View source for Canonical representation – Chapter 10/es

<languages/>

{{publication
 | name    = Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
 | image   = [[File:DigitalImaging.png|frameless|180px]]
 | width   = 120px
 | series  = Geophysical References Series
 | title   = Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
 | part    = 
 | chapter = 10
 | frompg  = 253
 | topg    = 288
 | author  = [[Special:MyLanguage/Enders A. Robinson|Enders A. Robinson]] and [[Special:MyLanguage/Sven Treitel|Sven Treitel]]
 | doi     = 10.1190/1.9781560801610
 | pdf     = 
 | store   = https://seg.org/shop/products/detail/898
 | isbn    = 9781560801481
}}
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.

[[file:Ch10_fig4.png|thumb|{{figure number|4.}} A minimum-delay wavelet convolved with an all-pass filter gives a mixed-delay wavelet. Both wavelets have the same autocorrelation.]]

<blockquote>
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).
</blockquote>

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)<ref name=ch10r8>Jensen, O. G., T. J. Ulrych, J. P. Todeschuk, W. S. Leaney, and C. Walker, 1988, Bluenesscompensation in deconvolution for the reflectivity sequence: 58th Annual International Conference, SEG, Expanded Abstracts, 939-942.</ref> made a blueness compensation in deconvolution for the reflectivity sequence. Marschall and Knecht (1986)<ref name=ch10r12>Marschall, R., and M. Knecht, 1986, Reflectivity-corrected deconvolution and its influence on inversion: Presented at the Research Workshop on Deconvolution and Inversion, EAGE.</ref> devised a reflectivity-corrected deconvolution and described its influence on inversion. Saggaf and Toksöz (1999)<ref name=ch10r29>Saggaf, M. M., and N. Toksöz, 1999, An analysis of deconvolution, modeling reflectivity by fractionally integrated noise: Geophysics, '''64''', 1093-1107.</ref> presented an analysis of deconvolution in which they modeled reflectivity by fractionally integrated noise. Saggaf and Robinson (2000)<ref name=ch10r28>Saggaf, M. M., and E. A. Robinson, 2000, A unified framework for the deconvolution of traces of nonwhite reflectivity: Geophysics, '''65''', 1660-1676.</ref> provided a unified framework for the deconvolution of traces with nonwhite reflectivity. Porsani and Ursin (1998)<ref name=ch10r17>Porsani, M. J., and B. Ursin, 1998, Mixed-phase deconvolution: Geophysics, '''63''', 637-647.</ref> devised methods of mixed-phase deconvolution. Carrion (1986)<ref name=ch10r3>Carrion, P. M., 1986, A layer-stripping technique for the suppression of water-bottom multiple reflections: Geophysical Prospecting, '''34''', 330-342.</ref> gave a layer-stripping technique to suppress multiples, and Carrion and de Pinto Braga (1990)<ref name=ch10r4>Carrion, P. M., and A. de Pinto Braga, 1990, Iterative trace deconvolution and noncausal transform for processing band-limited data: Geophysics, '''55''', 1549-1557.</ref> developed methods of iterative trace deconvolution. Mendel (1983)<ref name=ch10r10>Mendel, J. M., 1983, Optimal seismic deconvolution, an estimation based approach: Academic Press Inc.</ref> used an estimation-based approach for optimal [[Special:MyLanguage/seismic deconvolution|seismic deconvolution]]. Mendel (1990)<ref name=ch10r11>Mendel, J. M., 1990, Maximum-likelihood deconvolution, a journey into model-based signal processing: Springer-Verlag.</ref> also pioneered the effective and advantageous method called ''maximum-likelihood deconvolution''.

