A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations |

In Figure 1, we illustrate image rays in 2-D and a *forward* mapping from depth coordinate
to time coordinate (Hubral, 1977). Throughout this paper stands for one-way time,
while time migration usually produces images in two-way time (Yilmaz, 2001), i.e. . Under the
assumption of no caustics, for each subsurface location we consider the image ray from to the
surface, where it emerges at point , with slowness vector normal to the surface. Here is the
location of the image ray at the earth surface and is a scalar. is the traveltime along this image ray between
and . The forward mapping and can be done with a knowledge of interval
velocity . A unique *inverse* mapping and also exists that enables
us to directly map the time-migrated image to depth.

imageray
Image ray in (left) the depth-domain
can be traced with a source at location with slowness vector normal
to the earth surface. Each depth coordinate along this image ray
is then mapped into (right) the time coordinate by using its
corresponding traveltime and source location .
Figure 1. |
---|

The counterpart for in the time-domain is the time-migration velocity , which is
commonly estimated in prestack Kirchhoff time migration (Yilmaz, 2001; Fomel, 2003). In a medium,
corresponds to the root-mean-square (RMS) velocity:

followed by performing a simple conversion from to according to .

In equations 1 and 2, there is no dependency on or , because image rays are
all vertical. For an arbitrary medium, image rays will bend as they travel through the medium (Larner et al., 1981).
Therefore, in general, is a function of both and
and no longer satisfies the simple expression 1, which limits the applicability of the Dix
formula. Cameron et al. (2007) proved that the seismic velocity and the Dix velocity in this case are connected through
geometrical spreading of image rays:

Combining equations 3 and 4 results in

Finally, since remains constant along each image ray, there is an orthogonality condition between gradients of and (Cameron et al., 2007):

Equations 5, 6 and 7 form a system of nonlinear PDEs for and
. The input is , estimated from by equation 3. Solving a
boundary-value problem for the PDEs should provide , as well as and .
Because seismic acquisitions are
limited to the earth surface, we can only use boundary conditions at the surface. For a rectangular Cartesian
domain with being the surface, the boundary conditions are

In Appendix A, we show that the time-to-depth conversion is an ill-posed problem because it requires solving a Cauchy-type problem for an elliptic PDE. The missing boundary conditions on sides of the computational domain other than those in equation 8 can induce numerical instability when extrapolating in (or, equivalently, ). Instead, we consider an alternative formulation of the problem in the following section.

A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations |

2015-03-25