Difference between revisions of "Maximum dip to migrate"

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During [[migration]], we can specify the maximum dip we want migrated in the section. This may be useful when we need to suppress the steeply dipping coherent noise. Figure 4.2-8 shows [[migration|migrations]] of the dipping events with four different maximum allowable dips. For a 4 ms/trace dip limit, events with dips greater than this value are suppressed. Similarly, for an 8 ms/trace dip value, events with dips greater than this value are suppressed. When the dip value is 12 ms/trace, no suppression occurs, since all events in the input section have dips less than this value. Limiting the dip parameter is a way to reduce computational cost, since it is related to [[aperture width]] (equation {{EquationNote|1}}), which determines the cost.
 
During [[migration]], we can specify the maximum dip we want migrated in the section. This may be useful when we need to suppress the steeply dipping coherent noise. Figure 4.2-8 shows [[migration|migrations]] of the dipping events with four different maximum allowable dips. For a 4 ms/trace dip limit, events with dips greater than this value are suppressed. Similarly, for an 8 ms/trace dip value, events with dips greater than this value are suppressed. When the dip value is 12 ms/trace, no suppression occurs, since all events in the input section have dips less than this value. Limiting the dip parameter is a way to reduce computational cost, since it is related to [[aperture width]] (equation {{EquationNote|1}}), which determines the cost.
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{{NumBlk|:|<math>d_x=\frac{v^2t}{4}\frac{\Delta t}{\Delta x},</math>|{{EquationRef|1}}}}
  
 
From Figure 4.2-1, note that the [[Kirchhoff migration]] impulse response can be limited to various maximum dips. The smaller the maximum allowable dip, the smaller the aperture. This combination of maximum [[aperture width]] and maximum dip limit determines the actual effective [[aperture width]] used in [[migration]]. In particular, diffraction hyperbolas along which summation is done are truncated beyond the specified maximum dip limit.
 
From Figure 4.2-1, note that the [[Kirchhoff migration]] impulse response can be limited to various maximum dips. The smaller the maximum allowable dip, the smaller the aperture. This combination of maximum [[aperture width]] and maximum dip limit determines the actual effective [[aperture width]] used in [[migration]]. In particular, diffraction hyperbolas along which summation is done are truncated beyond the specified maximum dip limit.
  
 
A [[field data examples|field data example]] of testing the maximum dip parameter is shown in Figure 4.2-9. Some steep dips are lost on the section that corresponds to the 2 ms/trace maximum allowable dip. The 8 ms/trace dip appears to be optimum. The maximum dip parameter must be chosen carefully so that the steep dips of interest in the input section are preserved. Finally, dip value can be changed spatially and in time; however, practical implementation can be cumbersome.
 
A [[field data examples|field data example]] of testing the maximum dip parameter is shown in Figure 4.2-9. Some steep dips are lost on the section that corresponds to the 2 ms/trace maximum allowable dip. The 8 ms/trace dip appears to be optimum. The maximum dip parameter must be chosen carefully so that the steep dips of interest in the input section are preserved. Finally, dip value can be changed spatially and in time; however, practical implementation can be cumbersome.
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<gallery>
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file:ch04_fig2-1.png|{{figure number|4.2-1}} [[Migration]] can be confined to a range of dips present on a seismic section. The impulse response for the dip-limited [[migration]] operator is a truncated semicircle. Dip angle ''θ'' is measured from the vertical axis.
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file:ch04_fig2-8.png|{{figure number|4.2-8}} Tests for maximum dip to migrate in [[Kirchhoff migration]]: (a) a zero-offset section that contains a diffraction hyperbola with 2500-m/s velocity, (b) desired [[migration]]; [[Kirchhoff migration]] using (c) 4-ms/trace, (d) 8-ms/trace, (e) 12-ms/trace, and (f) 24-ms/trace maximum dip.
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file:ch04_fig2-9.png|{{figure number|4.2-9}} Tests for maximum dip to migrate in [[Kirchhoff migration]]: A low value for maximum dip to migrate can be hazardous. All dips of interest must be preserved during [[migration]].
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</gallery>
  
