4-D Seismic Tomography for the Complex System of Strong Earthquakes: Formulation of a Problem

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1 4-D Seismic Tomography for the Complex System of Strong Earthquakes: Formulation of a Problem Tatyana A. Smaglichenko and Ingi Th. Bjarnason Abstract Geodynamic processes are acting in the Earth s interior and they cause earthquakes of various intensity. Earthquakes occur randomly and they are often in clusters. Sometimes it happens that before strong earthquakes there is a seismic quiescence that is characterized by the absence of significant seismic events. This may indicate that Earth s geological system prepares itself for a catastrophe. Complexity theory describes regularities of the behavior of dynamical systems before the occurrence of a disaster. The main part of this chapter is formulating a problem to investigate the behavior of a geophysical parameter, namely seismic velocity before the occurrence of the strong earthquake. Considering that velocity is a random variable, we apply the distribution function to estimate the dynamic state of the strong earthquakes complex system. Keywords Seismic tomography Velocity model Statistics Geodynamics 1 Introduction The Earth is a dissipative system of heterogeneous geological structures. The state of structures can be changed under the influence of various physical and chemical factors. This leads e.g. to the unpredictable occurrences of strong earthquakes. In his work Nicolis and Prigogine [8] examined various states of stability of dissipative systems. The states can be described via the Lyapunov function [6]. However this is not always possible. For example, it is difficult to describe the un-insulated T. A. Smaglichenko (B) Research Oil and Gas Institute of Russian Academy of Sciences, Moscow, Russia t.a.smaglichenko@gmail.com I. Th. Bjarnason Institute of Earth Sciences, University of Iceland, Reykjavík, Iceland ingib@hi.is A. Sanayei et al. (eds.), ISCS 2013: Interdisciplinary Symposium on Complex Systems, 379 Emergence, Complexity and Computation 8, DOI: / _37, Springer-Verlag Berlin Heidelberg 2014

2 380 T. A. Smaglichenko and I. Th. Bjarnason systems via an ideal function. According to Nicolis and Prigogine [8] there are several important control parameters. Critical values of these parameters are responsible for the instability of the system. Once parameters pass their critical values the system enhances fluctuations and the system can cause a catastrophic state. Nowadays various seismic parameters and measurements are used in order to investigate a behavior of the complex system of strong earthquakes with the aim to predict strong earthquakes. Forecast models are constructed using catalog information, which contain magnitudes of earthquakes prior and after the main shock. In this approach the time window, within which a forecast is made, is often selected by taking into account information on historical and pre-historical earthquakes that occurred in the region of study [7, 16]. Other factors to determine of the forecast time window are based on the seismic activity that is instrumentally recorded in a region. Then forecasts are given for 1 day up to couple of years depending on the detailed information in the seismic catalog data [4, 15]. The probability and magnitude of a catastrophic event can be estimated by studying complex earthquake networks and constructing the so called chain graphs [1]. The other line of approach in studying the earthquake forecast problem is based on determining the physical parameters and the wider geodynamical effects of the earthquake source. The response of an elastic medium to the seismic wave propagation is analyzed. One of the main observations for this analysis is the polarity of the first arriving P-wave as a function of azimuth. From the P-wave polarity distribution it is possible to estimate the earthquake mechanism. The stress/strain fields in the Earth s crust are estimated by inverting the earthquake mechanisms. For example it was shown that after the 2011 Tohoku earthquake (M = 9.1) in Japan, the main axis of the stress field was rotated with respect to Pacific plate boundary [3]. In order to map in detail the stress field, Rebetskii [9, 10] has developed the method of cataclastic analysis of discontinuous displacement. It has been shown that the fault rupture in the 2004 Sumatra Andaman earthquake (M = 9.1), the 2003 Tokachi Ochi earthquake (M = 8.3), the 2006 Simushir earthquake (M = 8.3), and the 2010 Chile Earthquake (M = 8.3) was started on the border of areas of high and low pressure levels, i.e. in areas with high pressure gradient [10]. These results have an important implication in understanding the likely geographical location of large earthquakes. The second main parameter to consider is a time. It is already established that accumulation of stresses in the crust is characterized by time cycles. Cycles can be estimated by measuring movements of the crust with GPS data (both the sudden movement in earthquakes and the gradual drift of the crust) and by account of historical seismicity and paleoseismical data that can be determined with methods of field-geology. In a region a normal cycle may e.g. be defined for earthquakes with magnitudes 7 and less and a mega cycle for earthquakes with magnitudes greater than 8. In other regions mega cycle may be defined to consist of earthquakes of magnitude 9. Careful evaluation of the geological field data suggests that the 2011 Tohoku earthquake may have been part of a 500-year-long cycle of mega earthquakes in this region of Japan. The previous earthquake in this 500-year cycle was the 1611 Keicho earthquake, which also caused a devastating tsunami in Northeast Japan [5].

