3D plane-wave migration in tilted coordinates

A 3D plane-wave source is specified by two ray-parameters, p_x and p_y. Given the velocity at the surface v_z0, we can calculate the propagation direction of the plane-wave source at the surface from the two ray-parameters. For each plane-wave source, we rotate the Cartesian coordinates, so that the extrapolation direction of the new coordinates is close to the propagation direction of the plane-wave. In 3D, the propagation direction of a plane-wave source is defined by two angles: the azimuth angle $\alpha$ and the take-off angle $\theta$. Given the velocity at the surface v_z0 and a plane-wave source with a ray parameter pair (p_x p_y), its propagation direction at the surface is defined by the vector (q_x,q_y,q_z), where

$$\begin{eqnarray} q_x&=&p_xv_{z_0},\ q_y&=&p_yv_{z_0},\ q_z&=&\sqrt{1-(q_x^2+q_y^2)}. \end{eqnarray}$$ (13) (14) (15)

The azimuth angle and take-off angle of the plane-wave source are calculated from the vector (q_x,q_y,q_z) as follows:

$$\begin{eqnarray} \alpha&=&\arcsin(q_y/q_x) ,\ \theta&=&\arccos(q_z).\end{eqnarray}$$ (16) (17)

 

rotazimuth Figure 1 Rotation about the axis z by $\alpha$

Rotations in 3D are specified by the axis of rotation and rotation angle. They can be described by a rotation matrix. For example, a rotation about the z-axis by an angle $\alpha$ (Figure ) is  

$$\left( \begin{array} {c} x_1\ y_1\ z_1 \end{array}\right... ...ght) \left( \begin{array} {c} x\ y\ z \end{array} \right).$$ (18)

To design a coordinate system with an extrapolation direction parallel the propagation direction of the plane-wave source (q_x,q_y,q_z), we rotate the coordinates in two steps. First we rotate about the z-axis by an angle $\alpha$ (equation 18) shown in Figure . Second we rotate about the y₁-axis by an angle $\theta$ (Figure ) as follow:

 

rottilt Figure 2 Rotation about the axis y1 by $\theta$

 

$$\left( \begin{array} {c} x_2\ y_2\ z_2 \end{array}\right... ...left( \begin{array} {c} x_1\ y_1\ z_1 \end{array} \right).$$ (19)

Combining the two rotations, we have the rotation from original coordinates to the new tilted coordinates as follows,  

$$\left( \begin{array} {c} x_2\ y_2\ z_2 \end{array}\right)... ...\right) \left( \begin{array} {c} x\ y\ z \end{array}\right).$$ (20)

It is easy to verify that the depth axis of the new coordinates parallels the propagation direction (q_x,q_y,q_z). In practice, we do not use the direction exactly paralleling the propagation of the plane-wave source at the surface. Considering that the velocity usually increases with the depth, the propagation direction of a plane-wave source becomes increasingly horizontal. So we usually choose a tilting angle $\theta$ that is a little bigger than $\arccos(q_z)$.

 

coordinates
Figure 3 A plane-wave source and its coordinates to image the salt body.(x,y,z) is the original Cartesian coordinates. (x₂,y₂,z₂) is the new tilted coordinates. The dashed line with arrows are the source and receiver rays of the plane-wave source. The extrapolation direction of the new coordinates z₂ axis is closer to the propagation direction of the plane-wave source.

Figure shows a typical coordinate system used for a plane-wave source to image a salt dome. The dashed lines with arrows show the propagation direction of the source and receiver waves for the plane-wave source. (x,y,z) is the Cartesian coordinate system, and (x₂,y₂,z₂) is the tilted coordinates for the plane-wave source. The extrapolation direction of the new coordinates, which parallels z₂-axis, is much closer to the propagation direction of the plane-wave source than the conventional vertical extrapolation direction.

For conical-wave source migration, we can similarly design tilted coordinates for each conical source. In contrast to the 3D plane-wave source migration, we apply the second rotation directly without rotating the azimuth of the data. Given the conical-wave source with a ray parameter p_x and the surface velocity v_z0, we rotate the velocity and the surface data along y-axis by a angle of $\theta=\arcsin(p_xv_{z0})$ and migrate the data in the new coordinates. This usually works when the in-line direction is the predominant dip direction in the subsurface. When there are steep dips in the cross-line direction, it is still difficult to image these dips with conical-wave migration in tilted coordinates. In contrast, 3D plane-wave migration in tilted coordinates rotate the data and model into a general direction dependent on the propagation direction and it can image these cross-line direction dips.

 

ppanel
Figure 4 The dots represent all possible plane-wave sources. The circles are the contour of their take-off angles and the smallest circle represents plane-wave sources with a take-off angle of $15^\circ$. The radial lines show the contour of azimuth angles. All possible plane-wave sources are divided into small cells. The plane-wave sources in a cell have a similar take-off and azimuth angle and share a coordinate system.