Propagation of Instrumental Self-Noise Through Causal Response Correction: Implications for Seismic-Network Detectability and Completeness

Abstract. Seismic detectability is commonly interpreted as a consequence of network geometry, station density, and environmental noise. However, instrumental self-noise and causal instrument correction can substantially modify the effective station noise floor and therefore influence earthquake detectability and completeness magnitude. Despite their importance in instrumental seismology, these effects are rarely incorporated explicitly into network-scale detectability analyses. This study develops a unified theoretical and computational framework linking instrumental response, causal response correction, ground-referred self-noise, environmental seismic noise, station-level signal-to-noise ratio, probabilistic detection, and network-scale completeness magnitude. The framework propagates instrumental self-noise through the inverse instrumental response into corrected ground-motion units and integrates this contribution into a probabilistic detectability formalism. Global broadband station geometries derived from publicly available FDSN metadata are combined with physically informed environmental-noise fields and parameterized instrumental self-noise models to investigate large-scale detectability behavior. The results demonstrate that seismic detectability emerges from the coupled interaction among network geometry, environmental noise, and propagated instrumental self-noise. Three principal operational regimes arise naturally from the framework: geometry-limited, ambient-noise-limited, and self-noise-limited detectability conditions. Sparse networks are dominated primarily by geometric effects, whereas densely instrumented low-noise networks become increasingly sensitive to instrumental self-noise after causal response correction. The results further show that the marginal detectability gain associated with network densification progressively decreases as station density increases, while the relative importance of instrumental quality correspondingly grows. The framework additionally demonstrates that instrumental self-noise may be substantially amplified near the limits of the instrumental passband after response correction, producing detectability degradation even when raw instrumental-noise levels appear comparatively small. Consequently, seismic-network performance cannot be interpreted exclusively as a geometrical property, but instead reflects a multiscale interaction among instrument physics, environmental conditions, and network organization. The proposed methodology provides a computationally reproducible and physically interpretable basis for analyzing detectability limits and regime-dependent network optimization in modern seismic monitoring systems, including dense broadband arrays and emerging low-cost MEMS-based networks. The proposed methodology is intended primarily as a physically interpretable framework for investigating first-order detectability controls and regime-dependent network optimization rather than as an operational estimator of real-world global completeness magnitude.