Second-order Central Difference Finite-difference Wave Equation Datuming Formulation

Daisi Israel Komolafe

doi:doi:10.11648/j.earth.20221105.16

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Second-order Central Difference Finite-difference Wave Equation Datuming Formulation

Daisi Israel Komolafe

Published in Earth Sciences (Volume 11, Issue 5)

Received: 13 June 2022 Accepted: 15 August 2022 Published: 8 October 2022

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Abstract

Several methods of referencing seismic reflection data to a new datum have been researched and presented in the literature, that differ in the approach used to implement the calculation of datumed wavefields. The implementation time determines, in part, the efficiency of such a method. In this paper, a simple finite-difference approach for extrapolating seismic data from one datum to another for seismic near-surface irregularity correction is derived. Kirchhoff’s integral forms the basis on which the formulation is based. The Kirchhoff summation method is used, in a novel way, in agreement with summation theory by considering a single trace as an input wavefield to the algorithm. The approach is considered time-saving in computing the output wavefields as only a single trace serves as an input in any occasion of producing every output trace. For prestack data, the method consists of two steps; calculating the wavefields referenced to the output datum for a source and receivers and time- shifting the resulting data along the mean travel-time ray paths. The algorithm is successfully applied to real data and synthetic shot gathers. For a horizontally layered earth model with a planar surface, mathematical analysis shows that the gradient of the locus of the reflection travel time increases with offset, for a shot-on-end reflection profiling, in conformity with the synthetic data obtained for such a model. In the said analysis, considering parabolic, hyperbolic, and nonhyperbolic travel-time approximations, the gradient forms a useful tool for normal moveout velocity determination. The implementation of the algorithm on field data samples suppressed coherent noise, including the long-wavelength type such as ground roll, in contrast with the static correction method. The normal moveout corrections show that shift along ray-path produced less normal moveout stretch than shift along vertical for near-surface events for the same normal moveout parameters.

Published in	Earth Sciences (Volume 11, Issue 5)
DOI	10.11648/j.earth.20221105.16
Page(s)	289-306
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Datuming, Wave-equation, Central Difference, Finite-difference, Extrapolation, Second-order, Static, NMO Stretch

References

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[2]	Adegoke, K., 2016. Interpreting the summation notation when the lower limit is greater than the upper limit. Downloaded from https://vixra.org/pdf/1601.0207v2.pdf
[3]	Alkhalifah, T., and Tsvankin, I., 1995, Velocity analysis for transversely isotropic media: Geophysics, 60, 1550- 1566.
[4]	Barison, E.; Brancatelli, G.; Nicolich, R.; Accaino, F.; Giustiniani, M.; Tinivell, U. Wave equation datuming applied to marine OBS data and to land high-resolution seismic profiling Journal of Applied Geophysics 73, 267- 277.
[5]	Berryhill, J. R 1979. Wave-equation datuming. Geophysics 44, 1329-1344.
[6]	Berryhill, J. R 1984. Wave-equation datuming before stack. Geophysics 49, 2064-2066.
[7]	Bevc, D 1997. Flooding the topography: Wave equation datuming of land data with rugged acquisition topography. Geophysics 62, 1558-1569.
[8]	Causon, D. M and Mingham, C. G. 2010. Introductory Finite difference Methods for PDEs. Ventus Publishing ApS, ISBN 978-87-7681-642-1.
[9]	Claerbout, J. F., and Doherty, S., 1972, Downward continuation of Moveout-corrected seismograms: Geophysics, 37, no. 5, 741-768.
[10]	Jimenez-Tejero, C. E., Ranero, C. R., Sallares, V, and Gras, C., 2022. Downward continuation of marine seismic reflection data: an undervalued tool to improve velocity models. Geophys. J. Int. 230, 831-848.
[11]	Ke, B., Zhao, B., Liu, C and Chen, B., 1999 Improve image quality by wave-equation datuming based on a single shot gather SEG Technical Program Expanded Abstracts, DOI: 10.1190/1.1845210.
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[16]	Rasheed, M. 2004. Fifty years of stacking Acta Geophysica vol. 62, no. 3, pp. 505-528, DOI: 10.2478/s11600-013-0191-4.
[17]	Schneider, S. A, 1978. Integral formulation for migration in two and three dimensions. Geophysics 43, 49-76.
[18]	Shearer, P. M. 2009. Introduction to SEISMOLOGY Second Edition. Cambridge University Press. ISBN-13 978-0-511-58010-9.
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Daisi Israel Komolafe. (2022). Second-order Central Difference Finite-difference Wave Equation Datuming Formulation. Earth Sciences, 11(5), 289-306. https://doi.org/10.11648/j.earth.20221105.16

