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REFLECTION AND TRANSMISSION AT Geophysical Institute

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REFLECTION AND TRANSMISSION AT Geophysical Institute Powered By Docstoc
					    Reflection and transmission
       at curvilinear interfaces
        in multilayered media
    in terms of surface integrals

1   K. Klem-Musatov, 1 A. Aizenberg, 2 H.B. Helle,
            2 J. Pajchel, 3 M. Aizenberg


      1 Institute of Geophysics SB RAS (Novosibirsk, Russia)
      2 O&E Research Centre, Norsk Hydro (Bergen, Norway)

       3 IPT NTNU (Trondheim, Norway)
         I.1. Reflection and transmission phenomena
   A general formulation of reflection and transmission problems for inhomogeneous
    media in terms of the wave field expansion in the basis functions of the lateral
    coordinates is given in [Kennett, Geophys. J. Int., 1994, 118, 344-357].

   Integral [De Santo 1983; Kennett 1984] and differential [Weston 1989] equations
    are introduced for determing of reflection/transmission operators.

   For the analytical evaluation of these operators there are some heuristic
    approaches:
     •    approximation by the plane-wave reflection/transmision coefficients (based on an assumption of
          plane incident wave and plane interface)
     •    convolution of Green function with the plane-wave reflection/transmision coefficients
          [Wenzel, Stenzel and Zimmermann 1990; Sen and Frazer 1991]
    Because all these approaches don’t have theoretical foundation, they can’t provide
    correct description of the reflected and transmitted wave fields.

   Below we introduce transmission operator for simple scalar transmission problem.
   I.2. Scalar transmission problem

                                                                                        1
                                                                                       ...
                                                                                       m-1
é     w2 ù m
ê +
D          ú f (R ) = - j   m
                                (R )
ê   cm (R )ú
     2
ë          û
                                       m
f m (R )Î (Radiation Conditions )                                        nm
                                                                          m
                                                                                        m
                                   m           m+ 1
                                  f (rm )= f     (rm )
                        ¶ f m (rm )/ ¶ nm = - ¶ f m+ 1 (rm )/ ¶ nm + 1
                                        m                        m
                                                                              n m+ 1
                                                                                m

                                                                                       m+1
                                                                                       ...
                                                                                        M
II.1. Generalized spectral representation of Weyl type
                                                                x3 p                  
      wm   R; rm 0   F 1
       p
                                               wm  rm 0  exp   qm  x3 ; rm 0  dx3 
                                               ˆ p
                                                                   ˆ
                                                               0
                                                                                       
                                                                                        
                                                                        
                                                    dx dx                                                     d 1d  2
        F             exp i  1 x1 +  2 x2  1 2
                                                   2     F   1
                                                                              exp i  1 x1 +  2 x2 
                                                                                                         
                                                                      
                                                                                                                 2



                    R  x1 , x2 , x3   rm  x1 , x2  + x3 n m  x1 , x2 
                                                               p




                               drm         dx12 + 2 g12  rm  dx1 dx2 + dx2
                                        2                                     2



       g12  rm0   g12  rm0  / x1  g12  rm0  / x2  0                        rm0  rm  0, 0


               2
                    2  2                        2  p                   
                                                                              
              2   + 2 + 2  2 H m  rm 0 
                                  p
                                                + 2           w m  R; rm 0         0
             
               x1 x2 x3                 x3 cm  rm 0  
                                                                             R rm 0
                                                                              
II.2. Matrix form of interface conditions


       fm  rm   U fm  rm 
       0 J             fm 
                            m

     U=         f m   m+1         m  1, 2,    ,M
        J 0           f m 
           

       1 0                    f p  rm  
                                               
     J          f m  rm    p
                    p
                                              p
                                                    p  m, m + 1
         0 1                 f  rm  / nm 
                                               
            
        II.3. Reflection/transmission transform

       fm  rm0   Km  rm0 , rm  fm  rm 
                                                                          m  1, 2,   ,M


                 Km J
                     mm
                             K m m +1 J 
                                m
                                                                               J     J 
Km  r0 , r    m +1m                      U + K m  rm 0 , r 
                                                       mm
                                                                                        
                K m          
                          J K mm +1m +1 J                                  J     J 
                                                                                      

             m m +1
         K   m          1K           mm
                                       m
                                                                      
                                                     K mm +1m  1  K mm +1m +1

                                   3pm  rm0    32m +1 p  rm0 
                                                       m

