Publications

Seismic event location : dealing with multi-dimensional uncertainty, model non-linearity and local minima

Ballard, Sanford B.; Ballard, Sanford B.

Seismic event location is made challenging by the difficulty of describing event location uncertainty in multidimensions, by the non-linearity of the Earth models used as input to the location algorithm, and by the presence of local minima which can prevent a location code from finding the global minimum. Techniques to deal with these issues will be described. Since some of these techniques are computationally expensive or require more analysis by human analysts, users need a flexible location code that allows them to select from a variety of solutions that span a range of computational efficiency and simplicity of interpretation. A new location code, LocOO, has been developed to deal with these issues. A seismic event location is comprised of a point in 4-dimensional (4D) space-time, surrounded by a 4D uncertainty boundary. The point location is useless without the uncertainty that accompanies it. While it is mathematically straightforward to reduce the dimensionality of the 4D uncertainty limits, the number of dimensions that should be retained depends on the dimensionality of the location to which the calculated event location is to be compared. In nuclear explosion monitoring, when an event is to be compared to a known or suspected test site location, the three spatial components of the test site and event location are to be compared and 3 dimensional uncertainty boundaries should be considered. With LocOO, users can specify a location to which the calculated seismic event location is to be compared and the dimensionality of the uncertainty is tailored to that of the location specified by the user. The code also calculates the probability that the two locations in fact coincide. The non-linear travel time curves that constrain calculated event locations present two basic difficulties. The first is that the non-linearity can cause least squares inversion techniques to fail to converge. LocOO implements a nonlinear Levenberg-Marquardt least squares inversion technique that is guaranteed to converge in a finite number of iterations for tractable problems. The second difficulty is that a high degree of non-linearity causes the uncertainty boundaries around the event location to deviate significantly from elliptical shapes. LocOO can optionally calculate and display non-elliptical uncertainty boundaries at the cost of a minimal increase in computation time and complexity of interpretation. All location codes are plagued by the possibility of having local minima obscuring the single global minimum. No code can guarantee that it will find the global minimum in a finite number of computations. Grid search algorithms have been developed to deal with this problem, but have a high computational cost. In order to improve the likelihood of finding the global minimum in a timely manner, LocOO implements a hybrid least squares-grid search algorithm. Essentially, many least squares solutions are computed starting from a user-specified number of initial locations; and the solution with the smallest sum squared weighted residual is assumed to be the optimal location. For events of particular interest, analysts can display contour plots of gridded residuals in a selected region around the best-fit location, improving the probability that the global minimum will not be missed and also providing much greater insight into the character and quality of the calculated solution.