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Efficient data-driven strategy for 3D model-preconditioning FWI P.T. - - PowerPoint PPT Presentation

Oct 25, 2022 991 likes •1.49k views

Efficient data-driven strategy for 3D model-preconditioning FWI P.T. Trinh 1,2 , R. Brossier 1 , L. Mtivier 1 , L. Tavard 1 , J. Virieux 1 and P. Wellington* 1 ISTerre/LJK/GRICAD Univ. Grenoble Alpes, CNRS Total E&P * Now at Chevron

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SLIDE 1

Efficient data-driven strategy for 3D model-preconditioning FWI

P.T. Trinh1,2, R. Brossier1, L. Métivier1, L. Tavard1, J. Virieux1 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

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SLIDE 2

Seismic data

Nov 6-10 WS - Seismic modeling & inversion - ICERM 2

SEISMICS Dense acquisition Towards continuous recording SEISMOLOGY Towards dense recording

Absolute time: date

Honoring a simple PDE!

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SLIDE 3

Outline

Nov 6-10 WS - Seismic modeling & inversion - ICERM 3

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

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SLIDE 4

Model/Physical parameter hunting?

Nov 6-10 WS - Seismic modeling & inversion - ICERM 4

Micro-scale Meso-scale Macro-scale

Model parameters: velocities, attenuation, anisotropy, density for seismic waves Inference parameters: one solid skeleton and one fluid. Gassmann rheology Porosity, saturation, tortuosity, consolidation parameter … Physical parameters: mineral composition, gas, liquid … Upscaling Downscaling Important parameters at the macro-scale level ? Attenuation, Elasticity, Anisotropy, Density

100*mm

> 10*m < or ~ m

(Investigated by FWI)

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SLIDE 5

High-resolution seismic imaging

Nov 6-10 WS - Seismic modeling & inversion - ICERM 5

  • 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
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SLIDE 6

Macro-scale imaging

Nov 6-10 WS - Seismic modeling & inversion - ICERM 6

Operto & Miniussi (2017)

FWI provides high-resolution capacity

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SLIDE 7

High-resolution seismic imaging

Nov 6-10 WS - Seismic modeling & inversion - ICERM 7

  • 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
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SLIDE 8

Which physics to consider at macro-scale?

Nov 6-10 WS - Seismic modeling & inversion - ICERM 8

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
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SLIDE 9

High-resolution seismic imaging

Nov 6-10 WS - Seismic modeling & inversion - ICERM 9

  • 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

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SLIDE 10

Which physics to consider?

Nov 6-10 WS - Seismic modeling & inversion - ICERM 10

Physical interpretation = Many model parameters?

Gassmann’s equation: porosity 𝝔 and consolidation parameter 𝒅𝒕

Pride (2005); Chopra & Marfurt (2007); Mavko et al. (2009); Dupuy et al. (2016)

Model parameters are now the data used for downscaling …

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SLIDE 11

Visco-elastic FWI: challenges

Nov 6-10 WS - Seismic modeling & inversion - ICERM 11

  • Model parameters reconstruction
  • FWI pros and cons
  • Non-linearity of FWI
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SLIDE 12

High-resolution seismic imaging

Nov 6-10 WS - Seismic modeling & inversion - ICERM 12

Cycle-skipping issue Local minimum challenge Multiple-parameters reconstruction

We face different difficulties …

  • Initial model design is a key step …
  • Model parameter trade-off …
  • Uncertainty quantification …
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SLIDE 13

Outline

Nov 6-10 WS - Seismic modeling & inversion - ICERM 13

1.Motivation 2.FWI: single scattering 3.PDE visco-elastic wave propagation 4.Model discretization & preconditioning 5.3D elastic SEAM II Foothills application

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SLIDE 14

FWI = simple wave-matter interaction

Nov 6-10 WS - Seismic modeling & inversion - ICERM 14

  • 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 𝜄 – Aperture or illumination angle 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) (Devaney, 1982)

Controlling parameters of the model velocity spectrum

𝒍 = 2𝜌𝑔𝒓 = 4𝜌𝑔 𝑑 cos 𝜄 2 𝒐 𝒍 = 4𝜌 𝜇 cos 𝜄 2 𝒐

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SLIDE 15

Scattering diagram

Nov 6-10 WS - Seismic modeling & inversion - ICERM 15

Weak interaction: transmission regime Strong interaction: reflection regime Intermediate interaction

How waves interact with matter!

