1. Depth imaging using slowness-averaged Kirchoff extrapolators
Slowness-averaged Kirchhoff extrapolators
Depth imaging using slowness-averaged Kirchoff extrapolators
Hugh Geiger, Gary Margrave, Kun Liu
and Pat Daley
CREWES Nov
Hugh Geiger

2. POTSI* Sponsors: C&C Systems
*Pseudo-differential Operator Theory in Seismic Imaging

3. Overview motivation wave equation depth migration simplified
our approach
recursive Kirchhoff extrapolators
conceptual
theory
PSPI, NSPS, SNPS, and Weyl-type extrapolators
PAVG or slowness-averaged extrapolator
2D tests
towards “true-amplitude” depth migration
accurate source modeling
extrapolator aperture size and taper width
modified deconvolution imaging condition
depth imaging of Marmousi dataset
conclusions

4. Sigsbee image - “Kirchhoff” diffraction stack
J. Paffenholz - SEG 2001

5. Sigsbee image - recursive “wave-equation”
J. Paffenholz - SEG 2001

6. The standard terminology
“Kirchhoff” migration
- synonym: weighted diffraction stack
- typically nonrecursive (Bevc semi-recursive)
- diffraction surface defined by
ray-tracing or eikonal solvers
- first arrival, maximum energy, multi-arrivals
- more efficient/flexible, common-offset possible
“wave equation” migration
– synonyms: up/downward continuation
and forward/inverse wavefield extrapolation
with an imaging condition
- typically recursive
- typically Fourier or finite difference or combo
- less efficient/flexible, common-offset difficult
but all extrapolators are based on the wave equation!

7. Wave equation depth migration = wavefield extrapolation + imaging condition
a) forward extrapolate b) backward extrapolate
source wavefield (modeling) receiver wavefield t t x x z z
horizontal reflector (blue)
(figures a and b courtesy J. Bancroft)
c) deconvolution of receiver wavefield by source wavefield each extrapolated (x,t) depth plane yields depth image x z
image of horizontal reflector

8. z=0
z=2
z=3
z=4
z=1 x t
reflector
In constant velocity, 2D forward (green) and backward (red)
extrapolators sum over a hyperbola and output to a point

9. recursive Kirchhoff vs. non-recursive Kirchhoff
- operator varies laterally v(x,z)
does not vary with time
output to next depth plane
- operator varies laterally
also varies with time
output to depth image x x t t
Recursive extrapolation can be implemented in space-frequency domain

10. Our approach shot-record “wave-equation” migration
recursive downward extrapolation (forward modeling) of the source wavefield using one-way recursive Kirchhoff extrapolators
recursive downward extrapolation (backpropagation) of the receiver wavefield using one-way recursive Kirchhoff extrapolators
modified stabilized deconvolution imaging condition at optimal image resolution (Zhang et al, 2003)
why Kirchhoff extrapolators?
Applications: resampling the wavefield, rough topography, and complex near surface velocities
POTSI goal: excellent imaging of 2D and 3D land data

11. Extrapolator tests
For the 2D forward extrapolator tests that follow, there are two impulses at x=1728m and x=2880m, t=1.024s, and z=0m. Output (e.g. green curve above – a hyperbola in constant velocity) lies on the x-t plane at z=-200m.

12. V=3000m/s
V=2000m/s

13. V=( i)*2000m/s
V=( i)*3000m/s

14. Kirchhoff/k-f PSPI extrapolator
velocity defined at output point
wavenumber-frequency domain PSPI
has wrap-around that can also be
reduced using a complex velocity
velocity
Input pts
Output pts
vi(x)
vo(x) x2 x3 x1 x4 x5
vo(x3)

15. V=3000m/s
V=2000m/s
cos taper 70°-87.5°

16. V=( i)*2000m/s
V=( i)*3000m/s

17. Kirchhoff/k-f NSPS extrapolator
velocity defined at input points
wavenumber-frequency domain NSPS
has wrap-around that can also be
reduced using a complex velocity
velocity
Input pts
Output pts
vi(x)
vo(x) x2 x3 x1 x4 x5
vi(x1)
vi(x2)
vi(x3)
vi(x4)
vi(x5)

18. V=2000m/s
V=3000m/s
cos taper 70°-87.5°

19. V=( i)*2000m/s
V=( i)*3000m/s

20. SNPS as cascaded k-f PSPI/NSPS
velocity defined using input points
but output point formulation possible
wavenumber-frequency domain SNPS
has wraparound that can reduced
using complex velocities
velocity
Input pts
Output pts
vi(x)
vo(x) x2 x3 x1 x4 x5
vi(x1)
vi(x2)
vi(x3)
vi(x4)
vi(x5)

21. V=( i)*2000m/s
V=( i)*3000m/s

22. Kirchhoff WEYL extrapolator
velocity defined as average of
velocities at input and output points
no wavenumber-frequency domain
equivalent for Weyl extrapolator
velocity
Input pts
Output pts
vi(x)
vo(x) x2 x3 x1 x4 x5
0.5*[vi(x1)+vo(x3)]
0.5*[vi(x2)+vo(x3)]
0.5*[vi(x3)+vo(x3)]
0.5*[vi(x5)+vo(x3)]
0.5*[vi(x4)+vo(x3)]

23. V=2000m/s
V=3000m/s
cos taper 70°-87.5°

24. Kirchhoff averaged slowness
velocity defined using average of
slownesses along straight ray path
best performance of all extrapolators
based on kinematics and amplitudes
velocity
Input pts
Output pts
vi(x)=1/pi(x) x2 x3 x1 x4 x5
4/[pi(x1)+pi(x2)+po(x2)+po(x3)]
2/[pi(x2)+po(x3)]
2/[pi(x3)+po(x3)]
2/[pi(x4)+po(x3)]
vo(x)=1/po(x)
4/[pi(x5)+pi(x4)+po(x4)+po(x3)]

25. V=2000m/s
V=3000m/s
cos taper 70°-87.5°

26. V=( i)*2000m/s
V=( i)*3000m/s

27. Comments on extrapolator tests
The recursive Kirchhoff “averaged slowness” method should compare well against other wide-angle methods, such as Fourier finite-difference.
We plan to compare performance and accuracy between our new method and other methods
Note that the recursive Kirchhoff method has advantages over methods requiring a regularized grid, for example when dealing with resampling and rough topography.

