Transdimensional Approaches to Geophysical Inverse Problems

For the past forty years seismologists have built models of the Earth's seismic structure over local, regional and global distance scales using derived quantities of a seismogram covering the frequency spectrum. A feature common to (almost) all cases is the objective of building a single `best’ Earth model, in some sense. This is despite the fact that the data by themselves often do not require, or even allow a single best fit Earth model to exist. It is widely recognized that many seismic inverse problems are ill-posed and non-unique and hence require regularization or additional constraints to obtain a single structural model. Interpretation of optimal models can be fraught with difficulties, particularly when formal uncertainty estimates become heavily dependent on the regularization imposed. An alternative approach is to embrace the non-uniqueness directly and employ an inference process based on parameter space sampling. Instead of seeking a best model within an optimization framework one seeks an ensemble of solutions and derives properties of that ensemble for inspection. While this idea has itself been employed for more than 30 years, it is not commonplace in seismology. Recent work has shown that transdimensional and hierarchical sampling methods may have some considerable benefits for seismological problems involving multiple parameter types, uncertain data errors and/or uncertain model parametrizations. Rather than being forced to make decisions on parametrization, level of data noise and weights between data types in advance, as is often the case in an optimization framework, these choices can be relaxed and instead constrained by the data themselves. Limitations exist with sampling based approaches in that computational cost is often considered to be high for large scale structural problems, i.e. many unknowns and data. However there is a surprising number of areas where they are now feasible.