# Parameterization of A Priori Geological Knowledge in Seismic Inversion

**Abstract**—An approach to parameterization of prior geological knowledge concerning the changes in depositional environment in space and geological time for their quantitative use in the workflow of seismic inversion is presented. The idea is to describe the observed or expected facies diversity in terms of a few statistically independent factors (generalized geological variables). The topology and metrics of the model are determined by the set of basic depositional environments and the statistics of facies transitions. The introduced parameters make it possible to estimate the occurrence probability of different facies at each model point. The proposed technique can be applied for regions with various degree of detail of the existing geological knowledge and amount of available well logging data.

## Keywords:

seismic inversion depositional environments facies conditional probability a priori knowledge Bayes’ theorem## Notes

### ACKNOWLEDGMENTS

I am deeply grateful to Doctor of Science in geology and mineralogy, Professor of the Faculty of Geology of the Lomonosov Moscow State University Valentina Alekseevna Zhemchugova. The main ideas reflected in this paper resulted from our long-standing collaboration and fruitful discussions during the work on the real well and seismic data integrated interpretation projects on various oil and gas fields.

I am also grateful to Doctor of Science in physics and mathematics, corresponding member of the Russian Academy of Sciences Sergei Andreevich Tikhotskii for his valuable comments during the discussions and preparation of this publication.

