Wave equation techniques for attenuating multiple reflections

Transcription

1 Wave equation techniques for attenuating multiple reflections Fons ten Kroode Shell Research, Rijswijk, The Netherlands Wave equation techniques for attenuating multiple reflections p.1/41

2 Contents 1. Statement of the problem 2. Basic assumptions and notation 3. Surface multiples Integral equation Power series solution Solution by deconvolution 4. Internal multiples (a la Weglein) Forward scattering 1st order term: Born modelling and inversion 3rd order term: internal multiples Wave equation techniques for attenuating multiple reflections p.2/41

3 Statement of the problem Present day imaging algorithms are based on linear relationships between data and scattering potential. Single scattering Thereby, they neglect scattering processes of the form Surface multiples Internal multiples Multiple reflections show up as artifacts in the images. Wave equation techniques for attenuating multiple reflections p.3/41

4 Statement of the problem, example How to turn this Courtesy of Shell Geoscience Services Wave equation techniques for attenuating multiple reflections p.4/41

5 Statement of the problem, example into this? Courtesy of Shell Geoscience Services Wave equation techniques for attenuating multiple reflections p.5/41

6 Basic assumptions and notation Earth is a constant density acoustic medium; wave propagation described by ( ) 1 2 c 2 ( x) t G( x, x 2 s, t) = δ( x x s )δ(t). where G( x, x s, t) is Green s function, the response of the medium at location x due to an instantaneous point source at location x s. This equation needs to be supplied with additional initial/boundary/radiation conditions to determine the solution uniquely. Wave equation techniques for attenuating multiple reflections p.6/41

7 Basic assumptions and notation, continued The temporal Fourier transform of G will be denoted by Ĝ( x, x s, ω). The data are obtained from the Green s function by convolving with a (generally unknown) wavelet: ˆd( x, x s, ω) = ŵ(ω)ĝ( x, x s, ω) Wave equation techniques for attenuating multiple reflections p.7/41

8 Surface multiples, derivation integral equation Let G mf and G be two solutions for the acoustic wave equation in the region z > 0 satisfying the conditions G( x, x s, t) z=0 = 0, G mf ( x, x s, t) O( x x s 1 ) for x x s (radiation condition) The condition for G means vanishing pressure at z = 0, as is the case at an air/water interface. G is therefore a wavefield that has reflections against z = 0 in it. G mf on the other hand, is the desired multiple-free solution. Wave equation techniques for attenuating multiple reflections p.8/41

9 Derivation integral equation, continued Assume that z s, z r ɛ > 0. Consider the domain V bounded by the plane z = ɛ and the half sphere x 2 + y 2 + (z ɛ) 2 < R 2, z > ɛ, which encloses both x s and x r. Using reciprocity, G( x, y, t) = G( y, x, t), we find that for all x inside V ) (Ĝ( x, xr, ω) Ĝmf ( x s, x, ω) Ĝmf ( x s, x, ω) Ĝ( x, x r, ω) = Ĝ( x, x r, ω) Ĝmf ( x s, x, ω) Ĝmf ( x s, x, ω) Ĝ( x, x r, ω) = Ĝ( x, x r, ω)δ( x x s ) + Ĝmf ( x s, x, ω)δ( x x r ) Wave equation techniques for attenuating multiple reflections p.9/41

10 Derivation integral equation, continued Integrating over V, we find (using Gauss theorem) Ĝ mf ( x s, x r, ω) Ĝ( x s, x r, ω) = = ds n (Ĝ( x, xr, ω) Ĝmf ( x s, x, ω) δv ) Ĝmf ( x s, x, ω) Ĝ( x, x r, ω). Letting R and using the radiation condition, we derive Ĝ mf ( x s, x r, ω) Ĝ( x s, x r, ω) = = dxdy (Ĝ( x, xr, ω) z Ĝ mf ( x s, x, ω) z=ɛ ) Ĝmf ( x s, x, ω) z Ĝ( x, x r, ω). Wave equation techniques for attenuating multiple reflections p.10/41

