Wave Propagation in Heterogeneous Media: Born and Rytov Approximations. Chris Sherman

Transcription

1 Wave Propagation in Heterogeneous Media: Born and Rytov Approximations Chris Sherman

2 Stochastic Scalar Wave Equation Wave Equation: &! " %& 1 # t V x ' ) () u(x,t) = 0 (*) Velocity Perturbation: = V o 1!! ( x) V x 1 V ( x)! 1 V o ( 1+! x ) for small ξ (**) Inserting (**) into (*): &! " 1 % V o ( ' 1+! ( x) )# t ) u(x,t) = 0 ( Sato, Haruo, Michael C. Fehler, and Takuto Maeda. 01. Seismic Wave Propagation and Scattering in the Heterogeneous Earth : Second Edition. Springer.

3 Born Approximation Rearranging: &! " 1 V # t % o Born Approximation: u = u o + u ( 1 u 1 << u ) o ' ) u(x,t) = " ( V! ( x )# t u(x,t) o & %! " 1 V # t o ' ) u +! " 1 o & ( V # t % o ' ) u = " 1 ( V! ( x )# t u o " o V! ( x )# t u 1 o Reference Solution Perturbation Solution Higher Order Term

4 Reference Solution Homogeneous scalar wave equation: (Incident wave) & %! " 1 V # t o ' ) u = 0 o ( Plane wave solution: u o = A o e i( kx!!t) Point source solution: u o = 1 4!r! " t! r # V o % & ' General solution: u o = G o ( x,t) Lay, Thorne, and Terry C. Wallace Modern Global Seismology. Academic Press.

5 Born Approximation Inserting the reference solution: & %! " 1 V # t o ' ) u = " 1 ( V! x # t o ( A o e i ( kxe 3"!t) )! = "A o V! ( x )e i ( kxe 3"!t) o = "A o k! ( x)e i kxe 3"!t = S( x,t)*! ( x)! ( t) Point Source Terms Applying the point-source solution and representation theorem: u 1 = G( x,t)! S( x,t)

6 Born Approximation # A o e i kx'!!t Expanding: u 1 = " k! x' G x! x',t! t ' dx'dt '!#""" =!k A o! =!k A o! e!i"t =!k A o! e!i"t! t! t '! x! x' ' " ( x' )e i ( kx'e 3!!t) % & V o ( ) # "!#""" dx'dt ' 4! x! x' e i kx'e! ' 3+ x!x' % & V o ( x! x' )! x' """ dx' e ik ( x'e 3+ x!x' )! x' """ dx' x! x' # Fraunhofer zone: x! x' " r! x'e r r >> 1 &! L k % ' (

7 Born Approximation Expanding: u 1 =!k A o! e!i"t! x' r! x'e r """ e ik x'e 3+r!x'e r dx' =!k A o!r ekr!i"t """ # ( x' )e ikx' ( e 3!e r ) dx' u 1 =!k A o!r!! ( ke 3! ke r )e kr!i!t Exchange wavenumber

8 Applicability of the Born Approximation } Outside of Fraunhoffer zone } Perturbation size is small } Low velocity contrast r >> 1! L k a! < " 4 } Commonly used for backscattering problems ke r Scattered wave (u 1 ) Incident wave (u 0 ) ke 3

9 Rytov Approximation Stochastic wave equation: &! " 1 V # t % o ' ) u = " ( V!# t o u z Rytov Approximation: u = e i( kz!!t) e " ( x #,z,! ) = e i( kz!!t) U ( x #, z,t) = e i( kz!!t) e ln (")+i# x Log(A) fluctuations Phase fluctuations

10 Rytov Approximation Derivatives of u: ( ( )) = % z U + ik% z U k U +! & U!u = "# " Ue i kz!t % t u = % + % Ue i kx!t % t, - % t ( ) '(. / 0 = '( % t U i!% t U! U Substituting:! z U + ik! z U " k U + # U " 1 V o = " V %&! t U " i!! t U "! U' ( o Fourier Transform:! z!u + ik! z!u + " #!U =!k!u ) * e i( kz!t) ) * e i( kz!t) %&! t U " i!! t U "! U' (