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Daisi Israel Komolafe. Second-order Central Difference Finite-difference Wave Equation Datuming Formulation. Earth Sci. 2022, 11(5), 289-306. doi: 10.11648/j.earth.20221105.16

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AMA Style

Daisi Israel Komolafe. Second-order Central Difference Finite-difference Wave Equation Datuming Formulation. Earth Sci. 2022;11(5):289-306. doi: 10.11648/j.earth.20221105.16

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@article{10.11648/j.earth.20221105.16,
  author = {Daisi Israel Komolafe},
  title = {Second-order Central Difference Finite-difference Wave Equation Datuming Formulation},
  journal = {Earth Sciences},
  volume = {11},
  number = {5},
  pages = {289-306},
  doi = {10.11648/j.earth.20221105.16},
  url = {https://doi.org/10.11648/j.earth.20221105.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.earth.20221105.16},
  abstract = {Several methods of referencing seismic reflection data to a new datum have been researched and presented in the literature, that differ in the approach used to implement the calculation of datumed wavefields. The implementation time determines, in part, the efficiency of such a method. In this paper, a simple finite-difference approach for extrapolating seismic data from one datum to another for seismic near-surface irregularity correction is derived. Kirchhoff’s integral forms the basis on which the formulation is based. The Kirchhoff summation method is used, in a novel way, in agreement with summation theory by considering a single trace as an input wavefield to the algorithm. The approach is considered time-saving in computing the output wavefields as only a single trace serves as an input in any occasion of producing every output trace. For prestack data, the method consists of two steps; calculating the wavefields referenced to the output datum for a source and receivers and time- shifting the resulting data along the mean travel-time ray paths. The algorithm is successfully applied to real data and synthetic shot gathers. For a horizontally layered earth model with a planar surface, mathematical analysis shows that the gradient of the locus of the reflection travel time increases with offset, for a shot-on-end reflection profiling, in conformity with the synthetic data obtained for such a model. In the said analysis, considering parabolic, hyperbolic, and nonhyperbolic travel-time approximations, the gradient forms a useful tool for normal moveout velocity determination. The implementation of the algorithm on field data samples suppressed coherent noise, including the long-wavelength type such as ground roll, in contrast with the static correction method. The normal moveout corrections show that shift along ray-path produced less normal moveout stretch than shift along vertical for near-surface events for the same normal moveout parameters.},
 year = {2022}
}

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TY  - JOUR
T1  - Second-order Central Difference Finite-difference Wave Equation Datuming Formulation
AU  - Daisi Israel Komolafe
Y1  - 2022/10/08
PY  - 2022
N1  - https://doi.org/10.11648/j.earth.20221105.16
DO  - 10.11648/j.earth.20221105.16
T2  - Earth Sciences
JF  - Earth Sciences
JO  - Earth Sciences
SP  - 289
EP  - 306
PB  - Science Publishing Group
SN  - 2328-5982
UR  - https://doi.org/10.11648/j.earth.20221105.16
AB  - Several methods of referencing seismic reflection data to a new datum have been researched and presented in the literature, that differ in the approach used to implement the calculation of datumed wavefields. The implementation time determines, in part, the efficiency of such a method. In this paper, a simple finite-difference approach for extrapolating seismic data from one datum to another for seismic near-surface irregularity correction is derived. Kirchhoff’s integral forms the basis on which the formulation is based. The Kirchhoff summation method is used, in a novel way, in agreement with summation theory by considering a single trace as an input wavefield to the algorithm. The approach is considered time-saving in computing the output wavefields as only a single trace serves as an input in any occasion of producing every output trace. For prestack data, the method consists of two steps; calculating the wavefields referenced to the output datum for a source and receivers and time- shifting the resulting data along the mean travel-time ray paths. The algorithm is successfully applied to real data and synthetic shot gathers. For a horizontally layered earth model with a planar surface, mathematical analysis shows that the gradient of the locus of the reflection travel time increases with offset, for a shot-on-end reflection profiling, in conformity with the synthetic data obtained for such a model. In the said analysis, considering parabolic, hyperbolic, and nonhyperbolic travel-time approximations, the gradient forms a useful tool for normal moveout velocity determination. The implementation of the algorithm on field data samples suppressed coherent noise, including the long-wavelength type such as ground roll, in contrast with the static correction method. The normal moveout corrections show that shift along ray-path produced less normal moveout stretch than shift along vertical for near-surface events for the same normal moveout parameters.
VL  - 11
IS  - 5
ER  -

