1 Riemannian Wavefield Migration: Imaging non-conventional wavepaths and geometries Jeff Shragge Geophysics Department University Oral Qualification Exam

2 Agenda What is my problem? my solution? –Statement of Problem –Proposed Solution –Thesis Objectives –Potential Impact Part I Part II How am I going to solve it? –Riemannian Wavefield Extrapolation –Completed Work –Work to do

3 3-D Seismic Imaging Seismic targets increasingly complex Improved 3-D Seismic acquisition Improved 3-D seismic imaging Improved 3-D interpretation

4 3-D Seismic Imaging Seismic targets increasingly complex Improved 3-D Seismic acquisition Improved 3-D seismic imaging Improved 3-D interpretation

5 Wave-Equation Migration – Review Downward continuation Wavefield U(t,s,g,z = 0) Wavefield U(t,s,g,z = nΔz) Apply recursive filter e (-ik z Δz) n times Imaging Condition

6 3-D Imaging is Successful… Distance – x axis Depth Distance – y axis Distance – x axis From: 3-DSI, Biondi (2004)

7 Distance Depth …so why are we needed? From: 3-DSI, Biondi (2004) Seismic imaging science is good, but not perfect –Irregular and sparse data –Illumination limitations – physical and imaging procedure –Topographic surface complexity –…

8 Distance Depth …so why are we needed? From: 3-DSI, Biondi (2004) Seismic imaging science is good, but not perfect –Irregular and sparse data –Illumination limitations – physical and imaging procedure –Topographic surface complexity –…

9 Distance Depth Illumination Steep Dip reflector Weak reflector Subsalt Imaging Complex Structure From: 3-DSI, Biondi (2004) Poor illumination of subsurface because of : –poor physical illumination (Acquisition) –incomplete imaging procedure (Processing)

10 Topographic surface complexity How do we deal with topography directly in wave-equation imaging?

11 Why do we have Imaging Limitations? Imaging illumination limitations Topographic surface limitations Coordinate system not conformal to propagation direction or acquisition surface Migration physics decoupled from geometry

12 Evidence – I Problem Non-conformal coordinate systems Migration physics decoupled from geometry Resulting Limitations Inaccurate imaging of steep dips –Downward continuation inaccurate at high angles –Overturning waves not used Extrapolation from acquisition surface topography –Hard to define extrapolation axis –Free-surface topography –Deviated well VSP geometry

13 Steep Dip Imaging Accuracy of wavefield extrapolation decreases as propagating waves tend to horizontal Extrapolation Direction

14 Using Overturning Waves Currently do not use potentially useful information provided by overturning waves Extrapolation Direction

15 Proposed Solution Use coordinate system conformal with wavefield propagation Extrapolation Direction

16 Tilted Cartesian Example ELF North-Sea Dataset From: Shan and Biondi (2004) Distance [km] Depth [km] 0 8

17 Tilted Cartesian Example Nmo stack of near offsets (0-1000m) From: Shan and Biondi (2004) Distance [km] Time [s] 8

18 Tilted Cartesian Example Downward ContinuationReverse-time Plane-wave w/ dipping Cartesian coordinates From: Shan and Biondi (2004)

19 Evidence – II Problem Non-conformal coordinate systems Migration physics decoupled from geometry Resulting Limitations Inaccurate imaging of steep dips –Downward continuation inaccurate at high angles –Overturning waves not used Extrapolation from acquisition surface topography –Hard to define extrapolation axis –Free-surface topography –Deviated well VSP geometry

20 Free Surface Topography How to define extrapolation surface orthogonal to free surface?

21 VSP Deviated Well Topography Receiver wavefield acquired in well deviated in 3-D How to define wavefield extrapolation from borehole surface?

