Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness

TL;DR

The study tackles the generalization and robustness challenges of data-driven full waveform inversion (FWI) by introducing Fourier-DeepONet, a Fourier-enhanced DeepONet whose trunk is conditioned on source parameters and whose decoder employs Fourier/U-Fourier layers to map seismic data and source variations to velocity models. It constructs three benchmark datasets (FWI-F, FWI-L, FWI-FL) with varied frequencies and locations to probe generalization beyond fixed-source settings. Across these datasets, Fourier-DeepONet achieves higher accuracy and markedly better robustness to noise, missing traces, and source noise than pretrained or retrained baselines such as InversionNet and VelocityGAN. The results demonstrate strong generalizability to diverse source configurations, with architectural choices (one Fourier layer plus three U-Fourier layers and multiplication-based fusion) balancing accuracy and efficiency. The work advances practical data-driven FWI and suggests future extensions to non-Ricker sources and varying receiver placements, including unsupervised physics-informed approaches.

Abstract

Full waveform inversion (FWI) infers the subsurface structure information from seismic waveform data by solving a non-convex optimization problem. Data-driven FWI has been increasingly studied with various neural network architectures to improve accuracy and computational efficiency. Nevertheless, the applicability of pre-trained neural networks is severely restricted by potential discrepancies between the source function used in the field survey and the one utilized during training. Here, we develop a Fourier-enhanced deep operator network (Fourier-DeepONet) for FWI with the generalization of seismic sources, including the frequencies and locations of sources. Specifically, we employ the Fourier neural operator as the decoder of DeepONet, and we utilize source parameters as one input of Fourier-DeepONet, facilitating the resolution of FWI with variable sources. To test Fourier-DeepONet, we develop three new and realistic FWI benchmark datasets (FWI-F, FWI-L, and FWI-FL) with varying source frequencies, locations, or both. Our experiments demonstrate that compared with existing data-driven FWI methods, Fourier-DeepONet obtains more accurate predictions of subsurface structures in a wide range of source parameters. Moreover, the proposed Fourier-DeepONet exhibits superior robustness when handling data with Gaussian noise or missing traces and sources with Gaussian noise, paving the way for more reliable and accurate subsurface imaging across diverse real conditions.

Figures (16)

• Figure 1: Schematic illustration of forward modeling and FWI. Forward modeling generates seismic data from velocity maps, while the purpose of FWI is to infer velocity maps from seismic data measurements. The five red stars in the velocity map are the five point sources utilized to generate seismic data. The seismic data can be clean, noisy, or with missing traces.
• Figure 2: Examples of velocity maps (top row) and seismic data (bottom row) for FVB, CVA, CFA and STA datasets. The seismic data comes from the source C in Fig. \ref{['fig:fwi']}.
• Figure 3: Fourier-DeepONet architecture. (A) Branch net and trunk net are two linear transformations lifting inputs to high dimensional space. Green circle represents the merger operation which denotes point-wise multiplication. (B) Fourier layer, adapted from li2020fourier. (C) U-Fourier layer, adapted from wen2022u. (D) Projection layer $Q$. (E) Shapes of outputs $\textbf{z}_i$ in one channel, $i \in \{0, 1, 2, 3, 4\}$.
• Figure 4: Performance of three methods on four datasets (FVB, CVA, CFA, and STA) of different sources frequencies. Fourier-DeepONet is trained on FWI-F; InversionNet and VelocityGAN are trained on OpenFWI. Fourier-DeepONet performs better for all datasets across a wide range of frequencies, while InversionNet and VelocityGAN can only give accurate predictions when the frequency is 15 Hz.
• Figure 5: Examples of velocity maps predicted by three methods on four datasets with source frequencies from 13 to 17 Hz. (A) Ground truth of examples from FVB, CVA, CFA, and STA datasets. (B) Predictions for the FVB case. (C) Predictions for the CVA case. (D) Predictions for the CFA case. (E) Predictions for the STA case. The seismic data comes from the source C in Fig. \ref{['fig:fwi']}.