𝜄~0∘ 𝜄~𝜌 𝜄~𝜌/2

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SLIDE 16

FWI strategy

Nov 6-10 WS - Seismic modeling & inversion - ICERM 16

Forward modeling

Data misfit 𝑫 𝐧 = 𝟐 𝟑 ‖𝐞𝑝𝑐𝑡 − 𝐞𝑑𝑏𝑚‖𝟑

Inverse problem

Model estimation 𝐧 = 𝐧 + Δ𝐧

  • Gradient estimation 𝐡 𝐲 = 𝜖𝑫(𝐧)/𝜖𝐧
  • Gradient smoothing 𝐭 𝐲 = 𝐂 𝐲 ∗ 𝐡(𝐲)
  • Model update Δ𝐧 = 𝛽 × 𝐭(𝐲)

Initial guess

𝐞𝑑𝑏𝑚 = ℱ(𝐧) Data-fitting technique

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SLIDE 17

FWI strategy

Nov 6-10 WS - Seismic modeling & inversion - ICERM 17

Forward modeling

Data misfit 𝑫 𝐧 = 𝟐 𝟑 ‖𝐞𝑝𝑐𝑡 − 𝐞𝑑𝑏𝑚‖𝟑

Inverse problem

Model estimation 𝐧 = 𝐧 + Δ𝐧

  • Gradient estimation 𝐡 𝐲 = 𝜖𝑫(𝐧)/𝜖𝐧
  • Gradient smoothing 𝐭 𝐲 = 𝐂 𝐲 ∗ 𝐡(𝐲)
  • Model update Δ𝐧 = 𝛽 × 𝐭(𝐲)

Initial guess

𝐞𝑑𝑏𝑚 = ℱ(𝐧) Data-fitting technique

1. SEM-based modeling & inversion kernels

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SLIDE 18

FWI strategy

Nov 6-10 WS - Seismic modeling & inversion - ICERM 18

Forward modeling

Data misfit 𝑫 𝐧 = 𝟐 𝟑 ‖𝐞𝑝𝑐𝑡 − 𝐞𝑑𝑏𝑚‖𝟑

Inverse problem

Model estimation 𝐧 = 𝐧 + Δ𝐧

  • Gradient estimation 𝐡 𝐲 = 𝜖𝑫(𝐧)/𝜖𝐧
  • Gradient smoothing 𝐭 𝐲 = 𝐂 𝐲 ∗ 𝐡(𝐲)
  • Model update Δ𝐧 = 𝛽 × 𝐭(𝐲)

Initial guess

𝐞𝑑𝑏𝑚 = ℱ(𝐧) Data-fitting technique

1. SEM-based modeling & inversion kernels 2. Bessel FWI gradient smoothing for SEM mesh

Model preconditioning

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SLIDE 19

FWI gradient: often all you need

Nov 6-10 WS - Seismic modeling & inversion - ICERM 19

𝑣𝑢 𝜖[𝑄𝐸𝐹] 𝜖𝑛 𝑠𝑒

S.K.

𝑣𝑢 𝜖[𝑄𝐸𝐹] 𝜖𝑛 𝑠

𝑒

Zero-lag cross-correlation of incident 𝑣𝑢and adjoint 𝑠

𝑒 fields

through interlaced backward-incident and adjoint integration

Sensitivity kernel

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SLIDE 20

Outline

Nov 6-10 WS - Seismic modeling & inversion - ICERM 20

1.Motivation 2.FWI: single scattering 3.PDE visco-elastic wave propagation 4.Model discretization & preconditioning 5.3D elastic SEAM II Foothills application

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SLIDE 21

Designing PDE solver

Nov 6-10 WS - Seismic modeling & inversion - ICERM 21

Complex topography

  • Simple geometry representation.
  • Accurate boundary free-surface conditions.

