3#3 | (1) |
where 4#4 is the reflection angle, 5#5 is the geologic dip, and 6#6 is the velocity function in depth and CMP location.
The inversion procedure used in this paper can be expressed as fitting goals as follows:
7#7 | (2) | |
In the data fitting goal, 9#9 is the input data and 10#10 is the image obtained through inversion. 11#11 is a linear operator, in this case it is the adjoint of the angle-domain wave-equation migration scheme summarized above and explained thoroughly by (). In the model styling goal, 8#8 is, as has already been mentioned, a regularization operator. 12#12 controls the strength of the model styling.
Unfortunately, the inversion process described by Equation can take many iterations to produce a satisfactory result. We can reduce the necessary number of iterations by making the problem a preconditioned one. We use the preconditioning transformation 13#13 (, ) to give us these fitting goals:
14#14 | (3) | |
The question now is what the preconditioning operator 15#15 is. We have chosen to make this operator from steering filters (, ) which tend to create dips along chosen reflectors. This paper includes results from two different preconditioning schemes. One is called the 1-D preconditioning scheme and simply acts horizontally along the offset ray parameter axis. The 2-D scheme acts along chosen dips on the CMP axis and horizontally along the offset ray parameter axis. To construct the preconditioning operator along the CMP axis, we pick ``reflectors'' that represent the dip we believe should be in a certain location, then interpolate the dips between the picked reflectors to cover the whole plane.