# Difference between revisions of "Principles of dip-moveout correction"

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{{NumBlk|:|<math>\Delta t_{DMO} = t_n\left(1-\frac{1}{A}\right).</math>|{{EquationRef|9b}}}} | {{NumBlk|:|<math>\Delta t_{DMO} = t_n\left(1-\frac{1}{A}\right).</math>|{{EquationRef|9b}}}} | ||

− | Finally, as sketched in Figure 5.1-1, the reflection point dispersal Δ = ''NR'' is defined by the distance along the dipping reflector between the normal-incidence points ''N'' and ''R'' associated with midpoints ''y<sub>n</sub>'' and ''y''<sub>0</sub>, respectively. By way of equations ({{EquationNote|E-18}}) and ({{EquationNote|8a}}) it follows that ([[Topics in Dip-Moveout Correction and Prestack Time | + | Finally, as sketched in Figure 5.1-1, the reflection point dispersal Δ = ''NR'' is defined by the distance along the dipping reflector between the normal-incidence points ''N'' and ''R'' associated with midpoints ''y<sub>n</sub>'' and ''y''<sub>0</sub>, respectively. By way of equations ({{EquationNote|E-18}}) and ({{EquationNote|8a}}) it follows that ([[Topics in Dip-Moveout Correction and Prestack Time Migration#E.1 Reflection point dispersal|Section E.1]]) |

{{NumBlk|:|<math>y_0=y_n-\frac{\Delta}{\cos\phi}</math>|{{EquationRef|E-18}}}} | {{NumBlk|:|<math>y_0=y_n-\frac{\Delta}{\cos\phi}</math>|{{EquationRef|E-18}}}} |

## Latest revision as of 12:44, 8 October 2014

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 objective we want to achieve with the combination of normal-moveout and dip-moveout correction is mapping nonzero-offset data to the plane of zero-offset section. Once each common-offset section is mapped to zero-offset, the data can then be migrated either before or after stack using the zero-offset theory for migration as described in migration.

Figure 5.1-1a depicts the nonzero-offset recording geometry associated with a dipping reflector. The nonzero-offset traveltime *t = SRG/v* is measured along the raypath from source *S* to reflection point *R* to receiver *G*, where *v* is the velocity of the medium above the dipping reflector. This arrival time is depicted on the time section in Figure 5.1-1b by point *A* on the trace that coincides with midpoint *y _{n}*. We want to map the amplitude at time

*t*denoted by the sample

*A*on the trace at midpoint

*y*of the common-offset section with offset 2

_{n}*h*to time

*τ*

_{0}denoted by the sample

*C*on the trace at midpoint

*y*

_{0}of the zero-offset section. We achieve this mapping in two steps:

*Normal-moveout correction*that maps the amplitude at time*t*denoted by the sample*A*on the trace at midpoint*y*of the common-offset section with offset 2_{n}*h*to time*t*denoted by the sample_{n}*B*on the same trace at midpoint*y*of the same common-offset section._{n}*Dip-moveout correction*that maps the amplitude at time*t*denoted by the sample_{n}*B*on the trace at midpoint*y*of the moveout-corrected common-offset section with offset 2_{n}*h*to time*τ*_{0}denoted by the sample*C*on the trace at midpoint*y*_{0}of the zero-offset section.

Zero-offset migration then maps the amplitude at time *τ*_{0} denoted by the sample *C* on the trace at midpoint *y*_{0} of the zero-offset section to the amplitude at time *τ* denoted by the sample *D* on the trace at midpoint *y _{m}* of the migrated section. Note that the combination of NMO correction, DMO correction, and zero-offset migration achieves the same objective as direct mapping of the amplitude at time

*t*denoted by the sample

*A*on the trace at midpoint

*y*of the common-offset section with offset 2

_{n}*h*to the amplitude at time

*τ*denoted by the sample

*D*on the trace at midpoint

*y*of the migrated section. This direct mapping procedure is the basis of algorithms for migration before stack (prestack time migration).

_{m}The important point to note is that the normal-moveout correction in step (a) is performed using the velocity of the medium above the dipping reflector.

The NMO equation (3-8) defines the traveltime *t* from source location *S* to the reflection point *R* to the receiver location *G*. This equation, written in prestack data coordinates, is

**(**)

where 2*h* is the offset, *v* is the medium velocity above the reflector, *ϕ* is the reflector dip, and *t*_{0} is the two-way zero-offset time at midpoint location *y _{n}*.

