Modeling 32D scalar waves using the Fourier finite2difference method

Transcription

1 CHINESE JOURNAL OF GEOPHYSICS Vol. 50, No. 6 Nov., 007,,..,007,50 (6) : Zhang J H,Wang W M, Zhao L F,et al. Modeling 3D scalar waves using the Fourier finitedifference method. Chinese J. Geophys. (in Chinese), 007, 50 (6) : ,,,, (FFD),,. Fourier,. FFD,. French,,,.,,, (007) P , Modeling 3D scalar waves using the Fourier finitedifference method ZHANGJinHai,WANG WeiMin,ZHAO LianFeng, YAO ZhenXing Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 10009, China Abstract Fourier finitedifference (FFD) operator has both of the advantages of Fourier and finitedifference methods, it can handle the wave propagations in complex media with large velocity contrasts and wide propagation angles. In 3D case, however,twoway splitting may cause artificial azimuthal anisotropy. In this paper,we handle the remnants of the conventional twoway splitting technique by Fourier transforms, and the azimuthal anisotropy is removed. Based on the dualdomain scheme of the FFD method,the proposed algorithm significantly reduces computational cost caused by error correction. Compared with the finitedifference method by the zerooffset records and shot profiles based on 3D modified French model, the proposed method can properly model the primary reflected waves and is much more attractive in both computational cost and storage demand. Keywords 3D modeling, Fourier finitedifference, Twoway splitting, Wave equation 1.,,,,., ( ).,, ( KZCXYW101).,,1978,007,. igcas. ac. cn

2 6 : 1855.,,,.,,.,,.,, [1 3 ] Fourier [4 9 ] (FFD) [10,11 ] [1 ]. [13,14 ].,.,,. Fourier,,, 90.,Fourier., [18 ] (Twoway splitting, [19 ] ) x y FFD., [1,,0 ] (,azimuthal anisotropy), FFD., [1,,17,19 1 ],,. (ADIPI) [19 ],, [ ]., ADIPI [19 ], FFD. ADIPI FFD, x y, FFD,,. ADIPI FFD 11 FFD FFD [16,17 ] b k z k 0 z + s a 9 9x + 9 9y 9 9x + 9 9y, (1), k 0 z = Πv Π9x + 9 Π9y, s = 1Πv - 1Πv 0, v, v 0, a = 015 ( v + vv 0 + v 0 )Π, b = 015 ( v - v 0 )Π. [15 ] (1). Fourier,, FFD [16,17 ] ;,,,, ;, [3 ],. [5,6 ],,,,.. FFD z, P ( x, y, z ; ) (FFT), P ( kx, k y,z ; ), () : P ( kx, k y, z + z ; ) = e i k0 z z P( k x, k y, z ; ), (). P ( k x, k y, z + z ; ), P ( x, y, z + z ; ),, P( x, y, z + z ; ) = e i s z P ( x, y, z + z ; ). (3), P ( x, y, z + z ; ) (4), 9P( x, y, z + z ; ) 9z

3 1856 (Chinese J. Geophys. ) 50 9 i b 9x + 9 9y = a 9x + 9 9y 1 ADIPI P( x, y, z + z ; ). (4) (4),., FFD., [0 ],., ADIPI FFD [],, FFD., P ( x, y, z ; ) P, P ( k x, k y, z ; ) P, k x k y x y, z P( x, y, n z ; ) P n, 9 Π9x 9 Π9y D xx D yy, 9Π9z D z, (4) [1 + a ( D xx + D yy ) ] D z P = i b( D xx + D yy ) P. (5) (5) D z P D z P Pn+1 - P n. (6) z CrankNicolson,,,, P (6) (7) (5) P Pn+1 + P n. (7) (1 + cd xx + cd yy ) P n+1 = (1 + c D xx + c D yy ) P n, (8), c = a g i b zπ, c = a i b zπ., (8) (1 + cd xx ) (1 + cd yy ) P n+1 = (1 + c D xx ) (1 + c D yy ) P n +, (9),, = c D xx D yy P n+1 - c D xx D yy P n. (10), x y, (1 + cd xx ) P n+1 33 = (1 + c D xx ) P n, (11) (1 + cd yy ) P n+1 3 = (1 + c D yy ) P n+1 33, (1), P n x, y P n , (10) = c k x k y P n+1 - c k x k n y P. (13) [,13 ] (13) = sin 4 cos sin ( c k 4 n+1 P - c k 4 n P ), (14), k = Πv, k xπk = sin cos, k yπk = sin sin.,,, (14) = 45.,,, (14). (14),., (9) FFD, : n+1 P = (1 - c k x ) (1 - c k y ) n (1 - ck x ) (1 - ck P y + - c k x k y (1 - ck x ) (1 - ck y ) P n+1 c k x k y (1 - ck x ) (1 - ck y ) P n. (15) (15) n+1 P 3 = (1 - c k x ) (1 - c k y ) n (1 - ck x ) (1 - ck P. (16) y ) (16) (15) P n + 1 n+1 P = c k x k y 1 - ck x - ck y c k x k y 1 - ck x - ck y n+1 P 3 n P. (17) (17) : P n + 1 P n FFD n + 1 P 3.,,, ADIPI FFD. (17), (1 - ck x - ck y ),,, ( Evanescent waves) [13 ],., (16) (17).,

