Comparison of linear and non-linear earthquake locations for the 1995 Ventimiglia sequence

A.Lomax1, M.Cattaneo2, N.Bethoux1, A.Deschamps1,
F.Courboulex1, J.Déverchère1, J.Virieux1

1UMR Géosciences Azur, Nice, France
2Universita di Genova, Genova, Italy

Poster presentation at:
European Geophysical Society, XXII General Assembly
Nice, France, 20-24 April, 1998

1. Introduction

The ML=4.7 1995 Ventimiglia earthquake (French-Italian border) caused some damage and was widely felt. Because the Vintimiglia sequence occurred near a coastline, with no permanent stations to the south, there is a relatively large north-south uncertainty in the event locations (Figure 1). In addition, for events with few S arrival times there is a large uncertainty in event depth. For notification of civil authorities and for later seismicity and hazard studies, it is important to have both accurate event locations and comprehensive uncertainty estimates.

In this paper we compare event locations and uncertainty estimates (probabiltiy density function (PDF) and maximim likelihood hypocenter) obtained with a non-linear, Grid-search approach to the those (68% confidence ellipsoid and optimal hypocenter) produced by Hypoellipse, an iterative, linear algorithm (Lahr, 1989).

We find that the hypocentral coordinates obtained by the linear and non-linear approaches are nearly identical, but that the uncertainty estimates can differ significantly. In particular, for a location with few S wave arrivals, the 68% confidence ellipsoid obtained by the linear approach is smaller than the equivalent confidence region of the spatial PDF for the non-linear location and does not include a shallow, high-likelihood region for the hyocenter location.

2. Linear earthquake location - Hypoellipse

The program Hypoellipse (Lahr, 1989) estimates an event hypocenter by linearized, iterative application of Gieger's method - the minimization of a sum of weighted-squared residuals between observed and predicted arrival times. Hypoellipse produces a 68%, joint hypocentral confidence ellipsoid based on the arrival-time errors and the spatial partial derivatives at the final, hypocentral (xyz) location. The iterative, linear approach converges rapidly to a solution and requires little computer memory. It is particularly useful for real-time location and for re-location of large numbers of events.

3. Non-linear earthquake location - Grid-search

The Grid-search algorithm produces an estimate of the spatial probability density function (PDF) and maximum likelihood origin time for earthquake location at all points on a 3D, xyz spatial grid.

In the first stage of this algorithm, the traveltimes between each station of a network and all nodes of an xyz spatial grid are calculated once, using an Eikonal finite-difference scheme (Podvin and Lecomte, 1991), and then stored on disk. Next, for each event, the weighted-squared residuals are obtained on successively finer grid-searches over xyz space to produce a probabilistic representation of the event location, following the approach of Tarantola and Valette (1982) and Moser, van Eck and Nolet (1992).

The Grid-search earthquake location is expressed probabilistically through a 3D, spatial PDF. In contrast to the single, "optimal" location and associated, local, ellipsoidal (Gaussian) hypocentral statistics produced by linear location methods, the non-linear PDF can define multiple, maximum-likelihood hypocenters and highly non-ellispoidal (non-Gaussian) errors volumes. The grid-search approach does not require partial-derivatives, but does use significantly more computing time and computer memory than linear methods.

The non-linear, gird-search approach allows the use of heterogeneous, 3D velocity models, while the comprehensive, non-Gaussian uncertainty information given by the PDF is important for studies of hypocentral depth, and for location of events outside of a station network.

4. Comparison of Linear and Non-linear locations

We first examine the Hypoellipse and Grid-search locations for the Vintimiglia mainshock. The same phase readings and layered velocity model are used for both location methods.

The non-linear, Grid-search location is represented by a "cloud" of samples (red dots) drawn from the spatial PDF and by the maximum likelihood point (yellow dot) of the PDF (Figure 2). The extent and density of this PDF cloud indicates the non-linear confidence region for the event location. Notice the large vertical extent of the PDF cloud - the event depth is poorly constrained, there is significant likelihood that it is between 0 and about 15 km. Secondly, note the large North-South uncertainty in epicenter - this reflects the poor distribution of stations (green dots), which were not available off-shore or above the sequence at the time of the mainshock.

