Abstract
For conventional single-event, nonlinear, least-squares hypocentral estimates, I show that the total error is expressible as a linear combination of three terms: (1) measurement error; (2) modeling errors caused by inadequacy of the travel-time tables; and (3) a nonlinear term. Errors in calculating travel-time partial derivatives are shown to have no effect, provided a stable solution can be found. This is in contrast to linear problems where errors in calculating matrix elements can distort the solution drastically. The error appraisal technique developed here examines each of the three error terms independently. The first can be analyzed by standard confidence ellipsoids with critical values based on measurement error statistics. The second can cause conventional error ellipsoid calculations that derive a critical value from an estimate based on rms residuals, to give misleading results. I introduce an alternative extremal bound procedure for appraising such errors. Travel-time modeling errors are bounded as the product of ray arc length and an estimate of the nominal scale of slowness errors along the ray path. These are used to derive an upper bound on systematic errors in each hypocentral coordinate based on a novel bounding criteria. Finally, I show that, for errors of a reasonable scale, the nonlinear error term can be estimated adequately using a second-order approximation. Given an upper bound on the total location error, bounds on the travel-time error induced by nonlinearity can be calculated from the spectral norm of the Hessian for each measured arrival time. The systematic errors in each hypocentral coordinate due to nonlinearity can then be bounded using the same criteria used for constructing modeling error bounds. This overall procedure is complete because it allows one to independently appraise the relative importance of all sources of hypocentral errors. It is practical because the required computational effort is small.
