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 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.

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

n m Transfer matrix expression Modified from chen xiaofei (1993)

Symbol Inversion via element of Transfer matrix

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 (8) which is the spherical reflection coefficient.

If the incident wave can be read as: Method (9) the scattering wave can be read as: (10) (11)

Outline  Introduction  Method  Numerical 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)

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

snapshots modelt=0.16s t=0.32st=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

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( ) MOST:National Hi-Tech Research and Development Program of China..(863 Program),Grant No 2006AA09A 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 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, The inward wave-field can be read as: Where is the inward scattering coefficient, is the second kind n-order Hankel function. (1) (2)

we build up the transfer matrix : chen xiaofei (1993) give different formular on scattering matrix Method Where: (3)

It should be noted that eq. (3) can be adapted to calculate any convex inclusions. By the differential operators, we can get: Method (4) Where the global scattering matrix can be read as: (5) (6) (7)