Plane-wave migration in tilted coordinates |

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tiltcoordinate1
Coordinate system rotation:
are conventional vertical Cartesian coordinates,
are tilted coordinates, represents the source location,
and
represent receiver locations. The source and receivers are on regular grids in vertical Cartesian coordinates.
Figure 1. | |
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As in the vertical Cartesian coordinates, the up-going and down-going one-way wave equations can be obtained by splitting the acoustic wave equation
in the tilted coordinate system
:

The extrapolation direction of equations 12 and 13 parallels the axis, which is from the vertical direction. Figure 1 illustrates the coordinate transformation, where are vertical Cartesian coordinates and are tilted coordinates, represents the source location and represent the corresponding receiver locations. The accuracy of the one-way wavefield extrapolators is still very important for wavefield extrapolation in tilted coordinates. The more accurately we design the wavefield extrapolator, the less sensitive the migration is to the coordinates. With an extrapolator that is not very accurate, such as the equation, waves well handled in one coordinate system are not handled in one that is slightly rotated. In contrast, with an accurate extrapolator, waves can be handled in both tilted coordinate systems. Since one-way wave equations in tilted coordinates are exactly the same as those in vertical Cartesian coordinates, all the methods used to improve the accuracy in the conventional Cartesian coordinates still work in tilted coordinates.

tiltcoordinate2
Source and receivers in grids of a tilted coordinate
system: are conventional vertical Cartesian coordinates,
are tilted coordinates, represents the source location, and
represent receiver locations.
Neither source nor receiver locations are on regular grids in the tilted coordinate system.
Their wavefield values must be interpolated onto regular grids around the slanted line in tilted coordinates. The wavefield on is interpolated onto the grids a, b, c, and d.
Figure 2. | |
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To extrapolate wavefields in a tilted coordinate system, it is necessary to interpolate
the surface dataset, velocity model and image between the coordinate systems and migrate the dataset on a slanted line in implementation.
In Figure 1, the source and receivers are on regular grids in conventional Cartesian coordinates. Figure 2 shows the source and receivers in
meshes in the tilted coordinates
. Source and receivers are on an inclined line defined by the equation

Figure 3 shows a velocity model revised from the Sigsbee 2A model (Sava, 2006). The sediment part of the model is extended vertically and horizontally to receive the overturned waves from the overhanging salt flank at the surface. The rays correspond to the overturned waves from the overhanging flanks on opposite sides of the salt. Figure 4 shows the model and rays in a tilted coordinate system with a tilting angle of . Figures 3 and 4 illustrate that the waves that overturn in vertical Cartesian coordinates do not overturn in a tilted coordinate system with a well-chosen tilting direction.

zigvelwithraycart
A velocity model revised from Sigsbee 2A. The sediment parts of the model are extended to
allow the overturned waves from the overhanging salt flanks to be received at the surface.
The rays represent the overturned waves
from the overhanging salt flank.
Figure 3. | |
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Plane-wave migration in tilted coordinates |

2007-09-18