Download Xiao Dong Shi and Hong Liu, The integral expression and numerical

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
Introduction
Method
Numerical examples
Conclusions
Outline
Introduction
Method
Numerical examples
Conclusions
Introduction
 Biot theory (1956)
 Eshelby (1957) proposed the classical formulas about the nonuniform 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.
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 crack’s scattering:
 coupling among the spherical harmonic mode
 the multiple scattering
Outline
Introduction
Method
Numerical 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
Transfer matrix expression
 un 
1 
i
 
 d n  8  m 
d  r  


n 
dnH
(2)
n
 dW W
r 
n

 kr  exp  in 
(1)
m
 um 
 
 dm 
u  r  


n 
un Hn(1)  kr  exp in  
n
m

 Hn(2)  kr  exp  in 
Wn  
 H(1)  kr  exp in 
n








  n Hn(2)  kr  exp  in    W

(1)
  n Hn  kr  exp in    1
 


(1)
m
  n Hm(1)  kr  exp im      n Hm(2)  kr  exp  im   


(2)
 Hm(1)  kr  exp im    
Hm  kr  exp  im   

0  T  0 1

 Wm 

0

1


 1 0
Modified from chen xiaofei (1993)
Symbol Inversion via element of Transfer
matrix
un  k2  
s nm = d
m 

m 
s nm dm  k1 
1  exp  i2  k1  k2  a  b cos2   R  k2 , k1 , , n  m 
1  exp  i2  k1  k2  a  b cos2   R  k2 , k1 , , n  m 
b sin2
n  m
a

b
cos2



R  k2 , k1 , , n  m  
b sin2
k2  k1 
n  m
 a  b cos2 
exp  i  n  m  
dm
un
k2  k1 

k2 
v2
ab
a b
Method
In fact, R is an evolutional form of the Sphere Reflection
Coefficient, n-m is the Mode Coupling Coefficient, and
the factor b sin2  a  b cos2  is depending on the shape of
the crack. If b=0, R can be expressed as:
R
k 2  k1
k 2  k1
which is the spherical reflection coefficient.
(8)
Method
If the incident wave can be read as:
1 
d1  u1 
i  Jn  kr  Hn(1)  krs  exp in    s  
4  n 
(9)
the scattering wave can be read as:
u  r,rs , , s  
f  r , , s  
1
2


i
exp  ik1rs  i  f  r, , s 
8  k1rs
4


H
n 
(1)
n


m 

 k1r  exp in    snm  exp  im(

(10)

   

  s )     exp  in(   s )  
2
   2

(11)
Outline
Introduction
Method
Numerical examples
Conclusions
the global scattering matrix
(a)
(b)
(c)
(d)
snm
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).
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).
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
Introduction
Method
Numerical examples
Conclusions
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 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
that the velocity anomaly becomes smaller while
the absorption anomaly become larger.
 New quasi static approximation should be given
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.
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
Method
u1+d1
Figure 1 is the crack
model. The length of the
crack is a+b and the
thickness of it is a-b.
u(n)
d(n)
a+b
Fig1: the crack model
a-b
Method
The outward wave-field can be written as:
u  r  

u
n 
n
Hn(1)  kr  exp in  
(1)
Where un is the outward scattering coefficient, H (kr ) 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,
1
n
The inward wave-field can be read as:
d  r  

d
n 
n
Hn(2)  kr  exp  in  
Where d n is the inward scattering coefficient, H n2 (kr )
is the second kind n-order Hankel function.
(2)
Method
we build up the transfer matrix : chen xiaofei (1993) give
different formular on scattering matrix
u
 un 
1 
(1)  m 
i   drWnWm  
 
 dn  8  m r
 dm 
(3)
Where:

 Hn(2)  kr  exp  in 
Wn  
 H(1)  kr  exp in 
n



Wm(1)





  n Hn(2)  kr  exp  in   

  n Hn(1)  kr  exp in  



  n Hm(1)  kr  exp im      n Hm(2)  kr  exp  im   


(1)
(2)
 Hm  kr  exp im    
Hm  kr  exp  im   

Method
It should be noted that eq. (3) can be adapted to calculate
any convex inclusions. By the differential operators, we can
get:
un 
m 

m 
snm dm
(4)
Where the global scattering matrix snm can be read as:
snm   A(k2 , k1 , , n  m)exp  i  n  m   d
(5)
A(k2 , k1 , , n  m) 
1  exp  i2  k1  k2  a  b cos2   R  k2 , k1 , , n  m 
1  exp  i2  k1  k2  a  b cos2   R  k2 , k1 , , n  m 
b sin 2
n  m 
a  b cos 2 

R  k2 , k1 , , n  m  
b sin 2
k2  k1 
n  m 
 a  b cos 2 
k2  k1 
(6)
(7)