The dataset on which we test the nonlinear location algorithm is the same as that used
for the joint velocity/hypocentre inversion, described in ch. 6. Full details of the selection
criteria, phase identification and observational errors are described in section 6.1.
Essentially the dataset consist of relatively well constrained events with an even distribution
across the region (see Fig. 6.2). In an effort to obtain the highest quality set of arrival times
as possible the entire earthquake catalogue of the A.N.U. was re-examined for the present
work and only a small fraction of the original 6000 events were finally chosen.
Initially 286 events were relocated using both the original linearized matrix inversion
Earthquake Relocation 3.2
the simple DEH model described in the previous chapter, which seems to be a reasonable
one for events spread across the entire region. Fig. 3.1 shows a comparison of the misfit
function at the hypocentres determined from the two procedures. The solutions have been
ordered according to the value of the solution misfit obtained from the linearized algorithm.
The most noticeable feature of this diagram is that the solid line, representing the nonlinear
misfit curve, lies below the linearized misfit curve (dotted) for nearly every event located. It
appears therefore that the nonlinear algorithm is indeed much better suited to finding the
minimum of a given data misfit function than the corresponding linearized scheme. This is
especially so in cases where the events are not constrained very well or when the errors in
the velocity model produce large residuals, indicated by the right hand portion of the curve.
The only contradictions to this rather satisfying result are given by the three events for which
the nonlinear misfit is greater than the linear one. Upon inspection of these events it was
found that in all three cases a premature convergence had occurred due to a 'sticking' of the
algorithm during the temporal minimization. This is due to a poor choice of initial origin
time bounds and is easily identified and rectified by an observer. For the events in Fig. 3.1
the inversions were performed consecutively, and so the solutions were not scrutinized in
detail before being accepted for display. Hence the three rouge 'solutions' have weaved
their way into Fig. 3.1.
Although a general increase in the nonlinear misfit is observed with event number, it
is interesting to note that the size of oscillation also increases. This would seem to suggest
that the ordering of events, based on the size of linearized misfit, places the more ill-
constrained events on the right hand side on the diagram. However it also moves a large
number of well-constrained events there too. These are picked out by the nonlinear scheme
and correspond to the places where the solid curve falls down close to the abscissa. An
interesting question is why, for events where the linearized scheme results in poor solutions
(large misfits), is the nonlinear scheme a lot more successful in some cases than others ? A
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Earthquake Relocation 3.3
portion of Fig. 3.1 correspond to events for which the linearized solution is unstable because
of errors in the Frechet derivatives (which are avoided in the nonlinear scheme), whereas the
increasing band of misfit values corresponds to events which are more genuinely ill-
constrained, perhaps due to their position in relation to the network, or the inadequacy of the
velocity model in these regions.
Fig. 3.2 shows the distribution of residuals resulting from the two inversion
procedures. The open circles represent the residuals obtained from linearized hypocentral
solutions while the solid dots are the corresponding nonlinear solutions. Neither set is very reminiscent of a Gaussian distribution. In fact they both appear to be closer to an
distribution i.e. an exponential decay away from the central peak around zero, than any other
standard type of error distribution. The difference between the linear and nonlinear
inversions is readily observed. The nonlinear distribution of residuals has a central peak of
larger magnitude and two side lobes of a consistently smaller magnitude than the
corresponding linearized residuals. This reflects the general reduction in residual size
obtained from the nonlinear algorithm which was prominent in Fig. 3.1. The spread of
these two distributions is also quite revealing. The half-width of both distributions is
approximately 0.3 - 0.4 seconds which is significantly larger than the observational picking
errors estimated at 0.1 - 0.2 seconds for P and S-waves. It seems reasonable to suppose
that this difference is largely due to errors introduced by the forward modelling, primarily
through the inadequacy of the laterally homogeneous velocity model itself.
Since the actual residual distribution in Fig. 3.2 bares little resemblance to a bell
shaped Gaussian then it would appear that one of the underlying premises of the 'least
squares' inversion is quite invalid for our hypocentral location problem (at least with the data
available to us). Consequently the resulting earthquake locations will be adversely affected
by this dubious assumption of Gaussian errors. Ideally, then, we should take a more
accurate account of the error distribution when performing the inversion. Since the
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Figure 3.2 The diagram shows the distributions of arrival time residuals found at the hypocentral solutions of the linear and nonlinear inversions. The solid dots represent nonlinear solutions and the open circles the linear ones. Note that the trend of residuals from the nonlinear solutions lies above that of the linearized solutions in the central peak and lower in both side lobes.
L a ti tu d e
Figure 3.3 The diagram shows I he movements in epicentral position between the linear and nonlinear inversion solutions. The hexagons are placed at the position of the linearized solution and vectors have been drawn in the direction of the corresponding nonlinear solutions. The length of each vector has been magnified by a factor of 9 in order to make differences more visible.
