Multidimensional seismic data processing: Difference between revisions

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'''Multidimensional seismic data processing''' forms a major component of [[Vertical seismic profile | seismic profiling]], a technique used in geophysical exploration. The technique itself has various applications, including mapping ocean floors, determining the structure of sediments, mapping [[subsurface currents]] and [[hydrocarbon exploration]]. Since geophysical data obtained in such techniques is a function of both space and time, [[multidimensional signal processing]] techniques may be better suited for processing such data.

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'''Multidimensional seismic data processing''' forms a major component of [[Vertical seismic profile | seismic profiling]], a technique used in geophysical exploration. The technique itself has various applications, including mapping ocean floors, determining the structure of sediments, mapping [[subsurface currents]] and [[hydrocarbon exploration]]. Since geophysical data obtained in such techniques is a function of both space and time, [[multidimensional signal processing]] techniques may be better suited for processing such data.

== Data acquisition ==

[[File:Offset VSP.jpg|thumb|Offset VSP]]

There are a number of data acquisition techniques used to generate seismic profiles, all of which involve measuring acoustic waves by means of a source and receivers. These techniques may be further classified into various categories,<ref>{{Cite encyclopedia|title = Vertical Seismic Profiling | encyclopedia = Encyclopedia of Solid Earth Geophysics|last = Rector|first = James|publisher = Springer|year = 2010|isbn = 978-90-481-8702-7|pages = 430–433|last2 = Mangriotis|first2 = M. D.}}</ref> depending on the configuration and type of sources and receivers used. For example, zero-offset vertical seismic profiling (ZVSP), walk-away VSP etc.

The source (which is typically on the surface) produces a wave travelling downwards. The receivers are positioned in an appropriate configuration at known depths. For example, in case of [[Vertical seismic profile|vertical seismic profiling]], the receivers are aligned vertically, spaced approximately 15 meters apart. The vertical travel time of the wave to each of the receivers is measured and each such measurement is referred to as a “check-shot” record. Multiple sources may be added or a single source may be moved along predetermined paths, generating seismic waves periodically in order to sample different points in the sub-surface. The result is a series of check-shot records, where each check-shot is typically a two or three-dimensional array representing a spatial dimension (the source-receiver offset) and a temporal dimension (the vertical travel time).

== Data processing ==

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=== Multichannel filtering ===

{{see also | Multidimensional Filterfilter Designdesign}}

Multichannel filters may be applied to each individual record or to the final seismic profile. This may be done to separate different types of waves and to improve the signal-to-noise ratio. There are two well-knowknown methods of designing velocity filters for seismic data processing applications.<ref>{{cite journal|last1=Tatham|first1=R|last2=Mangriotis|first2=M|title=Multidimensional Filtering of Seismic Data|journal=Proceedings of the IEEE|date=Oct 1984|volume=72|issue=10|pages=1357-13691357–1369|doi=10.1109/PROC.1984.13023}}</ref>.

==== Two-dimensional Fourier transform design ====

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<math>

F(\underline{k},\omega) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(\underline{x},t)e^{-j(\omega t - kx\underline{k} \underline{x})}dxd\underline{x} dt

</math>

where <math>\underline{k}</math> is the spatial frequency (also known as wavenumber) and <math>\omega</math> is the temporal frequency. The two-dimensional equivalent of the frequency ___domain is also referred to as the <math>\underline{k} -\omega</math> ___domain. There are various techniques to design two-dimensional filters based on the Fourier transform, such as the minimax design method and design by transformation. One disadvantage of Fourier transform design is its global nature; it may filter out some desired components as well.

==== ''τ-p'' transform design ====

The ''τ-p'' transform is a special case of the [[Radon transform]], and is simpler to apply than the Fourier transform. ThisIt isallows dueone to thestudy factdifferent thatwave the ''τ-p'' transform represents a signalmodes as a superpositionfunction of plane-wavetheir componentsslowness instead of sinusoids. Application of this transform involves summing (stacking) all traces in a record along a slope (slant)values, which results in a single trace (called the ''p'' value or the ray parameter).<math>

p

</math>.<ref>{{Cite journal|url = http://www.crewes.org/ForOurSponsors/ResearchReports/1995/1995-11.pdf|title = Seismic reconstruction using a 3D tau-p transform|last = Donati|first = Maria|date = 1995|journal = CREWES Research Report|volume = 7}}</ref> Application of this transform involves summing (stacking) all traces in a record along a slope (slant), which results in a single trace (called the ''p'' value, slowness or the ray parameter). It transforms the input data from the space-time ___domain to intercept time-slowness ___domain.

