Comparison of Earthquake Travel-time Residuals How to Compare Apples to Oranges

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 mn degrees of freedom, where mn 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. INTRODUCTION Much information can often be obtained from refraction data by simply looking at travel-time anomalies as, e.g., in the now classical time-term method (Wilmore and Bancroft, 1960). Analysis of travel-time residuals from earthquakes, however, is complicated by the fact that the residuals one calculates are inseparably linked to the hypocenter location process. This would not be a significant problem if it were not for the fact that no two earthquakes are ever really alike. That is, one earthquake may be recorded with more and/or better quality arrivals than another. The interaction of this data heterogeneity with the location process makes the direct comparison of the residuals from two different earthquakes akin to comparing apples to oranges. In this paper, I show that for any two earthquakes, one can construct an operator that projects the residuals from each earthquake onto a subspace in which they can be compared without being biased by the heterogeneity of the data. The resulting process is akin to comparing apples and oranges by only asking if they are the same kind of fruit. The projected residuals obtained using the methods described below can, unfortunately, only be compared in what one might call a global sense. That is, we can only ask if there is a significant difference between the residuals of two given earthquakes. It would be nice if we could discriminate differences in residuals for individual stations, but that turns out to be fundamentally impossible. The projected residuals, like the more familiar raw earthquake residuals, are inevitably projections onto a lower dimensional subspace. Just as components of a vector in 3 space and 2245