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

Figure  shows how the horizontal double elliptic approximation works for different ray angles measured from the horizontal. When comparing this case with the VSP case (Figure ), we see that the approximation works better for ray angles around the horizontal than around the vertical, even when the angles are large. The estimated impulse responses agree well with the given ones, and the errors in the elastic constants are negligible for angles less of fewer than 30 degrees.

 

xw
Figure 3 Left: impulse response for P- and SV-waves (continuous lines) compared with their elliptical approximations around the vertical (dashed lines). Center: given impulse responses (continuous) compared to the ones calculated from the estimated elastic constants (dashed). Right: absolute value of the error made in the estimation of the elastic constants when using the horizontal double elliptic approximation. The elastic constants are A = W₁₁, F = W₁₃, C = W₃₃, and L = W₄₄.

The reason why the horizontal double elliptic approximation works better than the vertical double elliptic approximation is that the elliptical approximation for P-wave impulse response is more adequate at larger angles around the horizontal than around the vertical, where the group velocity is lower. Notice that the elliptic approximations around the vertical are almost circular.

Figure  shows that when small and large ray angles are used simultaneously to calculate the elliptical approximations around the horizontal, the result is roughly an average of the results shown in Figure . The agreement between given and estimated impulse responses is excellent and the error in the estimation of the elastic constants is $\approx 2\%$ for W₁₃ (F) and $\approx 1\%$for W₃₃ (C).

 

xw-4angles
Figure 4 Left: impulse response for P- and SV-waves (continuous lines) compared with their elliptical approximations around the vertical (dashed lines). All ray angles shown are used simultaneously to calculate the elliptical approximations. Center: given impulse responses (continuous) compared to the ones calculated from the estimated elastic constants (dashed). Right: absolute value of the error made in the estimation of the elastic constants when using the horizontal double elliptic approximation. The elastic constants are A = W₁₁, F = W₁₃, C = W₃₃, and L = W₄₄.