As we discussed in class, seismic waves can lose energy through reflection, geometrical spreading and intrinsic attenuation, also referred as anelasticity.

Geometrical spreading depends on the distance r the wave has propagated from the source. In a uniform material, seismic waves propagate away from their source as spherical wave front of increasing area. Because of the conservation of energy, the energy per unit area of wave front decreases as the distance from the source increases. For surface waves, in the case of a homogeneous flat earth, the energy per unit area of wave front decreases as 1/r and hence the amplitude, which is proportional to the root square of the energy, decreases as 1/√r. For body waves, the energy per unit area of wave front decreases as 1/r² and hence the amplitude decreases as 1/r.

On the other hand, anelasticity reduces seismic wave amplitudes by converting part of their kinetic energy to frictional heat by permanent deformation of the medium. Anelasticity is characterized by the frequency-dependent quality factor Q, which is a measure of the the energy lost per oscillation of the seismic wave : Q = 2πE/∆E . So the smaller Q, the larger the energy loss.The loss of energy will lead to exponential decay of the seismic wave amplitude : A(t) = A₀*e^-πft/Q . The smaller Q and the larger the frequency (i.e. more oscillations per second), the larger the attenuation and the seismic wave amplitude decay.

Two other processes can also reduce seismic wave amplitudes: multipathing and scattering. Multipathing and scattering can be thought of as elastic processes. They conserve energy and decrease or increase the amplitude of an incoming wave by shifting its energy to an earlier or later arrival.

Seismic wave multipathing is caused by velocity variations within the medium of propagation. According to Fermat’s principle, seismic waves follow the least-time path of propagation between two points in a medium. Lateral velocity variations in the medium will then cause seismic waves to focus in high velocity regions and defocus in low velocity regions.The spacing between seismic rays in a region represents the energy density in this region. The further apart the rays are, the lower the amplitudes of the recorded wave. By contrast, the closer the rays are, the larger the wave amplitudes. So the seismic waves arriving at a station have usually followed different ray paths in addition to the ideal, direct path and the region of the earth they sampled forms a volume called Fresnel zone. Multipathing can be a significant attenuation effect because most seismic activity occur at plate boundaries and velocity heterogeneities are important in these regions.

Likewise, heterogeneities within the propagation medium cause a propagating wave field to be scattered. These heterogeneities can be velocity anomalies but also material heterogeneities such as mineral boundaries, pore edges, cracks… Scattering will cause part of the energy released by an earthquake to arrive later at a receiver (i.e. after the initial pulse) as a coda (i.e. tail of incoherent energy that decays over a few seconds to a few minutes). Whether a seismic wave will be scattered or not when encountering a heterogeneity depends on the ratio of the heterogeneity size to the wavelength and the propagation distance in the heterogeneous medium. If the heterogeneity is large compared to the wavelength, the seismic energy will follow a different ray path (i.e. multipathing effect). However if the heterogeneity and the wavelength have the same order of magnitude, the seismic energy will be scattered. Heterogeneities much smaller than the wavelength will just change the medium’s “bulk” properties. Scattering can be significant in the continental crust because of the presence of many small-scale geologic structures that can significantly affect short wavelength waves (i.e. tens of kilometers or smaller).

Reference : Stein, S., & Wysession, M. (2009). An introduction to seismology, earthquakes, and earth structure. John Wiley & Sons.