Using P- and SV-wave narrow aperture phase velocities

Elastic constants from phase velocities near the vertical.- When the phase angles are close to zero, it is possible to estimate elastic constants from the corresponding phase velocities by using equations (4), (5), (7), and (8), a system of four equations and four unknowns. The independent term (that I haven't explained how to obtain yet) is formed by W_P,z, W_P,xnmo, W_SV,z, and W_SV,xnmo. The solution of this system of equations is

$$\begin{eqnarray} W_{33} & = & W_{P,z}, \\ W_{44} & = & W_{SV,z}, \\ W_{13} & = &... ...- W_{SV,z}, \\ W_{11} & = & W_{SV,xnmo} + W_{P,xnmo} - W_{SV,z}.\end{eqnarray}$$ (1) (2) (3) (4)

In (22d), W₁₁ is a linear combination of elliptical parameters independent of the horizontal P-wave phase velocity, unlike W₁₁ estimated from equation (21b). Roughly speaking, (22d) says that summing P- and SV-wave NMO velocities (squared) is the same as summing the elastic constants that control the horizontal wave propagation. The examples in the final section of the paper show the range of angles near the vertical for which these equations are valid.

Since this approximation simultaneously uses the elliptical parameters of two ellipses fitted near the vertical, I call it vertical double elliptic approximation, analogous to Muir's double elliptic approximation that uses horizontal and vertical ellipses to approximate the slowness surface and impulse response for all angles (Muir, 1990a; Dellinger et al., 1992). It is important to point out the differences between these two approximations. On the one hand, the vertical double elliptic approximation is used to estimate elastic constants from phase velocities near the vertical axis. Slowness surfaces and impulse responses can be calculated from these elastic constants using (1a) and the exact relationships between phase and group velocities. On the other hand, Muir's double elliptic approximation is used to estimate directly slowness surfaces and impulse responses from data near both axes without having to know the elastic constants.

Fitting P- and SV-wave phase velocities with ellipses near the vertical is not the same as using the vertical double elliptic approximation. However, the fitting is a necessary intermediate step in the estimation of elastic constants using equations (22). When the elliptical fitting is done near the horizontal axis the result is the horizontal double elliptic approximation, as follows.

Elastic constants from phase velocities near the horizontal.- When the phase angle is close to 90 degrees, the expressions for the elastic constants as a function of P- and SV-wave phase velocities are obtained by solving the system of equations (13), (14), (16), and (17), with the following result:

$$\begin{eqnarray} W_{11} & = & W_{P,x}, \\ W_{44} & = & W_{SV,x}, \\ W_{13} & = &... ...- W_{SV,x}, \\ W_{33} & = & W_{SV,znmo} + W_{P,znmo} - W_{SV,x}.\end{eqnarray}$$ (1) (2) (3) (4)

This set of equations forms the horizontal double elliptic approximation. It uses elliptical P- and SV-wave phase velocities near the horizontal to approximate the elastic constants. Notice that the estimation of W₃₃ is independent of the P-wave phase velocity along the vertical axis. Equations (23) can be obtained from (22) by interchanging W₁₁ and W₃₃, and x and z.

The estimation of W₃₃ from near-horizontal phase velocities (equation (23d)) and W₁₁ from near-vertical phase velocities (equation (22d)) is in both cases the sum of NMO velocities minus W₄₄. Michelena et al. (1992) and Michelena (1992b) show that when estimating velocities tomographically, NMO velocities correspond to the smallest singular values of the problem. The largest singular values correspond to velocities estimated from rays that travel along the axes. Therefore, as expected, estimating W₃₃ from cross-well traveltimes alone is a harder problem than estimating W₁₁ from the same data. The opposite is true when estimating W₃₃ and W₁₁ from VSP measurements.