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My great amigo

I sincerely thank. My great amigo. Zoli. for being Zoli. I sincerely thank. this amigo. and. this amigo. for the kind invite. I wish to, humbly and most sincerely, thank the. The Society that publishes 2 of the very best. and holds meetings in exotic places.

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My great amigo

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  1. I sincerely thank My great amigo Zoli for being Zoli

  2. I sincerely thank this amigo and this amigo for the kind invite

  3. I wish to, humbly and most sincerely, thank the

  4. The Society that publishes 2 of the very best

  5. and holds meetings in exotic places

  6. for honoring me with this marvelous and unique adventure

  7. This tour would have been a rout without Judy Wall who organised it ALL !!! Imagine. Angels do exist in the sky.

  8. With additonal thanks to Tury Taner, what can I say?, he who has done it all. Enders Robinson, he was and is, numero uno. Sven Treitel, there are no words, except, Sven. Arthur Weglein, my friend, my teacher. Mauricio Sacchi, without whom Tad would be Tad who? Those marvelous friends, colleagues, students, who must assume full responsibility for making me who I have become.

  9. Join Today! Membershiphas its Advantages • Scholarly Journals in Print and Online • Networking Opportunities • Receive Membership Discounts on: • Professional Development Courses • Publications • Workshops and Meetings Need more information about joining SEG? SEG Membership Brochures and Applications are available today! Join Online  http://membership.seg.org

  10. and now the taaaalk

  11. The role of AmplitudeandPhasein Processingand InversionTadeusz Ulrych

  12. I have chosen this title, because I can talk about ANYTHING !!

  13. This presentation was prepared while partying in the local bar, illustrated in the next slide

  14. Definitions

  15. A brief story Doug Foster arranges a presentation for Monday Dr. Doug J. Foster This is Me Sunday evening is slightly brutal I cannot remember [1] How many participants? [2] Where is my presentation? I have a Canadian cell with enough credit for ONE question

  16. What question do I ask? How many participants? or Where is the presentation?

  17. The answer to HOW MANY? is AMPLITUDE (goodbye presentation and future invitation)

  18. The answer to WHERE? is PHASE (Oblivious to the number, I blindly carried on)

  19. WHERE ? HOW BIG ? x

  20. Relative “Importance” of

  21. ? Original

  22. INTRODUCTION • Mathematics is Beautiful. • However, it is tiresome to digest. • Therefore, this talk contains aslittle of this beauty as possible. • Please remember, that the magic of mathematics lies in its physical interpretation. For example ….

  23. Question Why is it true that = (-1)1/2 x

  24. Because, as is well known (-1)1/2 = i and i is an operator that rotates by 90o

  25. Amplitude & Phase in blind deconvolution The Enders example

  26. The Man Enders Robinson

  27. The canonical model for the seismogram is the seismogram is the source signature is the Greens function, the reflectivity is ‘everything else’, the noise

  28. This equation, is 1 equation with 2 unknowns. This is akin to 7= a + b and what is a and b uniquely ?

  29. This, of course, is an impossible problem unless a priori constraints are known or, at least, assumed

  30. THIS IS A YOU Some more thoughts regarding Phase

  31. OUTLINE for the next few slides • POCS and only-phase reconstruction • Phase and cepstral processing • Summary

  32. POCS Projection onto convex sets POCS attempts to solve an underdetermined, generally nonlinear, inverse problem G[x]+n=d where G is a nonlinear operator

  33. A convex set, A, is one for which the line joining any two points, x and y, in the set, is totally within the set. In other words, a set A in a vector space is convex, iff x and y Є A λx + (1 - λy) Є A  0 ≤ λ ≤ 1

  34. Illustrating convex and non-convex sets A convex setA non-convex set

  35. Alternating POCS Iterative projection onto convex sets

  36. Possible stagnation point when one of the sets is non-convex

  37. Application of alternating POCS to the problem of reconstruction from phase-only to obtain the only-phase image

  38. The image, of finite support , is a convex set. The set of constraints, the thresholded image, is also another convex set.

  39. Phase-only

  40. Only-phase

  41. Original

  42. Phase in Cepstral analysis Phase is fundamental in cepstral processing • Phase must be unwrapped • Phase must be detrended • A serious problem is additive noise

  43. The cepstrum (complex) is defined as C(n) = {ln[A(ω)] + iΦ(ω)} where is the inverse Fourier transform

  44. Application of cepstral analysis to • thin bed blind deconvolution • Compute cepstrum for each trace • Stack the cepstra • Transform back to the time domain • Deconvolve with estimated wavelet

  45. The original synthetic section

  46. The original reflectivity

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