Principles of dip-moveout correction

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 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 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 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.
2. 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 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 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).

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 y_n − t to y_n − t_n by mapping amplitude A at time t to B at time t_n on the same trace. DMO correction involves coordinate transformation from y_n − t_n to y₀ − τ₀ by mapping amplitude B at time t_n on the trace at midpoint location y_n 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 y₀ − τ₀ to y_m − 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 y_m of the migrated section. Migration before stack involves direct mapping of amplitude A at time t on the trace at midpoint location y_n of the common-offset section to amplitude D at time τ on the trace at midpoint location y_m 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

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

(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 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:

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

(2)

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 t₀ are related as follows (Section E.2)

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

(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 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.
2. 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 t₀ 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 t₀.

Our goal, however, is to map event time t not to t₀ — the two-way zero-offset time associated with midpoint y_n 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 (y_n, t_n) coordinates of the normal-moveout-corrected data and (y₀, τ₀) coordinates of the dip-moveout-corrected data are given by (Section E.2):

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

(4a)

and

$\tau _{0}={\frac {t_{n}}{A}},$

(4b)

where

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

(5)

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

$t_{0}=t_{n}A.$

(6)

Note from equation (5) that A ≥ 1; therefore, τ₀ ≤ t_n (equation 4b) and t₀ ≥ 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

$\Delta t_{NMO}=t-t_{n},$

(7a)

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

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

(7b)

where

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

(7c)

Again, refer to Figure 5.1-1 and note that the dip-moveout correction maps the amplitude at sample B with coordinates (y_n, t_n) 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

$\Delta y_{DMO}=|y_{n}-y_{0}|,$

(8a)

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

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

(8b)

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

$\Delta t_{DMO}=t_{n}-\tau _{0},$

(9a)

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

$\Delta t_{DMO}=t_{n}\left(1-{\frac {1}{A}}\right).$

(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 y_n and y₀, respectively. By way of equations (E-18) and (8a) it follows that (Section E.1)

$y_{0}=y_{n}-{\frac {\Delta }{\cos \phi }}$

(E-18)

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

(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.

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

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