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

Min Zhu; Shihang Feng; Youzuo Lin; Lu Lu

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

Min Zhu, Shihang Feng, Youzuo Lin, Lu Lu

arXiv:2305.17289v2May 26, 2023cs.LGphysics.comp-phphysics.geo-ph

TL;DR

FWI requires accurate subsurface imaging despite difficult optimization and mismatches between training and field source functions. The paper introduces Fourier-DeepONet, which combines a Fourier-based DeepONet decoder with source-parameter inputs and evaluates it on variable-source benchmarks. It reports higher accuracy across source parameters and stronger robustness to noisy or incomplete data than existing data-driven baselines.

Problem
Data-driven FWI remains limited by source-function discrepancies and insufficiently studied robustness to noisy or incomplete inputs.
Method
Fourier-DeepONet combines a Fourier neural operator decoder with source parameters as inputs and is evaluated on three variable-source FWI benchmark datasets.
Results
Fourier-DeepONet is more accurate than InversionNet and VelocityGAN across varying source parameters and remains robust to Gaussian noise and missing traces.
Takeaways & Limitations
The method supports more reliable subsurface imaging across diverse source parameters and real-condition perturbations.
Takeaways & Limitations
The study leaves non-Ricker source wavelets and source-amplitude inputs for future investigation.

Abstract

from arXiv · show

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.

1 Introduction

FWI is a computationally demanding, non-convex inverse problem, and data-driven methods face unresolved generalization and robustness challenges, especially when seismic sources vary. Fourier-DeepONet addresses these issues by combining DeepONet with a Fourier-based decoder and source-parameter inputs.

FWI is an ill-posed, nonlinear, computationally complex non-convex optimization problem for imaging subsurface structures.
Data-driven FWI has improved efficiency and accuracy across CNN, RNN, and GAN architectures, but source generalization remains largely unexplored.
Small source-wavelet disturbances can produce large discrepancies in inverted models, making source-wavelet generalization crucial for successful inversion.
Robustness to noise or outliers is another underexplored FWI challenge, with accuracy–robustness trade-offs still requiring study.
Fourier-DeepONet uses a U-FNO decoder and source parameters as trunk inputs to support clearer predictions with variable source frequencies and locations.
Three benchmark datasets—FWI-F, FWI-L, and FWI-FL—evaluate Fourier-DeepONet against InversionNet and VelocityGAN across varying source conditions.

2 Full waveform inversion

FWI infers subsurface velocity maps from seismic measurements governed by an acoustic wave equation, using multiple surface-source recordings. The section describes OpenFWI and three new datasets that vary source frequency, location, or both.

2.1 Problem setup: The acoustic formulation uses a two-dimensional wave equation with pressure wavefield p, velocity map c, source function s, time t, and spatial coordinates.The source is a near-surface point source generated by a Ricker wavelet.
2.1 Problem setup: FWI learns subsurface velocity maps from surface seismic data generated by multiple source functions.The forward model maps velocity maps and source functions to pressure wavefields, while FWI reverses this mapping using surface measurements.
2.1 Problem setup: The problem setup includes five point sources, surface receivers spaced 10 m apart, 0.001-second temporal resolution, and clean, noisy, or missing-trace seismic data.The total recording time is one second.
2.2.1 OpenFWI: OpenFWI supplies velocity maps from four datasets—FVB, CVA, CFA, and STA—with training/testing sizes of 24K/6K, 24K/6K, 48K/6K, and 60K/7K.These datasets span flat layers, curved layers, faults, and style-transferred geological structures.
2.2.2 FWI-F, FWI-L, and FWI-FL: The new FWI-F, FWI-L, and FWI-FL datasets vary source frequency, source location, or both, whereas OpenFWI keeps source parameters fixed.FWI-F uses frequencies from 5 to 25 Hz, while FWI-L permits source-location variation and FWI-FL combines both variations.

3 Methods

Fourier-DeepONet extends DeepONet for FWI by encoding seismic data and source parameters separately, then decoding their merged representation with Fourier-based layers. The architecture targets variable-source inversion while producing clearer predictions than vanilla inner-product decoding.

