Note that given , , and , the incident
and reflected rays and the event element *s*,*p*_{s},*r*,*p*_{r},*t* are
completely determined: therefore the latter are functions of
, , and .

The angle transform of a data set is

where is an appropriate weighting function
and *s*,*r*,*t* are also functions of .

The principle of stationary phase shows that an event in a single
angle panel, i.e. a position and an *angle domain
dip vector* , arise when incident and reflected rays meet
at and are bisected by ; these rays determine
once again an event element *s*,*p*_{s},*r*,*p*_{r},*t*, *and this
event must have been present in the data for the event in question
to be present in the angle domain.* Of course the event element
*s*,*p*_{s},*r*,*p*_{r},*t* completely determines the rays in the subsurface
carrying the energy of the event. We assume the *Traveltime
Injectivity Condition* ten Kroode et al. (1999): a pair
of rays and a total (two-way) time determines at most one
reflector element. In that case, the event in the angle domain
is compatible with at most one reflector element ().

Note the contrast with the constant offset domain as described in the preceding section where an event element in the data could correspond kinematically to more than one reflector element.

The velocity field used to generate the rays used in the formation of the angle transform does not necessarily need to be the same as the velocity field which gave rise to the moveout in the data - which is fortunate, as we don't know the latter at the outset of the migration/velocity analysis process, and have only an approximation of it at the end! When the two velocity fields are different, the angle transform events will not necessarily match the reflectors in the Earth: the two will differ by a residual migration. When the two velocity fields are the same, the image is perfect, i.e. .

4/20/1999