3D (visco)elastic modeling & FWI

  • Complete and accurate physics seen by waves
  • Simultaneous design of modeling/adjoint/gradient

Time-domain

  • Signal muting and multi-frequencies processing
  • Data-component hierarchy FWI, thanks to the causality

Integrated approach: FWI design should not be reduced to wave propagation design

Memory requirement Simulation accuracy Numerical efficiency

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SLIDE 22

Attenuation: Efficient implementation

Nov 6-10 WS - Seismic modeling & inversion - ICERM 22

Complex seismic data (i.e. land data):

  • Acoustic might not be enough!
  • Elastic neither: Attenuation is required when fitting phase &

amplitude!  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.

Visco-elastic 3D aniso-elastic reconstruction

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SLIDE 23

Wave propagation: lossy medium

Nov 6-10 WS - Seismic modeling & inversion - ICERM 23

𝜍𝜖𝑢𝑢𝐯 = 𝐸𝝉 + 𝐠 𝜻 = 𝐸𝑢𝐯 𝝉 = 𝐷𝜻 − 𝐷𝑆 ෍

𝑚=1 𝑀

𝝎𝑚 + 𝓤 𝜖𝑢𝝎𝑚 + 𝜕𝑚𝝎𝒎 = 𝜕𝑚𝑧𝑚𝜻, 𝑚 = 1, … , 𝑀

Time domain

Wavefield conditions

  • Medium at rest at initial time (zero initial conditions)
  • Unbounded domain (free surface condition -zero

stress- and absorbing boundary conditions) Standard Linear Solid: Generalized Maxwell model or Generalized Zener model

  • Attenuation is carried by 𝑀 sets of memory variables 𝝎𝑚 = non-physical parameters.

Quantities 𝜕𝑚 and 𝑧𝑚 are uniform inside the medium (resonance frequencies and relative weights)

  • Attenuation = SLS − 𝑹-constant approx. over frequencies.
  • Memory variables obey a 1st order equation.

Additional needs: storing decimated boundaries (inside nearby PML) and few snapshots for backpropagation (Yang at al, 2016a,2016b)

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SLIDE 24

Wave propagation: FWD modeling OK

Nov 6-10 WS - Seismic modeling & inversion - ICERM 24

𝜍𝜖𝑢𝑢𝐯 = 𝐸𝝉 + 𝐠 𝜻 = 𝐸𝑢𝐯 𝝉 = 𝐷𝜻 − 𝐷𝑆 ෍

𝑚=1 𝑀

𝝎𝑚 + 𝓤 𝜖𝑢𝝎𝑚 + 𝜕𝑚𝝎𝒎 = 𝜕𝑚𝑧𝑚𝜻, 𝑚 = 1, … , 𝑀

Time domain

Heterogeneities inside the medium described by 𝐷 – Unrelaxed (elastic) stiffness tensor (anisotropic); 𝐷𝑆 – Relaxed stiffness tensor (isotropic);

  • Elastic system is conservative: self-adjoint structure of PDE

𝜍𝜖𝑢𝑢𝐯 = 𝐸𝐷𝐸𝑢𝐯 + 𝐠 ⟹ Stable backpropagation of the wavefield.

  • With attenuation, the system is no more conservative!