22 Proposed Solution Use coordinate system conformal with borehole geometry

23 Summary of Problem Imaging Illumination Limitations Topographic Surface Limitations Coordinate system not conformal to propagation direction or acquisition surface Migration Physics decoupled from Geometry Difficult Steep Dip Imaging No use of Overturning waves Extrapolation from complex free-surface Extrapolation from deviated boreholes

24 Summary of Solution Reduce Imaging Illumination Limitations Enable W.E. imaging directly from Topographic Surfaces Perform Migration on Coordinate systems conformal to propagation direction/acquisition surface Couple Migration Physics with Geometry Improve Steep Dip Imaging Use Overturning waves Eliminate need for free-surface datuming W.E. Imaging for massive 3-D VSP data

25 What am I going to do? Handle multipathing/ triplication Applicable to 3-D field data Handle arbitrary geometry Handle large data volumes Improve steep dip Imaging Eliminate need for free-surface datuming W.E. imaging for massive 3-D VSP data Requirements AVA studies Physical property analysis Impact 3-D curvilinear coordinate wave-equation migration method

26 End of Part I Handle multipathing/ triplication Applicable to 3-D field data Handle arbitrary geometry Handle large data volumes Improve steep dip imaging Eliminate need for free-surface datuming W.E. imaging for massive 3-D VSP data Requirements AVA studies Physical property analysis Impact 3-D curvilinear coordinate wave-equation migration method

27 Potential Impact Improved steep dip imaging Better interpretation geologic structure Assist downstream processing tasks –AVA studies –Physical property analysis Massive 3-D VSP imaging W.E. imaging directly applicable even in deviated boreholes Provide additional tools –Angle-domain CIGs, etc.

28 Part II – Agenda Riemannian Wavefield Extrapolation (RWE) Completed Work –Generating Coordinate systems –Extrapolation examples –Deal with or avoid Triplication Work to do –Tackling the Geophysical Imaging problem –Technical issues

29 Part II – Agenda Riemannian Wavefield Extrapolation (RWE) Completed Work –Generating Coordinate systems –Extrapolation examples –Deal with or avoid Triplication Work to do –Tackling the Geophysical Imaging problem –Technical issues

30 RWE in 2-D ray-coordinates RiemannianCartesian Extrapolation Direction Orthogonal Direction

31 RWE: Helmholtz equation (associated) metric tensor Laplacian Coordinate system Sava and Fomel (2004)

32 RWE: (Semi)orthogonal coordinates

st order2 nd order 1 st order RWE: Helmholtz equation Ray-coordinate Interpretation α = velocity function J = geometric spreading or Jacobian

34 RWE: Dispersion relation Riemannian Cartesian

35 RWE: Dispersion relation Riemannian Cartesian

36 RWE: Wavefield extrapolation Riemannian Cartesian

37 Part II – Agenda Riemannian Wavefield Extrapolation (RWE) Completed Work –Generating coordinate systems –Extrapolation examples –Deal with or avoid Triplication Work to do –Tackling the Geophysical Imaging problem –Technical issues

38 Generating Coordinate Systems Single Arrival Multiple Arrival Monochromatic ray tracing To what degree can wave propagation be modeled in a coordinate system? Broad-band ray tracing Cartesian Coordinates (SEP-114) Ray Coordinates (SEP-115)

39 Monochromatic ray tracing Use local velocity and WAVEFIELD PHASE information to calculate coordinate system Create a coordinate system conformal with wavefield propagation direction

40 Monochromatic ray tracing 0 Distance Depth Distance Depth NOTE: Gradient of monochromatic wavefield phase shows orientation of propagation direction

41 What is a phase-ray? Distance Depth Phase-ray

42 Calculating phase-rays Cartesian ray equations Decoupled system of 1 st order ODEs Ray solution: –Specify an initial point –Numerically integrate dx and dz –Output x and z coordinates of ray Rayfield explicitly dependent on frequency

43 Phase-ray example Distance Depth

44 Part II – Agenda Riemannian Wavefield Extrapolation (RWE) Completed Work –Generating Coordinate systems –Extrapolation examples –Deal with or avoid Triplication Work to do –Tackling the Geophysical Imaging problem –Technical issues

