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12 Waves in layered media: frequency-dependent anisotropy, dispersion, and attenuation

12 Waves in layered media: frequency-dependent anisotropy, dispersion, and attenuation

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" cos P



ỵ 2 "2 2 1 þ "1 À "2 Þ sin3 P

" À



"

#

"2



À À 2 "2

2 1 ỵ "1 "2 ị sin3 P

" cos S



"

#

"2

ỵ 2 "2

1 2 ị sin3 P

" cos S

2

"

#

"2

ỵ 2 "2

2 1 ỵ "1 "2 ị sin5 P

" À cos S





110



Seismic wave propagation



Transversely isotropic media: HTI

In HTI (transversely isotropic with horizontal symmetry axis) media, reflectivity will

vary with azimuth, z, as well as offset or incident angle y. Ruăger (1995, 1996) and

Chen (1995) derived the P-wave reflection coefficient in the symmetry planes for

reflections at the boundary of two HTI media sharing the same symmetry axis. At a

horizontal interface between two HTI media with horizontal symmetry axis x1 and

vertical axis x3, the P-wave reflectivity for propagation in the vertical symmetry axis

plane (x1–x3 plane) parallel to the x1 symmetry axis can be written as

"

#

 "? 2 ?

1 ÁZ 1

2 

Vị





RPP  ẳ 0; ị %

ỵ 

"

"



"?

2 Z"

2

!

1

2

Vị

ỵ "

sin2  tan2 

sin  ỵ

"

2

"

#



 "2 

1 Z 1

2



Vị





2
ỵ 

%

"

"



"

2 Z"

2

!

1

2

Vị

sin2  tan2 

sin  ỵ

ỵ "

"

2

where the azimuth angle z is measured from the x1-axis and the incident angle y is

defined with respect to x3. The isotropic part RPP-iso(y) is the same as before. In the

preceding expression

sffiffiffiffiffiffiffi

C33

C11 C33



;

"Vị ẳ



2C33

s

C44

C13 ỵ C55 ị2 C33 C55 ị2



;

Vị ¼



2C33 ðC33 À C55 Þ

sffiffiffiffiffiffiffi

C55

C66 À C44



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12 Waves in layered media: frequency-dependent anisotropy, dispersion, and attenuation

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