3.1 Vanilla DeepONet: DeepONet approximates operators between function spaces using a branch network for discretized input functions and a trunk network for output coordinates.The branch network receives pointwise evaluations of the input function, while the trunk network receives output coordinates.
3.2 Fourier-DeepONet in parameter spaces: Fourier-DeepONet uses seismic data in the branch network and source parameters in the trunk network, defining FWI in parameter space rather than the spatial-temporal domain.Source parameters represent either frequency or source locations.
3.2 Fourier-DeepONet in parameter spaces: Each source’s seismic data is discretized at 70 surface receivers over 1000 recording times, producing a tensor with shape (1000, 70, 5).The five source functions are sampled at predetermined receiver positions and times.
3.2 Fourier-DeepONet in parameter spaces: The branch and trunk outputs are merged by pointwise multiplication after tensor broadcasting before passing through the merger network.The branch and trunk transformations increase the channel dimension to C = 64.
3.2 Fourier-DeepONet in parameter spaces: The merger network contains one Fourier layer followed by three U-Fourier layers, with this configuration selected for its balance of performance and efficiency.The Fourier and U-Fourier layers use FFT-based operations, U-Net layers, weight matrices, and biases.

4 Results

The experiments evaluate Fourier-DeepONet across varying source frequencies, locations, and combined source parameters, followed by robustness tests and architecture analysis. Model performance is assessed using normalized data and velocity maps with four reconstruction metrics.

4 Results: Experiments cover varying source frequencies, locations, or both, followed by robustness evaluation and architecture analysis.These tests are organized across Sections 4.1–4.5.
4 Results: Performance is evaluated with MAE, RMSE, SSIM, and L2 relative error after normalizing seismic data and velocity maps to the range [-1, 1].MAE and RMSE quantify differences between predicted and actual values.

4.1 Source frequencies

Fourier-DeepONet generalizes across variable source frequencies, whereas pretrained and improved baseline models are most reliable near their training frequency. Across datasets, it maintains accurate velocity-map predictions over broad frequency ranges.

Comparison with pretrained baseline models: Source frequency affects FWI resolution and convergence, while fixed-frequency datasets limit evaluation of generalization to realistic variable-frequency conditions.The study therefore uses FWI-F with frequencies randomly distributed from 5 Hz to 25 Hz.
Comparison with pretrained baseline models: At 10 Hz on FVB, Fourier-DeepONet has about 10% L2 relative error, while pretrained InversionNet and VelocityGAN reach 150%.At 15 Hz, all three models have L2 relative errors around 10%.
Comparison with pretrained baseline models: When frequency changes, the pretrained baselines tend to underestimate velocity at higher frequencies, overestimate it at lower frequencies, and sometimes split layers incorrectly.Fourier-DeepONet remains accurate in the shown cases.
Comparison with improved baseline models: Fourier-DeepONet accurately predicts velocity maps from 5 to 25 Hz, outperforming both improved baseline models across all datasets and frequencies.The improved baselines were also trained on FWI-F, yet Fourier-DeepONet remained superior throughout the tested range.

4.2 Source locations

Fourier-DeepONet generalizes to varying source locations, while the baseline models are accurate mainly when sources retain their default uniform positions. The source-location experiments evaluate this behavior using controlled shifts and realistic FWI-L data.

A particular test case for demonstration: Source-location variability changes subsurface illumination and can affect velocity-model accuracy, coverage, and inversion convergence.Uniform source distributions may be impractical under topographical, environmental, or logistical constraints.
A particular test case for demonstration: The source-location test reduces five independent source positions to one variable by moving four sources equidistantly toward the fixed central source.The variable is the shift distance from the default uniform locations.
A particular test case for demonstration: Fourier-DeepONet performs well across source locations, whereas InversionNet and VelocityGAN are accurate only at zero shift and deteriorate as locations deviate.The result is reported across the FVB, CVA, CFA, and STA datasets.
A particular test case for demonstration: On FWI-L datasets, Fourier-DeepONet significantly outperforms InversionNet and VelocityGAN across all datasets.Table 3 summarizes the comparison, with bold values marking the best performance for each dataset.

