# 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

a) forward extrapolate b) backwardcondition extrapolate source wavefield (modeling) x receiver wavefield 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 x depth plane - output to next 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) 4500 -200 0

z (m) 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)+p o(x2)+po(x3)] 2/[pi(x4)+po(x3)] 2/[pi(x3)+po(x3)] 4/ [pi(x5)+pi(x4)+p o(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 wideangle 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 zero-phase 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 Greens function - then forward extrapolate source wavefield using one-way operator - ideal for marine air-gun source (constant velocity Greens 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 Greens 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 x=400m x=0m v=1500m/s =1000kg/m3 v=1549m/s =1478kg/m3

0m 28m 32m v=1598m/s =1955kg/m3 220m v=1598m/s =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 0 and ~45 degree incidence receiver array @ 0 receiver array @ 45 upgo ing r eflec ted wa v e

reflector ave w ed t t i m ans r t ing o

g n dow 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 Deconvoluti on

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 Greens 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 The reflectivity at each depth level is determined using a condition modified deconvolution imaging condition expressed as a crosscorrelation over autocorrelation, which ensures that the stability factor does not contaminate the phase response. S 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 ; ) S 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 1 such 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 90 aperture with no taper deconvolution imaging condition PSPI-type extrapolator: smoothed velocities and a 90 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 84.5 aperture with 1.75 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

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