11 Derivation integral equation, continued Letting ɛ 0 and using the vanishing of G at z = 0, this becomes Ĝ mf ( x s, x r, ω) Ĝ( x s, x r, ω) = dxdy Ĝmf ( x s, x, ω) z Ĝ( x, x r, ω). z=0 Approximating z Ĝ( x, x r, ω) = ( z) 1 Ĝ( x, x r, ω), z=0 z= z Ĝ mf ( x s, x, ω) = Ĝ mf ( x s, x, ω), z=0 z= z Wave equation techniques for attenuating multiple reflections p.11/41

12 Derivation integral equation, continued we finally obtain the integral equation we are after Ĝ mf ( x s, x r, ω) Ĝ( x s, x r, ω) = ( z) 1 dxdy Ĝmf ( x s, x, ω)ĝ( x, x r, ω). z= z Using the (unknown) wavelet, we can rewrite the integral equation as ˆd mf ( x s, x r, ω) ˆd( x s, x r, ω) = (ŵ(ω) z) 1 z= z dxdy ˆd mf ( x s, x, ω) ˆd( x, x r, ω). Since the data is non-singular at x = x r or x = x s we can and will assume from now on that z = z r = z s z acq. Wave equation techniques for attenuating multiple reflections p.12/41

13 Power series solution Define the linear integral operator (M ˆd mf ) ( x s, x r, ω) := (ŵ(ω)z acq ) 1 z=z acq d x ˆd mf ( x s, x, ω) ˆd( x, x r, ω). Then the integral equation can be rewritten as (1 M) ˆd mf = ˆd. The formal solution expressing the multiple free data in the measured data is ˆd mf = ˆd + M ˆd + M 2 ˆd +. Wave equation techniques for attenuating multiple reflections p.13/41

14 Power series solution, first order term x s x x r M ˆd z=z acq d x ˆd( x s, x, ω) ˆd( x, x r, ω) The integral over x is stationary when Snell s law is obeyed in the picture above. Note that no velocity information is required to calculate the multiples. Wave equation techniques for attenuating multiple reflections p.14/41

15 Power series solution, second order term x s x 1 x 2 x r M 2 ˆd z 1 =z 2 =z acq d x 1 d x 2 ˆd( xs, x 1, ω) ˆd( x 1, x 2, ω) ˆd( x 2, x r, ω) The integrals over x 1 and x 2 are stationary when Snell s law is obeyed in the picture above. Again, no velocity information is required. Wave equation techniques for attenuating multiple reflections p.15/41

16 Practical implementation To attenuate e.g. the first order surface multiple, one first predicts the kinematics by evaluating the first order term (ŵ(ω)z acq ) 1 z=z acq d x ˆd( x s, x, ω) ˆd( x, x r, ω). Since we do not know the wavelet, we cannot directly add this to the data. Instead, we use an ad hoc energy minimization criterion to attenuate the first order multiple: min ˆf dω ˆd( x s, x r, ω) + ˆf(ω) d x ˆd( x s, x, ω) ˆd( x, x r, ω) z=z acq 2. Wave equation techniques for attenuating multiple reflections p.16/41

17 Solution by deconvolution, principle The integral equation relating multiple free data and measured data can also be solved by multi-dimensional deconvolution. To this end, rewrite the integral equation in the form (1 + D) ˆd = ˆd mf, where D is defined as the integral operator ( D ˆd )( x s, x r, ω) := (z acq ) 1 d x Ĝmf ( x s, x, ω) ˆd( x, x r, ω) = dω e iωt d xdτ F ( x s, x, τ)d( x, x r, t τ), with F ( x s, x, t) := (z acq ) 1 G mf ( x s, x, t). Wave equation techniques for attenuating multiple reflections p.17/41

18 Solution by deconvolution, principle We now try to find a multi dimensional deconvolution filter F ( x s, x, t) which minimizes the energy in (1 + D) ˆd, d x s d x r dt d( x s, x r, t) + d x dτ F ( x s, x, τ)d( x, x r, t τ) T gap T max 2. The time T gap is introduced to avoid the trivial solution F ( x s, x, t) = δ( x s x)δ(t) for which the energy would be zero. Wave equation techniques for attenuating multiple reflections p.18/41

