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Wavefield reconstruction inversion via physics-informed neural networks

Chao Song, Tariq Alkhalifah

arXiv:2104.06897v2physics.geo-ph

TL;DR

FWI can suffer cycle skipping, and WRI wavefield reconstruction can require expensive matrix inversions. This paper uses two PINNs sequentially to reconstruct scattered wavefields and predict velocity; on two model examples, it reports a reasonably smooth velocity with limited iterations and frequencies that can serve as an initial model for FWI.

  • Problem

    FWI can suffer cycle skipping, and WRI reconstruction of frequency-domain wavefields requires costly large-matrix inversions, especially for high frequencies and large 3D models.

    Source: §1 · pp. 1–2

  • Method

    Two independent PINNs first reconstruct scattered wavefields using physics and data-fitting losses, then predict velocity using the reconstructed fields and a scattered-wave-equation loss.

    Source: §2.3 · p. 4, §5 · p. 17

  • Results

    The Sigsbee2A and modified Marmousi examples yield a reasonably smooth velocity with very limited iterations and frequencies, which can be used as an initial velocity for FWI.

    Source: §5 · p. 18

  • Takeaways & Limitations

    The inverted velocity can provide a starting model for subsequent FWI, which recovered details in the true velocity in the Marmousi example.

    Source: §3.2 · p. 14, §5 · p. 18

    For small 2D isotropic acoustic models, PINN training remains more computationally costly overall than finite-difference matrix inversion, and reconstructed wavefields limit velocity-model detail.

    Source: §4 · p. 17

Abstract

from arXiv · show

Wavefield reconstruction inversion (WRI) formulates a PDE-constrained optimization problem to reduce cycle skipping in full-waveform inversion (FWI). WRI often requires expensive matrix inversions to reconstruct frequency-domain wavefields. Physics-informed neural network (PINN) uses the underlying physical laws as loss functions to train the neural network (NN), and it has shown its effectiveness in solving the Helmholtz equation and generating Green's functions, specifically for the scattered wavefield. By including a data-constrained term in the loss function, the trained NN can reconstruct a wavefield that simultaneously fits the recorded data and satisfies the Helmholtz equation for a given initial velocity model. Using the predicted wavefields, we rely on a small-size NN to predict the velocity using the reconstructed wavefield. In this velocity prediction NN, spatial coordinates are used as input data to the network and the scattered Helmholtz equation is used to define the loss function. After we train this network, we are able to predict the velocity in the domain of interest. We develop this PINN-based WRI method and demonstrate its potential using a part of the Sigsbee2A model and a modified Marmousi model. The results show that the PINN-based WRI is able to invert for a reasonable velocity with very limited iterations and frequencies, which can be used in a subsequent FWI application.

1 Introduction

FWI can suffer cycle skipping, while WRI mitigates its nonlinearity by reconstructing a data-fitting wavefield but requires costly matrix inversions. The paper motivates PINNs as a way to reconstruct wavefields and predict velocity using physics-based losses.

1 Introduction

  • WRI mitigates FWI nonlinearity by relaxing wave-equation accuracy for data fitting, but its augmented-wavefield step requires a costly large-matrix inverse.The cost rises for high frequencies, large 3D models, and complex physics.Source: §1 · p. 2
  • Prior ML velocity-inversion methods described here use time-domain data and require substantial training data, while SeisInvNets are reliable only near their training data.The cited introduction also notes that a physics-based NN FWI method still suffers cycle skipping.Source: §1 · p. 2
  • PINNs use physical laws in their loss and coordinate inputs to approximate PDE solutions, including Helmholtz wavefields, without relying solely on data-mapping losses.Prior geophysical applications use spatial coordinates as inputs and treat velocity and anisotropic parameters as implicit loss variables.Source: §1 · p. 2
  • The proposed approach reconstructs scattered wavefields with one PINN conditioned on horizontal source location, then uses a smaller independent NN to predict velocity.The source-location input allows one network to generate wavefields for multiple sources at a fixed depth.Source: §1 · pp. 2–3

2 Theory

The method uses PINNs to reconstruct scattered wavefields that fit recorded data and satisfy a wave-equation constraint, then predicts velocity with a separate network. Its four stages define the physics and data losses, the scattered-wave formulation, the augmented-WRI reconstruction, and the velocity update.

2.1 Physics-informed neural network

  • PINN training combines a physics-constrained loss with a data-constrained loss so the wavefield approximation satisfies the equation and fits recorded observations.The PINN uses coordinate inputs and automatic differentiation to evaluate PDE derivatives.Source: §2.1 · p. 3

2.2 The scattered wavefield

  • The scattered-wave formulation uses the difference between total and background wavefields; its equation is exact rather than a Born approximation.The homogeneous-background wavefield can be solved analytically, and the scattered equation's source term is associated with that background field.Source: §2.2 · pp. 3–4

2.3 The augmented wave equation

  • For an initial velocity, the reconstruction loss fits scattered data at receivers while regularizing the scattered-wave equation; the network takes coordinates and source locations as inputs.Its predicted scattered wavefields partially satisfy the equation and fit observations simultaneously, carrying information about the true velocity into the inversion.Source: §2.3 · p. 4

2.4 The velocity inversion

  • A second network predicts squared slowness from grid coordinates using reconstructed scattered fields, with total-variation regularization; each frequency uses reconstruction followed by velocity prediction.The method treats those two tasks as one iteration and uses multiple frequencies to refine the model.Source: §2.4 · pp. 4–5

3 Numerical Tests

The numerical tests apply PINN-based WRI to Sigsbee2A and modified Marmousi models, using reconstructed scattered wavefields to predict velocity. In both examples, the inverted models are evaluated as starting points for conventional FWI.