28. Towards “true-amplitude” depth migration
“True-amplitude” depth migration depends on preprocessing, velocity model, extrapolators, source modeling, and imaging condition
a more correct term is “relative amplitude preserving” depth migration, because a number of effects are not typically considered, such as transmission losses (including mode conversions), attenuation, and reflector curvature
our approach includes preprocessing towards a zero-phase response (possibly Gabor deconvolution to address attenuation), accurate source modeling, tapered recursive Kirchhoff extrapolators, and a modified deconvolution imaging condition

29. Accurate source amplitudes
- seed a depth level with a bandlimited analytic Green’s function
then forward extrapolate source wavefield using one-way operator
ideal for marine air-gun source (constant velocity Green’s function)
simple to model source arrays and surface ghosting (e.g. Marmousi)
might be useful for land seismic (the Green’s function is complicated)
z = 0
z = 2dz dx
z = dz
z = 0
z = 2dz dx
z = dz

30. free surface
source array
rec array
hard water bottom
Complications for Marmousi imaging:
free-surface and water bottom
ghosting and multiples modify wavelet
source and receiver array directivity
two-way wavefield, one-way extrapolators
heterogeneous velocity

31. x=0m
x=400m 0m
v=1500m/s ρ=1000kg/m3
28m
v=1549m/s ρ=1478kg/m3
32m
v=1598m/s ρ=1955kg/m3
220m
v=1598m/s ρ=4000kg/m3
Marmousi source array: 6 airguns at 8m spacing, depth 8m
receiver array: 5 hydrophones at 4m spacing, depth 12m

32. upgoing reflected wave
Modeled with finite difference code (courtesy Peter Manning) to examine response of isolated reflector at 0º and ~45º degree incidence
receiver 0º
receiver 45º
upgoing reflected wave
reflector
downgoing transmitted wave

33. Marmousi airgun wavelet
desired 24 Hz
zero-phase
Ricker wavelet
~60ms
~60ms
normal incidence reflection
~45 degree incidence reflection
After free-surface ghosting and water-bottom multiples, the Marmousi airgun wavelet propagates as ~24 Hz zero-phase Ricker with 60 ms delay.

34. Deconvolution
The deconvolution chosen for the Marmousi data set is a simple spectral whitening followed by a gap deconvolution (40ms gap, 200ms operator)
this yields a reasonable zero phase wavelet in preparation for depth imaging

35. the receiver wavefield is then static shifted by -60ms to create an approximate zero phase wavelet
if the receiver wavefield is extrapolated and imaged without compensating for the 60ms delay, focusing and positioning are compromised, as illustrated using a simple synthetic for a diffractor
diffractor imaging with no delay
diffractor imaging with 60ms delay

36. Shot modelling
the shot can be seeded at depth using finite difference modeling or constant velocity Green’s functions. This accounts for source directivity and inserts the correct zero-phase wavelet
Marmousi shot wavefield seeded at 24m depth with ghost amplitudes
Seeded shot wavefield propagated to 400m depth – phase preserved

37. Adaptive extrapolator taper
an adaptive taper minimizes artifacts from data truncation and extrapolator operator truncation

38. Modified deconvolution imaging condition
The reflectivity at each depth level is determined using a modified deconvolution imaging condition expressed as a crosscorrelation over autocorrelation, which ensures that the stability factor does not contaminate the phase response.
Estimate of true-amplitude reflectivity
Upgoing receiver wavefield backward extrapolated to depth z
Downgoing source wavefield forward extrapolated to depth z
Optimal chi-squared weighting function, where
is a good estimator of the signal to noise ratio at each frequency, normalized such that:
( is the source spectrum)

39. Marmousi velocity model (m/s)

40. Marmousi reflectivity model
calculated for vertical incidence

41. Marmousi model shallow image
deconvolution imaging condition
PAVG-type extrapolator: slowness-averaged velocities and a 90º aperture with no taper

42. deconvolution imaging condition
PSPI-type extrapolator: smoothed velocities and a 90º aperture with no taper

43. accurate prestack imaging requires good lateral and vertical propagation of source and receiver wavefields

44. deconvolution imaging condition
PAVG-type extrapolator: slowness-averaged velocities and a 84.5º aperture with 1.75º taper (10dx/5dx per dz)
reduced extrapolator aperture can result in inaccurate imaging of steeper dips

45. Conclusions
Kirchhoff extrapolators can be designed to mimic a variety of explicit extrapolators (e.g. PSPI, NSPS)
Kirchhoff extrapolators can provide flexibility in cases of irregular sampling and rough topography
the slowness averaged Kirchhoff extrapolator appears to have excellent wide-angle accuracy in cases of strongly varying lateral velocity
when combined with a modified deconvolution imaging condition, Kirchhoff extrapolators can be used for true amplitude imaging