. Then ${\displaystyle w_{M}}$ is called the minimum-phase counterpart of w. The canonical representation' states that any causal wavelet w can be represented as the convolution of its minimum-phase counterpart ${\displaystyle w_{M}}$ and a causal all-pass wavelet p; that is,

 {\displaystyle {\begin{aligned}&w=w_{M}*p.\end{aligned}}} (58)

Because the inverse of a minimum-phase wavelet is minimum phase, it follows that ${\displaystyle W_{M}^{-1}}$ is minimum phase and hence is causal. From the canonical representation, we see that the inverse wavelet is given by

 {\displaystyle {\begin{aligned}&w^{-1}=w_{M}^{-1}*p^{-{l}}.\end{aligned}}} (59)

Two cases can occur. In the first case, the causal wavelet w is itself minimum phase. Then the causal all-pass wavelet is trivial, so the inverse wavelet is simply ${\displaystyle w^{-{1}}=w_{M}^{-{1}}}$. In this case, the inverse wavelet is minimum phase and causal.

In the second case, the causal wavelet w is not minimum phase. Then the causal all-pass wavelet p is not trivial and therefore its inverse ${\displaystyle p^{-1}=p^{R}}$ is a one-sided noncausal wavelet. In this case, the inverse wavelet ${\displaystyle w^{-1}}$ is a two-sided noncausal wavelet.