Dip-moveout correction of step (b) is preceded by zero-dip normal-moveout correction of step (a) using the dip-independent velocity *v*:

**(**)

where *t _{n}* is the event time at midpoint

*y*after the NMO correction. Event time

_{n}*t*after the NMO correction and event time

_{n}*t*

_{0}are related as follows (Section E.2)

**(**)

At first glance, equations (**2**) and (**3**) suggest a two-step approach to moveout correction:

- Apply a
*dip-independent*moveout correction using equation (**2**) to map the amplitude at time*t*denoted by the sample*A*on the trace at midpoint*y*of the common-offset section with offset 2_{n}*h*to time*t*denoted by the sample_{n}*B*on the same trace at midpoint*y*of the same common-offset section._{n} - Apply a
*dip-dependent*moveout correction using equation (**3**) to map the amplitude at time*t*denoted by the sample_{n}*B*on the trace at midpoint*y*of the moveout-corrected common-offset section with offset 2_{n}*h*to time*t*_{0}denoted by the sample*B′*on the same trace at midpoint*y*of the same common-offset section._{n}

This two-step moveout correction is equivalent to the one-step moveout correction using equation (**1**) to map event time *t* directly to event time *t*_{0}.

Our goal, however, is to map event time *t* not to *t*_{0} — the two-way zero-offset time associated with midpoint *y _{n}* between source

*S*and receiver

*G*, but to

*τ*

_{0}— the two-way zero-offset time at midpoint location

*y*

_{0}associated with the normal-incidence reflection point

*R*(Figure 5.1-1). The relationships between (

*y*) coordinates of the normal-moveout-corrected data and (

_{n}, t_{n}*y*

_{0},

*τ*

_{0}) coordinates of the dip-moveout-corrected data are given by (Section E.2):

**(**)

and

**(**)

where

**(**)

For completeness, the relationship between event times *t _{n}* and

*t*

_{0}is given by (Section E.2)

**(**)

Note from equation (**5**) that *A* ≥ 1; therefore, *τ*_{0} ≤ *t _{n}* (equation

**4b**) and

*t*

_{0}≥

*t*(equation

_{n}**6**).

Refer to Figure 5.1-1 and note that the normal-moveout correction that precedes the dip-moveout correction maps the amplitude at sample *A* with coordinates (*y _{n}*,

*t*) to sample

*B*with coordinates (

*y*,

_{n}*t*). So,

_{n}*the midpoint coordinate is invariant under NMO correction*. The difference between the input time

*t*and the output time

*t*is defined by

_{n}

**(**)

which can be expressed by way of equation (**2**) as follows

**(**)

where

**(**)

Again, refer to Figure 5.1-1 and note that the dip-moveout correction maps the amplitude at sample *B* with coordinates (*y _{n}*,

*t*) to sample

_{n}*C*with coordinates (

*y*

_{0},

*τ*

_{0}). So,

*the midpoint coordinate is variant under DMO correction*. The lateral excursion associated with the DMO correction is given by

**(**)

which can be expressed by way of equations (**4a**) and (**5**) as

**(**)

The difference between the input time *t _{n}* and the output time

*τ*

_{0}is defined by

**(**)

which can be expressed by way of equations (**4b**) and (**5**) as

**(**)

Finally, as sketched in Figure 5.1-1, the reflection point dispersal Δ = *NR* is defined by the distance along the dipping reflector between the normal-incidence points *N* and *R* associated with midpoints *y _{n}* and

*y*

_{0}, respectively. By way of equations (

**E-18**) and (

**8a**) it follows that (Section E.1)

**(**)

**(**)

Note from equation (**10**) that reflection point dispersal is nill for zero offset, and increases with the square of the offset. Also, the larger the dip and shallower the reflector, the larger the dispersal.

A direct consequence of equation (**10**) is that a reflection event on a CMP gather is associated with more than one reflection point on the reflector. Following DMO correction, reflection-point dispersal is eliminated and, hence, the reflection event is associated with a single reflection point at normal-incidence (point *R* in Figure 5.1-1). While prestack data before DMO correction can be associated with *common midpoints*, and thus sorted into *common-midpoint (CMP) gathers*; after DMO correction, the data can be associated with *common reflection points*, and thus can be considered in the form of *common-reflection-point (CRP) gathers*.

## See also

- Introduction to dip-moveout correction and prestack migration
- Dip-moveout correction in practice
- Prestack time migration
- Migration velocity analysis
- Exercises
- Topics in Dip-Moveout Correction and Prestack Time Migration