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.