Comparing residuals from two earthquakes is difficult because earthquake locations interact with differences in three variables: (1) the spatial location of the source; (2) the set of stations that record each earthquake; and (3) the precision of individual arrival time measurements. Effective comparisons can be made by looking only at the components the residuals from two different earthquakes have in common. This is done with an operator that projects the residual vector for each event onto a subspace defined by the intersection of two linear manifolds related to the matrices of partial derivatives used to locate each earthquake. If one assumes the measurement error process is Gaussian, then under a null hypothesis of no differences in the travel-time anomalies between the two earthquakes, the sum of the squared differences in the projected residuals, has a chi-squared distribution with m∩ degrees of freedom, where m∩ is the dimension of the subspace of intersection. This is a the basis for a formal statistical test used to discriminate such anomalies.
Examples using synthetic data demonstrate the projection operator does work as predicted, but is limited to some extent by complications due to nonlinearity. The method is also applied to data from three earthquake swarms recorded by the Coso seismic network, in east-central California. These calculations reveal detectable differences in travel-time anomalies associated with these clusters of earthquakes. Apparently, significant lateral velocity variations exist at Coso over scales ≈ 2 km.