Acoustic Wave Equation

Transcription

1 Acoustic Wave Equation Sjoerd de Ridder (most of the slides) & Biondo Biondi January 16 th 2011

2 Table of Topics Basic Acoustic Equations Wave Equation Finite Differences Finite Difference Solution Pseudospectral Solution Stability and Accuracy Green s function Perturbation Representation Born Approximation

3 Basic linearized acoustic equations in lossless, isotropic, non flowing media Linearized - Linear for small perturbation on a static state. Lossless - Material parameters are independent of time. Isotropic - Material response independent of direction. Non flowing - No material derivative Equation of motion (three equations for three components) Acoustic stress-strain relationship (a pressure-rate strain-rate relation) ρ t υ i + i p = f i (1) ρ t p + i υ i = q (2)

4 Fields p = p(x, t) υ i = υ i (x, t) pressure i component of velocity Sources q = q(x, t) f i = f i volume injection rate i component of external force Medium Parameters κ = κ(x) ρ = ρ(x) compressibility density

5 Wave Equation Solve equations (1) and (2) for pressure ρ i ρ 1 i p ρκ t 2 p = ρ i ρ 1 f i ρ t q, (3) or i 2 p ρκ t 2 p = ρ i ρ 1 f i ρ t q + ρ 1 i ρ i p. (4) Thus in a constant density and sourceless medium i 2 p c 2 t 2 p = 0, (5) with wave velocity c = c(x) = κρ, κ = κ(x), ρ = ρ 0.

6 Finite Differences Derivation of finite difference stencils for F (s) s Expand F (s + s) in Taylor series F (s + s) = F (s) + 1 1! sf (s) s + i=2 1 i! i sf (s) { s} i (6) Express F (s) s as a function of... s F (s) = 1 s {F (s + s) F (s)} i=2 1 i! i sf (s) { s} i 1 (7) this is a forward finite difference stencil.

7 Expand F (s + s) and F (s s) in Taylor series F (s + s) = F (s) + 1 1! sf (s) s + i=2 1 i! i sf (s) { s} i (8) F (s s) = F (s) 1 1! sf (s) s + i=2 Substract equations (9) from (8), express s F (s) = 1 2 s {F (s + s) F (s s)} this is a centered finite difference stencil. i=1 1 i! i sf (s) { s} i (9) F (s) s as a function of... 1 (1 + 2i)! 1+2i s F (s) { s} 2i (10)

8 or last, s F (s) in a backward finite difference stencil from equation (9) as s F (s) = 1 s {F (s) F (s s)} i=2 1 i! i sf (s) { s} i 1 (11)

9 Derivation of finite difference stencils for 2 s F (s) Add equation (9) and (8), express 2 s F (s) as a function of... s 2 F (s) = 1 {F (s s) 2F (s) + F (s + s)} + ( s) 2 1 (2 + 2i)! 2+2i s F (s) { s} 2i (12) i=1 This is a centered finite difference. Forward and backward finite difference stencils for 2 s F (s) can be obtained from combinations of Taylor series for F (s + s) and F (s + 2 s), or F (s s) and F (s 2 s) respectively.

10 Finite Difference Solution of WE Wave equation, FD 2 nd -order in space h = y = h 2 P(x, t) 1 c 2 (x) 2 t P(x, t) = 1 ( h) P x (t) 1 c 2 (x) 2 t P x (t) (13) Laplacian, FD 4 th -order in space 2 P(x, t) 1 c 2 (x) 2 t P(x, t) = 1 12( h) c 2 (x) 2 t P x (t) (14) P x (t)

11 Laplacian, FD 4 th -order in space, isotropic 2 P(x, t) = 1 α 12( h) P x (t) + 1 β 12( h) P x (t) (15) β = 1 α 2, for example: α = 1 β = 0, α = 1/2 β = 1/4 or α = 2/3 β = 1/6.

12 Wave equation, FD 2 nd -order time stepping 2 t P(x, t) c 2 (x) 2 P(x, t) = 1 t P t (x) c 2 (x) 2 P t (x) Solve for P t+1 (x) (16) P t+1 (x) = 2P t (x) P t 1 (x) + t 2 c 2 (x) 2 P t (x) (17) Wave equation, FD 4 th -order time stepping Include the 4 th -order derivative from equation (12), by substituting the wave equation (Dablain, 1986), as 4 t P(x, t) = 2 t 2 t P(x, t) = 2 t c 2 (x) 2 P(x, t) (18)

13 Pseudospectral (Fourier) methods Laplacian computed using FFTs: c 2 (x) 2 P t (x) c 2 (x) FFT { 1 } k 2 FFT [P t (x)] Wave equation, FD 2 nd -order time stepping and pseudospectral Laplacian: P t+1 (x) = 2P t (x) P t 1 (x) + t 2 c 2 (x) FFT { 1 } k 2 FFT [P t (x)]

