Natural Hazards and Earth System Science www.nat-hazards-earth-syst-sci.net 1561-8633 1684-9981 8 2 2008 10.5194/nhess-8-369-2008 http://www.nat-hazards-earth-syst-sci.net/8/369/2008/ http://www.nat-hazards-earth-syst-sci.net/8/369/2008/nhess-8-369-2008.html http://www.nat-hazards-earth-syst-sci.net/8/369/2008/nhess-8-369-2008.pdf 369 376 2008-04-24 Automatic estimation of optimal autoregressive filters for the analysis of volcanic seismic activity P. Lesage lesage@univ-savoie.fr Laboratoire de Géophysique Interne et Tectonophysique, CNRS, IRD: R157, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France Instituto de Geofísica, Universidad Nacional Autónoma de México, Mexico, D.F., Mexico Long-period (LP) events observed on volcanoes provide important information for volcano monitoring and for studying the physical processes in magmatic and hydrothermal systems. Of all the methods used to analyse this kind of seismicity, autoregressive (AR) modelling is particularly valuable, as it produces precise estimations of the frequencies and quality factors of the spectral peaks that are generated by resonance effects at seismic sources and, via deconvolution of the observed record, it allows the excitation function of the resonator to be determined. However, with AR modelling methods it is difficult to determine the order of the AR filter that will yield the best model of the signal. This note presents an algorithm to overcome this problem, together with some examples of applications. The approach described uses the kurtosis (fourth order cumulant) of the deconvolved signal to provide an objective criterion for selecting the filter order. This approach allows the partial automation of the AR analysis and thus provides interesting possibilities for improving volcano monitoring methods. Adnet, C.: Unification des méthodes d'analyse spectrale (Fourier et haute résolution) en vue de la réalisation d'un système expert d'aide à l'analyse. Ph.D. Thesis, INPG, Grenoble, France, 1990. Brenguier, F., Shapiro, N. M., Campillo, M. et al.: Towards forecasting volcanic eruptions using seismic noise, Nature Geoscience, 1, 126–130, doi:10.1038/ngeo104, 2008. Carniel, R., Di Cecca, M. and Rouland, D.: Ambrym, Vanuatu (July–August 2000): spectral and dynamical transitions on the hours-to-days timescale. J. Volcanol, Geotherm. Res., 128, 1–13, doi:10.1016/S0377-0273(03)00243-9, 2003. Chouet, B.: Dynamics of a fluid-driven crack in three dimensions by the finite difference method, J. Geophys. Res., 91(B14), 13 967–13 992, 1986. Chouet, B.: A seismic model for the source of long-period events and harmonic tremor, in: Volcanic seismology, edited by: P. Gasparini, R. Scarpa, and K. Aki, Berlin, 133–156, 1992. Chouet, B.: Long-period volcano seismicity: its source and use in eruption forecasting, Nature, 380, 309–316, 1996a. Chouet, B.: New methods and future trends in seismological volcano monitoring, In: Monitoring and mitigation of volcano hazards, edited by: R. Scarpa and R. I. Tilling, 23–97, 1996b. Chouet, B., Sacorrotti, G., Dawson, P., Martini, M., Scarpa, R., De Luca, G., Milana, G. and Cattaneo, M.: Broadband measurements of sources of explosions at Stromboli volcano, Italy. Geophys. Res. Lett., 26(13), 1937–1940, 1999. Donoho, D.: On minimum entropy deconvolution, Applied time-series analysis II, Academic Press, 565–609, 1981. Endo, E.T . and Murray, T.: Real-time seismic amplitude measurement (RSAM): a volcano monitoring and prediction tool, Bull. Volcanol., 53, 533–545, 1991. Fukao, Y. and Suda, N.: Core modes of the Earth's free oscillations and structure of the inner core, Geophys. Res. Lett., 16, 401–404, 1989. Hori, S., Fukao, Y., Kumazawa, M., Furumoto, M. and Yamamoto, A.: A new method of spectral analysis and its application to the Earth's free oscillations: The "Sompi" method, J. Geophys. Res., 94(B6), 7535–7553, 1989. Jaquet, O. and Carniel, R.: Stochastic modelling at Stromboli: a volcano with remarkable memory. J. Volcanol. Geotherm. Res., 105, 249–262, 2001. Jousset, P., Neuberg, J., and Sturton, S.: