Figure 3.6 A diagram showing the epicentral differences between two sets of inversion solutions. The lexagons represent the solutions obtained using the nonlinear algorithm, incorporating the original 'fixed descent angle' DEH velocity model. Vectors are drawn in the direction of a second set of solutions, also obtained with the nonlinear algorithm, only using a variable descent angle version of the DEH model (see

Earthquake Relocation 3.7

Linearized location algorithms usually give indications of confidence ellipses by using local

curvature information (i.e. of the function being minimized). These estimates can be rather

optimistic in cases where the velocity structure is not too well known and the station

coverage is poor. During the routine location of events in south-east Australia the linearized

algorithm, used in the above trials, frequently indicated depth uncertainties of ± 2 - 4 km,

even for events near the edge of the network. In the previous chapter we examined the

process of estimating confidence ellipses and showed how the nonlinear algorithm may be

used to map out non-ellipsoidal confidence regions which incorporated some measure of

uncertainty in the velocity model. Using only modest estimates for the errors in the velocity

model the corresponding nonlinear confidence ellipses are somewhat larger in the depth

direction. On average between 1.5- 2.5 times, with the greatest difference occurring for

events that have been located at depths greater than 10 km.

The ML 4.3 Oolong earthquake of 1984 (McCue et al. 1988) shows a 95 % confidence region, obtained from the nonlinear algorithm, extending at least 10 km into the

Crust (see Fig. 3.7) although other evidence (i.e intensity reports and previous seismicity in

the region) seems to suggest that the event was actually quite shallow. The deployment of

additional recorders in the Dalton/Gunning area by the Bureau of Mineral Resources in

1985/86, and also records from portable digital recorders, have all indicated that the

seismicity occurs predominantly within the upper 4 km of the Crust (McCue et al. 1988). The preferred depth of this event obtained from the nonlinear algorithm is 3.8 km, which

would appear to be rather satisfactory. We might even suggest that, at least for this event,

the size of confidence ellipse is unduly pessimistic. (Possibly due to an over-estimation of

the errors induced by the Finlayson & McCracken velocity model). In general, however, we

cannot expect this to be the case. Errors in the depth locations induced by either a poor

distribution of stations about the event, or through errors in the assumed velocity model, are

likely to be represented better by the size of the 'nonlinear' confidence ellipses than the linear


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Figure 3.7 The diagram shows the spatial solution obtained for the Oolong earthquake of 1984 (McCue et al. 1988). The misfit contours and the estimated 95 % confidence interval (stippled) are plotted on three orthogonal planes intersecting the proposed solution.

Earthquake Relocation 3.8

considerable, and therefore making responsible statements about the likely depth distribution

of events in south-east Australia based solely on these histograms is very difficult Perhaps

the most cautious statement we can make is that the earthquakes seems to occur largely, if

not entirely, in the upper half of the Crust. This seems reasonable especially since the

number of events located at depths > 1 8 km have estimates of errors which increase with


One way in which it is possible to overcome the effects of velocity model errors, at

least to a limited extent, is to compare the above depth distributions with that obtained from a

relocation of controlled sources. The success of this type of approach will depend on the

geographical distribution and quality of the controlled source data available. Although the

position of quarries in relation to the south-east Australian network is not ideal, a large

number of blasts are regularly detected. However only a very small proportion of these have

ever had their arrival times recorded, and hence only a small fraction of this potentially

useful dataset is available. The results of relocating 30 quarry blasts with magnitudes large

enough to be detected at 5 or more stations are summarized in Figs. 3.8, 3.9 and 3.10.

The epicentral errors, shown in Fig. 3.8, seem to be rather irregularly distributed.

The most alarming feature of this diagram is possibly the distance over which the epicentral

errors are spread. Unfortunately a large proportion of the blasts used in this study were

derived from the two largest quarries in the region, Marulan and Ravensworth, the latter of

which lies outside of the network to the north, and the former on the eastern edge. The

range of epicentral errors is probably a result of this lack of geographical constraint rather

than any velocity model effects. The distribution of depth errors, shown in Fig. 3.9 also

contains a large spread of values. However here we observe that events seem to be located

with more or less equal frequency throughout the first 10 km (even though their actual

location is with ± 1 km of the sea level reference zero). Comparison with the earthquake

depth histogram of Fig. 3.4c would tend to suggest that in general the earthquakes are not all

V c o 3 Ü “ ts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Epicentral error range (km)

Figure 3.8 A histogram showing the distribution of epicentral errors obtained by relocating quarry blasts using readings from the A.N.U. network of seismic stations.

