Leveraging Deep Operator Networks (DeepONet) for Acoustic Full Waveform Inversion (FWI)

TL;DR

This paper addresses the computational and non-uniqueness challenges of Full Waveform Inversion by employing DeepONet, a neural operator that maps surface seismic traces to the subsurface velocity field. The approach blends a branch-trunk DeepONet architecture with a UNet-enhanced branch to capture multi-scale features and enforces the operator via a loss on velocity outputs, while also proposing a hybrid workflow where DeepONet outputs initialize conventional FWI. Results across noise-free, noisy, and incomplete data demonstrate competitive accuracy to InversionNet on clean data and superior robustness to noise and missing measurements, with additional evidence that the DeepONet output can accelerate FWI convergence. The study highlights practical implications for faster, more robust seismic inversion and suggests future work on improving discontinuity resolution, advanced training strategies, and transfer learning to broaden generalization to diverse velocity fields.

Abstract

Full Waveform Inversion (FWI) is an important geophysical technique considered in subsurface property prediction. It solves the inverse problem of predicting high-resolution Earth interior models from seismic data. Traditional FWI methods are computationally demanding. Inverse problems in geophysics often face challenges of non-uniqueness due to limited data, as data are often collected only on the surface. In this study, we introduce a novel methodology that leverages Deep Operator Networks (DeepONet) to attempt to improve both the efficiency and accuracy of FWI. The proposed DeepONet methodology inverts seismic waveforms for the subsurface velocity field. This approach is able to capture some key features of the subsurface velocity field. We have shown that the architecture can be applied to noisy seismic data with an accuracy that is better than some other machine learning methods. We also test our proposed method with out-of-distribution prediction for different velocity models. The proposed DeepONet shows comparable and better accuracy in some velocity models than some other machine learning methods. To improve the FWI workflow, we propose using the DeepONet output as a starting model for conventional FWI and that it may improve FWI performance. While we have only shown that DeepONet facilitates faster convergence than starting with a homogeneous velocity field, it may have some benefits compared to other approaches to constructing starting models. This integration of DeepONet into FWI may accelerate the inversion process and may also enhance its robustness and reliability.

Figures (17)

• Figure 1: A schematic representation of the problem setup illustrating the forward problem (predicting seismic waveforms at receivers for a given subsurface velocity model and source) and the inverse problem (estimating the subsurface velocity field from recorded seismic waveforms). The forward problem models the data generation process, while the inverse problem focuses on reconstructing the velocity profile from the recorded seismic data. In the present study, our focus is the inverse problem of velocity field prediction using recorded seismic waveforms.
• Figure 2: Schematic diagram of the proposed DeepONet architecture: considered in the present inverse problem. The trunk network consists of a DNN (Fully Connected feed-forward Network) with two input neurons, which take the coordinate $(x, y)$ as input where the output velocity field needs to be predicted. The branch network consists of a UNet block followed by multiple CNN layers and a DNN (Fully Connected feed-forward Network). We also added a skip connection for the input of the branch after the UNet (concatenated with the output of UNet in the channel). The input to the branch network is the pre-processed (e.g. gain function) seismic waveforms of full-waveform acoustic data for all sources recorded at stations located at the surface. The size of the input is $n_\text{station}\times n_\text{time}\times n_\text{source}$, where $n_\text{source}$ is the number of channels in the UNet / CNN. The predicted velocity field is given by the dot product of the output of the branch and trunk as shown in \ref{['Eq:DeepONet']}.
• Figure 3: Schematic showing a hybrid method of DeepONet and existing FWI. (a) A schematic showing a conventional FWI scheme, where an initial velocity field is predicted using a method like travel-time analysis for the FWI process. (b) A schematic showing the proposed operator-infused hybrid FWI scheme. In the first step, we predict the velocity field using the recorded seismic waveforms using a trained DeepONet. This predicted velocity field is considered as the informed velocity for existing implementation of FWI workflow, e.g., here we used the method proposed by virieux2009overviewvirieux2009overview as inversion traditional FWI
• Figure 4: Scatter plot of relative $L_2$ error, Clean seismic waveforms: Scatter plot of relative $L_2$ error for the predicted individual test samples when predicted using DeepONet and InversionNet. The x-axis shows the individual 6000 test samples and the y-axis shows the corresponding relative $L_2$ error. The overall result is comparable. We observed that in many cases, the error for DeepONet is smaller than that of InversionNet and vice versa. The violin plots for the relative $L_2$ error are shown in \ref{['Fig:Clean data:Violin']}, showing the qualitative distribution of the relative $L_2$ error.
• Figure 5: Violin plot of relative $L_2$ error, Clean seismic waveforms: Violin plot of relative $L_2$ error for predicted velocity model for test samples when predicted using DeepONet and InversionNet (InvNet).