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Earthquakes and Ground Motion

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Ground Motion Seismology

Part of the book series: Advances in Geological Science ((AGS))

Abstract

In Sect. 1.1, basic concepts and terms in “ground motion seismology” are defined. In Sect. 1.2, the fundamentals of “elastodynamics” are explained since “seismology” is built on a foundation of elastodynamics. The “equation of motion” and “wave equation” are then obtained for ground motion, and their nature as seismic waves with attenuation are discussed. In Sect. 1.3, important principles in ground motion seismology are explained. The final topic described in this chapter, the “representation theorem”, demonstrates that ground motion seismology consists in deriving the effect of the earthquake source and the effect of propagation separately from the observed ground motions, and in evaluating them quantitatively.

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Notes

  1. 1.

    “Subsurface structure” is also used, but its frequency of use seems to be lower than “velocity structure”. The term “underground structure” is used mostly for engineering structures in the ground.

  2. 2.

    In engineering, \(\displaystyle \gamma _{ij}=\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j.}\) is mostly used for shear strain rather than (1.5). Love [16] and Takeuchi and Saito [23] also used this definition.

  3. 3.

    See Chap. 12 of Fung [5].

  4. 4.

    Defined by Hudson [6]. Note that \(\rho \,\mathbf {f}\rightarrow \mathbf {f}\) in Aki and Richards [1], because their body force works on a unit volume.

  5. 5.

    Box 6.5 of Aki and Richards [1]. However, although the point is that (1.43) is equivalent to (1.42), they did not prove this equivalence. Kennett’s proof using \(u_V\) and \(u_H\) [7] can be considered to correspond to the point.

  6. 6.

    By Hudson [6]. He noted in the footnote on Page 194 of his book that L. Boltzmann made this formulation in 1876 for the first time.

  7. 7.

    Kreyszig [12] called it the damping constant, but Aki and Richards [1] and Utsu [26] used this term for a quantity related to \(\eta \) (Sect. 4.1).

  8. 8.

    In Hudson [6], the numerator of the equation corresponding to (1.65) has a minus sign. The reason for this is that the definition of the Fourier transform is different and \(\mathrm{i}\omega \) in the second equation of (1.56) was \(-\mathrm{i}\omega \) there.

  9. 9.

    This book deals mainly with isotropic media, and (1.30) may be used as the equation of motion. However, since the most general expression can be written formally in a shorter form, we use this expression here.

  10. 10.

    See also Sect. 4.2.2. In this book, the integral variable \(\tau \) is used consistently for the convolution in the time domain, and should not be confused with the stresses \(\tau _ {ij}\).

  11. 11.

    This is equal to (5.13) of Hudson [6]. The reciprocity theorem is known as Betti’s theorem, but Hudson noted that the equation in the form of a convolution as in (1.96) was published by D. Graffi in 1947.

  12. 12.

    From Dictionary of Mathematics, 2nd Edition [17]. The Physics Dictionary, Revised Edition [20] mentions only partial differential equations of elliptic type.

  13. 13.

    Equation (1.108) agrees with Equation (3.1) of Aki and Richards [1] in the case of no internal surface.

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Author information

Authors and Affiliations

  1. Earthquake Research Institute, The University of Tokyo, Bunkyo-ku, Tokyo, Japan

    Kazuki Koketsu

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  1. Kazuki Koketsu
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Correspondence to Kazuki Koketsu .

Problems

Problems

1.1

‘The medium is symmetric about the z-axis’ (Sect. 1.2.3) means that the elastic properties are invariant even if the Cartesian coordinate system is rotated about the z-axis [13]. In other words, the generalized elastic constants \(C_{i'j'k'l'}\) in the \(x'y'z'\) coordinate system, obtained by rotating the original xyz coordinate system around the z-axis by an arbitrary angle \(\theta \), are equal to \(C_{ijkl}\) in the xyz coordinate system. Show that Eqs. (1.18) of Love [16] are obtained from this equality.

1.2

In addition to the z-axis symmetry in Problem 1.1, obtain the equations from the x-axis symmetry. Show that these equations and Eqs. (1.18) result in Eqs. (1.19).

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Koketsu, K. (2021). Earthquakes and Ground Motion. In: Ground Motion Seismology. Advances in Geological Science. Springer, Singapore. https://doi.org/10.1007/978-981-15-8570-8_1

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