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Author Information

Daisi Israel Komolafe

Science Department, Greensprings School, Lagos, Nigeria

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Daisi Israel Komolafe. (2022). Second-order Central Difference Finite-difference Wave Equation Datuming Formulation. Earth Sciences, 11(5), 289-306. https://doi.org/10.11648/j.earth.20221105.16

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Daisi Israel Komolafe. Second-order Central Difference Finite-difference Wave Equation Datuming Formulation. Earth Sci. 2022, 11(5), 289-306. doi: 10.11648/j.earth.20221105.16

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AMA Style

Daisi Israel Komolafe. Second-order Central Difference Finite-difference Wave Equation Datuming Formulation. Earth Sci. 2022;11(5):289-306. doi: 10.11648/j.earth.20221105.16

Copy | Download

@article{10.11648/j.earth.20221105.16,
  author = {Daisi Israel Komolafe},
  title = {Second-order Central Difference Finite-difference Wave Equation Datuming Formulation},
  journal = {Earth Sciences},
  volume = {11},
  number = {5},
  pages = {289-306},
  doi = {10.11648/j.earth.20221105.16},
  url = {https://doi.org/10.11648/j.earth.20221105.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.earth.20221105.16},
  abstract = {Several methods of referencing seismic reflection data to a new datum have been researched and presented in the literature, that differ in the approach used to implement the calculation of datumed wavefields. The implementation time determines, in part, the efficiency of such a method. In this paper, a simple finite-difference approach for extrapolating seismic data from one datum to another for seismic near-surface irregularity correction is derived. Kirchhoff’s integral forms the basis on which the formulation is based. The Kirchhoff summation method is used, in a novel way, in agreement with summation theory by considering a single trace as an input wavefield to the algorithm. The approach is considered time-saving in computing the output wavefields as only a single trace serves as an input in any occasion of producing every output trace. For prestack data, the method consists of two steps; calculating the wavefields referenced to the output datum for a source and receivers and time- shifting the resulting data along the mean travel-time ray paths. The algorithm is successfully applied to real data and synthetic shot gathers. For a horizontally layered earth model with a planar surface, mathematical analysis shows that the gradient of the locus of the reflection travel time increases with offset, for a shot-on-end reflection profiling, in conformity with the synthetic data obtained for such a model. In the said analysis, considering parabolic, hyperbolic, and nonhyperbolic travel-time approximations, the gradient forms a useful tool for normal moveout velocity determination. The implementation of the algorithm on field data samples suppressed coherent noise, including the long-wavelength type such as ground roll, in contrast with the static correction method. The normal moveout corrections show that shift along ray-path produced less normal moveout stretch than shift along vertical for near-surface events for the same normal moveout parameters.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Second-order Central Difference Finite-difference Wave Equation Datuming Formulation
AU  - Daisi Israel Komolafe
Y1  - 2022/10/08
PY  - 2022
N1  - https://doi.org/10.11648/j.earth.20221105.16
DO  - 10.11648/j.earth.20221105.16
T2  - Earth Sciences
JF  - Earth Sciences
JO  - Earth Sciences
SP  - 289
EP  - 306
PB  - Science Publishing Group
SN  - 2328-5982
UR  - https://doi.org/10.11648/j.earth.20221105.16
AB  - Several methods of referencing seismic reflection data to a new datum have been researched and presented in the literature, that differ in the approach used to implement the calculation of datumed wavefields. The implementation time determines, in part, the efficiency of such a method. In this paper, a simple finite-difference approach for extrapolating seismic data from one datum to another for seismic near-surface irregularity correction is derived. Kirchhoff’s integral forms the basis on which the formulation is based. The Kirchhoff summation method is used, in a novel way, in agreement with summation theory by considering a single trace as an input wavefield to the algorithm. The approach is considered time-saving in computing the output wavefields as only a single trace serves as an input in any occasion of producing every output trace. For prestack data, the method consists of two steps; calculating the wavefields referenced to the output datum for a source and receivers and time- shifting the resulting data along the mean travel-time ray paths. The algorithm is successfully applied to real data and synthetic shot gathers. For a horizontally layered earth model with a planar surface, mathematical analysis shows that the gradient of the locus of the reflection travel time increases with offset, for a shot-on-end reflection profiling, in conformity with the synthetic data obtained for such a model. In the said analysis, considering parabolic, hyperbolic, and nonhyperbolic travel-time approximations, the gradient forms a useful tool for normal moveout velocity determination. The implementation of the algorithm on field data samples suppressed coherent noise, including the long-wavelength type such as ground roll, in contrast with the static correction method. The normal moveout corrections show that shift along ray-path produced less normal moveout stretch than shift along vertical for near-surface events for the same normal moveout parameters.
VL  - 11
IS  - 5
ER  -

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