<math>

p = \frac{1}{v} = \frac{dt}{dx}

</math>

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<math>

F(p,\tau) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, \tau + px) dx dt = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,t) \delta(t - \tau - xpx) dx dt

</math>

The ''τ-p'' transform converts seismic records into a ___domain where all these events are separated. ForSimply exampleput, each point in the ''τ-p'' ___domain is the sum of all the points in the ''x-t'' plane lying across a straight line with a slope ''p'' and intercept ''τ''.<ref>{{cite journal|last1=McMechan|first1=G. A.|last2=Clayton|first2=R. W.|last3=Mooney|first3=W. D.|title=Application of Wave Field Continuation to the Inversion of Refraction Data|journal=Journal of Geophysical Research|date=10 February 1982|volume=87|doi=10.1029/JB087iB02p00927 | pages=927–935|url=https://authors.library.caltech.edu/34941/1/McMechan1982.pdf}}</ref> That also means a point in the ''x-t'' ___domain transforms into a line in the ''τ-p'' ___domain, reflectionshyperbolae transform into pointsellipses and reflectionso hyperbolae transform into ellipseson. Similar to the Fourier transform, a signal in the ''τ-p'' ___domain can also be transformed back into the ''x-t'' ___domain.

=== Deconvolution ===

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During data acquisition, various effects have to be accounted for, such as near-surface structure around the source, noise, wavefront divergence and reverbations. It has to be ensured that a change in the seismic trace reflects a change in the geology and not one of the effects mentioned above. Deconvolution negates these effects to an extent and thus increases the resolution of the seismic data.

Seismic data, or a [[seismogram]], may be considered as a convolution of the source wavelet, the reflectivity and noise.<ref>{{cite journal|last1=Arya|first1=V|title=Deconvolution of Seismic Data - An Overview|journal=IEEE Transactions on Geoscience Electronics|date=April 1984|volume=16|issue=2|pages=95-9895–98|doi=10.1109/TGE.1978.294570|doi-access=free}}</ref>. Its deconvolution is usually implemented as a convolution with an inverse filter. Various well-known deconvolution techniques already exist for one dimension, such as predictive deconvolution, [[Kalman filter | Kalman filtering]]ing and deterministic deconvolution. In multiple dimensions, however, the deconvolution process is iterative due to the difficulty of defining an inverse operator. The output data sample may be represented as:

<math>

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<math>

X_F_{n+1}(\underline{k},\omega) = \lambda Y(\underline{k},\omega) + X_nF_n(\underline{k},\omega) - \lambda X_nF_n(\underline{k},\omega)R(\underline{k},\omega)

</math>

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<math>

X_nF_n(\underline{k},\omega) = \frac{Y(\underline{k},\omega)}{R(\underline{k},\omega)}\lbrace1-[1 - \lambda R(\underline{k},\omega)]^{n+1}\rbrace u(n)

</math>

The above equation can be approximated toas

<math>

X_nF_n(\underline{k},\omega) = \frac{Y(\underline{k},\omega)}{R(\underline{k},\omega)}

</math>,

if <math>n \rightarrow \infty</math> and <math>|1 - \lambda R(\underline{k},\omega)| < 1</math>

Note that the output is the same as the output of an inverse filter. An inverse filter does not actually have to be realized and the iterative procedure can be easily implemented on a computer.<ref>{{cite book|last1=Mersereau|first1=Russell|last2=Dudgeon|first2=Dan|title=Multidimensional Digital Signal Processing|publisher=Prentice-Hall|pages=350-352350–352}}</ref>.

=== Stacking ===

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{{see also | Seismic migration}}

Assuming a seismic wave <math>s(x,z,t)</math> travelling upwards towards the surface, where <math>x</math> is the position on the surface and <math>z</math> is the depth. The wave's propagation is described by:

[[File:Migration_Filter_Response.jpg|thumb|Evanescent and propagation regions for migration filter]]

<math>

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<math>

H(\omega_1, \omega_2) = e^{j\sqrt{\alpha^2 {\omega_2}^2 - {\omega_1}^2}}

\begin{cases}

e^{j\sqrt{\alpha^2 {\omega_2}^2 - {\omega_1}^2}}, & \text{ for } |\omega_1| < |\alpha \omega_2| \\

0, & \text{ else}

\end{cases}

</math>

where <math>\omega_1</math> is the x component of the wavenumber, <math>kk_x</math>, <math>\omega_2</math> is the temporal frequency <math>\Omega</math> and

<math>

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</math>

For implementation, ana all-passcomplex fan filter is used to approximate the ideal filter described above. It must allow propagation in the region <math>|\alpha \omega_2 | > |\omega_1 | </math> (called the propagating region) and attenuate waves in the region <math>|\alpha \omega_2 | < |\omega_1 | </math> (called the evanescent region). The ideal frequency response is shown in the figure below.<ref>{{cite book|last1=Mersereau|first1=Russell|last2=Dudgeon|first2=Dan|title=Multidimensional Digital Signal Processing|publisher=Prentice-Hall|pages=359-363359–363}}</ref>.

[[File:Filter response for Seismic Migration.jpg|thumb|center|Filter response for Seismic Migration]]

== References ==

<!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags, these references will then appear here automatically -->

{{Reflist}}

== External links ==

* [http://www.otago.ac.nz/geology/research/geophysics/controlled-source-seismology/tau-p.html Tau-P Processing of Seismic Refraction Data]

* [http://csegrecorder.com/articles/view/reflections-on-the-deconvolution-of-land-seismic-data Reflections on the Deconvolution of Land Seismic Data]

* [http://petrowiki.org/Seismic_profiling Seismic profiling]

* [http://www.reproducibility.org/RSF/book/bei/vela/paper_html/node6.html COMMON-MIDPOINT STACKING]

[[Category:Geophysics]]