4.3 Source frequencies and locations

The combined source-frequency and source-location test evaluates Fourier-DeepONet on FWI-FL with both parameters varying. It produces satisfactory velocity maps and accuracy similar to its separate-variable evaluations.

Source frequencies and locations: Baseline models are not tested on FWI-FL because their performance was poor on both the FWI-F and FWI-L datasets.The combined-variable evaluation therefore reports Fourier-DeepONet alone.
Source frequencies and locations: Fourier-DeepONet provides satisfactory predictions with source frequencies from 10 to 20 Hz and five sources randomly distributed on the surface.The examples are shown for frequencies of 10, 15, and 20 Hz.

4.4 Robustness evaluation

Fourier-DeepONet is more robust than the baseline models to noisy inputs, missing seismic traces, and noisy source wavelets. Its predictions remain accurate under tested disturbances that cause the baselines to fail or degrade substantially.

Testing input data with noise: With input-noise standard deviation up to 0.1, Fourier-DeepONet’s accuracy is almost unaffected, while InversionNet and VelocityGAN fail.Fourier-DeepONet performs much better across all four datasets and tested noise levels.
Testing input data with missing traces: Fourier-DeepONet significantly outperforms the baselines as missing traces increase from 5 to 35 of 70 receivers.Missing traces are represented by zero-filled seismic-data entries, with the strongest advantage reported on CVA, CFA, and STA.
Ricker wavelet source with noise: Under noisy Ricker-wavelet sources, Fourier-DeepONet consistently outperforms both baselines and still provides satisfactory predictions at standard deviation 0.1.Both InversionNet and VelocityGAN fail under that condition.

4.5 Discussion about the architecture of Fourier-DeepONet

Fourier-DeepONet’s architecture choices trade accuracy against computational cost. Pointwise multiplication is retained as the merger, while U-Fourier layers improve high-frequency predictions over Fourier-only layers and Fourier decoding sharpens interfaces over vanilla DeepONet.

Comparison of different merger operations: Pointwise multiplication after tensor broadcasting performs slightly better than addition and concatenation, although the differences are not significant.The study therefore adopts multiplication, following the vanilla DeepONet convention.
Comparison of Fourier and U-Fourier layers: U-Fourier layers capture higher-frequency features missed by Fourier layers, improving accuracy while increasing computational cost.Fourier-only networks are faster because they omit the computationally expensive U-Net component.
Comparison of Fourier and U-Fourier layers: Fourier-only networks perform worst and produce blurry interfaces because truncating higher-frequency modes loses sharp structural features.The prediction examples are shown in Figure 15.
Comparison between Fourier-DeepONet and vanilla DeepONet: Fourier-DeepONet outperforms vanilla DeepONet across all FWI-L datasets, while vanilla predictions blur interfaces in FVB, CVA, and CFA.Vanilla DeepONet remains visually satisfactory on the smoother STA dataset.

5 Conclusions

Fourier-DeepONet improves FWI accuracy and robustness across varying source parameters, outperforming baseline models under broader source conditions. Its current scope uses Ricker-wavelet sources, with non-Ricker sources and additional receiver-related extensions left for future work.

5 Conclusions: Fourier-DeepONet is more accurate than InversionNet and VelocityGAN across a wide range of source parameters and is more robust to noisy or incomplete inputs.When source frequencies or locations vary, the baselines’ L2 relative errors are about 10 times greater than Fourier-DeepONet’s.
5 Conclusions: The current study uses Ricker-wavelet sources, while sources with other shapes remain future work.The authors propose generating seismic data with varied source shapes for subsequent evaluation.
5 Conclusions: Fourier-DeepONet outperforms vanilla DeepONet on all FWI-L datasets, with clearer interfaces in predictions.The comparison is reported in Table 6 and illustrated in Figure 16.
5 Conclusions: Future extensions will use receiver locations or source amplitudes as trunk inputs and may reduce training data through unsupervised learning.These directions are proposed as extensions of the current framework.