In the inversion part of the full-waveform inversion (FWI) that brings high resolution in finding a convergence point in the model space, a localized numerical optimization technique reduces the objective function employing the \(\ell_{2}\) norm in a least-squares approach. Given the \(\ell_{2}\) norm's vulnerability to outliers and noise, this strategy might frequently yield imprecise imaging outcomes. Consequently, it's essential to introduce a novel regulatory approach that incorporates a more applicable form of relaxation to address the issue of overfitting, specifically, the K-support norm, characterized by its more logical and stringent restrictions. Unlike the least-squares method that targets the \(\ell_{2}\) norm reduction, the \(\ell_{1}\) norm is celebrated for its ability to ensure sparsity, durability, and superior noise reduction capabilities; thus, we consider a new regularization form, the K-support norm, which combines the \(\ell_{2}\) and the \(\ell_{1}\) norms in minimization. Subsequently, a quadratic penalty approach is utilized to expedite the convergence process and prevent entrapment in local minima, linearizing the nonlinear issue to reduce computational demands. This study presents the K-support norm concept, incorporating it with the quadratic penalty approach to enhance both convergence speed and resilience to ambient noise. In the numerical demonstration, three synthetic models are evaluated to demonstrate the K-support norm's superiority over Tikhonov regularization using two distinct noise data sets. Experimental results indicate that the modified FWI improves inversion accuracy by enhancing lateral resolution in deeper parts even with data with a low signal-to-noise ratio.
