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:

$$\begin{eqnarray} &&DSR(k_{m},k_{h},z,\omega) = \nonumber \\ && \sqrt{\frac{\ome... ...{v_{(mG,z)}^2} - \frac{1}{4}((k_{mx}+k_{hx})^2+(k_{my}+k_{hy})^2)}\end{eqnarray}$$ (1)

Applying the split-step approximation to the DSR operator, we obtain

$$\begin{eqnarray} &&DSR(k_{m},k_{h},z,\omega) \cong \nonumber \\ & &\sqrt{\frac{... ...er \\ &+& (\frac{\omega}{v_{(mG,z)}} - \frac{\omega}{v_{Gref,z}}),\end{eqnarray}$$ (2)

where k_mx is the CDP in-line wavenumber, k_my is the CDP cross-line wavenumber, k_hx is the offset in-line wavenumber and ${\overline k_{hy}}$ is the offset cross-line wavenumber, v_mG,z and v_mS,z are the velocity fields for source and geophone midpoint-coordinates and v_Sref,z and v_Gref,z are the set of the reference velocities for every depth.

The vertical wavenumber k_z in a conventional downward continuation is a real number (equation 2) avoiding the evanescent energy. Therefore, in the 2-D prestack case, k_z is calculated by setting ${\frac{\omega^2}{v^2}\gt\frac{1}{4}(k_{mx} \pm k_{hx})^2}$. This condition guarantees that k_z is real, and it rejects the energy that corresponds to imaginary k_z 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 k_z 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 k_z multiplied by the depth step. The negatives k_z 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 k_z is linear interpolated between two different reference velocities.

In addition, Figure 1 helps to define the evanescent energy nesccesary to image steep events. The k_z 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 k_z helps to save computer time.

In order to preserve very steep dips during the linear interpolation between two downward continued wavefields, the k_z 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 k_z for values calculated using a straight line tangent to a k_z 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 (k_mx) and cross-line CDP wavenumbers (k_my) and in-line offset wavenumbers (k_hx) Biondi and Palacharla (1996).