illumination, resolution, and incidence-angle in psdm: a tutorial

12006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Illumination, resolution, and incidence-angle in PSDM:A tutorial

Isabelle LecomteNORSAR, R&D Seismic Modelling, P.O.Box 53, 2027 Kjeller, Norway

isabelle@norsar.no

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22006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Hubble telescope: space-variant PSF*

http://huey.jpl.nasa.gov/mprl

*Point-Spread Functions

Space-variant PSF!

32006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Point-Spread Functions in Marmousi*

*Marmousi model courtesy of IFP

Seismics: PSF may be very space-variant!

42006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Resolution, illumination, …etc!

*http://www.lenna.org , **Liner (2000), and Monk (2002)

*

Acoustic/elastic impedance Reflection ~ contrasts!

**

PSDM … at best!

!

Not 1D convolution!

52006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Content

• Introduction

• Image formation in PSDM

• Scattering wavenumber: the key!

• Resolution

• Illumination

• Examples

• Controlling imaging

• Conclusions

62006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Waves!

Scattering

Imaging in PSDM: K is the key!

Getting data

Incident wave

Waves!

Imaging

Key information:

Scattering Wavenumber!

Imaging2Focusing

?

Back propagation

Waves!

1Wave propagation corrections

G,G: GF(*)

●: GF-node

(*)GF: Green’s FunctionMigration

72006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Scattering isochrones

Common shot(x = 0)

■Common offset(0 m)

■Model: constant velocityData: point scatterer

data data

ellipse circle

■

● point scatterer

PSDMPSDM

82006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

PSDM and point scatterer

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■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■●

Common offset (0 m)

Common shot (x = 0)

1 trace

1 trace

∑ traces

∑ traces

Same point scatterer…

…different PSDM images!

92006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

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?

102006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Content

• Introduction

• Image formation in PSDM

• Scattering wavenumber: the key!

• Resolution

• Illumination

• Examples

• Controlling imaging

• Conclusions

112006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

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

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122006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Scattering Wavenumber K

Incident wavenumber

Definition at a local “Scattering Object” (diffraction, reflection, ..)

scattered wavenumber

Easy to calculate with ray tracingand similar

Calculation performed in thePSDM velocity model

132006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

K: which parameters?

g sK k k

ˆ ˆ2

g s

g sK

V V

s sP SV V orV V 2

ˆs

sk sV

source s

g gP SV V orV V 2

ˆg

gk gV

geophone g

- Vs: incident wave velocity- Vg: scattered wave velocity- ŝ and ĝ: unit vectors- frequency

- VP: P-velocity- VS: S-velocity

If Vs = Vg (no wave conversion)

2ˆ ˆ( )K g s

V

”incidence” angle = 0║ĝ – ŝ║ = 2

●●ŝ ĝ

K

●ŝ ĝ

K

”incidence” angle ≠ 0║ĝ – ŝ║ < 2

142006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

From K to PSF using FFT

2D FFT-1

PSF

Z

X

Green’s Functionsat one GF-node

●

Marmousi

●

-Kx max. +Kx max.-Kz max.

0.

max./2

0m

odu

le

no data!

●

2D FFT

●

152006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

K and scattering isochrones

K corresponds to a localplane wavefront

approximation of thescattering isochrone

K is perpendicular to thescattering isochrone

║K ║ = f() : pulse effect

[K] PSF

162006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Content

• Introduction

• Image formation in PSDM

• Scattering wavenumber: the key!

• Resolution

• Illumination

• Examples

• Controlling imaging

• Conclusions

172006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

truetruegest mRmGGm .).(. 1+2

.. . obsgest dGm Inverse problem

2trueobs mGd .. Direct problem

1

Resolution of an inverse problem!

GGR g

1 !!!gG G

Your model! Generalized Inverse

d: datam: parametersobs.: observedest.: estimated

Data independent!

Resolution!

