# Canonical representation – Chapter 10

Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 10 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

the canonical representation of a seismic trace states that a nonminimum-delay wavelet can be represented uniquely by the convolution of an all-pass filter (which has a flat magnitude spectrum) and a minimum-delay wavelet. Both wavelets have the same magnitude spectrum (i.e., the same color). This representation is called the canonical representation (Figure 4). Let us call white anything that has a flat magnitude spectrum, whereas we call colored anything that has a curved, or nonwhite, magnitude spectrum. The canonical representation then states that a nonminimum-delay wavelet is equal to the convolution of a minimum-delay wavelet of the same color as the given nonminimum-delay wavelet with an all-pass system, which is white. Figure 4.  A minimum-delay wavelet convolved with an all-pass filter gives a mixed-delay wavelet. Both wavelets have the same autocorrelation.

1) The trace model is: seismic trace = mixed-delay wavelet * reflectivity.

2) A mixed-delay wavelet has both a colored and a white component: mixed-delay wavelet = minimum-delay wavelet * all-pass wavelet.

3) Thus, we can write our seismic trace as: seismic trace = minimum-delay wavelet * (all-pass filter * reflectivity).

On the right-hand side of the equation in (3), two members are expressed in parentheses. These parentheses contain the white components (i.e., the all-pass filter and the reflectivity function). Because the convolution of two white waveforms is itself white, the parentheses enclose a white component.

How does deconvolution work in such a case? The process of predictive deconvolution separates on the basis of two fundamental criteria: “minimum-delay” and “white.” Thus, the method of predictive convolution attempts to remove the minimum-delay component to yield an estimate of the white component. That is, if we deconvolve the seismic trace according to the above model we obtain as the output the white component: (all-pass filter * reflectivity function).

Thus, we do not obtain the desired reflectivity function; instead, we obtain the reflectivity function passed through an unknown all-pass filter. If the source wavelet was close to a minimum-delay wavelet, the action of the all-pass filter will not be severe, and the above result will be acceptable. Otherwise, the desired reflectivity function can be distorted appreciably by the presence of this unknown and unwanted all-pass filter.

Over the years, hundreds of papers have been written in geophysics on deconvolution, and many more have been published in other fields of science. Much research has been devoted to extensions of the simple convolutional model by assuming that the source wavelet is mixed delay and/or that the reflectivity is not white but, say, blue. A blue reflectivity is one that is still random but is richer in the higher, or “blue,” frequencies than in the lower, or “red,” frequencies. Jensen et al. (1988) made a blueness compensation in deconvolution for the reflectivity sequence. Marschall and Knecht (1986) devised a reflectivity-corrected deconvolution and described its influence on inversion. Saggaf and Toksöz (1999) presented an analysis of deconvolution in which they modeled reflectivity by fractionally integrated noise. Saggaf and Robinson (2000) provided a unified framework for the deconvolution of traces with nonwhite reflectivity. Porsani and Ursin (1998) devised methods of mixed-phase deconvolution. Carrion (1986) gave a layer-stripping technique to suppress multiples, and Carrion and de Pinto Braga (1990) developed methods of iterative trace deconvolution. Mendel (1983) used an estimation-based approach for optimal seismic deconvolution. Mendel (1990) also pioneered the effective and advantageous method called maximum-likelihood deconvolution.

Fokkema and Ziolkowski (1987) gave a proposition that they called the critical-reflection theorem, which can be used for deconvolution. Kelamis and Chiburis (1988) originated the method of postcritical deconvolution, which emphasizes reliance on postcritical reflection data. Ulrych (1971) and Buttkus (1975) gave an outstanding development of homomorphic deconvolution in seismology. Ulrych and Bishop (1975) treated maximum-entropy spectral analysis and autoregressive decomposition. Ooe and Ulrych (1979) extended the efficacy of minimum entropy deconvolution by use of an exponential transformation. Ulrych and Matsuoka (1991) gave insights on the character of the output of predictive deconvolution. Robinson and Treitel (1980) discussed maximum entropy and the relationship of the partial autocorrelation to the reflection coefficients of a layered system. Carrion and Foster (1985) treated the inversion of seismic data using precritical reflection and refraction data. Ulrych and Walker (1982) discussed analytic minimum-entropy deconvolution.

Many important papers on deconvolution are found in the SEG reprint volumes: Webster (1978), Osman and Robinson (1996), and Robinson and Osman (1996). Excellent examples of deconvolution are given in Yilmaz (2001).