Transcript PPT - KAUST
Angle-domain Wave-equation Reflection Traveltime Inversion 1 2 Sanzong Zhang, Yi Luo and Gerard Schuster (1) KAUST, (2) Aramco 1 Outline Introduction Theory and method Numerical examples Conclusions Outline Introduction Theory and method Numerical examples Conclusions Velocity Inversion Methods Data space (Tomography) Ray-based tomography Wave-equ. Reflection traveltime inversion Full Waveform inversion Inversion Image space (MVA) Ray-based MVA Wave-equ. Reflection traveltime inversion Wave-equ. MVA Problem The waveform (image) residual is highly nonlinear with respect to velocity change. 2 e= - ∆𝜏 Pred. data – Obs. data ∆𝜏 Model Parameter The traveltime misfit function enjoys a somewhat linear relationship with velocity change. Angle-domain Wave-equation Reflection Traveltime Inversion Traveltime inversion without high-frequency approximation Misfit function somewhat linear with respect to velocity perturbation. Wave-equation inversion less sensitive to amplitude Multi-arrival traveltime inversion Beam-based reflection traveltime inversion Outline Introduction Theory and method Numerical examples Conclusions Wave-equation Transmission Traveltime Inversion 1). Observed data 𝑝𝑜𝑏𝑠 0 Time (s) 5 2). Calculated data 𝑝𝑐𝑎𝑙𝑐 0 Time (s) 5 3). 𝑝𝑜𝑏𝑠 -1.5 0 Lag time (s) 𝑝𝑐𝑎𝑙𝑐 ∆𝜏 1.5 4). Smear time delay ∆𝜏 along wavepath Angle-domain Wave-equation Reflection Traveltime Inversion Suboffset-domain crosscorrelation function : 𝑓 𝑥, 𝑧, ℎ, 𝜏 = 𝑑x𝑠 𝑝𝑓 (𝑥 − ℎ, 𝑧, 𝑡 + 𝜏|x𝑠 )𝑝𝑏 (𝑥 + ℎ, 𝑧, 𝑡|x𝑠 )𝑑𝑡 s 𝑝𝑓 (𝑥 − ℎ, 𝑧, 𝑡|x𝑠 ) ℎ: 𝑠𝑢𝑏𝑜𝑓𝑓𝑠𝑒𝑡 𝜏: time shift g 𝑝𝑏 (𝑥 + ℎ, 𝑧, 𝑡|x𝑠 ) x-h x x+h 𝑝𝑓: 𝑓𝑜𝑤𝑎𝑟𝑑 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑒𝑑 𝑑𝑎𝑡𝑎 𝑝𝑏: 𝑏𝑎𝑐𝑘𝑤𝑎𝑟𝑑 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑒𝑑 𝑑𝑎𝑡𝑎 Angle-domain Crosscorrelation Angle-domain CIG decomposition (slant stack ): 𝑓 𝑥, 𝑧, 𝜃, 𝜏 = angle-domain 𝑓 𝑥, 𝑧 + ℎ tan 𝜃 , ℎ, 𝜏 𝐱𝑠 𝑑ℎ suboffset-domain Angle-domain crosscorrelation function : 𝑓 𝑥, 𝑧, 𝜃, 𝜏 = 𝑑𝐱 𝑠 𝑑ℎ 𝑝𝑓 𝑥 − ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡 + 𝜏 𝐱 𝑠 𝑝𝑏 (𝑥 + ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡|𝐱𝑠 ) 𝑑𝑡 Angle-domain Crosscorrelation: physical meaning 𝑑𝐱 𝑠 𝑝𝑓 𝑥 − ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡 𝐱 𝑠 𝑝𝑏 (𝑥 + ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡|𝐱𝑠 ) 𝑑𝑡 𝑧 𝑧 𝜃 𝑑ℎ 𝑥 𝑝𝑓 (𝑥 − ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡|𝐱 𝑠 ) Local plane wave 𝜃 𝑓 𝑥, 𝑧, 𝜃, 𝜏 = 0 = 𝑥 𝑝𝑏 (𝑥 + ℎ, 𝑧 + ℎ tan 𝜃 , 𝑡|𝐱 𝑠 ) Local plane wave Angle-domain crosscorrelation is the crosscorrelation between downgoing and upgoing beams with a certain angle. The time delay for multi-arrivals is available in angle -domain crosscorrelation function . Angle-domain Wave-equation Reflection Traveltime Inversion Objective function: 𝜀 = 12 ∆𝜏(𝐱, 𝜃) 𝐱 Velocity update: 𝟐 𝜽 𝑐𝑘+1 (x)= 𝑐𝑘 (x) + 𝛼𝑘 ∙ 𝛾𝑘 (x) Gradient function: 𝜕𝜀 𝛾𝑘 (x)= − =− 𝜕𝑐(𝐱) 𝐱 𝜽 𝜕(∆𝜏) ∆𝜏 𝜕𝑐(𝐱) Traveltime wavepath Traveltime Wavepath Angle-domain time delay ∆𝜏 𝜃 𝑓 𝑥, 𝑧, 𝜃, ∆𝜏 = max 𝑓 𝑥, 𝑧, 𝜃, 𝜏 −𝑇<𝜏<𝑇 𝑓 𝑥, 𝑧, 𝜃, ∆𝜏 = m𝑖𝑛 𝑓 𝑥, 𝑧, 𝜃, 𝜏 −𝑇<𝜏<𝑇 Angle-domain connective function 𝜕𝑓(𝑥, 𝑧, 𝜃, 𝜏) 𝑓∆𝜏 = =0 𝜕𝜏 Traveltime wavepath 𝜏=∆𝜏 𝜕(∆𝜏) 𝜕𝑓∆𝜏 𝜕𝑓∆𝜏 =− 𝜕𝑐(𝑥) 𝜕𝑐(𝑥) 𝜕(∆𝜏) Transforming CSG Data Xwell Trans. Data reflection = Src-side