Time migration velocity analysis by velocity continuation |

where index corresponds to the time dimension, index corresponds to the velocity dimension, is a vector along the midpoint direction, is the identity matrix, represents the finite-difference approximation to the second-derivative operator in midpoint, and .

In the two-dimensional case, equation 5 reduces to a tridiagonal system of linear equations, which can be easily inverted. In 3-D, a straightforward extension can be obtained by using either directional splitting or helical schemes (Rickett et al., 1998). The direction of stable propagation is either forward in velocity and backward in time or backward in velocity and forward in time as shown in Figure 1.

vlcfds
Finite-difference scheme for the
velocity continuation equation. A stable propagation is either forward
in velocity and backward in time (a) or backward in velocity
and forward in time (b).
Figure 1. | |
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In order to test the performance of the finite-difference velocity continuation method, I use a simple synthetic model from Claerbout (1995). The reflectivity model is shown in Figure 2. It contains several features that challenge the migration performance: dipping beds, unconformity, syncline, anticline, and fault. The velocity is taken to be constant .

mod
Synthetic model for testing
finite-difference migration by velocity continuation.
Figure 2. | |
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Figures 3-6(b) compare invertability of different migration methods. In all cases, constant-velocity modeling (demigration) was followed by migration with the correct velocity. Figures 3 and 4 show the results of modeling and migration with the Kirchhoff (Schneider, 1978) and - (Stolt, 1978) methods, respectively. These figures should be compared with Figure 5, showing the analogous result of the finite-difference velocity continuation. The comparison reveals a remarkable invertability of velocity continuation, which reconstructs accurately the main features and frequency content of the model. Since the forward operators were different for different migrations, this comparison did not test the migration properties themselves. For such a test, I compare the results of the Kirchhoff and velocity-continuation migrations after Stolt modeling. The result of velocity continuation, shown in Figure 6, is noticeably more accurate than that of the Kirchhoff method.

vlckaa
Result of modeling and migration with the
Kirchhoff method. Top left plot shows the reconstructed model. Top
right plot compares the average amplitude spectrum of the true model
with that of the reconstructed image. Bottom left is the
reconstruction error. Bottom right is the absolute error in the
spectrum.
Figure 3. |
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vlcsto
Result of modeling and migration with the
Stolt method. Top left plot shows the reconstructed model. Top
right plot compares the average amplitude spectrum of the true model
with that of the reconstructed image. Bottom left is the
reconstruction error. Bottom right is the absolute error in the
spectrum.
Figure 4. |
---|

vlcvel
Result of modeling and
migration with the finite-difference velocity continuation. Top left plot
shows the reconstructed model. Top right plot compares the average amplitude
spectrum of the true model with that of the reconstructed image. Bottom left
is the reconstruction error. Bottom right is the absolute error in the
spectrum.
Figure 5. |
---|

vlcstk,vlcstv
(a) Modeling with Stolt method, migration with the Kirchhoff
method. (b) Modeling with Stolt method, migration with the
finite-difference velocity continuation. Left plots show the
reconstructed models. Right plots show the reconstruction errors.
Figure 6. |
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These tests confirm that finite-difference velocity continuation is an attractive migration method. It possesses remarkable invertability properties, which may be useful in applications that require inversion. While the traditional migration methods transform the data between two completely different domains (data-space and image-space), velocity continuation accomplishes the same transformation by propagating the data in the extended domain along the velocity direction. Inverse propagation restores the original data. According to Li (1986), the computational speed of this method compares favorably with that of Stolt migration. The advantage is apparent for cascaded migration or migration with multiple velocity models. In these cases, the cost of Stolt migration increases in direct proportion to the number of velocity models, while the cost of velocity continuation stays the same.

Time migration velocity analysis by velocity continuation |

2013-03-03