PS common-azimuth downward-continuation

Chapter 2 discussed the basics for downward-continuation migration in 2-D using the concept of survey-sinking. For 3-D the sinking or downward continuation of the wavefield at depth z (U_z), in midpoint-offset coordinates and in the frequency-wavenumber domain [$U_{z} \left (\omega,{\bf k_m},{\bf k_h} \right )]$,to a different depth level $z+\Delta z$ can be expressed as:

$$U_{z+\Delta z} \left (\omega,{\bf k_m},{\bf k_h} \right ) = ... ...} \left (\omega,{\bf k_m},{\bf k_h} \right ) e^{ik_z \Delta z},$$ (52)

the variable $\omega$ is the Fourier pair for the time axis, and the variables ${\bf k_m}$,${\bf k_h}$ and k_z are the representation in the Fourier domain (wavenumber) for the midpoint, offset and depth axes. After each depth-propagation step, the resulting wavefield represents a synthesized dataset with the source and receiver distribution at the new depth level, $z+\Delta z$,Schultz and Sherwood (1980). For converted-wave data, the downgoing wavefield is propagated using the P-velocity ($v_p({\bf s},z)$), whereas the upgoing wavefield is propagated applying the S-velocity ($v_s^2({\bf r},z$)). The basic downward-continuation step, for converted waves, is performed by applying the Double-Square-Root (DSR) equation that in midpoint-offset coordinates is

$$\begin{eqnarray} k_z \left (\omega,{\bf k_m},{\bf k_h} \right ) &=&\mbox {DSR} ... ...},z)}-\frac{1}{4}({\bf k_m}+{\bf k_h})\cdot({\bf k_m}+{\bf k_h})}.\end{eqnarray}$$ (53)

The common-azimuth downward-continuation operator can then be expressed as follows Biondi and Palacharla (1996):

$$\begin{eqnarray} U_{z+\Delta z} \left (\omega,{\bf k_m},k_{h_x},h_y=0 \right ) ... ...t (\omega,{\bf k_m},k_{h_x}\right ) e^{-i\widehat{k_z}\Delta z}.\end{eqnarray}$$ (54)

Since common-azimuth data is independent of k_hy, the integral can be pulled inside and analytically approximated by the stationary-phase method Bleinstein (1984). The application of the stationary-phase method is based on a high-frequency approximation, that is, it assumes that $\omega$ tends toward infinity.

The expression for $\widehat{k_z}$ in equation  comes from substituting the stationary-path approximation into the expression for the full DSR equation :

 

$$\widehat{k_z}=\mbox {DSR} \left [\omega,{\bf k_m},k_{h_x},\widehat{k}_{h_y}(z),z \right ],$$ (55)

where

 

$$\widehat{k}_{h_y}(z)=k_{m_y}\frac{\sqrt{\frac{\omega^2}{v_s^... ...{\omega^2}{v_p^2({\bf s},z)} -\frac{1}{4}(k_{m_x}-k_{h_x})^2}},$$ (56)

represents the PS stationary path approximation for the double square root equation as a function of k_hy. To obtain the PS stationay path approximation I follow a geometric interpretation for the PS common-azimuth downward-continuation operator. I adhere to Biondi and Palacharla (1996) demonstration for the equivalence of the stationary phase derivation for the common-azimuth operator and the constraint on the propagation directions of the source rays and receiver rays for the single-mode case.

Figure  shows the ray geometry for the common-azimuth downward-continuation operator. Notice that the source ray (p_sx,p_sy,p_sz) and the receiver ray (p_rx,p_ry,p_sz) lie on the same plane. To fulfill this condition the components for both rays along the crossline y axis must be equal.

I follow simple geometry to obtain the two crossline y axis components for the source and receiver rays. Figure  represents one of the slanted planes on Figure , and the elements for this geometric derivation. From the definition of ray parameter, the component p_sy for the source ray and the component p_ry for the receiver ray are

 

comaz-down2
Figure 1 Schematic representation of the ray geometry for common-azimuth downward-continuation. For each pair of source ray and receiver ray, the two rays must lie on the same slanted plane. This figure is taken from Biondi and Palacharla (1996).

 

planes2
Figure 2 Schematic representation of the ray geometry for the PS-CAM operator. For each pair of source and receiver rays, the two rays lie on the same slanted plane. All the components for the geometric derivation ray parameter representation of the stationary path approximation.

$$\begin{eqnarray} p_{sy} & = & \frac{1}{v_p({\bf s},z)} \frac{dy_s}{dl_s}, \nonumber\ p_{ry} & = & \frac{1}{v_s({\bf r},z)} \frac{dy_r}{dl_r},\end{eqnarray}$$ (57)

where dl_s, dl_r are the differential raypath lengths for the source and receiver ray, respectively, dy_s and dy_r are the source and receiver crossline components, as they are illustrated on Figure . The components p_sz for the source ray and p_rz for the receiver ray are

$$\begin{eqnarray} p_{sz} & = & \frac{1}{v_p({\bf s},z)} \frac{dz}{dl_s}, \nonumber\ p_{rz} & = & \frac{1}{v_s({\bf r},z)} \frac{dz}{dl_r},\end{eqnarray}$$ (58)

where dz is the component along the depth axis. This component must be the same for both the source and the receiver ray. This condition also holds for converted waves. The final image forms when all the energy is focused at zero at time equal zero. This is true if both the P-velocity and the S-velocity models are correct. This is a key element for the PS-CAM operator, because of this the converted-wave stationary path does not incorporates any additional parameters.

The final crossline y components for both the source and receiver rays are the result of combining the set of equations  with the ones in , such that the differential raypath lengths, dl_s and dl_r, are eliminated. The components dy_s and dy_r are

$$\begin{eqnarray} dy_s & = & \frac{p_{sy} dz}{p_{sz}}, \nonumber\ dy_r & = & \frac{p_{ry} dz}{p_{rz}}.\end{eqnarray}$$ (59)

The condition for the source and receiver rays lie in the same plane is that both dy_s and dy_r are the same. By equating the two equations in I obtain the mathematical expression of this condition

 

$$\frac{p_{sy}}{p_{sz}} = \frac{p_{ry}}{p_{rz}}.$$ (60)

To eliminate both p_sz and p_rz, from equation , I use the following relationship among the ray parameters:

$$\begin{eqnarray} p^2_{sx} + p^2_{sy} + p^2_{sz} & = & \frac{1}{v_p({\bf s},z)}, ... ...r\ p^2_{rx} + p^2_{ry} + p^2_{rz} & = & \frac{1}{v_s({\bf r},z)}.\end{eqnarray}$$ (61)

After some mathematical simplifications the stationary path relationship in terms of ray parameters is

 

$$(p_{ry} - p_{sy}) = (p_{ry} + p_{sy}) \frac{\sqrt{\frac{1}{v... ...\bf r},z)}-p^2_{rx}}+\sqrt{\frac{1}{v_p({\bf s},z)}-p^2_{sx}}},$$ (62)

this equation transforms into equation  by substituting the wave numbers for the ray parameters.

This process just validates the PS common-azimuth downward-continuation operator, I implement PS common-azimuth migration using a split-step algorithm Kessinger (1992). The main difference between the implementation of the PS-CAM operator and the conventional CAM operator is the need of two velocity models.