Comparative analysis of formulations for crossed artifact elimination in the frequency–Bessel spectrograms

The frequency–Bessel transformation method has significantly advanced the extraction of multimodal surface waves in seismic research. However, the presence of crossed artifacts in frequency–Bessel spectrograms, particularly when stations are regularly distributed, presents a persistent challenge. Various methods have been proposed to mitigate these artifacts, yet their diverse formulations often lead to confusion about their practical application and interrelations. This study aims to demystify these ambiguities by analyzing the existing formulations within a unified framework. We uncover that the apparent discrepancies among these methods primarily originate from the differing conventions across various studies. Consequently, we establish explicit mathematical relationships among these existing formulations. Moreover, we demonstrate that the reliance on numerical Hilbert transform can be avoided by maintaining only the causal component of cross-correlation functions. This approach simplifies the artifact removal process, enhancing the practical utility of frequency–Bessel spectrograms in geophysical analysis.