The inverse NMO stack operator is based on the hyperbolic moveout equation

(1) |

Here, *t* is the nonzero-offset traveltime, zero-offset traveltime.
*x* is the shot-geophone offset and is the RMS
depth-variable velocity.

The model **m** is the stack trace, while the CMP gather is the data
**d**. Correspondingly, **m** is located
in the *model space* and **d** in the *data space*. The hyperbolic
moveout
operator **L** links the model **m** with the data **d** by

(2) |

Equation (2) is a set of simultaneous equations, which is typically
over-determined. It is very hard to find the
exact solution **m** for known data **d**. Therefore, we have to
reformulate this
problem as an optimization problem and try to find the optimal solution in
some sense, such as least square. The conjugate-gradient algorithm is used
to search for the optimal solution by minimizing the residual

(3) |

11/11/1997