4. Markov chain Monte Carlo inversions
The Markov chain Monte Carlo is an inversion technique ideally suited
for non-linear problems with fast forward solvers (here the finite
volume model described in section 3). Its implementation includes
computing hundreds of thousands of forward model simulations given a
suite of parameters of interest (Allan et al. , 2013, Florez et al. , 2021). For the calculations in this study, the parameters
of interest include \(\phi_{m}\) and \(\xi_{\text{ref}}\). The forward
model is comprised of eq. (10) and (11), which are solved numerically
using a finite volume approach based on an explicit upwind scheme and
centered difference, respectively.
Over successive iterations of the MCMC, each of the parameters is chosen
such that \(L_{0}\) is minimized, where \(L_{0}=\ \left(\Pi-\vartheta\ \right)^{2}\) and is the square of
the misfit between the model, \(\vartheta\), and observed data, \(\Pi\).
Here, \(\Pi\) is provided by the load versus displacement data of Hoyos et al. (2022) or by the melt fraction profiles from the
centrifuge experiments summarized by Connolly and Schmidt (2022). The
former records mechanical data resolved over time but porosity data
integrated over space, while the latter provides depth dependent melt
fraction profiles at one instance in time. To compare the model to the
experimental data of Hoyos et al. (2022), eq. (13) is solved
using a midpoint Riemann sum using eqs. (10) and (11).
When applying the inversion model to the olivine centrifuge data, we use
two different MCMC schemes. For the first, the random starting values
for each parameter can be written in vector form as:
\({\left(17\right)\text{\ m}}^{\text{ini}}\mathbf{=\ }\left[{{\phi_{m}}^{\text{ini}}}_{\text{ol}},\ {{\xi_{\text{ref}}}^{\text{ini}}}_{\text{ol}}\right]\).
For this scheme, \(\xi_{\text{ref}}\) is treated explicitly as a
material parameter. \(\phi_{m}\), on the other hand, is likely a history
dependent parameter, but is similarly treated here as a material
parameter (see section 6 for more details).
The random starting values for each parameter for the second scheme can
be written in vector form as:
\(\left(18\right)\mathbf{\ }m^{\text{ini}}\mathbf{=\ }\left[{{\phi_{m}}^{\text{ini}}}_{\text{ol}},\ {{\xi_{\text{ref}}}^{\text{ini}}}_{p}\right]\),
where \(p\) is an index that refers to an individual experiment and
indicates that a value is unique to that experiment. For this scheme,
only \(\phi_{m}\) is treated explicitly as a material parameter that
applies to each mineralogy, while each experiment has a unique \(\xi_{\text{ref}}\). These different schemes allow comparison between
inversions that fit the data presented by each individual experiment, as
well as the data of a particular group as a whole. When in agreement,
this gives confidence to the model because optimizations for individual
experiments are similar to the optimization for the dataset as a whole.
Finally, when applying the model to the experiments of Hoyos et
al. (2022), the random starting values for each parameter can be
written in vector form as:
\(\left(19\right)\mathbf{\ }m^{\text{ini}}\mathbf{=\ }\left[{{\phi_{m}}^{\text{ini}}}_{p},\ {{\xi_{\text{ref}}}^{\text{ini}}}_{p}\right]\).
The median and standard deviation of all the parameters converged to for
each inversion are tabulated in tables 3 and 4. More
details on the MCMC inversion technique implemented in this study can be
found in the supplements.
Table 3 – Summary of the MCMC analysis of experimental data of
olivine centrifuge experiments using repacking rheology.