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.. , .. , .. - () . .. . E-mail:voevodin@parallel.ru . . . . . . . . 1. . , . , , - , , . , [1-5]. , , [6, 7]. [8, 9], , . .


. , .. . [10-12]. , , , . . . «» . 2. . , ( ). , , . e it , : (1) u (r , q, ) 2 ( r , )u (r , q, ) f (r , q, ), r R 3 - , - , u (r , q, ) - . R . , ( q ) Q , u (r , q, ) P , R . , P Q . f ( r , q, ) , Q . , q Q

) ( r q ) . R ( r , ) . ,
f ( r , q,


c( r ) , (r , ) / c (r ) . R ( r , ) / c0 , c0 const .


(r ) c

2 0

c 2 (r ) . (1)

u (r , q, )

2 u ( r , q, ) f ( r , q, ) 2 (r )u (r , q, ) (2) 2 c0
1 exp( x y ), 4 x y c0
(3)



G ( x, y , )



2 G ( x, y , ) 2 G ( x, y , ) ( x y ) . c0
(2)

u ( r , q, ) = u 0 ( r , q , ) +

2


R

G (r ' , r , ) (r ' )u (r ' , q, )dr ' , (4)
Q

u 0 ( r , q, ) = G ( r , r , ) f ( r , q,
' '



)dr

'

(4) R P, [7]

, . . (5), U ( p, q, ) .

u (r , q, ) = u 0 (r , q, ) + 2 G (r ' , r , ) (r ' )u (r ' , q, )dr ' , r R R (5) U ( p, q, ) = 2 G (r ' , p, ) (r ' )u (r ' , q, )dr ' , p P R


3. . (5) ( ), u ( r ' , q, )

u 0 ( r ' , q, ) .
,






'


'

(..

f ( r , q, ) = - ( r - q ) ) u 0 ( r , q, ) = G (r , q, ) . U ( p , q, ) = u ( p, q , ) - G ( p, q , ) =

2



G (r , p, )G (r , q, ) (r )dr ,(6)

G (r , q, ) - (3); p P ; q Q - ; u ( p, q, ) - , , P ; ( r ) - , ; r - , , 3- R . (6)
1 . . , P, Q z = 0 , OXYZ Z , OXY . (3)

R

G ( r , p, ) = -

1 4 ( p1 - r1 ) 2 + ( p 2 - r2 ) 2 + r3
2

exp(

2 ( p1 - r1 ) 2 + ( p 2 - r2 ) 2 + r3 ) , c0

r = ( r1 , r2 , r3 ) R , p = ( p1 , p 2 ,0) P . .

(7)

2 .

G (r , p, ) = H ( p1 - r1 , p 2 - r2 , r3 , ) .





, , r3 , r3 (6). ( r ) , . , , (, ,


), . , (6)

U ( p1 , q1 ) = H ( p1 - r1 , r3 ) H (q1 - r1 , r3 ) (r1 , r3 )dr1 ,
R ( p1 , q1 ) ,

(8)

(r ) = 0

1 .

~ ~ ~ U (v1 , w1 ) = H (v1 , r3 ) H ( w1 , r3 ) exp{-i (v1 + w1 )r1 } ( r1 , r3 ) dr1 , ~~
~ U (v1 , w1 ) ~ ~ = H (v1 , r3 ) H (w1 , r3 ) 1 = exp{i ( xv1 + yw1 )} exp{- i (v1 + w1 )r1} (r1 , r3 )dr1 dv1dw1 = 2
-1

U , H - U , H . ,

= ( x - r1 ) ( y - r1 ) (r1 , r3 )dr1 = ( x - y ) ( x, r3 )
-1 - (9) , ( r )

,

(9)

(v1 , w1 ) . , r3 U

H . , (9) . (9) ( ). , , . Q

^^ ( q1 , q 2 ,0) . , .


, (9) ( r1 , r2 ) r3

-1


~ U (v1 , v 2 ) = (r1 , r2 , r3 ) ~ H (v1 , v 2 , r3 )

(10)



^ ^ ( r1 , r2 , r3 ) = H (q1 - r1 , q 2 - r2 , r3 ) (r1 , r2 , r3 ) . , ( r1 , r2 , r3 ) (r1 , r2 , r3 )
. (10) (r ) r3 . , r3 . : (10) r3 . . 4. . , . . ­ . . : - ( ), - , - , - . , , . , ( ),


. 679 . . . , . 1 40 . 5 100 . 1 20 Ѕ . , 5-10 . , -, ( - ), , -, , , . 8 20-25 . , . , . 5. . .1.

. 1.

. 2. .

( ) 6 . 15 . ,


. 50 . . , Z , . (7.5; 0.; 50.) (-7.5; 0.; 50.). Z = 0 . 0.3 , 0.3 . , . Y 1 . , , 24 . 1300 . 6. . , . 2. 24 . , . , ­ . , . . .3 OXY 0.3 1 Y.

. 3. OXY ( ).

. 4. ( OXY).


, , (). , , 5000 24 OXY. (10). r3 . .4 . , , . , XY , - . .4 , . z . 7. . , , , . , , . , (, , ), , . ( ) . , ( ) . . , .


. , . , . . 1. J. Claerbout "Imaging the Earth's interior", Blackwell Scientific Publishers, 1986. 2. E. Robinson, T.S. Durrani, L.G. Peardon "Geophysical signal processing". Prentice Hall, 1986. 3. J. Gazdag, P. Squazzero "Migration of seismic data" // Proceedings of the IEEE, 72:1302-1315, 1984. 4. O. Yilmaz, "Seismic Data Processing" // Tulsa: Society of Exploration Geophysicists, 1987. 5. E. Baysal, D.D. Kosloff, J.W.C. Sherwood "Reverse time migration" //Geophysics. 1983. 48. 1514-1524. 6. Bakushinsky A.B. , Goncharsky A.V. "Ill-posed Problems. Theory and Applications" // Dordrect: Kluwer Academic Publ., 1994. 7. Bakushinsky A.B. , Goncharsky A.V. , Romanov S.Yu. , S. Seatzu "On the identification of velocity in seismic and in acoustic sounding " // Firenze, 1994. 8. .. , .. // . 2000. 40. - 9. ­ .1364-1367. 9. .. , .. // , 2006. - .7. - 1. .40-45. 10. .. , .. , .. // " ". ­ : - , 2007. - .1. ­ .109-114. 11. .. , .. , .. // , 2006. - .7. - 1. - .117-126.


12. .. , .. // " ". ­ : - , 2006. - .172-175.