A Wavelet-Based Method for Simulation of Seismic Wave Propagation

Transcription

1 AGU Fall Meeting, A Wavelet-Based Method for Simulation of Seismic Wave Propagation Tae-Kyung Hong & B.L.N. Kennett Research School of Earth Sciences The Australian National University

2 Abstract Seismic wave propagation (e.g., both P-SV and SH in -D) can be modeled using wavelets. The governing elastic wave equations are transformed to a first-order differential equation system in time with a displacement-velocity formulation. Spatial derivatives are represented with a wavelet expansion using a semigroup approach. The evolution equations in time are derived from a Taylor expansion in terms of wavelet operators. The wavelet representation allows high accuracy for the spatial derivatives. Absorbing boundary conditions are implemented by including attenuation terms in the formulation of the equations. The tractionfree condition at a free surface can be introduced with an equivalent force system. Irregular boundaries can be handled through a remapping of the coordinate system. The method is based on a displacement-velocity scheme which reduces memory requirements by about % compared to the use of velocity-stress. The new approach gives excellent agreement with analytic results for simple models including the Rayleigh waves at a free surface. A major strength of the wavelet approach is that the formulation can be employed for highly heterogeneous media and so can be used for complex situations.

3 Wavelets & Applications (I-) Signal processing Phase picking (P, S, Lg) Estimation of a time-varying spectral density matrix for three component seismic data Measurement of the anisotrophy at a given area using a relative phase between components Numerical analysis Simplification of governing equations by applying a wavelet function Composite use of a FD scheme and an interpolating wavelet transform Application of wavelets in differentiation of functions

4 Wavelets & Applications (I-).5 ψ(x) x.5 ϕ(x) x Wavelet and scaling functions Example of signal decomposition f(t) Q f(t) Q f(t) Q f(t) Q f(t) 4 Q f(t) 5 Q f(t) 6 Q f(t) 7 Q f(t) 8 P f(t) 8 Input signal f(t) Decomposition using wavelets t

5 Wavelets & Applications (I-) Discrete wavelet transform (DWT) Daubechies (99) built the foundation of a DWT Based on a multiresolution analysis that decomposes a signal into components of different scales Wavelets & parabolic PDEs Beylkin & Keiser (997) introduced a numerical scheme for parabolic PDEs using semigroup approach and wavelets Semigroup approach t g = Lg N + f (g) g n+ = e δtl g n + δt ( γ N n+ + R m= β m N n m )

6 A Wavelet-Based Method (II-) Spatial derivative operators in elastic wave equations are treated through wavelet transforms in a physical domain The resulting second order differential equations for time evolution are solved via a system of first-order differential equations using a displacement-velocity formulation, which can reduce memory requirements by % compared to a velocity-stress formulation With the combined aid of a semigroup representation and spatial differentiation using wavelets, a uniform numerical accuracy of spatial differentiation can be maintained across the domain

7 A Wavelet-Based Method (II-) Numerical difficulties Numerical formulation of wave equations for the parabolic PDE scheme Representation of matrix operators in wavelet bases Treatment of artificial boundary conditions Treatment of inherent periodic boundary conditions Implementation of external boundary conditions (e.g. free-surface conditions, rigid boundary conditions) Expansion to complex structure problems

8 A Wavelet-Based Method (II-) Corresponding remedies Introduction of a displacement-velocity formulation Application of a Taylor expansion for the simplification of matrix operators Introduction of absorbing boundary conditions Adjustment of physical parameters around the artificial boundaries Application of equivalent force terms in the system of equations for external boundary conditions Introduction of grid generation scheme

9 Formulation of Equations ( Formulation of Equations ( Formulation of Equations ( Formulation of Equations (III) -D elastic wave equations (P-SV) F LU U + = + + = + + = t z zxz xz z x xz xx x f z x t u f z x t u σ σ ρ σ σ ρ = = = z x zz zx xz xx z z x x f f L L I L L I v u v u ρ F L U where, : vector variable : matrix operator : force vector U L F

10 Test of Equivalent Forces (IV) -D acoustic waves. t=6 s. t= s.8 analytic direct(point) direct(band) semigroup.8 analytic direct(point) direct(band) semigroup u(x).4 u(x) The equivalent force terms for boundary conditions can simulate exactly the effects which are caused by the presence of the boundaries x u(x) analytic direct(point) direct(band) semigroup x t=9 s x

11 Unbounded Medium ( Unbounded Medium ( Unbounded Medium ( Unbounded Medium (V) Source H H H H R R R R m 598 m α =.5 km/s β =.8 km/s ρ =. g/cm Time (s) Comparisons with analytic solutions X Z X Z X Z R R R X Z R 4 numerical analytic Model Comparisons with analytic solutions P-SV waves

12 Media with a Free Surface ( Media with a Free Surface ( Media with a Free Surface ( Media with a Free Surface (VI-) Time (s) Comparisons with analytic solutions S SS d=. km d=.6 km d=4.5 km numerical analytic Model Comparisons with analytic solutions SH waves H R H R H R S Ɣ R Ɣ L Ɣ B Ɣ T m m α =.5 km/s β =.8 km/s ρ =. g/cm z x

13 Media with a Free Surface ( Media with a Free Surface ( Media with a Free Surface ( Media with a Free Surface (VI-) P-SV waves Comparisons with analytic solutions H R H R H R H R 4 S Ɣ R Ɣ L Ɣ B Ɣ T m m α =.5 km/s β =.8 km/s ρ =. g/cm z x Model Time (s) Comparisons with analytic solutions X Z X Z X Z d=. km d=.5 km d=4.6 km X Z d=. km P S P H S+R numerical analytic

