# Principles of dip-moveout correction

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Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 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 yn. We want to map the amplitude at time t denoted by the sample A on the trace at midpoint yn of the common-offset section with offset 2h to time τ0 denoted by the sample C on the trace at midpoint y0 of the zero-offset section. We achieve this mapping in two steps:

1. Normal-moveout correction that maps the amplitude at time t denoted by the sample A on the trace at midpoint yn of the common-offset section with offset 2h to time tn denoted by the sample B on the same trace at midpoint yn of the same common-offset section.
2. Dip-moveout correction that maps the amplitude at time tn denoted by the sample B on the trace at midpoint yn of the moveout-corrected common-offset section with offset 2h to time τ0 denoted by the sample C on the trace at midpoint y0 of the zero-offset section.

Zero-offset migration then maps the amplitude at time τ0 denoted by the sample C on the trace at midpoint y0 of the zero-offset section to the amplitude at time τ denoted by the sample D on the trace at midpoint ym 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 yn of the common-offset section with offset 2h to the amplitude at time τ denoted by the sample D on the trace at midpoint ym of the migrated section. This direct mapping procedure is the basis of algorithms for migration before stack (prestack time migration).

Figure 5.1-1  (a) The geometry of a nonzero-offset recording of reflections from a dipping layer boundary; (b) a sketch of the time section depicting the various traveltimes. NMO correction involves coordinate transformation from yn − t to yn − tn by mapping amplitude A at time t to B at time tn on the same trace. DMO correction involves coordinate transformation from yn − tn to y0τ0 by mapping amplitude B at time tn on the trace at midpoint location yn of the moveout-corrected common-offset section to amplitude C at time τ0 on the trace at midpoint location y0 of the zero-offset section. Zero-offset migration involves coordinate transformation from y0τ0 to ym − τ by mapping amplitude C at time τ0 on the trace at midpoint location y0 of the zero-offset section to amplitude D at time τ on the trace at midpoint location ym of the migrated section. Migration before stack involves direct mapping of amplitude A at time t on the trace at midpoint location yn of the common-offset section to amplitude D at time τ on the trace at midpoint location ym of the migrated section. See text for the relationships between the coordinate variables.

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

 ${\displaystyle t^{2}=t_{0}^{2}+{\frac {4h^{2}{\rm {cos}}^{2}\phi }{v^{2}}},}$ (1)

where 2h is the offset, v is the medium velocity above the reflector, ϕ is the reflector dip, and t0 is the two-way zero-offset time at midpoint location yn.

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

 ${\displaystyle t^{2}=t_{n}^{2}+{\frac {4h^{2}}{v^{2}}},}$ (2)

where tn is the event time at midpoint yn after the NMO correction. Event time tn after the NMO correction and event time t0 are related as follows (Section E.2)

 ${\displaystyle t_{n}^{2}=t_{0}^{2}-{\frac {4h^{2}{\rm {sin}}^{2}\phi }{v^{2}}}.}$ (3)

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

1. 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 yn of the common-offset section with offset 2h to time tn denoted by the sample B on the same trace at midpoint yn of the same common-offset section.
2. Apply a dip-dependent moveout correction using equation (3) to map the amplitude at time tn denoted by the sample B on the trace at midpoint yn of the moveout-corrected common-offset section with offset 2h to time t0 denoted by the sample B′ on the same trace at midpoint yn 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 t0.

Our goal, however, is to map event time t not to t0 — the two-way zero-offset time associated with midpoint yn between source S and receiver G, but to τ0 — the two-way zero-offset time at midpoint location y0 associated with the normal-incidence reflection point R (Figure 5.1-1). The relationships between (yn, tn) coordinates of the normal-moveout-corrected data and (y0, τ0) coordinates of the dip-moveout-corrected data are given by (Section E.2):

 ${\displaystyle y_{0}=y_{n}-{\frac {h^{2}}{t_{n}A}}\left({\frac {2{\rm {sin}}\phi }{v}}\right),}$ (4a)

and

 ${\displaystyle \tau _{0}={\frac {t_{n}}{A}},}$ (4b)

where

 ${\displaystyle A={\sqrt {1+{\frac {h^{2}}{t_{n}^{2}}}\left({\frac {2{\rm {sin}}\phi }{v}}\right)^{2}}}.}$ (5)

For completeness, the relationship between event times tn and t0 is given by (Section E.2)

 ${\displaystyle t_{0}=t_{n}A.}$ (6)

Note from equation (5) that A ≥ 1; therefore, τ0tn (equation 4b) and t0tn (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 (yn, t) to sample B with coordinates (yn, tn). So, the midpoint coordinate is invariant under NMO correction. The difference between the input time t and the output time tn is defined by

 ${\displaystyle \Delta t_{NMO}=t-t_{n},}$ (7a)

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

 ${\displaystyle \Delta t_{NMO}=t_{n}\left(A_{n}-1\right),}$ (7b)

where

 ${\displaystyle A_{n}={\sqrt {1+{\frac {h^{2}}{t_{n}^{2}}}\left({\frac {2}{v}}\right)^{2}}}.}$ (7c)

Again, refer to Figure 5.1-1 and note that the dip-moveout correction maps the amplitude at sample B with coordinates (yn, tn) to sample C with coordinates (y0, τ0). So, the midpoint coordinate is variant under DMO correction. The lateral excursion associated with the DMO correction is given by

 ${\displaystyle \Delta y_{DMO}=|y_{n}-y_{0}|,}$ (8a)

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

 ${\displaystyle \Delta y_{DMO}={\frac {h^{2}}{t_{n}A}}\left({\frac {2{\rm {sin}}\phi }{v}}\right).}$ (8b)

The difference between the input time tn and the output time τ0 is defined by

 ${\displaystyle \Delta t_{DMO}=t_{n}-\tau _{0},}$ (9a)

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

 ${\displaystyle \Delta t_{DMO}=t_{n}\left(1-{\frac {1}{A}}\right).}$ (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 yn and y0, respectively. By way of equations (E-18) and (8a) it follows that (Section E.1)

 ${\displaystyle y_{0}=y_{n}-{\frac {\Delta }{\cos \phi }}}$ (E-18)

 ${\displaystyle \Delta ={\frac {h^{2}}{t_{n}A}}\left({\frac {{\rm {sin2}}\phi }{v}}\right).}$ (10)

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.