49 2 2006 3 CHINESE JOURNAL OF GEOPHYSICS Vol. 49, No. 2 Mar., 2006,,..,2006,49 (2) :577 583 We C, Zhu P M, Wang J Y. Quantum annealng nverson and ts mplementaton. Chnese J. Geophys. (n Chnese), 2006,49 (2) : 577 583 1,2, 1, 1 1 ( ), 430074 2, 100029,.,.,,,,,,.,, 0001-5733(2006)02-0577 - 07 P631 2005-01 - 17,2005-11 - 25 Quantum annealng nverson and ts mplementaton WEI Chao 1,2, ZHU Pe-Mn 1, WANGJa- Yng 1 1 Insttute of Geophyscs and Geomatcs, Chna Unversty of Geoscences, Wuhan, 430074, Chna 2 Insttute of Geology and Geophyscs, Chnese Academy of Scences, Bejng 100029, Chna Abstract Quantum annealng s a new method to solve nonlnear nverse problems based on the quantum tunnelng effect of state transtons. Compared wth smulated Annealng, the method of quantum annealng nverson has a potental advantage n some aspects such as the annealng convergent rate and avodng to be trapped n a local mnmum. So the study of quantum annealng nverson s of mportant academc sgnfcance and practcal value. In ths paper, some model tests and a practcal data nverson show that quantum annealng has hgher effcency and better accuracy. It s the frst tme to apply the quantum annealng method to contnuous geophyscal nverse problems. Keywords Quantum annealng, Smulated annealng, Geophyscal nverson 1 ( ), [1 ].,,..,, Newton Newton.,. [2 ] (Monte Carlo method), (40174033,40304004,40437018).,,1981,,. E-mal :cug - we @1631com
578 (Chnese J. Geophys. ) 49,,. (Smulated Annealng,SA) ( Genetc Algorthm, GA ) ( Artfcal Neural Networks,ANN) [3 ].,,,, [4 ] (Quantum Annealng,QA). Brooke et al. [5 ] LHo 0144 Y 0156 F 4,, 2 30,. [6 ],.,. 1996 Tsalls [4 ], ;Tadash [7 ], ( Isng), ; Huntsman [8 ] ;Santoro et al. [9 ], [10 ] ; Trugenberger [11 ] ;Martonak et al. [6 ],.,.,,.,,.,.,. 2,., D.,.,., T, r (Boltzmann) r { E( r) } = 1 Z ( T) exp - E( r) kt, (1), E( r) r, k > 0, [12 ]. Z ( T), ( Partton Functon ), ( State Summaton), [13 ] : Z ( T) = exp - s D E( s) kt. (2) (1),,, ;,,., T ().,,., [1 ]. T,, ;,,,.,.,,. Hamlton H 0 ( Hamlton H ( t) = U ( r, t) + K, [14 ] ).,,.,,,, [15 ]. H ( t),, ( Transverse Feld) [5 11] ( t) x, Hamlton H( t) = H 0 + H ( t), H( t) = H 0 + ( t) x, (3)
2 : 579 ( t), T ; x x ( Paul Representaton). (3) H 0, ( t) x. U = H 0, K = ( t) x, (3) H( t) = U + K. (4) T ( k = 1) [10 ] Z = Tr (e - HΠT ) = Tr (e - HΠPT ) P = Tr (e - ( K+ U)ΠPT ) P,, P,Tr (5). = (e - H ΠPT ) P ΠZ. (6),, Hamlton. K( ),. m (0), m ( = 1,2,, P), d j ( j = 1,2,, M). l, M E( m ( l) ) = ( d j - j = 1 d j ( m ( l) ) ) 2. (7),m ( l + 1), E( m ( l + 1) ), E = E ( m ( l + 1) ) -,Hamlton E( m ( l ) H = E + C ( t), (8),, x C.,,(6)., Z, : ( H) = (e - HΠPT ) P, (9) (8),,,. [16, 17 ],, : (1) m (0), 0,( t) = 0 t (, t, ( t) 0). ( = 1,2,, P), ; (2). l, m ( l) = m ( l) 1, m ( l) 2,, m ( l) P E( m ( l) ) ;, (3) [ 0,1 ] ( = 1,2,, P),l + 1 m ( l + 1) ( m 1 ) l +1 = ( m 1 ) l + (2 1-1) 1 ( m 2 ) l +1 = ( m 2 ) l + (2 2-1) 2 ( m P ) l +1 = ( m P ) l + (2 P - 1) P ; (10) (4) m ( l + 1) E ( m ( l + 1) ) E = E( m ( l + 1) ) - E ( m ( l) ), (8) Hamlton. E < 0, l + 1 m ( l + 1) m ( l) ; E > 0, (9) m ( l + 1) m ( l) ; (5) m ( = 1,2,, P),., ( t). ( t),, m., 0, ( t). 