Inverse Problems

What is an inverse problem with incomplete information?

A forward model is a mathematical construct capable of predicting observational data given a set of parameters, i.e.:

(1)\[\text{observational_data} = f(\text{unknown_parameters, known_parameters}).\]

When some of the forward model parameters are unknown, they comprise the incomplete information to be obtained by solving the inverse problem, which expresses the unknown parameters as a function of the observational data and known parameters, i.e.:

(2)\[\text{unknown_parameters} = f^{-1}(\text{observational_data, known_parameters}).\]

In thermobarometry the forward model is:

(3)\[\text{observed_mineralogy} = f(\text{pressure, temperature, bulk_composition, thermodynamic_data})\]

where, under the equilibrium assumption, the function \(f\) is free energy minimization. Pressure and temperature are by definition unknown parameters, and for a local equilibrium problem bulk composition is formally undefined (Appendix B: Bulk Composition?), in which case the inverse thermobarometry problem is:

(4)\[\{\text{pressure, temperature, bulk_composition}\} = f^{-1}(\text{observed_mineralogy, thermodynamic_data}).\]

More broadly, the unknown parameters of the thermobarometry problem may include unmeasured compositional information and unknown thermodynamic data, and the known parameters may include bulk compositional information. A goal in the development of MC_fit was to provide an interface that would be versatile enough to allow the user to freely choose which parameters are treated as known or unknown.

The challenge in solving the inverse thermobarometry problem is that the inverse function \(f^{-1}\) generally does not exist. Thus, the inverse problem must be solved by iteratively guessing the unknown parameters of Eq 1 and comparing the predicted and actual observational data. In some sense, this description of the inverse problem amounts to little more than the reality of modern thermobarometry, where bulk compositions and unmeasured components are adjusted or guessed to obtain a match with the observed mineralogy. The justification for using a fancy name, i.e., solving an inverse problem with incomplete information, for what MC_fit does, is that MC_fit, and other programs of its ilk (Table 1), reduces guessing and adjusting to a set of objective rules, with some theoretical basis, that can be implemented in a computer program.

To avoid logically awkward constructions, in the remainder of this document the unknown parameters of the inverse problem are referred to as the inversion parameters.