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**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.

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**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

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**Point-Spread Functions in Marmousi***

Seismics: PSF may be very space-variant! *Marmousi model courtesy of IFP

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**Resolution, illumination, …etc!**

* Acoustic/elastic impedance Reflection ~ contrasts! ** PSDM … at best! ! Not 1D convolution! *http://www.lenna.org , **Liner (2000), and Monk (2002)

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**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.

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**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 ?

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**Scattering isochrones**

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

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**PSDM and point scatterer**

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

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**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?

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**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.

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**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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**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

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**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

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**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 ●

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**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

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**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.

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**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!

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**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

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**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)

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**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.

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**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.

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**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

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**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.

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**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

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**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

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**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°

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**● ● Overburden effects A B K PSF Good resolution Good illumination K**

Poor resolution Bad illumination Not illuminated!

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**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

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**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.

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**Image and survey sampling**

K PSF SimPLI Dshot: 12.5 m K PSF SimPLI Dshot: 125 m K PSF SimPLI Dshot: 625 m

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**Controlling imaging: check local K!**

”blind!” automatic corrections Irregular Sampling! Blind! Controlled!

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**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

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Acknowledgements Research Council of Norway (projects /420, /43, and /420) Statoil (Gullfaks), IFP (Marmousi), Seismic Unix, and the “Svalex” project (www.svalex.net, 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!

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