Untitled Document
Dynamic objective functions in seismic tomography
N. Rawlinson1, M. Sambridge1 and E. Saygin2
1 Research School of Earth Sciences, Australian
National University, Canberra, ACT 0200, Australia
2 Geoscience Australia, Symonston ACT 2609, Australia
Figure 1.Schemactic
diagram demonstrating the principle of the dynamic objective function
method. When each new solution is located by the gradient based method,
a “hump” is introduced in model space at that point to disuade future
solutions from investigating this region.
A new technique designed for generating multiple solutions to seismic
tomography problems using gradient based inversion has been developed.
The basic principle is to exploit information gained from previous solutions
to help drive the search for new models. This is achieved by adding a
feedback or evolution term to the objective function that creates a local
maximum at each point in parameter space occupied by the previously computed
models (Figure 1). The advantage of this approach is that it only needs
to produce a relatively small ensemble of solutions, since each model
will substantially differ from all others to the extent permitted by
the data. Common features present across the ensemble are therefore likely
to be well constrained. A synthetic test using surface wave traveltimes
and a highly irregular distribution of sources and receivers shows that
a range of different velocity models are produced by the new technique.
These models tend to be similar in regions of good path coverage, but
can differ substantially elsewhere. A simple measure of the variation
across the solution ensemble, given by one standard deviation of the
velocity at each point, accurately reflects the robustness of the average
solution model. Comparison with a standard bootstrap inversion method
unequivocally shows that the new approach is superior in the presence
of inhomogeneous data coverage that gives rise to under or mixed-determined
inverse problems. Estimates of posterior covariance from linear theory
correlate more closely with the dynamic objective function results, but
require accurate knowledge of a priori model uncertainty.
Application of the new method to traveltimes derived from long term
cross-correlations of ambient noise contained in passive seismic data
recorded in the Australian region demonstrates its effectiveness in practice,
with results well corroborated by prior information (Figure 2). The dynamic
objective function scheme has several drawbacks, including a somewhat
arbitrary choice for the shape of the evolution term, and no guarantee
of a thorough exploration of parameter space. On the other hand, it is
tolerant of non-linearity in the inverse problem, is relatively straightforward
to implement, and appears to work well in practice. For many applications,
it may be a useful addition to the suite of synthetic resolution tests
that are commonly used.
Figure 2: (a) Stations
used in the cross-correlation of ambient noise data; (b)
path coverage through the initial model; (c) average solution model computed
from an ensemble of 25. VR denotes Rayleigh wave group velocity; (d) variation
of the model ensemble as represented by one standard deviation of the distribution.