Regularized Nonstationary Phase Estimation via Proximal Maximization of Skewness and Kurtosis

Abstract

Wavelet phase is a critical parameter in seismic processing, where zero-phase wavelets are essential for maximizing temporal resolution and ensuring accurate interpretation of subsurface structures. In practice, however, the seismic wavelet is often nonstationary, exhibiting a phase that varies in space and time due to physical factors such as attenuation, dispersion, and thin-bed tuning effects. Higher-order statistical measures-specifically kurtosis and skewness-are traditionally maximized to drive the signal toward a maximally non-Gaussian or maximally asymmetric zero-phase state. This paper addresses the computational and stability challenges inherent in nonstationary estimation by casting the problem as a regularized non-convex optimization task. We propose a robust framework based on the Alternating Direction Method of Multipliers (ADMM) that eliminates the instability and artifacts associated with traditional piecewise-stationary windowed approaches. The core of our contribution is the derivation of the first closed-form proximity operators for the scale-invariant inverse kurtosis and inverse skewness functionals. By exploiting the signed permutation invariance of these statistical measures, we reduce the high-dimensional proximal subproblems to efficient one-dimensional root-finding tasks. We provide a detailed geometric interpretation of the optimality conditions, demonstrating that the global minimizer is governed by a branch-separation property. Furthermore, we derive an explicit critical threshold parameter which provides a theoretical rule for identifying the global minimum among multiple stationary points. Numerical validations on synthetic and real seismic data demonstrate that the proposed proximal algorithms achieve linear computational complexity and superior stability compared to traditional methods, effectively enabling nonstationary phase correction.