<p>The thermal evolution of terrestrial planets depends strongly on several parameters and initial conditions that are poorly constrained. Often, direct or indirect observables from planetary missions such as elastic lithospheric thickness, crustal thickness and duration of volcanism are inverted to infer the unknown parameter values and initial conditions. The non-uniqueness and non-linearity of this inversion necessitates a probabilistic inversion framework. However, due to the expensive nature of forward dynamic simulations of thermal convection , Markov Chain Monte Carlo methods are rarely used. To address this shortcoming, some studies have recently shown the effectiveness of Mixture Density Networks (MDN) (Bishop 1995) in being able to approximate the posterior probability using only the dataset of simulations run prior to the inversion (Meier et al. 2007, de Wit et al. 2013, K&#228;ufl et al. 2016, Atkins et al. 2016).</p><p>Using MDNs, we systematically isolate the degree to which a parameter can be constrained using different &#8220;present-day&#8221; synthetic observables from 6130 simulations for a Mars-like planet. The dataset &#8211; generated using the mantle convection code GAIA (H&#252;ttig et al. 2013)- is the same as that used by Agarwal et al. (2020) for a surrogate modelling study.</p><p>The loss function used to optimize the MDN (log-likelihood) provides a single robust quantity that can be used to measure how well a parameter can be constrained. We test different numbers and combinations of observables (heat flux at the surface and core-mantle boundary, radial contraction, melt produced, elastic lithospheric thickness, and duration of volcanism) to constrain the following parameters: reference viscosity, activation energy and activation volume of the diffusion creep rheology, an enrichment factor for radiogenic elements in the crust, and initial mantle temperature. If all observables are available, reference viscosity can be constrained to within 32% of its entire range (10<sup>19</sup>&#8722;10<sup>22</sup> Pa s), crustal enrichment factor (1&#8722;50) to within 15%, activation energy (10<sup>5</sup>&#8722;5&#215;10<sup>5</sup> J mol-1 ) to within 80%, and initial mantle temperature (1600&#8722;1800K) to within 39%. The additional availability of the full present-day temperature profile or parts of it as an observable tightens the constraints further. The activation volume (4&#215;10<sup>-6</sup> &#8722;10&#215;10<sup>-6</sup>&#160; m<sup>3</sup> mol<sup>-1</sup>) cannot be constrained and requires research into new observables in space and time, as well as fields other than just temperature. Testing different levels of uncertainty (simulated using Gaussian noise) in the observables, we found that constraints on different parameters loosen at different rates, with initial temperature being the most sensitive. Finally, we present how the marginal MDN proposed by Bishop (1995) can be modified to model the joint probability for all parameters, so that&#160; the inter-parameter correlations and the associated degeneracy can be capture, thereby providing a more comprehensive picture of all the evolution scenarios that fit given observational constraints.</p>
Risk Assessment of Heavy Metals in Basmati Rice: Implications for Public Health
