BASIC IDEA

Assuming a v(z) earth with flat reflectors, we can define the theoretical RMS velocities by

$$\bold C\bold u \quad = \quad\bold d .$$ (71)

where

$\bold C$

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

$\bold D$

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

$\bold u$

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

$\bold d$

is 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.

Start from a CMP gather q(t,i) moveout corrected with velocity v. A good starting guess for our RMS velocity function is the maximum ``instantaneous stack energy''

$${\rm stack(t,v)} = \sum_{i=0}^n q(t,i) .$$ (72)

An alternate starting guess for our RMS velocity function is the conventional one of maximum semblance

$${\rm semb(t,v)} = {{1\over n}\sum_{i=1}^n q(t,i) \over \sqrt{ {1\over n} \sum_{i=1}^n q(t,i)^2}}.$$ (73)

In addition to the RMS velocities we need a diagonal weighting matrix $\bold W$,again found from stack energy or semblance, that differentiates between RMS velocities which we have confidence in (at reflectors) and ones that are more a function of noise in the data. Our data fitting goal is to minimize the residual

$$\bold 0 \quad\approx\quad \bold W \left[ \bold C\bold u - \bold d \right] .$$ (74)

Because we are multiplying our RMS function by $\tau$, we must must make a slight change in our weighting function to give early times approximately the same priority as later times.

$$\bold{W'} = {\Delta \tau \over \tau} \bold{W}$$ (75)

To find the interval velocity where there is no data (where the stack power theoretically vanishes) we have the ``model damping'' goal to minimize the wiggliness $\bold p$of the squared interval velocity $\bold u$ where $\bold p$ equals

$$\bold 0 \quad\approx\quad \bold D \bold u \quad = \quad\bold p .$$ (76)

To speed convergence we ``precondition'' these goals () by changing the optimization variable from interval velocity squared $\bold u$ to its wiggliness $\bold p$.Substituting $\bold u=\bold C\bold p$ gives the two goals expressed as a function of wiggliness $\bold p$.

$$\begin{eqnarray} \bold 0 &\approx& \bold W' \left[ \bold C^2\bold p - \bold d \right] \\ \bold 0 &\approx& \epsilon \bold p .\end{eqnarray}$$ (77) (78)