Modelling the time-frequency content of low-frequency volcanic earthquakes, J. Volcanol. Geotherm. Res., 128, 201–223, 2003. Kay, S. M.: Modern spectral estimation, theory and application, Academic Press, 1981. Kumagai, H. and Chouet, B. A.: Acoustic properties of a crack containing magmatic or hydrothermal fluids, J. Geophys; Res., 105(B11), 25 493–25 512, 2000. Kumagai, H., Chouet, B. A., and Nakano, M.: Temporal evolution of a hydrothermal system in Kusatsu-Shirane Volcano, Japan, inferred from complex frequencies of long-period events, J. Geophys. Res., 107(B10), 2236, doi:10.1029/2001JB000653, 2002. Kumazawa, M., Imanishi, Y., Fukao, Y., Furumoto, M., and Yamamoto, A. A.: Theory of spectral analysis based on the characteristic property of a linear system, Geophys. J. Int., 101, 613–630, 1990. Lacoume, J.-L., Amblard, P.-O., and Comon, P.: Statistiques d'ordre supérieur pour le traitement du signal, Masson, Paris, 1997. Lesage, P. and Surono: Seismic precursors of the 10~February~1990 eruption of Kelut volcano, Java. J. Volcanol. Geotherm. Res., 65, 135–146, 1995. Lesage, P., Glangeaud, F. and Mars, J.: Applications of autoregressive and time-frequency analysis to the study of volcanic tremor and LP events, J. Volcanol. Geotherm. Res., 114, 391–417, 2002. Lesage, P., Mora, M., Alvarado, G., Pacheco, J. F., and Métaxian, J.-P.: Complex behavior and source model of the volcanic tremor at Arenal volcano, Costa Rica, J. Volcanol. Geotherm. Res., 157, 49–59, 2006. Marple, S. L.: Digital spectral analysis with applications, Prentice Hall, Englewood Cliffs, 1987. Mars, J., Lacoume, J.-L., Mari, J.-L., and Glangeaud, F.: Traitement du signal pour géologues et géophysiciens – Vol 3, Techniques avancées, Technip, 2004. McNutt, S.: Observations and analysis of B-type earthquakes, explosions, and volcanic tremor at Pavlof volcano, Alaska. Bull. Seis. Soc. Am., 76, 153–175, 1986. McNutt, S. R.: Seismic monitoring and eruption forecasting of volcanoes: A review of the State-of-the-Art and case histories, in: Monitoring and mitigation of volcano hazards, edited by: R. Scarpa and R.I. Tilling, Springer-Verlag, Berlin, 99–146, 1996. Nakano, M., Kumagai, H., Kumazawa, M., Yamaoka, K., and Chouet, B.: The excitation and characteristic frequency of the long-period volcanic event: an approach based on an inhomogeneous autoregressive model of a linear dynamic system, J. Geophys. Res., 103, 10 031–10 046, 1998. Nakano, M., Kumagai, H., and Chouet, B.: Source mechanism of long-period events at Kusatsu-Shirane volcano, Japan, inferred from waveform inversion of the effective excitation functions, J. Volcanol. Geotherm. Res., 122, 139–164, 2003. Ohminato, T., Chouet, B. A., Dawson, P., and Kedar, S.: Waveform inversion of very long period impulsive signals associated with magmatic injection beneath Kilauea volcano, Hawaii, J. Geophys. Res., 103(B10), 23 839–23 862, 1998. Richard, G. and Lesage, P.: Autoregressive models of resonators for volcanic tremors using Kurtosis, Phys. Signal Image Processing, Marseille, France, 2001. Schlindwein, V., Wassermann, J., and Scherbaum, F.: Spectral analysis of harmonic tremor signals at Mt Semeru volcano, Indonesia, Geophys. Res. Lett., 22, 1685–1688, 1995. Seidl, D., Kirbani, S. B., and Brüstle, W.: Maximum entropy spectral analysis of volcanic tremor using data from Etna (Sicily) and Merapi (central Java), Bull. Volcanol., 52, 460–474, 1990. Sturton, S. and Neuberg, J.: The effects of conduit length and acoustic velocity on conduit resonance: implications for low-frequency events, J. Volcanol. Geotherm. Res., 128, 187–199, 2006. Vandemeulebrouck, J., Sabroux, J.-C., Halbwachs, M., Surono, Poussielgue, N., Grangeon, J., and Tabbagh, J.: Hydroacoustic noise precursors of the 1990 eruption of Kelut volcano, Indonesia, J. Volcanol. Geotherm. Res., 97, 443–456, 2000. Vila, J., Macia, R., Kumar, D., Ortiz, R., Moreno, H., and Correig, A. M.: Analysis of the unrest of active volcanoes using variations of the base level noise seismic spectrum, J. Volcanol. Geotherm. Res., 153(1–2), 11–20, 2006.