3 2015 12 GLOBAL GEOLOGY Vol. 3 No. Dec. 2015 100 5589 2015 0 1106 07 L BFGS Q 130026 Q 2D L BFGS Marmousi Q L BFGS P631. 3 A doi 10. 3969 /j. issn. 1005589. 2015. 0. 02 Method of Q through full waveform inversion based on L BFGS algorithm SUN Huiqiu HAN Liguo XU Yangyang GAO Han ZHOU Yan ZHANG Pan College of Geoexploration Science and Technology Jilin University Changchun 130026 China Abstract Estimating the stratigraphic quality factor Q is important for the description of tectonic distribution and the prediction of oil and gas. Considering the viscous property of the medium when the wave propagation is modeled 2D viscoacoustic equation is used to full waveform inversion. The observed wave field and the other wave field obtained by forward modeling construct an objective function and L BFGS algorithm is used to inverse seismic quality factor. To test the efficiency of this method L BFGS in FWI an abnormal model and a Marmousi model are examined from surface seismic data. The results show that the inversion model is closer to the theorical model high in the computational accuracy and good in inversion effect. Key words quality factor Q inversion L BFGS method viscoacoustic equation full waveform inversion 0 Q Q Q Q 2013 1 BFGS 20150527 20151020 201AA06A605. 1961. Email hanliguo@ jlu. edu. cn
L BFGS Q 1107 S Q ω 2 2 ρ ( x z) v ~ ( x ) S 1 2D P( x z ω ) 1 P x z x( ( ω) z 2 ρ( x z) x ) ( ) 201 3 1 P ( x z ω) =?( x z ω) 1 z ρ( x z) z Q Malinowski 2D ρ x z 珓 v x y 珓 v x z = v x z i ( 1 2Q x z ) v x z Q 5 Q x z P ( ω) ω? x z ω 6 1 PML 1 5 ω2 c 1 a b i 1 /2 j1 /2 1 b i 1 /2 j1 /2 1 b i 1 /2 j1 /2 1 b i 1 /2 j1 /2 Δx 2 ξ xi ξ xi 1 /2 Δx 2 ξ xi ξ xi 1 /2 Δz 2 ξ xi ξ xi 1 /2 Δz 2 ξ xi ξ xi 1 /2 1 b i 1 /2 j1 /2 1 b i 1 /2 j1 /2 1 b i 1 /2 j1 /2 1 b i 1 /2 j1 /2 a b i 1 /2 j Δx 2 ξ zj ξ zj1 /2 Δx 2 ξ zj ξ zj1 /2 Δz 2 ξ zj ξ zj1 /2 Δz 2 ξ zj ξ zj1 /2 Δx 2 ξ xi ξ xi 1 /2 a Δx 2 b i 1 /2 j a ξ xi ξ xi 1 /2 Δz 2 1 a 1 b i 1 /2 j1 /2 ΔxΔz ξ xi ξ xi 1 /2 ω2 d a b i j1 /2 Δz 2 ξ zj ξ zj1 /2 1 b i 1 /2 ΔxΔz ξ zj ξ zj1 /2 1 b i 1 /2 j1 /2 ΔxΔz ξ zj ξ zj1 /2 j1 /2 P i j1 b i j1 /2 a ξ zj ξ zj1 /2 Δz 2 1 b i 1 /2 ΔxΔz ξ zj ξ zj1 /2 b i j1 /2 1 b i 1 /2 j1 /2 ΔxΔz ξ xi ξ xi 1 /2 1 a 1 b i 1 /2 j1 /2 ΔxΔz ξ xi ξ xi 1 /2 ω2 d a Δz 2 j1 /2 P i 1 b i 1 /2 j1 /2 ΔxΔz ξ xi