Applied Geophysics

, Volume 15, Issue 3–4, pp 466–480 | Cite as

Elastic modulus extraction based on generalized pre-stack PP–PS joint linear inversion

  • Qi-Qi Ma
  • Zan-Dong SunEmail author


Joint PP–PS inversion offers better accuracy and resolution than conventional P-wave inversion. P- and S-wave elastic moduli determined through data inversions are key parameters for reservoir evaluation and fluid characterization. In this paper, starting with the exact Zoeppritz equation that relates P- and S-wave moduli, a coefficient that describes the reflections of P- and converted waves is established. This method effectively avoids error introduced by approximations or indirect calculations, thus improving the accuracy of the inversion results. Considering that the inversion problem is ill-posed and that the forward operator is nonlinear, prior constraints on the model parameters and modified low-frequency constraints are also introduced to the objective function to make the problem more tractable. This modified objective function is solved over many iterations to continuously optimize the background values of the velocity ratio, which increases the stability of the inversion process. Tests of various models show that the method effectively improves the accuracy and stability of extracting P and S-wave moduli from underdetermined data. This method can be applied to provide inferences for reservoir exploration and fluid extraction.


Pre-stack joint PP–PS inversion P-and S-wave moduli exact Zoeppritz equation generalized linear inversion reservoir and fluid prediction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors thank the National Science and Technology Major Project (No. 2016ZX05047-002-001) for providing the support. The authors are grateful to editors and reviewers for their valuable comments on and also would like to thank Dr. Zhang Fengqi for his help in the writing of this article.


