Event locations: speeding up grid searches using quadratic interpolation

SUMMARY The grid search method is a common approach to estimate the three spatial coordinates of event hypocentres. However, locating events in large search spaces with small grid spacings is computationally prohibitive. This study accelerates the grid searches over large search spaces using a quadratic interpolation technique. We start with the coarse-grid-estimated location, where we have the minimum value of the difference in the traveltimes between S and P waves summed over all receivers. Then, we select the neighbouring grid points and build a 3D quadratic function. The unknown coefficients of the 3D quadratic function are computed by solving a system of linear equations. After that, we interpolate the location by solving partial derivatives of the quadratic function. The quadratic interpolation technique performs well on both synthetic and real microseismic data examples, typically leading to similar event locations as those obtained using 10 times smaller grid spacings in all three directions, at a minor additional computational expense, and without the need to generate traveltimes at new spatial positions.