Earthquake Relocation 3.4
this purpose. By taking the log of frequency values in Fig. 3.2 and plotting these against
frequency we obtain the appropriate log-likelihood function, analogous to the quadratic
misfit function (2.2). Fitting a one-dimensional spline to this function allows us to calculate
the new misfit function given any residual. This procedure has not been attempted in this
work and so the effect it would have on the distribution of hypocentres is unknown. We do
not expect any major change in the resulting epicentral earthquake distribution, since this
was a rather stable feature of the earlier work involving different misfit functions (see ch. 2).
The effect on the depth distribution is more intriguing however, and perhaps warrants a
future experiment along this line.
The movements in the epicentral position between the linearized and nonlinear
inversion solutions are displayed in Fig. 3.3. The hexagons are placed at the position of the
linearized solution and vectors have been drawn in the direction of the corresponding
nonlinear solutions. The length of each vector has been magnified by a factor of 9 in order
to make differences more visible. We observe from Fig. 3.3, rather unsurprisingly, that the
larger epicentral movements are confined to the regions surrounding the network, while the
well constrained events within the network show very little change. Another interesting
feature of the diagram is that in the region surrounding the network the update vectors tend
to point away from the centre of the network. This seems to suggest that the linearized
solutions are biased towards the station positions i.e. solutions are pulled towards the
network. The nonlinear solutions, on the other hand, are placed closer to the actual
minimum of the misfit function which lies further away from the network.
The depth distribution of the two solution sets is show in Fig. 3.4a & 3.4b. There
does not seem to be a substantial difference between these two sets of solutions. The only
noticeable shift in depths is a slight movement towards the shallow end of the depth range
going from linear to nonlinear solutions, although no great physical significance is attached
to this observation. Fig. 3.5 shows the lateral distribution of depth changes between the two
Earthquake Relocation 3.5
The regions in which the nonlinear solutions are deeper than the linear ones seem to lie
largely outside the network and conversely the more shallow nonlinear solutions he within
the network. The major exception here is the diagonal strip running approximately north
west south-east, where the movement seems to be in the opposite direction. It is likely that
this effect is caused by the linear distribution of stations situated nearby.
A more interesting situation arises when we compare the two depth histograms in
Figs. 3.4a & 3.4b, to that in Fig. 3.4c which is the result of a second nonlinear inversion,
using, this time, a slightly modified form of the DEH velocity model. The modification
takes the form of a correction to the take-off angle of all Pn rays, which are modeled as head
wave, or critically refracted, arrivals (see Fig. 6.9). The original version of the DEH model
used in the earlier relocations does not allow for a variation in the take-off angle as the P-
wave velocity in the second layer is altered when raypaths are modeled at large epicentral
distances (see section 6.2). The corrected 'variable' angle DEH model consists of raypaths
which are consistent with Snell’s law at the Moho interface. Although this appears to be
only a slight change in the model itself, the effect on the depth distribution is evident from
Fig. 3.4c. A marked increase in the number of events located at the more shallow depths is
readily observed although little change occurs in epicentral positions (see Fig. 3.6). As with
the example of the Biala event discussed in ch. 2, we find that even though the nonlinear
algorithm 'improves' the location in the sense of finding a better set of fits to the observed
data, it is the velocity model which has the greatest influence on the earthquake location,
especially in the depth parameter.
We notice also that the more consistent variable angle velocity model produces a
decrease in the number of depths located around the 8 - 10 km depth interval (labeled as 10
in Fig. 3.4c) relative to its neighbours. This feature does not appear in either of the two
previous histograms, however it was noticed by Lambeck et al. (1985) as being a general trend of the complete set of earthquake locations for the region. A possible explanation of
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Figure 3.4 (a) Depth distribution of the events located using the linearized inversion algorithm and the DEH velocity model.
c-> c 4> 3 cr V u fa 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 Depth range (km)
Figure 3.4 (b) Depth distribution of the events located using the nonlinear inversion algorithm and the DEH velocity model.
Earthquake Relocation 3.6
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44
Depth range (km)
Figure 3.4 c) Depth distribution of the events located using the nonlinear inversion algorithm and the 'variable descent angle' version of the DEH velocity model.
where stress is released both by creep and brittle fracture. However our enthusiasm for the
observation must be tempered somewhat since the errors in depth location are likely to be
much larger than the width of the seismic gap, which is approximately 2 - 4 km. Indeed the
movement in the main peak of the histogram brought about by the modified velocity model is
of the order of 6 - 8 km, which suggests that the modelling errors themselves can account
for shifts with a size equivalent to that of the anomalous gap. We must submit therefore that
although this tantalizing feature seems to persist the evidence for a low velocity layer based
on the depth distribution of earthquakes is still rather inconclusive.
The peak in the depth distribution of these events seems to be positioned between 8 -
14 km for the variable angle model and deeper for the earlier models. However we must be
very careful in our interpretation of these results since we really have no clear and reliable
estimates of the depth accuracy of locations. (In ch. 6 we examine this problem in more