𝜍𝜖𝑢𝑢𝐯 = 𝐸𝐷𝐸𝑢𝐯 − 𝐸𝐷𝑆 σ𝑚=1

𝑀

𝝎𝑚 + 𝐠 ⟹ Unstable backpropagation of the wavefield! Tracking the total energy for detecting the instability during the backpropagation: if divergence is observed, use stored snapshots to restart the backpropagation from them (assisted checkpointing strategy) Relaxed « Lamé » coefficients: 𝜈𝑆 = 1 3 𝑅𝑡

−1 ෍ 𝑘=4 6

𝐷

𝑘𝑘

𝜇𝑆 + 2𝜈𝑆 = 1 3 𝑅𝑞

−1 ෍ 𝑗=1 3

𝐷𝑗𝑗 ;

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SLIDE 25

Incident + Adjoint propagation

Nov 6-10 WS - Seismic modeling & inversion - ICERM 25

𝜍𝜖𝑢𝑢𝐯 = 𝐸𝐷𝐸𝑢𝐯 − 𝐸𝐷𝑆 ෍

𝑡=1 𝑀

𝝎𝑡 + 𝐓 𝜖𝑢𝝎𝑡 + 𝜕𝑡𝝎𝑡 = 𝜕𝑡𝑧𝑡𝐸𝑢𝐯

Incident field

𝜍𝜖𝑢𝑢ഥ 𝐯 = 𝐸𝐷𝐸𝑢ഥ 𝐯 − 𝐸𝐷𝑆 ෍

𝑡=1 𝑀

𝝎𝑡 − 𝑆†𝛦𝑒𝐯 𝜖𝑢𝝎𝑡 − 𝜕𝑡𝝎𝑡 = −𝜕𝑡𝑧𝑡𝐸𝑢ഥ 𝐯

Adjoint field

ഥ 𝐯 – Displacement; 𝝎𝑡 – Memory variables;

𝛦𝑒𝐯– Data residual;

𝐯 – Displacement; 𝝎𝑡 – Memory variables; 𝐓 – Source term;

  • Similar but not identical structure and equations for incident and adjoint fields
  • Computing incident field from initial time with zero initial conditions 
  • Computing adjoint field from final time with zero final conditions 

+ recomputing incident field backward  but  using Lagrange formulation (final and boundary conditions!)

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SLIDE 26

Inversion workflow

Nov 6-10 WS - Seismic modeling & inversion - ICERM 26

Acoustic case (Yang et al., 2016c)

1 2 𝐞𝑝𝑐𝑡 − 𝐞𝑑𝑏𝑚(𝐧) 2

  • Least-squares norm:
  • All gradients = Adjoint-state approach (Plessix, 2006):

Directly accumulated during the backpropagation of the incident field while computing adjoint fields. ⟹ No I/O Incident field Adjoint field

Backward reconstruction Adjoint propagation

Xcross 𝑜𝑢 = 𝑂 Xcross 𝑜𝑢 = 𝑂 − 1 Xcross 𝑜𝑢 = 2 Xcross 𝑜𝑢 = 1

Affordable numerical cost

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SLIDE 27

FWI gradients

Nov 6-10 WS - Seismic modeling & inversion - ICERM 27

  • 𝑴𝟑 FWI gradient:

𝜖(Data misfit) 𝜖𝐷𝑗𝑘 = ത 𝜻, 𝜖𝐷 𝜖𝐷𝑗𝑘 𝜻

Ω,𝑢

− ത 𝜻, ෍

𝑚=1 𝑀 𝜖𝐷𝑆

𝜖𝐷𝑗𝑘 𝝎𝑚

Ω,𝑢

Elastic rheology Attenuation mechanism ⟹ Attenuation affects the velocity estimation 𝜖(Data misfit) 𝜖𝑅𝑞,𝑡

−1

= − ത 𝜻, ෍

𝑚=1 𝑀

𝜖𝐷𝑆 𝜖𝑅𝑞,𝑡

−1 𝝎𝑚 Ω,𝑢

; 𝜖(Data misfit) 𝜖𝜍 = ഥ 𝒗, 𝜖𝑢𝑢𝒗

Ω,𝑢

  • Separate the elastic rheology 𝐷 and the attenuation mechanism 𝐷𝑆 → 𝑹𝒒, 𝑹𝒕 .

Isotropic attenuation Anisotropic attenuation: VSP data?

