The determination of earth velocity from head waves is an old subject in seismology. Travel times are hard to pick out on noisy data. On gathers you have the extra information not on a section that downward continuation focuses energy on zero offset. The focus is not a featureless point. Take original data to consist of a head wave only, with no reflection.
Downward continuation yields a focus at zero offset.
Or the entire calculation could be done in depth z directly instead of in travel-time depth . Clayton and McMechan actually do the downward continuation and the slant stack in the opposite order. They slant stack first and then downward continue. In principle these processes can be done in either order. Remember that we are bootstrapping to the correct earth velocity.