Propagation-Constrained Stochastic Modeling and Robust Recursive Estimation of Air-to-Underwater ELF Signals Under Depth-Evolving Alpha-Stable Disturbances

Air-to-underwater extremely low frequency (ELF) signal recovery requires a receiver that accounts for both cross-medium attenuation and impulsive interference. Seawater weakens the desired electromagnetic waveform while altering the tail behavior that remains observable within the receiver bandwidth. Conventional recursive robust filters normally choose their linearity parameters and output scale from empirical or data-driven rules and therefore do not explicitly incorporate this depth-dependent physical–statistical coupling. This work formulates the propagation-guided scaled recursive weighted myriad (PG-SRWMy) filter as a mathematically structured framework that links a propagation-evolving stochastic process model with robust nonlinear recursive estimation for reference-normalized underwater ELF recovery. The propagation state determines a depth-evolving effective stable-like description through a characteristic exponent and a dispersion parameter, and these stochastic descriptors are transformed into branch-specific initial linearity parameters of the recursive myriad estimator. The same propagation model yields a positive and bounded normalization transform for the reconstruction scale, while first-order sensitivity relations characterize the local effect of environmental-state uncertainty. The original SRWMy sample-wise recursion is retained for data-driven adaptation. Numerical experiments show that PG-SRWMy accelerates bilinear-parameter stabilization, lowers waveform-reconstruction error, and improves end-to-end reference-normalized signal-to-noise ratio (SNR) gain across changes in receiver depth and surface-side impulsiveness. The results support propagation-aware initialization as a mathematically structured and statistically interpretable route to depth-adaptive underwater ELF reception.