Inverse NMO stack operator

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

 

$$t = \sqrt{\tau ^2 + \frac{x^2}{v^2(\tau)}},$$ (1)

Here, t is the nonzero-offset traveltime, $\tau$ zero-offset traveltime. x is the shot-geophone offset and $v(\tau)$ 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

 

$${\bf d} = {\bf L} {\bf m}.$$ (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

 

$${\bf 0} \approx {\bf r} = {\bf L} {\bf m} - {\bf d}.$$ (3)