## REFERENCES

- 1.Ajdukiewicz, J.M. and Lander, R.H., Sandstone reservoir quality prediction: the state of the art,
*AAPG Bull*., 2010, vol. 94, no. 8, pp. 1083–1091.CrossRefGoogle Scholar - 2.Amari, Sh. and Nagaoka, H.,
*Methods of Information Geometry*, Vol. 191 of Transl. Math. Monogr., Providence: Am. Math. Soc. and Oxford Univ., 2000.Google Scholar - 3.Avseth, P., Mukerji, T., and Mavko, G.,
*Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk*, Cambridge: Cambridge Univ. Press, 2005.CrossRefGoogle Scholar - 4.Ball, V., Tenorio, L., Schiott, C., Blangy, J.P., and Thomas, M., Uncertainty in inverted elastic properties resulting from uncertainty in the low-frequency model,
*The Leading Edge*, 2015, vol. 34, no. 9, pp. 1028–1030, 1032, 1034–1035.CrossRefGoogle Scholar - 5.Bhattacharyya, A., On a measure of divergence between two statistical populations defined by their probability distribution,
*Bull. Calcutta Math. Soc*., 1943, vol. 35, pp. 99–110.Google Scholar - 6.Bohacs, K. and Suter, J., Sequence stratigraphic distribution of coaly rocks: fundamental controls and paralic examples,
*AAPG Bull*., 1997, vol. 81, no. 10, pp. 1612–1639.Google Scholar - 7.Borg, I. and Groenen, P.J.F.,
*Modern Multidimensional Scaling: Theory and Applications*, New York: Springer, 2005.Google Scholar - 8.Bosch, M., Mukerji, T., and Gonzalez, E.F., Seismic inversion for reservoir properties combining statistical rock physics and geostatistics: a review,
*Geophysics*, 2010, vol. 75, no. 5.CrossRefGoogle Scholar - 9.Buland, A., Kolbjørnsen, O., Hauge, R., Skjœveland, O., and Duffaut, K., Bayesian lithology and fluid prediction from seismic prestack data,
*Geophysics*, 2008, vol. 73, no. 3, pp. C13–C21.CrossRefGoogle Scholar - 10.Carroll, A.R. and Bohacs, K.M., Stratigraphic classification of ancient lakes: balancing tectonic and climatic controls,
*Geology*, 1999, vol. 27, no. 2, pp. 99–102.CrossRefGoogle Scholar - 11.Catuneanu, O., Galloway, W.E., Kendall, C.G.S. t.C., Miall, A.D., Posamentier, H.W., Strasser, A., and Tucker, M.E., Sequence stratigraphy: methodology and nomenclature,
*Newslett. Stratigr.*, 2011, vol. 44, no. 3, pp. 173–245.CrossRefGoogle Scholar - 12.Chentsov, N.N., On the systematic theory of exponential families of probability distributions,
*Teor. Veroyat. Prim*., 1966, vol. 11, no. 3, pp. 483–494.Google Scholar - 13.Chernoff, H., A measure of asymptotic efficiency for test of a hypothesis based on the sum of observations,
*Ann. Math. Stat*., 1952, vol. 23, no. 4, pp. 493–507.CrossRefGoogle Scholar - 14.Colombera, L., Mountney, N.P., and Mccaffrey, W.D., A quantitative approach to fluvial facies models: methods and example results,
*Sedimentology*, 2013, vol. 60, pp. 1526–1558.Google Scholar - 15.Davison, M.,
*Multidimensional Scaling*, New York: Wiley, 1983.Google Scholar - 16.Dubrule, O.,
*Geostatistics for seismic data integration in Earth models: 2003 Distinguished Instructor Short Course*, Distinguished Instructor Series, no. 6, Tulsa: Soc.*Explor. Geophys.*, 2003.Google Scholar - 17.Duijndam, A.J.W., Bayesian estimation in seismic inversion. Part I: Principles,
*Geophys. Prospect.,*1998, vol. 36, no. 8, pp. 878–898.CrossRefGoogle Scholar - 18.Duijndam, A.J.W., Bayesian estimation in seismic inversion. Part II: Uncertainty analysis,
*Geophys. Prospect.,*1998, vol. 36, no. 8, pp. 899–918.CrossRefGoogle Scholar - 19.Dvorkin, J., Gutierrez, M.A., and Grana, D.,
*Seismic Reflections of Rock Properties*, Cambridge: Cambridge Univ. Press, 2014.CrossRefGoogle Scholar - 20.Eidsvik, J., Mukerji, T., and Switzer, P., Modeling lithofacies alternations from well logs using Hierarchical Markov Chains,
*SEG Technical Program Expanded Abstracts*, 2002, vol. 21, pp. 2463–2466.Google Scholar - 21.Eidsvik, J., Avseth, P., More, H., Mukerji, T., and Mavko, G., Stochastic reservoir characterization using prestack seismic data,
*Geophysics*, 2004, vol. 69, no. 4, pp. 978–993.CrossRefGoogle Scholar - 22.Gogonenkov, G.N.,
