The boundary condition correction

We tested the amplitude correction at the boundary condition by downward propagating a shot and picking the amplitudes of the wavefield recorded at a certain depth. Since our particular purpose of using WEMVA is finding velocity anomalies that generate focusing-effect AVO Vlad and Biondi (2002), we used the same focusing-generating velocity model as that presented in the lower right panel of Figure 6 of Vlad (2002). For convenience, we present it in the upper panel of Figure 4, also tracing wavefronts through it for a better visualization of the kinematics of propagation. The time delays induced by the presence of the low-velocity slab have not been shown because they are under the common picking threshold - this is a typical focusing-effect AVO case.

To obtain the curves in the lower panel of Figure 4, for each of the three methods, we propagated a wavefield through the velocity model in the upper panel of the figure (with the low-velocity slab), and another wavefield through the constant-velocity background only. For each x location, we picked the maximum amplitudes of each wavefield at the depth of 6000m, and we divided the amplitudes obtained from the model with the slab by the amplitudes obtained from the constant-velocity background. A deviation from the value of 1 indicates the presence of the wavefield scattered by the slab. We performed this procedure using three different algorithms: (1) - Linearized downward continuation with no amplitude correction applied; (2) - Linearized downward continuation with boundary condition amplitude correction applied at the surface; (3) - Pseudospectral wave propagation Biondi (2002), for reference. The application of the boundary condition correction has brought the values closer to those of the reference curve. A possible shot-profile formulation of WEMVA would therefore benefit from the application of the boundary condition correction.

 

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Figure 4 Top panel: Velocity model for testing the amplitude behavior of the linearized downward continuation operator. A shot has been generated at (0,0) and the wavefield is recorded at a depth of 6 km. The thin slab has a velocity of 1647 m/s, contrasting to the background of 1830 m/s. Wavefronts have been traced for better visualization of propagation kinematics. Bottom panel: Curve ``-L'' - Linearized downward continuation with no amplitude correction applied; Curve ``+L'' - Linearized downward continuation with boundary condition amplitude correction; Curve ``Ref'' - Pseudospectral wave propagation.