4 6 : 1857 FFT, FFD FFT.,, FFT., P n P n, FFT., FFD,,, FFT. 13 FFD ADIPI FFD, [,13], k z = Πvcos. k z = k 0 z + s - b 1 - a - b 1 - a, (18), k 0 z = Πv 0 - -, = k cos sin, = k sin sin., FFD R (, ) = k z - k z k z 100. (19) (18) 45 k z = k 0 z + s - b 1 - a - b 1 - a, (0), = k cos ( + Π4) sin, = k sin ( + Π4) sin, R (, ) = k z - k z k z 100. (1), FFD R 4 (, ) = R (, ) + R (, ). (), xπy,, ( ), FFD.,.,. v 0 Πv = 90 %, xπy ( 0 90 ), % ; ( ), 7 %, 3 ; v 0 Πv = 30 %( ), 6 % 17 %, 3,. 1 FFD ( 60 ) Fig. 1 Relative errors of conventional twoway splitting versus the azimuth angle with dip angle equal to 60 in polar coordinates xπy.,,. ( ) ( xπy ),,,,., ( ) xπy ( ), 90 % 30 %.,,. 3 1, v 0 Πv = 30 % ( 1 ).,.,

5 1858 (Chinese J. Geophys. ) 50 xπy Fig. Relative errors of conventional twoway splitting versus dip angle in the diagonal and inlineπcrossline directions, 3 ( IFFD)., ADIPI FFD. 14 ( ),,, Π.,. Π,,,,.,Graves Clayton Π [1 ],. [4 ].,,,,. Π, [5 ],.,,,, Π [10,11 ], [5 ]. 3 Fig. 3 Relative error contrasts between the improved FFD,fourway splitting and conventional twoway FFD in polar coordinates 1 %, 17 % 1 %,, xπy 6 % 1 %. xπy, xπy,,., 6 %, 3 311, x, y, z , 10 m m s - 1, 1000 m s - 1, v 0 Πv 3313 %. (180,180,0), Ricker, 0 Hz, ms, 1 s s. (a1,a,a3), ( b1, b,b3), (a1,b1) (a,b) (a3,b3) 50,60,70., x y,

6 6 : s Fig. 4 Comparison of the time slice between the conventional twoway splitting FFD method (a1,a,a3) and the proposed method (b1,b,b3),,,.,. FFD Intel Xeon GHzΠGDDR 56 s ; ADIPI FFD 87 s, 1 %,. 31 French, 5, , 10 m. 45 ; 600 m, (700,1100,1300), (1600,1800,1300) ; 600 m 1000 m 1500 m. 000 m s - 1, 4000 m s - 1. (180,180,0), Ricker 0 Hz, s. [6], 1 ms. [6 ] [10,11 ] y, 6a 6b.,,,,,, 5 French Fig. 5 The modified 3D French model

7 1860 (Chinese J. Geophys. ) 50 [1,5 ]., Intel Xeon GHzΠ GDDR 58 s 1496 s, 315.,,, , 0 m, ms,, 141 s, ( 6c) 6b,., [6 ],. 10 m (58 s) (0 m ) (141 s) 37. 6c, 4 s, 6d 0 s,, 88 s., ,., [6 ], Ricker 0 Hz,,., Ricker 15 Hz. 6e 6f French, 6 Fig. 6 The modeling results by 3D finitedifference method and the proposed method

8 6 : 1861 x, 5394 s 78 s, 69.,,,.,, [8,10,11 ]., 6e 6f,., 15 Hanning,, x y z N x N y N z, N w,, N x N y N z (1 + 3) 4 ; FFD, N x N y N z 8 + N x N y N w 8.,,, N x N y 4 + N x N y N w 8.,.,,. :1) ; ),, ;3),, ;4), ;5).,,,,,., FFD,,.,, ;.,. 41 Wang ADIPI [19 ],,,, FFT., FFD FFT, FFT. [1 3,13 ], [16 ],. Claerbout 1Π6 [13 ] [7 ],. 413 FFD [0 ],.,,,, xπy., x y,,,., xπy.,,,., xπy,., ADIPI FFD FFD, FFD.

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