Recovering locations and takeoff angles of earthquakes in clusters based on the difference in their S-P intervals

It is demonstrated in [Gnyp & Malytskyy, 2023] that only the difference in the intervals between the first P- and S-waves can be used to relocate a cluster of similar earthquakes. The advantage of using only the difference is that it is measured by cross-correlation within a window containing the corresponding arrivals, eliminating the need to know their exact timing. Another advantage is that relative locations can be recovered regardless of source times and, therefore, of often inaccurate arrival picks or velocity models. It is assumed in [Gnyp & Malytskyy, 2023] that the cluster size is significantly smaller than the distance to the stations, and that the takeoff angles of the first P- and S-waves, as well as station azimuths, are known for at least one reference earthquake. Under these conditions, the relationship between the locations and the difference becomes purely geometrical and linear, allowing for a straightforward solution of the corresponding system. However, if both the locations and takeoff angles are unknown, the system becomes nonlinear and singular, making it nearly impossible to solve. In the current version of the algorithm, we propose circumventing the singularity by optimizing the locations and takeoff angles separately. First, we determine the locations for some initial angles, then adjust the angles, re-evaluate the locations, and repeat this process. To evaluate the effectiveness of this approach, we conduct a series of synthetic experiments, focusing primarily on the ability to achieve complete recovery of locations and takeoff angles using a damped least-squares solution, depending on the accuracy of the initial angles, the number and configuration of stations, and the damping applied. To reduce the impact of local minima, we propose estimating the median of solutions obtained for an ensemble of randomly perturbed initial angles. The tests demonstrate the effectiveness of the algorithm and its potential applicability to real data. The algorithm can be combined with other relocation techniques, which makes it possible to link the poorly recorded events to well-constrained ones. This is particularly important for clearer imaging of fault structures in intraplate areas with low seismicity, improving our understanding of local seismic activity and earthquake hazard.