1. Reduced-Time Migration of Converted Waves
David Sheley
University of Utah

2. Outline Motivation Migration Theory Error Analysis
Synthetic Data Results
Field Data Result
Conclusions & Future Work

3. PP vs PS Transmission Migration
Depth
Source Well
Receiver Well
This crude example is for crosswell and most specifically for orebody delineation. If you are more interested in salt flank imaging with VSP let use a bit of imagination. Z
Offset X

4. PP Reflection MigrationP
P = ? =?
Depth
Source Well
Receiver Well
PP Migration allow us to image sub-horizontal interfaces. The migration algorithm is highly dependent on the velocity information. Z
Offset X

5. Converted Wave Migration
Vp,Vs = ? P
Depth P PS
Source Well
Receiver Well PS
converted wave migration allows us to image subvertical boundaries giving us information about what could be happening parallel to the boreholes. Unfortunately it is not only dependent on the P-wave velocity model but also on the S-wave velocity model. To add to our grief, for deep holes there are often positioning errors. So you’re probably thinking to yourself “this kid thinks converted wave migration may actually be useful for real geologic problems. Perhaps during the course of this talk I can remove some …. Z
Offset X =?

6. Outline Motivation Migration Theory Error Analysis
Synthetic Data Results
Field Data Result
Conclusions & Future Work

7. Conventional Migration
m(r) = S(zg, tsr + trg) g
The green line is an isocron. s g
tsr
trg r

8. PS Transmission Migration
m(r) = S(zg, dsr/Vp + drg/Vs) g
This equtaion and figure model PS converstion SP conversion is also possible to migrate simply by swapping Vp and Vs. Again the green line is an isocron.
If we examine this equation a bit we can see that if the image point r is at the geophone the green isochron will collapse about the receiver and the time argument will be equal to the travel time of the direct P-wave from the source to the receiver. s g
drg/Vs
dsr/Vp r

9. Problem Receiver Source Well Well 20 40 60 50 km/sec Depth (m)
km/sec
6.0
5.0
7.0
Receiver
Source
Well
Well 20
Depth (m) 40
I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60
Offset (m) 50

10. Reduced-Time Migration
Data time shift
S’(g, t) = S(zg, t + tsg )
tsg = Observed direct-P time
obs
obs

11. Data Shift Original Data Shifted Muted Data SP PS P S Depth (m) 114 20
114 20 35
Time (ms)
Original Data
Depth (m)
114 8 2
Time (s)
Shifted Muted Data SP PS P S

12. Reduced-Time Migration
Data time shift
S’(zg, t) = S(zg, t + tsg )
tsg = Observed direct-P time
obs
obs
Modify the migration equation
m(r) = S(zg, tsr + trg - tsg + tsg )
calc
obs g
calc
m(r) = S’(zg, tsr + trg – tsg ) g

13. Outline Motivation Migration Theory Error Analysis
Synthetic Data Results
Field Data Result
Conclusions & Future Work

14. m(r) = S(zg, tsr + trg psr)
Error Analysis -- CWM
Assumptions:
Single trace
Homogeneous media
True velocity = c
Migration velocity
c’ = c + dc
Vp/Vs = psr
m(r) = S(zg, tsr + trg psr) g l
m(r) = S(zg, (dsr + drg psr)/c’ ) g l

15. Error Analysis Conventional Migration
(dsr + drg psr)/c’ l
(dsr + drg psr)/(c + dc) = ~
(dsr + drg psr)(s – s dc) l 2
e = - (dsr + drg psr) s dc 2 l cm

16. Error Analysis Reduced-Time Migration
m(r) = S(g, g
obs
calc
tsr + trg - tsg + tsg )

17. Error Analysis Reduced-Time Migration
obs
calc
tsr + trg - tsg + tsg = l
(dsr + drg psr - dsg)(s – s dc) + dsg s 2
e = - (dsr + drg psr - dsg) s dc
rtm 2 l

18. Error Functions CWM vs. RTMcm
e = - (dsr + drg psr) s dc 2 l
rtm
e = - (dsr + drg psr - dsg) s dc 2 l

19. e e Imaging-Time Error cm rtm Offset (m) 500 16 Depth (m) 12 Imaging
Offset (m)
500 16 cm e
Depth (m) 12
Imaging
Error
(ms)
250 8
rtm e
Depth (m) 4
250
Offset (m)
500

20. Outline Motivation Migration Theory Error Analysis
Synthetic Data Results
Field Data Result
Conclusions & Future Work

21. Crosswell Model V V V = 5000 m/s V = 5500 m/s Vp /Vs = 1.5
V = 5500 m/s 2 V 2
Vp /Vs = 1.5
Source = 1500 Hz
Depth (m)
ds = 2 m V 1
dg = 2 m
Well Separation
= 100 m
114
114
Offset (m)

22. Synthetic Data Original Data Shifted Muted Data P PS S SP SP PS
Depth (m)
114 20 35
Time (ms)
Original Data
Depth (m)
114 8 2
Time (s)
Shifted Muted Data P PS S SP SP PS

23. PS Transmission Migration True Velocity
Depth (m)
114
Offset (m)
114

24. Conventional PS Migration + 10 % Velocity
Depth (m)
114
Offset (m)
114

25. Reduced-Time PS Migration + 10% Velocity
Depth (m)
114
Offset (m)
114

26. Outline Motivation Migration Theory Error Analysis
Synthetic Data Results
Field Data Result
Conclusions & Future Work

27. Data Problems
Time Delay = 3 ms ?
Well location
Velocity Model

28. Kidd Creek Receiver Source Well Well 20 40 60 50 km/sec Depth (m)
km/sec
6.0
5.0
7.0
Receiver
Source
Well
Well 20
Depth (m) 40
I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60
Offset (m) 50

29. Time Shifted CRG 20
Depth (m) 40 60
Time (ms) 6

30. Conventional PS Migration20
Depth (m) 40
I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60
Offset (m) 50

31. Reduced-Time PS Migration20
Depth (m) 40
I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60
Offset (m) 50

32. RTM-PS CRG #8 20 40 60 50 Depth (m) Offset (m)20 40 60 50
Depth (m)
Offset (m)
I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever.

33. Kidd Creek 20 40 60 50 50 Offset (m) Offset (m)20 40
I will prove that mapping the direce P-wave to the receiver will acomplish the goal later. First, I will discuss how, even in the presence of velocity errors the direct-P is mapped to the reciever. 60
Offset (m) 50
Offset (m) 50

34. Outline Motivation Migration Theory Error Analysis
Synthetic Data Results
Field Data Result
Conclusions & Future Work

35. Discussion & Conclusions
PS migration can image structure invisible to reflection migration.
Reduced-time migraton decreases the error of an incorrect velocity model.
Converted wave reduced-time migration can successfully image a transmitting boundary.

36. Future Work
Model and migrate salt proximity VSP data with converted wave RTM.
Model and test PP RTM.
Search for other applications of RTM.
Graduate.