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,

or its equivalent in the Fourier domain The one-way wave equation is obtained by solving for(1) |

Here, 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 .There are two main classes of finite difference scheme that are used to
do this: Explicit and Implicit schemes.

- Explicit schemes
- Implicit schemes
- Conjugate gradient implicit schemes
- Predictor-corrector method
- Other options

12/18/1997