Unveiling the Mechanism of Continuous Representation Full-Waveform Inversion: A Wave Based Neural Tangent Kernel Framework

Full-waveform inversion (FWI) estimates physical parameters in the wave equation from limited measurements and has been widely applied in geophysical exploration, medical imaging, and non-destructive testing. Conventional FWI methods are limited by their notorious sensitivity to the accuracy of the initial models. Recent progress in continuous representation FWI (CR-FWI) demonstrates that representing parameter models with a coordinate-based neural network, such as implicit neural representation (INR), can mitigate the dependence on initial models. However, its underlying mechanism remains unclear, and INR-based FWI shows slower high-frequency convergence. In this work, we investigate the general CR-FWI framework and develop a unified theoretical understanding by extending the neural tangent kernel (NTK) for FWI to establish a wave-based NTK framework. Unlike standard NTK, our analysis reveals that wave-based NTK is not constant, both at initialization and during training, due to the inherent nonlinearity of FWI. We further show that the eigenvalue decay behavior of the wave-based NTK can explain why CR-FWI alleviates the dependency on initial models and shows slower high-frequency convergence. Building on these insights, we propose several CR-FWI methods with tailored eigenvalue decay properties for FWI, including a novel hybrid representation combining INR and multi-resolution grid (termed IG-FWI) that achieves a more balanced trade-off between robustness and high-frequency convergence rate. Applications in geophysical exploration on Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP model, and the more realistic 2014 Chevron models show the superior performance of our proposed methods compared to conventional FWI and existing INR-based FWI methods.

Initial Model Incorporation for Deep Learning FWI: Pretraining or Denormalization?

Subsurface property neural network reparameterized full waveform inversion (FWI) has emerged as an effective unsupervised learning framework, which can invert stably with an inaccurate starting model. It updates the trainable neural network parameters instead of fine-tuning on the subsurface model directly. There are primarily two ways to embed the prior knowledge of the initial model into neural networks, that is, pretraining and denormalization. Pretraining first regulates the neural networks' parameters by fitting the initial velocity model; Denormalization directly adds the outputs of the network into the initial models without pretraining. In this letter, we systematically investigate the influence of the two ways of initial model incorporation for the neural network reparameterized FWI. We demonstrate that pretraining requires inverting the model perturbation based on a constant velocity value (mean) with a two-stage implementation. It leads to a complex workflow and inconsistency of objective functions in the two-stage process, causing the network parameters to become inactive and lose plasticity. Experimental results demonstrate that denormalization can simplify workflows, accelerate convergence, and enhance inversion accuracy compared with pretraining.