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 20 2003 Hugh Geiger

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 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

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

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

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 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=3000m/s V=2000m/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 Input pts Output pts vi(x) vo(x) x2 x3 x1 x4 x5 vo(x3)

V=3000m/s V=2000m/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 Input pts Output pts vi(x) vo(x) x2 x3 x1 x4 x5 vi(x1) vi(x2) vi(x3) vi(x4) vi(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 Input pts Output pts vi(x) vo(x) x2 x3 x1 x4 x5 vi(x1) vi(x2) vi(x3) vi(x4) vi(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 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)]

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 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)]

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 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 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

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

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

upgoing reflected wave 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 upgoing reflected wave reflector downgoing transmitted wave

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.

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. 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)

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