Velocity analysis is posed in this paper as an optimization problem. The paper begins by defining an objective function, which is the measure of how well a velocity model explains the recorded data. The objective function I chose is the power in a common-midpoint stack. This stack is formed by a summation along offset-dependent trajectories, determined by the velocity model. I propose two forms of this objective function: one with trajectories that honor the calculated traveltimes, the other with hyperbolic trajectories. Having defined an objective function, I then look more carefully at its calculation.
The calculation is divided into two steps. The first relates a velocity model to the corresponding traveltimes and stacking slownesses. This step can be easily linearized, so it can be rapidly calculated. The second step uses these traveltimes or stacking slownesses to define a summation trajectory through the data. In this step the traveltime and stacking slowness approaches diverge: the stacking slownesses lead to much simpler and faster algorithms than do the traveltimes.
Then, I develop a steepest-descent algorithm, based on stacking slownesses, that does not require any velocity picking. I apply this algorithm to a field dataset that has a large, near-surface, low-velocity anomaly.
STANFORD