Caifeng Zou

Accurate and efficient wavefield modeling underpins seismic structure and source studies. Traditional methods comply with physical laws but are computationally intensive. Data‐driven methods, while opening new avenues for advancement, have yet to incorporate strict physical consistency. The principle of reciprocity is one of the most fundamental physical laws in wave propagation. We introduce the reciprocity‐enforced neural operator, a transformer‐based architecture for modeling seismic wave propagation that hard‐codes the reciprocity principle. The model leverages the cross‐attention mechanism and commutative operations to guarantee invariance under swapping source and receiver positions. Beyond improved physical consistency, the proposed architecture supports simultaneous realizations for multiple sources. This yields an order‐of‐magnitude inference speedup at a similar memory footprint over a conventional neural operator on a realistic multisource configuration. We demonstrate the functionality using the reciprocity relation for particle velocity fields under single forces. This architecture is also applicable to pressure fields under dilatational sources and travel‐time fields governed by the eikonal equation, paving the way for encoding more complex reciprocity relations.

Numerical simulations of seismic wave propagation are crucial for investigating velocity structures and improving seismic hazard assessment. However, standard methods such as finite difference or finite element are computationally expensive. Recent studies have shown that a new class of machine learning models, called neural operators, can solve the elastodynamic wave equation orders of magnitude faster than conventional methods. Full waveform inversion is a prime beneficiary of the accelerated simulations. Neural operators, as end-to-end differentiable operators, combined with automatic differentiation, provide an alternative approach to the adjoint-state method. State-of-the-art optimization techniques built into PyTorch provide neural operators with greater flexibility to improve the optimization dynamics of full waveform inversion, thereby mitigating cycle-skipping problems. In this study, we demonstrate the first application of neural operators for full waveform inversion on a real seismic data set, which consists of several nodal transects collected across the San Gabriel, Chino, and San Bernardino basins in the Los Angeles metropolitan area.

Numerical simulations of seismic wave propagation in heterogeneous 3-D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is particularly problematic for full-waveform inversion (FWI), which typically involves numerous runs of the forward process. In machine learning there has been considerable recent work in the area of operator learning, with a new class of models called neural operators allowing for data-driven solutions to partial differential equations. Recent work in seismology has shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3-D time domain equations may be prohibitive. In this study, we show that these limitations can be overcome by solving the wave equations in the frequency domain, also known as the Helmholtz equations, since the solutions for a set of frequencies can be determined in parallel. The 3-D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We use a Helmholtz neural operator for 2-D and 3-D elastic wave modelling, achieving two orders of magnitude acceleration compared to a baseline spectral element method. The neural operator accurately generalizes to variable velocity structures and can be evaluated on denser input meshes than used in the training simulations. We also show that when solving for wavefields strictly at the free surface, the accuracy can be significantly improved via a graph neural operator layer. In leveraging automatic differentiation, the proposed method can serve as an alternative to the adjoint-state approach for 3-D FWI, reducing the computation time by a factor of 350.

We show reflectivity cross sections for the San Gabriel, Chino, and San Bernardino basins north of Los Angeles (LA), California, determined from autocorrelations of ambient noise and teleseismic earthquake waves. These basins are thought to channel the seismic energy from earthquakes on the San Andreas fault to LA, and a more accurate model of their depth is important for hazard mitigation. We use the causal side of the autocorrelation function (ACF) to determine the zero‐offset reflection response. To minimize the smoothing effect of the source time function, we remove the common mode from the autocorrelation to reveal the zero‐offset reflection response. We apply this to 10 temporary nodal lines consisting of a total of 758 geophones with an intraline spacing of 250–300 m. We also show that the ACF from teleseismic events can provide illumination on the subsurface that is consistent with ambient noise. Both autocorrelation results compare favorably to receiver functions.