Data-Driven Pseudo-spectral Full Waveform Inversion via Deep Neural Networks

TL;DR

This work introduces a data-driven, pseudo-spectral full waveform inversion (FWI) framework that recasts FWI as a neural-network mapping from seismic data to subsurface velocity, enabling a two-stage inversion via Fourier coefficients and velocity profiles. A five-module hour-glass DNN architecture is trained on large synthetic datasets, with two data-generating pipelines (FFT-based and continuous wavelet transform-based) and a Conv1D-Adadelta configuration emerging as the best performer. On Marmousi-like data, the data-driven pseudo-spectral FWI delivers improved imaging in deeper and over-thrust regions compared to classical FWI, while shallow inversions remain comparable. The approach shows robustness to noise and highlights potential as a priors-based accelerator or de-noising tool for traditional deterministic FWI, suggesting practical utility in complex geologies and limited-geometry scenarios. Future work includes expanding data generators, integrating real-data priors, and extending the framework to broader frequency bands and multi-azimuth settings.

Abstract

FWI seeks to achieve a high-resolution model of the subsurface through the application of multi-variate optimization to the seismic inverse problem. Although now a mature technology, FWI has limitations related to the choice of the appropriate solver for the forward problem in challenging environments requiring complex assumptions, and very wide angle and multi-azimuth data necessary for full reconstruction are often not available. Deep Learning techniques have emerged as excellent optimization frameworks. These exist between data and theory-guided methods. Data-driven methods do not impose a wave propagation model and are not exposed to modelling errors. On the contrary, deterministic models are governed by the laws of physics. Application of seismic FWI has recently started to be investigated within Deep Learning. This has focussed on the time-domain approach, while the pseudo-spectral domain has not been yet explored. However, classical FWI experienced major breakthroughs when pseudo-spectral approaches were employed. This work addresses the lacuna that exists in incorporating the pseudo-spectral approach within Deep Learning. This has been done by re-formulating the pseudo-spectral FWI problem as a Deep Learning algorithm for a data-driven pseudo-spectral approach. A novel DNN framework is proposed. This is formulated theoretically, qualitatively assessed on synthetic data, applied to a two-dimensional Marmousi dataset and evaluated against deterministic and time-based approaches. Inversion of data-driven pseudo-spectral DNN was found to outperform classical FWI for deeper and over-thrust areas. This is due to the global approximator nature of the technique and hence not bound by forward-modelling physical constraints from ray-tracing.

Figures (6)

• Figure 1: An example of a fully connected neural network with 2 hidden layers. All weights $w$ and bias $b$ are learned during the training phase. The $1$'s connected to each hidden layer represents bias nodes which help the neural network learn patterns by allowing the output of an activation function to be shifted. Adapted from Lucas2018.
• Figure 2: Pseudo-spectral FWI DNN frameworks to invert for Fourier Transform and CWT. $X$ is the input time domain, $Y$ is the output $v_p$ velocity and $\mathcal{M}$ is the Fourier domain, with magnitude $\zeta$ and phase $\phi$. Each component (blue or grey box) is a network.
• Figure 3: Comparison of velocity profiles with and without pre-processing.
• Figure 4: Comparison of data-driven DNN and Classical FWI on Marmousi.
• Figure 5: Improved FWI result with DNN as initial model.