14 Stability and accuracy of explicit methods Courant number : Cour = maximum velocity. c max t min( x, y, z) where c max is the Courant-Friedrichs-Lewy (CFL) condition : Cour 1 it is a necessary, but not sufficient condition for a stable explicit extrapolator. Numerical dispersion causes c P c, where c P = ω/ k is the effective phase velocity of numerically propagated waves

15 Stability and accuracy analysis of pseudospectral methods Substitute [ ( a generic )] plane wave solution: exp i kx + ωt Dispersion relation: ω = «2 sin 1 ± c t k 2 t Phase velocity: c P = ω 2 sin 1 k = «± c t k 2 t k For stability it must be c t k 2 1: 1D: Maximum k equal to Nyquist wavenumber k Nyq = π/ x stability requires Cour 2/π D: k max = 2k Nyq stability requires Cour 2/π D: k max = 3k Nyq stability requires Cour 2/ 3π 0.367

16 Stability and accuracy of 2 nd -order in time and space Substitute [ ( a generic )] plane wave solution: exp i kx + ωt Dispersion relation: ω =» 2 sin 1 c t x q sin 2 kx x ( 2 )+sin 2 kz z ( 2 ) t Phase velocity (worst case at k x = 0 or k z = 0): c P = ω k x = 2 sin 1 [ c t kx x sin( x 2 )] tk x For stability the argument of sin 1 must be between -1 and 1: 1D: Cour 1 2D: Worst case at k x = k z = k Nyq : Cour 2/

17 Observations Stability Stability constraint becomes more stringent with higher dimensions FD more stable than pseudospectral because errors in the spatial derivatives slows down high frequencies. Dispersion Pseudospectral High frequencies (wavenumbers) arrive before low frequencies (wavenumbers). Dispersion gets worse as the Courant number increases. FD High frequencies (wavenumbers) tend to arrive after low frequencies (wavenumbers). Dispersion gets better as the Courant number increases.

18 Frequency dispersion with finite-differences schemes

19 Frequency dispersion with finite-differences schemes

20 Frequency dispersion with finite-differences schemes

21 Frequency dispersion with finite-differences schemes

22 Frequency dispersion with pseudspectral Laplacian

23 Wavelength dispersion with finite-differences schemes

24 Wavelength dispersion with finite-differences schemes

25 Wavelength dispersion with finite-differences schemes

26 Wavelength dispersion with finite-differences schemes

27 Wavelength dispersion with pseudspectral Laplacian

28 Green s function Introduce Green s function for a constant density and sourceless medium equation (5) by a point source term acting at t = 0 and x = x s 2 i G c 2 2 t G = δ(x x s )δ(t), (19) where G = G(x, x s, t) is the Green s function. The solution for pressure to another forcing function for example s = sx, t can be represented as p(x, t) = G(x, x, t t )s(x, t )dx dt (20)

29 Perturbation Representation Represent the medium velocity as a background velocity and a perturbation c 2 (x) = c 2 b (x) [1 + α(x)] (21) Substitution into equation (19) gives i 2 G(x, x s, t) c 2 b (x) 2 t G(x, x s, t) = (22) δ(x x s )δ(t) + α(x)c 2 b (x) 2 t G(x, x s, t),

30 Introducing G b (x, x s, t) as a solution to 2 i G b (x, x s, t) c 2 b (x) 2 t G b (x, x s, t) = δ(x x s )δ(t), (23) we see that is we represent the full solution as a sum of the background solution plus a perturbed solution as G(x, x s, t) = G b (x, x s, t) + G p (x, x s, t). (24)

31 Equation (22) can be thus written as 2 i G p (x, x s, t) c 2 b (x) 2 t G p (x, x s, t) = α(x)c 2 b (x) 2 t G(x, x s, t). (25) Note the forcing function dependent on medium parameter α. Thus using a representation as (20) for G p (x, x s, t) we find for G(x, x s, t) G(x, x s, t) = G b (x, x s, t) (26) G b (x, x, t t )α(x )c 2 b (x ) 2 t G(x, x s, t )dx dt

32 Born Approximation The Born approximation is made in the perturbation representation by substituting the total field under the integral for the background field. G(x, x s, t) = G b (x, x s, t) (27) G b (x, x, t t )α(x )c 2 b (x ) 2 t G b (x, x s, t )dx dt This is an explicit representation for G(x, x s, t). The perturbation can represent a (single additional) scattered wavefield as G s (x, x s, t) = d(x, x s, t) = G b (x, x, t t )α(x )c 2 b (x ) t 2 G b (x, x s, t )dx dt. (28)