Seismic coda due to non-linear elasticity

Abstract

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...

Three‐dimensional modeling of near‐fault ground motions with nonplanar rupture models and topography: Case of the 2004 Parkfield earthquake

“…The existence of a fault zone structure is one of the most important effects to consider in further investigations. Furthermore, directional site effects [ Bonamassa and Vidale , 1991], medium anisotropy [ Cochran et al , 2006], or nonlinear behavior [ Bataille and Calisto , 2008; Karabulut and Bouchon , 2007; Wu and Chen , 2009] might also contribute to better explain the observed data.…”

Localized stationary seismic waves predicted using a nonlinear gradient elasticity model

“…This is a nonlinear second-order ordinary differential equation, which can be solved for stationary wave solutions of Eq. (10). Specific stationary solutions are determined in Sect.…”

“…Employing the commonly used hyperbolic constitutive model [4,5], softening behavior and super-harmonic resonances were recently demonstrated for a superficial soil layer under uniform harmonic excitation at the lower boundary [6]. However, the possibility for localized stationary waves such as solitons to propagate through the soil column and reach the surface has not been widely recognized in the seismological literature and neither in the geo-technical literature, although a number of publications hint at the possibility of Lovetype surface solitary waves, solitary waves along faults, and the importance of nonlinearity in general [7][8][9][10][11][12][13][14].…”

Evidence that non-linear elasticity contributes to the seismic coda

“…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).…”

“…Helbig (1998) considered an anisotropic and non-linear medium when expanding the stiffness tensor in a Taylor series. Recently, Bataille & Calisto (2008) used the second and fourth order of the strain tensor. In the latter work the theoretical strength of the scattered field for this special form of strain energy is shown to be proportional to the amplitude of the incoming wave up to the third power, while for a heterogeneous medium it is proportional to the first power.…”