           K m  rm0 , rm   F01 p
             pp
                                                                      F   p  m, m + 1
                                   3m  rm0  +  3m  rm0 
                                                     2 m +1 p
                  III.1. Total wavefield
                                                       m
f   m
        R                f
                                m 0 
                                         R  +               s   m
                                                                   p   R 
                                                    p  m 1
                                                                       m  1, 2,   , M +1



                  g m  R, r  m                     f pm  r  
    s  R    
        m
                                f p  r   g  R, r 
                                             m
                                                             m 
                                                                     dS p
                  n p                                  n p 
        p                 m
                                                                  



              f
                  m 0
                           R    g m  R, R m  R dV m

            g m  R, R  ( Absorption Conditions)m
                      III.2. Modified system of BIE
                                                                                   0
                                       f  PK f +f
      P11 K1       P12 K 2        0                        0                         0                      0       
     P K           P22 K 2    P23 K 3                      0                         0                       0      
      21 2                                                                                                          
      0            P32 K 2    P33 K 3                      0                         0                       0      
                                                                                                                    
PK                                                                                                                 
      0               0           0             P M 2 M 2 K M  2    P M 2 M 1 K M 1            0      
                                                                                                                    
      0               0           0             P M 1 M  2 K M  2   P M 1 M 1 K M 1    P M 1 M K M 
                                                                                                                    
      0
                      0           0                        0                PM  M 1 K M 1         PMM K M      

                  0 Pm m 1 
                      m
                                                       Pmm
                                                         m
                                                                 0                                 0           0
                                                                                                   m+1
                                                                                                             0
                                                                                     Pm m+1
                                                                 m+ 
     Pm m 1                                 Pmm
                  0
                      0                             0       Pmm 1                              Pm m+1
                                                                                                                 
                                                                                                                  

         Pmq,2p
                                 Pmq,1 
                                    p
                                                  p  m, m + 1
                                                                                           f  f1 , f 2 ,       , fM 
                                                                                                                      T
    P   p ,2
     p
                                         p      
        Pmq / nm           Pmq,1 / nm      q  m  1, m, m + 1
     mq           p              p
                                          
       III.3. Multiple scattered wavefield
                                                N
                  f f           0
                                        + f               n
                                                                      +f     N +1
                                              n 1
                  f     P K f                                     P K
                                         0
                                                         f
                     n              n                         N +1              N +1
                                                                                        f

                    g m  r, rp 0  m n                               f p    rp 0  
                                                                           m n

f p    r                     f p    rp 0   g m  r, rp 0                     dS p   p  m  1, m
  m n

                        n pm
                                                                             nm          
                                                                                p
                                                                                          


           f pm n   rm 0   K p f pp  n 1  rp  + K p  p +1 f p p +1 n 1  rp 
                                   mp                        m

          
           m n 
           f p  rm 0                f p    rp                      f p    rp 
                                              p n 1                               p +1 n 1

                                                            Kp 
                                                                  m p +1
                               K pmp

          
               n m  p                         n p p
                                                                                     n p +1
                                                                                          p
                                                  IV.1. Effective reflection coefficient

                               Generalization of plane-wave reflection coefficient for non-plane incident wave,
                                curvilinear reflecting interface and seismic frequencies


                          1,4                                                                                                     1,4
                          1,3                                                                                                     1,3
                          1,2                                                                                                     1,2
Modulus of acoustic ERC




                          1,1                                                                                                     1,1




                                                                                                         Modulus of elastic ERC
                          1,0                                                                                                     1,0
                          0,9                                                                                                     0,9
                          0,8                                                                                                     0,8
                          0,7                                                                                                     0,7
                                                                                                                                                                                   cP1 = 2.0 km/s
                          0,6                                                                                                     0,6
                                                                                                                                                                                   cS1 = 1.2 km/s
                          0,5                                                        c1 = 2.0 km/s                                0,5                                                              3
                                                                                                     3
                                                                                                                                                                                   1 = 2.0 g/cm
                          0,4                                                        1 = 4.0 g/cm                                0,4
                          0,3                                                                                                     0,3                                              cP2 = 4.0 km/s
                          0,2                                                        c2 = 4.0 km/s                                0,2                                              cS2 = 2.4 km/s
                                                                                                     3
                          0,1                                                        2 = 2.0 g/cm                                0,1                                              2 = 2.6 g/cm
                                                                                                                                                                                                   3


                          0,0                                                                                                     0,0
                                0   5   10   15   20   25   30   35   40   45   50   55   60   65   70                                  0   10   20   30       40        50   60      70      80
                                                             (degrees)                                                                                     (degrees)
                IV.2. Reflection from flexure-shape interface
                                                                                                                   Tip Wave Superposition Method with effective reflection coefficient
                                                                                                                                          Elasticity
                                                                                                                                 Vertical displacement field
                                                                                               1,2
          0,2
                                                                                               1,0
          0,1
                                        Source
          0,0                                                                                  0,8
         -0,1
         -0,2                                                                                  0,6
                                                                                                                                                  P-SV
         -0,3




                                                                                 Offset (km)
                                                                                               0,4
         -0,4
         -0,5                                                                                  0,2          P-P
         -0,6
z (km)




         -0,7
                                        Data window                                            0,0

