Efficient data-driven strategy for 3D model-preconditioning FWI P.T. Trinh 1,2 , R. Brossier 1 , L. Métivier 1 , L. Tavard 1 , J. Virieux 1 and P. Wellington* 1 ¹ ISTerre/LJK/GRICAD Univ. Grenoble Alpes, CNRS ² Total E&P * Now at Chevron Australia http://seiscope2.osug.fr Nov 6-10 WS - Seismic modeling & inversion - ICERM 1
Seismic data SEISMOLOGY SEISMICS Absolute time: date Honoring a simple PDE! Dense acquisition Towards continuous recording Towards dense recording Nov 6-10 WS - Seismic modeling & inversion - ICERM 2
Outline 1.Motivation 2.FWI: single scattering 3.PDE visco-elastic wave propagation 4.Model discretization & preconditioning 5.3D elastic SEAM II Foothills application Ebook of SEG: encyclopedia of exploration geophysics http://library.seg.org/doi/abs/10.1190/1.9781560803027.entry6 Nov 6-10 WS - Seismic modeling & inversion - ICERM 3
Model/Physical parameter hunting? Physical parameters: mineral Micro-scale composition, gas, liquid … 100* m m Upscaling Inference parameters: one solid skeleton and one fluid. Meso-scale Gassmann rheology Porosity, saturation, tortuosity, consolidation parameter … < or ~ m Downscaling Model parameters: velocities, attenuation, anisotropy, density Macro-scale for seismic waves (Investigated by FWI) > 10*m Important parameters at the macro-scale level ? Attenuation, Elasticity, Anisotropy, Density Nov 6-10 WS - Seismic modeling & inversion - ICERM 4
High-resolution seismic imaging Macro-scale imaging: FWI provides high-resolution capacity Vertical components or 4C data Body waves versus surface waves Diving waves versus reflected waves Which physics to consider at this scale? Visco-elastic anisotropic propagation Related model parameters … Medium interpretation: which physics to consider? Downscaling using biphasic model (Gassmann relation) Upscaling from multi-phases rock description related to physical parameters … Inference step between downscaling and upscaling Nov 6-10 WS - Seismic modeling & inversion - ICERM 5
Macro-scale imaging FWI provides high-resolution capacity Operto & Miniussi (2017) Nov 6-10 WS - Seismic modeling & inversion - ICERM 6
High-resolution seismic imaging Macro-scale imaging: FWI provides high-resolution capacity Vertical components or 4C data Body waves versus surface waves Diving waves versus reflected waves Which physics to consider at this scale? Visco-elastic anisotropic propagation Related model parameters … Medium interpretation: which physics to consider? Downscaling using biphasic model (Gassmann relation) Upscaling from multi-phases rock description related to physical parameters … Inference step between downscaling and upscaling Nov 6-10 WS - Seismic modeling & inversion - ICERM 7
Which physics to consider at macro-scale? Anisotropic visco-elastic propagation True 𝑊 𝑡 • Highly dispersive surface waves • Waves conversion P-S, body-surface • Transmission/Reflection regimes • Back-scattering due to steep slopes at the free surface Nov 6-10 WS - Seismic modeling & inversion - ICERM 8
High-resolution seismic imaging Macro-scale imaging: FWI provides high-resolution capacity Vertical components or 4C data Body waves versus surface waves Diving waves versus reflected waves Which physics to consider at this scale? Visco-elastic anisotropic propagation Related model parameters … Medium interpretation: which physics to consider? Downscaling using biphasic model (Gassmann relation) Upscaling from multi-phases rock description related to physical parameters … Inference step between downscaling and upscaling ⟹ Towards reservoir interpretation and monitoring Nov 6-10 WS - Seismic modeling & inversion - ICERM 9
Which physics to consider? Physical interpretation = Many model parameters? Model parameters are now the data used for downscaling … Pride (2005); Chopra & Marfurt (2007); Gassmann’s equation: porosity 𝝔 and consolidation parameter 𝒅 𝒕 Mavko et al. (2009); Dupuy et al. (2016) Nov 6-10 WS - Seismic modeling & inversion - ICERM 10
Visco-elastic FWI: challenges Model parameters reconstruction FWI pros and cons Non-linearity of FWI Nov 6-10 WS - Seismic modeling & inversion - ICERM 11
