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3 Gassmann's relations: isotropic form

3 Gassmann's relations: isotropic form

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0ị;



2  ẳ B  0ị;



2  ẳ B  0ị



The real part of the effective vertical wavenumber for pressure waves in acoustic

media is (neglecting higher than second-order powers of the fluctuations)

Z 1

stat

2

2

kz ¼ kz À k0 cos 

d BðÞ sinð2k0  cos ị

0



kzstat ẳ k0 cos 1 ỵ 2 =2 ỵ 2 = cos2 ị

Bị ẳ B ị ỵ 2B ị= cos2  ỵ B ị= cos4 

For waves in an elastic layered medium, the real part of the vertical wavenumber

for P-, SV-, and SH-waves is given by

Z 1

P

2

^

kz ẳ la ỵ oAP o

d ẵBP ị sin2 la ị ỵ BBB ị sinl ị ỵ BDD ị sinlỵ ị

0



kzSV



ẳ lb ỵ oA^SV o2



kzSH ẳ lb ỵ oA^SH o2



Z

Z



1



d ẵBSV ị sin2lb ị BBB ị sinl ị ỵ BDD ị sinlỵ ị



0

1



d ẵBSH ị sin2lb ị



0



and the imaginary part of the vertical wavenumber (which is related to the attenuation

coefficient due to scattering) is



140



Seismic wave propagation



Z

P ẳ o2



1



d ẵBP ị cos2la ị ỵ BBB ị cosl ị ỵ BDD ị coslỵ ị



0



Z



1



SV ẳ o



2



Z



d ẵBSV ị cos2lb ị ỵ BBB ị cosl ị ỵ BDD ị coslỵ ị



0

1



SH ẳ o2



d ẵBSH ị cos2lb ị



0



p

p

where la ẳ o 1= 20 p2 ; lb ẳ o 1= 02 p2 ; lỵ ẳ la ỵ lb , and l ẳ lb la

(only real-valued la,b are considered). The other quantities in the preceding expressions are

BP ị ẳ X2 20 ị1 ẵB ịC21 ỵ 2B  ịC1 ỵ 2B  ịC1 C2

ỵ 2B ịC2 ỵ B ị ỵ B ịC22

BSV ị ẳ Y 2 02 ị1 ẵB ịC27 ỵ 2B

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3 Gassmann's relations: isotropic form

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