On the mathematics of using difference operators to relocate earthquakes

This article examines the properties of difference operators that are used to relocate earthquakes and remove path anomaly biases. There are presently three established algorithms based on such techniques: (1) the method of Jordan and Sverdrup (1981), (2) the double-difference method of Got et al. (1994), and (3) the modified double-difference method of Waldhauser and Ellsworth (2001). We show that the underlying mathematics of these three methods are similar, although there are distinct contrasts in how each is adapted. Our results provide insight into the performance of individual methods. Both the Jordan and Sverdrup (1981) and double difference methods (Got et al., 1994; Waldhauser and Ellsworth, 2001) remove the average path anomaly bias in a set of events, but the equation weighting is more ideal in the first method. Distance dependent weighting in the Waldhauser and Ellsworth (2001) method does not reduce earthquake location-dependent path anomaly bias unless damping is applied, but damping causes the locations between earthquakes spaced far apart to be less well resolved. Alternatively, the results using Jordan and Sverdrup (1981) and Got et al. (1994) only remove a constant bias across a model subregion and cannot resolve the relative locations between subregions. The results of this study indicate that differencing operators contain the fundamental limitation that when the path anomalies from velocity heterogeneity change stongly with earthquake position, the bias effects can be reduced in the relative locations between closely spaced earthquakes, but the effects cannot be reduced in the relative locations between earthquakes spaced far apart.