19 Solution by deconvolution, example Left: input stack, right: stack after 2D deconvolution (Courtesy of Shell Geoscience Services) Wave equation techniques for attenuating multiple reflections p.19/41

20 Contents 1. Statement of the problem 2. Basic assumptions and notation 3. Surface multiples Integral equation Power series solution Solution by deconvolution 4. Internal multiples (a la Weglein) Forward scattering 1st order term: Born modelling and inversion 3rd order term: internal multiples Wave equation techniques for attenuating multiple reflections p.20/41

21 Forward scattering Let c 0 ( x) be a smooth approximation of the true velocity model c( x), G 0 the surface multiple free Green s function associated to the model c 0, G the surface multiple free Green s function for the true velocity model. Thus, G 0 and G are determined by ( ) 1 2 c 2 0( x) t G 2 0 ( x, x s, t) = δ( x x s )δ(t), ( ) 1 2 c 2 ( x) t G( x, x 2 s, t) = δ( x x s )δ(t), plus radiation conditions for x. Wave equation techniques for attenuating multiple reflections p.21/41

22 Forward scattering, continued Subtracting these two equations, we get ( ) 1 2 [ c 2 0( x) t G( x, x 2 s, t) G 0 ( x, x s, t)] This can be solved as = ( c 2 0 ( x) c 2 ( x) ) 2 G t 2. G( x s, x r, t) G 0 ( x s, x r, t) = d xdτg 0 ( x s, x, τ)v ( x) 2 t 2 G( x, x r, t τ), where we have introduced the scattering potential V ( x) := c 2 0 ( x) c 2 ( x). Wave equation techniques for attenuating multiple reflections p.22/41

23 Forward scattering, continued This is called the Lipmann-Schwinger equation. In the frequency domain it reads Ĝ( x s, x r, ω) Ĝ0( x s, x r, ω) = ω 2 d xĝ0( x s, x, ω)v ( x)ĝ( x, x r, ω). Introducing the integral operator ( ) C Ĝ ( x s, x r, ω) := ω 2 d xĝ0( x s, x, ω)v ( x)ĝ( x, x r, ω), we rewrite this as (1 C) Ĝ = Ĝ0. Wave equation techniques for attenuating multiple reflections p.23/41

24 Forward scattering, continued Formally inverting, we obtain the forward scattering series Ĝ( x s, x r, ω) = ( ) C k Ĝ0 ( x s, x r, ω) k 0 = ω 2 + ω 4 ω 6 + d xĝ0( x s, x, ω)v ( x)ĝ0( x, x r, ω) d x 1 d x 2 Ĝ 0 ( x s, x 1, ω)v ( x 1 )Ĝ0( x 1, x 2, ω) V ( x 2 )Ĝ0( x 2, x r, ω) d x 1 d x 2 d x 3 Ĝ 0 ( x s, x 1, ω)v ( x 1 )Ĝ0( x 1, x 2, ω)v ( x 2 ) Ĝ 0 ( x 2, x 3, ω)v ( x 3 )Ĝ0( x 3, x r, ω) Wave equation techniques for attenuating multiple reflections p.24/41

25 1st order term, Born modelling and inversion Assuming that Ĝ0 vanishes at the acquisition level (i.e. that the background medium is reflection free), and neglecting all terms after the first one, we find the Born approximation for the scattered field Ĝ( x s, x r, ω) = ω 2 d xĝ0( x s, x, ω)v ( x)ĝ0( x, x r, ω). This is a linear relation between deconvolved data Ĝ and the scattering potential V. To invert it, we use high frequency asymptotics Ĝ 0 ( x, y, ω) A( x, y)e iωφ( x, y), where A is an amplitude and φ( x, y) is the traveltime for a ray from x to y in the velocity model c 0 ( x). Wave equation techniques for attenuating multiple reflections p.25/41

