Interval Velocity Estimation

A basic daily problems in seismic processing, such as the estimation of interval velocities from RMS velocities, will be solved in this part.

The method used here was first introduced by Clapp et al. (1998). The method builds a velocity model from surface seismology while retaining the null-space. They start from fundamental concepts in Geophysical Estimation by Example Claerbout (1997) and define the simplest interval velocity estimation including the notion of null-space. Generally, Clapp et al. (1998) minimize interval velocities ``wiggliness'' where there are not good quality reflections.

In order to understand the method used in this part it is necessary to make some definitions (for further explanation the reader could refer to Clapp et al. (1998):

$\bold C$

as the matrix of causal integration, a lower triangular matrix of ones.

$\bold D$

as the matrix of causal differentiation, namely, $\bold D=\bold C^{-1}$.

$\bold u$

as a vector whose components range over the vertical traveltime depth $\tau$,and whose component values contain the interval velocity squared $v_{\rm interval}^2$.

$\bold d$

as a data vector whose components range over the vertical travel time depth $\tau$,and whose component values contain the scaled RMS velocity squared $\tau v_{\rm RMS}^2/\Delta \tau$where $\tau /\Delta \tau$ is the index on the time axis.

The theoretical (squared) RMS velocity is defined by  

$$\bold C\bold u \quad = \quad\bold d .$$ (1)

With imperfect data, our data fitting goal is to minimize the residual

$$\bold 0 \quad\approx\quad \bold W \left[ \bold C\bold u - \bold d \right] .$$ (2)

To find the interval velocity where there is no data, we have the ``model damping'' goal to minimize the ``wiggliness'' $\bold p$ of the squared interval velocity $\bold u$

$$\bold 0 \quad\approx\quad \bold D \bold u \quad = \quad\bold p .$$ (3)

These two goals are preconditioned by changing the optimization variable from interval velocity squared $\bold u$ to its wiggliness $\bold p$. Substituting $\bold u$= $\bold{Cp}$ gives the two fitting goals expressed as a function of wiggliness $\bold p$

$$\begin{eqnarray} \bold 0 &\approx& \bold W' \left[ \bold C^2\bold p - \bold d \right] \\ \bold 0 &\approx& \epsilon \bold p .\end{eqnarray}$$ (4) (5)

This method was tested on two synthetic CMP gathers and one real CMP gather from the Gulf of Mexico.