Estimating W_ij from cross-well and VSP P-wave traveltimes

For TI media, the elastic constants that control the P- and SV-wave propagation can be estimated from P-wave measurements near the axes, as equation (24) shows. Figure  shows an example. The left column shows the given P-wave impulse response (continuous line) and the approximating ellipses (dashed lines) around the horizontal and around the vertical for two different apertures. The central column shows the given and approximate impulse responses for P- and SV-waves, the approximate impulse responses calculated from the estimated elastic constants. The right column shows the absolute value of the error made in the estimation of the elastic constants. Unlike the previous approximations, the P-wave double elliptic approximation is valid only at very small angles ($\approx 2$ degrees). At greater angles it yields highly inaccurate results, particularly in the estimation of W₄₄ (L), which controls the shear wave propagation along the axes. Note that for an aperture of 10 degrees, the shear wave impulse response has been considerably underestimated.

 

xw-vsp
Figure 5 P-wave impulse response used to estimate the elastic constants from measurements around the axes. Left: given P-wave impulse response (continuous) and elliptical approximations around the axes (dashed). Center: given impulse response (continuous) and estimated ones (dashed) for P- and SV-waves. Right: absolute value of the error made in the estimation of the elastic constants when using the P-wave double elliptic approximation. The elastic constants are A = W₁₁, F = W₁₃, C = W₃₃, and L = W₄₄.

This approximation may be hard to use in practical situations because it works only for very small angles near the axes.