26 Born modelling and inversion, continued Inserting this, the Born approximation becomes Ĝ( x s, x r, ω) ω 2 d xa( x s, x)a( x, x r )V ( x)e iω(φ( x s, x)+φ( x, x r )). x s x r G 0 (x s, x, t) G 0 (x, x r, t) x Because of the high frequency approximation this is a relation between singularities in V and singularities in Ĝ. Wave equation techniques for attenuating multiple reflections p.26/41

27 Born modelling and inversion, continued Now fix the shot coordinates x s and y s, i.e., consider only the data from a single shot. In order to invert the resulting forward operator F ( x s ), one requires ideal illumination and the absence of caustics. The result is V ( x) z r =0 dx r dy r W ( x s, x, x r )G( x s, x r, t = φ( x s, x)+φ( x, x r )). Here W is a weight, which we will not specify further. This formula is usually referred to as common shot migration. Note that the left hand side does not depend on the shot coordinate. In other words: the image of the earth does not depend on the shot used. For this to be true, c 0 ( x) needs to be a good approximation of c( x). Wave equation techniques for attenuating multiple reflections p.27/41

28 3rd order term: internal multiples The third order term, ω 6 d x 1 d x 2 d x 3 Ĝ 0 ( x s, x 1, ω)v ( x 1 )Ĝ0( x 1, x 2, ω)v ( x 2 ) Ĝ 0 ( x 2, x 3, ω)v ( x 3 )Ĝ0( x 3, x r, ω), contains contributions, which can easily be identified as internal multiples. This is explained in the following figure. x s x r G 0 (x s, x 1, ω) G 0 (x 1, x 2, ω) x 2 G 0 (x 2, x 3, ω) G 0 (x 3, x r, ω) x 1 x 3 Wave equation techniques for attenuating multiple reflections p.28/41

29 3rd order term, other contributions The third order term also gets contributions of the form x s x r x s x r G 0 (x s, x 1, ω) x 2 x 3 G 0 (x 2, x 3, ω) G 0 (x 1, x 2, ω) G 0 (x 3, x r, ω) G 0 (x s, x 1, ω) G 0 (x 3, x r, ω) x 1 x 3 G 0 (x 1, x 2, ω) G 0 (x 2, x 3, ω) x 2 x 1 z 1 > z 2 > z 3 z 2 > z 1, z 2 > z 3 To avoid these, we restrict the integration domain to z 1 > z 2, z 3 > z 2. Wave equation techniques for attenuating multiple reflections p.29/41

30 Internal multiples, velocity dependent prediction The resulting formula, ˆd IM ( x s, x r, ω) = ŵ(ω)ω 6 d x 1 d x 2 d x 3 Ĝ 0 ( x s, x 1, ω)v ( x 1 ) z 1 >z 2 z 3 >z 2 Ĝ 0 ( x 1, x 2, ω)v ( x 2 )Ĝ0( x 2, x 3, ω)v ( x 3 )Ĝ0( x 3, x r, ω), models interbed multiples, but the prediction clearly requires the velocity model c 0 ( x). Wave equation techniques for attenuating multiple reflections p.30/41

31 Kinematics can be predicted independent of velocity Proposition 1 Under suitable assumptions, the velocity dependent formula is asymptotically equivalent to ˆd IM ( x s, x r, ω) = ŵ(ω) 2 d r 1 d r 2 dt 1 dt 2 dt 3 B d( x s, x r1, t 1 ) t 1 >t 2 t 3 >t 2 d( x r1, x r2, t 2 )d( x r2, x r, t 3 )e iω(t 1 t 2 +t 3 ). Here d( x s, x r, t) := 1 8π 3 d y r d k r dt dω ω 2 /c( x r ) 2 k 2 r d( x s, y r, t )e i k r ( x r y r ) iω(t t ) and B = B[c 0 ] is a velocity dependent amplitude, which reduces to 1 for a constant velocity background medium. Wave equation techniques for attenuating multiple reflections p.31/41

32 Velocity free prediction, assumptions Assumption 1 (Conormality assumption) The reflectivity of the earth is conormal, i.e., there is a smooth map x n( x) from R 3 to the unit sphere such that the function V ( x) is singular in the direction n( x) only. conormal reflectivity distribution non-conormal reflectivity distribution Wave equation techniques for attenuating multiple reflections p.32/41

