Estimating W_ij from VSP P- and SV-wave traveltimes

Figure  shows how the vertical double elliptic approximation works for different angles around the vertical. The parameters of the ellipses that approximate the impulse response have been calculated at the angles shown by the straight lines. The left column compares the given impulse responses for P- and SV-waves (continuous lines) with the elliptical approximations around the vertical (dashed lines). With the four elliptical parameters obtained for each aperture, I calculate the elastic constants of the medium by using equation (22). From the estimated elastic constants, I calculate the corresponding impulse responses for both P- and SV-waves. The result is shown in the central column (dashed lines) simultaneously with the given impulse responses. In most cases (except at 40-degree aperture) the agreement is excellent. At 40-degree aperture the horizontal P-wave group velocity has been overestimated and the shear wave triplication is larger than expected. The right column shows the absolute value of the percentage error made in the estimation of the elastic constants. For small angles ($\leq$ 10 degrees), the error is negligible. For angles between 10 and 30 degrees the error is smaller in W₁₁ (A) than in W₁₃ (F). At large angles the error in W₁₁ (A) is the largest, almost 30%. Notice that up to 30 degrees, even though the error in the estimation of the elastic constants is not zero but a few percent, the differences between given and estimated impulse responses are hard to see.

 

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Figure 1 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 vertical double elliptic approximation. The elastic constants are A = W₁₁, F = W₁₃, C = W₃₃, and L = W₄₄. The density is assumed to be unity.

In Figure , the elliptical approximations to the impulse responses are calculated using two ray angles at a time, one zero and one nonzero. In Figure , I show what happens when all these angles (or different VSP offsets) are used simultaneously to calculate the elliptical approximations and the elastic constants. The horizontal P-wave group velocity is now slightly overestimated and the shear velocities are retrieved well. The errors in the estimated elastic constants are 6% in W₁₁ (A) and 4% in W₁₃ (F). The errors made when using large angles only have been compensated by using also small angles.

 

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Figure 2 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 vertical double elliptic approximation. The elastic constants are A = W₁₁, F = W₁₃, C = W₃₃, and L = W₄₄.