Model calibration for ice sheets and glaciers dynamics: a general theory of inverse problems in glaciology

Numerical modelling of the dynamic evolution of ice sheets and glaciers requires the solution of discrete equations which are based on physical principles (e.g. conservation of mass, linear momentum and energy) and phenomenological constitutive laws (e.g. Glen’s and Fourier’s laws). These equations must be accompanied by information on the forcing term and by initial and boundary conditions (IBCs) on ice velocity, stress and temperature; on the other hand the constitutive laws involve many physical parameters, some of which depend on the ice thermodynamical state. The proper forecast of the dynamics of ice sheets and glaciers requires a precise knowledge of several quantities which appear in the IBCs, in the forcing terms and in the phenomenological laws. As these quantities cannot be easily measured at the study scale in the field, they are often obtained through model calibration by solving an inverse problem (IP). The objective of this paper is to provide a thorough and rigorous conceptual framework for IPs in cryospheric studies and in particular: to clarify the role of experimental and monitoring data to determine the calibration targets and the values of the parameters that can be considered to be fixed; to define and characterise identifiability, a property related to the solution to the forward problem; to study well-posedness in a correct way, without confusing instability with ill-conditioning or with the properties of the method applied to compute a solution; to cast sensitivity analysis in a general framework and to differentiate between the computation of local sensitivity indicators with a one-at-a-time approach and first-order sensitivity indicators that consider the whole possible variability of the model parameters. The conceptual framework and the relevant properties are illustrated by means of a simple numerical example of isothermal ice flow, based on the shallow-ice approximation.