The integral expression of the acoustic multiple scattering about cracks

1
The integral expression of the acoustic multiple
scattering about cracks

• Xiaodong Shi Hong Liu

Key Laboratory of Petroleum Resources,
Institute of Geology and Geophysics, Chinese
Academy of Sciences
2
Outline

• Introduction
• Method
• Numerical examples
• Conclusions

3
Outline

• Introduction
• Method
• Numerical examples
• Conclusions

4
Introduction

• Biot theory (1956)
• Eshelby (1957) proposed the classical formulas
  about the non-uniform media .
• HKT theory
• Hudson (1980,1981) proposed the expression on
  the velocity anisotropy caused by cracks and
  scattering absorption.
• Kuster and Toksoz(1979,1981) mainly presented
  the  equivalent velocity for the cracks with the
  Biot viscous fluid in it.
• Chen Xiaofei (1993), scattering matrix in
  wavenumber domain by means of continuation
  according to direction,

5
Introduction

• The defect about the HKT theory is that
• there is no analytical solution for the
• ellipsoidal seismic wave, because it lacks an
  orthogonal coordinate system to get
• the differential equation with coordinate
•   separation.

6
Introduction

• Characters of the integral expression which we
• proposed
• Via frequency wavenumber domain.
• Include the exponential function, separable
  approximation and fractional operators.

• two important characteristics of the cracks
  scattering
• coupling among the spherical harmonic mode
• the multiple scattering

7
Outline

• Introduction
• Method
• Numerical examples
• Conclusions

8
Method

• Modified from Chen Xiaofeis method(1993), so
  called continuation according to direction
• Difference Chen find scattering matrix, We give
  transfer matrix
• Based on transfer matrix, we inverse its element
  by Witt formula in pseudo differential operator
  theory

9
Transfer matrix expression
n
m
Modified from chen xiaofei (1993)
10
Symbol Inversion via element of Transfer matrix
11
Method

• In fact, R is an evolutional form of the
  Sphere Reflection Coefficient, n-m is the Mode
  Coupling Coefficient, and the factor
  is depending on the shape of the
  crack. If b0, R can be expressed as

(8)
which is the spherical reflection coefficient.
12
Method

• If the incident wave can be read as

(9)
the scattering wave can be read as
(10)
(11)
13
Outline

• Introduction
• Method
• Numerical examples
• Conclusions

14
the global scattering matrix
(a)
(b)
(d)
(c)

• the global scattering matrix changes with
  the value of incident frequency which is 5Hz,
  10Hz, 15Hz and 30Hz with respect to sub-picture
  (a), (b), (c) and (d).

15
the global scattering matrix
(a)
(b)
(c)
(d)

• the global scattering matrix changes with the
  value
• of the size about the crack which is
  10m,20m,40m
• and 80m corresponds to sub-picture
  (a),(b),(c)and (d).

16
incident wave
17
Wave-field for single wavenumber

• Angle.in0 Ka1.5

Angle.inpi/6 Ka1.5
18
snapshots

• model

t0.16s
t0.32s
t0.4s
19
Outline

• Introduction
• Method
• Numerical examples
• Conclusions

20
Conclusions

• two important characteristics of the scattering
  firstly
• spherical harmonic mode coupling which is
  different from the sphere scattering.
• it gives an expression about the multiple
  scattering which is distinct from Eshebys
  static field.
• Eshebys static field methods ignore the multiple
  scattering and the mode coupling,
• the equavalent theory based on the method is
  that the velocity anomaly becomes smaller while
  the absorption anomaly become larger.
• New quasi static approximation should be given

21
Conclusions(continued)

• Further works
• more comparision of our method to numerical
  calculation on single and more cracks
• Giving the integral expression of the elastic
  wave P-SV or P-SV-SH.

22
Thanks for your attention!!
welcome comments and suggestions!!
23
acknowledgements

• NSFC key project of National natural science
  foundation(40830424)
• MOSTNational Hi-Tech Research and Development
  Program of China..(863 Program),Grant No
  2006AA09A102-08
• MOSTNational Basic Research Program of
  China..(973 Program), Grant No2007CB209603

24
Method

• Figure 1 is the crack model. The length
  of the crack is ab and the thickness of it is
  a-b.

Fig1 the crack model
25
Method

• The outward wave-field can be written as

(1)
Where is the outward scattering
coefficient, is the first kind n-order
Hankel function, the subscript gt means
outward, is the outward angle between the
normal and the outgoing wave, k is the
wavenumber,
The inward wave-field can be read as
(2)
Where is the inward scattering
coefficient, is the second kind n-order
Hankel function.
26
Method

• we build up the transfer matrix chen xiaofei
  (1993) give different formular on scattering
  matrix

(3)
Where
27
Method

• It should be noted that eq. (3) can be
  adapted to calculate any convex inclusions. By
  the differential operators, we can get

(4)
Where the global scattering matrix can
be read as
(5)
(6)
(7)