3 Numerical Tests

  • On Sigsbee2A, proposed PINN-based WRI uses a single 4 Hz frequency to reconstruct scattered wavefield and invert velocity; modified Marmousi examples show 3 Hz scattered wavefields.Both use ten surface sources; Sigsbee2A has 401 receivers on a 401 × 101 grid, and modified Marmousi has 301 receivers on a 301 × 101 grid.Source: §3.1 · p. 5, §3.2 · pp. 10–11

3.1 Sigsbee2A model

  • The hierarchically decreasing network reconstructs scattered wavefields accurately and efficiently compared with uniform networks, balancing smaller differences against training runtime.The tested layout uses widths {64, 64, 32, 32, 16, 16, 8, 8}; uniform networks with more neurons reduce error but require more runtime, and unusually large networks may overfit.Source: §3.1 · p. 7, Figure 4 · p. 7, Figure 6 · p. 8
  • For Sigsbee2A, velocity predictions from larger uniform and hierarchically decreasing networks recover the shallow low-velocity zone better than the 20-neuron network.The hierarchical network produces a velocity closer to the true model, consistent with its smaller scattered-wavefield differences.Source: §3.1 · p. 9, Figure 8 · p. 9, Figure 9 · p. 10
  • On Sigsbee2A, PINN-based WRI models improve FWI starting velocities; the hierarchically decreasing network yields the lowest RMS model error.Direct FWI from the raw initial velocity does not converge to the true model, while the WRI-derived starts have lower RMS errors.Source: §3.1 · p. 10, Figure 12 · p. 12

3.2 2D Marmousi model

  • On modified Marmousi, reconstructed scattered wavefields are smooth and differ from numerical solutions across much of the model, with smaller differences near shallow receivers.The resulting WRI velocity remains smooth, reflecting general true-model structure while adding only mild detail at higher frequencies.Source: §3.2 · pp. 12, 14, Figure 18 · p. 14

4 Discussions

The discussion presents PINN-based WRI as a way to obtain smooth background velocities for subsequent FWI, while detailing constraints in wavefield reconstruction, frequency, and computational cost.

4 Discussions

  • Scattered-wavefield reconstruction mitigates point-source singularity by using an analytical homogeneous background wavefield and a source term extended across the domain.The authors recommend setting background velocity to the source-location velocity, or to water velocity for a source in a marine water layer.Source: §4 · pp. 14, 16
  • The figures track velocity-prediction training loss and show inverted models for specified iteration and frequency settings.Source: Figure 19 · p. 15, Figure 20 · p. 15
  • A hierarchically decreasing neuron count performed better than uniform-width hidden layers, while excessively large networks risk overfitting and added computational cost.The wavefield network is enlarged and deepened for the more complex Marmousi model; the velocity network is smaller because it needs fewer inputs and only first-order derivatives for TV regularization.Source: §4 · pp. 16–17
  • One network conditioned on source horizontal location reconstructs wavefields for all survey sources, but PINN training still costs more than FD matrix inversion for small 2D isotropic acoustic models.The authors propose cPINN domain partitioning as future work for parallelization and faster convergence.Source: §4 · p. 17
  • The study focuses on 2D models; 3D and complex-media applications remain untested, and 3D use would require a new network architecture.The discussion suggests modifying loss functions for different wave equations, while specifying 3D spatial and surface-source coordinates as network inputs.Source: §4 · p. 17
  • The smooth PINN-WRI velocity can initialize FWI, whose sequential application provides an accurate, high-resolution inverted velocity.Figures 21 and 22 compare FWI models and vertical profiles; their captions do not state the comparison outcome.Source: Figure 21 · p. 16, Figure 22 · p. 16, §4 · p. 17
  • The weighting factor α balances data and physics constraints, and trial-and-error tests are needed to find its optimal value.Source: §4 · p. 17
  • Higher-frequency components remain difficult to reconstruct and add little velocity detail; the authors recommend the lowest available frequency for background recovery.The discussion attributes this limitation to the sinusoidal complexity of high-frequency wavefields and points to sequential FWI for high-resolution inversion.Source: §4 · p. 17

5 Conclusion

The conclusion introduces a sequential two-PINN WRI method for reconstructing wavefields and estimating velocity. Tests on Sigsbee2A and modified Marmousi yield smooth velocities with few iterations and frequencies that can initialize FWI.

5 Conclusion

  • The method uses two independent PINNs sequentially: one reconstructs scattered wavefields with wave-equation and data-fitting losses, and the other predicts velocity using the scattered wave equation.Source: §5 · p. 17
  • The authors analyze network architectures and report better results with a hierarchically decreasing network.Source: §5 · p. 17
  • On part of Sigsbee2A and modified Marmousi, PINN-based WRI produces reasonably smooth velocity with very limited iterations and frequencies, suitable as an FWI initial model.Source: §5 · pp. 17–18

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