45 Adaptive phase-ray extrapolation Calculate 1 Phase-ray step Calculate 1 wavefield step Bootstrapping procedure

46 Example – Point Source Distance Depth

47 Example – Plane wave Depth Distance

48 Example – Salt Depth Distance

49 Part II – Agenda Riemannian Wavefield Extrapolation (RWE) Completed Work –Generating Coordinate systems –Extrapolation examples –Deal with or avoid Triplication Work to do –Tackling the Geophysical Imaging problem –Technical issues

50 Coordinate System Triplication Depth Distance

51 Coordinate System Triplication Smooth Velocity Ray-trace on single arrival Eikonal solvers Coordinate System Triplication: Avoid or Handle? Avoid Triplication Avoid Zero Division: Add ε (Sava & Fomel,2003) Iterative coord. updating SEP-115 Handle Triplication Numerically isolate triplication branches SEP-115 Regularization through Inversion

52 Regularization through Inversion 0 Time [s] X-Coordinate Time [s] Z-Coordinate Shooting Angle [deg]

53 Regularization through Inversion Time [s] X-Coordinate Shooting Angle [deg] 0 Time [s] Z-Coordinate Data: X d and Z d coordinates from ray-tracing Model parameters to be fit: X m and Z m X m and Z m are triplication-free if they satisfy Laplace equation ΔX m = 0 = ΔZ m

54 Regularization through Inversion Time [s] X-Coordinate Shooting Angle [deg] 0 Time [s] Z-Coordinate ΔX m = 0 = ΔZ m Geometric Regularization Minimize Curvature of X m and Z m

55 Fitting Goals Equation TypeX-CoordinateZ-Coordinate Physical Data Fitting W(J)(X m -X d )  0W(J)(Z m -Z d )  0 Geometric Regularization ΔX m  0ΔZ m  0 W(J) dependent on Jacobian of coordinate system, e.g., J 0 = initial spreading n = curve adjusting parameter

56 Part II – Agenda Riemannian Wavefield Extrapolation (RWE) Completed Work –Generating Coordinate systems –Extrapolation examples –Deal with or avoid Triplication Work to do –Tackling the geophysical imaging problem –Technical issues

57 Imaging – Overturning waves 3-D surveys consist of 1000s of shot + receiver locations –Shot-profile migration often prohibitively expensive Plane-wave migration (Liu et al., 2002) –Phase encoding scheme (Romero et al., 2000) independent of coordinate-system

58 Imaging – Overturning waves Source wavefield Receiver wavefield Imaging Condition

59 Imaging – Overturning waves

60 Imaging – Overturning waves Multiple plane-wave imaging 3-D Extension – use 2-D plane-wave filter –Cylindrical wave migration (Duquet et al., 2001)

61 Imaging – Massive 3-D VSP Experiment Setup –2-D areal surface source pattern –3-C downhole receivers –Record forward- and backscattering wavefield

62 VSP Imaging – Reciprocity Use reciprocity to create common receiver profiles –Large computational savings Source wavefield – plane wave coordinates Receiver wavefield – quasi-spherical coordinates Source wavefield Receiver wavefield

63 VSP Imaging – Phase Encoding Use algorithm similar to plane-wave –Random time shifts Source wavefield Receiver wavefield

64 VSP Imaging – Passive Seismic Mode Back-propagate wavefield from borehole to source Record P- and S-waves – multimode imaging –Permutations of S + R velocity models, causal/acausal propagation –Examine 6 candidate scattering modes (Shragge and Artman, 2003) Source wavefield Receiver wavefield

65 Part II – Agenda Riemannian Wavefield Extrapolation (RWE) Completed Work –Generating Coordinate systems –Extrapolation examples –Deal with or avoid Triplication Work to do –Tackling the Geophysical Imaging problem –Technical issues

