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

Outline

IntroductionMethodNumerical examples

Conclusions

Outline

IntroductionMethodNumerical examples

Conclusions

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,

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.

IntroductionCharacters 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 crack’s scattering: coupling among the spherical harmonic mode the multiple scattering

Outline

IntroductionMethodNumerical examples

Conclusions

Method

Modified from Chen Xiaofei’s 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

(1) expn nn

u u H kr inr

(2) expn nn

d d H kr inr

(2) (2)

(1) (1)

exp exp

exp exp

n n n

n

n n n

H kr in H kr inW

H kr in H kr in

(1) (2)

(1)

(1) (2)

exp exp

exp exp

1 0 0 1

0 1 1 0

n m n m

m

m m

Tm

H kr im H kr imW

H kr im H kr im

W

(1)1

8n m

n mmn m

u ui d WW

d dr

n

m

Transfer matrix expression

Modified from chen xiaofei (1993)

Symbol Inversion via element of Transfer matrix

nu

2 1

2 1

2 1

sin2cos2

, , ,sin2cos2

n mb

k ka b

R k k n mb

ka

nk mb

1 2

2 nm 1

nm2 1

1 2 2 1

exp 2 cos2 , , ,

exp

s

1s = exp

2 cos2 , , ,1

m

n mm

u k d k

i k k a b R k k n m

i k k a b R k k ni n

md m

md

a b

a b2

2

kv

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 b=0, R can be expressed as:

Method

sin2 cos2b a b

2 1

2 1

k kR

k k

(8)

which is the spherical reflection coefficient.

If the incident wave can be read as:

Method

(1)1 1

1exp

4 n n s sn

d u i J kr H kr in (9)

the scattering wave can be read as:

11

1 2, , , exp , ,

8 4s s s ss

u r r i ikr i f rkr

(10)

(1)1, , exp exp ( ) exp ( )

2 2s n nm s sn m

f r H k r in s im in

(11)

Outline

IntroductionMethodNumerical examples

Conclusions

the global scattering matrix

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).

(a)

(d)

(b)

(c)

nms

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).

(a) (b)

(c) (d)

the global scattering matrix

incident wave

Wave-field for single wavenumber

Angle.in=0 Ka=1.5 Angle.in=pi/6 Ka=1.5

snapshots

model t=0.16s

t=0.32s t=0.4s

Outline

IntroductionMethodNumerical examples

Conclusions

Conclusions

two important characteristics of the scattering: firstly spherical harmonic mode coupling which is different fro

m the sphere scattering. it gives an expression about the multiple scattering whic

h is distinct from Esheby’s static field. Esheby’s static field methods ignore the multiple

scattering and the mode coupling, the equavalent theory based on the method is t

hat the velocity anomaly becomes smaller while the absorption anomaly become larger.

New quasi static approximation should be given

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.

Conclusions(continued)

acknowledgements

NSFC: key project of National natural science foundation(40830424)

MOST:National Hi-Tech Research and Development Program of China..(863 Program),Grant No 2006AA09A102-08

MOST:National Basic Research Program of China..(973 Program), Grant No2007CB209603

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

Method

a+b

u(n)u1+d1

d(n) a-b

Fig1: the crack model

The outward wave-field can be written as:

Method

(1) expn nn

u u H kr inr

Where is the outward scattering coefficient, is the first kind n-order Hankel function, the subscript ‘>’ means ‘outward’, is the outward angle between the normal and the outgoing wave, k is the wavenumber,

nu1 ( )nH kr

The inward wave-field can be read as:

(2) expn nn

d d H kr inr

Where is the inward scattering coefficient, i

s the second kind n-order Hankel function. nd 2 ( )nH kr

(1)

(2)

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

Method

Where:

(1)1

8n m

n mmn m

u ui drWW

d dr

(2) (2)

(1) (1)

exp exp

exp exp

n n n

n

n n n

H kr in H kr inW

H kr in H kr in

(1) (2)

(1)

(1) (2)

exp exp

exp exp

n m n m

m

m m

H kr im H kr imW

H kr im H kr im

(3)

It should be noted that eq. (3) can be adapted to calculate an

y convex inclusions. By the differential operators, we can get:

Method

m

n nm mm

u s d (4)

Where the global scattering matrix can be read as: nms 2 1( , , , )expnms A k k n m i n m d

1 2 2 12 1

1 2 2 1

1 exp 2 cos2 , , ,( , , , )

1 exp 2 cos2 , , ,

i k k a b R k k n mA k k n m

i k k a b R k k n m

2 1

2 1

2 1

sin2cos2

, , ,sin2cos2

bk k n m

a bR k k n m

bk k n m

a b

(5)

(6)

(7)