ξ xi 1 /2 P ξ zj ξ i j ω2 d a zj1 /2 Δx 2 1 b i 1 /2 j1 /2 ΔxΔz ξ zj ξ zj1 /2 1 b i 1 /2 j1 /2 ΔxΔz ξ xi ξ xi 1 /2 b i 1 /2 j ξ xi ξ xi 1 /2 1 b i 1 /2 ΔxΔz ξ zj ξ zj1 /2 b i j1 /2 1 a ξ zj ξ zj1 /2 1 b i 1 /2 j1 /2 ΔxΔz ξ xi ξ xi 1 /2 j1 ω2 d a Δx 2 1 b i 1 /2 j1 /2 ΔxΔz ξ zj ξ zj1 /2 j1 /2 P i 1 j 1 b i 1 /2 j1 /2 ΔxΔz ξ zj ξ zj1 /2 1 b i 1 /2 j1 /2 ΔxΔz ξ xi ξ xi 1 /2 b i 1 /2 j 1 a ξ xi ξ xi 1 /2 1 b i 1 /2 j1 /2 ΔxΔz ξ xi ξ xi 1 /2 1 b i 1 /2 ΔxΔz ξ zj ξ zj1 /2 j1 /2 P i 1 j ω2 1 c d ω2 1 c d ω2 1 c d 1 a 1 a 1 a b i 1 /2 j1 /2 Δx 2 j1 /2 1 b i 1 /2 ξ xi ξ xi 1 /2 Δx 2 ξ zj ξ zj1 /2 b i 1 /2 j1 /2 Δz 2 j1 /2 1 b i 1 /2 ξ xi ξ xi 1 /2 Δz 2 ξ zj ξ zj1 /2 b i 1 /2 j1 /2 Δx 2 j1 /2 1 b i 1 /2 ξ xi ξ xi 1 /2 Δx 2 ξ zj ξ zj1 /2 P i 1 j1 P i 1 j1 P i 1 j1 ω2 1 c d 1 a b i 1 /2 j1 /2 Δz 2 j1 /2 1 b i 1 /2 ξ xi ξ xi 1 /2 Δz 2 ξ zj ξ zj1 /2 P i 1 j1 = iω F ξ xi i j 2
1108 3 2 a = 0. 56 1 c = 0. 62 8 d = 0. 0938 1 2 AP = F 3 m k 1 m k k 1 k g k 1 g k k 1 k A P F 2 Δm k = IH k g k IH k k a k a k = 1 m k a k Δm k C( m ) = 1 2 δpt δp * P obs s k = m k 1 m k y k = g k 1 g k 10 a k = 1 c ( 0 0. 5) 0 < l < u < 1 C m k a k m k > C m k ca k C m k T Δm k T * δp = P cal a k = εa k ε [ l u] P obs P cal 11 11 m k 1 = m k a k Δm k 5 2 a k k m k m k 1 Q k k 1 Δm k 1 k 101 201 25 m 12Hz 2. s 0. 000 6 s 6 5 m C = Re{ J T δp T } 6 10 J Jacobian T Re 3 L BFGS L 0. 5 ~ 1. 0 km Q 70 Q 90 Q 80 Q Q 2 L BFGS x = 2 150 BFGS m x = 3 050 m Q Q 3 Q 7 8 Q 500 ~ 1 000 m 90 m k 1 = m k a k H k g k 7 Q Q 600 ~ 800 m 0 H k Hessian ~ 300 m 500 ~ 1 000 m H k 1 = V T k H k V k ρ k s k s T k 8 Q Q ρ k = 1 V y T k = 1 ρ k y k s T k 9 k s k ε = Q' Q 2 Q 2 100% 12
L BFGS Q 1109 1 Fig. 1 Q Theorical model of Q Fig. 2 2 Q Model of Q after full waveform inversion Fig. 3 3 Q Q Contrast curve of real Q and Q of inversion from two different locations Q' Q Q 128 38 12 Q 10 10 Q 0. 010 0 50 25 m L BFGS 12Hz 2. s 0. 000 6 s Q 5 50 L Marmousi Q BFGS 6
1110 3 7 2. 7 km L BFGS 2. km Q x 1 = 1 250 m x 2 = 3 750 m x 3 = 6 250 m x = 8 750 m Q Q 8 1. 6 km 6 12 Marmousi Q Q 0. 063 0 Q Q Q Fig. Theorical model of Q 5 Fig. 5 Q Initial model of Q 6 L BFGS Q 7 Q Fig. 6 Model of Q after FWI based on L BFGS Fig. 7 Model of Q after FWI based on conjugate gradient
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