Shallow example

The preceding example assumed that typical reflection seismic data collection could be performed at the site of interest. But suppose instead that the region of interest is quite shallow, possibly very soft and/or compliant sediments or soils, and that, in particular, it is not possible to obtain shear wave data directly. Then what can be done?

One of the most common problems with traditional compressional wave surveys is ground roll. Ground roll is typically composed of Rayleigh and/or Love waves, and usually the Rayleigh wave component is the one we need to eliminate because it is contaminating the P-wave data near the shot point. The Rayleigh wave speed depends on both the compressional and shear wave speeds of the medium, and - being a surface wave - it is most strongly influenced by the topmost layers of the earth (usually those within about one wavelength from the surface). So for shallow imaging and analysis, why not consider using Rayleigh wave speed measurements together with P-wave speed measurements to infer the S-wave speed.? The pertinent S-wave speed in an anisotropic (VTI) medium is the shear wave speed in the symmetry plane (perpendicular to the axis of symmetry). So the formula shown previously (7) is pertinent, but it needs to be used in a different way to find the shear wave speed $v_s = \sqrt{c_{66}/\rho_0}$, when $v_p = \sqrt{c_{11}/\rho_0}$and v_R are known.

To accomplish this goal, I first square (6). The result is a quartic equation for q = (v_R/v_s)². In this case, v_R is known, but v_s is unknown (opposite of the earlier case). But this difference does not cause any difficulty in the analysis. The equation can be rearranged into the form:  

$$\frac{1}{16}q^4 - \frac{1}{2}q^3 + \frac{3}{2}q^2 - \left(1 + \frac{v_R^2}{v_p^2}\right)q + \frac{v_R^2}{v_p^2} = 0.$$ (21)

Equation (21) is straightforward to solve by iteration using a simple Newton-Raphson scheme (Hildebrand, 1956; Press et al., 1988). Generally a good starting value for the scheme will be $q \simeq 0.8$ as this corresponds roughly to a trial value of v_R = 0.9v_s.

Having once determined the value of $v_s = \sqrt{c_{11}/\rho_0}$ -- using the measured Rayleigh wave speed and the compressionial wave speed v_p -- in the symmetry plane, Thomsen parameter analysis can be combined with the Sayers and Kachanov (1991) method in order to deduce useful information about the nature of the heterogeneities causing the anisotropic at the macroscale. Once these wave speeds are known, the analysis for interpretation can proceed in essentially the same manner as in the previous example.

TABLE 1. Examples of Sayers and Kachanov (1991) parameters $\eta_1(\rho)$ and $\eta_2(\rho)$ when crack density $\rho << 1$for penny-shaped cracks. Four choices of effective medium theory are considered: NI (non-inteacting), DS, (differential scheme), CPA (coherent potential approximation), and SC (the Budiansky and O'Connell self-consistent scheme). Note that crack density is defined here as $\rho = Nr^3/V$, where N/V is number density of cracks, and $A = \pi r^2$ is the area of the circular crack face.

$$\eta_1 \eta_2$$

NI -0.000216 0.0287

DS -0.000216 0.0290

CPA -0.000258 0.0290

SC -0.0000207 0.0290