Velocity Model Sampling and Interpolation: The Gradient Problem

Transcription

1 January 11, 2013

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3 Kirchhoff migration uses raytracing equations for prestack depth migration. d x dt = v( x)2 p d p dt = v( x) v( x) These equations estimate the paths of seismic energy through the subsurface as a function of a known velocity model and its gradient. (1)

4 Depth (m) Boundary True gradient Computed gradient Offset (m) 2000

5 Depth (m) Boundary True gradient Computed gradient Offset (m) 2000

6 models are generally smoothed before Kirchhoff migration. Moser (2011) Uses a smoothed velocity model to compute raypaths, and then computes traveltimes using the original blocky model. The result is a sharper image with more continuous reflectors than using the smooth model everywhere.

7 Depth (m) v 2 v 1 θ 1 θ n θ 2 Snell s Law Offset (m) sin(θ 1 θ n ) v 1 = sin(θ 2 θ n ) v 2 (2)

8 Depth (m) v 2 v 1 p 1 Snell s Law Offset (m) d p dt = v( x) v( x) p 2 p = p 2 = p 1 t v (3) v

9 Processing delivers a velocity model sampled on a grid: Rays travel between these nodes, so we need to interpolate Sampling of the velocity grid makes it hard to accurately compute gradient vectors Some sort of regression would be ideal! N v(x,z) = α k ṽ k (x,z) (4) k=1

10 Depth (m) Offset (m)

11 Depth (m) Basis Function with Dipping Layer Offset (m)

12 Depth (m) Offset (m)

13 We can easily construct infinitely many basis functions: Dipping layers Anticlines Salt structures Random functions Key Observation: The dipping layers are invariant from trace to trace except for a constant dip. This is the same assumption Spitz interpolation works on.

14 Spitz Method the traces as a sum of L events with constant dip g n (k z ) = L G j (k z )e 2πikz(n 1)d j j=1 Unknown d j gives the dip of the event. Interpolate between known traces using PEFs, or solve for d j directly.

15 models are not band limited in z. Caveats For multiple dipping layers, this is a multiple nonlinear regression problem. For a single dipping interface, this is a simple linear regression, so we can recover the dip directly by least-squares. log(g n (f)) = log(g 1 (k z ))2πik z (n 1)d 1

16 log(g n (k z )) Spitz Method Time Shift Cross plot (Real Part) π i k (n 1) log(g (k )) z 1 z

17 4 Spitz Method Time Shift Cross plot (Imaginary Part) log(g n (k z )) π i k (n 1) log(g (k )) z 1 z

18 Recovered d 1 = Actual dip is (difference of π radians)

19 Depth (m) Reconstructed Data Offset (m)

20 Depth (m) Reconstructed Data Offset (m)

21 model interpolation is similar to seismic interpolation. Nonlinear regression may be able to recover multiple dipping events.

22 Future work Implement seismic interpolation techniques to upsample velocity models for better lateral continuity and gradients. Use accurate gradients to perform Kirchhoff migration, to see if we can improve on the results of Moser (2011).

23 Acknowledgements Hassan Khaniani Rob Ferguson CREWES Staff and Sponsors

24 Any Questions?