*Izuchenie detal’nogo stroeniya osadochnykh tolshch seismorazvedkoi*(Seismic Prospecting of the Detailed Structure of Sedimentary Sections), Moscow: Nedra, 1987.Google Scholar - 23.González, E.F., Mukerji, T., and Mavko, G., Seismic inversion combining rock physics and multiple-point geostatistics,
*Geophysics*, 2008, vol. 73, no. 1, pp. R11–R21.CrossRefGoogle Scholar - 24.González, E.F., Gesbert, S. and Hofmann, R., Adding geologic prior knowledge to Bayesian lithofluid facies estimation from seismic data,
*Interpretation*, 2016, vol. 4, no. 3, pp. 1A–Y1.CrossRefGoogle Scholar - 25.Grana, D., Bayesian inversion methods for seismic reservoir characterization and time-lapse studies,
*Ph. D. Dissertation*, Stanford University, 2013.Google Scholar - 26.Grana, D., Pirrone, M., and Mukerji, T., Quantitative log interpretation and uncertainty propagation of petrophysical properties and facies classification from rock-physics modeling and formation evaluation analysis,
*Geophysics*, 2012, vol. 77, no. 3, pp. WA45–WA63.CrossRefGoogle Scholar - 27.Gunning, J. and Glinsky, M., Delivery: an open-source model-based Bayesian seismic inversion program,
*Comput. Geosci*., 2004, vol. 30, pp. 619–636.CrossRefGoogle Scholar - 28.Gunning, J.S., Kemper, M., and Pelham, A., Obstacles, challenges and strategies for facies estimation in AVO seismic inversion,
*Extended Abstract of the 76th EAGE Conf. and Exhibition*, 2014.Google Scholar - 29.Haas, A. and Dubrule, O., Geostatistical inversion—a sequential method of stochastic reservoir modelling constrained by seismic data,
*First Break*, 1994, vol. 12, pp. 561–569.CrossRefGoogle Scholar - 30.Harper, C.W.J., Facies model revisited: an examination of quantitative methods,
*Geosci. Can*., 1984, vol. 11, no. 4, pp. 203–207.Google Scholar - 31.Kabanikhin, S.I. and Isakov, K.T.,
*Obratnye i nekorrektnye zadachi dlya giperbolicheskikh uravnenii*(Inverse and Ill-Posed Problems for Hyperbolic Equations), Almaty: Kazakh. nats. pedagog. univ. im. Abaya, 2007.Google Scholar - 32.Kemper, M.A.C. and Gunning, J., Joint impedance and facies inversion—seismic inversion redefined,
*First Break*, 2014, vol. 32, no. 9, pp. 89–95.Google Scholar - 33.Krasheninnikov, G.F.,
*Uchenie o fatsiyakh. Uchebnoe posobie*(Theory of Facies: A Tutorial), Moscow: Vysshaya shkola, 1971.Google Scholar - 34.Larsen, A.L., Ulvmoen, M., Omre, H., and Buland, A., Bayesian lithology/fluid prediction and simulation on the basis of a Markov-chain prior model,
*Geophysics*, 2006, vol. 71, no. 5, pp. R69–R78.CrossRefGoogle Scholar - 35.Mahalanobis, P.C., On the generalized distance in statistics,
*Proc. Natl. Inst. Sci. India*, 1936, vol. 2, no. 1, pp. 49–55.Google Scholar - 36.Mavko, G., Mukerji, T., and Dvorkin, J.,
*The Rock Physics Handbook*, New York: Cambridge Univ. Press, 2009.CrossRefGoogle Scholar - 37.Miall, A.D., Markov chain analysis applied to an ancient alluvial plain succession,
*Sedimentology*, 1973, no. 20, pp. 347–364.CrossRefGoogle Scholar - 38.Morad, S., Al-Ramadan, K., Ketzler, J.M., and De Ros, L.F., The impact of diagenesis on the heterogeneity of sandstone reservoirs: a review of the role of depositional facies and sequence stratigraphy,
*AAPG Bull*., 2010, vol. 94, no. 8, pp. 1267–1309.CrossRefGoogle Scholar - 39.Morozova, E.A. and Chentsov, N.N.,
*Natural geometry on families of probability laws, Itogi Nauki Tekhn. Sovrem. Probl. Matem. Fundam. Napravleniya.*Moscow: VINITI, 1991, pp. 133–265.Google Scholar - 40.Mukerji, T., Avseth, P., Mavko, G., Takahashi, I., and Gonzalez, E.F., Statistical rock physics: combining rock physics, information theory, and geostatistics to reduce uncertainty in seismic reservoir characterization,
*The Leading Edge*, 2001, vol. 20, pp. 313–319.CrossRefGoogle Scholar - 41.Myasoedov, D.N., The influence of the interpolation model on the result of geostatistical inversion,
*Geofizika*, 2015, no. 6, pp. 20–28.Google Scholar - 42.Nielsen, F., Chernoff information of exponential families. arXiv:1102.2684v1, 2011.Google Scholar
- 43.Nielsen, F., An information-geometric characterization of Chernoff information,
