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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.