 
==Frequency-wavenumber migration in practice==
 
==Frequency-wavenumber migration in practice==
The phase-shift method of [[migration]] ([[migration principles]] and Section D.7) allows vertical variations in velocity and is accurate for up to dips of 90 degrees. Figure 4.5-1 shows the impulse response of the phase-shift algorithm. Clearly, for a constant-velocity medium, this response is equivalent to that of the [[Stolt migration]]. The impulse response shown in Figure 4.5-1 is considered to be the desired impulse response for 2-D zero-offset [[migration]], and as such, responses of all [[migration algorithms]] discussed in this chapter are benchmarked against it.
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The phase-shift method of [[migration]] ([[migration principles]] and [[Mathematical foundation of migration#D.7 Frequency-wavenumber migration|Section D.7]]) allows vertical variations in velocity and is accurate for up to dips of 90 degrees. Figure 4.5-1 shows the impulse response of the phase-shift algorithm. Clearly, for a constant-velocity medium, this response is equivalent to that of the [[Stolt migration]]. The impulse response shown in Figure 4.5-1 is considered to be the desired impulse response for 2-D zero-offset [[migration]], and as such, responses of all [[migration algorithms]] discussed in this chapter are benchmarked against it.
  
<gallery>file:ch04_fig4-29.png|{{figure number|4.4-29}} From top to bottom, desired [[migration]] as in Figure 4.2-15b and frequency-space explicit [[migration]] with 30-degree, 50-degree, and 70-degree accuracy, and using interval velocities derived from 100 percent of the rms velocities. The input CMP stack is shown in Figure 4.2-15a.
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<gallery>
file:ch04_fig4-30.png|{{figure number|4.4-30}} From top to bottom, desired [[migration]] as in Figure 4.2-15b and tests for [[velocity errors]] in frequency-space explicit [[migration]] with 30-degree, 50-degree, and 70-degree accuracy, and using interval velocities derived from 90 percent of rms velocities. The input CMP stack is shown in Figure 4.2-15a.
 
file:ch04_fig4-31.png|{{figure number|4.4-31}} From top to bottom, desired [[migration]] as in Figure 4.2-15b and tests for [[velocity errors]] in frequency-space explicit [[migration]] with 30-degree, 50-degree, and 70-degree accuracy, and using interval velocities derived from 110 percent of rms velocities. The input CMP stack is shown in Figure 4.2-15a.
 
file:ch04_fig4-32.png|{{figure number|4.4-32}} Summary of the results of [[migration]] of a zero-offset section that contains a diffraction hyperbola with 2500-m/s velocity as in (a): (b) [[Kirchhoff migration]]; frequency-space implicit [[finite-difference migration]] with (c) 65-degree, (d) 80-degree, (e) 87-degree, and (f) 90-degree accuracy; frequency-space explicit [[migration]] with (g) 30-degree, (h) 50-degree, and (i) 70-degree accuracy; and (j) frequency-wavenumber [[phase-shift migration]].
 
file:ch04_fig4-33.png|{{figure number|4.4-33}} Summary of the results of [[migration]] of a zero-offset section that contains a set of dipping events with 3500-m/s velocity as in (a): (b) [[Kirchhoff migration]]; frequency-space implicit [[finite-difference migration]] with (c) 65-degree, (d) 80-degree, (e) 87-degree, and (f) 90-degree accuracy; frequency-space explicit [[migration]] with (g) 30-degree, (h) 50-degree, and (i) 70-degree accuracy; and (j) frequency-wavenumber [[phase-shift migration]].
 
 
file:ch04_fig5-1.png|{{figure number|4.5-1}} Impulse response of [[phase-shift migration]] has a semicircular shape.
 
file:ch04_fig5-1.png|{{figure number|4.5-1}} Impulse response of [[phase-shift migration]] has a semicircular shape.
 
file:ch04_fig5-2.png|{{figure number|4.5-2}} The impulse response of the ''f − k'' [[migration]] operator has a truncated semicircular shape when a maximum dip limit is imposed. For comparison, the desired response shape has been superimposed on the ''f − k'' responses.
 
file:ch04_fig5-2.png|{{figure number|4.5-2}} The impulse response of the ''f − k'' [[migration]] operator has a truncated semicircular shape when a maximum dip limit is imposed. For comparison, the desired response shape has been superimposed on the ''f − k'' responses.
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The dip-filtering action caused by imposing a dip limit on the impulse response also is visible on the results shown in Figure 4.5-3. Note that steep dips greater than the specified maximum dip to migrate have been annihilated. On the [[field data examples|field data example]] shown in Figure 4.5-4, severe dip filtering action of the 2 ms/trace maximum dip has caused smearing and eliminated virtually all of the signal contained in the section.
 
