1. Illumination, resolution, and incidence-angle in PSDM: A tutorial
Isabelle Lecomte
NORSAR, R&D Seismic Modelling, P.O.Box 53, 2027 Kjeller, Norway ?
Thank you mister chairman, and good morning everybody.

2. Hubble telescope: space-variant PSF*
*Point-Spread Functions
Space-variant PSF!
Let us first have a look at some images from the Hubble telescope. As highligthed here, a star appears as a cross-like pattern through the telescope. To understand why, the so-called Point-Spread functions, or PSF, i.e, the response of a point scatterer through the imaging system, are plotted here, and we recognize the cross-pattern. But Hubble was a poor telescope at the beginning due to irregularities on its mirror. As a consequence, the

3. Point-Spread Functions in Marmousi*
Seismics: PSF may be very space-variant!
*Marmousi model courtesy of IFP

4. Resolution, illumination, …etc!*
Acoustic/elastic impedance
Reflection ~ contrasts! **
PSDM … at best! !
Not 1D convolution!
* , **Liner (2000), and Monk (2002)

5. Content Introduction Image formation in PSDM
Scattering wavenumber: the key!
Resolution
Illumination
Examples
Controlling imaging
Conclusions
The content of this talk will be as following. After a short introduction about the motivations, I will give the key concept behind this study, to make us understand how a seismic image is formed. I am speaking here of PSDM images. A quick presentation of the methods I use will then be given, follow by examples to highlight illumination, resolution and incidence-angle effects. Such analyses are very useful, both prior and after acquisition and processing, as we will discuss.

6. Imaging in PSDM: K is the key!
Migration
Getting data
Back propagation
Waves! 1
Wave propagation corrections
G,G: GF(*)
●: GF-node
(*)GF: Green’s Function
Incident wave
Waves!
Imaging
Key information:
Scattering Wavenumber!
Waves!
Scattering
Imaging 2
Focusing ?

7. Scattering isochrones
Common shot
(x = 0) ■
Common offset
(0 m)
Model: constant velocity
Data: point scatterer
data
ellipse
circle ■
● point scatterer
PSDM

8. PSDM and point scatterer
Common offset (0 m) ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■ ● ■
1 trace
∑ traces
Same point scatterer…
…different PSDM images!
Common shot (x = 0) ■ ● ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
1 trace
∑ traces

9. PSF and PSDM: why? scattering structures = set of point scatterers
(e.g., exploding reflector concept, etc)
PSDM(point scatterer) = Point-Spread Function
If PSF known: PSDM image = Reflectivity * PSF
Question 1: how to get PSF without generating synthetic point scatterers at each image point?
Question 2: how to use PSF to understand and improve PSDM?

10. Content Introduction Image formation in PSDM
Scattering wavenumber: the key!
Resolution
Illumination
Examples
Controlling imaging
Conclusions
The content of this talk will be as following. After a short introduction about the motivations, I will give the key concept behind this study, to make us understand how a seismic image is formed. I am speaking here of PSDM images. A quick presentation of the methods I use will then be given, follow by examples to highlight illumination, resolution and incidence-angle effects. Such analyses are very useful, both prior and after acquisition and processing, as we will discuss.

11. Methods: ”ray-tracing” based
Green’s functions
Paraxial ray tracing
Wavefront Construction
Eikonal solver
PSDM (~Kirchhoff)
Diffraction Stack (DS)
Local Imaging (LI)
1 GF-node only!
”SimPLI” (*)
Simulated Prestack Local Imaging
No seismic records needed! ●
(*) patent pending

12. Scattering Wavenumber K
Definition at a local “Scattering Object” (diffraction, reflection, ..)
Incident wavenumber
Easy to calculate with ray tracing
and similar
scattered wavenumber
Calculation performed in the
PSDM velocity model

13. If Vs = Vg (no wave conversion)
K: which parameters?
source s
geophone g
- Vs: incident wave velocity
Vg: scattered wave velocity
ŝ and ĝ: unit vectors
n frequency
- VP: P-velocity
VS: S-velocity
”incidence” angle = 0
║ĝ – ŝ║ = 2 ● ŝ ĝ K
If Vs = Vg (no wave conversion) ● ŝ ĝ K
”incidence” angle ≠ 0
║ĝ – ŝ║ < 2

14. From K to PSF using FFT ● ● ● ● no data! 2D FFT Marmousi 2D FFT-1 Z X
-Kx max.
+Kx max.
-Kz max. 0.
max./2
module
no data! ●
Green’s Functions
at one GF-node ●
Marmousi ●
2D FFT ●

15. K and scattering isochrones
K is perpendicular to the
scattering isochrone
K corresponds to a local
plane wavefront
approximation of the
scattering isochrone
║K ║ = f(n) : pulse effect
[K]
PSF

16. Content Introduction Image formation in PSDM
Scattering wavenumber: the key!
Resolution
Illumination
Examples
Controlling imaging
Conclusions
The content of this talk will be as following. After a short introduction about the motivations, I will give the key concept behind this study, to make us understand how a seismic image is formed. I am speaking here of PSDM images. A quick presentation of the methods I use will then be given, follow by examples to highlight illumination, resolution and incidence-angle effects. Such analyses are very useful, both prior and after acquisition and processing, as we will discuss.

17. Resolution of an inverse problem!
Your model!
Generalized Inverse
Direct problem 1
Inverse problem 2
d: data
m: parameters
obs.: observed
est.: estimated
Resolution!
1+2
Data independent!

