Tilted Cartesian coordinates

In Cartesian coordinates (x,z), the one-way wave equation is  

$$\frac{\partial P}{\partial z}=\frac{i\omega}{v} \sqrt{1+\frac{v^2}{w^2}\frac{\partial^2}{\partial x^2}}P.$$ (1)

It can be obtained by factoring the two-way acoustic wave equation. If the coordinates are rotated by an angle $\theta$, the two-way acoustic wave equation doesn't change in the new coordinates $(x^\prime,z^\prime)$. Factoring the two-way wave equation in the tilted coordinates, we can obtain the same one-way wave equation. In the tilted coordinates, wavefields are now extrapolated in the $z^\prime$ direction, rather than in the downward direction as done in downward continuation. Also the source and receiver data are not on the line $z^\prime=0$, but on $x^\prime\sin\theta-z^\prime\cos\theta=0$ (Figure 1a ).

 

tilt
Figure 1 Tilted Cartesian coordinates: (a) sources and receivers are on the line $x^\prime\sin\theta-z^\prime\cos\theta=0$ and the extrapolation direction is the $z^\prime$ direction; (b) sources and receivers are usually very far from the reflection point for overturned waves and the opening angle is usually small.

For a point source, the waves propagate in all directions, making it impossible to cover all propagation directions with only one Cartesian coordinate system. We decompose the point source into plane waves and extrapolate each plane wave in tilted coordinates, whose extrapolation direction $z^\prime$ is determined by the propagation direction of the plane wave. The propagation direction of the plane wave at the surface can be used as the extrapolation direction $z^\prime$ of the tilted coordinates. In this coordinate system, the source and receiver wavefields of overturned waves can usually be caught, since the opening angle between them is usually small and the propagation directions of the source and receiver wavefields are close to each other (Figure 1b).

Another advantage of using a plane-wave rather than a point source is that no padding is needed to image steeply dipping reflectors. To catch the energy from steeply dipping reflectors, the locations of sources and receivers are usually far from the reflector point (Figure 1b). Large extrapolation aperture is required to image these reflectors using a point-source method. In contrast, plane waves inherently have large apertures, which can easily cover the locations of source, receiver and reflector points. This greatly reduces the cost of imaging the steeply dipping reflectors.