Direct Waveform Inversion by Iterative Inverse Propagation

Seismic waves are the most sensitive probe of the Earth’s interior we have. With the dense data sets available in exploration, images of subsurface structures can be obtained through processes such as migration. Unfortunately, relating these surface recordings to actual Earth properties is non-trivial. Tomographic techniques use only a small amount of the information contained in the full seismogram and result in relatively low resolution images. Other methods use a larger amount of the seismogram but are based on either linearization of the problem, an expensive statistical search over a limited range of models, or both. We present the development of a new approach to full waveform inversion, i.e., inversion which uses the complete seismogram. This new method, which falls under the general category of inverse scattering, is based on a highly non-linear Fredholm integral equation relating the Earth structure to itself and to the recorded seismograms. An iterative solution to this equation is proposed. The resulting algorithm is numerically intensive but is deterministic, i.e., random searches of model space are not required and no misfit function is needed. Impressive numerical results in 1D are shown for several test cases.

💡 Research Summary

The paper introduces a novel deterministic approach to full‑waveform inversion (FWI) that bypasses the conventional reliance on misfit functions, gradient calculations, and stochastic model searches. By starting from the acoustic (or elastic) wave equation, the authors recast the relationship between the Earth’s subsurface parameters and the recorded seismograms as a highly non‑linear Fredholm integral equation of the second kind. This integral formulation explicitly contains the self‑scattering of the medium, thereby naturally accounting for multiple scattering, mode conversion, and complex propagation paths that are usually linearized or ignored in traditional tomography.

The core of the method is an “iterative inverse propagation” scheme. An initial reference model—typically a homogeneous velocity—is used to forward‑model synthetic seismograms. The difference (residual) between synthetic and observed data is then back‑propagated through the same integral kernel, effectively projecting the data residual into the model space as a correction term. This back‑propagation is mathematically equivalent to applying the adjoint of the Fredholm operator, and the resulting correction is added to the current model. The updated model is again forward‑modeled, and the cycle repeats until the residual diminishes below a prescribed tolerance.

Because the algorithm directly solves the integral equation through successive Neumann‑type expansions, it does not require an explicit objective function or a line‑search strategy. Convergence is driven solely by the reduction of the data residual, and the process is deterministic: each iteration follows a uniquely defined path dictated by the physics of wave propagation. The authors demonstrate the method on a suite of one‑dimensional synthetic tests, including smooth velocity gradients, sharp discontinuities, and layered media with strong impedance contrasts. In all cases, convergence is achieved within a modest number of iterations (typically 5–10), and the recovered velocity profiles match the true models to within the discretization error.

The paper also discusses computational considerations. Each iteration involves a full forward simulation and a corresponding inverse propagation, which are both computationally intensive, especially when extended to two‑ or three‑dimensional domains. However, the authors argue that the deterministic nature of the algorithm eliminates the need for large ensembles of forward runs that are typical in stochastic or global‑optimization FWI, potentially offsetting the per‑iteration cost. They also note that the method’s performance hinges on accurate knowledge of the source wavelet and on sufficient spatial sampling of receivers; noise and incomplete coverage can degrade convergence, suggesting that regularization (e.g., Tikhonov, total variation) may be incorporated into the integral kernel to improve robustness.

Conceptually, the approach bears resemblance to Marchenko‑based inverse scattering techniques, which also employ backward‑propagated fields to retrieve internal reflectivity without explicit imaging conditions. The key distinction is that the present method utilizes the full recorded wavefield—including both transmitted and reflected energy—rather than relying solely on reflected arrivals. This broader data usage promises higher resolution but also imposes stricter data‑quality requirements.

In summary, the authors present a mathematically elegant and physically grounded framework for FWI that replaces the traditional misfit‑gradient paradigm with an iterative solution of a non‑linear Fredholm equation. The method’s deterministic convergence, inherent handling of multiple scattering, and avoidance of stochastic searches make it an attractive alternative for high‑resolution seismic imaging. While the current demonstration is limited to 1‑D synthetic examples, the authors outline a clear path toward 2‑D/3‑D implementation, emphasizing the need for efficient parallel solvers and robust regularization strategies. If successfully scaled, this inverse‑propagation FWI could significantly enhance the fidelity of subsurface models in exploration seismology, earthquake seismology, and related geophysical disciplines.