18. K and resolution: wavenumber coverage
[K] for [5-60] Hz
qs = [0-10] º
Marmousi model
Courtesy of IFP
DKx
DKZ
Lateral resolution ~ 2p / DKX
Vertical resolution ~ 2p / DKZ 1

19. PSDM of point scatterer and PSF
K and PSF: no data!
PSF
high R K
Kmean
low R
Common offset (0 m)
PSDM – data from point scatterer
Common shot (x = 0)

20. Content Introduction Image formation in PSDM
Scattering wavenumber: the key!
Resolution
Illumination
Examples
Controlling imaging
Conclusions
The content of this talk will be as following. After a short introduction about the motivations, I will give the key concept behind this study, to make us understand how a seismic image is formed. I am speaking here of PSDM images. A quick presentation of the methods I use will then be given, follow by examples to highlight illumination, resolution and incidence-angle effects. Such analyses are very useful, both prior and after acquisition and processing, as we will discuss.

21. K and reflection qs qg incident ray reflected ray Reflector Reflector
”P-to-P” reflection
”P-to-S” reflection
From source
incident ray
To geophone
reflected ray
qg scattering angle qg
qs incidence angle qs
Reflector
Reflector
In the PSDM velocity model:
A given couple (ks,kg) may correspond to an actual reflection.
it is the case IF there is a reflector perpendicular to K at the GF-node.

22. K and illumination: dip
[K] for [5-60] Hz
qs = [0-10] º
Marmousi model
Courtesy of IFP
Marmousi Model
Courtesy of IFP
Illuminated dips 2
~ 45 º
~ 25 º

23. Content Introduction Image formation in PSDM
Scattering wavenumber: the key!
Resolution
Illumination
Examples
Controlling imaging
Conclusions
The content of this talk will be as following. After a short introduction about the motivations, I will give the key concept behind this study, to make us understand how a seismic image is formed. I am speaking here of PSDM images. A quick presentation of the methods I use will then be given, follow by examples to highlight illumination, resolution and incidence-angle effects. Such analyses are very useful, both prior and after acquisition and processing, as we will discuss.

24. Playing with the pulse [K] Target model (Vp) Reflectivity 10 Hz SimPLI
Spectrum
10 Hz
SimPLI
20 Hz
SimPLI
30 Hz
SimPLI
40 Hz
SimPLI

25. Illumination and resolution: illustration
FFT+1
Reflectivity = 1
Fault
“Green’s Functions”
Fault
[K] incl. 20 Hz pulse
0 km offset
Fault
FFT-1
SimPLI – 0 km offset
PSF
FFT-1
Fault
Fault
4 km offset
FFT-1
SimPLI – 4 km offset
PSF
FFT-1

26. Incidence-angle in PSDM
Reflectivity : 05°-15°
Reflectivity : 00°-05°
Reflectivity : 15°-25°
Reflectivity: 25°-35°
Reflectivity: 35°-45°
[K] Filter : 25°-35°
[K] Filter : 15°-25°
[K] Filter : 00°-05°
[K] Filter : 05°-15°
[K] Filter: 35°-45°
Final SimPLI Image – 20 Hz Σ
SimPLI Image: 00°-05°
SimPLI Image: 05°-15°
SimPLI Image: 25°-35°
SimPLI Image: 35°-45°
SimPLI Image: 15°-25°

27. ● ● Overburden effects A B K PSF Good resolution Good illumination K
Poor resolution
Bad illumination
Not illuminated!

28. PSDM images: not a simple 1D convolution!
No illumination effects! KX KZ
2D Filter: 0 km offset
2D Filter: 4 km offset
This is PSDM effects!
Function of survey, overburden, pulse, wave-phases, local velocity.
Elastic impedance (x,z) KX KZ
“1D” PSDM

29. Content Introduction Image formation in PSDM
Scattering wavenumber: the key!
Resolution
Illumination
Examples
Controlling imaging
Conclusions
The content of this talk will be as following. After a short introduction about the motivations, I will give the key concept behind this study, to make us understand how a seismic image is formed. I am speaking here of PSDM images. A quick presentation of the methods I use will then be given, follow by examples to highlight illumination, resolution and incidence-angle effects. Such analyses are very useful, both prior and after acquisition and processing, as we will discuss.

30. Image and survey samplingK
PSF
SimPLI
Dshot: 12.5 m K
PSF
SimPLI
Dshot: 125 m K
PSF
SimPLI
Dshot: 625 m

31. Controlling imaging: check local K!
”blind!”
automatic corrections
Irregular
Sampling!
Blind!
Controlled!

32. Conclusions Define your PSDM velocity model…
Should be smooth in the imaging zone…
… but can have layers with contrast outside!
…then use the scattering wavenumbers!
Prior or after imaging
Survey planning mode
Resolution/illumination analyses
Controlling and improving imaging
Understanding image formation
Testing the validity of interpretation results
Flexible and fast!
Ray tracing based
FFT

33. Acknowledgements
Research Council of Norway (projects /420, /43, and /420)
Statoil (Gullfaks), IFP (Marmousi), Seismic Unix, and the “Svalex” project ( Storvola)
Håvar Gjøystdal, Åsmund Drottning and Ludovic Pochon-Guerin.
Thanks 
At last, we would like to thank the Research Council of Norway for supporting this research through different projects. Statoil for the Gullfaks model, IFP for the Marmousi model, Seismic Unix and the Svalex project, which is a multi-disciplinary education program grouping all Norwegian universities and sponsored by Statoil. I am especially grateful to my colleagues, Håvar Gjøystdal, Åsmund Drottning and Ludovic Pochon-Guerin, for their early support and enthousiasm. Thank you all for your attention!