The process of creating dispersion curves from raw surface wave data (distance vs. time plot) can be performed using five transformation processes. The first is known as the wave-field transformation (τ-p transformation), first performed by McMechan and Yedlin (1981). The second is a 2-dimensional wave-field transform (f-k transformation) performed by Yilmaz (1987). The third is a wave-field transform base on phase shift, performed by Park et al. (1998). The fourth is a modified wave-field transform base on frequency decomposition and slant stacking, performed by Xia et al. (2007). The fifth is a high-resolution Linear Radon transformation performed by Luo et al. (2008). In performing a wave-field transformation, a slant stack is done, followed by a
Fourier transform. The way in which a Fourier transform changes x-t data into x-ω (ω is angular frequency) data shows why phase velocity dominates surface wave inversion theory. Phase velocity is the velocity of each wave with a given frequency. The modified wavefield transform is executed by doing a Fourier transform first before a slant stack. Slant stacking is a process by which x-t (where x is the offset distance, and t is the time) data is transformed into
slowness versus time space. A linear move (similar to
normal move out (NMO)) out is applied to the raw data. For each line on a seismic plot, there will be a move out that can be applied that will make that line horizontal. Distances are integrated for each slowness and time composition. This is known as a slant stack because each value for slowness represents a slant in x-t space and the integration stacks these values for each slowness.
Modified wavefield transform 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,\omega)= \int u(x,t)e^{-i \omega t} \,dt U(x,ω) is then deconvolved and can be expressed in terms of phase and amplitude. :U(x,\omega)= P(x,\omega)A(x,\omega) 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, :U(x,\omega)= e^{-i \Phi x} A(x,\omega) 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: :V(\omega,\Phi)= \int e^{i \Phi x} \frac{U(x,\omega)}\,dx 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
Multilayer dispersion computation Haskell (1953) came up with an algorithm based on Haskell’s earlier work. Their method used an iterative technique that enabled the user to input parameters and the computer to find which exact parameters best fit the experimental data.
Knopoff’s method Knopoff’s method also uses Haskell’s equations to perform the surface wave data inversion, but it simplifies the equations for the fastest computation. The increased speed is mostly accomplished in programming as well as the lack of complex numbers in the calculations. In this algorithm, approximate layer thicknesses, compressional and shear velocities, as well as density values must be input for the model.
Direct search algorithm The direct search algorithm matches a data driven model to the synthetic dispersion curve (Wathelet et al., 2004). This algorithm creates a theoretical dispersion curve by guessing parameters such as shear wave velocity, compressional wave velocity, density, and thickness. After the theoretical curve is created, the computer then attempts to match this theoretical curve with the actual (experimental) dispersion curve. The values of the parameters are picked at random, with different permutations, and repeated continuously until matching curves are achieved. In some cases, while running the algorithm, different values of shear and compressional velocities, density, and thickness might produce the same dispersion curve. The algorithm calculates a value known as the misfit value as it generates each theoretical dispersion curve. The misfit value is simply a measure of how the generated model stacks up to a true solution. Misfit is given by, :Misfit = \sqrt{\sum_{i-\sigma}\frac{(x_{di}-x_{ci})^2}{\sigma^2_i n_F}} where xdi is the velocity of data curve at frequency fi, xci is the velocity of the calculated curve at frequency fi, σi is the uncertainty of the frequency samples considered and nF is the number of frequency samples considered. If no uncertainty is provided, σi is replaced by xdi.
High frequency Rayleigh wave inversion The high frequency Rayleigh wave inversion performed by Xia
et al. (1999) analyzed the earth using Knopoff’s method. By varying different properties used in creating the dispersion curve, it was discovered that different earth properties had significantly different effects on phase velocities. Changing the S-wave velocity input has a dramatic impact on Rayleigh wave phase velocities at high frequencies (greater than 5 Hz). A change in S-wave velocity of 25% changes the Rayleigh wave velocity by 39%. Conversely, P-wave velocity and density have a relatively small impact on Rayleigh wave phase velocity. A change in density of 25% will cause a less than 10% change in surface wave velocity. A change in P-wave velocity will have even less effect (3%).
Microtremor method The final inversion method, the refraction microtremor (ReMi) technique, makes use of a computer algorithm that forward models normal mode dispersion data obtained from a survey. This method uses regular P-wave and simple refraction acquisition equipment, and does not require an active source, hence the name. Pullammanapellil et al. (2003) used this method to accurately match the S-wave profile of the ROSRINE borehole drilled. The ReMi method accurately matched the overall shear wave velocity profile, but cannot match the detail provided by the
shear velocity well log. The discrepancy in overall detail should have no effect in evaluating the subsurface. ==Advantages/Disadvantages of Surface Wave Inversion==