33 Velocity free prediction, assumptions Assumption 2 (Traveltime Monotonicity Condition) Let τ( x s ; α, z) be the traveltime of a ray taking off at a source location x s at angle α with the normal, reflecting at depth z according to Snell s law with respect to the local normal and travelling back to the surface. Then x s, α τ( x s ; α, z 1 ) > τ( x s ; α, z 2 ) z 1 > z 2 x s z=z 2 α low velocity region high velocity region Possible violation of Assumption 2 z=z 1 Wave equation techniques for attenuating multiple reflections p.33/41

34 Sketch of the proof Replacing the scattering potentials occurring in the velocity dependent formula by common shot migrated data, we get ˆd IM ( x s, x r, ω) = ŵ(ω) 2 dt 1 dt 2 dt 3 d r 1 d r 2 z 1 >z 2 z 3 >z 2 ( d r 3 d x 1 d x 2 d x 3 dω 1 dω 2 dω 3 ) Bd( x s, x r1, t 1 )d( x r1, x r2, t 2 )d( x r2, x r3, t 3 )e iψ, where r i = (x ri, y ri ), B is a product of forward amplitudes A and migration weights W and the phase ψ is given by ψ := ω [φ( x s, x 1 ) + φ( x 1, x 2 ) + φ( x 2, x 3 ) + φ( x 3, x r )] ω 1 [φ( x s, x 1 ) + φ( x 1, x r1 ) t 1 ] ω 2 [φ( x r1, x 2 ) + φ( x 2, x r2 ) t 2 ] ω 3 [φ( x r2, x 3 ) + φ( x 3, x r3 ) t 3 ] Wave equation techniques for attenuating multiple reflections p.34/41

35 Sketch of the proof, continued Step 1: Integration with respect to ( x 1, ω 1 ). Relative part ψ: ω [φ( x s, x 1 ) + φ( x 1, x 2 )] ω 1 [φ( x s, x 1 ) + φ( x 1, x r1 ) t 1 ] s r 1 x 2 x 1 The result of this integration is 1. x 1 is on the ray connecting x 2 and r 1, 2. φ( x s, x 1 ) + φ( x 1, x r1 ) = t 1. Wave equation techniques for attenuating multiple reflections p.35/41

36 Sketch of the proof, continued Step 2: Integration with respect to ( x 2, ω 2 ). Relative part ψ: ω [ φ( x r1, x 2 ) + φ( x 2, x 3 )] ω 2 [φ( x r1, x 2 ) + φ( x 2, x r2 ) t 2 ] s r 2 r 1 x 2 x 3 x 1 The result of this integration is 1. x 2 must be on the ray connecting r 2 and x 3, 2. φ( x r1, x 2 ) + φ( x 2, x r2 ) = t 2. Wave equation techniques for attenuating multiple reflections p.36/41

37 Sketch of the proof, continued Step 3: Integration with respect to ( x 3, ω 3, r 3 ). Relative part ψ: ω [φ( x r2, x 3 ) + φ( x 3, x r )] ω 3 [φ( x r2, x 3 ) + φ( x 3, x r3 ) t 3 ] s r 2 r 1 r 3 = r x 2 x 3 x 1 The result of this integration is 1. r 3 = r, 2. φ( x r2, x 3 ) + φ( x 3, x r3 ) = t 3. Wave equation techniques for attenuating multiple reflections p.37/41

38 Velocity model Velocity model used in synthetic interbed multiple study Wave equation techniques for attenuating multiple reflections p.38/41

39 Images Image of the data with (left) and without (right) interbed multiples Wave equation techniques for attenuating multiple reflections p.39/41

40 Predictions Image of predicted multiples (black) on top of image of the data (color) Wave equation techniques for attenuating multiple reflections p.40/41

41 Cleaned images Image of the data (right) vs image of the data after adaptive multiple subtraction (right) Wave equation techniques for attenuating multiple reflections p.41/41