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Extended split-step using evanescent energy

For the downward continuation of the wavefield, the DSR in the ${\omega-k}$ domain is chosen to be always real to avoid the seismic evanescent energy. In the ${\omega-k}$ space the DSR operator is the following:
&&DSR(k_{m},k_{h},z,\omega) = \nonumber \\  && \sqrt{\frac{\ome...
 ...{v_{(mG,z)}^2} - \frac{1}{4}((k_{mx}+k_{hx})^2+(k_{my}+k_{hy})^2)}\end{eqnarray}
Applying the split-step approximation to the DSR operator, we obtain
&&DSR(k_{m},k_{h},z,\omega) \cong \nonumber \\  & &\sqrt{\frac{... \\ &+& (\frac{\omega}{v_{(mG,z)}} - \frac{\omega}{v_{Gref,z}}),\end{eqnarray}
where kmx is the CDP in-line wavenumber, kmy is the CDP cross-line wavenumber, khx is the offset in-line wavenumber and ${\overline k_{hy}}$ is the offset cross-line wavenumber, vmG,z and vmS,z are the velocity fields for source and geophone midpoint-coordinates and vSref,z and vGref,z are the set of the reference velocities for every depth.

The vertical wavenumber kz in a conventional downward continuation is a real number (equation 2) avoiding the evanescent energy. Therefore, in the 2-D prestack case, kz is calculated by setting ${\frac{\omega^2}{v^2}\gt\frac{1}{4}(k_{mx} \pm k_{hx})^2}$. This condition guarantees that kz is real, and it rejects the energy that corresponds to imaginary kz or the evanescent energy domain. In the case of an extended split-step migration, the evanescent energy is useful for interpolating two downward continued wavefields.

Figure 1 shows the values of kz for 10 different reference velocities associated with receiver locations for the zeror-offset case. The outer curve corresponds to a low reference velocity (1500m/s) and the inner curve corresponds to a high velocity (5000m/s). The phase correction to downward continue the wavefield in one depth step is the positive vertical wavenumbers kz multiplied by the depth step. The negatives kz multiplied by a depth step represent the argument of a damped exponential applied to the wavefield that is traveling with an angle sine greater than one. Figure 1 also shows that dips at a higher reference velocity correspond to smaller dips at a lower reference velocity, when kz is linear interpolated between two different reference velocities.

In addition, Figure 1 helps to define the evanescent energy nesccesary to image steep events. The kz values for the minimun velocity define the the evanescent energy necessary for imaging steep events for a depth z. Setting this new limit for the kz helps to save computer time.

In order to preserve very steep dips during the linear interpolation between two downward continued wavefields, the kz domain must be extended (Fig. 1). Therefore, in our extended split-step algorithm, instead of rejecting imaginary values of the 2-D DSR, we save and use those values in the interpolation of downward continued wavefields. Gazdag (1984) presented a different approach, where evanescent energy is used in a phase-shift-plus-interpolation migration by replacing the imaginary values of kz for values calculated using a straight line tangent to a kz curve from a specific angle.

The 3-D prestack downward continuation operator in common-azimuth is based on a stationary-phase approximation of the 3-D DSR (equation 1). This approximation reduces the dimensionality of the downward continuation operator from 5 to 4 dimensions, constraining the direction of propagation of source and receiver rays to the same plane and azimuth, and calculating the cross-line-offset wavenumbers (${\overline k_{hy}}$) from the in-line (kmx) and cross-line CDP wavenumbers (kmy) and in-line offset wavenumbers (khx) Biondi and Palacharla (1996).

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Next: Results Up: Malcotti & Biondi: Extended Previous: Introduction
Stanford Exploration Project