Migration velocity analysis using multifocusing/crs attributes based on paraxial raytracing

Migration velocity analysis using multifocusing/crs attributes based on paraxial raytracing Tijmen Jan Moser Fugro-Jason, The Netherlands 29 September 24 Charles University, Prague, Czech Republic

Biography 987 - masters degree Geophysics, State University of Utrecht, The Netherlands (Nolet) 992 - PhD, "Shortest path method for seismic ray tracing in complicated media" State University of Utrecht, The Netherlands (Helbig, Nolet) 992 - Postdoc Amoco, Tulsa, OK (Gray, Treitel) 992-4 - Institut Français du Pétrole and Institut de Physique du Globe, Paris (Lailly, Madariaga, Tarantola) 994-5 - Institute of Solid Earth Physics and Norsk Hydro (Hanyga, Pajchel) 996-7 - Alexander von Humboldt fellow -Karlsruhe (Hubral, Karrenbach, Shapiro) 997-2 - Geophysical Institute of Israel/Norsk Hydro (Landa, Keydar, Gelchinsky, Pajchel) 2-present - Fugro-Jason

. Prestack depth migration 2. Multifocusing/Common reflection surface stack 3. Migration velocity analysis 4. MVA using MF/CRS and paraxial raytracing

. Prestack depth migration Diffraction stack (A unified approach to 3-D seismic reflection imaging, Part II: Theory. [Geophysics 6, 759 (996)]. Tygel et al.) where V (x) image, x subsurface point x S, x R source, receiver points U(x S, x R, t) (prestack) seismic data A weighting factor V (x) = dx Sdx R AU(x S,x R,T(x S,x,x R )) T (x S, x, x R )(multivalued) traveltime from source to image point to receiver point in a presumed known Earth model Image formula from first Born approximation Background velocity model c(x), first order perturbation δ ( c(x) ). 2 ΣΣ A(x S, x, x R ) U(x S, x R, T (x S, x, x R )) V (x) = δ ( c(x) ) = x S xr 2 ΣΣ A(x S, x, x R ) 2 x S xr A(x S, x, x R )product of amplitudes from x S to x and from x to x R.

Marmousi model 2 4 6 8 2.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 km/s Smoothed version for ray tracing 2 4 6 8 2.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 km/s c(x, y, z) = ΣΣ i j Σ c ijk B i (x)b j (y)b k (z) k

Shot gather at 4 km offset(km).5..5 2. 2.5.5 time(s)..5 2. 2.5 Near-offset (2m) gather 4 6 8.5 time(s)..5 2. 2.5

Green s functions at four different locations (color scale: cos(5t)) 2 3 4 5 6 7 8 9 2 3 2 3 4 5 6 7 8 9 2 3 2 3 4 5 6 7 8 9 2 3 2 3 4 5 6 7 8 9 2 3

Marmousi prestack depth migrated image 8 6 4 2 2 3

2. Multifocusing (MF)/Common reflection surface (CRS) stack Purpose: optimal zero-offset stack in time domain by finding traveltime surface in supergather domain with highest stack/semblance + useful attributes Multifocusing (Landa et al., Applications of multifocusing method for subsurface imaging: J. Appl. Geoph., 42, 243-26, 99) S C G P β R N Multifocus move-out = traveltime difference SRG - CNC Traveltime difference SRP - CNP Traveltime difference PG - PC τ = τ + + τ - τ + = (R + ) 2 + 2R + X S sin β + XS 2 R + V τ - = (R - ) 2 2R - X S sin β + XG 2 R - V R + = + σ R N + σ, R NIP R - = σ σ R N R NIP σ = X S X G X S + X G + 2 X S X G R NIP sin β Exact for constant velocity and circular reflector

Common Reflection Surface stack (Jaeger et al., Common-reflection-surface stack: Image and attributes, Geophysics, 66, 97-9, 2) β β R NIP R N CRS move-out t 2 (x m, h) = (t + 2sin β (x V m x )) 2 + 2t cos 2 β ( (x m x ) 2 + h2 ) V R N R NIP Exact for constant velocity and planar reflector

x = (x S + x R )/2, h = (x R x S )/2 x h x S x R Second-order hyperbolic traveltime expansion in cdp x and offset h: T 2 (x, h) = T 2 + (T h) 2 T h + (T x) 2 T x + 2 ht (T hh)h 2 + x T (T xh)h 2 + 2 xt (T xx)x 2 = Drop linear terms in h: T 2 + 2T T x T x + 2 ht (2T h T h T + 2T T hh )h + 2 xt (2T x T x T + 2T T xx )x = T h = and T x = p: T 2 + 2T p T x + T h T T hh h + x T pp T x +T x T T xx x = T hh = M NIP and T xx = M N : (T + p T x) 2 + T h T M NIP h + T x T M N x = 2D: (T + p x x) 2 + m NIP T h 2 + m N T x 2 β, R NIP, R N or p x, m NIP, m N are found by a 3-parameter optimisation of signal semblance along the surface T (β, R NIP, R N )ort(p x,m NIP, m N )-for each x and each T. supergather R NIP = R N diffractions!

