Image amplitudes in reverse time migration/inversion

Transcription

1 Image amplitudes in reverse time migration/inversion Chris Stolk 1, Tim Op t Root 2, Maarten de Hoop 3 1 University of Amsterdam 2 University of Twente 3 Purdue University TRIP Seminar, October 6, 2010

2 RTM based linearized inversion Seismic imaging is mathematically treated as a linearized inverse problem Kirchhoff migration can correspond to a method for inverting discontinuities (Beylkin, 1985). There are conditions on the background medium. Single source: no multipathing. Multisource: Traveltime Injectivity Condition (Rakesh, 1988; Nolan and Symes 1997; Ten Kroode Et Al., 1998) RTM has the advantage that there are no approximations from ray-theory and one-way methods. We will do linearized inverse scattering by a modified reverse time migration (RTM algorithm

3 The linearized inverse problem Source field u src ( ) 1 v 2 2 t 2 x u src (x, t) = δ(x x src )δ(t) 1 Scattered field: Linearize in the velocity (1 + r(x)). v(x) 2 v the smooth velocity model, r contains singularities ( ) 1 v 2 2 t 2 x u scat (x, t) = r v 2 2 t u src. v(x) 2 1 Problem: Determine r(x) from the following data: u scat (x, t) for x = (x 1, x 2, x 3 ) in a subset of x 3 = 0, t [0, T max ]

4 A new view on Reverse Time Migration Numerically solve wave equation to obtain u src (x, t) = incoming (source) wave field u rtc (x, t) = reverse time continued receiver field New imaging condition I (x) = 2i ω2 û src (x, ω)û rtc (x, ω) v 2 û src (x, ω) û rtc (x, ω) 2πω 3 û src (x, ω) 2 dω. (cf. Kiyashchenko et al., 2007) Characterization of I I (x) = R(x, D x )r, Here R is a pseudodifferential operator that describes the aperture effect, i.e. R(x, ξ) = 1 for visible reflectors (cf. Beylkin, 1985).

5 Table of contents Model problem: constant coefficients Extension to variable background Numerical examples Conclusion and discussion

6 Model problem Notation x = (x 1, x 2, x 3 ). Source field is plane wave with velocity (0, 0, v) Scattered field slightly simplified u src (x, t) = A δ(t x 3 v ) ( 1 v 2 2 t 2 x)u scat (t, x) = A δ(t x 3 v )r(x). Perfect backpropagation: u rtc solution of a final value problem ( 1 v 2 2 t 2 x)u rtc (x, t) = 0 u rtc (x, T 1 ) = u scat (x, T 1 ), t u rtc (x, T 1 ) = t u scat (x, T 1 ), x R 3.

7 Solving the wave equation Spatial Fourier transform (v 2 2 t + ξ 2 ) û(ξ, t) = f (ξ, t). Duhamel s principle: Let ŵ(ξ, t, s) Result û(ξ, t) = (v 2 2 t + ξ 2 )ŵ(ξ, t, s, ) = 0, ŵ(ξ, s, s) = 0 t 0 t ŵ(ξ, s, s) = v 2 f (s, ξ). ( e iv ξ (t s) e iv ξ (t s)) v 2 f (ξ, s) 2iv ξ ds Needed: Fourier transform of r.h.s. A δ(t x 3 v )r(x)

8 Fourier transform f (x, t) = A δ(t x 3 v )r(x) f (ξ, t) = A δ(t x 3 v )r(x)e ix ξ dx = va F (x1,x 2 ) (ξ 1,ξ 2 )r(ξ 1, ξ 2, v 0 t)e iξ 3vt. Computations involve two similar terms, take only one of those. û scat (ξ, t) = t 0 v 3 2iv ξ e iv ξ (t s) iξ 3vs A F (x1,x 2 ) (ξ 1,ξ 2 )r(ξ 1, ξ 2, vs) ds + pos. freq. term = Av 2 2iv ξ e iv ξ t r(ξ 1, ξ 2, ξ 3 ξ ) + pos. freq. term, if t is after the incoming has passed supp(r) Back to space domain u scat (x, t) = 2 Re 1 (2π) 3 Av 2 2iv ξ eix ξ iv ξ t r(ξ (0, 0, ξ )) dξ

9 Interpretation Scattered field u scat (x, t) = 2 Re 1 (2π) 3 Av 2 2iv ξ eix ξ iv ξ t r(ξ (0, 0, ξ )) dξ Wave vectors ξ outgoing wave number (0, 0, ξ ) incoming wave number ξ (0, 0, ξ ) reflectivity wave number in!!"(0,! ) refl (0,! )

10 Modified excitation time imaging condition Same formula for the backpropagated field! Valid for all t u rtc (x, t) = 2 Re 1 (2π) 3 Basic image: I 0 (x) = u rtc (x, x 3 /v). Linearized inverse scattering: Try Av 2 2iv ξ eix ξ iv ξ t r(ξ (0, 0, ξ )) dξ I (x) = 2 v 2 A ( t + v x3 ) u rtc (x, x 3 /v) (meaning first take derivatives then insert (x, t) = (x, x 3 /v) ) Straightforward calculation yields ( 1 I (x) = 2 Re (2π) 3 1 ξ ) 3 r(ξ (0, 0, ξ )) dξ ξ R 3

11 A change of variables Copy from previous slide: 1 I (x) = 2 Re (2π) 3 R 3 ( 1 ξ ) 3 r(ξ (0, 0, ξ )) dξ ξ Change of variables Jacobian 1 ξ 3 ξ ξ = ξ (0, 0, ξ ) in!!"(0,! ) refl Domain ξ R 2 R <0 (0,! ) Result: I (x) = 2 Re 1 (2π) 3 r( ξ)e ix ξ d ξ = 1 R 2 R <0 (2π) 3 r( ξ)e ix ξ d ξ. R 2 R 0 Reconstruction except for ξ 3 = 0, corresponding to reflection over 180 degrees.

