Transcript True-amplitude seismic imaging using recursive Kirchhoff

Depth imaging using
slowness-averaged
Kirchoff extrapolators
Hugh Geiger, Gary Margrave, Kun Liu
and Pat Daley
CREWES Nov 20 2003
POTSI* Sponsors:
C&C Systems
*Pseudo-differential Operator Theory in Seismic Imaging
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
Sigsbee image - “Kirchhoff” diffraction stack
J. Paffenholz - SEG 2001
Sigsbee image - recursive “wave-equation”
J. Paffenholz - SEG 2001
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!
Wave equation depth migration = wavefield
extrapolation + imaging condition
a) forward extrapolate
source wavefield (modeling)
b) backward extrapolate
receiver wavefield
x
t
t
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
x
z=0
z=1
z=2
z=3
z=4
t
reflector
In constant velocity, 2D forward (green) and backward (red)
extrapolators sum over a hyperbola and output to a point
recursive Kirchhoff vs. non-recursive Kirchhoff
- operator varies laterally v(x,z)
- does not vary with time
- output to next depth plane
x
t
- operator varies laterally
- also varies with time
- output to depth image
x
t
Recursive extrapolation can be implemented in space-frequency domain
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
Extrapolator tests
x (m)
-200
0
z (m)
4500
0 2.048
t (s)
1.024
Band-limited
zero- phase
impulse
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.
V=2000m/s
V=3000m/s
•V=(1+0.001i)*2000m/s
•V=(1+0.001i)*3000m/s
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
vi(x)
vo(x)
Input pts
vo(x3)
vo(x3) vo(x3) vo(x3)
vo(x3)
Output pts
x1
x2
x3
x4
x5
V=2000m/s
V=3000m/s
cos taper 70°-87.5°
V=(1+0.001i)*2000m/s
V=(1+0.001i)*3000m/s
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
vi(x)
vo(x)
Input pts
vi(x1)
vi(x2)
vi(x3)
vi(x4)
vi(x5)
Output pts
x1
x2
x3
x4
x5
V=2000m/s
V=3000m/s
cos taper 70°-87.5°
V=(1+0.001i)*2000m/s
V=(1+0.001i)*3000m/s
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
vi(x)
vo(x)
Input pts
vi(x3)
vi(x3)
vi(x3)
vi(x3)
vi(x3)
vi(x1)
vi(x2)
vi(x3)
vi(x4)
vi(x5)
Output pts
x1
x2
x3
x4
x5
V=(1+0.001i)*2000m/s
V=(1+0.001i)*3000m/s
Kirchhoff WEYL extrapolator
• velocity defined as average of
velocities at input and output points
• no wavenumber-frequency domain
equivalent for Weyl extrapolator
velocity
vi(x)
vo(x)
Input pts
0.5*[vi(x2)+vo(x3)]
0.5*[vi(x4)+vo(x3)]
0.5*[vi(x3)+vo(x3)
0.5*[vi(x1)+vo(x3)]
]
0.5*[vi(x5)+vo(x3)]
Output pts
x1
x2
x3
x4
x5
V=2000m/s
V=3000m/s
cos taper 70°-87.5°
Kirchhoff averaged slowness
• velocity defined using average of
slownesses along straight ray path
• best performance of all extrapolators
based on kinematics and amplitudes
velocity
vi(x)=1/pi(x)
vo(x)=1/po(x)
Input pts
2/[pi(x2)+po(x3)]
4/[pi(x1)+pi(x2)
+po(x2)+po(x3)]
2/[pi(x4)+po(x3)]
2/[pi(x3)+po(x3)
]
4/[pi(x5)+pi(x4)
+po(x4)+po(x3)]
Output pts
x1
x2
x3
x4
x5
V=2000m/s
V=3000m/s
cos taper 70°-87.5°
•V=(1+0.001i)*2000m/s
•V=(1+0.001i)*3000m/s
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.
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 zerophase response (possibly Gabor deconvolution to
address attenuation), accurate source modeling,
tapered recursive Kirchhoff extrapolators, and a
modified deconvolution imaging condition
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)
z=0
z = dz
dx
z = 2dz
- might be useful for land seismic (the Green’s function is complicated)
z=0
z = dz
dx
z = 2dz
free surface
rec array
source 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
x=400m
x=0m
ρ=1000kg/m3
v=1500m/s
v=1549m/s ρ=1478kg/m3
0m
28m
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
Modeled with finite difference code (courtesy Peter Manning) to examine
response of isolated reflector at 0º and ~45º degree incidence
receiver array @ 0º
receiver array @ 45º
reflector
Marmousi airgun wavelet
desired 24 Hz
zero-phase
Ricker wavelet
~60ms
normal
incidence
reflection
~60ms
~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.
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
• 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
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
Adaptive extrapolator taper
• an adaptive taper minimizes artifacts from data
truncation and extrapolator operator truncation
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.
1
U ( x, y, z; ) D ( x, y, z;  )
R( x, y, z; ) 
F ( )
d


2
D( x, y, z; ) D ( x, y, z;  )  stab
R( x, y, z; )
U ( x, y, z; )
D( x, y, z; )
F ( )
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
F ( ) S  ( ) is a good estimator of the signal to
noise ratio at each frequency, normalized such
1
that:
F ( ) S  ( ) d  1
2 
F (() S  ( ) is the source spectrum)
Marmousi velocity model (m/s)
Marmousi reflectivity model
• calculated for vertical incidence
Marmousi model shallow image
• deconvolution imaging condition
• PAVG-type extrapolator: slowness-averaged velocities
and a 90º aperture with no taper
• deconvolution imaging condition
• PSPI-type extrapolator: smoothed velocities and a 90º
aperture with no taper
• accurate prestack imaging requires good lateral and
vertical propagation of source and receiver wavefields
• 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
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