High-resolution seismic imaging Cycle-skipping issue We face different difficulties … Local minimum challenge Multiple-parameters reconstruction Initial model design is a key step … Model parameter trade- off … Uncertainty quantification … Nov 6-10 WS - Seismic modeling & inversion - ICERM 12
Outline 1.Motivation 2.FWI: single scattering 3.PDE visco-elastic wave propagation 4.Model discretization & preconditioning 5.3D elastic SEAM II Foothills application Nov 6-10 WS - Seismic modeling & inversion - ICERM 13
FWI = simple wave-matter interaction (Devaney, 1982) FWI is an ill-posed problem based on a single-scattering formulation Model is described through a pixel structure (# from a blocky structure) The model wavenumber spectrum is probed through this pixel strategy 𝑔 – Frequency 𝒍 = 2𝜌𝑔𝒓 = 4𝜌𝑔 cos 𝜄 2 𝒐 𝜄 – Aperture or illumination angle 𝑑 𝒍 = 4𝜌 𝜇 cos 𝜄 2 𝒐 Controlling parameters of the model velocity spectrum Low 𝒍 – low frequency 𝑔 or aperture angle 𝜄 around 𝜌 (weak interaction) High 𝒍 – high frequency 𝑔 or aperture angle 𝜄 around 0 (strong interaction) Nov 6-10 WS - Seismic modeling & inversion - ICERM 14
Scattering diagram Strong interaction: 𝜄~𝜌/2 reflection regime Intermediate interaction 𝜄~0 ∘ 𝜄~𝜌 Weak interaction: transmission regime How waves interact with matter! Nov 6-10 WS - Seismic modeling & inversion - ICERM 15
FWI strategy 𝐞 𝑑𝑏𝑚 = ℱ(𝐧) Initial guess Forward Data-fitting modeling technique Data misfit Model estimation 𝑫 𝐧 = 𝟐 𝟑 ‖𝐞 𝑝𝑐𝑡 − 𝐞 𝑑𝑏𝑚 ‖ 𝟑 𝐧 = 𝐧 + Δ𝐧 Inverse problem Gradient estimation 𝐡 𝐲 = 𝜖𝑫(𝐧)/𝜖𝐧 Gradient smoothing 𝐭 𝐲 = 𝐂 𝐲 ∗ 𝐡(𝐲) Model update Δ𝐧 = 𝛽 × 𝐭(𝐲) Nov 6-10 WS - Seismic modeling & inversion - ICERM 16
FWI strategy 𝐞 𝑑𝑏𝑚 = ℱ(𝐧) Initial guess Forward Data-fitting modeling technique Data misfit Model estimation 𝑫 𝐧 = 𝟐 𝟑 ‖𝐞 𝑝𝑐𝑡 − 𝐞 𝑑𝑏𝑚 ‖ 𝟑 𝐧 = 𝐧 + Δ𝐧 1. SEM-based modeling & Inverse problem inversion kernels Gradient estimation 𝐡 𝐲 = 𝜖𝑫(𝐧)/𝜖𝐧 Gradient smoothing 𝐭 𝐲 = 𝐂 𝐲 ∗ 𝐡(𝐲) Model update Δ𝐧 = 𝛽 × 𝐭(𝐲) Nov 6-10 WS - Seismic modeling & inversion - ICERM 17
FWI strategy 𝐞 𝑑𝑏𝑚 = ℱ(𝐧) Initial guess Forward Data-fitting modeling technique Data misfit Model estimation 𝑫 𝐧 = 𝟐 𝟑 ‖𝐞 𝑝𝑐𝑡 − 𝐞 𝑑𝑏𝑚 ‖ 𝟑 𝐧 = 𝐧 + Δ𝐧 1. SEM-based modeling & Inverse problem inversion kernels 2. Bessel FWI gradient Gradient estimation 𝐡 𝐲 = 𝜖𝑫(𝐧)/𝜖𝐧 smoothing for SEM mesh Gradient smoothing 𝐭 𝐲 = 𝐂 𝐲 ∗ 𝐡(𝐲) Model preconditioning Model update Δ𝐧 = 𝛽 × 𝐭(𝐲) Nov 6-10 WS - Seismic modeling & inversion - ICERM 18
FWI gradient: often all you need 𝑣 𝑢 𝑠 𝑒 𝜖[𝑄𝐸𝐹] 𝜖𝑛 𝑣 𝑢 𝜖[𝑄𝐸𝐹] 𝑠 𝑒 S.K. 𝜖𝑛 Sensitivity kernel Zero-lag cross-correlation of incident 𝑣 𝑢 and adjoint 𝑠 𝑒 fields through interlaced backward-incident and adjoint integration Nov 6-10 WS - Seismic modeling & inversion - ICERM 19
Outline 1.Motivation 2.FWI: single scattering 3.PDE visco-elastic wave propagation 4.Model discretization & preconditioning 5.3D elastic SEAM II Foothills application Nov 6-10 WS - Seismic modeling & inversion - ICERM 20
Designing PDE solver Integrated approach : FWI design should not be reduced to wave propagation design Complex topography Simple geometry representation. Accurate boundary free-surface conditions. Numerical Memory efficiency requirement 3D (visco)elastic modeling & FWI Complete and accurate physics seen by waves Simultaneous design of modeling/adjoint/gradient Time-domain Simulation Signal muting and multi-frequencies processing accuracy Data-component hierarchy FWI, thanks to the causality Nov 6-10 WS - Seismic modeling & inversion - ICERM 21
Attenuation: Efficient implementation Complex seismic data (i.e. land data): Acoustic might not be enough! Elastic neither: Attenuation is required when fitting phase & amplitude! Visco-elastic 3D aniso-elastic reconstruction Tarantola (1988): Convolutional rheology with application by Charara et al. (2000) ⟹ Computationally intensive. Tromp (2005) & Liu and Tromp (2006): General multiparameter workflow with adjoint methods . Fichtner & van Driel (2014): Clarification of the Q parameter imaging of Tromp (2005) ⟹ Lowering the computational needs . Yang et al (2016): Explicit formulations for FWI gradients using visco-anisotropic elastic wave propagation based on standard linear solid (SLS) mechanisms ⟹ Straightforward numerical implementation . Nov 6-10 WS - Seismic modeling & inversion - ICERM 22
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