S E I S C O P E Using the Ensemble Transform Kalman Filter to estimate uncertainty in Full Waveform Inversion Julien Thurin 1 , Romain Brossier 1 and Ludovic M´ etivier 1 , 2 Wednesday 30 th May, 2018 - 13 th International ENKF workshop 1 Univ. Grenoble Alpes, ISTerre 2 Univ. Grenoble Alpes, CNRS, LJK
Introduction: Seismic Tomography S E I S C O P E Goal of seismic tomography: find physical parameters of the subsurface from seismic wavefield data. The recorded data are directly linked to the subsurface physical properties. Comparison of observed wavefield with synthetic wavefield allows formulating an inverse problem to find the model that gives the best data-fit. Illustration of the tomographic problem. ETKF-FWI 1
Introduction: Full Waveform Inversion S E I S C O P E In Full Waveform Inversion (FWI), we try to match the entire recorded wavefield ( d obs ) at receiver locations with the synthetic waveform data computed in a starting model ( d cal ). FWI allows to obtain higher resolution than ”classical” tomography techniques relying only on travel time. The FWI inverse problem is more difficult (more non-linear) as it attempts to fit an entire pressure recordings. Exemple of recorded wavefield data. ETKF-FWI 2
Outline S E I S C O P E Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions ETKF-FWI 3
Outline S E I S C O P E Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions ETKF-FWI 3
Full Waveform Inversion Workflow S E I S C O P E Courtesy of Isabella Masoni ETKF-FWI 4
Full Waveform Inversion Workflow S E I S C O P E Courtesy of Isabella Masoni ETKF-FWI 4
Full Waveform Inversion Workflow S E I S C O P E Courtesy of Isabella Masoni ETKF-FWI 4
Full Waveform Inversion Workflow - Forward modeling S E I S C O P E As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. min m 1 2 � d cal ( m ) − d obs � 2 The misfit of the FWI problem is : ETKF-FWI 5
Full Waveform Inversion Workflow - Forward modeling S E I S C O P E As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. min m 1 2 � d cal ( m ) − d obs � 2 The misfit of the FWI problem is : ETKF-FWI 5
Full Waveform Inversion Workflow - Forward modeling S E I S C O P E As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. min m 1 2 � d cal ( m ) − d obs � 2 The misfit of the FWI problem is : ETKF-FWI 5
Full Waveform Inversion Workflow - Forward modeling S E I S C O P E As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. min m 1 2 � d cal ( m ) − d obs � 2 The misfit of the FWI problem is : ETKF-FWI 5
Full Waveform Inversion Workflow - Forward modeling S E I S C O P E As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. min m 1 2 � d cal ( m ) − d obs � 2 The misfit of the FWI problem is : ETKF-FWI 5
Full Waveform Inversion Workflow - Forward modeling S E I S C O P E As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. min m 1 2 � d cal ( m ) − d obs � 2 The misfit of the FWI problem is : ETKF-FWI 5
Full Waveform Inversion Workflow - Forward modeling S E I S C O P E As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. min m 1 2 � d cal ( m ) − d obs � 2 The misfit of the FWI problem is : ETKF-FWI 5
Full Waveform Inversion Workflow - Forward modeling S E I S C O P E As we try to fit the recorded wavefield with synthetics, we need an appropriate forward modeling engine to reproduce accurately wave propagation physics. min m 1 2 � d cal ( m ) − d obs � 2 The misfit of the FWI problem is : ETKF-FWI 5
Full Waveform Inversion Workflow S E I S C O P E Courtesy of Isabella Masoni ETKF-FWI 5
Full Waveform Inversion Workflow S E I S C O P E Courtesy of Isabella Masoni ETKF-FWI 5
Full Waveform Inversion Workflow S E I S C O P E Courtesy of Isabella Masoni ETKF-FWI 5
Full Waveform Inversion Workflow S E I S C O P E Courtesy of Isabella Masoni ETKF-FWI 5
Full Waveform Inversion Workflow S E I S C O P E Courtesy of Isabella Masoni ETKF-FWI 5
Introduction : Why using FIW ? S E I S C O P E Traveltime tomography in the Valhall Oil Field. ETKF-FWI 6
