S U M M A R YNon-linear elastic response of rocks has been widely observed in laboratory, but very few seismic studies are reported in the literature, even though it is the most natural environment where this feature could be observed. Analytic solutions to the non-linear wave propagation phenomena are not readily available, and there is a need to use approximated techniques. It is clear that when a seismic wave propagates through a homogeneous non-linear elastic media, it will be perturbed by the non-linearity. This perturbation can be treated as a source of scattering, spreading the energy of the primary wave in space and time, contributing to the seismic coda. This is in some sense similar to the effect of heterogeneities. The properties of the coda due to the non-linearity depend on the amount of non-linearity and the seismic moment. Using a perturbation approach we calculate the amplitude of the scattered waves, and show that it can describe reasonably well the main features of real seismic codas.Seismology has shown that a linearly elastic response of rock provides a good description of a variety of phenomena related to earthquakes. Most observations related to large earthquakes can indeed be explained with the linear elastic theory, but it would be naive to believe that it provides a complete description, knowing that laboratory observations show that most rocks do behave non-linearly close to the rupture condition (Scholz 1990).The classical approach to relate ground deformation to source parameters is through the representation theorem, and the principle of superposition. These are only valid for a linear constitutive law. Thus, if non-linear elasticity is important, one can expect that this classical approach is not appropriate.Since non-linear elastic behaviour of most materials has been recognized as a significant factor for several centuries by physicists and material scientists (Bell 1984), it is important to investigate the effect of non-linear terms in the wave propagation phenomena, especially near fault zones. Among the studied features related to non-linear effects of rocks are the non-linear dependence of seismic velocities on stress (Birch 1961;Johnson & McCall 1994;McCall 1994), resonant peak shift (Winkler et al. 1979), effects of anisotropy and non-linearity (Zheng et al. 2006).The acquisition of high precision displacements near sources has improved the understanding of sources of deformation in a wide range of scales of space and time, especially with respect to earthquake and volcanic processes. However, since the quantity and quality of observations are increasing rapidly, it provides the opportunity to increase the understanding of the source process, structure and rheology of the media, including second order effects, such as the non-linearity of the media.In this paper, we consider a general non-linear constitutive relationship on the dynamic deformation near fault zones, showing that the effect of this non-linear term is to spread the seismic energy in space and time, even when the...
S U M M A R YDifferent factors might affect the propagation of seismic waves producing scattering, including heterogeneities and non-linear elasticity. A key difference between these two factors is the dependence of the strength of the scattered waves on the strength of the incident wave, being linear for the former and non-linear for the latter. A detailed study of the TIPTEQ data, where about a hundred explosions were recorded on 180 three-component stations in the distance range of approximately 0-100 km, shows that this dependence is non-linear. Data were analysed in the following way: (i) the envelope of bandpass filtered data between 10 and 40 Hz was obtained for a large number of stations from different distance ranges and charge sizes of shots, (ii) for these distances we modelled the envelope considering the non-linear elasticity. The shapes of the theoretical and observed envelopes were in general very similar. A scale factor for each case was obtained considering the best fit of its complete envelope and (iii) since this scale factor depends mainly on the size of the explosion, we computed the ratio (R) of the scale factor (sf ) for different sizes of explosions at the same distance. Finally, varying the distance between 0 and 50 km and (iv) we computed the power (p) of the dependence of the ratio (R) on the ratio of the charge sizesFor the complete data set we obtain a value of p = 2.5 ± 0.9, which is clearly greater than 1. This shows that non-linear elasticity is an important factor in the contribution to seismic wave scattering in the frequency range of 10-40 Hz.The principal contribution of scattering in coda waves was originally thought to be heterogeneities, which are present in the crust. The coda wave is the most worthwhile to study when considering scattering as it has the highest sensitivity to the changes in the media (Nikolaev 1988). However, since laboratory experiments show that a non-linear behaviour of rocks exists close to the rupture condition (Scholz 1990) then the presence of heterogeneities would not be the only explanation for the scattering in the envelope of seismic waves, and the effect of weak non-linearity of the elastic medium produces comparable observations even when the medium is homogeneous (Bataille & Calisto 2008).The interpretation of seismic results based on Hooke's model fails to account for certain observed effects (Lyakhovsky & Myasnikov 1988) such as, for example, a special experimental investigation which allowed the propagation of periodic seismic signals using seismic vibrators. During this investigation strong non-linear effects that are caused by physical non-linearity of the medium were observed (Nikolaev 1988).One can point out that, in general, how the amplitude of the scattered waves scales with the seismic moment depends directly on the constitutive law of the media. For the non-linear process, there is still no agreement upon the most appropriate form for the strain energy from the point of view of physical principles and direct observations.The equat...
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