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# BASIC IDEA

Assuming a v(z) earth with flat reflectors, we can define the theoretical RMS velocities by
 (1)
where
is the matrix of causal integration, a lower triangular matrix of ones.
is the matrix of causal differentiation, namely, .
is a vector whose components range over the vertical traveltime depth ,and whose component values contain the interval velocity squared .
is a data vector whose components range over the vertical travel time depth ,and whose component values contain the scaled RMS velocity squared where 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''
 (2)
An alternate starting guess for our RMS velocity function is the conventional one of maximum semblance
 (3)
In addition to the RMS velocities we need a diagonal weighting matrix ,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
 (4)
Because we are multiplying our RMS function by , we must must make a slight change in our weighting function to give early times approximately the same priority as later times.
 (5)
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 of the squared interval velocity where equals
 (6)

To speed convergence we precondition'' these goals Fomel et al. (1997) by changing the optimization variable from interval velocity squared to its wiggliness .Substituting gives the two goals expressed as a function of wiggliness .
 (7) (8)

Next: SIMPLE EXAMPLE Up: Velocity estimation: Clapp, et Previous: INTRODUCTION
Stanford Exploration Project
7/6/1998