11 Rytov Approximation Ignoring the first term and substituting U: ik! z " + # " =!k Fourier transform in perpendicular direction: ik! z! " # k! " =!!k Green s function solution: Convolving with the source term: G! ( k!, z,! ) = "i "i k e k k! z = k G! ( k", Z # z,! )!! k ", Z,! Z! ( k ", z)dz 0

12 Rytov Approximation: General Solution: u = e i( kz!!t) e " ( x #,z,! )!( x ", Z,! ) = k Z G! ( k", Z # z,! )! ( k! ", z)e #ik "x " dk " dz 0

13 Applicability of the Rytov Approximation } Small wavefield fluctuations } Low scattering angle ka > 1 } Commonly used for forward scattering problems Phase, Amplitude Distortions Incident wave (u 0 ) k

14 Media with Continuous Random Fluctuations Single scatterer vs. Continuous random medium

15 Media with Continuous Random Fluctuations } Assumptions: } Material fluctuations are random, stationary, and have zero mean } Characterized by an autocorrelation function! ( x) = 0! ( k )! k "! Born scattering amplitude (F) for a single scatterer: u 1 =!k A o!r!! ( ke 3! ke r )e kr!i!t = FA o r e kr!i!t Born scattering amplitude in an ensemble medium: " " " " " " # # # # #!"!"!"!"!"!" F = #! ( x' )! ( x'' ) e!i ( ke r!ke 3 )( x'!x'' ) dx'dx''

16 Most Probable Seismic Pulse JQ.& /& /+0%%"1 3 (& 4& (56718 amplitude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i)5.&l # )"&4"5+0"6-B O/" i kz! t 3ln)"&")0) )"&")0) &5*6"& *(#! +/&",*5.')%(#!"#%! & 5/*)*5+");"#,- * 'E9*0" 0" PQQQ! &!@L *( (E9*0" 0"65+- &!@ *(# * #"(&+3 SBR ' 5!!PB C( 1'B K (6- +/" /"+")'"("%& 4*)+ 3 +/"!#"6 & &/9(B C( +/" *5+%*6 1T!#"6L 1'B K & "!,"##"# ( * 5(&+*(+,*5.')%(#!"#%! 9+/ +/" 4)4")+"& #">("# time [s] *,0"B U" 5/&" +/"!#"6 '"!"+)- ( &%5/ * 9*- +/*+ %(#"&)"# )"7"5+(& 3)! +/"!#"6,)#")& *)" "D56%#"#B 1) 8-31)0 9" b!4*)&( 3 +/" 1T "D4")!"(+ *(# +/" '"(")*6;"# a^t/")+-c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at? 3)!*6&! '0"& * &!+/"# +/" &"5(# #")0*+0" 3 * Y5.") 9*0"6"+ 9+/ * #!(*(+ &"&!')*!L *6+/%'/ + & (+.(9( 9/*+.(# 3 *0")*'(' 3)"=%"(5-3 *,%+ ZR [; 8+/& 5))"&4(#& + * 9*0"6"('+/ 3 &/%6#," *446"# +/" Realizations ("D+ &"5+(L ( of 9/5/ +/" AQ!T. 3)M., +/" ' 9*0"<B 3%6>6 +/" &+*,6+*(# Müller, and S. 1%)+/")!)"L A. Shapiro.9"001. Most Probable Seismic Pulses8&"" in *6& Single Two)"6*+( + 5!!( *0")*'(' 4)5"#%)"& & #&5%&&"#<B #&4")&( 5)+")* )"=%)"# 3) +/" )+*+"# &+*''")"# ')# *+ and Three-dimensional Random Media. Geophysical Journal International 144 (1): u=e( ) e

17 Scattering Attenuation 0 lnfla(tu>)i); f=loo; data=. theory=fat line 0 I e! local =!!L ln A 0 I N 0 I w 0 I L 0 I VI ZOO distance [m] #! local =! s! k " cos (# 3D L / k)" n # d# 0 Shapiro, S. A., and G. Kneib Seismic Attenuation By Scattering: Theory and Numerical Results. Geophysical Journal International 114 ():