The dip-filtering action caused by imposing a dip limit on the impulse response also is visible on the results shown in Figure 4.5-3. Note that steep dips greater than the specified maximum dip to migrate have been annihilated. On the [[field data examples|field data example]] shown in Figure 4.5-4, severe dip filtering action of the 2 ms/trace maximum dip has caused smearing and eliminated virtually all of the signal contained in the section.
  
<gallery>file:ch04_fig5-3.png|{{figure number|4.5-3}} Tests for maximum dip to migrate in [[phase-shift migration]]: (a) a zero-offset section that contains dipping events with 3500-m/s velocity, (b) desired [[migration]]; [[phase-shift migration|phase-shift migrations]] using (c) 2-ms/trace, (d) 4-ms/trace, (e) 8-ms/trace, and (f) 16-ms/trace maximum dip limit.
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<gallery>
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file:ch04_fig5-3.png|{{figure number|4.5-3}} Tests for maximum dip to migrate in [[phase-shift migration]]: (a) a zero-offset section that contains dipping events with 3500-m/s velocity, (b) desired [[migration]]; [[phase-shift migration|phase-shift migrations]] using (c) 2-ms/trace, (d) 4-ms/trace, (e) 8-ms/trace, and (f) 16-ms/trace maximum dip limit.
 
file:ch04_fig5-4.png|{{figure number|4.5-4}} Tests for maximum dip to migrate in [[phase-shift migration]]: A low value for maximum dip to migrate can be hazardous. All dips of interest must be preserved during [[migration]].
 
file:ch04_fig5-4.png|{{figure number|4.5-4}} Tests for maximum dip to migrate in [[phase-shift migration]]: A low value for maximum dip to migrate can be hazardous. All dips of interest must be preserved during [[migration]].
 
</gallery>
 
</gallery>

Latest revision as of 10:59, 8 October 2014

Seismic Data Analysis
Seismic-data-analysis.jpg
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


Kirchhoff migration in practice

During migration, we can specify the maximum dip we want migrated in the section. This may be useful when we need to suppress the steeply dipping coherent noise. Figure 4.2-8 shows migrations of the dipping events with four different maximum allowable dips. For a 4 ms/trace dip limit, events with dips greater than this value are suppressed. Similarly, for an 8 ms/trace dip value, events with dips greater than this value are suppressed. When the dip value is 12 ms/trace, no suppression occurs, since all events in the input section have dips less than this value. Limiting the dip parameter is a way to reduce computational cost, since it is related to aperture width (equation 1), which determines the cost.


(1)

From Figure 4.2-1, note that the Kirchhoff migration impulse response can be limited to various maximum dips. The smaller the maximum allowable dip, the smaller the aperture. This combination of maximum aperture width and maximum dip limit determines the actual effective aperture width used in migration. In particular, diffraction hyperbolas along which summation is done are truncated beyond the specified maximum dip limit.

A field data example of testing the maximum dip parameter is shown in Figure 4.2-9. Some steep dips are lost on the section that corresponds to the 2 ms/trace maximum allowable dip. The 8 ms/trace dip appears to be optimum. The maximum dip parameter must be chosen carefully so that the steep dips of interest in the input section are preserved. Finally, dip value can be changed spatially and in time; however, practical implementation can be cumbersome.

Frequency-wavenumber migration in practice

The phase-shift method of migration (migration principles and Section D.7) allows vertical variations in velocity and is accurate for up to dips of 90 degrees. Figure 4.5-1 shows the impulse response of the phase-shift algorithm. Clearly, for a constant-velocity medium, this response is equivalent to that of the Stolt migration. The impulse response shown in Figure 4.5-1 is considered to be the desired impulse response for 2-D zero-offset migration, and as such, responses of all migration algorithms discussed in this chapter are benchmarked against it.

As with the Kirchhoff summation method, migration with the phase-shift method can be limited to smaller dips by truncating the semicircular wavefront (Figure 4.5-2). This dip filtering capability is useful in rejecting coherent noise from the stacked section while migrating the data. If migration is constrained to small dip values, then the steeply dipping reflectors may be filtered out unintentionally. Edge effects also are pronounced when a very narrow range of dips is passed. Note the linear streaks on the impulse response with a dip limit of 2 ms/trace (Figure 4.5-2).

The dip-filtering action caused by imposing a dip limit on the impulse response also is visible on the results shown in Figure 4.5-3. Note that steep dips greater than the specified maximum dip to migrate have been annihilated. On the field data example shown in Figure 4.5-4, severe dip filtering action of the 2 ms/trace maximum dip has caused smearing and eliminated virtually all of the signal contained in the section.

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Maximum dip to migrate
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