🌋Seismology Unit 4 – Surface Waves – Rayleigh and Love Waves

Key Concepts and Definitions

• Surface waves propagate along the Earth's surface and are distinct from body waves (P-waves and S-waves) that travel through the interior
• Rayleigh waves exhibit elliptical particle motion in the vertical plane, causing the ground to move in a rolling motion perpendicular to the direction of wave propagation
• Love waves exhibit transverse particle motion, causing the ground to move side-to-side horizontally, perpendicular to the direction of wave propagation
• Dispersion is the phenomenon where different frequencies of surface waves travel at different velocities, resulting in the separation of the wave packet over time and distance
• Group velocity refers to the velocity at which the energy of the wave packet propagates, while phase velocity refers to the velocity of individual wave components
• Attenuation is the decrease in amplitude and energy of surface waves as they propagate due to factors such as geometric spreading, intrinsic absorption, and scattering
• Seismic anisotropy refers to the directional dependence of seismic wave velocity in materials, which can affect the propagation and behavior of surface waves

Types of Surface Waves

• Rayleigh waves, named after Lord Rayleigh, are the most common type of surface wave and are often the most destructive in earthquakes
  • Rayleigh waves cause the ground to move in a rolling motion, with particles following an elliptical path in the vertical plane
  • They have both vertical and horizontal components of motion, but the vertical component is typically larger
• Love waves, named after A.E.H. Love, are the second type of surface wave and are observed in horizontally layered media
  • Love waves cause the ground to move side-to-side horizontally, perpendicular to the direction of wave propagation
  • They are purely transverse waves and do not have a vertical component of motion
• Stoneley waves are a type of interface wave that propagate along the boundary between two solid media or between a solid and a fluid
• Scholte waves are similar to Stoneley waves but propagate along the interface between a fluid and a solid, such as at the ocean bottom
• Pseudo-Rayleigh waves and pseudo-Love waves are surface waves that exist in layered media with a strong velocity contrast and exhibit dispersive behavior

Wave Characteristics and Behavior

• Surface waves have lower frequencies and longer wavelengths compared to body waves, typically ranging from a few seconds to several minutes
• Rayleigh waves have a retrograde elliptical particle motion at the surface, which transitions to prograde motion at depth
  • The amplitude of Rayleigh waves decreases exponentially with depth, with most of the energy confined within one wavelength of the surface
• Love waves have a particle motion that is transverse to the direction of propagation and is confined to the horizontal plane
  • Love waves are dispersive, meaning that their velocity depends on the frequency and the properties of the layered medium
• Surface waves exhibit dispersion, where different frequencies travel at different velocities, leading to the separation of the wave packet over time and distance
  • Dispersion is caused by the velocity variation with depth in the Earth's layered structure
• Surface waves can be influenced by seismic anisotropy, which refers to the directional dependence of seismic wave velocity in materials
  • Anisotropy can cause surface waves to propagate with different velocities in different directions, leading to azimuthal variations in their arrival times and amplitudes
• Attenuation of surface waves occurs due to geometric spreading, intrinsic absorption, and scattering, resulting in a decrease in amplitude and energy as they propagate

Generation and Propagation

• Surface waves are generated by shallow earthquakes, explosions, or other seismic sources that create disturbances at or near the Earth's surface
  • The source mechanism, depth, and size of the seismic event influence the characteristics and amplitudes of the generated surface waves
• Rayleigh waves are generated by the interaction of P-waves and SV-waves (vertically polarized S-waves) at the free surface
  • The constructive interference of these waves results in the elliptical particle motion characteristic of Rayleigh waves
• Love waves are generated by the interaction of SH-waves (horizontally polarized S-waves) with the Earth's surface and shallow velocity discontinuities
  • Love waves require the presence of a low-velocity layer over a high-velocity half-space to exist and propagate
• Surface waves propagate along the Earth's surface, following the curvature of the planet
  • The velocity of surface waves depends on the elastic properties and density of the medium they travel through
• As surface waves propagate, they are influenced by the Earth's lateral heterogeneities, such as variations in crustal thickness, sedimentary basins, and mountain ranges
  • These heterogeneities can cause scattering, focusing, or defocusing of surface wave energy, affecting their amplitudes and travel times
• Surface waves can undergo mode conversion, where energy is transferred between different types of waves (e.g., Rayleigh to Love waves) due to lateral variations in the Earth's structure

