Full wavefield migration: Utilization of multiples in seismic migration

B041

Full Wavefield Migration - Utilization of Multiples

in Seismic Migration

A.J. Berkhout* (Delft University of Technology) & D.J. Verschuur (Delft University of Technology)

SUMMARY

The next generation migration technology considers multiple scattering as vital information, allowing the industry to generate significantly better images of the subsurface. The proposed full wavefield algorithm (FWM) makes use of two-way wave theory that is formulated in terms of one-way wavefields. We show that the current migration algorithms for primary reflections can be easily extended to handle multiple scattering as well.

Introduction

Until today, seismic imaging requires primary reflection data as input (wavefields with a single bounce in the earth). This means that multiple scattering is considered as noise and need be removed. In the last decade, a lot of progress has been made on the removal of multiple scattering events (wavefields with more than one bounce). This is particularly true for multiples that cannot be easily distinguished from primaries with respect to traveltime behavior (’move-out’). Here, the data-driven approach in terms of multi-record convolution (Berkhout, 1982) has shown excellent results in practice (Verschuur, 1991) and this technology is now being applied worldwide (see e.g. Dragoset, 1999; Hadidi et al., 2002; van Borselen et al., 2005; Moore and Bisley, 2005; van Groenestijn and Verschuur, 2009). Today, however, it is realized that multiples should not be qualified as noise but as important signal components. Using this new paradigm, we already see algorithms being developed to utilize the large amount of information that is hidden in multiple scattering (see e.g. Berkhout and Verschuur, 1994; Whitmore et al., 2010). Similarly, a lot of progress has also been made on the development of seismic migration algorithms. An example is model-based reverse time migration (Baysal et al., 1983; Biondi and Shan, 2002), where time-reversed seismic measurements - after multiple removal - are fed into an advanced modeling program. A major challenge in migration algorithms is to maintain wide-angle (even beyond 90◦

) reflections in the wavefield extrapolation process. For this reason, we now observe that two-way finite difference algorithms have gained a lot popularity. Application of reverse time migration (RTM) to shot records is an attractive way of using the two-way technology.

In this presentation, the one-way wave theory is extended to the two-way theory in terms of one-way wavefields. This approach allows us to utilize internal multiples in the migration process without hard-wiring multiple-generating boundaries in the solution.

Utilization of surface multiples

There is increasing evidence that multiples contain a lot of valuable subsurface information. Hence, multiples should not only be removed but they should also be used to improve seismic migration and inversion results. This important property can be easily seen from the feedback model:

P−= XS+ = X0S++ R ∩ P− (1) or P−= X0S++ X0R ∩ P−= P− 0 + M − , (2)

P−0 representing the full waveform primary reflections (including their coda due to internal multiple

scattering) and M−

representing the surface-related multiples. Note that Q+= S+

+ R∩

P−

(double illumination) (3)

represents the total downgoing wavefield at the surface. Examples of migration with surface multiples are shown in Verschuur and Berkhout (2011).

Utilization of internal multiples

So far, the wavefield model does not include transmission effects and internal multiples. If we look at the inhomogeneities at depth level zm, then the two-way scattered wavefield can be written as (Figure

1a):

δP= R∪

P++ R∩

P−

(two-sided illumination), (4)

at each gridpoint, leading to the total wavefield at that depth level (Figure 1b):

Q−= P−

! R i ∪ P_ij+

δ

δ

!

Q

=

!

P

+

δ

_P

!

!

P

!

Q

₌

!

P

₊

δ

_P

!

!

P

R

R

z

Figure 1 Full wavefield theory: up- and downgoing wavefields (a) at gridpoint i and (b) at depth level

zm, showing two-way scattering in terms of elastic reflection and transmission.

Q+= P++ δP (total downgoing wavefield). (5b)

Note that the difference between the Q and P wavefields is given by the scattered wavefields δP, assur-ing that the elastic boundary conditions are fulfilled. Hence, we may write Q = P at a homogeneous

depth level only (δP = 0). Note also that if we move from depth level to depth level, the two-way

scattering term δP automatically takes care of the internal multiple reflection process (Figure 2):

Q−(zm, z0) = W ∗ (zm, zm−1)P − (zm−1, z0) (6a) P+(zm, z0) = W(zm, zm−1)Q+(zm−1, z0). (6b)

Note that equations (4), (5a,b) and (6a,b) represent a new approach to two-way wavefield propagation: not in terms of total pressure and particle velocity, being coupled by the acoustic parameters ρ and κ at each time step, but in terms of up- and downgoing wavefields, being coupled by the elastic reflectivities

R∪

and R∩

at each depth level. This has the significant advantage that the migration algorithm for primaries can be easily extended to the migration algorithm for primaries and multiples:

