John C. Bancroft and grad students AVO, migration apertures Fresnel zones, stacking, Q, anisotropy, and fast inversions John C. Bancroft and grad students University of Calgary CREWES 2003 NSERC

Grad students Shuang Sun Pavan Elapavuluri Kun Liu (and Hugh Geiger) John Millar Xiang Du Zhihong (Nancy) Cao Chunyan (Mary) Xiao

Outline Differentiators Fresnel zones AVO migration apertures Q Wavefield extrapolators Multiples and radon transforms Inversion

Digital differentiator Accuracy Speed Beauty (1, -1) 1, 0, -1) better Theoretical shape?

Inverse FFT of (j)

Derivative assumes: Maximum bandwidth Ideal interpolator Sinx/x operator Derivative of sinx/x SincFn.m

What is the basic unit of laryngitis? Shuang Sun Stacking specula energy Fresnel zone for offset and dipping reflectors Balanced AVO in areas of poor geometry Caution: model based migration One hoarsepower What is the basic unit of laryngitis?

What size hole can we see with seismic?

What happens at the edge of a reflector?

Fresnel zone Fresnel Size: frequency depth velocity Claerbout

Depends on the size of the Fresnel radius Only for stacked data!!!

Analytic function for zero-offset case (no wavelet) Solution to the acoustic wave equation for zero-offset, the diffraction response can be written as: Diffraction operator D0: te : minimum two-way travel time to the edge of the reflector; t’=t-te: time measured after onset time te ; e : angle between the normal to the reflector and raypath of the minimum travel time to the edge of the reflector. Berryhill, 1977

Diffraction amplitude Time Angle

Diffraction amplitude

Scaled amplitude (evaluate phase)

Fresnel zone with Ricker wavelet source Source signature with dominant frequency 50hz Reflected signal amplitude

Fresnel zone with fixed bandwidth Frequency from 0-150 hz Frequency from 0-120 hz

Reflector smaller than Fresnel zone: width 50m

Reflector width: 1000m

Amplitude (no wavelet)

Diffraction modelling

Valid area for AVO ?

Comments Locations for AVO analysis before migration should be at least a Fresnel radius away from the target edge. The amplitudes will be in error if the target size is smaller than or equal to the Fresnel zone. Wavelet will contribute to the size of contamination area. Only zero offset was considered Should consider a prestack migration to perform AVO analysis.

Offset traveltime 2h DSR eqn.

Cheop’s pyramid - DSR eqn. Hyperbola CMP at T0 Zero off. DSR eqn. Const. Off CMP gath.

Prestack data, flat… 7 Cheop’s . Specula energy

Reflections from dipping event Reflecting element

Prestack data, dipping... Cheop’s p. Defines exact shape of surface, specula energy

Cheop’s summation: specula energy Fresnel zones

Prestack Fresnel zone horizontal layers

P. S. Fresnel zone for dipping event

Contours : equal angle of incidence Offset increases with x Angle 1

Constant offset migration

What about dipping specula energy? Smear of dips Still OK ???

CMP gather

Limited aperture EO gather

Comments … Prestack specula energy can be identified, Limited aperture horizontal and dipping Limited aperture faster, better SNR, may preserve amplitudes Optimum size of a super-CMP gather can be defined using Fresnel zones Should use prestack migration gathers for AVO Use model based migration with caution

Comments Constant offset migration EOM smears AVO energy (OK) requires velocity information EOM also smears AVO energy (OK) Stacking within the Fresnel zone may reduce acquisition geometry artifacts

What is one millionth of a mouth wash? Pavan Elapavuluri Q What is one millionth of a mouth wash? One microscope

Wavelets with varying Q and Time Build Q wavelet table: [Q, time] Q: 30, 50, 70, 90, 110, … T: 0 to 6.0 sec Cross-correlate (coherence) trace with all wavelets At a given time, peak corresponds to Q

9 wavelets T=1000 T=3000 T=5000 Q=90 Q=70 Q=50

3 Spikes on trace Q = 70 T=1000 T=3000 T=5000 Q=90 Q=70 Q=50

Noise on trace Q = 70 T=1000 T=3000 T=5000 Q=90 Q=70 Q=50

Kun Liu (and Hugh Geiger) Wavefield extrapolation What is 2000 pounds of Chinese soup? Won ton

Input

200m depth extrapolation …with adaptive taper

What is half a large intestine? Xiang Du Finite element method integrated with Finite difference migration A semicolon What is half a large intestine?

Comparison: FE-FDM with FX-FDM Result of FE-FDM Result of FX-FDM (SU)

Trace comparisons

Steep oblique model Velocity model Seismic section

Comparison of methods Result of FE-FDM Result of FX-FDM (SU)

Runtime and memory requirements FE-FD 26 secs 2.2 meg FK-FD 31 secs 1.3 meg.

What is one thousand aches? Zhihong (Nancy) Cao Chunyan (Mary) Xiao Tutorials Review of Multiple suppression techniques Multiple attenuation using the Radon transform What is one thousand aches? A kilohurtz

after Kabir and Marfurt, 1999

Semblance plot is a Radon transform

Lorraine Bloom David Henley Larry Lines The model and inverse

Modelling process Inversion expert

The Model (Not the inverse)

The Inverse

The artifacts (of inversion) Before modelling After inversion

Of course, … some things are easier to model and invert.

Inversion Mathematical process Geophysical terminology Used in many geophysical applications: decon., statics, tomography, tau-p, … Geophysical terminology Estimation of rock properties Modelling <-> Inversion

What is the ratio of an igloo’s circumference to its diameter? Millar time Multigrid inversion Borrowed from fluid dynamics Recursive Variable grid size Solutions are frequency dependent Fast What is the ratio of an igloo’s circumference to its diameter? Eskimo pi

Iterative solution

Laplace’s equation

Variable grid size

Analogous to FFT ???

Sorting a 2-D array into a vector

Extractions, smoothings and interpolations S = Solution with one iterative step 33x33 17x17 3x3 9x9 5x5 3x3 9x9 5x5 Pass defects down and errors up

Performance

Conclusions: Migration apertures speed, SNR, balanced amplitudes CMP AVO: consider Fresnel zones Finite element / Finite difference migration Multiples and Radon transforms Q and differential operators Multi grid inversions

The end