Coarse-to-fine multi-resolution hash encoding for implicit full waveform inversion

Abstract: Implicit full waveform inversion (IFWI) reparameterizes the velocity model with a neural network, but it suffers from spectral bias and requires many iterations to converge. Multi-resolution hash encoding uses a learnable feature grid to recover high wavenumber details at multiscales and provides strong local representations. However, activating all resolution levels from the start of the inversion introduces high wavenumber components, which can exacerbate cycle skipping. In this paper, we propose a coarse-to-fine IFWI (C2F-IFWI) that activates the multi-resolution hash levels progressively rather than all at once. Specifically, only a few coarse base levels are kept active at initialization, while the remaining levels are introduced progressively. A frozen hash level contributes no gradient until it is activated, so the coarse levels resolve the long-wavelength background before the finer levels are introduced regardless of the frequency range of the data. A continuous activation index increases with iterations and determines how many levels are active at any point in training. Each newly admitted level is then faded in gradually through a smooth weighting rather than switched on abruptly, which keeps the optimization stable during transition. This scheme allows model capacity to grow synchronously with wavenumber without adding any additional trainable parameters and with negligible computational overhead. Experiments on the synthetic Overthrust and BP 2004 models and on the Viking marine field dataset show that, compared with the all-resolution hash baseline, the coarse-to-fine scheme converges faster, attains higher accuracy, and is markedly more robust to the choice of initial model and to strong data noise.