Q L -BFGS. Method of Q through full waveform inversion based on L -BFGS algorithm. SUN Hui-qiu HAN Li-guo XU Yang-yang GAO Han ZHOU Yan ZHANG Pan

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1 GLOBAL GEOLOGY Vol. 3 No. Dec L BFGS Q Q 2D L BFGS Marmousi Q L BFGS P A doi /j. issn 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 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 BFGS AA06A hanliguo@ jlu. edu. cn

2 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 ) ( ) 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

3 a = c = d = 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 < 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 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 m 12Hz 2. s 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 = BFGS m x = m Q Q 3 Q 7 8 Q 500 ~ 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 ~ 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.

3 3 Q Q Contrast curve of real Q and Q of inversion from two different

4 L BFGS Q 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 Q Q m L BFGS 12Hz 2. s s Q 5 50 L Marmousi Q BFGS 6

6 km 6 12 Marmousi Q Q 0. 063 0 Q Q Q Fig. Theorical model of Q 5 Fig.

5 km L BFGS 2. km Q x 1 = m x 2 = m x 3 = m x = m Q Q km 6 12 Marmousi Q Q 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

6 L BFGS Q Fig Q Q Contrast curve of real Q and Q of inversion from four different locations 3 L BFGS YU Qinglu HAN Liguo ZHANG Ying et al. 1. BFGS J Inver Q sion of stratigrarhic quality factor based on BFGS algorithm J. Global Geology Q J S

7 FU Xunxun ZHANG Junxue CHEN Zhen et al. Seeking Q value of formation quality factor based on S transform J. Geophysical & Geochemical Exploration Q J CAO Siyuan TAN Jia GAO Ming et al. Inversion of Q by logarithmic spectrum radical method J. Oil Geophysical Prospecting Malinowski M Operto S Ribodetti A. Highresolution seismic attenuation imaging from wideaperture onshore data by viscoacoustic frequencydomain fullwaveform inversion J. Geophysical Journal International D GAO Fengxia. Multiparameter inversion from wave e quations by frequency domain full waveform inversion doctor's degree thesis D. Changchun Jilin University Song Z M Williamson P R Pratt R G. Frequencydomain acousticwave modeling and inversion of crosshole data Part ⅡInversion method synthetic experiments and realdata results J. Geophysics D DENG Wubing. Study of blended acquisition based frequency domain full waveform inversion master's degree thesis D. Changchun Jilin University Q J HUANG Fei HAN Liguo ZHANG Bo et al. Extraction of Q value from prestack PSwave and a stable and PS Q efficient inverse Q filtering algorithm J. Global Geology

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