1. Lasse Amundsen, Arne Reitan, and Børge Arntsen
Classification: Statoil internal Status: Draft
DATA-DRIVEN INVERSION/DEPTH IMAGING DERIVED FROM APPROXIMATIONS TO 1-D INVERSE ACOUSTIC SCATTERING ”DATA HAVE A STORY TO TELL IF THEY’RE ASKED THE RIGHT QUESTIONS” (Art Weglein)
Lasse Amundsen, Arne Reitan, and Børge Arntsen

2. CONTENTS INTRODUCTION FORWARD SCATTERING MODELS WKBJ, Eikonal and Born
TWO INVERSE SCATTERING SOLUTIONS
Implicit solution = ”Squeeze”, ”Scale”, and ”Stretch”
Explicit/Data-driven solution = ”Scale” and ”Stretch”
TWO PRESENTATION SOLUTIONS
Show the mathematics
Show how to guess the form of the solution

3. REFERENCES WKBJ APPROXIMATION
GENERAL: Schiff (1955) Morse and Feshbach (1953) Frøman and Frøman (1965) Bender and Orszag (1978) Bransden and Joachain (1989)
GEOPHYSICAL APPLICATIONS: Bremmer (1951) Aki and Richards (1980) Clayton and Stolt (1981) Robinson (1982, 1986) Bleistein (1984) Ursin (1984, 1987) Stolt and Weglein (1985) Amundsen (1994)

4. REFERENCES EIKONAL APPROXIMATION
Glauber (1959) in his 1958 Boulder lectures: ”High-energy collision theory”
Reitan (1979) ”Relativistic Glauber amplitudes for elastic electron and positron scattering by hydrogen atoms and hydrogenlike ions in the ground state”: Phys. Rev. A20, 1385.

5. VELOCITY, POTENTIAL, AND PRIMARIES
Depth
Depth
Time R1 R2 R3 R4 R5

6. FORWARD SCATTERING
Forward model (dimensionless scattering amplitude = designatured seismic data measured at z=0)
The influence of the potential a on the incident and scattered waves is contained in the ”shift function” n.
The forward model takes into account discontinuities (interfaces) by coupling of the incident and scattered fields.

7. FORWARD SCATTERING Forward model WKBJ
Eikonal (closely related to WKBJ)
Born

8. INVERSE SCATTERING
Born: Linear migration-inversion/constant-velocity imaging
WKBJ/Eikonal
INVERSION GOAL: Given aB, find a on closed-form
PROCEDURE: (1) Calculate the derivatives, (2) disregard non-WKBJ terms, (3) cleverly reorganize and sum remaining terms (and hope that the infinite times infinite series correspond to algebraic and exponential functions), and (4) use properties of Fourier transforms

9. CONSTANT-VELOCITY DEPTH IMAGING
~R1+R2
~R1 R1 R2 R3 R4 R5
~R1+R2+R3
~R1+R2+R3+R4
~R1+R2+R3+R4+R5
The Born potential is far from the actual potential, both in layer amplitude and reflector depth

10. ”SQUEEZE” AND ”STRETCH”

11. GUESS ON IMPLICIT SOLUTION: ”SQUEEZE”, ”SCALE” AND ”STRETCH”
THE SOLUTION FORMALLY CAN BE DERIVED FROM WKBJ INVERSE SCATTERING!
WKBJ CORRECTION AMPLITUDE:

12. WKBJ THEORY GIVES Relationship between Born and ”squeezed” potentials
Replace ”squeeze”
by ”scale”
”Scale” precise when O(R3) ^ R

13. WKBJ DATA-DRIVEN SOLUTION: ”SCALE” AND ”STRETCH” (INDEPENDENT OPERATIONS)
No information other than Born potential required!

14. EIKONAL DATA-DRIVEN SOLUTION: ”SCALE” AND ”STRETCH”
No information other than Born potential required!

15. EXAMPLE: ”HIGH-VELOCITY” CONTRAST
WKBJ
Eikonal

16. ”LOW-VELOCITY” CONTRAST MODEL

17. EXAMPLE: ”LOW-VELOCITY” CONTRAST
WKBJ
Eikonal

18. FUTURE WORK BENEFIT FROM M-OSRP WORK
ACOUSTIC AND ELASTIC MULTIDIMENSIONAL
SYNTHETIC DATA TESTS
PRACTICAL QUESTIONS RELATED TO DATA COLLECTION AND PRE-PROCESSING
INVERSE SCATTERING = INVERSE SCATTERING
What is the relationship to LOIS & HOIS ?

19. ACKNOWLEDGEMENTS STATOIL FOR ALLOWING US TO PUBLISH THIS WORK
ART FOR SHARING HIS VISIONS ON ”VELOCITY INDEPENDENT DEPTH IMAGING” WHICH TRIGGERED THE CURRENT WORK
M-OSRP FOR SIGNIFICANT PROGRESS ON PROJECTS THAT WILL IMPACT SEISMIC EXPLORATION AND PRODUCTION