Variable velocity

Seismic Data Analysis

Series	Investigations in Geophysics
Author	Öz Yilmaz
DOI	http://dx.doi.org/10.1190/1.9781560801580
ISBN	ISBN 978-1-56080-094-1
Store	SEG Online Store

The preceding discussion was based on a constant-velocity assumption. To be of practical use, DMO must be applicable to data with velocity gradients. Figure 5.1-17 shows the depth model we will use to investigate DMO correction in the case of vertical velocity variation. The depth model consists of three point scatterers buried beneath the center midpoint (CMP 32) in a medium with a horizontally layered velocity-depth model. Mathematical aspects of the variable-velocity DMO theory is quite involved and we shall only refer to the results of the experiments with the synthetic data associated with the depth model in Figure 5.1-17.

Selected common-offset sections and CMP gathers associated with this subsurface model are shown in Figures 5.1-18a and 5.1-18b. The same processing sequence is followed as that used for the constant-velocity model (Figure 5.1-3). The NMO correction (Figure 5.1-18c) before DMO correction is done using the rms velocity function indicated in Figure 5.1-17. Selected moveout-corrected common-offset sections are shown in Figure 5.1-19a. We shall apply both constant-velocity DMO correction and variable-velocity DMO correction to these data. The corresponding impulse responses are shown in Figures 5.1-19b,c. Consider the common case of velocities increasing with depth in practice. As noted earlier, the higher the velocity the less the action of the DMO operator. Note that at late times the lateral extent of the impulse response of the variable-velocity DMO operator is less than that of the constant-velocity DMO operator. This is equivalent to modifying the offset value for the common-offset section under consideration — making it smaller than it is so as to decrease the action of the DMO operator.

Results of constant-velocity and variable-velocity DMO corrections are shown in Figures 5.1-20 and 5.1-21, respectively. Events on the selected CMP gathers are better flattened with the variable-velocity DMO correction. The corresponding stacked sections accompanied with the zero-offset conventional stacked sections without DMO correction are shown in Figure 5.1-22. The improvement with the depth-variable velocity also is evident on the stacked section (Figure 5.1-22d). Specifically, note that the flanks of the diffraction events are enhanced with the depth-variable velocity, making it resemble much more closely the zero-offset section (Figure 5.1-22a) as compared to the constant-velocity DMO stack (Figure 5.1-22c).

In practice, constant-velocity DMO correction aften yields acceptable results so long as the vertical velocity gradient is reasonably small and does not change rapidly in depth. The constant-velocity DMO correction also has the bonus effect of attenuating coherent linear noise as demonstrated in the next section.

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  Figure 5.1-3  Depth model of six point scatterers buried in a constant-velocity medium. The asterisks indicate the positions of the point scatterers.

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  Figure 5.1-17  Depth model of three point scatterers buried in a vertically varying velocity medium. The asterisks indicate the positions of the point scatterers.

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  Figure 5.1-18  Intermediate results from DMO processing the nonzero-offset synthetic data derived from the depth model in Figure 5.1-17: (a) common-offset sections with offset range from 50 to 1550 m and an increment of 300 m; (b) CMP gathers sorted from the common-offset sections as in (a) at midpoint locations from 32 to 63 as denoted in Figure 5.1-17 with an increment of 3; (c) the CMP gathers as in (b) after NMO correction and muting.

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  Figure 5.1-19  Intermediate results from DMO processing the nonzero-offset synthetic data derived from the depth model in Figure 5.1-17: (a) common-offset sections with offset range from 50 to 1550 m and an increment of 300 m sorted from the NMO-corrected gathers as in Figure 5.1-4c; (b) impulse responses of the constant-velocity DMO operators applied to the common-offset gathers; (c) impulse responses of the variable-velocity DMO operators applied to the common-offset gathers.

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  Figure 5.1-20  Intermediate results from DMO processing the nonzero-offset synthetic data derived from the depth model in Figure 5.1-17: (a) common-offset sections as in Figure 5.1-19a after the constant-velocity DMO correction; (b) CMP gathers sorted from the common-offset sections as in (a) at midpoint locations from 32 to 63 as denoted in Figure 5.1-17 with an increment of 3.

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  Figure 5.1-21  Intermediate results from DMO processing the nonzero-offset synthetic data derived from the depth model in Figure 5.1-17: (a) common-offset sections as in Figure 5.1-19a after the variable-velocity DMO correction; (b) CMP gathers sorted from the common-offset sections as in (a) at midpoint locations from 32 to 63 as denoted in Figure 5.1-17 with an increment of 3.

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  Figure 5.1-22  (a) Zero-offset section associated with the depth model in Figure 5.1-17; (b) stack derived from the CMP gathers as in Figure 5.1-18c; (c) DMO stack derived from the CMP gathers as in Figure 5.1-20b; (d) DMO stack derived from the CMP gathers as in Figure 5.1-21b.

See also

• Prestack partial migration
• Frequency-wavenumber DMO correction
• Log-stretch DMO correction
• Integral DMO correction
• Velocity errors
• Turning-wave migration

External links

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