Flux-normalized wavefield decomposition and migration of seismic data Bjørn Ursin 1, Ørjan Pedersen and Børge Arntsen 1 1 NTNU Statoil China, July 011
Overview Wave equation Up-down separation Amplitude normalization Flux normalization One-way wave equations Heterogeneous media Data examples Conclusions
3 Fourier-transformed wave equations (1-D medium) 3 b = iωab, The system matrix A is given by [ ] 0 ρ A = ( ) 1 1 ρ k c 1 +k, 0 ω and the field vector b by [ ] P b =. V 3
4 Up/down separation gives w = 3 w = [ ] U = L 1 b D ( ) iωλ L 1 3 L w
5 Amplitude normalization (Claerbout, 1976) gives where L 1 = [ ] 1 Z, 1 Z [ ] [ ] p3 0 1 1 3 w = iω w γ (x 0 p 3 ) w, 3 1 1 is the reflectivity function. γ(x 3 ) = 1 γ (x 3 ) = 1 3 log Z (x 3 ) [ 1 ρ + 1 ρ x 3 Z (x 3 ) = ρω k 3 = 1 c cos θ ρc cos θ ] c x 3
6 Flux-normalization (Ursin, 1983) Z [ L 1 = 1 1 1 Z ] Z, Z gives We have [ ] [ ] p3 0 0 1 3 w = iω w γ (x 0 p 3 ) w. 3 1 0 w = Z w = k 3 ρω w.
7 One-way wave equation x 3 with interface conditions [ ] [ ] [ ] U ik = 3 + γ 0 U D 0 ik 3 + γ D [ ] U = Tu 1 D + [ ] U D
8 One-way wave equation Exact solutions with D(x 3 ) = D(x 0 3 )T (x 3) exp U(x 3 ) = U(x 0 3 )T (x 3) exp Z (x 3 ) T (x 3 ) = Z (x3 0) Z (x 3 ) = Z (x3 0) [ [ i i 0<x 3k <x 3 T 1 0<x 3k <x 3 T 1 x3 x3 0 x3 x 0 3 k 3 (ζ)dζ ] k 3 (ζ)dζ ] u (x 3k ) Z (x 3k ) Z (x 3k+ ) u (x 3k )
9 Flux-normalization and [ D(x 3 ) = D(x ] x3 1 30 ) T u (x 3k ) exp ik 3 (ζ)dζ 0<x 3k <x x 0 3 3 [ Ũ(x 3 ) = Ũ(x 3 0 ) 1 T u (x 3k ) exp 0<x 3k <x 3 x3 x 0 3 ik 3 (ζ)dζ ]. Exactly the same solution with w = k 3 ρω w.
10 Heterogeneous media Flux normalization: D 0 (ω, k 1, k, 0) = πi S(ω) ρωk 3 Ũ 0 = Z U 0 Marine seismics (Amundsen and Ursin, 1991): D 0 (ω, k 1, k, 0) = πi (exp [ ik 3 x3 s ρωk ] R 0 exp [ik 3 x3 s ]) S(ω) 3 Ũ 0 = 1 Z [P ZV 3 ]
11 One-way wave equations [ ] w iĥ1 0 = w x 3 0 iĥ1 where ( ω ) Ĥ 1 Ĥ 1 = + ρ c x 1 ( 1 ρ x 1 ) + ρ ( ) 1 x ρ x Interface correction T 1 u = 1 [ Z + + Z Z Z + ]
1 Imaging conditions Imaging condition (Vivas et al., 009) shots Ũk D I(x) = k dω Dk dω shots AVP-analysis R(p 1, p, x) = shots Ũk (ωp 1, ωp, x, ω)) D k (ωp 1, ωp, x, ω)dω shots Dk (ωp 1, ωp, x, ω) dω
13 One-way wave equations To extract AVP or AVA information at a reflector position, we need information from more than one shot. Receivers Source Depth θ θ 1 x m, x m,1
I(x, p) (in gray-scale) overlaid the source correction term (in color-scale) at midpoints x m,1 = 0.5 km, x m, = 0.0 km, and x m,3 = 0.5 km. 1 Depth (km) 3-0.5 0 0.5-0.5 0 0.5-0.5 0 0.5 Slowness (s/km) Slowness (s/km) Slowness (s/km)
By simulating more shots over one midpoint location x m, we can extract angle information for larger angle coverage. Sources Depth θ x m
We simulate 100 shots with a shot-distance of 10 m on both sides of x m, in addition to one shot just above x m. We plot I(x, p) overlaid the corresponding source illumination for the fixed midpoint location x m. 1 Depth (km) 3 4-80 -60-40 -0 0 0 40 60 80 Angle (deg) Figure: Angle coverage I(x, p) (gray-scale) from one spatial location x m overlaid the corresponding source correction (color-scale) where the contribution from multiple shots are included.
In this example (constant velocity, varying densities), we expect an angle-independent reflectivity, and from the result we see that the reflectivity is recovered relatively accurately for a wide range of angles. Due to a limited aperture, edge effects impact the results, and the largest angles on each reflector are affected. Amplitude 1.5 1 0.5 0 0 5 10 15 0 5 30 35 40 45 50 Angle (deg) Figure: Peak amplitudes at each reflector at 1 km (red), km (green), and 3 km (blue) for one spatial location x m.
18 Marine field data example We apply both conventional pressure-normalized and the derived flux-normalized methods to a field dataset from the Nordkapp Basin. The basin is located offshore Finnmark, in the Norwegian sector of the Barents Sea.
Distance (km) 4 6 8 10 1 14 16 18 0 Depth (km) 4 6 8 10 Figure: Migrated image of the Nordkapp field example with the flux-normalized wavefield decomposition and migration approach.
Distance (km) 4 6 8 10 1 14 16 18 0 Depth (km) 4 6 8 10 Figure: Migrated image of the Nordkapp field example with the pressure-normalized wavefield decomposition and migration approach.
Distance (km) 4 6 8 10 1 14 16 18 0 Depth (km) 4 6 8 10 Figure: Difference of the absolute value of the migrated flux-normalized wavefield decomposition and the pressure-normalized wavefield. Red indicates that flux-normalized image is dominating, black indicates that the pressure-normalized image is dominating.
1 1 3 3 Depth (km) 4 5 6 Depth (km) 4 5 6 7 7 8 8 9 9 10-0.5 0 0.5 Slowness (s/km) 10-0.5 0 0.5 Slowness (s/km) (a) (b) Figure: Slowness-gather at a distance of 14.4 km from the Nordkapp basin field data example. (a) with pressure-normalized and (b) flux-normalized variables.
1 Depth (km) -0.1375 0 0.1375 Peak amplitudes 0.8 0.6 0.4 0. -0.1375 0 0.1375 0.0 Depth (km) Difference 0.01 0-0.01-0.1375 0 0.1375 Slowness (s/km) (a) -0.0-0.1375 0 0.1375 Slowness (s/km) (b) Figure: (a) Extracted event at 7.8 km depth (top) flux-normalized (bottom) pressure-normalized. (b) Normalized peak amplitudes along event (top) flux-normalized (red) and pressure-normalized (blue). Difference (bottom), where positive and negative values corresponds to higher and lower peak amplitudes in using flux-normalized variables.
4 Conclusions Use flux-normalized variables Image with normalized cross-correlation in the local wave-number domain
5 Acknowledgments Hans-Kristian Helgesen and Robert Ferguson for useful discussions VISTA and The Norwegian Research Council (ROSE) for financial support.