Principles of dip-moveout correction
Series | Investigations in Geophysics |
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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_{n} of the common-offset section with offset 2h 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_{n} of the common-offset section with offset 2h to time t_{n} denoted by the sample B on the same trace at midpoint y_{n} of the same common-offset section.
- Dip-moveout correction that maps the amplitude at time t_{n} denoted by the sample B on the trace at midpoint y_{n} of the moveout-corrected common-offset section with offset 2h 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_{n} of the common-offset section with offset 2h to the amplitude at time τ denoted by the sample D on the trace at midpoint y_{m} of the migrated section. This direct mapping procedure is the basis of algorithms for migration before stack (prestack time migration).
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 2h 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_{n} after the NMO correction. Event time t_{n} after the NMO correction and event time 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_{n} of the common-offset section with offset 2h to time t_{n} denoted by the sample B on the same trace at midpoint y_{n} of the same common-offset section.
- Apply a dip-dependent moveout correction using equation (3) to map the amplitude at time t_{n} denoted by the sample B on the trace at midpoint y_{n} of the moveout-corrected common-offset section with offset 2h to time t_{0} denoted by the sample B′ on the same trace at midpoint y_{n} of the same common-offset section.
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_{n}, t_{n}) coordinates of the normal-moveout-corrected data and (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_{n} (equation 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_{n}). So, the midpoint coordinate is invariant under NMO correction. The difference between the input time t and the output time t_{n} is defined by
( )
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_{n}) to sample 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