Numerical Examples

Figure 2 shows from left to right: a two-spike synthetic model, modeling with equations (1) and (2). It is possible to observe the decay of amplitude (energy) with respect to time when we include the damping factor ($\epsilon$) in our modeling equation. The same effect is observable in Figure 3, which shows the migration results using the results of Figure 2.

 

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Figure 2 Modeling Comparison. From left to right: synthetic model; modeling with $k_z=\sqrt{\frac{\omega^2}{v^2}-{k_x}^2}$; modeling with $R = ik_z = \sqrt{({-i\omega+\epsilon})^2 + {vk_x}^2}$

 

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Figure 3 Modeling-Migration Comparison. From left to right: synthetic model; results with $k_z=\sqrt{\frac{\omega^2}{v^2}-{k_x}^2}$;results with $R = ik_z = \sqrt{({-i\omega+\epsilon})^2 + {vk_x}^2}$

The energy decay is a function of the damping factor ($\epsilon$). Figure 4 presents experimental results of the energy variability with respect to $\epsilon$.The experiment consists of modeling and migration of the two-spike model with equation (2) for different values of $\epsilon$.Each sample in the plot (Figure 4) represents the energy of each modeling-migration result for $\epsilon$ ranging between 0.002 and 0.04 every 0.002. It is possible to observe that the energy decays exponentially with respect to the damping factor; this decrease is due to the omission of causality in the redefinition of ik_z.

 

energy Figure 4 Energy decay with respect to the damping factor $\epsilon$

For comparison, Figure 5 shows three results of the experimental procedure described before. It is possible to observe that the bigger the damping factor, the less energy is present in the final result.

 

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Figure 5 Three different values of $\epsilon$ for the original model, right with $\epsilon=0.00312$, center $\epsilon=0.008$, left $\epsilon=0.04$