182006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

[K] for [5-60] Hz

s = [0-10] º

K and resolution: wavenumber coverage

Kx

KZ

Lateral resolution ~ 2 / KX

Vertical resolution ~ 2 / KZ

1

Marmousi modelCourtesy of IFP

192006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

K and PSF: no data!

PSF

PSF

K

K

PSDM of point scatterer and PSF

PSDM – data from point scatterer

Common offset (0 m)

Common shot (x = 0)

high R

high Rlow R

low R

Kmean

Kmean

202006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Content

• Introduction

• Image formation in PSDM

• Scattering wavenumber: the key!

• Resolution

• Illumination

• Examples

• Controlling imaging

• Conclusions

212006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

K and reflection”P-to-S” reflection”P-to-P” reflection

ReflectorReflector

From sourceFrom source

incident ray

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.

s incidence angle

s s

To geophoneTo geophone

reflected ray

g scattering angle

g g

222006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

[K] for [5-60] Hz

s = [0-10] º

Illuminated dips2

~ 45 º ~ 25 º

K and illumination: dip

Marmousi ModelCourtesy of IFP

Marmousi modelCourtesy of IFP

232006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Content

• Introduction

• Image formation in PSDM

• Scattering wavenumber: the key!

• Resolution

• Illumination

• Examples

• Controlling imaging

• Conclusions

242006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

0 Hz

120 Hz

[K]

Playing with the pulse

Spectrum

Target model (Vp) Reflectivity

10 Hz

SimPLI

20 Hz

SimPLI

30 Hz

SimPLI

40 Hz

SimPLI

252006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Fault

4 km offset

Reflectivity = 1

FFT+1

“Green’s Functions”

Illumination and resolution: illustration

Fault

[K] incl. 20 Hz pulse

0 km offset

Fault

Fault

PSF

FFT-1

PSF

FFT-1

FFT-1

SimPLI – 0 km offset

FFT-1

SimPLI – 4 km offset

Fault

262006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Incidence-angle in PSDM

SimPLI Image: 00°-05°SimPLI Image: 05°-15°SimPLI Image: 15°-25°SimPLI Image: 25°-35°SimPLI Image: 35°-45°

[K] Filter : 00°-05°[K] Filter : 05°-15°[K] Filter : 15°-25°[K] Filter : 25°-35°[K] Filter: 35°-45°Reflectivity : 00°-05°Reflectivity : 05°-15°Reflectivity : 15°-25°Reflectivity: 25°-35°Reflectivity: 35°-45°

Final SimPLI Image – 20 Hz

Σ

272006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Overburden effects

●A

K PSF

Good resolutionGood illumination

●B

K PSF

Poor resolutionBad illumination

Not illuminated!

282006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

PSDM images: not a simple 1D convolution!

Elastic impedance (x,z)

KX

KZ

“1D” PSDM

!

No illumination effects!

KX

KZ

2D Filter: 0 km offset

KX

KZ

2D Filter: 4 km offset

This is PSDM effects!

Function of survey, overburden, pulse, wave-

phases, local velocity.

292006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Content

• Introduction

• Image formation in PSDM

• Scattering wavenumber: the key!

• Resolution

• Illumination

• Examples

• Controlling imaging

• Conclusions

302006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Image and survey sampling

K

PSF

SimPLI

shot: 12.5 m

K

PSF

SimPLI

shot: 125 m

K

PSF

SimPLI

shot: 625 m

312006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

”blind!”

automatic corrections

Controlling imaging: check local K!

IrregularSampling!

Blind! Controlled!

322006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

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

332006 SEG 76th Annual Meeting – 10/5/2006 – SPMI6-6

Acknowledgements

• Research Council of Norway (projects 131341/420, 128440/43, and 153889/420)

• Statoil (Gullfaks), IFP (Marmousi), Seismic Unix, and the “Svalex” project (www.svalex.net, Storvola)

• Håvar Gjøystdal, Åsmund Drottning and Ludovic Pochon-Guerin.

• Thanks