Kinematics of Multiarrivals in the Angle Domain

We confine our discussion to the 2D case; the 3D case is similar, provided that complete surface coverage is available. Specular reflection connects a reflector element, consisting of a subsurface position ${\bf x}$ containing the midpoint location ${\bf m}$ and the depth z and dip vector ${\bf \nu}$ representing the reflector normal at that position with an event element s,p_x,r,p_r,t consisting of source and receiver positions s and r, source and receiver horizontal slownesses p_s and p_r, and two way time t. The connection via incident and reflected rays is depicted in Figure 4, which also shows the opening angle $\theta$.

Note that given ${\bf x}$, $\nu$, and $\theta$, 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 ${\bf x}$, $\nu$, and $\theta$.

The angle transform of a data set $\{d(s,r,t)\}$ is

$$a({\bf x},\theta) = \int \,d\nu w(\nu,\theta,{\bf x}) d(s,r,t)$$

where $w(\nu,\theta,{\bf x})$ is an appropriate weighting function and s,r,t are also functions of $\nu,\theta,{\bf x}$.

The principle of stationary phase shows that an event in a single angle panel, i.e. a position ${\bf y}$ and an angle domain dip vector $\eta$, arise when incident and reflected rays meet at ${\bf y}$ and are bisected by $\eta$; 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 ($\bf x, \nu$).

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. $({\bf x},\nu) = ({\bf y},\eta)$.