Deep learning has been widely applied to denoising ground-penetrating radar (GPR) signals. However, most existing methods lack physical constraints consistent with GPR data characteristics, especially in the frequency domain, leading to the loss of weak reflections and blurred reconstruction. Conventional networks also treat GPR denoising as a generic image restoration task without explicit weak-signal enhancement. To address these issues, this paper proposes a frequency-domain multi-scale loss function to introduce physical constraints into network training. Combined with traditional loss functions, the proposed method effectively improves the fidelity of weak reflection recovery. A multi-domain constrained TransUNet is further developed for GPR noise reduction. Experiments on synthetic data and field GPR data demonstrate that the proposed method achieves stronger robustness and competitive denoising performance.

In penetrating radar waveform inversion, the Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm only needs to store the most recent gradients and model parameters to approximate the Hessian matrix, which effectively keeps computational costs and memory usage in check. However, when the L-BFGS algorithm is combined with multi-scale inversion and dual-parameter inversion strategies, model updates become slow, and the update directions can easily get stuck in local extrema. This issue is especially pronounced in dual-parameter inversion, where L-BFGS doesn’t perform that well. Still, the idea of building the Hessian matrix using limited stored information can be extended to other Newton-type algorithms. The traditional truncated Newton method is mainly applied in the frequency domain, as the computational cost in the time domain is too high, although the resolution of time-domain imaging is higher than that of the frequency domain. Therefore, this paper proposes a limited-memory truncated Newton method carried out in the time domain. By storing the gradients and model information of recent iterations, a direct approximation of the Hessian matrix is constructed. This approximate matrix is then multiplied by the current gradient to approximate the product form of the Hessian matrix and a known vector in the truncated Newton method. Subsequently, by combining the linear CG algorithm from the conjugate gradient method to solve the Newton equation, the model increment is obtained, thereby making fuller use of the Hessian operator information. In quantitative analysis, the results of L-BFGS and LMTN algorithms in terms of objective function convergence and structure error parameters show that the LMTN algorithm can achieve more accurate inversion results.