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SLIDE 28

SEM46 for modeling and inversion

Nov 6-10 WS - Seismic modeling & inversion - ICERM 28

Time-domain Spectral Element Method

SEM-based implementation

  • Topography & simple geometry representation.
  • Accurate boundary free-surface conditions.

3D (visco)elastic modeling & FWI

  • Complete and accurate physics seen by waves.
  • Simultaneous design of modeling/adjoint/gradient.

Time-domain

  • Signal muting and multi-frequencies processing.
  • Data-component hierarchy FWI (causality).
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SLIDE 29

Cartesian-based deformed mesh

Nov 6-10 WS - Seismic modeling & inversion - ICERM 29

  • Combine the accuracy of FE mesh with the easiness of implementation of FD grid.
  • Avoid the heavy searching operator over the global mesh.
  • Efficient domain-decomposition in a parallel scheme.

Numerical cost vs. simulation accuracy

  • Vertical deformed elements to follow the topography.
  • High-order presentation of the topography
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SLIDE 30

Variable element-size for modeling

Nov 6-10 WS - Seismic modeling & inversion - ICERM 30

Variable element-size

  • Respect the theoretical

resolution of FWI (𝟏. 𝟔𝝁𝒕).

  • Follow the velocity variation.

Constant element-size Reduce 6 times the number of elements, thus 6 times the computational cost. Mesh design is constrained by ≥ 5 GLL points /min (wavelength)

  • Same element-size everywhere.
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SLIDE 31

Outline

Nov 6-10 WS - Seismic modeling & inversion - ICERM 31

1.Motivation 2.FWI: single scattering 3.PDE visco-elastic wave propagation 4.Model discretization & preconditioning 5.3D elastic SEAM II Foothills application

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SLIDE 32

Model discretization

Nov 6-10 WS - Seismic modeling & inversion - ICERM 32

Model meshing adapted to the expected FWI resolution (few ls). Modeling meshing adapted to the required local sampling of wavelengths for wave propagation (fractions of 𝜇)

Pixel-oriented FWI: which sampling strategy for this ill-posed problem ? Expensive back and forth projections, especially in 3D

  • Same mesh for forward/inverse problems ⟹ Efficient computation.
  • Mathematically ill-posed features of FWI: expected low-wavenumber content.
  • Preconditioning and/or regularization is mandatory in FWI.

Inversion mesh An alternatrive could be the ROM strategy

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SLIDE 33

Necessary of preconditioning & regularization

Nov 6-10 WS - Seismic modeling & inversion - ICERM

Gradient example Complex geometry

Why?

  • Suppress high-wavenumber artifacts
  • Acquisition footprints
  • Poor illumination
  • Guide the inversion towards a desired solution

Need?

  • Nonstationary & anisotropic operator
  • Anisotropic coherent lengths
  • Local 3D rotation
  • Numerical efficiency
  • SEM mesh compatible: Non-regular grid points

Smoothing the FWI gradient!

33

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SLIDE 34

Bessel smoothing for SEM mesh

Nov 6-10 WS - Seismic modeling & inversion - ICERM 34

(Trinh et al, 2017b; Wellington et al, 2017)

  • Considering the sparse inverse operator:

𝐶3𝐸

−1 𝐲

∗ ฑ 𝐭 𝐲

𝐓𝐧𝐩𝐩𝐮𝐢𝐟𝐞 𝐡𝐬𝐛𝐞𝐣𝐟𝐨𝐮

= ฑ 𝐡(𝐲)

𝐒𝐛𝐱 𝐡𝐬𝐛𝐞𝐣𝐟𝐨𝐮

1 − 𝑀𝑨

2 𝜖2

𝜖𝑨2 + 𝑀𝑦

2 𝜖2

𝜖𝑦2 + 𝑀𝑧

2 𝜖2

𝜖𝑧2 𝐭 𝐲 = 𝐡 𝐲

  • 𝟏° rotation, homogeneous

coherent lengths:

