Under-Balcony Acoustic Diagnosis Using FOA-Based Directional Metrics: Early–Late Entropy and Vertical-Energy Discrepancy at 125 Hz, 1 kHz, and 4 kHz

Traditional concert-hall evaluations primarily rely on ISO 3382-1 scalar parameters (e.g., C₅₀ and C₈₀), which summarize temporal energy behavior but provide limited insight into the directional composition of early reflections, particularly in geometrically shadowed seating zones. This paper presents a first-order Ambisonics (FOA)-based 3D acoustic sensing framework to diagnose under-balcony directional imbalance, with emphasis on early vertical-reflection deficiency. Scene-based FOA impulse responses (WXYZ) were measured at 11 audience positions (P1-P11) in the National Concert Hall (Taipei) and analyzed using intensity-based direction-of-arrival (DoA) proxies, axis-resolved directional energy build-up, and a distributional descriptor based on directional spatial entropy. Results are presented at three representative frequencies (125 Hz, 1 kHz, and 4 kHz) and analyzed within full (0-200 ms), early (0-80 ms), and late (80-200 ms) windows. While the magnitude proxy ∣pmeas(f)∣ exhibits strong seat-to-seat variability and does not support a uniform attenuation assumption under the balcony, direction-resolved metrics reveal a consistent under-balcony signature. Specifically, the early-late vertical energy discrepancy ΔRz=Rzearly-Rzlate is persistently negative at under-balcony positions (P7-P11) across all three frequencies, indicating a selective reduction in early vertical contribution relative to the late field. Directional entropy analysis further shows predominantly negative ΔHn=Hnearly-Hnlate, with more negative values in the under-balcony group, consistent with stronger early directional constraint in shadowed seats. Spatial trend maps are provided via Gaussian RBF interpolation within the audience domain for visualization only. The proposed FOA-based diagnostic framework provides a practical and physically interpretable approach to identify direction-specific early-reflection deficits that remain masked in conventional scalar evaluations, supporting mechanism-oriented assessment and targeted intervention in geometrically constrained listening areas.