Slant stack is a transformation of the offset axis. It is like steering a beam of seismic waves. I believe I introduced the term slant stack (Schultz and Claerbout ) as a part of a migration method to be described next. I certainly didn't invent the slant-stack concept! It has a long history in exploration seismology going back to Professor Rieber in the 1930s and to Professor Riabinkin in the Soviet Union. Mathematically, the slant-stack concept is found in the Radon  transformation.
The slant-stack idea resembles the Snell trace method
of organizing data around emergent angle.
The Snell trace idea selects data based
on a hypothetical velocity predicting the local stepout p= dt/dx.
Slant stack does not
the stepout, but
it by filtering.
Thus slant stack does its job correctly whether or not the
velocity is known.
When the velocity of the medium is known,
slant stack enables immediate downward continuation
even when mixed apparent velocities are present
as with diffractions and multiple reflections.