On a single random walk, the set of *N*_{s} footsteps explore the
model space in the vicinity of the initial interval velocity model by making
random perturbations to that initial model. For a single footstep,
all the interval velocities *v*_{i} are perturbed simultaneously. I use
a random uniform distribution with a standard deviation of about 10% of
the interval velocity value at that depth (time). The next step is
performed by returning to the initial model and repeating the process.
This means that all steps *for a single walk* are within about 20%
of the initial interval velocity model. I tried letting the steps
concatenate such that each step takes you farther away from the initial
model, but this seemed to be a waste of time without some directional
guidance in the step direction. I use multiple walks to achieve this goal.

At the end of *N*_{s} steps, the best fitting interval velocity model is
retained as the initial model for the next random walk. Then the footstep
procedure is repeated until *N*_{w} walks have been completed, or
some measure of ``convergence'' has been achieved. The final best fit model
at the end all random walks is the Monte Carlo optimal interval velocity
*v*_{i}^{*}, and its associated rms curve is the optimal stacking velocity
trajectory .

11/17/1997