3 311 MT 1 H MT ( 1a ), 1b MT Bostck SA QA 1 MT Table 1 Comparson among varous nverse methods for MT model data 1 Π m 2 Π m 3 Π m h 1 Πm h 2 Πm 801000 301000 801000 2001000 4001000 Bostck 571695 341754 801729 1951756 4991796 SA 791484 291784 791992 2021530 3941220 QA 791728 291966 801048 2001730 3991300
580 (Chnese J. Geophys. ) 49,0 10 % 20 %. 2 2.,,,, ;,,,. 4 1 MT (a) MT ; (b). Fg. 1 Comparson of results of modelng MT sound curves usng three methods (a) Modelng MT sound curves ; (b) Inverse results. ( 1 ), Bostck., ( 1, 2, 3, h 1, h 2 ) ( 12010 m, 5010 m, 10010 m,35010m,50010m)., 14501, 10001. 312,. [18, 19 ].,, ( - ) -, - -, 215ms ;,, ;,. 3. 3,., 172801, 1 56, 01000215 ;12mn 1000000,,01358. 10 % 20 %,. 3a, 3b 3a, 45Hz Rcker. 4 5 0 2 Table 2 Comparson between nverse results of QA and SA ( ( ) ( ) QA 1201 0150 0100027 2161 0150 0101890 6241 0180 0100620 SA 6721 0150 0100215 18361 0170 0101247 155041 0195 0100917 10 % 20 % QA 1201 0150 0102060 2161 0150 0101937 6241 0180 0103783 SA 6721 0150 0102442 21241 0175 0102865 161761 0195 0106520 QA 1201 0150 0104294 2521 0160 0105888 5281 0180 0106940 SA 6721 0150 0104016 30241 0180 0106270 > 200000 0199
2 : 581 2 (A) ; (B) ; (C). (a 1, a 2, a 3 ) ; (b 1, b 2, b 3 ) 10 % ; (c 1, c 2, c 3 ) 20 %. Fg. 2 Inverson results of dfferent mpedance models at dfferent nose levels (A) Model ; (B) Model ; (C) Model. (a 1, a 2, a 3 ) No nose ; (b 1, b 2, b 3 ) 10 %nose ; (c 1, c 2, c 3 ) 20 %nose. 3 Table 3 Comparson of nverse results of real acoustc mpedance between QA and SA 0 10 % 20 % QA 172801 1 56 2115 10-4 172801 1 54 2127 10-4 SA > 1000000 > 12 3158 10-1 > 1000000 > 12 3180 10-1 QA 345601 3 48 1125 10-3 345601 3 46 1136 10-3 SA > 1000000 > 12 3164 10-1 > 1000000 > 12 3173 10-1 QA 552961 6 02 4111 10-3 587521 6 22 4135 10-3 SA > 1000000 > 12 3175 10-1 > 1000000 > 12 3197 10-1. 4a 5a, ( ),.
582 (Chnese J. Geophys. ) 49 4 (a) ; (b) ; (c) 10 % ; (d) 20 %. Fg. 4 Inverson results of real acoustc mpedance at dfferent nose levels for mode (a) Intal wave mpedance model ; (b) Inverse result wthout nose ; (c) Inverse result wth 10 % nose ; (d) Inverse result wth 20 % nose. 5 (a) ; (b) ; (c) 10 % ; (d) 20 %. Fg. 5 Inverson results of real acoustc mpedance at dfferent nose levels for mode (a) Intal wave mpedance model ; (b) Inverse result wthout nose ; (c) Inverse result wth 10 % nose ; (d) Inverse result wth 20 % nose.
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