Low frequency passive seismic interferometry for land data

Spatial variability and dispersion

Stacking is required to further analyze the obtained virtual seismic survey. First, each spatial axis in the virtual seismic survey is transformed from source-geophone coordinates $(s,g)$ , to midpoint (half)offset coordinates $(m,h)$ . Then, under a local 1D approximation, the offset coordinates $(h_x,h_y)$ are transformed into cylindrical coordinates and stacked over azimuth. The result is a virtual seismic survey at each midpoint, as a function of radial offset $h_r$ . This result is analogous to the spatial auto-correlation of Aki (1957) (Asten, 2006; Yokoi and Margaryan, 2008). To investigate if there is any spatial variability despite the local 1D approximation, the retrieved Green's function for a full radial offset of $2h_r=1640$ m are displayed in Figure 9 for two midpoint slices at $m_y=100$ m and $m_x=100$ m. Notice the Rayleigh waves arrive earlier for midpoints with negative $m_x$ and positive $m_y$ than for midpoints with positive $m_x$ and negative $m_y$ . This is consistent with the observation of higher velocities on the east side than on that the west end of the array, see previous section. Notice how the dominant phase of the arrival-train travels with a different velocity than the group velocity, indicating dispersion.

Mvariability
Figure 9. Two slices through a cube of retrieved Green's functions as a function of midpoints, for fixed full radial offset of $2h_r=1640$ m. Left: $m_y=100$ m, right: $m_x=100$ m.

The fundamental mode of a Rayleigh surface wave is represented by a zero-order Bessel function of the first kind, stretched by phase velocity, station distance and frequency (Aki, 1957; Okada, 2003). In Figure 10, this is observed in a frequency range between $1.5$ Hz and $6$ Hz, for retrieved Rayleigh waves averaged over all midpoints in the array. The jump in the dispersion curve at $4.5$ Hz is caused by a poor interpolation technique in the rotation from Cartesian to cylindrical offsets. The lowest retrieved frequencies travel with a wavelength of approximately $1250$ m, suggesting sensitivity to a depth of approximately $1250$ m. If more data was analyzed, convergence could have been achieved without averaging over azimuth or midpoint, preserving greater detail in subsurface properties for further study, and potentially yielding sufficient energy below $1$ Hz and thus sensitivity to greater depth.

CSdispersion
Figure 10. Left; real part of the retrieved Green's functions in the frequency-domain, shown as function of full radial offset, averaged over all midpoints. Right; estimated dispersion curve for the retrieved Green's functions shown in the left panel.

Low frequency passive seismic interferometry for land data