Random walks

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 $\bar{v}_s^*$.