1. Recursive one-way migration (OWM)

ˆ

P−= Q−

−R∪P+ = minimum, (single minimization) (7)

yielding R∪

at each depth level zm(m= 1, 2, ...). Application of equation (7) means that

trans-mission effects and internal multiples are neglected (R∩

= 0 and Q+ = P+ at zm).

z

z

W z

(

, z

)

W z

(

, z

)

Q

P

P

Q

Figure 2 Up- and downgoing wavefields in depth layer (zm−1, zm), connecting the coupled wavefields

at depth level zm−1 (P

−

, Q+) with the coupled wavefields at depth level zm (Q

−

2. Recursive full wavefield migration (FWM, see Figure 1):

P−= Q−

−R∪P+−R∩P−= minimum, (double minimization) (8a)

meaning that both the reflections (R∪

P+_{) and transmission effects (R}∩

P−

) at depth level zmare

removed from the total upgoing wavefield (from Q−

to P−

). This minimization yields R∪

and

R∩

at each depth level zm(m= 1, 2, ...). In addition (see Figure 1):

Q+= P+

+ R∪

P+R∩

P−

. (double illumination) (8b)

Application of equation (8b) means that transmission effects and multiples are included in the downgoing illuminating wavefields.

Equations 8a,b show that the FWM process recursively moves down in the subsurface by downward extrapolating the incident and reflected wavefields - including multiple scattering - to each depth level

zm (m = 1, 2, . . . , M ). The double reflectivity (representing operators for upward and downward

re-flection) at depth level zm is obtained by minimizing the residue between the reflected wavefields and

the weighted incident wavefields at zm. For the next recursion (from zmto zm+1) the upward scattering

generated by depth level zm (primary + multiple tail) is subtracted from the reflection response, and

the downward scattering generated by depth level zmis added to the incident wavefields. Hence, while

moving down in the subsurface, the upgoing reflected wavefield is recursively decreased and the down-going reflected wavefield is recursively increased with higher-order reflectivity terms. When arriving at the maximum depth level (z = zM), the reflection response contains the minimum number of terms

(from z ≥ zM) and the illuminating wavefield contains the maximum number of terms: the direct source

wavefield + the downward scattering of the entire overburden (z < zM). 0 0.5 1.0 1.5 2.0 2.5 3.0 Time (s) -2000 -1000 0 1000 2000 Offset (m) 0 0.5 1.0 1.5 2.0 2.5 3.0 Time (s) -2000 -1000 0 1000 2000 Offset (m) 0 0.5 1.0 1.5 2.0 2.5 3.0 Time (s) -2000 -1000 0 1000 2000 Offset (m) 0 0.5 1.0 1.5 2.0 2.5 3.0 Time (s) -2000 -1000 0 1000 2000 Offset (m) a) ~P− (z0, z0) b) ~Q+(z0, z0) c) ~P − (z3, z0) d) ~Q+(z3, z0)

Figure 3 a) Upgoing wavefield for one shot record in a three-reflector medium with all multiples

in-cluded. b) Total downgoing wavefield at the surface. c) Upgoing wavefield below z3, theoretically being

zero. d) Total downgoing wavefield below z3. Note that in one-way migration ~P

−

(z3, z0) consists of the

overburden coda, giving false images, and ~Q+_(z

3, z0) equals the forward extrapolated source wavefield

only. Example

Figure 3 shows a simple example of FWM based on the three-reflector subsurface model in Figure 4a. At the surface (a) the response P−

and (b) the incident wavefield Q+equals the source wavefield (S+) combined with the reflected response (R∩

P−

). By applying the FWM process we see that just below the last boundary (c) the response equals zero and (d) the incident wavefield equals the propagated source wavefield (W+S+

) + the propagated surface-related multiples (W+R∩

P−

) + all internal multiples (P W+_δP).

If we compare these wavefields for the one-way method, then we see for the OWM-algorithm a totally different picture due to the wrong handling of the higher-order terms (representing transmission effects and internal multiples). Figure 4 shows the corresponding (b) FWM and (c) OWM image. Note the superior performance of FWM. z0 z1 z2 z3 a) b) c)

Figure 4 a) Three-reflector model. b) Result of full wavefield migration (FWM). c) Result of one-way migration (OWM). Note the false images in c).

Conclusions

The two-way wavefield theory is formulated with one-way concepts, allowing the current migration algorithms for primary wavefields to be extended to multiple scattering (FWM). In FWM subsurface boundaries are illuminated by both downgoing and upgoing wavefields (two-sided illumination). FWM utilizes all these wavefields in a constrained minimization process, leading to a full wavefield image of the subsurface.

Acknowledgments

The authors thank the DELPHI consortium sponsors for their support. References

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