3 4-D Seismic Tomography 381 We propose to bring in a new type of seismic observation to aid the forecasting of earthquakes, namely by using travel times of seismic waves, which were recorded by seismic stations from local earthquakes in areas of large earthquakes. From travel times the inherent seismic velocity and other elastic properties of the Earth between the earthquake source and the recording station can be determined. In a number of studies information on velocity is applied for monitoring of stress field in areas of strong earthquakes (see e.g [11]). However the time dependence of the velocity parameter behavior before disaster is not much explored. In this chapter we formulate the research problem to study the behavior of this parameter, which can be reconstructed by using the travel times of P-waves and by applying tomography methods. Thus additional observations can be involved to solve the earthquake forecasting problem. It is assumed that 3-D velocity images can be analyzed during a long time period before the appearance of large event. The time window can be selected on the base of particularities of seismic activity in a region. Such 4-D imaging will allow us study the complex system of strong earthquakes. 2 A Method to Construct the 3-D Velocity Model for Each Time Window The Differentiated Approach (DA) is a flexible recently developed method to resolve a complex velocity structure [14]. It is based on an algorithm called the Modification of Gaussian Elimination (MGE) which is a new numerical scheme [13]. The advantage of DA is its stability, in spite of data errors, that characterize all measurements (in the seismic case they include observation and modeling errors that lead to errors in hypocenter determination). Additional advantage is the ability of the MGE method to effectively solve large sparse linear systems with the possibility to illuminate structures in details almost down to their inherent physical measuring resolution, which is e.g. governed by the frequency range of the observed data (Fresnel zone radius). In other words, there is no numerical limitation in DA. With conventional methods the smaller the block-size of a modeled structure, the larger is a size of the initial system of equations, which is computationally intensive to solve. With the MGE any small size of model block (cell) can be chosen, because MGE divides the initial system into a set of subsystems and thus estimates the resolution of smaller subsystems. An estimation of the resolving power of the least squares method for a large initial set of equations is a highly challenging problem, even for the best of modern computers. In selecting a robust solution the DA method applies statistical evaluation of the solution. The method, however, requires relatively good coverage of the study area by seismic rays. The greater the number of seismic rays that intersect a target block from different directions, the higher the probability of a reliable estimate. Let us consider an example of the DA method. Figure 1 shows the P-wave velocity image of the Tjörnes Fracture Zone (TFZ) in Northeast Iceland that has been

4 382 T. A. Smaglichenko and I. Th. Bjarnason Fig. 1 P-wave velocity structure derived by DA for TFZ along the Grímsey lineament in Iceland. Low (high) velocity zones are colored in white (black). Hypocenters of local earthquakes are denoted by open circles. The black arrow shows neighboring low and high velocities that correspond to the greatest jump in the distribution function (see Sect. 3) constructed with travel times from local earthquakes recorded between 1986 and 1988 [12]. Thus the length of the time-window was 3 years. In Sect. 3 we will analyze the given velocity model in detail. 3 4-D Velocity Tomography via Distribution Function The construction of tomographic images in various regions of the world shows that 3-D velocity images of the same area are different for different time windows. It could be due to random character of seismic events. For a given time window, the 3-D image is determined by set of velocity values that were found in blocks of the geophysical model. These values are real numbers v 1, v 2,...,v k, which can be considered as the result of a random experiment. The vector v = (v 1, v 2,...,v k ) has a dimension of k, which is defined by the number of blocks of the medium and it is a random variable. The probability of distribution of the random variable is characterized by the distribution function F(x), which plays a fundamental role in mathematical statistics. According to the definition, F(x) = P(v x), 0 F(x) 1

5 4-D Seismic Tomography 383 Fig. 2 The horizontal axis is the velocity random value. The vertical axis is the distribution function value. The black arrow shows the greatest jump of the distribution function where P denotes a probability or the repetition frequency of an event {v x}. P = k1/k, where k1 is a number of the repetitions of an event in a given random experiment. We will apply this function in order to estimate the velocity field that determines a structure of the geophysical medium for a given time window. Jumps in the distribution function correspond to discontinuities in velocity values or contrasts in the velocity distribution. Figure 2 shows the distribution function that was constructed using velocity values, which correspond to the image in Fig. 1. The distribution function can be decomposed into a discontinuous (stepped) part and a continuous part that can be obtained by approximating discrete values via a curve. The greatest jump of the function is denoted by the arrow in Fig. 2. Location of the largest velocity jump is indicated in Fig. 1. This jump corresponds to a low velocity zone (colored in white) that is sandwiched between two high velocity anomalies (colored in black). In total, the distribution function has 5 discontinuity points: x = 6.4; 6.5; 6.7; 7.0; 7.4. The dotted line shows the position of each point. From the point of view of statistics the probability mass is concentrated at discontinuity points [2]. This means that the listed velocity values characterize P-wave velocity with a high probability for the given time window along Grímsey lineament in TFZ, Northeast Iceland. We have analyzed a behavior of the distribution function for a single sample, which corresponds to a fixed time window. The time behavior of this function can be determined, if it is inspected in multiple time windows, which together make up a longer period of time. The problem can be reduced to the construction of the sequence of distribution functions. We assume that a series of random experiments will allow us to get a stable result, which determines a behavior of the geophysical parameter (velocity of seismic waves) before the onset of strong earthquakes.

