Consider a set of image rays coming to the surface.
Suppose we are tracing them all backwards in time
together with the quantities
and
.
Let us eliminate the unknown velocity
in system 9
using equation 10.
Moreover, let us eliminate the differentiation
in
using the definition
of
and rewrite it in the time-domain
coordinates
.
Indeed,
, hence
.
Therefore, system 9 becomes
(13)
Eliminating
in system 13, we get
the following partial differential equation (PDE) for
(14)
The initial conditions are
,
.
Equation 14 simplifies in terms of the negative
reciprocal of
. Introduce
. Then
equation 14 becomes