Optimized Workflows with a Single Phase-Only Response Correction for Building Empirical Green's Functions for Ambient-Noise Tomography

We present simple and optimized workflows for computing empirical Green's functions in ambient-noise tomography that enhance computational efficiency and numerical stability. A key improvement is a phase-only instrument-response correction applied only once after stacking instead of to the raw data before correlation. This prevents instability in spectral division, simplifies computations, and reduces execution time. While some of the additional optimizations we employ are already in use within the ambient-noise tomography community, we provide a detailed description along with systematic benchmarks that quantify their actual impact on runtime and stability. Key improvements include reducing redundant Fourier transforms and combining spectral equalization, cross-correlation, and stacking into a single frequency-domain step. An additional optimization reuses spectral representations of individual stations across multiple station pairs, maintaining linear complexity. We also propose a completely new optimization: applying a phase-only instrument-response correction only once after stacking instead of before correlation. This prevents instability in spectral division, simplifies computations, and reduces execution time. We validate the workflows using datasets from Southern California, Brazil, and Uganda. For individual station pairs, our primary optimized workflow (WF2) reduces execution time by approximately 67–75% (speed-up factors of 3.0–3.9), closely matching theoretical expectations (~5.1). A more scalable variant (WF3) achieves speed-up factors of 15–60 for moderate-sized networks. Furthermore, we demonstrate that a partial implementation into existing codes, requiring only minimal modifications, yields about 10% execution-time savings and improved numerical stability. The proposed workflows produce EGFs nearly indistinguishable from conventional methods and are particularly suitable for large-scale ambient-noise tomography in computationally limited environments.