- 1.
__Downward continuation__The first step of migration consists of downward continuation of the wavefield measured at the surface (a.k.a. the data), which is done by the recursive application of the equation:

(10) (11) - is the wavefield at depth
*z*, - is the wavefield at the surface
*z*=0, - is the downward continuation operator at depth
*z*, - is the data, i.e.. the wavefield at the surface, and
- is a frequency-dependent scale factor for the data.

The recursion in Equations (A-1) and (A-2) can be also rewritten in matrix form as

(12) - is a square matrix containing the downward continuation operator for all depth levels,
- is a column vector containing the wavefield at all depth levels, and
- is a column vector containing the scaled
data.
^{}

Equation (A-3) represents the downward continuation recursion written for a given frequency. We can write a similar relationship for each of the frequencies in the analyzed data, and group them all in the matrix relationship

(13) - is a diagonal matrix containing the downward continuation operators for all the frequencies in the data,
- is a column vector containing the wavefield data for all the frequencies, and
- is a column vector containing the scaled data at all
frequencies.
^{}

It follows from Equation (A-4) that the background wavefield () can be computed as a function of the measured data (), as follows:

(14) - is the wavefield at depth
- 2.
__Imaging__The second step of the migration by downward continuation is imaging. In the exploding reflector concept, the image is found by selecting the wavefield at time

*t*=0 or, equivalently, by summing over the frequencies :(15) - is the image (reflectivity) corresponding to a given
depth level
*z*.

We can write the Equation (A-6) in matrix form as

(16) - is an operator performing the summation over frequency for
every depth level
*z*, and - is a column vector containing the image at every depth
level.
^{}

Therefore, the image (), corresponding to the background velocity field, can be computed from the measured data () using the summation () and the downward continuation operators () as

(17) - is the image (reflectivity) corresponding to a given
depth level

6/1/1999