2. Agenda
1. Where does Reverse Time Migration (RTM)
algorithm fit within other algorithms
2. The three assumptions behind the RTM
migration
3. How RTM works: mathematics
4. Conclusion and modern RTM migration
applications
Page 2 of 14
3. • RTM = Reverse Time Migration
• FDM=Finite Difference Modelling
• ZO=zero offset time section
• f(x,z,t)=seismogram in x,z,t domain
• F(x,z,t)=Fourier Transform of seismogram
• kx=wavenumber Fourier Transform of
𝜕𝑃
𝜕𝑥
• kz=wavenumber Fourier Transform of
𝜕𝑃
𝜕𝑍
• P=pressure amplitude
• C=velocity (constant velocity field)
• V(x,z)=velocity (variable velocity field)
• 𝑡 = 𝑡𝑓 final time of the ZO at z=0
• 𝑡 = 𝑡0 exploding time within exploding reflector concept
• X-T domain = offset – time domain
• X-Z domain = offset – depth domain
Definition of Terms
Page 3 of 14
4. Where does RTM fit in Migration’s world ?
Page 5 of 14
Sources: Bancroft, 2007
Main types of migration methods:
• Kirchhoff migration
• FK migration
• Downward Continuation Migration (Phase Shift)
Downward Migration RTM Migration
• RTM
5. Sources: http://sep.stanford.edu/oldreports/sep40/40_01.pdf //
Loewenthal, 1983
RTM Assumptions: Exploding Reflectors
Page 6 of 14
Three Assumptions are:
• CDP stacked data time section ( Zero Offset Time section) in X-
Z plan that fits exploding reflection models/concept
Explosion at 𝑡 = 𝑡0Resulting CDP stacked
section at 𝑡 = 𝑡𝑓
6. Sources: Urosevic, Curtin University Lectures Notes, 2015
RTM Assumptions: backward time marching schemes
Page 7 of 14
Three Assumptions are:
• Exploding Reflector concept
• Solution by FDM of wave equation can be driven either forward or
backward in TIME and keep spatial frame X-Z fixed
7. RTM Assumptions: backward time marching schemes
Sources: Urosevic, Curtin University Lectures Notes, 2015
Page 7 of 14
Three Assumptions are:
• Exploding Reflector concept
• Solution by FDM of wave equation can be driven either forward or
backward in TIME and keep spatial frame X-Z fixed
8. RTM Assumptions: backward time marching schemes
Sources: Urosevic, Curtin University Lectures Notes, 2015
Page 7 of 14
Three Assumptions are:
• Exploding Reflector concept
• Solution by FDM of wave equation can be driven either forward or
backward in TIME and keep spatial frame X-Z fixed
9. RTM Assumptions: backward time marching schemes
Sources: Urosevic, Curtin University Lectures Notes, 2015
Page 7 of 14
Three Assumptions are:
• Exploding Reflector concept
• Solution by FDM of wave equation can be driven either forward or
backward in TIME and keep spatial frame X-Z fixed
10. RTM Assumptions: backward time marching schemes
Sources: Urosevic, Curtin University Lectures Notes, 2015
Page 7 of 14
Three Assumptions are:
• Exploding Reflector concept
• Solution by FDM of wave equation can be driven either forward or
backward in TIME and keep spatial frame X-Z fixed
11. Sources: Loewenthal, 1983
RTM Assumptions: Exploding Reflectors
Page 8 of 14
CDP section =U𝑛𝑚𝑖𝑔𝑟𝑎𝑡𝑒𝑑 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 =
𝑓 𝑥, 𝑧, 𝑡 = 𝑡𝑓 = 𝑓(𝑥′
, 𝑧′
, 𝑡′
)
CDP section =𝑚𝑖𝑔𝑟𝑎𝑡𝑒𝑑 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 =
𝑓 𝑥, 𝑧, 𝑡 = 𝑡0
𝑡′
= 2. 𝑡𝑖,𝑗 = 2. 𝑡𝑗,𝑖
Three Assumptions are:
• Exploding Reflector concept
• Solution of wave equation can be driven either forward or
backward in TIME and keep spatial frame fixedby FDM
12. RTM Assumptions: backward time marching schemes
Whole point of the RTM migration within the
exploding reflector concept is to move energy (wave
propagation) back in time to its reflection (i.e. its
explosion) time .