Xwell Data transmission transmission + source Redatuming data Rec-side Xwell Data Observed data Redatuming source Workflow Forward propagate source to trial image points and get downgoing beams Backward propagate observed reflection data from geophonses to trial image points , and get upgoing beams Crosscorrelate downgoing beam and upgoing beam, and pick angledomain time delay Smear time dealy along wavepath to update velocity model Outline Introduction Theory and method Numerical examples Simple Salt Model Sigsbee Salt Model Conclusions Simple Salt Model 0 0 z (km) t (s) (a) True velocity model 0 x (km) 8 5 5 0 (c) Initial Velocity Model 8 0 1 z (km) z (km) 0 4 x (km) (d) RTM image 0 x (km) 8 4 0 V(km/s) 4 (b) CSG x (km) 8 Angle-domain Crosscorrelation (b) Angle-domain Crosscorrelation (a) Initial Velocity Model z (km) 0 4 0 x (km) 8 𝑓 𝑧, 𝜃, ∆𝜏 (c) Angle-domain Crosscorrelation ∆𝜏 = 𝛼(tan 𝜃)2 ∆𝝉 ∆𝜏: time delay 𝑓 𝑥, 𝑧, 𝜃, ∆𝜏 𝛼: curvature 𝜃: reflection angle Inversion Result (a) Initial velocity model z (km) 0 4 5 0 x (km) (b) Inverted velocity model Velocity(km/s) 8 0 z (km) 1 4 0 x (km) 8 Inversion Result (a) RTM image z (km) 0 4 0 x (km) (b) RTM image 0 x (km) 8 z (km) 0 4 8 Outline Introduction Theory and method Numerical examples Simple Salt Model Sigsbee Salt Model Conclusions Sigsbee Model (b) Initial velocity model z(km) 0 z(km) 0 (a) True velocity model Vinitial = 0.85 Vtrue 6 0 x(km) (c) RTM image 12 x(km) 4.5 0 Velocity (km/s) z(km) 6 0 6 0 1.5 x(km) 12 12 Initial Velocity Model z(km) 0 CIG 0 z(km) 0 x(km) Semblance 6 -50° 𝜃 +50° 6 -0.04 -0.2 ∆𝜏(𝑠) 0 z(km) 6 𝛼 12 Crosscorrelation ∆𝜏 = 𝛼(tan 𝜃)2 0.2 0.04 -50° 𝜃 +50° Initial Velocity Model z(km) 0 CIG z(km) 0 x(km) Semblance 12 Crosscorrelation 0 -0.2 ∆𝜏(𝑠) 0 z(km) 6 ∆𝜏 = 𝛼(tan 𝜃)2 6 -50° 𝜃 +50° 6 -0.04 𝛼 0.2 0.04 -50° 𝜃 +50° Initial Velocity Model z(km) 0 CIG 0 z(km) 0 x(km) Semblance 6 -50° 𝜃 +50° 6 -0.04 12 Crosscorrelation -0.2 ∆𝜏(𝑠) 0 z(km) 6 𝛼 ∆𝜏 = 𝛼(tan 𝜃)2 0.2 0.04 -50° 𝜃 +50° Inverted Velocity Model z(km) 0 CIG 0 z(km) 0 x(km) Semblance 6 -50° 𝜃 +50° 6 -0.04 12 Crosscorrelation -0.2 ∆𝜏(𝑠) 0 z(km) 6 𝛼 0.04 0.2 -50° ∆𝜏 = 𝛼(tan 𝜃)2 𝜃 +50° Inverted Velocity Model z(km) 0 CIG 6 -50° -0.2 0 z(km) 0 𝜃 12 Crosscorrelation x(km) Semblance +50° 6 -0.04 ∆𝜏(𝑠) 0 z(km) 6 𝛼 0.04 0.2 -50° ∆𝜏 = 𝛼(tan 𝜃)2 𝜃 +50° Inverted Velocity Model z(km) 0 CIG z(km) 0 x(km) Semblance 6 -50° 𝜃 +50° 12 Crosscorrelation 0 -0.2 ∆𝜏(𝑠) 0 z(km) 6 6 -0.04 𝛼 0.04 ∆𝜏 = 𝛼(tan 𝜃)2 0.2 -50° 𝜃 +50° RTM Image (b) RTM image using inverted model 0 z(km) z(km) (a) RTM image using initial velocity 0 6 0 6 x(km) 12 0 x(km) 12 Outline Introduction Theory and method Numerical examples Conclusions Velocity Inversion Methods Data space (Tomography) Ray-based tomography Wave-equ. traveltime inversion Full Wavform inversion Inversion Image space (MVA) Ray-based MVA Wave-equ. traveltime inversion Wave-equ. MVA Angle-domain Wave-equation Reflection Traveltime Inversion Traveltime inversion without high-frequency approximation Misfit function somewhat linear with respect to velocity perturbation. Wave-equation inversion less sensitive to amplitude Multi-arrival traveltime inversion Beam-based reflection traveltime inversion Thank you for your attention