Flux-normalized wavefield decomposition and migration of seismic data

Transcription

1 Flux-normalized wavefield decomposition and migration of seismic data Bjørn Ursin 1, Ørjan Pedersen and Børge Arntsen 1 1 NTNU Statoil China, July 011

2 Overview Wave equation Up-down separation Amplitude normalization Flux normalization One-way wave equations Heterogeneous media Data examples Conclusions

3 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 4 Up/down separation gives w = 3 w = [ ] U = L 1 b D ( ) iωλ L 1 3 L w

5 5 Amplitude normalization (Claerbout, 1976) gives where L 1 = [ ] 1 Z, 1 Z [ ] [ ] p w = iω w γ (x 0 p 3 ) w, 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 6 Flux-normalization (Ursin, 1983) Z [ L 1 = Z ] Z, Z gives We have [ ] [ ] p w = iω w γ (x 0 p 3 ) w w = Z w = k 3 ρω w.

7 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 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 9 Flux-normalization and [ D(x 3 ) = D(x ] x ) T u (x 3k ) exp ik 3 (ζ)dζ 0<x 3k <x x [ 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 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 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 + ]

12 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 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

14 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) Slowness (s/km) Slowness (s/km) Slowness (s/km)

15 By simulating more shots over one midpoint location x m, we can extract angle information for larger angle coverage. Sources Depth θ x m

16 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) 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.

17 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 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 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.

19 Distance (km) Depth (km) Figure: Migrated image of the Nordkapp field example with the flux-normalized wavefield decomposition and migration approach.

20 Distance (km) Depth (km) Figure: Migrated image of the Nordkapp field example with the pressure-normalized wavefield decomposition and migration approach.

21 Distance (km) Depth (km) 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.

22 Depth (km) Depth (km) Slowness (s/km) 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.

23 1 Depth (km) Peak amplitudes Depth (km) Difference Slowness (s/km) (a) 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.

24 4 Conclusions Use flux-normalized variables Image with normalized cross-correlation in the local wave-number domain

25 5 Acknowledgments Hans-Kristian Helgesen and Robert Ferguson for useful discussions VISTA and The Norwegian Research Council (ROSE) for financial support.