Seismic inversion involves the set of methods which seismologists use to infer properties through physical measurements. Surface-wave inversion is the method by which elastic properties, density, and thickness of layers in the subsurface are obtained through analysis of surface-wave dispersion. The entire inversion process requires the gathering of seismic data, the creation of dispersion curves, and finally the inference of subsurface properties.
Surface waves Surface waves are seismic waves that travel at the surface of the earth, along the air/earth boundary. Surface waves are slower than P-waves(compressional waves) and S-waves(transverse waves). Surface waves are classified into two basic types, Rayleigh waves and Love waves. Rayleigh waves travel in a longitudinal manner (the wave motion is parallel to the direction of wave propagation) with particle motion in a retrograde elliptical motion (Figure 1). The Rayleigh waves result from the interaction between P-waves and vertically polarized S-waves. Conversely, Love waves travel in a traverse manner (Figure 1) (the wave motion is perpendicular to the direction of wave propagation), consisting of horizontally polarized S-waves. In seismology, surface waves are collected along with other seismic data, but are traditionally considered noise and an impedance in interpreting deeper reflection and refraction information. Seismologists usually modify seismic equipment and experimental procedures to remove surface wave information from the data. Earthquake seismologists however require the information seismic surface waves provide and thus design their equipment to amplify and gather as much information on these waves as possible. The work by early earthquake seismologists to extract substantial information from surface wave data was the basis for surface wave inversion theory.
Dispersion The usefulness of surface waves in determining subsurface elastic properties arises from the way in which they disperse. Dispersion (geology) is the way in which surface waves spread out as they travel across the surface of the earth. Basically, if ten waves travel along the surface of the earth at the same speed, there is no dispersion. If several of the waves start to travel faster than the others, dispersion is occurring. Surface waves of varying wavelengths penetrate to different depths (Figure 2) and travel at the velocity of the mediums they are travelling through. Figure 2 was generated by plotting the amplitude of surface waves against depth. This was done for two different wavelengths. Both waves have the same total energy, but the longer wavelength has its energy spread out over a larger interval. If earth materials’ elastic parameters yield higher velocities with depth, longer wavelength surface waves will travel faster than those with shorter wavelengths. The variation of velocities with wavelength makes it possible to infer critical information about the subsurface. Dobrin (1951) uses a water disturbance example to illustrate the phenomenon that longer wavelengths tend to travel faster. This increase in speed with wavelength is seen for both group velocities and phase velocities. A wave group consists of waves at varying wavelengths and frequencies. Individual waves of a wave group are usually generated at the same time, but tend to spread out within the group because each wavelet travels at a different speed. A group velocity is basically the speed at which a wave group travels. A phase velocity is the speed at which an individual wave travels, having its own characteristic wavelength and frequency. Fourier theory tells us that a sharp impulse is made up of infinite frequency content in phase at one point. If each frequency travels at the same speed, that peak will remain intact. If each frequency travels at a different speed, that peak will spread out (Figure 3). This spreading out is dispersion. Phase and group velocity are both dependent on wavelength and are related by the equation
V g r o u p = V p h a s e − λ δ V p h a s e δ λ {\displaystyle V_{\mathrm {group} }=V_{\mathrm {phase} }-\lambda {\frac {\delta V_{\mathrm {phase} }}{\delta \lambda }}}
where Vgroup is the group velocity, Vphase is the phase velocity, and λ is the wavelength. When attempting surface wave inversion, phase velocities are used more often than group velocities because it is easier to create a dispersion curve of phase velocities. A dispersion curve is a plot of velocity versus frequency or wavelength. After the dispersion curve has been generated, a surface wave inversion process is performed to calculate the subsurface elastic properties. The accuracy of the dispersion curve is crucial in obtaining the correct subsurface elastic parameters from inversion.
Elastic Properties Elastic properties of the earth are those properties which affect the propagation of elastic waves. These properties are Lamé parameters and are used to relate stress to strain in isotropic media through Hooke’s law. Density is also related to elastic parameters through velocity equations for compressional and shear waves.
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