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\title{Trans-dimensional trees for parsimonious geophysical inversion}

\authors{
\underline{R. Hawkins}$^1$, M. Sambridge$^1$, J. Dettmer$^1$, E. Saygin$^1$, J. Stip\v{c}evi\'{c}$^1$ \& H. Tkal\v{c}i\'{c}$^1$
}

\affiliations{
$^1$Research School of Earth Sciences, Australian National University, Canberra, Australia\\
}

\maketitle

\begin{summary}
Bayesian, Inverse methods, Seismic tomography \\
\end{summary}


Geophysical inversion consists of choosing a model parameterization
and a function that quantifies the fit of data predictions to
observations. Within such framework, inferences of the Earth
can be obtained from optimal parameter estimates and 
uncertainties\cite{bib:Rawlinson2014A}.  In classical approaches,
model parameterization is fixed at the outset and optimization is
performed using a minimization of some combination of norms that
attempt to balance the inherent trade off between model complexity and
the spatial resolving power of the observations. Uncertainties are
then estimated from linearized approximations from the optimization
process. The initial choice of some fixed model parameterization has
dramatic effects on the resulting inferences and equally on the
estimated uncertainty.

In contrast, sampling based approaches using Markov chain Monte Carlo
(McMC) techniques offer a way to obtain robust uncertainty estimates
through numerically approximating the posterior probability density
with large ensembles of solutions that are consistent with data and
prior information. While computationally expensive, these approaches
require no linearization assumptions.  With advances in McMC
techniques \cite{bib:Geyer1994B,bib:Green1995A} that allow the model
parameterization to change, called reversible jump or
trans-dimensional sampling in the literature, model parameterization
can remain a variable whose complexity is data driven rather than
subjectively fixed from the outset.

We have developed a novel Bayesian trans-dimensional approach which,
at its core\cite{bib:Hawkins2015A}, uses an abstract tree to subdivide
any space (e.g. 2D, 3D, spherical, Cartesian, temporal), allowing the
model parameterization to vary spatially or temporally to better suit
the problem at hand. Compared to previous work\cite{bib:Bodin2009A},
our approach is more flexible in terms of the problem geometry and the
ease with which various basis functions can be used.  In our initial
synthetic experiments with the new formalism, we have used simple
functions such as box-car, triangle, and Lanczos functions, as well as
many families of wavelets.

We present two recent applications of our new approach with wavelet
parameterizations to geophysical inversion problems. First, we show
the results of the inversion of ambient noise tomography data for the
Iceland region. Second, we highlight the generality of the approach
with a non-linear tsunami-source inversion to determine the
spatio-temporal evolution of the sea surface displacement after the
great 2011 Japan tsunami.

Recent experiments into truncated polynomial representations show
great promise in the solution of geophysical imaging problems due to
their greater utility. We demonstrate these through some early
simulated and real data examples. These include large scale seismic
tomography inversions using Finite Frequency kernels, and small scale
inversions using full-waveform simulations. These may provide a
template for Bayesian full waveform and joint inversions which will
become more feasible as computational power increases.

\begin{thebibliography}{1}

\bibitem{bib:Rawlinson2014A}
  N. Rawlinson, A. Fichtner, M. Sambridge M. and M. Young,
  \textit{Seismic tomography and the assessment of uncertainty},
  Advances in Geophysics,
  \textbf{55},
  Academic Press,
  2014
  
\bibitem{bib:Geyer1994B}
  C. J. Geyer and J. M{\o}ller,
  \textit{Simulation procedures and likelihood inference for spatial point processes},
  Scandinavian Journal of Statistics,
  \textbf{21},
  1994,

\bibitem{bib:Green1995A}
  P. J. Green, \textit{Reversible jump {M}arkov chain {M}onte {C}arlo computation and {B}ayesian model determination},
  Biometrika,
  \textbf{4}(82),
  1995

\bibitem{bib:Hawkins2015A}
  R. Hawkins and M. Sambridge, \textit{Geophysical imaging using trans-dimensional trees}, Geophysical Journal International, \textbf{203} (2), 2015.
  
\bibitem{bib:Bodin2009A}
  T. Bodin and M. Sambridge
  \textit{Seismic Tomography with the reversible jump algorithm},
  Geophysical Journal International
  \textbf{178},
  2009
  
\end{thebibliography}

\end{document}