Introduction : Why using FIW ? S E I S C O P E Full Waveform Inversion in the Valhall Oil Field. ETKF-FWI 7
Motivations S E I S C O P E • FWI is generally applied in a deterministic fashion from a starting model • FWI relies on local optimization (quasi-Newton) • FWI results are generally difficult to assess Only a few recent papers propose to tackle the uncertainy problem in FWI : still no systematic applications. We propose an approach relying on a mixed-method based on an Ensemble Transform Kalman Filter and the classic quasi-Newton optimization scheme to evaluate uncertainty in the FWI results. ETKF-FWI 8
Outline S E I S C O P E Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions ETKF-FWI 8
Adapting the FWI problem to the EnKF S E I S C O P E We define the FWI problem as a non-linear operator F 1 2 || d cal ( m ) − d obs || 2 F ( m ) = min m • m is the model containing the n physical parameters • d cal ( m ) the synthetic wavefield data computed in m • d obs the observed data • || . || the Euclidean distance in the data space Applying F on an initial model m 0 → unique solution with a local optimization scheme. But how do we apply an EnKF to this static problem? ETKF-FWI 9
Multi Scale Approach S E I S C O P E Full Waveform data fitting is ill-posed by nature. Non-unique, its cost function can be strongly non-convex. The cost function convexity is primarily dominated by the data frequency content (due to the nature of cycle skipping problem) We can use the multi-scale frequency strategy as a proxy for evolution in the frequency domain. 1D waveform cost function, at different frequency content from (Bunks et al., 1995) ETKF-FWI 10
The ensemble approach and dynamic axis S E I S C O P E We can recast our problem as an ensemble representation. Our ensemble m is a collection of N e models m i , with i = 1 , 2 , . . . , N e . In place of the typical DA forecast forward modeling problem we have : i = F ( m i m f k − 1 ) (1) k We decompose the d obs in K frequency bands → solve FWI independently on each of N e models, at a given frequency k . Allowing to consider a dynamic axis in frequency, with k = 1 , . . . , K instead of temporal evolution ETKF-FWI 11
Scheme specificities S E I S C O P E The EnKF scheme we follow : the Ensemble Transform Kalman Filter ( ETKF ) (Bishop et al., 2001). We apply FWI in the frequency domain → complex wavefield data. We consider all measurements as uncorrelated → measurement noise operator is diagonal whose values are calibrated on the data noise level. ETKF-FWI 12
ETKF-FWI Scheme S E I S C O P E dobs , k ⋆ Observation (modeling) Forecast (FWI) mf df • k +1 k +1 • ma × k • • ⋆ • • ⋆ • × × • ⋆ • ⋆ • ⋆ ⋆ • • ⋆ • × × • • ⋆ • • • ⋆ • • • • ma ma mk − 1 ⋆ × × × k +1 k +1 • • • • mf df • • k k • • Forecast (FWI) Observation dobs , k +1 ⋆ (modeling) step k − 1 k k + 1 (modeling frequency) ETKF-FWI 13
Outline S E I S C O P E Characteristics of FWI From FWI to ETKF-FWI Application on a Synthetic Case Conclusions ETKF-FWI 13
Applying ETKF to FWI S E I S C O P E Application on 2D Marmousi model : • Fixed spread surface acquisition (144 sources, 660 receivers) • Noisy signal (SNR = 5) • 15 ETKF-FWI cycles from 3 to 10Hz. • Initial gaussian repartition ETKF-FWI 14
Applying ETKF to FWI S E I S C O P E 0 4500 500 1000 4000 1500 2000 2500 3500 3000 Velocity (m/s) 3500 3000 0 500 1000 2500 Depth (m) 1500 2000 2000 2500 3000 1500 3500 0 2000 4000 6000 8000 10000 12000 14000 16000 Offset (m) Numerical test setting. Top : True model, bottom : Initial model ETKF-FWI 15
Generating the initial ensemble S E I S C O P E 0 500 100 Depth (m) 1000 1500 50 Velocity (m/s) 2000 2500 0 0 500 1000 −50 1500 2000 −100 2500 0 2500 5000 7500 10000 12500 0 2500 5000 7500 10000 12500 Offset (m) Exemple of random perturbation selected to build the initial gaussian repartition. ETKF-FWI 16
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