𝛼

𝑨,𝑦,𝑧 = 𝜖𝑨, 𝜖𝑦, 𝜖𝑧 𝑢

1 − 𝛼

𝑨,𝑦,𝑧 𝑢

𝐐 𝐲 𝐐𝑢 𝐲 𝛼

𝑨,𝑦,𝑧 𝐭 𝐲 = 𝐡(𝐲)

  • Fully anisotropic & nonstationary filter:

→ Self-adjoint PDE  Variable coherent lengths and angles  3D rotation Geological prior information

Azimuth 𝜄 Dip 𝜒

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SLIDE 35

Parallel implementation

Nov 6-10 WS - Seismic modeling & inversion - ICERM 35

  • Linear numerical complexity 𝓟 𝐃𝐩𝐢𝐟𝐬𝐟𝐨𝐮 𝐦𝐟𝐨𝐡𝐮𝐢

In FD scheme: as cheap as tensorized Gaussian convolution.

  • Smoothing ≈ 0.4 % cost of 1 iteration
  • Self-adjoint PDE = Symmetric, well-conditioned, positive-definite linear system

⟹ Efficiently solved by a matrix-free parallel conjugate-gradient

1 − 𝛼

𝑨,𝑦,𝑧 𝑢

𝐐 𝐲 𝐐𝑢 𝐲 𝛼

𝑨,𝑦,𝑧 𝐭 𝐲 = 𝐡(𝐲)

𝐁𝐭 = 𝐡

Coherent lengths 𝑀 = 𝑀𝑨 = 𝑀𝑦 = 𝑀𝑧 (m)

Bessel smoothing Windowed explicit convolution

(Trinh et al, 2017b; Wellington et al, 2017)

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SLIDE 36

Structure-oriented preconditioning

Nov 6-10 WS - Seismic modeling & inversion - ICERM 36

(Trinh et al, 2017d)

Nonstationary & anisotropic Bessel gradient preconditioning

a) True 𝑊

𝑡

model b) Initial 𝑊

𝑡

model c) Raw gradient d) Smoothed gradient

Prior information?

𝑀𝑥 = 25m and 𝑀𝑣, 𝑀𝑤 = 25~100m; Dip & azimuth from true models.

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SLIDE 37

Preconditioning for FE mesh

Nov 6-10 WS - Seismic modeling & inversion - ICERM 37

Accuracy Efficiency Nonstationarity Projection between SEM & Cartesian meshes Explicit truncated convolution ถ 𝐭 𝐲

𝐓𝐧𝐩𝐩𝐮𝐢𝐟𝐞 𝐡𝐬𝐛𝐞𝐣𝐟𝐨𝐮

≈ 𝐶3𝐸 𝐲 ∗𝛁𝐬 ถ 𝐡(𝐲)

𝐒𝐛𝐱 𝐡𝐬𝐛𝐞𝐣𝐟𝐨𝐮

Bessel smoothing 𝐶3𝐸

−1 𝐲 ∗

ถ 𝐭 𝐲

𝐓𝐧𝐩𝐩𝐮𝐢𝐟𝐞 𝐡𝐬𝐛𝐞𝐣𝐟𝐨𝐮

= ถ 𝐡(𝐲)

𝐒𝐛𝐱 𝐡𝐬𝐛𝐞𝐣𝐟𝐨𝐮

 

?

  

? ? ?

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SLIDE 38

Outline

Nov 6-10 WS - Seismic modeling & inversion - ICERM 38

1.Motivation 2.FWI: single scattering 3.PDE visco-elastic wave propagation 4.Model discretization 5.3D elastic SEAM II Foothills application

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SLIDE 39

3D elastic example: subset of SEAM II

Nov 6-10 WS - Seismic modeling & inversion - ICERM 39

  • Significant topography variation: Δ𝑎 ≈ 800 m.
  • 3D surface acquisition:

True 𝑊

𝑡

  • 𝑇 𝑢 = Ricker wavelet centered at 3.5 Hz
  • Meshing: P4 high-order topography representation.
  • Initial models = Smoothed version of true model.
  • Simultaneous inversion for 𝑊

𝑞 and 𝑊 𝑡

  • Smoothed density is kept unchanged.
  • 60 FWI iterations using the l-BFGS optimization method.