Depth error range (km)

Figure 3.9 A histogram of the depth errors found by relocating quarry blasts using the A.N.U. network.

Earthquake Relocation 3.9

increase over the first 10 km. Since this conclusion is drawn from a purely statistical

argument it is dependent on the number of observations available for comparison. In truth

the number of blasts used here is not enough to produce a well defined trend over the first 10

km depth range, and so we must treat this hypothesis with caution. If the blast dataset could

be increased by an order of magnitude or so then the comparison between depth histograms

may prove to be a lot more enlightening.

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Residual range (secs)

Figure 3.10 Distribution of arrival time residuals obtained from the relocation of quarry blasts. Note the histogram is highly peaked around the +0.1 s to -0.1 s interval (labeled 0.0) even though the distribution of epicentral and depth errors is rather sporadic (see Figs. 3.8 & 3.9). This indicates that the algorithm locates a minimum in the data misfit accurately, however other systematic errors, presumably the inadequacy of the velocity model used, force a mis-location of the source.

A plot of the residual distribution resulting from the relocated blast hypocentres is

displayed in Fig. 3.10. The curve is peaked in the range 0.0 - 0.2 s and shows a similar

distribution to the set of earthquake residuals (Fig. 3.2). Again the half width is greater than

Earthquake Relocation 3.10

errors on the calculated traveltimes. An interesting feature of this diagram is the slight bias

towards positive residuals (shown by the asymmetry in the positive tail). This suggests that

velocities, in the DEH model, are on the average too fast over the region covered by the blast

raypaths. Between the range ± 1.4 s a relatively symmetric distribution is observed with a

high concentration of residuals in the central peak. This shows that the nonlinear algorithm

has again been successful in locating the minimum of misfit function, however the spread in

both epicentral and depth errors gives another indication of the adverse effect that forward

modelling errors can have on the inversion solution.

3.3 Discussion

From the relocation of events in south-east Australia it is clear that the nonlinear

algorithm achieves a better fit to the observed data than the corresponding linearized

technique. The hypocentral solutions from the linearized inversion appear to be favourably

disposed towards the receiver positions, which is most likely an effect of the iterative nature

of the algorithm. The nonlinear solutions are placed closer to the actual minimum of the

misfit function than the linearized solutions and hence the confidence regions obtained are

more representative of its local curvature. In order to make a direct comparison with matrix

inversion techniques we have applied the nonlinear direct search algorithm to the same

velocity models and Gaussian misfit function as used in the nonlinear study. The versatility

of the algorithm has therefore not been exploited to the full. A complete relocation of the

earthquake dataset using the Jeffreys misfit function (2.7) has not been performed to date.

However experience shows that little difference occurs between the Gaussian and Jeffreys

solutions for events in the centre of the network where most residuals are small i.e. relative

to the standard deviation of the background distribution v, in the Jeffreys expression. (This

is to be expected since in this case the Jeffreys function approximates the underlying least

Earthquake Relocation 3.11

The ability of the algorithm to accurately locate the minimum of any given misfit

function is a necessary tool in the location of earthquake hypocentres. However it must be

stressed that finding the 'best fitting' hypocentre is only useful if that hypocentre is

representative of the actual source parameters. The use of a fixed velocity model will

inevitably introduce systematic errors into the problem. We have attempted to overcome

these errors by relocating controlled sources using the same set of receivers and earth model

as used in the earthquake locations. This process was encouraging and is limited only by the

amount of blast data available. The influence which the earth model exerts on hypocentral

locations has been demonstrated in the last two chapters. It appears that the hypocentral

solutions are more dependent on the earth model than either the type statistics employed or

the inversion algorithm itself. To improve earthquake locations then we must also improve

the velocity model. It is to this end that the practical application of the work in the next five




In document Seismic inversion for earthquake location and 3-D velocity structure (Page 62-70)