Non-linear methods for event location in a global context

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Abstract

Directed search methods in four-dimensional hypocentre space do not require the differentiation of travel times and are particularly suitable for the use of information from multiple seismic phases including joint use of primary and depth phases (e.g. pP) for a reference model such as a k135. These approaches can be used with a wide range of measures of misfit between observed and estimated times. A very effective method of this class is the use of the neighbourhood algorithm that allows all previous hypocentre estimates to be employed to estimate the shape of the misfit surface. When used with a robust measure of misfit such as the sum of the absolute values of the residuals (L1 norm), the neighbourhood algorithm provides rapid convergence, and a measure of the uncertainty in the hypocentral estimate from the concentration of the points with low misfit in four-space. Such a procedure is well suited to the first pass analysis of observations since it is not distracted by occasional readings associated with other events. Phase association can then be improved and a more conventional procedure used to provide error ellipsoids around a well-defined depth.

Introduction

Most methods employed for global location of seismic events are based on a local linearisation of the travel time behaviour about a trial solution, and a least squares fit to data (e.g., Adams et al., 1982, Storchak, this issue). The actual distribution of residuals is longer tailed than the Gaussian implicit in the least squares procedure (Buland, 1986), but the discrepancy can be controlled by the method of uniform reduction introduced by Jeffreys (1932). A further problem arises from the convergence behaviour of the Gauss–Newton scheme. The hypocentre estimate can get trapped in a local minimum of the misfit function rather than the desired global minimum, or fail to converge at all. To guard against these possibilities, major agencies such as the US National Earthquake Information Center (NEIC) and the International Seismological Centre (ISC) use a scheme of default depths when convergence is suspect (see Bolton, this issue). A further feature of the linearisation is that there is a very strong trade-off between event depth and origin time.

In recent years a number of approaches have been introduced that seek to overcome some of these difficulties by working directly with the arrival times, rather than needing partial derivatives. Most procedures can be regarded as a directed search in four-dimensional hypocentre space. This is explicit in the contracting grid procedure of Kennett (1992), and implicit in the application of more general methods such as genetic algorithms (Kennett and Sambridge, 1992) and the neighbourhood algorithm (Sambridge and Kennett, 2001). The most rapid convergence can be achieved with a suitably tuned neighbourhood algorithm, in which the entire history of tested hypocentres is employed to provide a representation of the misfit surface. Because no differentiation is required, any appropriate misfit function can be applied. It is often convenient to use an L1 norm, i.e., the sum of the absolute value of the residuals, because of its robustness to large differences between some observations and the corresponding predictions.

Large deviations between the observed arrival times and those predicted from some seismic model can arise from blunders in picking or timing, and from discrepancies between the reference model and the real situation. Such “model error” is likely out to 25° epicentral distance when a single one-dimensional reference model is employed, because of the variations in lithospheric structure that also influence phases returned from the transition zone.

One of the particular merits of the hypocentre search techniques is the ability to constrain the depth of an event, through direct minimisation of the misfit function. There is still some unavoidable trade-off between depth and origin time, but, as demonstrated by Billings et al. (1994), the dependence is non-linear. A counter intuitive requirement for the use of directed search methods is that the initial region should not be too small. It is important to sense the large scale structure of the misfit surface before attempting refinement, since otherwise the potential for side-tracking to local misfit minima is enhanced.

In this work we employ the neighbourhood algorithm procedure of Sambridge and Kennett (2001), with the one-dimensional wavespeed model ak135(Kennett et al., 1995) which reflects the continental location of most seismic stations, for a set of 155 reference events distributed across the globe (Storchak, this issue; based on Bondár et al., 2004). The advantage of the use of travel times calculated with the ak135 model is that they are mutually consistent since they are calculated from a single isotropic velocity model. ak135 was constructed to provide a good fit to the travel times of 19 seismic phases as well as differential times for core phases, and provides a good description of average wavespeed structure in the lower mantle and core. Further, the upper mantle structure of ak135 provides a good representation of the depth phases such as pP, sP (Engdahl et al., 1998) so that these can be used directly in source location. Regional phases are less well represented because of the heterogeneity of the lithosphere, with the largest residuals for the very fast Precambrian shields.

A composite L1 misfit function is employed using weighted residuals to allow the inclusion of a broad range of phases for both P and S. A common initial domain is applied to each event, using the ISC hypocentre for the reference events as the centre of the four-dimensional region. We use ±2° in latitude and longitude, ±20  s in origin time, and ±60  km in depth (retaining only positive depths). In the majority of trials the depth is free and the information regarding a likely explosion enters solely through a somewhat smaller span of acceptable values of depth. With sufficient sampling of the hypocentral space the global minimum of misfit will always be found. However, a modest number of cycles of the neighbourhood algorithm (e.g., 30) is sufficient to delineate the region of best fit. Fortunately the position of the centre of the search space is not critical, displacements of the centre of the domain by ±1° in horizontal position normally produce less than a 0.5 km shift in the hypocentre with the lowest misfit for the test events, and very little difference in the distribution of the region of best fit.

Section snippets

Neighbourhood algorithm location

We illustrate the application of the neighbourhood algorithm with two examples from the set of reference events (Fig. 1, Fig. 2 ). The first is a shallow explosion in Western Russia, and the second is a well-controlled intermediate depth event in the Windward Islands of the Caribbean. In each case we display the region surrounding the best-fitting hypocentre and the progress of the algorithm towards convergence.

For each of the locations of the reference events we have used a weighting

Systematic application to reference events

The neighbourhood algorithm has been applied in a uniform style to all of the reference events, using the weighting scheme for phase type and quality outlined above, with two different choices for the phase selection used to produce the location estimate. In the first case, all P phases are accepted (as in Table 2), but S arrivals are excluded. In the second case, we use both the set of P phases and any reported S arrivals, but not later S phases. The phase identifications were taken directly

Discussion

The fully non-linear procedure using the neighbourhood algorithm is shown to be an effective tool for obtaining hypocentral estimates. A standard approach with a reasonably broad domain in hypocentral space provides rapid convergence in depth, without the need for introducing constraints on the solution. With a robust measure of misfit, it has been possible to use all submitted data without prior processing. Blunders in readings or transcription have little influence on the hypocentre. As a

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