  1. Aki, K., and Richards, P. G., 1980, Quantitative Seismology Theory and Method: W. H. Freeman and Company, San Francisco, 296–297.Google Scholar
  2. Alemie, W., and Sacchi, M. D., 2011, High–resolution three–term AVO inversion by means of a Trivariate Cauchy probability distribution: Geophysics, 76, 43–55.CrossRefGoogle Scholar
  3. Avseth, P., Mukerji, T. and Mavko, G., 2005, Quantitative Seismic Interpretation: Cambridge University Press, Cambridge, 236–237.CrossRefGoogle Scholar
  4. Backus, G. E., and Gilbert, F., 1968, The resolving power of gross earth data: Geophysical Journal International, 16, 169–205.CrossRefGoogle Scholar
  5. Bortfeld, R., 1978, Approximation to the reflection and transmission coefficients of plane longitudinal and transverse waves: Geophysical Journal of the Royal Astronomical Society, 53, 467–496.CrossRefGoogle Scholar
  6. Buland, A., and Omre, H., 2003a, Joint AVO inversion, wavelet estimation an noise–level estimation using a spatially coupled hierarchial Bayesian model: Geophysical Prospecting, 51, 531–550.CrossRefGoogle Scholar
  7. Buland, A., and Omre, H., 2003b, Bayesian linearised AVO inversion: Geophysics, 68, 185–198.CrossRefGoogle Scholar
  8. Downton, J. E., 2005, Seismic parameter Estimation from AVO Inversion: PhD thesis, University of Calgary, Calgary.Google Scholar
  9. Du, Q. Z., and Yan, H. Z., 2013, PP and PS joint AVO inversion and fluid prediction: Journal of Applied Geophysics, 90, 110–118.CrossRefGoogle Scholar
  10. Fang, Y., Zhang, F. Q., and Wang, Y. C., 2016, Generalized linear joint PP–PS inversion based on two constraints: Applied Geophysics, 13, 103–115.CrossRefGoogle Scholar
  11. Fatti, J. L., Smith, G. C, Vail, P. J., Strauss, P. J., and Levitt, P. R., 1994, Detection of gas in sandstone reservoirs using AVO analysis: A 3–D seismic case history using the Geostack technique: Geophysics, 59, 1362–1376.CrossRefGoogle Scholar
  12. Goodway, B., Chen, T. W., and Downton, J., 1997, Improved AVO fluid detection and lithology discrimination using Lame petrophysical parameters: “λρ”, “µρ”, and “λ/µ fluid stack”, from P and S inversions: 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 16, 183–186.Google Scholar
  13. Goodway, W., 2001, AVO and Lamé constants for rock parameterization and fluid detection: CSEG Recorder, 26, 39–60.Google Scholar
  14. Hansen, T. M., Mosegaard, K., Radmila, P. T., Uldall A., and Jacobsen, N. L., 2008, Attribute–guided welllog interpolation applied to low–frequency impedance estimation: Geophysics, 73, 83–95.CrossRefGoogle Scholar
  15. Hou, D. J., Liu, Y., Hu, G. Q., Wei, X. C., and Chen, T. S., 2014, Prestack multiwave joint inversion for elastic moduli based on Bayesian theory: Chinese Journal of Geophysics, 57, 1251–1264.Google Scholar
  16. Hou, D. J., 2017, PP and PS AVO joint inversion for Pand S–wave moduli based on Bayes theorem: 87th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 768–772.Google Scholar
  17. Huang, H. D., Wang, Y. C., Guo, F., Zhang, S., Ji, Y. Z. and Liu, C. H., 2015, Zoeppritz equation–based prestack inversion and its application in fluid identification: Applied Geophysics, 12, 199–211.CrossRefGoogle Scholar
  18. Jaime M., and Douglas, R. S., 2016, A comparative study of the anisotropic dynamic and static elastic moduli of unconventional reservoir shales: Implication for geomechanical investigations: Geophysics, 81, 245–261.CrossRefGoogle Scholar
  19. Klaas, V., Gregor, E., Gregor B., and Ralf, W., 2010, Effect of carbonate pore structure on dynamic shear moduli: Geophysics, 75, 1–8.Google Scholar
  20. Kurt, H., 2007, Joint inversion of AVA data for elastic parameters by bootstrapping: Computers and Geosciences, 33, 367–382.CrossRefGoogle Scholar
  21. Lu, J., Yang, Z., Wang, Y., and Shi, Y., 2015, Joint PP and PS AVA seismic inversion using exact Zoeppritz equations: Geophysics, 80, 239–250.CrossRefGoogle Scholar
  22. Martin, G. S., Wiley, R., and Martfurt, K. J., 2006, Marmousi2: an elastic upgrade for Marmousi: The Leading Edge, 25, 156–166.CrossRefGoogle Scholar
  23. Mavko, G., Mukerji, T., and Dvorkin, J., 1998, The rock physics handbook: Tools for seismic analysis in porous media: Cambridge University Press, Cambridge, 329.Google Scholar
  24. Ostrander, W. J., 1984, Plane–wave reflection coefficients for gas sands at non–normal angles of incidence: Geophysics, 49, 1637–1648.CrossRefGoogle Scholar
  25. Quakenbush, M., Shang, B., and Tuttle, C., 2006, Poisson impedance: The Leading Edge, 25, 128–138.CrossRefGoogle Scholar
  26. Rabben, T. E., and Ursin, B., 2000, Non–linear least–squares inversion with data–driven Bayesian regularization: Geophysical Journal International, 142, 43–72.Google Scholar
  27. Russell, B. H., Gray, D., and Hampson, D. P., 2011, Linearized AVO and poroelasticity: Geophysics, 76, 19–29.CrossRefGoogle Scholar
  28. Smith, G. C., and Gidlow, P. M., 1987, Weighted stacking for rock property estimation and detection of gas: Geophysical Prospecting, 35, 993–1014.CrossRefGoogle Scholar
  29. Stewart, R. R., 1990, Joint P and P–SV inversion: CREWES Research Report, 2, 112–115.Google Scholar
  30. Tang, J. and Wang, Y. F., 2017, PP and PS joint inversion with a posterior constraint and with particle filtering: Journal of Geophysics and Engineering, 14, 1399–1412.CrossRefGoogle Scholar
  31. Viere, H. H., and Landro, M., 2006, Simultaneous inversion of PP and PS seismic data: Geophysics, 71, 1–10.CrossRefGoogle Scholar
  32. Wang, B. L., Yin, X. Y., and Zhang, F. C., 2006, Lame parameters inversion based on elastic impedance ans its application: Applied Geophysics, 3, 174–178.CrossRefGoogle Scholar
  33. Yilmaz, O., 2001, Seimic data analysis: Processing, Inversion, and Interpretation of seismic data: Society of Exploration, I, 133–141.CrossRefGoogle Scholar
  34. Yin, X. Y., Sun, R. N., Wang, B. L., and Zhang, G. Z., 2014, Simultaneous inversion of petrophysical parameters based on geostatistical a priori information: Applied Geophysics, 11, 311–320.CrossRefGoogle Scholar
  35. Zhang, F. Q., Wei, F. J., Wang, Y. C., Wang, W. J., and Li, Y., 2013, Generalized linear AVO inversion with the priori constraint of trivariate cauchy distribution based on Zoeppritz equation: Chinese Journal of Geophysics, 56, 2098–2115.Google Scholar
  36. Zhi, L. X., Chen, S. Q., Song, B. S., and Li, X. Y., 2018, Nonlinear PP and PS joint inversion based on the exact Zoeppritz equations: a two–stage procedure: Journal of Geophysics and Engineering, 15, 397–410.CrossRefGoogle Scholar
  37. Zhou, L., Li, J., Chen, X., Liu, X., and Chen, L., 2017, Prestack AVA inversion of exact Zoeppritz equations based on modified trivariate Cauchy distribution: Journal of Applied Geophysic, 138, 80–90.CrossRefGoogle Scholar
  38. Zoeppritz, K., 1919, On reflection and propagation of seismic waves: Gottinger Nachrichten, I: 66–84.Google Scholar
  39. Zong, Z. Y., Yin, X. Y., and Wu, G. C., 2012, AVO inversion and poroelasticity with P–and S–wave moduli: Geophysics, 6, 17–24.CrossRefGoogle Scholar
  40. Zong, Z. Y., Yin, X. Y. and Wu, G. C., 2013, Elastic impedance parameterization and inversion with Young’s modulus and Poisson’s ratio: Geophysics, 78, 35–42.CrossRefGoogle Scholar

Copyright information

© Editorial Office of Applied Geophysics and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lab for Integration of Geology and GeophysicsChina University of PetroleumBeijingChina

Personalised recommendations