6 384 T. A. Smaglichenko and I. Th. Bjarnason 4 Conclusion The seismic wave velocity of the crust is directly related to the state of the stress field [11] and it is an important parameter for the complex system of strong earthquakes. We assume that a change in the stress field before the occurrence of large earthquakes will influence this parameter, and that it may be possible to determine its critical values. Hence it is necessary to establish its critical values. In this chapter we propose to analyze a seismic velocity as a random variable and to study it via a distribution function. We will formulate the forecasting of strong earthquakes by means of constructing of series of 3-D velocity structures and distribution functions respectively in order to understand complex dynamics of a geophysical medium before the earthquake. Statistical analysis of distribution functions will reveal the stable characteristics of tomographic models, which may include time dependent features. Acknowledgments We thank the DAAD foundation (Germany) for support, due of which tomography image that we used in this chapter has been constructed. Our thanks go to Prof. Wolfgang Jacoby (Mainz University, Germany) for fruitful discussion of tomography results. References 1. Abe, S., Suzuki, N.: Main shocks and evolution of complex earthquake networks. Braz. J. Phys. 39(2A), (2009) 2. Cramer, H.: Mathematical Methods of Statistics. University of Stockholm (1946) 3. Hasegawa, A., Yoshida, E., Okada, T.: Nearly complete stress drop in the 2011 Mw 9.0 off the Pacific Coast of Tohoku Earthquake. Earth Planet. Space. 63, (2011) 4. Hirata, N., Yokoi, S., Nanjo, K.Z., Tsuruoka H.: A forecast experiment of earthquake activity in Japan under collaboratory for the study of earthquake predictability (CSEP). In: Geophysical Research Abstracts of EGU General Assembly. 14. EGU EGU, Vienna, Austria (2012) 5. Koketsu, K., Yokota, Y., Kato, N., Kato, T.: Identification and simulation of seismic supercycles along the Japan trench including the 2011 Tohoku Earthquake. In: Geophysical Research Abstracts of EGU General Assembly. 14. EGU EGU, Vienna, Austria (2012) 6. Lyapunov, A.M.: Works in 3 Volumes (in Russian). Publishing House of the USSR Academy of Sciences, Moscow-Leningrad ( ) 7. Maccaferri, F., Rivalta, F., Passarelli, L., Jonsson, S.: The stress shadow induced by the Krafla Rifting Event. In: Geophysical Research Abstracts of EGU General Assembly. 14. EGU EGU, Vienna, Austria (2012) 8. Nikolis, G., Prigogin, I.: Exploring Complexity, an Introduction. W.H. Freedman and Co., New York (1989) 9. Rebetskii, Y. L.: Estimation of the stress field values in the method of cataclastic analysis of shear fractures. Dokl. Earth Sci. 428(7), 1202 (2009) 10. Rebetsky, YuL, Kuchai, O.A., Sycheva, N.A., Tatevossian, R.F.: Development of inversion methods on fault slip data stress state in orogenes of the Central Asia. Tectonophysics 581, (2012) 11. Slavina L.B., Myachkin V.V., Kuzmina T.A.: On the location and time of travel time precursors before large earthquakes. J. Earthq. Predict. Res. 2(4), (1993) 12. Smaglichenko, T., Jacoby, W., Fedorova, T., Wallner, H.: Stable estimate of velocity anomalies around Grimsey Lineament (Tjornes Fracture Zone, Iceland) with differentiated

7 4-D Seismic Tomography 385 tomography. In: Geophysical Research Abstracts. EGU General Assembly. 11, EGU EGU, Vienna, Austria (2009) 13. Smaglichenko, T. A.: Modification of Gaussian Elimination for the Complex System of Seismic Observations. Founded by Stephen Wolfram, vol. 20. Issue 3, pp Complex Systems Publications, Inc., USA (2012) 14. Smaglichenko, T. A., Shigeki, H., Kaori, T.: A differentiated approach to the seismic tomography problem: method, testing and application to the Western Nagano fault area (Japan). Int. J. Appl. Earth Obs. Geoinf. (Elsevier) 16, (2012) 15. Stefánsson R.: Advances in Earthquake Prediction: Seismic Research and Risk Mitigation, p Spinger-Verlag, Berlin (2011) 16. Zöller, G., Hainzl, S., Holschneider, M.: Recurrence of large earthquakes: Bayesian inference from catalogs in the presence of magnitude uncertainties. Pure Appl. Geophys. 167(6 7), (2012)