Full Waveform Inversion (FWI) Overview

• Full Waveform Inversion (FWI) is a nonlinear, PDE-constrained technique that recovers subsurface parameters by minimizing discrepancies between observed and simulated waveforms.
• It employs adjoint-state methods and gradient-based optimization (e.g., L-BFGS, Adam) to iteratively update model parameters through forward and backward PDE solves.
• Recent developments integrate extended formulations, operator learning, and probabilistic approaches to address nonlinearity, cycle skipping, and incomplete data challenges.

Searching arXiv for recent and foundational Full Waveform Inversion papers to ground the article. Full Waveform Inversion (FWI) is a nonlinear PDE-constrained inverse problem that estimates subsurface or material parameters by minimizing the discrepancy between recorded waveforms and waveforms simulated from a forward model. In geophysics it is described as a high-resolution technique for evaluating physical parameters and constructing subsurface models, while in broader wave-based imaging it reconstructs material fields from sparsely measured data obtained by wave propagation (Lima et al., 2022, Singh et al., 2024). Across acoustic, elastic, borehole, and non-destructive testing settings, FWI combines a wave-equation solver, an objective functional, and gradient-based or sampling-based updates to infer velocity, slowness, density, Lamé parameters, or related parameterizations from full seismograms rather than reduced kinematic picks (Rojas-Gómez et al., 2020, Tang et al., 2020).

1. Mathematical definition

At its most common level of abstraction, FWI seeks a model $m$ such that synthetic data $d_{\mathrm{sim}}$ predicted by a wave equation match observed data $d_{\mathrm{obs}}$. A standard least-squares misfit is

$$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$

Here, $s$ indexes sources, $r$ indexes receivers, and $T$ is the recording duration. In acoustic settings a common parameterization is squared slowness, $m(x)=1/c(x)^2$, with $c(x)$ the wave speed; in elastic settings the parameter vector may include $V_p$, $d_{\mathrm{sim}}$0, density $d_{\mathrm{sim}}$1, or the Lamé parameters $d_{\mathrm{sim}}$2 and $d_{\mathrm{sim}}$3 (Singh et al., 2024, Lima et al., 2022, Rojas-Gómez et al., 2020).

A prototypical constant-density acoustic forward model is

$d_{\mathrm{sim}}$4

with zero initial conditions and absorbing boundaries. The predicted data are samples of the wavefield at receiver positions. In isotropic elasticity, the governing equation may be written as

$d_{\mathrm{sim}}$5

with the standard relations

$d_{\mathrm{sim}}$6

These formulations define the forward operator $d_{\mathrm{sim}}$7 or $d_{\mathrm{sim}}$8 that maps model parameters to synthetic waveforms (Rojas-Gómez et al., 2020, Lima et al., 2022).

The definition of FWI is not restricted to surface seismic velocity inversion. The same structure appears in borehole acoustic logging, where the inversion is posed in cylindrical coordinates for a fluid-filled borehole, and in defect imaging, where a scalar wave equation is used to detect high-contrast voids in a specimen (Tang et al., 2020, Bürchner et al., 2022).

2. Adjoint-state gradients and optimization

The computational core of FWI is the adjoint-state method. For each source, one first solves the forward PDE to obtain the wavefield and predicted data, then solves an adjoint PDE driven by the receiver residuals, and finally correlates forward and adjoint fields to obtain the gradient. In a constant-density acoustic formulation with $d_{\mathrm{sim}}$9, an adjoint equation takes the form

$d_{\mathrm{obs}}$0

integrated backward in time, where $d_{\mathrm{obs}}$1. A corresponding gradient is

$d_{\mathrm{obs}}$2

Equivalent acoustic expressions also appear in terms of $d_{\mathrm{obs}}$3 and the adjoint field $d_{\mathrm{obs}}$4, and elastic formulations yield componentwise kernels for $d_{\mathrm{obs}}$5, $d_{\mathrm{obs}}$6, and $d_{\mathrm{obs}}$7 (Singh et al., 2024, Londoño et al., 2022, Rojas-Gómez et al., 2020).

