page_title: Surface wave inversion
section_title: Modified wavefield transform
hierarchical_section_title: Surface wave inversion / Dispersion curves / Modified wavefield transform
caption_attribution_description: English: Model of what a dispersion curve might look like.
context_page_description: 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.
context_section_description: A Fourier transform is applied to raw surface wave data plotted x-t. u(x,t) represents the entire shot gather, and the Fourier transformation results in U(x,ω).
  U(x,ω) is then deconvolved and can be expressed in terms of phase and amplitude.
  where P(x,ω) is the phase portion of the equation that holds information containing the waves’ dispersion properties, including arrival time information and A(x,ω) is the amplitude portion that contains data pertaining to the attenuation and spherical divergence properties of the wave. Spherical divergence is the idea that as a wave spreads out, the energy in the wave spreads out over the surface of the waveform. Since P(x,ω) contains the dispersion property information,
  where Φ=ω/cω, ω is the frequency in radians, and cω is the phase velocity for frequency ω. This data can then be transformed to give velocity as a function of frequency:
  This will yield a dispersion curve showing a variety of frequencies travelling at different phase velocities.
  The surface wave inversion process is the act of inferring elastic properties such as density, shear wave velocity profile, and thickness from dispersion curves created. There are many methods (algorithms) that have been utilized to perform inversion including:
  Multilayer dispersion computation
  Least squares curve fitting program
  Knopoff’s method
  Direct search algorithm
  High frequency Rayleigh wave inversion
  Refraction microtremor method
caption: Figure 4. Example of a dispersion curve where velocity increases with depth. The blue area represents experimental data, while the red line represents an experimental curve fit to the data.
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