Low-Frequency Properties and Parameterization of Layered Medium - ORT Interpretation
Explore the low-frequency properties of a layered medium, including ORT stiffness coefficients, system matrix, upscaling techniques, and the BCH series. Understand dispersion in terms of ORT parameters and zero-frequency approximation for physical mediums. Discover insights by Alexey Stovas and IGP from NTNU.
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Low-frequency ORT Alexey Stovas, IGP, NTNU
OUTLINE Low-frequency properties of layered medium ORT medium and parameterization BCH series for ORT Eigenvalues, multipliers and frequency dependent velocities Interpretation of dispersion in terms of ORT parameters Conclusions
Low-frequency properties of the medium Zero- and infinite-frequency limits Given frequency w=w0 (non-physical medium) Low-frequency approximation + =
ORT: stiffness coefficient matrix c c c c c c 11 12 13 22 23 33 = C c 44 c 55 c 66 ( ) ij c ( ) , , , , , , , , v v Tsvankin, 1997 0 0 1 2 1 2 3 1 2 P S
System matrix for ORT 1 c c c c 13 23 p p 1 2 c p p 33 c c c c 33 33 1 2 M 0 N 1 = = = A N 13 p s s M 0 p 1 11 12 1 0 c 33 55 1 23 p s s 0 p 2 12 22 2 c 33 44 2 13 2 23 c c 13 23 c c c c c = + = + = + 2 1 2 2 2 2 2 1 , , . s c p c p s c c p p s c p c p 11 11 66 12 12 66 1 2 22 22 66 33 33 33
Upscaling (replacement of Schoenberg-Muir) = A A 1 c c c c 2 2 B B B B B B = = = , , , 13 23 B B B = + = + = , , , 2 B 3 c B c B B c 2 1 2 3 c 11 7 12 8 6 13 33 33 33 1 1 1 B 1 1 1 2 3 B B B B = = = , , , = + = = B B B c , , , 3 c B c c 4 5 6 66 22 9 23 33 c c 44 55 1 1 1 1 B 1 B 2 13 2 23 c c 13 23 c c c c c = = = , , . c c c B = = + = , , B c B c c B c 44 55 66 6 7 11 8 12 66 9 22 4 5 33 33 33 Zero-frequency limit
The BCH series ( ) ( ) ( ) 2 = + i H + i H + A F F F ... 0 1 2 Roganov and Stovas, 2012
The BCH series ( ) = + F A A 1 , 0 1 2 1 2 1 12 ) ( = F A A 1 , , 1 2 1 ) ( ( ) = + F A A A A A A 1 1 , , , , , 2 2 2 1 1 1 2 [x,y] is a commuting operator is a volume fraction Roganov and Stovas, 2012
The BCH series M N M N 0 2 1 1 2 = A A , , 2 1 N M N M 0 2 1 1 2 M M N M M N 0 2 1 2 2 2 1 = A A A , , 2 , 2 2 1 N M N N M N 0 2 2 1 2 1 2 M M N M M N 0 1 2 1 1 1 2 = A A A , , 2 . 1 1 2 N M N N M N 0 1 1 2 1 2 1 Roganov and Stovas, 2012
+ m m m m m m = 2 , m 2 1 Weak contrast 2 1 = m 2 1 a a a Isotropic background Weak contrast in elastic and anisotropy parameters 1 v ( ) ( ) 1 2 1 2 2 0 2 0 p p p p 1 2 2 P M 0 0 1 2 0 0 ( ) ( ( ) ) ( ) ( ) 1 v 0 = A = 1 2 1 4 1 2 3 4 2 0 2 0 2 1 2 S 2 2 2 S 2 S 2 0 N p p v p v p p v = M 0 p 0 1 0 0 0 0 1 2 0 0 0 1 2 S N 0 ( ) 0 0 ( ) ( ) 0 1 2 3 4 2 0 2 S 2 0 2 1 2 S 2 0 2 2 2 S 1 4 1 2 p p p v p v p v 1 v 2 0 1 2 0 0 0 0 0 p 2 2 S 0 0 ( ) ( ) 2 2 2 P 2 S 2 1 2 2 2 2 2 2 3 2 1 2 2 , , , , , , , , , o o d dv dv 1 = = + + + 2 A A A A , A A 1 0 Matrix series with respect to contrast 2 , A A 2 0
Weak contrast ( ) ( ) 2 = 1 2 + + 2 0 F A o A A 0 ) ( 11 6 = F 1 , , A A 1 0 ( ) ( ) = 1 2 + F A , , , , , A A A A A 2 0 0 0
Weak contrast ( ) = + + A R R R Matrix series with respect to contrast 0 1 2 = R A , 0 0 ( )( 6 ) 1 2 1 ) ( ( ) ( ) 2 = 1 2 + i H i H R 1 , , , , A A A A A A 1 0 0 0 ( ) 1 ( ) 2 = + i H 2 R , , . A A A A 2 0 6 No second-order contrasts in dispersion terms!