The optimization loop is then built around repeated forward and adjoint solves. The literature represented here uses gradient descent, L-BFGS, Barzilai–Borwein, Adam, RMSProp, and Gauss–Newton or truncated Newton updates, depending on whether the model is parameterized directly on the grid or through a neural network (Londoño et al., 2022, Singh et al., 2024, Goren et al., 2024). In borehole acoustic FWI, the inversion updates $d_{\mathrm{obs}}$8 and $d_{\mathrm{obs}}$9 through adjoint-state gradients and then recomputes $$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$0 and $$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$1 from

$$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$2

with density held fixed (Tang et al., 2020).

This forward–adjoint structure explains the characteristic computational profile of FWI. Each iteration requires one or more PDE solves per source and frequency, and the dominant cost is almost always wave simulation rather than the algebra of the outer optimization. A large fraction of current methodological work therefore targets either the misfit landscape or the cost of the wave solver.

3. Nonlinearity, cycle skipping, and extended formulations

FWI is repeatedly characterized as nonlinear and ill-posed. Multiple models can fit band-limited, noisy data, and the problem becomes especially difficult when the initial model is inaccurate or low frequencies are missing (Lima et al., 2022, Treister et al., 2016). In the conventional $$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$3 formulation, a sufficient condition for cycle skipping is

$$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$4

so phase errors beyond half a period can trap the inversion in a local minimum (He et al., 2 Apr 2025).

Several methodological families address this difficulty by altering either the data term or the constraint structure. Joint inversion with travel time tomography supplements waveform fitting with first-arrival information that carries low-frequency features, and the cited work combines this with a two-stage schedule and high-order regularization to promote low spatial frequencies in the early iterations (Treister et al., 2016). Alternative misfits include the multiscale structural similarity index measure, where the loss is built from local luminance, contrast, and structure terms over Gaussian windows at scales $$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$5, $$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$6, and $$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$7, and is combined with anisotropic total $$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$8-variation regularization to preserve sharp interfaces (He et al., 2 Apr 2025).

A second major line of development is model extension. Extended FWI can be written as

$$J(m) = \frac{1}{2} \sum_s \sum_r \int_0^T \left| d_{\mathrm{obs}}(s,r,t) - d_{\mathrm{sim}}(s,r,t;m) \right|^2 dt.$$9

so that uncertainty in the physics is explicitly penalized rather than excluded. One reformulation shows that this is equivalent to a conventional FWI objective with a medium-dependent residual weight,

$s$0

which gives a direct interpretation of why extended formulations can reduce nonlinearity even when the physics is modeled accurately (Leeuwen, 2019).

A related construction is full waveform inversion by model extension, or FWIME, which augments the predicted data by an extended modeling term and removes it progressively through an annihilator. Its joint objective is

$s$1

and variable projection eliminates $s$2 to yield a reduced objective in $s$3 alone. The stated role of this construction is to combine the robustness of wave-equation migration velocity analysis with the high resolution of FWI, thereby mitigating cycle skipping and reducing sensitivity to initial models and low-frequency long-offset data (Barnier et al., 2022).

A third family uses augmented Lagrangian and multiplier methods. In one reduced-space formulation, the augmented Lagrangian is

$s$4

so the primal step is standard FWI with modified data and the dual step updates the Lagrange multipliers. Multiplier-oriented and dual formulations emphasize that Lagrange multipliers can be interpreted as effective scattering or source-extension terms and that accurate multiplier estimation improves the linearization of the inverse problem (Gholami et al., 2021, Gholami, 2023, Aghazade et al., 2024).

Formulation	Representative objective	Stated role
Conventional FWI	$s$5	Direct data fitting
Extended FWI	$s$6	Relax PDE exactness
FWIME	$s$7	Combine data correction and annihilation

A common misconception is that all robust FWI variants merely change the optimizer. The formulations above instead modify the inverse problem itself: they alter the residual metric, relax the state equation, add extended variables, or transfer part of the nonlinear burden to multipliers and annihilators.

