Wave Propagation in Heterogeneous Media: Born and Rytov Approximations Chris Sherman
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.
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
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. 1995. Modern Global Seismology. Academic Press.
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)
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 % ' (
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
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
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
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' (
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
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
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
Media with Continuous Random Fluctuations Single scatterer vs. Continuous random medium
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''
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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 0 100 ZOO 300 400 500 800 distance [m] #! local =! s! k " cos (# 3D L / k)" n # d# 0 Shapiro, S. A., and G. Kneib. 1993. Seismic Attenuation By Scattering: Theory and Numerical Results. Geophysical Journal International 114 (): 373 391.
Phase Dispersion v = L dt =!L! = "L! o L +! ( L) 1.4 0.8 0.6 0.4 theory (10%) - expenment (10%) 0 theory (5%) experiment (5%) + theory (3%) - expenment (3%) 0 1 1.5.5 3 3.5 4 4.5 5 5.5 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. 1996. The Effect of Random Isotropic Inhomogeneities on the Phase Velocity of Seismic Waves. Geophysical Journal International 17 (3): 783 794.
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): 1833 1853.
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.
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. 008. Monte Carlo Simulation of Radiative Energy Transfer in Continuous Elastic Random Media three-component Envelopes and Numerical Validation. GJI173 (): 566 576.
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
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. 013. An Efficient Finite-Difference Method for Simulating 3D Seismic Response of Localized Basin Structures.. BSSA Vol 94 (9): 1690-1705.