*IEEE Signal Process. Lett*., 2013, vol. 20, no. 3, pp. 269–272.CrossRefGoogle Scholar - 44.Nielsen, F. and Garcia, V., Statistical exponential families: a digest with flash cards. arXiv:0911.4863v2, 2011.Google Scholar
- 45.Nurgalieva, N.G., Vinokurov, V.M., and Nurgaliev, D.K., The Golovkinsky strata formation model, basic facies law and sequence stratigraphy concept: historical sources and relations,
*Russ. J. Earth Sci*., 2007, vol. 9, ES 1003.Google Scholar - 46.Odesskii, I.A., Spatio-temporal meaning of the rock-layer boundaries,
*Zap. Gorn. Inst.*, 1992, vol. 134, pp. 116–123.Google Scholar - 47.Rao, C.R., Differential metrics in probability spaces, Chapter 5 in
*Differential Geometry in Statistical Inference*by Amari, S., Barndorff-Nielsen, O., Kass, R., Lauritzen, S., and Rao, C.R., IMS Lecture Notes—Monograph Series, vol. 10, Hayward:*Inst. Math. Stat.*, 1987, pp. 217–240.Google Scholar - 48.Reading, H.G., Clastic facies models, a personal perspective,
*Bull. Geol. Soc. Denmark*, 2001, vol. 48, pp. 101–115.Google Scholar - 49.Riegl, B. and Purkis, S.J., Markov models for linking environments and facies in space and time (recent Arabian Gulf, Miocene Paratethys), in
*Perspectives in Carbonate Geology: A Tribute to the Career of Robert Nathan Ginsburg*(Special Publication Number 41 of the International Association of Sedimentologists), Swart, P.K., Eberli, G.P., and McKenzie, J.A., Eds., Chicester: Wiley, 2009, pp. 337–360.CrossRefGoogle Scholar - 50.Rimstad, K. and Omre, H., Impact of rock-physics depth trends and Markov random fields on hierarchical Bayesian lithology/fluid prediction,
*Geophysics*, 2010, vol. 75, no. 4, pp. R93–R108.CrossRefGoogle Scholar - 51.Russell, B.H.,
*Introduction to Seismic Inversion Methods*, Course Notes Series, no. 2, Tulsa: Society of Exploration Geophysicists, 1988.Google Scholar - 52.Sams, M. and Saussus, D., Practical implications of low frequency model selection on quantitative interpretation results,
*SEG Technical Program Expanded Abstracts*, 2013, pp. 3118–3122.Google Scholar - 53.Saussus, D. and Sams, M., Facies as the key to using seismic inversion for modelling reservoir properties,
*First Break*, 2012, vol. 30, no. 7, pp. 45–52.CrossRefGoogle Scholar - 54.Scales, J.A. and Tenorio, L., Prior information and uncertainty in inverse problems,
*Geophysics*, 2001, vol. 66, no. 2, pp. 389–397.CrossRefGoogle Scholar - 55.Sen, M.K.,
*Seismic Inversion*, Dallas: Soc. Petrol. Engineers, 2006.Google Scholar - 56.Sen, M.K. and Stoffa, P.L., Nonlinear one-dimensional seismic waveform inversion using simulated annealing,
*Geophysics*, 1991, vol. 56, no. 10, pp. 1624–1638.CrossRefGoogle Scholar - 57.Tarantola, A.,
*Inverse Problem Theory and Methods for Model Parameter Estimation*, Philadelphia: Soc. Ind. Appl. Math., 2005.CrossRefGoogle Scholar - 58.
*The Sedimentary Record of Sea-Level Change*, Coe, A.L., Ed., Cambridge: Cambridge Univ. Press and the Open University, 2003.Google Scholar - 59.Tikhonov, A.N. and Arsenin, V.Ya.,
*Metody resheniya nekorrektnykh zadach*(Methods for Solving Ill-Posed Problems), Moscow: Nauka, 1979.Google Scholar - 60.Torgerson, W.S., Multidimensional scaling: I. Theory and method,
*Psychometrica*, 1952, vol. 17, pp. 401–419.CrossRefGoogle Scholar - 61.Ulvmoen, M. and Omre, H., Improved resolution in Bayesian lithology/fluid inversion from prestack seismic data and well observations: Part 1—Methodology,
*Geophysics*, 2010, vol. 75, no. 2.CrossRefGoogle Scholar - 62.Walker, M., Grant, S., Conolly, P., and Smith, L., Stochastic inversion for facies: a case study on the Schiehallion field,
*Interpretation*, 2016, vol. 4, no. 3, pp. 1A–Y1.CrossRefGoogle Scholar - 63.Yakovlev, I.V., Ampilov, Yu.P., and Filippova, K.E., Almost all about seismic inversion, part 2,
*Tekhnol. Seismorazvedki*, 2011, no. 1, pp. 5–15.Google Scholar - 64.Zavala, C., Carvajal, J., Mercano, J., and Delgado, M., Sedimentological indices: a new tool for regional studies of hyperpycnal systems,
*Sediment Transfer from Shelf to Deepwater—Revisiting the Delivery Mechanisms, AAPG Hedberg Conf. Proc.*, Ushuaia–Patagonia, March 3–7, 2008, AAPG Search and Discovery Article no. 50076.Google Scholar