Take successive steps BACKWARD in TIME
to go from 𝒕 = 𝒕 𝒇 𝒕𝒐 𝒕 = 𝒕 𝟎
Page 9 of 14
Sources: Loewenthal, 1983 // Levin,1985
13. RTM Assumptions: backward time marching schemes
Page 10 of 14
𝐹 𝑘 𝑥, 𝑘 𝑧, 𝑡 = 𝑓 𝑥, 𝑧, 𝑡 . exp 𝑖(𝑘 𝑥. 𝑥 + 𝑘 𝑧. 𝑧)
𝑍𝑋
Sources: Loewenthal, 1983
Three Assumptions are:
• Exploding Reflector concept
• Take Successive Steps Backward
• Easier and Faster to work in the space frequency domain
take Fourier transform of the whole zero offset time section
Zero offset time section
at time t
Fourier domain
representation of Zero
offset time section at time t
14. RTM Assumptions: backward time marching schemes
Page 10 of 14
Inverse Fourier Transform: we can compute the field at any
desired time level t from the corresponding value at t’
𝐹 𝑘 𝑥𝑝, 𝑘 𝑧𝑞, 𝑡′
= 𝑓 𝑚. ∆𝑥, 𝑛. ∆𝑧, 𝑡′
. exp 𝑖(𝑘 𝑥𝑝. 𝑚∆𝑥 + 𝑘 𝑧𝑞. 𝑛∆𝑧)
𝑁
𝑛=0
𝑀
𝑚=0
f 𝑚∆𝑥, 𝑛∆𝑧, 𝑡 = 𝐹 𝑘 𝑥𝑝, 𝑘 𝑧𝑞, 𝑡′
. exp −𝑖 𝑘 𝑥𝑝. 𝑚∆𝑥 + 𝑘 𝑧𝑞. 𝑛∆𝑧 + 𝜔 𝑝𝑞(𝑡 − 𝑡′
)𝑁
𝑞=0
𝑀
𝑝=0
adding some shift in time
Sources: Loewenthal, 1983
𝑀 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑎𝑐𝑒𝑠
𝑁 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒 𝑠𝑎𝑚𝑝𝑙𝑒𝑠 𝑎𝑙𝑜𝑛𝑔 𝑡𝑟𝑎𝑐𝑒
∆𝑥 = 𝑡𝑟𝑎𝑐𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
∆𝑡 = 𝑡𝑖𝑚𝑒 𝑠𝑎𝑚𝑝𝑙𝑖𝑛𝑔 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝑣 𝑥, 𝑧 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑓𝑖𝑒𝑙𝑑
∆𝑧 = 𝑑𝑒𝑝𝑡ℎ 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
𝜔 𝑝,𝑞 = 𝑡𝑒𝑚𝑝𝑜𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑐𝑒
𝑘 𝑥𝑝 =
2𝜋𝑝
(𝑀.∆𝑥)
=wavenumber x direction=
𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑠𝑝𝑎𝑡𝑖𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑎𝑙𝑜𝑛𝑔 𝑥 𝑎𝑥𝑖𝑠
𝑝 = 0,1,2 … . 𝑀
2
𝑘 𝑧𝑞 =
2𝜋𝑞
(𝑁.∆𝑧)
=wavenumber z direction=
𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑠𝑝𝑎𝑡𝑖𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑎𝑙𝑜𝑛𝑔 𝑥 𝑎𝑥𝑖𝑠
𝑞 = 0,1,2 … . 𝑁
2
∆𝑡 = 𝑡 − 𝑡′
> 0
16. RTM problem solved: backward time marching schemes
Page 12 of 14
f 𝒎∆𝒙, 𝒏∆𝒛, 𝒕 𝒇 − ∆𝒕 = 𝑭 𝒌 𝒙𝒑, 𝒌 𝒛𝒒, 𝒕 𝒇 . 𝐞𝐱𝐩 −𝒊 𝒌 𝒙𝒑. 𝒎∆𝒙 + 𝒌 𝒛𝒒. 𝒏∆𝒛 − 𝝎 𝒑𝒒(∆𝒕)𝑵
𝒒=𝟎
𝑴
𝒑=𝟎
Take 𝑡′ = 𝑡𝑓 𝑎𝑛𝑑 𝑡 = 𝑡𝑓 − ∆𝑡
removing some shift ∆𝒕 in time
RTM Migration
Sources: Loewenthal, 1983
17. RTM problem solved: backward time marching schemes
Loewenthal synthetic applications:
• dipping layer 29°
• dipping layer 33°
• dipping layer 35°
• syncline
Ok errror < 1%
OK error <1%
Ok error 5%
OK
Sources: Loewenthal, 1983
Page 13 of 14
18. RTM problem solved: backward time marching schemes
Overall RTM do work well !
Sources: Loewenthal, 1983
Page 13 of 14
19. RTM : Conclusion
RTM advantages:
• Used today (in prestack mode) to improve imaging in complex
geology (e.g Salt Dome)
• Resulting images are usually easier to interpret
• Can handle severe combination of structural dip (steep and
even overturned reflectors) with high velocity contrast
RTM disadvantages:
• Computer intensive (was ?? )
Sources: Ion, 2015 // CGG, 2015 // Paradigm, 2015 // CSEG, recorder, 2009
Page 14 of 14
24. What is Migration: an example
Page 4 of 14
Sources: Yilmaz, 1987
Focusing effect of
a sharp syncline
Migration
algorithm
25. What is Migration ?
Page 4 of 11
Post Stack Migration = Dealing with circles
𝑇𝑎𝑛 𝛼 = 𝑆𝑖𝑛 (𝛽)
Sources: Bancroft, 2007 // CSEG recorder, 2012
• The mathematical process of re-arranging the reflection
events at their correct locations in the seismogram
• Migration will shortens dipping events and increase the dip
• Can be in X-T or X-Z domain and pre-stack or post stack
30. RTM Assumptions: backward time marching schemes
From 𝑡 = 𝑡𝑓 = 𝑡′ 𝑡𝑜 𝑡 = 𝑡0
Take successive steps BACKWARD in TIME
until the desired migration is achieved
Page 9 of 11
Sources: Loewenthal, 1983 // Levin, 1985