4. Parameterizations, geometry, and numerical solvers

FWI is highly sensitive to parameterization. Besides direct grid-based updates of velocity or squared slowness, several works parameterize the model through auxiliary fields or implicit functions. In NN-based FWI, the material field is represented as

$s$8

and the wave-equation gradient $s$9 is propagated to the network weights by

$r$0

This replaces a high-dimensional pixel basis with a structured function class, which the cited work uses to improve robustness and reconstruction quality (Singh et al., 2024).

For high-contrast void imaging, an immersed-boundary parameterization introduces a dimensionless scaling function $r$1 with $r$2 in intact material and $r$3 in voids. In the density-scaling variant,

$r$4

so the forward PDE becomes

$r$5

The associated Fréchet kernel is

$r$6

The cited comparison concludes that scaling density with $r$7 is the most promising parameterization for voids (Bürchner et al., 2022).

The same emphasis on geometry appears in isogeometric finite-cell FWI, where the physical domain is embedded in a larger computational domain through an indicator function $r$8 and the unknown defect field is described by a separate scaling $r$9. The semi-discrete system

$T$0

is paired with an adaptive multi-resolution description of $T$1, refined locally by an image-processing-based refinement indicator. The paper stresses that the resolution of the unknown material density is decoupled from the knot-span grid, which is significant for memory-efficient defect identification in complex geometries (Bürchner et al., 2023).

On the solver side, the forward model may be posed in time or frequency domain. Frequency-domain FWI using the Helmholtz equation has been implemented with Gaussian RBF-FD weights on uniform hexagonal grids and a single-frequency multiscale continuation strategy on Marmousi and the 2004 BP velocity model (Londoño et al., 2022). A complementary line of work learns an encoder–solver CNN preconditioner for the discretized Helmholtz operator and retrains it between inversion iterations as the medium evolves; the reported purpose is to keep forward simulations effective throughout the process because performance deteriorates when the medium changes and the preconditioner is not retrained (Goren et al., 2024). In borehole acoustics, the forward problem is formulated in 2D axisymmetric cylindrical coordinates to honor the borehole geometry and guided-wave physics, and a diagonal preconditioner is applied to suppress the dominance of Stoneley and pseudo-Rayleigh waves in the gradient (Tang et al., 2020).

5. Probabilistic, operator-learning, and hybrid approaches

A widespread misconception is that FWI is intrinsically deterministic. Bayesian acoustic FWI based on Hamiltonian Monte Carlo instead treats the inverse problem as posterior sampling with likelihood

$T$2

and Hamiltonian

$T$3

In the cited reflection setup, a geometry-informed depth-dependent diagonal mass matrix improves sampling efficiency, increases acceptance, and reveals posterior mean, variance, and skewness rather than only a single inverted model (Lima et al., 2022).

Neural and hybrid variants are equally diverse. A physics-consistent data-driven inversion method for elastic FWI uses an encoder–decoder CNN trained with mean absolute error,

$T$4

and augments its training set by passing predicted models through the elastic forward solver to create physics-consistent pairs. In synthetic Kimberlina experiments, this adaptive augmentation improves inversion accuracy and generalization across leak-plume distributions (Rojas-Gómez et al., 2020).

Operator learning has been used in multiple ways. A DeepONet framework learns a mapping from multishot shot gathers to a continuous 2D acoustic velocity field,

$T$5

with a branch network processing waveforms and a trunk network processing coordinates. Its output can be used as a starting model for adjoint-based FWI, and the paper reports faster convergence than a homogeneous starting model (Nath et al., 14 Apr 2025). Physics-informed waveform inversion with pretrained Fourier neural operators instead freezes a neural operator $T$6 that predicts scattered wavefields and optimizes the velocity through a total loss

$T$7

where the physics term is a Helmholtz residual evaluated on the predicted wavefield. The stated effect is reduced noise and fewer artifacts than a vanilla operator-based inversion (Huang et al., 10 Sep 2025).

