DOWNWARD CONTINUATION IN THREE DIMENSIONS

Zero-offset migration is usually implemented as a downward continuation operation. The exploding reflector model is used to form an image of the subsurface. The wavefield at the surface is extrapolated downwards in depth using a one-way wave equation and the zero-time slice is extracted at each depth to form an image of the reflectivity as a function of depth.

The derivation of the method starts with the scalar wave equation for constant velocity media,

$${\partial^2 P\over\partial x^2}+{\partial^2 P\over\partial y... ...\partial z^2} = {1 \over v^2}{\partial^2 P\over\partial t^2}\,.$$

or its equivalent in the Fourier domain

$$k_x^2 + k_y^2 +k_z^2 P = {\omega^2 \over v^2} P\,.$$

The one-way wave equation is obtained by solving for k_z or its space-domain equivalent, $\partial/{\partial z}$:

 

$${\partial P\over\partial z} = - i { \omega \over v} \sqrt{ 1 + {v^2\over\omega^2} \nabla_{xy}^2}P \,.$$ (1)

Here, $\nabla_{xy}^2$ represents the 2-D Laplacian operator in x and y.

To perform migration we wish to extrapolate the wavefield at some depth z to another depth $z+\delta z$.There are two main classes of finite difference scheme that are used to do this: Explicit and Implicit schemes.