Recovery of event boundaries and Brillouin frequency shift in STFT-BOTDR using variable projection with spatial response constraint

Brillouin optical time-domain reflectometry based on the short-time Fourier transform (STFT-BOTDR) enables rapid acquisition of Brillouin gain spectra (BGSs). However, the spatial response determined by the probe pulse and the STFT window causes mixing between the background and event spectra near event boundaries. At small Brillouin frequency shift (BFS) differences, the mixed BGS remains single-peaked, leading to boundary broadening and BFS underestimation by conventional peak search. To overcome these challenges, we propose a variable projection method with a spatial response constraint (SR-VP). For candidate event boundaries, the equivalent spatial response determines the event weight along the fiber, while the event BGS is recovered from the continuous BGS sequence by solving a linear least-squares problem. Variable projection eliminates the event spectrum from the nonlinear optimization, leaving only the event boundaries to be determined. This enables recovery of the event boundaries and BFS without requiring resolvable double peaks in the mixed BGS. Experiments showed that, for a 5 m event with BFS differences ranging from 5 to 8 MHz, the maximum errors in boundary localization and the recovered BFS difference were 0.52 m and 0.14 MHz, respectively. The applicability of the method was evaluated for different event lengths, STFT window characteristics, probe pulse widths, and signal quality levels. Comparison with double-peak fitting further demonstrated the advantage of SR-VP under strongly overlapped spectral conditions. These results show that spatially correlated BGS information can overcome the limitations of single-spectrum analysis for event recovery in distributed Brillouin sensing.