Semblance S S = N ΣU(x i, h i, T (β, R NIP, R N )) 2 i N N ΣU(x i, h i, T (β, R NIP, R N )) 2 i x i, h i all cdp s and offsets close to the central ray Properties. S 2. S = ifall U are equal. 3. the more random U, the closer S to. 4. S is singular for U. Typically, the result of MF/CRS optimisation is three attribute sections:. R NIP (x, T ) 2. R N (x, T ) 3. β (x, T ) Stacking over the optimal traveltime surfaces results in enhanced stack.

Marmousi supergather - S =. 9. trace # 5 5.5. time(s).5 2. 2.5

3. Migration velocity analysis Common Image Gathers offset depth distance common offset gather common image gather h S2 S R R2 far z z z depth CIG offset Flatness of events in CIG Stacking power Σ V (x) 2 maximal image

offset(km) -2 - offset(km) -2 -.5.5...5 2..5 2. 2.5 2.5 3. x=4. km 3. x=6.4 km offset(km) -2 - offset(km) -2 -.5.5...5 2..5 2. 2.5 2.5 3. x=5.2 km 3. x=8. km

Practical aspects of migration velocity analysis layer stripping horizon/event picking local global CPU adequate parametrisation

4. Migration velocity analysis from CRS/MF attributes Invert {R N (x, t), R NIP (x, t), β (x, t)}-sections into a smooth velocity model c(x). Problems Theoretical relationship between c(x) and {R N (x, t), R NIP (x, t), β (x, t)}? Based on raytheory? Partial differential equation for c(x)? {R N (x, t), R NIP (x, t), β (x, t)}-sections contaminated with numerical noise Sharp focusing criterion for velocity/image quality + efficient way to compute it + efficient way to optimize it

Proposals combination of CRS/MF analysis and inversion into one step optimize focusing criterion for smooth velocity distributions using paraxial N and NIP rays. assumption: velocity contours are locally parallel to reflecting interfaces instead of picking, do this for each point of a raster

"rays.clip" "contourlines" -.5 - -.5-2 -2.5-3 2 3 4 5 6 7 8 9.5 "rays" "contourlines" -.5 - -.5-2 -2.5 2 3 4 5 6 7 8 9

Initial conditions at a point x Central ray p = c c 2, x T = H p, p T = H x, Π= x x p x x p p p = I O O I. Velocity contour c(x) = c(x ) x = x + g u + 2 g 2u 2, where g unit vector perpendicular to c, and = g T c g + c T g 2. Paraxial NIP rays Q = (x u, x T ) = (, x T ), P = p u, p T = g, p T. Paraxial normal rays p(u) = Rx u c(x(u)) x u Q = (x u, x T ) = g, x T, P = p u, p T. "brute force" differentiation p u = u Rx u c(x(u)) x u + Rx u u c(x(u)) x u + Rx u c(x(u)) u x u better: define Then W: = p T g 2 g T p T P = Q T W g T p T x T T p T.

Observations accuracy is an issue in Marmousi model (for small degrees of smoothing) ray tracing algorithms preserving symplecticity (implicit Runge-Kutta) testing of paraxial quantities cumbersome automatic differentiation of computer code

-.5 - -.5-2.5 2 2.5 3 3.5 4 "rays" "contourlines" time(s).8.7.6.5.4.3.2..99.98.8 2 2.2 2.4 2.6 2.8 3 "tt_nip_ray" "tt_nip_parax" log(time) -2-4 -6-8 - -2-4 -3.5-3 -2.5-2 -.5 - -.5 log(offset) "tt_nip_parax-ray" "ttpow2" "ttpow3"

-.5 "rays" "contourlines" - -.5-2 -2.5.5 2 2.5 3 3.5 4 time(s).945.94.935.93.925.92.95 2.5 2. 2.5 2.2 2.25 2.3 2.35 "tt_n_parax" "tt_n_ray" log(time) -4-6 -8 - -2-4 -6-8 -6-5.5-5 -4.5-4 -3.5-3 -2.5-2 log(offset) "tt_n_parax-ray" "ttpow2" "ttpow3"

Marmousi supergather - S =. 53 -max. offset =.5 km trace # 2 4 6.5. time(s).5 2. 2.5

Algorithm I. for each raster point, compute central ray from velocity contour to surface + M N + M NIP (diverging rays rejected). 2. collect all traces from the prestack data set with source and receiver point "around" the central ray emergence point. 3. construct MF/CRS traveltime surface 4. evaluate stack/semblance 5. compute "total" stack/semblance Algorithm II. generate velocity fields 2. apply Algorithm I to evaluate "total" stack/semblance 3. optimise stack/semblance

Average stacking power 3e+7 "REPORT_SMOOTHING_Stacking_power" 2.5e+7 2e+7.5e+7 e+7 5e+6 2 3 4 5 6 7 8 9 Degree of smoothing Average semblance.2 "REPORT_SMOOTHING_Semblance".8.6.4.2..8.6 2 3 4 5 6 7 8 9 Degree of smoothing Average stacking power 3e+7 "REPORT_PERTURBATION_Stacking_power" 2.5e+7 2e+7.5e+7 e+7 5e+6 2 3 4 5 % of perturbation Average semblance.2.8.6.4.2. "REPORT_PERTURBATION_Semblance".8 2 3 4 5 % of perturbation