Figure 6.23 e) Depth histogram resulting from a 6-D subspace inversion.
position at around 8.03 - 8.07 km s '1. The Moho parameters resulting from the 6-D
subspace algorithm, shown in Fig. 6.23d, also indicates the success of the 6-D inversion
over the 2-D, since a model similar to that obtained when the model covariances were
3-D Linear Inversion 6.54
Fig. 6.23e shows the depth distribution resulting from the 6-D inversion.
Comparing this to the 2-D inversion we see that the number of events in the 0 - 7 km range,
i.e. the first 4 intervals, have increased at the expense of those in the 7 - 13 km range. The
number of events at greater depths has also decreased more or less uniformly. The
epicentral relocation map is shown in Fig. 6.23f and is virtually identical to the original 2-D
map in Fig. 6.13b. 31500 30500 29500 28500 27500 26500 25500 24500 Iteration
Figure 6.24 Misfit reduction curves for four inversions; Curve A is from the original 2-D inversion using 312 events, while curve B is from the 2-D inversion which first introduces the Moho parameters. Curves C and D are obtained from the 6-D subspace algorithm, C being the first 6-D inversion and D from one which uses two Moho parameters per Pn ray (see text). Note the 6-D inversions reduce the misfit more quickly in both cases.
Comparing the rates of estimated misfit decline between the 2-D and 6-D inversions,
shown in Fig. 6.24, we see that the 6-D subspace algorithm is more efficient in reducing the
misfit. This is essentially because in one step of the 6-D algorithm we find the combination
of all six subspace vectors which produces the greatest decrease in the objective function (as
3-D Linear Inversion 6.55
combinations in the 2-D subspace step are, by definition, included within the range of 6-D
subspace steps, the latter will produce a step which is at least as efficient as the former, and
probably a lot more.
Fig. 6.24 also contains the convergence curves for the 2-D subspace inversion which
first introduces the Moho parameters i.e. that responsible for Fig. 6.21a, and that of a
second 6-D subspace inversion in which the Moho parameter representation has been altered
slightly (see below). An interesting point to note from these curves is the actual size of
misfit function at which these functions level off (~ 25 000 - 27 000), which represents
only an 18 - 20 % reduction in misfit. The reason for this is unclear. One possibility is that
the model parameterisation itself is of too broad a scale i.e. too inflexible, to allow the data to
be fit to a high degree. This would suggest that a more flexible parameterisation is required
in the nonlinear inversion. Another possibility is that the misfit value itself is misleading because our estimates of the observational errors, and hence the data covariances c^, are too
small. If this is the case, and we assume that the observed misfit value actually represents
the statistical expectation value of the dataset given the real error distribution, then we find
that the data covariances need to be increased by a factor of 2 - 2.5, giving a standard
deviation in the observed earthquake arrival times of between 0.2 - 0.25 s for the P readings
and 0.46 - 0.7 s for the S readings. These values do seem to be on the large side and so we
expect that neither factor is solely responsible for the large misfit, and it is more likely that a
combination of the two are involved.
A closer examination of the role played by the Moho parameters in the inversion
yields some interesting facts. The number of rays affecting the Moho parameters is closely
related to the position and direction of the ray paths. Since a given Pn ray interacts with the
first order velocity discontinuity at only two points, only two Moho parameters are
constrained by any one ray. However several tens of P or S-wave slowness parameters may
influence the arrival time of a ray. In the inversions discussed above we actually neglected
the depth parameter furthest from the source in an effort to concentrate on the Moho in the
Moho D e p t h s 6 D S (M)
3 2. r-
Figure 6.2SMoho depth map oblained from the 6-D inversion with the model covariances of the Moho parameters weighted inversely proportional to the number of rays striking the interface.
3-D Linear Inversion 6.56
Pn ray. In this case the form of the subspace vector corresponding to the Moho parameters
will be heavily influenced by the raypath distribution. This was found to be the case. In fact
the relative sizes of the subspace vector components reflected the number of rays striking the
Moho almost exactly. Consequently the perturbations of all depth parameters must be
heavily influenced by the Pn raypath density.
In an attempt to alleviate the problem we applied weights to the components of the Moho subspace vector (i.e. adjust the terms along the diagonal of C44), making them
inversely proportional to the number of rays striking each section of the surface. An
inversion using this transformed Moho subspace vector resulted in the depth model in Fig.
6.25. Rather surprisingly there does not seem to be a major difference in the actual
inversion results. The primary effect of the transformation is to enlarge the zone of shallow
depths, which may indicate that the distribution of Pn rays is in fact too poor to provide
much constraint on the Moho parameters.
Restricting the Pn rays to influencing a single depth parameter was, upon reflection,
a rather poor move, since we have effectively introduced an inconsistency in the overall
Moho representation and also reduced the level of constraint on the Moho parameters.
Correcting this feature led to the depth map shown in Fig. 6.26a and the upper Mantle P-
velocities shown in Fig. 6.26b. The velocity map has not altered noticeably from the
previous case (Fig. 6.23c) the relative highs and lows remain in the same position.
However the amplitudes do seem to have changed slightly, probably because the depth
parameters have reduced the misfit function to a greater degree and removed the need to
perturb the velocities as much. In contrast the depth map seems to have changed somewhat,
the sharp highs in the central part of Fig. 6.23d have been replaced by a plateau, which
indicates that the previous feature was an artifact produced by neglecting the second depth
parameter. The overall size of the uplifted region has diminished again and now remains
largely in the central portion where the data density is highest. Also the range of depth