Direct and Inverse Modeling of Gas Diffusion in Polar Firn with Variable Diffusion Coefficients

Abstract

The integration of data analysis with partial differential equation (PDE) models provides a powerful framework for estimating key parameters governing complex physical phenomena, particularly in climate and environmental sciences. In the study of historical climate change, accurate reconstruction of atmospheric gas concentrations, most notably carbon dioxide, is essential to understand global warming trends. Polar firn (compacted snow) serves as a natural archive of past atmospheric conditions, as gases diffuse through its open porosity before being permanently trapped in ice.

This thesis addresses the identification of gas parameters in polar firn, specifically the diffusion and volume fraction coefficients by formulating and solving a direct and inverse problem in which the diffusion volume coefficients are allowed to vary with depth, in line with the work done in MoufawadNassifTriki2024. Specifically, both coefficients are modeled as spatially dependent functions of the depth variable 'z', rather than being restricted to a constant value, allowing a more realistic representation of the firn heterogeneity.

The study begins with the analysis of a direct problem, consisting of a time-dependent diffusion-convection partial differential equation, which describes the gas diffusion in firn for given gas coefficients, D(z), f(z), respectively, the diffusion and volumetric fraction coefficients.

We consider then the problem of existence and uniqueness of solutions to the direct problem, followed by its numerical solution using semi-variational discretizations in space and finite-difference in time, that are implemented in Matlab.

Building then on the direct model, the inverse problem seeks to reconstruct the unknown coefficients from present-day firn concentration measurements. The resulting inverse formulation leads to a constrained optimization problem that is solved using efficient numerical optimization algorithms. Numerical experiments for both the direct and the inverse problems demonstrate the stability and accuracy of the proposed methodology. The results confirm the feasibility of recovering depth-dependent coefficients in the firn and highlight the effectiveness of inverse modeling techniques for characterizing gas transport processes in polar ice.