Abstract
We consider the problem of optimal statistical estimation of micro-seismic source parameters using multichannel data from surface arrays of seismometers affected by a strong seismic noise. The problem is treated as a statistical task of parameter estimation for a general type multidimensional linear model with random or completely unknown input time functions. The maximum-likelihood generic estimators are derived and their relationship with the well-known seismic emission tomography (SET) algorithm is established. The proposed estimation algorithms perform processing of multichannel discrete observations in the frequency domain and can be implemented in on-line mode. Using the method of successive independent trials (Monte-Carlo), we demonstrate that a proposed statistical estimation algorithm provides much higher accuracy of the micro-seismic event source location in real noise conditions than the widely used SET algorithm and, as a consequence, is more reliable in practical applications.
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Notes
Symbols \(\overline{1,m}\) denote the set of all integers with values between 1 and m.
We call a “segment” of the stationary time series \(\eta _k, \; k\in \mathbb Z \), the set of this time series values with indexes \(k\) belonging to the discrete time interval \(k\in \overline{1,n} \).
References
Aki, K., Richards, R.: Quantitative Seismology. University Science Books, CA, USA (2002)
Bartlett, M.S.: An inverse matrix adjustment, arising in discriminant analysis. Ann. Math. Stat. 22(1), 107–111 (1951)
Botev, Z.I., Grotowski, J.F., Kroese, D.P.: Kernel density estimation via diffusion. Ann. Math. Stat. 38(5), 2916–2957 (2010)
Brandstein, M., Ward, D. (eds.): Microphone Arrays. Springer, Berlin (2001)
Brigham, E.O.: The Fast Fourier Transform and its Applications. Prentice Hall, Englewood Cliffs (1988)
Brillinger D.: Time series. Data Analysis and Theory. Holt, Reinhart and Winston, Inc., New York (1975)
Capon, J., Greenfield, R.J., Kolker, R.J.: Multidimensional maximum-likelihood processing of a large aperture seismic array. Proc. IEEE 55(2), 192–211 (1967)
Champeney, D.C.: A Handbook of Fourier Theorems. Cambridge University Press, Cambridge (1987)
Devis, R.D.: Asymptotic inference on stationary Gaussian time series. Adv. Apl. Prob. 5, 469–497 (1973)
Duncan, P.M., Lakings, J.D., Flores, R.A.: Method for passive seismic emission tomography. US Patent #7,663,970 (2010)
Duncan, P., Eisner, L.: Reservoir characterization using surface micro-seismic monitoring. Geophysics 75, 139–146 (2010)
Eisner, L., Williams-Stroud, S., Hill, A., Duncan, P., Thornton, M.: Beyond the dots in the box: microseismicity-constrained fracture models for reservoir simulation. Lead. Edge 29(2), 326–333 (2010)
Graybill, F.A.: Matrices with Applications in Statistics, 2nd edn. Wadsworth International Group, Belmont, California (1983)
Hannan, E.J.: Multiple Time Series Analysis. John Willey and Sons, Inc., New York (1970)
Haykin, S.: Array Signal Processing. Prentice-Hall, Inc., Englewood Cliffs (1985)
Husebye, E., Dainty, A. (eds.): Monitoring a Comprehensive Test Ban Treaty. Kluwer Academic Publishers, Dortrecht (1995)
Ibragimov, I.A., Has’minskii, R.Z.: Statistical Estimation, Asymptotic Theory. Applications of Mathematics, vol. 16, Springer, New York (1981)
Kiselevitch, V.L., Nikolaev, A.V., Troitskiy, P.A., Shubik, B.M.: Emission tomography: main ideas, results, and prospects. In: Proceedings of the 61st annual international meeting, SEG, expanded abstracts, 1602 (1991)
Kushnir A.F., Rozhkov M.B., Tagizade, T.T.: Method for determination of microseismic source coordinates. Patent N 2451307 (2012a) (In Russian)
Kushnir A.F., Rozhkov, M.V., Varypaev A.V., Dricker I.G.: Evaluation of location capabilities of statistically optimal algorithms for microseismic monitoring. In: Proceedings of the 74th conference and exhibition, EAGE, extended abstracts, P025 (2012b)
Kushnir, A.F., Pinsky, V.I.: Asymptotically Efficient Estimation of Linear Systems Parameters. I. Local Asymptotic Normality. Mathematical methods in seismology and geodynamics (Computational seismology, 19), pp. 101–118. Nauka, Moscow (1986) (In Russian)
Kushnir, A.F.: Algorithms for linear systems identification in case of correlated noise at input and output. Probl. Inf. Trans. 23(2):61–74 (1987) (In Russian)
Kushnir A.F.: Statistical and Computational Methods of Seismic Monitoring. URSS, Moscow (2012) (In Russian)
Le Cam, L.M.: Locally asymptotically normal fami-lies of distributions. Univ. CA Publ. Statist. 3(2), 37–99 (1960)
Le Cam, L.M., Yang, G.L.: Asymptotics in Statistics. Some Basic Concepts. Springer Series in Statistics. Shpringer, New York (1990)
Lehmann, E.L., Casella, G.: Theory of Point Estimation, 2nd edn., Springer texts in statistics. Springer, Berlin (1998)
Marple, S.: Digital Spectral Analysis. Prentice-Hall, Inc., Englewood Cliffs (1985)
Maxwell, S.: Microseismic: growth born from success. Lead Edge 29(3), 338–343 (2010)
Nussbaum, M.: An asymptotic minimal risk for estimation of a linear functional relationship. J. Multivar. Anal. 4(3), 300–314 (1984)
Rozhkov, M., Kushnir, A., Rojkov, N., Dricker, I., Hellman, S.: Statistical analysis of microseismic noise during hydraulic fracturing. In: Proceedings of the 74th conference and exhibition, EAGE, extended abstracts, P092 (2012)
Shao, J.: Mathematical Statistics. Springer, New York (1998)
Van Trees, H.: Detection, Estimation and Modulation Theory, Part IV: Optimum Array Processing. Wiley, New York (2002)
Zhang, C., Florêncio, D.E., Ba Demba, E., Zhang, Z.: Maximum Likelihood Sound Source Localization and Beamforming for Directional Microphone Arrays in Distributed Meetings. IEEE Trans. Multim. 10(3):538–548 (2008)
Acknowledgments
The authors are grateful to the companies Synapse Science Center Inc. (Russia) and Earth Imaging Inc. (USA) for their support and assistance during the research and to the company ISTI Inc. (USA) for permission to use the records of surface seismic noise registered during hydraulic fracturing in the hydrocarbon field.
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Kushnir, A., Rozhkov, N. & Varypaev, A. Statistically-based approach for monitoring of micro-seismic events. Int J Geomath 4, 201–225 (2013). https://doi.org/10.1007/s13137-013-0049-6
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DOI: https://doi.org/10.1007/s13137-013-0049-6
Keywords
- Maximum-likelihood estimator
- Micro-seismic source
- Multichannel signal processing
- Hydraulic fracturing
- Asymptotically efficient estimator