2.4 Attenuation and dispersion of seismic waves

Seismic waves don't just travel through the Earth unchanged. They lose energy and change shape as they go. This happens because of things like friction, scattering, and how waves spread out.

Different parts of seismic waves can travel at different speeds too. This causes waves to stretch out or bunch up as they move. Understanding these effects helps scientists figure out what's inside the Earth.

Attenuation Mechanisms

Intrinsic and Scattering Attenuation

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• Intrinsic attenuation occurs due to energy conversion from seismic waves into heat within the Earth's materials
• Involves friction between particles and viscous damping in fluids within rock pores
• Results in a decrease of wave amplitude as it propagates through the medium
• Scattering attenuation happens when seismic waves encounter heterogeneities in the Earth's structure
• Causes energy to be redirected in multiple directions, reducing the amplitude of the primary wave
• Depends on the size and distribution of heterogeneities relative to the wavelength
• Both intrinsic and scattering attenuation contribute to the overall decay of seismic wave energy
• Can be quantified using the seismic quality factor Q, which measures the energy loss per cycle

Geometrical Spreading and Q Factor

• Geometrical spreading refers to the decrease in wave amplitude as it propagates outward from the source
• Follows an inverse square law in three-dimensional space, where amplitude decreases proportionally to 1/r (r = distance from source)
• Results in a natural decay of wave energy even in the absence of other attenuation mechanisms
• Q factor, also known as the quality factor, quantifies the energy loss in a medium
• Defined as the ratio of stored energy to dissipated energy per cycle of oscillation
• Higher Q values indicate lower attenuation and more efficient wave propagation
• Q factor varies with rock type, temperature, and pressure conditions in the Earth
• Typical Q values range from 20-200 for near-surface sediments to 1000+ for the lower mantle
• Relates to the damping ratio ζ through the equation $Q = \frac{1}{2ζ}$

Frequency-Dependent Effects

Frequency-Dependent Attenuation and Dispersion

• Frequency-dependent attenuation describes how different frequencies of seismic waves attenuate at different rates
• Higher frequency waves generally attenuate more rapidly than lower frequency waves
• Results in a change of waveform shape as it propagates through the Earth
• Leads to the phenomenon of pulse broadening, where sharp pulses become more spread out over time
• Dispersion occurs when different frequency components of a wave travel at different velocities
• Causes wave packets to spread out in time as they propagate
• Normal dispersion involves higher frequencies traveling faster than lower frequencies
• Anomalous dispersion occurs when lower frequencies travel faster than higher frequencies
• Dispersion curves graphically represent the relationship between frequency and wave velocity
• Used to analyze and interpret seismic data for subsurface characterization

Phase and Group Velocities

• Phase velocity represents the speed at which a specific phase of a wave (crest or trough) propagates
• Calculated using the equation $c = \frac{\omega}{k}$, where ω is angular frequency and k is wavenumber
• Varies with frequency in dispersive media, leading to wave shape changes during propagation
• Group velocity describes the speed at which the overall envelope of a wave packet travels
• Defined mathematically as $v_g = \frac{d\omega}{dk}$, the derivative of angular frequency with respect to wavenumber
• Represents the velocity of energy transport in the wave
• In non-dispersive media, phase and group velocities are equal
• In dispersive media, phase and group velocities differ, causing wave packet distortion
• Surface waves (Rayleigh and Love waves) exhibit strong dispersion effects in layered media
• Analysis of phase and group velocities helps in determining subsurface structure and properties