A Deep Learning-based time shift objective function for Full Waveform Inversion

TL;DR

Cycle-skipping in Full Waveform Inversion is mitigated by a neural network–based time-shift estimator that predicts tracewise shifts $\boldsymbol{\tau}_p(t)$ to form a differentiable misfit $\mathcal{J}(\mathbf{m}) = \tfrac{1}{2} \| \boldsymbol{\tau}_p(t) \|_2^2$. The method uses a two-stage framework: supervised training of a 1D UNet on synthetic Gaussian-sum time shifts $\boldsymbol{\tau}(t) = \sum_{i=0}^{N} \alpha_i e^{-\frac{(t - \mu_i)^2}{2\sigma_i^2}}$ and deployment within the FWI adjoint-state workflow, enabling smooth gradients and fast adjoint-source computation. Losses combine $L_{\text{MAE}}$ for the time shift and $L_{\text{MSE}}$ for data consistency, with a final fine-tuning step to handle phase/waveform differences. Validation on Marmousi II and Chevron datasets shows inversion convergence comparable to SoftDTW but with orders-of-magnitude speedups and lower memory usage, highlighting the approach's potential for robust, scalable FWI initialization and broader waveform-matching applications.

Abstract

Full Waveform Inversion (FWI) is a powerful technique for estimating high-resolution subsurface velocity models by minimizing the discrepancy between modeled and observed seismic data. However, the oscillatory nature of seismic waveforms makes point-wise discrepancy measures highly prone to cycle-skipping, especially when the initial velocity model is inadequate. To address this challenge, various alternative misfit functions have been proposed in the literature, each with unique strengths and limitations. Dynamic Time Warping (DTW) is a popular technique in signal processing for aligning time series using dynamic programming. While a differentiable variant of DTW has been recently proposed, its use in FWI is hindered by high-frequency artifacts in the adjoint source and the substantial computational cost of gradient evaluations. In this study, we propose a neural network-based approach to learn the time shifts that align two time series in a supervised manner. The trained network is then utilized to compare traces from observed and modeled seismic data, offering a stable and computationally efficient alternative to DTW. Furthermore, the inherent differentiability of neural networks via backpropagation enables seamless integration into the FWI framework as a misfit function. We validate this approach on two synthetic datasets, namely the Marmousi model and the Chevron blind test dataset, demonstrating in both cases a similar convergence behavior to that of SoftDWT whilst drastically reducing the computational time of the adjoint source calculation.

Figures (3)

• Figure 1: Example of the process used to generate time-shift functions.
• Figure 2: Workflow for training the neural network to predict time shifts.
• Figure 3: Schematic of the proposed method for approximating time shifts within the FWI workflow.