On a single random walk, the set of Ns 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 vi 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 Ns 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 Nw 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 vi*, and its associated rms curve is the optimal stacking velocity trajectory .