Mathematical Models and Equations

• The mathematical description of surface waves involves solving the elastodynamic wave equation with appropriate boundary conditions
  • The wave equation is derived from the conservation of momentum and the stress-strain relationship in elastic media
• For a homogeneous, isotropic, and elastic half-space, the Rayleigh wave velocity ($c_R$) is given by the solution to the equation: $\frac{c_R^6}{c_S^6} - 8\frac{c_R^4}{c_S^4} + (24 - 16\frac{c_S^2}{c_P^2})\frac{c_R^2}{c_S^2} + 16(1 - \frac{c_S^2}{c_P^2}) = 0$
  • where $c_S$ is the S-wave velocity and $c_P$ is the P-wave velocity
• The Love wave velocity ($c_L$) in a single layer over a half-space is given by the dispersion relation: $\tan(kH\sqrt{1 - \frac{c_L^2}{c_{S1}^2}}) = \frac{\mu_2\sqrt{\frac{c_L^2}{c_{S2}^2} - 1}}{\mu_1\sqrt{1 - \frac{c_L^2}{c_{S1}^2}}}$
  • where $k$ is the wavenumber, $H$ is the layer thickness, $c_{S1}$ and $c_{S2}$ are the S-wave velocities in the layer and half-space, and $\mu_1$ and $\mu_2$ are the shear moduli of the layer and half-space
• Surface wave dispersion can be modeled using normal mode theory, which describes the wave propagation in terms of discrete modes with specific frequencies and velocities
  • Each mode represents a different way the medium can vibrate and is characterized by its own dispersion curve
• The Green's function for surface waves in a layered medium can be computed using the reflectivity method or the propagator matrix method
  • These methods allow the calculation of synthetic seismograms and the modeling of surface wave propagation in complex Earth structures

Detection and Measurement Techniques

• Surface waves are detected and recorded by seismometers, which are instruments that measure ground motion in three orthogonal components (vertical, north-south, and east-west)
  • Broadband seismometers are particularly useful for studying surface waves due to their ability to record a wide range of frequencies
• Array techniques, such as seismic arrays or gradiometers, can be used to enhance the signal-to-noise ratio and to estimate the direction and velocity of incoming surface waves
  • Seismic arrays consist of multiple seismometers arranged in a specific geometry, allowing for the application of beamforming and stacking techniques
• Surface wave tomography is a technique used to image the Earth's interior structure by measuring the velocity variations of surface waves at different periods
  • By analyzing the dispersion characteristics of surface waves, seismologists can construct 3D models of the Earth's crust and upper mantle
• Polarization analysis is used to determine the type and orientation of surface waves by examining the particle motion recorded by three-component seismometers
  • Rayleigh waves exhibit elliptical particle motion in the vertical plane, while Love waves have transverse horizontal motion
• Frequency-time analysis (FTAN) is a technique used to measure the group velocity dispersion of surface waves by applying a series of narrow-band filters to the seismic record
  • FTAN allows the identification of different surface wave modes and the construction of dispersion curves
• Amplitude measurements of surface waves can provide information about the attenuation properties of the Earth's interior and the source characteristics of the seismic event
  • Surface wave magnitude scales, such as Ms and Msz, are based on the amplitude of Rayleigh waves at specific periods

Impacts on Seismic Activity

• Surface waves are responsible for most of the shaking and damage caused by earthquakes, particularly in areas far from the epicenter
  • The slow velocity and low attenuation of surface waves allow them to propagate over long distances and cause prolonged shaking
• The amplification of surface waves in sedimentary basins and valleys can lead to increased ground motion and damage during earthquakes
  • This is due to the trapping and focusing of seismic energy within the low-velocity sediments
• Surface waves can trigger secondary hazards, such as landslides, liquefaction, and tsunamis, depending on the local geological conditions and the characteristics of the seismic event
  • Landslides can be triggered by the strong shaking and destabilization of slopes caused by surface waves
  • Liquefaction occurs when water-saturated sediments lose their strength and behave like a liquid due to the cyclic loading induced by surface waves
• The interaction of surface waves with man-made structures, such as buildings and bridges, can lead to resonance and increased damage risk
  • The natural frequencies of structures may coincide with the dominant frequencies of surface waves, leading to amplified oscillations and potential structural failure
• Surface waves can also have long-term effects on the Earth's surface, such as the generation of seismic microzonation maps and the identification of areas prone to site amplification
  • These maps are used for seismic hazard assessment and risk mitigation purposes, guiding land-use planning and building design

Applications in Geophysics and Engineering

• Surface wave dispersion analysis is widely used in geophysics to study the Earth's interior structure, particularly the crust and upper mantle
  • By measuring the velocity of surface waves at different frequencies, seismologists can construct 1D and 3D velocity models of the subsurface
• Surface wave tomography is a powerful tool for imaging lateral variations in the Earth's structure, such as tectonic boundaries, cratonic roots, and mantle plumes
  • Global and regional surface wave tomography models provide valuable insights into the dynamics and evolution of the Earth's interior
• Site characterization for engineering projects, such as buildings, bridges, and critical infrastructure, often involves the use of surface wave methods
  • Techniques like the Multichannel Analysis of Surface Waves (MASW) and the Spectral Analysis of Surface Waves (SASW) are used to determine the shear wave velocity profile and the dynamic properties of the soil
• Seismic microzonation studies rely on surface wave measurements to assess the local site effects and to identify areas prone to ground motion amplification
  • This information is crucial for seismic hazard assessment and the development of building codes and design guidelines
• Surface waves can be used for non-destructive testing and monitoring of engineered structures, such as buildings, dams, and pipelines
  • By analyzing the propagation and attenuation of surface waves, engineers can detect and locate damage, cracks, or other structural defects
• In exploration geophysics, surface waves are sometimes considered as noise that masks the desired body wave reflections
  • Advanced processing techniques, such as surface wave attenuation and separation, are used to remove the surface wave energy and enhance the quality of seismic reflection data
• The study of surface wave propagation and attenuation is also relevant for understanding the seismic noise environment and for designing seismic isolation systems for sensitive equipment and facilities