Characteristic equation (eigenvalues) ( ) = A I det 0 q ( ) ( ) ( ) + + + = 6 4 2 0 q a q a q a 4 2 0
Characteristic equation (eigenvalues) ( ) = + + 2 j 2 j 2 2 3 2, q q H d o 0 j ( ) 2 2 2 1 3 ( ) ( ) 0 j ( ) 0 j ( ) 0 j ( ) 0 j = 1 d q k q A A I A 0 j j ( ) 0 j ( ) 0 j k = j
Characteristic equation (P-eigenvalues) ( ) 0 P ( ) 0 P k = P ( ( ) ( ( ) ) ( ) ( ) T ( ) 0 P ( ) 0 P ( ) 0 P ( ) 0 P = 1 2 + = 2 1 2 2 2 S , , , p p v p p 1 0 0 1 2 1 2 ) ( ) 0 P ( ) 0 P ( ) 0 P ( ) 0 P = , 2 , 2 0 P 0 P 2 S 0 P 2 S = q q p v q p v , 2 1 0 0 2 0 0 2 1 ( ) 0 P = 2 k q 0 P
Slowness surface dispersion P S1 S2 Frequency
P wave (down, up) S1 wave (down, up) S2 wave (down, up) ( ) ( ) e , , i p p = = 1 2 , , 0 0 p p 1 2 Multipliers p1=0.1 p2=0.1 p1=0.05 p2=0.2 p1=0.2 p2=0.05
Frequency-dependent phase velocity P-wave Phase velocity, km/s S1-wave S2-wave = = 0.1 p p s km 1 2 Frequency, Hz
Wave mode selection Trial series for dispersion coefficient: ( ) 2 ( ) 00 ( ) 20 ( ) 02 ( ) 40 ( ) 22 ( ) 04 j j j j j j = + + + + + + 2 1 2 2 4 1 2 1 2 2 4 2 j d a a p a p a p a p p a p o Three solutions for a00 that give the wave mode selection.
Quadratic form ( )( 4 0 3 P v = m D 2 2 1 ( ( ( ) , ( ( 2 ) , = = = = = + + + ) ( 00 ) m m , , , , , , , , v v v v 2 qP = + , a v 20 2 02 1 1 2 S P S P P ) ) m m , , , , , , , , v v v v 40 2 2 04 1 2 1 1 S P S P ( ) ) 2 2 m , , , , , . v v 1 ( 20 ) ( 2 ) 22 1 1 2 3 2 S P qP qP T m , a 20 20 2 S 12 v 0 ( ) 2 2 1 ( 02 ) ( 2 ) qP qP = T m D m , a 02 02 2 S 12 v 0 ( 12 ) 2 2 1 ( 40 ) ( 4 ) qP qP = T m D m , a 40 40 2 0 ( 12 ) 2 2 1 ( 04 ) ( 4 ) qP qP = T m D m , a 04 04 2 0 D4 D2 ( ) 2 2 1 6 ( 22 ) ( 22 ) qP qP = T m D m , a 22 22 2 0
Conclusions We derive the low frequency approximation for waves propagating in multi-layered orthorhombic model. The weak-contrast approximation is introduced. We show that the stop-bands are the result of interaction of different wave modes (P, S1 and S2). The stop-bands are illustrated by multipliers. By defining the low-frequency effective anisotropic parameters, we perform the sensitivity analysis for intrinsic anisotropy parameters.
