the integral expression of the acoustic multiple scattering about cracks
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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)