12 Modified ratio imaging condition We had I (x) = 2 v 2 A ( t + (0, 0, v) x ) u rtc (x, x 3 /v) Rewrite into modified ratio imaging condition Use that (0, 0, v) = v 2 T (x), and that Insert factors v 2 ω 2 This results in I (x) = 2i 2πω 3 û src (x, ω) = Ae iωt (x), x û src (x, ω) iω x T (x)ae iωt (x). left out in model problem ω2 û src (x, ω)û rtc (x, ω) v 2 û src (x, ω) û rtc (x, ω) û src (x, ω) 2 dω.

13 Extension to variable background Local vs. global analysis A local analysis generalizes the explicit formulas to variable coefficients, and leads to explicit inversion formulas A global analysis takes into account the of global effect of curved rays. No source-multipathing is allowed to exclude kinematic artifacts ( cross talk ). Cf. Rakesh (1988), Nolan and Symes (1997). Tools: microlocal analysis Fourier integral operators (FIO s), pseudodifferential operators (ΨDO s) Description of the mapping properties of operators for localized plane wave components. Work in (x, ξ) or (x, t, ξ, ω) domain.

14 Variable coefficients: The wave equation Local analysis Solve the wave equation using WKB with plane wave initial values. Result is an FIO u(y, t) = 1 (2π) 3 e iα(y,t s,ξ) a(y, t s, ξ) f (ξ, s) ds dξ+pos. freq. term, α satisfies the eikonal equation; a a transport equation. Formula is valid for t, s in a bounded time interval Global analysis Propagation of localized plane wave components along rays, properly described as curves in (x, t, ξ, ω) space Time reversal property is globally valid

15 Continued scattered field and linearized forward map Define the continued scattered field u h, as the perfect backpropagated field from a fixed time, full position space Local result For a localized contribution to r u h,a (y, t) = 1 (2π) 3 e iϕ T (y,t,x,ξ) A(y, t, x, ξ) r(x) dξ dx. With phase function ϕ T (y, t, x, ξ) = α(y, t T (x), ξ) ξ x Global result The map F : r u h is a FIO. Mapping of wave components (x, ζ) (y, t, η, ω): see picture ξ n s (x, T s(x)) ζ ξ n s ξ (y, t 1) η

16 RT continuation from the boundary Preprocess data: ΨDO cutoff Ψ M (y 1, y 2, t, η 1, η 2, ω) with three effects Smooth taper near acquisition boundary Remove direct waves Remove tangently incoming waves Normalize wave field to get true amplitude time reversal Characterize mathematically the aperture effect Result There are ΨDO s P, P +, such that u rtc (x, t) = P (t, x, D x )u h, + P + (t, x, D x )u h,+ in which u h,± is the part of u h with ±ω > 0. To highest order P ± (s, x, ξ) = Ψ M (y 1, y 2, t, η 1, η 2, ω) if there is a ray with ±ω > 0, connecting (x, s, ξ, ω) with (y 1, y 2, t, η 1, η 2, ω) reverse time cont. acquisition

17 Inverse scattering Modified excitation time imaging condition 2 I (x) = A src (x) t n+1 2 [ t + v 2 T x ] urtc (x, t) From the global analysis, there are no nonlocal contributions and I (x) = ΨDO r(x) From the local analysis To highest order I (x) = (R (x, D x ) + R + (x, D x )) r(x) R ± (x, ζ) = P ± (x, ξ) if ζ and ξ are related through ζ = ξ ± ω T (x). " x #!! inc y Modified ratio imaging condition derived similarly as above

18 Example: A vertical gradient medium stolkcc, Sun Apr 5 13:57 stolkcc, Sun Apr 5 14:02 Example 1: Velocity perturbation, reconstructed velocity perturbation, and three traces for background medium v = z (z in meters and v in km/s). Error 8-10 %. stolkcc, Sun Apr 5 14:27 stolkcc, Sun Apr 5 14:27 stolkcc, Sun Apr 5 14:27

19 Example: Horizontal reflector with receiver side caustics Velocity, rays and fronts z (m) x (m) stolkcc, Tue Apr 7 16:04 stolkcc, Sun Apr 5 18:11 stolkcc, Sun Apr 5 18:11 Example 2: (a) A velocity model with some rays; (b) Simulated data, with direct arrival removed; (c) Velocity perturbation; (d) Reconstructed velocity perturbation. Error 0-10%

20 Conclusion and discussion We modified reverse time migration to become a linearized inverse scattering method We characterized the resolution operator as a partial inverse, a ΨDO with symbol describing aperture effects The modified imaging condition suppresses low-frequency artifacts Good numerical results in (bandlimited) examples Straightforward generalization of the imaging condition to downward propagation methods