Fokkema and Ziolkowski (1987)<ref name=ch10r6>Fokkema, J. T., and A. Ziolkowski, 1987, The critical reflection theorem: Geophysics, '''52''', 965-972.</ref> gave a proposition that they called the ''critical-reflection theorem'', which can be used for deconvolution. Kelamis and Chiburis (1988)<ref name=ch10r9>Kelamis, P. G., and E. F. Chiburis, 1988, Post-critical wavelet estimation and deconvolution: Geophysical Prospecting, '''36''', 504-522.</ref> originated the method of ''postcritical deconvolution'', which emphasizes reliance on postcritical reflection data. Ulrych (1971)<ref name=ch10r31>Ulrych, T. J., 1971, Application of homomorphic deconvolution to seismology: Geophysics, '''36''', 650–660.</ref> and Buttkus (1975)<ref name=ch10r2>Buttkus, B., 1975, Homomorphic filtering, theory and practice: Geophysical Prospecting, '''23''', 712-748.</ref> gave an outstanding development of homomorphic deconvolution in seismology. Ulrych and Bishop (1975)<ref name=ch10r32>Ulrych, T. J., and T. N. Bishop, 1975, Maximum entropy spectral analysis and autoregressive decomposition: Reviews of Geophysics and Space Physics, '''13''', 183-00.</ref> treated maximum-entropy spectral analysis and autoregressive decomposition. Ooe and Ulrych (1979)<ref name=ch10r14>Ooe, M., and T. J. Ulrych. 1979, Minimum entropy deconvolution with an exponential transformation: Geophysical Prospecting, '''27''', 458-473.</ref> extended the efficacy of minimum entropy deconvolution by use of an exponential transformation. Ulrych and Matsuoka (1991)<ref name=ch10r33>Ulrych, T. J., and T. Matsuoka, 1991, The output of predictive deconvolution: Geophysics, '''56''', 371-377.</ref> gave insights on the character of the output of predictive deconvolution. Robinson and Treitel (1980)<ref name=ch10r25>Robinson, E. A., and S. Treitel, 1980, Maximum entropy and the relationship of the partial autocorrelation to the reflection coefficients of a layered system: IEEE Transactions on Acoustics, Speech, and Signal Processing, '''ASSP-28''', 224-235.</ref> discussed maximum entropy and the relationship of the partial autocorrelation to the reflection coefficients of a layered system. Carrion and Foster (1985)<ref name=ch10r5>Carrion, P. M., and D. J. Foster, 1985, Inversion of seismic data using the precritical reflection and refraction data: Geophysics, '''50''', 759-765.</ref> treated the inversion of seismic data using precritical reflection and refraction data. Ulrych and Walker (1982)<ref name=ch10r34>Ulrych, T. J., and C. Walker, 1982, Analytic minimum entropy deconvolution: Geophysics, '''47''', 1295-1302.</ref> discussed analytic minimum-entropy deconvolution.

Many important papers on deconvolution are found in the SEG reprint volumes: Webster (1978)<ref name=ch10r36>Webster, G. M., 1978, Deconvolution: SEG Geophysics Reprint Series No. 1.</ref>, Osman and Robinson (1996)<ref name=ch10r15>Osman, O. M., and E. A. Robinson 1996, Seismic source signature estimation and measurement: SEG Geophysics Reprint Series No. 18.</ref>, and Robinson and Osman (1996). Excellent examples of deconvolution are given in Yilmaz (2001)<ref name=ch10r37>Yilmaz, Ö., 2001, Seismic data analysis: Processing, inversion, and interpretation of seismic data, 2 v.: SEG.</ref>.
__NOTOC__

==Referencias==

{{reflist}}

== Sigue leyendo ==

{| border="1" cellpadding="5" cellspacing="0" align="center"
|-
! style="background: #426580; color: white;" | Sección previa
! style="background: #E5D75F;" | Siguiente sección
|-
| align="center" | [[Implementing deconvolution/es|Implementación de la deconvolución]]
| align="center" | [[Appendix J: Exercises/es|Apéndice J: Ejercicios]]
|-
! style="background: #426580; color: white;" | Capítulo previo
! style="background: #E5D75F;" |Siguiente capítulo
|-
| align="center" | [[:Category:Wavelet Processing - Chapter 9| Procesamiento de la ondícula]]
| align="center" | [[:Category:Fine Points|Algunas consideraciones]]
|-
| align="center" colspan="2" |
[[Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing/es|Tabla de contenido]]
|}


== También en este capítulo ==

 
* [[Model used for deconvolution/es|Modelos utilizados para la deconvolución]]
* [[Least-squares prediction and smoothing/es|Predicción de mínimos cuadrados y suavizamiento]]
* [[The prediction-error filter/es|El filtro de error predicción]]
* [[Spiking deconvolution - book/es|Deconvolución de pico]]
* [[Gap deconvolution/es|Deconvolución predictiva]]
* [[Tail shaping and head shaping/es| Modelado de la cola y modelado de la cabeza]]
* [[Seismic deconvolution/es|Deconvolución sísmica]]
* [[Piecemeal convolutional model/es|Modelo convolucional por partes]]
* [[Time-varying convolutional model/es|Modelo convolucional variable en tiempo]]
* [[Random-reflection-coefficient model/es|Modelo de coeficientes de reflexión aleatorios]]
* [[Implementing deconvolution/es|Implementación de la deconvolución]]
* [[Appendix J: Exercises/es|Apéndice J: Ejercicios]]


== Vínculos externos ==

{{search}}
 

[[Category:Digital Imaging and Deconvolution]]
[[Category:Publications]]