14 Media with a Free Surface (VI-) Snapshots of wavefields SH waves t=.5 s SS X comp. H PS P-SV waves t=. s Z comp. -5 SS S -5 SP S PP P

15 Two Two Two Two-Layered Media ( Layered Media ( Layered Media ( Layered Media (VII-) Time (s) Comparisons with analytic solutions R R R R R 4 5 S Sr S H+Sr S H+Sr numerical analytic SH waves Model Comparisons with analytic solutions S Ɣ R Ɣ L Ɣ B Ɣ T H R H R H R H R4 H R5 m m m α =.5 km/s β =.8 km/s ρ =. g/cm α = 6. km/s β =.6 km/s ρ =. g/cm z x

16 Two-)) Two-Layered Media (VII( Snapshots of SH wavefields SH waves S H t=. s Sr I -5 5 x( m) St ) z (m -5 5

17 Two-Layered Media (VII-) Snapshots of P-SV wavefields -5 X comp. PPr PS PSr PP PSt P-SV waves t=.5 s P Hrp,Hrs Ht -5 Z comp. PPt 5 5

18 Gradient Velocity Media (VIII-) Model & Time responses * Source Range (m) Range (m) P -5 Time (s) P H S+R Time (s) PP PS S SS Slant : 8.7 r:.~.85 g/cm a:.5~7.88 km/s b:.8~4.5 km/s 6 Z comp. of receivers on a free surface Vertical components of receivers on a free surface 6 Z comp. of receivers at depth 5 m Vertical components of receivers at depth 5 m

19 Gradient Velocity Media (VIII-) Snapshots of P-SV wavefields -5 X comp. SP+SS PS PP S P-SV waves t=.5 s -5 Z comp. P 5 SP SS t=.5 s 5 H -5 PS -5 S PP 5 5

20 Media with Topographic Surface (IX-) Application to topographic problems Introduce a grid generation scheme : mapping a rectangular grid system to a curved grid system with consideration of physical topography Numerical model A homogeneous medium with a sinusoidal free surface Three different positions of sources; at a trough, a hill and a crest beneath a free surface 5 5 * S α=.5 km/s β=. km/s ρ=. g/cm * * S S

21 Media with Topographic Surface (IX-) Snapshots of P-SV wavefields (t=.9 s) Trough source Hill source Crest source X comp. X comp. X comp Z comp. Z comp. Z comp

22 Random Heterogeneous Media (X-) Numerical models % Perturbation Pointwise random heterogeneous medium with Gaussian probability distribution and standard deviation of % in velocity Perturbation Stochastic heterogeneous medium using a Von Karman autocorrelation function with a correlation distance of 5 km %

23 Random Heterogeneous Media (X-) Snapshots of P-SV wavefields (t=.9 s, X comp.) Homogeneous Random heterogeneity Von Karman ACF One-layered One-layered One-layered Two-layered Two-layered Two-layered

24 Random Heterogeneous Media (X--)) Time responses (one-layered media, X comp.) Range (m) Range (m) Time (s) X comp. of receivers at depth 5 m Range (m) X comp. of receivers on a free surface X comp. of receivers on a free surface X comp. of receivers at depth 5 m Range (m) Time (s) X comp. of receivers on a free surface Range (m) Von Karman ACF 8 Time (s).5 Range (m) Time (s) Random heterog. Time (s) Homogeneous.5 Time (s) X comp. of receivers at depth 5 m

25 Media with Cavities ( Media with Cavities ( Media with Cavities ( Media with Cavities (XI-) Numerical model S Ɣ R Ɣ L Ɣ B Ɣ T m m α =.5 km/s β =.8 km/s ρ =. g/cm z x Implementation of boundary conditions One or five cavities inside a medium -via additional equivalent force terms (i.e. traction-free boundary conditions) -can implement various boundary conditions at various locations at the same time

26 Media with Cavities (XI-) Snapshots in one cavity model (Z comp.) t=. s t=. s t=.8 s PS -5 PP -5 PS -5 PS P PP

27 Media with Cavities (XI-) Snapshots in five cavities model (Z comp.) t=. s t=. s t=.8 s PS -5 PP -5 PS -5 PS P PP

28 Tool for Quantitative Study (XII-) Comparison of accuracy with 4 th -order FDM Apply the first-order differentiation to the input signal which is corresponding to the quick variation of physical parameters in random heterogeneous media 5 5 input signal 4 analytic analytic 4th-order FD wavelet where, N X =

29 Tool for Quantitative Study (XII-) Application to Q estimation Q - /ε Von Karman ACF (n=.5) - ka o o 45 o 9 o Cf. Other Researches -Frankel & Clayton (986) : qmin = -45 -Roth & Korn (99) : qmin = -4 -Jannaud et al. (99) : qmin = 9

30 Conclusions (XIII) Can obtain high accuracy of numerical responses of media for wave propagation due to almost no loss of accuracy during spatial differentiation and third order accuracy for time derivatives Stable in highly perturbed velocity media Numerically stable scheme Can implement various complex boundary conditions at various locations easily using equivalent force terms Energy is conserved can be applied in seismic quantitative studies (e.g. estimation of amount of energy loss during wave propagation)