Initial 𝑊

𝑡

  • 82600 receivers,12.5m, 3C

Δ𝑇𝑦 = 320 m Δ𝑇𝑧 = 500 m

  • 4 × 20 sources
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SLIDE 40

Complex wavefield

Nov 6-10 WS - Seismic modeling & inversion - ICERM 40

True 𝑊

𝑡

  • Highly dispersive surface waves
  • Waves conversion P-S, body-

surface

  • Back-scattering due to steep-slope

at the surface.

Early-body waves Back-scattering waves All wavefields Dispersive surface waves

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SLIDE 41

Simple FWI data-driven strategy

Nov 6-10 WS - Seismic modeling & inversion - ICERM 41

Two-steps data-component hierarchy

Use early-body waves (arriving before the surface waves) Use all wavefields (surface + body waves + back-scattering) 𝑾𝒒 𝑾𝒕 ⟹ Main features resolved ⟹ Refine near-surface & Enhance deep structures

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SLIDE 42

Data comparison

Nov 6-10 WS - Seismic modeling & inversion - ICERM 42

𝐞𝑝𝑐𝑡

𝐞𝑝𝑐𝑡 𝐞𝑑𝑏𝑚

Initial True Inverted

  • 0. Initial models
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SLIDE 43

Data comparison

Nov 6-10 WS - Seismic modeling & inversion - ICERM 43

𝐞𝑝𝑐𝑡

𝐞𝑝𝑐𝑡 𝐞𝑑𝑏𝑚

Initial True Inverted

  • 1. FWI with early-body waves
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SLIDE 44

Data comparison

Nov 6-10 WS - Seismic modeling & inversion - ICERM 44

𝐞𝑝𝑐𝑡

𝐞𝑝𝑐𝑡 𝐞𝑑𝑏𝑚

Initial True Inverted

  • 2. FWI with all wavefields
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SLIDE 45

Numerical efficiency

Nov 6-10 WS - Seismic modeling & inversion - ICERM 45

Memory estimation Elapsed time 1st gradient Elapsed time 60 FWI iterations 44 Gb/shot 20 min 20.8 h 74 Gb/shot 1.2 h 75 h

  • Deformed mesh: 32 × 68 × 28 elements (129 × 273 × 113 dofs).
  • 6 sec recording time (10 000 time-steps).
  • 1600 cores (20 cores/shot)

From the 3D elastic example Viscoelastic Elastic

  • 80 checkpoints for incident wavefield reconstruction.
  • Recomputation ratio ≈ 𝟒.

Extrapolation for viscoelastic case

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SLIDE 46

Conclusion I

Nov 6-10 WS - Seismic modeling & inversion - ICERM 46

  • Moving to 3D visco-aniso-elastodynamics FWI is now possible for crustal land

data (PhD topic of P.T. Trinh).

  • Application to real datasets: multi-parameters images?
  • Which macro-scale parameters are important for meso-scale downscaling

investigation for micro-scale interpretation: 𝑅 attenuation factor is important!

Cautiousness in interpretation as FWI results seem often quite realistic.

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SLIDE 47

Conclusion II

Nov 6-10 WS - Seismic modeling & inversion - ICERM 47

  • Different families: what is the « best » set ???
  • Velocity – slowness-square of slowness
  • Density- Buoyancy-Impedance
  • Attenuation-Inverse of attenuation
  • Log; tanh (or any non-linear transform) …
  • Hints: mitigate the leakage between parameters …
  • Model parameters # inference parameters # physical parameters …
  • FWI reconstructs model parameters … at the macro-scale level …
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SLIDE 48

End …

Nov 6-10 WS - Seismic modeling & inversion - ICERM 48

FWI=l/2

Thank you very much!

  • Cycle skipping problem: under control.
  • Local minima issue: better mitigation.
  • Multiple parameter issue: important for apps.