abstract

Mathematical instability in four-parameter least squares hypocenter solutions arises primarily from the fact that the four computed variables—origin time (T0), focal depth (h), latitude (θ), and longitude (λ)—are not strictly independent. Specifically, T0 exhibits a non-independent relationship with the geometric parameters. For small networks (< 10–15 stations), the lack of independence between T0 and the other variables results in unstable least-squares solutions. This instability is manifest most clearly by the fact that different station subsets of the observational network produce significantly different solutions for the same earthquake. The instability can be eliminated by computing T0 independently for each station using the formula

 
(T0)i=(Tp)iVk(Tsp)iVp,

where Tp = P-wave arrival time, Vk = S-P velocity, Vp = P-wave velocity, and Ts-p = time interval between P and S arrivals. An average value of T0 can be obtained from the individually calculated origin times and the P-wave travel times calculated. The variables ϕ, λ and z are then computed by the usual least-squares procedure using P-wave travel times only. The method is iterative and an average T0 is recalculated in the course of each iteration.

Fundamental properties of travel times within the Earth impose definite limitations upon the accuracy of the locations. Low values of the derivative dTp/dh at epicentral distances of a few degrees introduce a large uncertainty in focal depth, particularly for shallow (0–60 km) earthquakes. There is normally little error in epicenter, however, even for solutions in which depth is poorly determined. The dimensions and geometric configuration of the network in relation to the epicenter and the proximity of the epicenter to any one station are controlling factors in predicting the minimum uncertainty for any given hypocenter solution.

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