66 Technical Issues - I 3-D ray-coordinate system non-orthogonality –May be problematic because have cross partial differential terms –Solution – Use a Nth order polynomial to approximate the coordinates –Analytical derivatives Time [s] X-Coordinate Shooting Angle [deg] Time [s] Z-Coordinate

67 Technical Issues - II Code and test different extrapolation operators –Ray-coordinate 15 degree equation used –Examine more accurate operators, e.g., 45 degree, split-step Fourier Fourier Finite Differencing –2-D methods that approximate 3-D solutions, e.g., Splitting –Examine stability of operators in 2-D and 3-D

68 Field Data Verification Overturning Waves –3-D ELF North Sea synthetic + field data sets SEP) –3-D Exxon/Mobil data set SEP) Massive 3-D VSP –Paulsson Geophysical Inc. has agreed to make available a data set acquired in Long Beach, CA 5 wells – straight to deviated in 3-D 240 receivers, 1000s of shot Adequate wavefield sampling ~ 50 feet/receiver

69 Timeline for Proposed Work Activity 2004 Test 2-D code on synthetic/field data Develop other extrapolation operators Develop 2-D migration code Test methods for handling triplications Test non-orthogonality of 3-D coords. 3-D field tests – Overturning/VSP Develop 3-D migration code Graduate 3-D synthetic tests – Overturning/VSP Write Thesis Internship

70 Questions?

71 Depth Topography Example

72 Summary of Problem Problem Coordinate system not conformal to wavefield propagation Migration physics decoupled from geometry Resulting Limitations Inaccurate imaging of steeply dipping structure Difficult to extrapolate from topographic surfaces –free-surface topography, deviated well VSP Extrapolation Direction

73 Summary of Solution Proposed Solution Define Migration on generalized coordinate systems Couple physics of migration to geometry –Curvilinear coordinate system more conformal with: orientation of propagating wavefield topography of acquisition surface Extrapolation Direction

74 Potential Imaging Improvements Use more wavepaths to improve steep dip imaging –More information to contribute to imaging –Increase migration aperture Extrapolate direction from surfaces with topography –Directly apply wave-equation migration –Eliminate need for datuming Extrapolation Direction

75 Potential Impact Improved steep dip imaging Better interpretation geologic structure Assist downstream processing tasks –AVA studies –Physical property analysis Massive 3-D VSP imaging W.E. imaging directly applicable even in deviated boreholes Provide additional tools –Angle-domain CIGs, etc.

76 Objectives of Proposed Work Geoscience –3-D migration method improves steep dip imaging handles complex acquisition surfaces Technological –3-D curvilinear coordinate wave-equation migration method –Applicable to real 3-D seismic data

77 What is a phase-ray? Distance Depth = Ray magnitude = Ray vector

78 Calculating a phase-ray Cartesian coordinate phase-ray equations: Isolate phase-gradient:

79 Adaptive phase-ray extrapolation Recall: 1. Generate rayfield from wavefield phase gradient 2. Extrapolate wavefield on ray coordinate system Calculate 1 Phase-ray step Calculate 1 wavefield step

80 But…what coordinate system? Questions: –To what degree should wave propagation effects be modeled into the coordinate system? METHOD COMPLEXITY High Low Current Numerical Stability Possibility of ray coordinate triplication Cartesian Tilted Cartesian RWE: smooth velocity model RWE: rough velocity model

81 Mission Statement Why should wave propagation effects be incorporated into the coordinate system? High 0 Coordinate system conforming with wavefront High 0 Possibility of Extrapolating overturning waves Increasing method complexity Cartesian Tilted Cartesian RWE: smoothed velocity model RWE: true velocity model ADAPTIVE PHASE-RAY EXTRAPOLATION

82 Wavefield triplication Separated triplication branches Triplications Use appropriate one-sided derivatives at discontinuities Wavefield triplication Ray-coordinate triplication Numerical instability

83 Triplication: Gaussian Example Depth Error Distance

84 Triplication: Gaussian Example Depth Distance

85 Distance Depth Triplication: Sigsbee 2A