Recent data-driven work has also targeted the scale of the model and the training corpus. One large-model study reports coordinated scaling along model capacity, data diversity, and training strategy, using a non-causal transformer of about one billion parameters, a diffusion-augmented corpus of about five million synthetic velocity–seismic pairs, RL post-training, and physics-guided latent refinement

$T$8

Across six benchmarks the paper reports an overall SSIM improvement from $T$9 to $m(x)=1/c(x)^2$0, which it presents as a narrowing of the generalization gap between OpenFWI-style training data and more realistic geology (Feng et al., 27 Feb 2026).

A different pretraining-free route is Self-Flow-Matching assisted FWI, which learns a time-conditioned transport field online during inversion. Its proposal model is

$m(x)=1/c(x)^2$1

with loss

$m(x)=1/c(x)^2$2

The reported effect is a stable coarse-to-fine evolution without a predefined noise schedule or Gaussian initialization, and better robustness than standard FWI, TV-regularized FWI, and deep image prior baselines on several synthetic benchmarks (Huang et al., 13 Mar 2026).

Physics-driven GAN formulations occupy yet another point in the design space. In one such approach, a U-Net predicts a velocity model from observed shot gathers, a traditional FWI block corrects it under the acoustic wave equation, and a discriminator enforces adversarial consistency at the data level. The cited Marmousi and Overthrust experiments report higher SSIM and SNR than a prior physics-informed GAN baseline, both in noise-free and noisy settings (Zhang et al., 16 Mar 2026).

6. Applications, benchmarks, and contemporary scope

FWI is now applied across a wide range of acquisition geometries and target classes. The benchmark literature represented here includes Marmousi, Marmousi II, the 2004 BP velocity model, SEG/EAGE Salt, Overthrust, Sigsbee, and SEAM Phase I in surface seismic inversion (Treister et al., 2016, Londoño et al., 2022, Feng et al., 27 Feb 2026). It also includes carbon sequestration monitoring at Kimberlina, borehole acoustic logging in a fluid-filled cylindrical borehole, and non-destructive testing of circular or elliptical voids in solids (Rojas-Gómez et al., 2020, Tang et al., 2020, Bürchner et al., 2022).

These applications expose recurring practical limits. Reflection-only acquisition provides strong shallow illumination and weaker sensitivity at depth, which appears explicitly in the posterior uncertainty maps of Bayesian FWI (Lima et al., 2022). Acoustic approximations neglect elastic effects, anisotropy, and attenuation, and several papers identify extension to elastic or 3D settings as future work (Rojas-Gómez et al., 2020, Huang et al., 10 Sep 2025). Computational burden remains central: each FWI iteration typically requires one forward and one adjoint solve per source, and 3D scaling sharply increases memory and runtime (Goren et al., 2024, Bürchner et al., 2023).

At the same time, the current literature suggests that FWI is no longer a single algorithmic template. It is a family of inversion methods organized around a common forward operator and waveform misfit but differentiated by parameterization, constraint handling, uncertainty representation, and solver design. A plausible implication is that contemporary practice is increasingly hybrid: travel-time information may regularize the macro model, extended or multiplier formulations may reshape the nonconvexity, learned operators may accelerate or initialize the inversion, and probabilistic samplers may quantify uncertainty where a single deterministic model is insufficient (Treister et al., 2016, Barnier et al., 2022, Lima et al., 2022, Nath et al., 14 Apr 2025).

The contemporary scope of FWI is therefore defined less by a single objective than by a shared principle: infer physically consistent models from full waveforms by coupling measurement operators, wave propagation, and optimization or sampling. Within that principle, current research spans deterministic least-squares inversion, extended and dual formulations, Bayesian sampling, operator learning, physics-informed surrogates, and multi-resolution immersed geometries, all directed toward the same central difficulty: recovering informative model updates from band-limited, incomplete, and computationally expensive wavefield data.