         -0,8                                                                                  -0,2
         -0,9
         -1,0                                                                                  -0,4
         -1,1
                                                                                               -0,6
         -1,2
         -1,3   Interface                                                                      -0,8
         -1,4                                                                                         0,6         0,7    0,8      0,9     1,0      1,1       1,2   1,3    1,4     1,5    1,6
         -1,5                                                                                                                                   Time (sec)
         -1,6
                  -1,0      -0,5       0,0             0,5           1,0   1,5
                                                                                                                  Tip Wave Superposition Method with plane-wave reflection coefficient
                                             x1 (km)
                                                                                                                                          Elasticity
                                                                                                                                  Vertical displacement field
                                                                                               1,2

                                                                                               1,0

                                   cP1  3 km s                                                0,8                                          Artefacts


                                   cS 1  1.7 km s                                             0,6
                                                                                 Offset (km)


                                                                                               0,4


                                   1  2 g cm               3                                 0,2
                                                                                                            P-P
                                                                                               0,0

                                                                                               -0,2


                                   cP 2  4 km s
                                                                                               -0,4                                                 P-SV

                                                                                               -0,6

                                   cS 2  2.6 km s                                             -0,8
                                                                                                      0,6         0,7     0,8     0,9     1,0      1,1       1,2   1,3    1,4      1,5   1,6
                                                                                                                                            Time (sec)
                                   2  2.6 g cm                 3
                        IV.3. Reflection from anticline interface
                                                                                                          Tip Wave Superposition Method with effective reflection coefficient
                                                                                                                                 Elasticity
                                                                                                                        Vertical displacement field
                                                                                            2,0

         0,2                                                                                1,8

                                                                                            1,6
                        Source
         0,0                                                                                1,4
                                       Data window                                                                                    Head
                                                                                                                                      waves




                                                                              Offset (km)
                                                                                            1,2         P-P
         -0,2
                                                                                            1,0

                                                                                            0,8
z (km)




         -0,4                                                                                                               P-SV
                                                                                            0,6

         -0,6                                                                               0,4

                                                                                            0,2

         -0,8                                                                               0,0

                                                                 Interface                        0,9   1,0     1,1   1,2   1,3    1,4     1,5   1,6    1,7   1,8   1,9   2,0   2,1   2,2
         -1,0                                                                                                                            Time (sec)


         -1,2
                                                                                                         Tip Wave Superposition Method with plane-wave reflection coefficient
                -0,5   0,0       0,5       1,0       1,5   2,0          2,5
                                                                                                                                 Elasticity
                                         x1 (km)                                                                         Vertical displacement field
                                                                                            2,0

                                                                                            1,8


                                 cP1  2 km s                                               1,6

                                                                                            1,4

                                 cS1  1.2 km s                                             1,2
                                                                                                              P-P
                                                                                                                                  Artefacts
                                                                              Offset (km)


                                                                                            1,0

                                 1  2.4 g cm3                                             0,8
                                                                                                                                                       P-SV
                                                                                            0,6

                                                                                            0,4

                                 cP 2  4 km s                                              0,2

                                                                                            0,0

                                 cS 2  2.4 km s                                                  0,9   1,0     1,1   1,2   1,3    1,4    1,5    1,6   1,7    1,8   1,9   2,0   2,1   2,2
                                                                                                                                         Time (sec)
                                 2  2.4 g cm3
                                Research state
   Scalar wave case
    •   K. Klem-Musatov, A. Aizenberg, H. B. Helle, J. Pajchel. Reflection and transmission at
        curvilinear interface in terms of surface integrals. Wave Motion, 2004, 39, 1, 77-92.
    •   K. Klem-Musatov, A. Aizenberg, H. B. Helle, J. Pajchel. Reflection and transmission in
        multilayered media in terms of surface integrals. Wave Motion, 2005, 41, 4, 293-305.


   Acoustic wave case
    •   A.M. Aizenberg, M.A. Aizenberg, H.B. Helle, K.D. Klem-Musatov, J. Pajchel. Modeling
        of single reflection by tip wave superposition method using effective coefficient.
        Extended Abstracts, 66th Meeting of EAGE, 2004, Paper P187.
    •   A.M. Aizenberg, M.A. Aizenberg, H.B. Helle, J. Pajchel. Reflection and transmission of
        acoustic wave fields at curvilinear interface between two inhomogeneous media.
        Dynamics of solid mechanics, Proceedings of Lavrentiev’s Institute of Hydrodynamics
        SB RAS, “Acoustics of inhomogeneous media”, Novosibirsk, 2004, V. 123.


   Elastic wave case
    •   M.A. Aizenberg. Research of properties of reflection and transmission operators in the
        transmission problem at interface between two homogeneous half-spaces. Master
        thesis, Novosibirsk State University, Novosibirsk, 2003.

				
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