18 Phase Dispersion v = L dt =!L! = "L! o L +! ( L) theory (10%) - expenment (10%) 0 theory (5%) experiment (5%) + theory (3%) - expenment (3%) Wa Figure 8. Velocity shift of an initially plane wave in a -D exponential random medium versus normalized travel distance for different standard deviations of fluctuations and constant frequency (A/a = 0.6). & sin " L / k v 3D! c o ( 1" 4k! % '(!L 0 3D ) # n (!)d! +k" ln k +!, % * + k "! -. 0 / 3D # n (!)! d! 1 01 "1 Shapiro, S. A., R. Schwarz, and N. Gold The Effect of Random Isotropic Inhomogeneities on the Phase Velocity of Seismic Waves. Geophysical Journal International 17 (3):

19 Inverting for Distribution Parameters } Estimate fractal exponent (β) and amplitude (ε) from travel time deviations } Inverting for ε is stable and fast } Inverting for β is difficult and highly dependent on starting parameters Figure Deviations of field travel times from the reference travel-time curve. Klimeš, L. 00. Estimating the Correlation Function of a Self-affine Random Medium. Pure and Applied Geophysics 159 (7-8) (July 1):

20 Issues with Born and Rytov Approximation } Single scattering } Does not conserve energy } Does not take into account near-field effects } Effective shear energy ~10% of compressional wave energy is not accounted for (My research is looking into this) } Other methods: } Radiative Transfer Theory } Finite Difference / Finite Element Sato, Haruo, Michael C. Fehler, and Takuto Maeda. 01. Seismic Wave Propagation and Scattering in the Heterogeneous Earth : Second Edition. Springer.

21 Radiative Transfer Theory } Heuristic solution developed to model scattering of light through the Earth s atmosphere } Models energy transport through a heterogeneous medium } Conserves energy } Multiple scattering } Efficient for calculating seismogram envelopes Energy transfer equations: 1 I P (x, k, t) + k gradi P (x, k, t) α 0 t = 1 ( g ) pp k, k I ( P x, k, t ) dk g 0 pp 4π I P (x, k, t) + 1 ( g ) sp k, k I ( S x, k, t ) dk g 0 ps 4π I P (x, k, t) + Q P (x, k, t) Scattering Coefficient Energy 1 I S (x, k, t) + k gradi S (x, k, t) β 0 t = 1 ( g ) ss k, k I ( S x, k, t ) dk g 0 ss 4π I S (x, k, t) + 1 ( g ) ps k, k I ( P x, k, t ) dk g 0 sp 4π I S (x, k, t) + Q S (x, k, t). (1) Przybilla, Jens, and M. Korn Monte Carlo Simulation of Radiative Energy Transfer in Continuous Elastic Random Media three-component Envelopes and Numerical Validation. GJI173 ():

22 Finite Difference Newton s nd : Hooke s Law: Isotropy:!!u i = T ij, j + f i = ( C ijkl! ) ij, j + f i = (!" ij u k,k + µ ( u i, j + u )) j,i + f i, j Taylor Expansion: u( x +!x) = =! ( ij!, j u k,k +!u ) k,kj + µ ( u i, jj + u ) j,ij + µ (, j u i, j + u ) j,i + f i u ( 1) x N!x " i ( x) + O!x N+1 i=0 i! u(i) = u ( x +!x ) " u( x "!x)!x + O (!x 3 ) u ( x) = u ( x +!x ) " u( x) + u( x "!x)!x + O!x 4

23 Finite Difference } Limitations: } Time step size (Courant) } Grid size (>10 points/wavelength) } Numerical Dispersion } High frequency problem } Boundary Effects } Reflection from quiet boundaries } Source representation } Computational Resources T simulation! 1 "t T simulation,3d! 1 3 # & % "x' ( Kang, Tae-Seob, and Chang-Eob Baag An Efficient Finite-Difference Method for Simulating 3D Seismic Response of Localized Basin Structures.. BSSA Vol 94 (9):