At each stage of the inversion procedure, the hypocentral parameter space is partitioned into a series of convex polygons called Voronoi cells. Each cell surrounds a previously generated hypocentre for which the fit to the data has been determined. As the algorithm proceeds new hypocentres are randomly generated in the neighbourhood of those hypocentres with smaller data misfit. In this way all previous hypocentres guide the search, and the more promising regions of parameter space are preferentially sampled.
The NA procedure makes use of just two tuning parameters. It is possible to choose their values so that the behaviour of the algorithm is similar to that of a contracting irregular grid in 4-D. This is the feature of the algorithm that we exploit for hypocentre location.
In experiments with different events and data sources, the NA approach has been shown to achieve comparable or better levels of data fit than a range of alternative methods; linearised least-squares, genetic algorithms, simulated annealing and a contracting grid scheme. Moreover, convergence was achieved with a substantially reduced number of travel-time/slowness calculations compared with other nonlinear inversion techniques. Even when initial parameter bounds are very loose, the NA procedure produced robust convergence with acceptable levels of data fit.
Figure 1: The distribution of stations used in the hypocentral estimates for theSulawesi event. Polar projection centred on the hypocentre.
The NA location scheme is illustrated for an event in Sulawesi in June 1999 as recorded by stations reporting to the Prototype International Data Centre of the Comprehensive Nuclear-Test-Ban treaty. The intermediate depth event is reasonably well controlled with a sparse network of stations (see figure 1) The NA location procedure is applied for a region 2 degrees across in both latitude and longitude, with a depth interval of 60 km and a time interval of 40 s centred on the PIDC location. Figure 2 illustrates the progress of the NA procedure with points coded by the misfit to the data using an L1 norm i.e. the sum of the absolute values of the time residuals scaled by the estimates of picking error.
Figure 2: The central portion of the parameter space showing the sampling of model misfit in the NA procedure. The redder symbols indicate better fit and their distribution indicates the reliability of the estimated hypocentre. The environs of the best fitting model are well explored and this information is available for assessment of the probability distribution of the hypocentral parameters.
The neighbourhood algorithm initially explores much of the allowed domain but quite quickly shifts attention to a zone with improved misfit. The convergence towards a cluster of well-fitting location estimates is rapid. Each of the symbols in Fig. 2 represents one of the location estimates which has been assessed in the progress of the NA inversion. The symbols are coded in greytone by the level of misfit with darker tones indicating better fit. The display in Fig. 2 shows only a portion of the original search region and so shows the nature of the misfit function in the neighbourhood of the best fitting model. The lightest symbols have a misfit approximately twice that of the best. The immediate neighbourhood of the best locations shows relatively slow variation but away from this region the misfit in arrival times grows quite rapidly. The distribution of misfit in hypocentral space can be used directly as an indication of the reliability of the postulated hypocentre e.g. in fig 2 the cluster of the reddest symbols (indicating least misfit) occupy a zone about 7 km E/W by 8 km N/S.
The sampling of the misfit function in the course of the NA inversion can also be put to use in a retrospective assessment of the probability distribution for the hypocentre parameters.
Comments on the maintenance of these frames to Brian Kennett:
brian@rses.anu.edu.au