Generalized Bancroft algorithm for locating earthquakes with P- and S-wave arrival times
Because of similarities between locating an earthquake with seismic stations and locating a Global Positioning System (GPS) receiver from satellites, the Bancroft algorithm developed for GPS processing can be used to locate earthquakes. Such an approach to earthquake location differs from the conventional method of choosing an initial or trial solution and then iteratively improving the solution until convergence. The Bancroft algorithm has the advantage of being a direct, noniterative solution but with the disadvantage of only being able to accommodate a homogeneous velocity model. An additional limitation of the standard Bancroft algorithm is that it considers arrival times in a medium with a single propagation velocity. This poses no problem for GPS processing because electromagnetic waves travel at the speed of light; however, for seismic waves it means the algorithm can be applied to collections of either P ‐ or S ‐wave arrival times. Here, I show how the Bancroft algorithm can be generalized to handle both P ‐ and S ‐wave arrival‐time measurements simultaneously. I also show how to accommodate depth‐varying P ‐ and S ‐wave velocity models. I apply the generalized Bancroft algorithm to microearthquakes beneath Tanaga Volcano in Alaska and compare standard locations from the widely used HYPOINVERSE location code to Bancroft locations and to the output of HYPOINVERSE when setting the trial location to the Bancroft location. I find the Bancroft locations outperform the results from the other methods for shallow earthquakes near sea level, where a quantity known as the geometric dilution of precision is large and linearized approaches such as HYPOINVERSE are expected to struggle.