REFLECTION AND TRANSMISSION AT Geophysical Institute by alicejenny

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```									    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
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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