The linear, Hypoellipse location is indicated in Figure 2 by the joint, hypocentral 68% confidence ellipsoid (green ovals); the optimal hypocenter (not shown) is located at the center of this ellipsoid. Notice (1) the general agreement of the orientation of the Hypoellipse ellipsoid with that of the Grid-search PDF cloud, but (2) the extent of the ellisoid is much smaller than would be expected from the extent of a 68% sub-volume of the PDF cloud, and (3) the ellipsoid is positioned in the lower part of the PDF cloud and does not reflect the zone of high likelihood extending to the surface.

5. Why are the error estimates different?

The error ellipsoid produced by Hypoellipse is based on the partial derivatives for the spatial location evaluated at the final hypocenter, and on the estimates of standard error for the arrival times. This ellispoid is a local estimate of a joint 3D, xyz confidence ellipsoid, i.e. the integral over origin time of a 4D (xyz-t) ellispoidal (Gaussian) representation of the full, non-linear, 4D PDF for the event location (Figure 3). The full 4D PDF will typically be non-ellipsiodal (non-Gaussian) and may have multiple maxima. The 3D, joint error ellipsoid may be larger or smaller than the 3D, joint PDF (depending on the derivatives at the hypocenter point), and is always smaller than the projection of the 4D ellipsiod into the 3D, xyz space.

In contrast, the 3D, xyz PDF produced by the Grid-search is an estimate of the full, non-linear, 3D, xyz joint PDF (Figure 3). This 3D PDF is be smaller than the projection of the 4D PDF into the 3D, xyz space, but will reflect the irregularities of the full, 4D PDF. For locations where the full, 4D PDF is approximately ellispoidal, the 68% confidence region of the Grid-search, spatial PDF should match the Hypoellipse, spatial, 68% confidence ellipsiod.

But the full PDF for the Vintimiglia mainshcok apparently differs from an ellipsoid, as indicated by the zone of high likelihood extending to the surface in the Grid-search PDF cloud and the offset of the Hypoellipse ellipsoid to the lower part of this cloud (Figure 2). The non-ellipsoidal components of the PDF are not reflected in the Hypoellipse error analysis, which includes only a Gaussian expansion of partial derivatives taken at the optimal hypocenter point.

6. A locations with many S phases

The Hypoellipse and Grid-search locations for a Vintimiglia aftershock with many S readings and readings from nearby temporary stations are shown in Figure 4. The Grid-search PDF cloud is nearly ellipsoidal and matches closely the position, size and orientation of the Hypoellipse ellipsoid. Thus, for this well located event with little depth and origin time uncertainty, the non-linear and linear hypocenters and error indicators are the same. However, the North-South uncertainty in epicenter, which is primarily determined by the station geometry, remains similar to that for the mainshock.

7. Discussion

Though the Grid-search PDF for the Vintimiglia mainshock is only weakly non-ellipsoidal, there are significant differences between the linear Hypoellipse and non-linear Grid-search error estimates. Most importantly, the asymmetry in station distribution and lack of S arrivals for the mainshock lead to a large uncertainty in depth which is not fully reflected in the Hypoellipse error ellipsoid. The differences between non-linear and linear error estimates can be much larger for other network/event geometries.

For the aftershock with close stations and many S readings there is much better constraint of the event location and origin time, and the linear and non-linear error estimates are very similar. However, the results of both methods remain dependent on the velocity model - if there is large uncertainty in the model, the errors in event location, particularly in depth, can be underestimated.


This work was supported in part by a grant from IPSN, Fontenay-aux-Roses, France


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Podvin, P. and Lecomte, I., 1991. Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools, Geophys. J. Int., 105, 271-284.

Tarantola, A. and Valette, B., 1982. Inverse problems = quest for information, J. Geophys., 50, 159-170.

Anthony Lomax - UMR Géosciences Azur -