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In the process of tunnel excavation, advance geological prediction technology has become indispensable for safe, economical, and efficient tunnel construction. Although traditional methods such as drilling and geological analysis are effective, they typically involve destructive processes, carry high risks, and incur significant costs. In contrast, non-destructive geophysical exploration offers a more convenient and economical alternative. However, the accuracy and precision of these non-destructive methods can be severely compromised by complex geological structures and environmental noise. To address these challenges effectively, a novel approach using frequency domain full waveform inversion, based on a penalty method and Sobolev space regularization, has been proposed to enhance the performance of non-destructive predictions. The proposed method constructs a soft-constrained optimization problem by restructuring the misfit function into a combination of data misfit and wave equation drive terms to enhance convexity. Additionally, it semi-extends the search space to both the wavefield and the model parameters to mitigate the strong nonlinearity of the optimization, facilitating high-resolution inversion. Furthermore, a Sobolev space regularization algorithm is introduced to flexibly adjust the regularization path, addressing issues related to noise and artefacts to enhance the robustness of the algorithm. We evaluated the modified full waveform inversion with several tunnel fault models by comparing the results of the enhanced method with those of traditional least squares-based Tikhonov regularization and total variation regularization full waveform inversion methods. The verification results confirm the superior capabilities of the proposed method as expected., Comment: 60 pages, 21 figures

Full Waveform Inversion (FWI) is an advanced geophysical inversion technique. In fields such as oil exploration and geology, FWI is used for providing images of subsurface structures with higher resolution. The conventional algorithm minimises the misfit error by calculating the least squares of the wavefield solutions between observed data and simulated data, followed by gradient direction and model update increment. Since the gradient is calculated by forward and backward wavefields, the high-accuracy model update relies on accurate forward and backward wavefield modelling. However, the quality of wavefield solutions obtained in practical situations could be poor and does not meet the requirements of high-resolution FWI. Specifically, the low-frequency wavefield is easily affected by noise and downsampling, which influences data quality, while the high-frequency wavefield is susceptible to spatial aliasing effects that produce imaging artefacts. Therefore, we propose using an algorithm called Sparse Relaxation Regularised Regression to optimise the wavefield solution in frequency domain FWI, which is the forward and backward wavefield obtained from the Helmholtz equation, and thus improve the accuracy of the FWI. The sparse relaxed regularised regression algorithm combines sparsity and regularisation, allowing the broadband FWI to reduce the effects of noise and outliers, which can provide data supplementation in the low-frequency band and anti-aliasing in the high-frequency band. Our numerical examples demonstrate the wavefield optimization effect of the sparse relaxed regularised regression-based algorithm in various cases. The accuracy and stability of the improved algorithm are verified in comparison to the Tikhonov regularisation algorithm., Comment: 58 Pages, 22 figures

Full Waveform Inversion (FWI) is a modeling algorithm used for seismic data processing and subsurface structure inversion. Theoretically, the main advantage of FWI is its ability to obtain useful subsurface structure information, such as velocity and density, from complex seismic data through inversion simulation. However, under complex conditions, FWI is difficult to achieve high-resolution imaging results, and most of the cases are due to random noise, initial model, or inversion parameters and so on. Therefore, we consider an effective image processing and dimension reduction tool, randomized singular value decomposition (rSVD) - weighted truncated nuclear norm regularization (WTNNR), for embedding FWI to achieve high-resolution imaging results. This algorithm obtains a truncated matrix approximating the original matrix by reducing the rank of the velocity increment matrix, thus achieving the truncation of noisy data, with the truncation range controlled by WTNNR. Subsequently, we employ an inexact augmented Lagrangian method (iALM) algorithm in the optimization to compress the solution space range, thus relaxing the dependence of FWI and rSVD-WTNNR on the initial model and accelerating the convergence rate of the objective function. We tested on two sets of synthetic data, and the results show that compared with traditional FWI, our method can more effectively suppress the impact of random noise, thus obtaining higher resolution and more accurate subsurface model information. Meanwhile, due to the introduction of iALM, our method also significantly improves the convergence rate. This work indicates that the combination of rSVD-WTNNR and FWI is an effective imaging strategy which can help to solve the challenges faced by traditional FWI., Comment: 55 Pages, 21 Figures

Full-waveform inversion (FWI) is known as a seismic data processing method that achieves high-resolution imaging. In the inversion part of the method that brings high resolution in finding a convergence point in the model space, a local numerical optimization algorithm minimizes the objective function based on the norm using the least-square form. Since the norm is sensitive to outliers and noise, the method may often lead to inaccurate imaging results. Thus, a new regulation form with a more practical relaxation form is proposed to solve the overfitting drawback caused by the use of the norm,, namely the K-support norm, which has the form of more reasonable and tighter constraints. In contrast to the least-square method that minimizes the norm, our K-support constraints combine the and the norms. Then, a quadratic penalty method is adopted to linearize the non-linear problem to lighten the computational load. This paper introduces the concept of the K-support norm and integrates this scheme with the quadratic penalty problem to improve the convergence and robustness against background noise. In the numerical example, two synthetic models are tested to clarify the effectiveness of the K-support norm by comparison to the conventional norm with noisy data set. Experimental results indicate that the modified FWI based on the new regularization form effectively improves inversion accuracy and stability, which significantly enhances the lateral resolution of depth inversion even with data with a low signal-to-noise ratio (SNR)., Comment: 54 pages, 21 figures