Consistent Point Data Assimilation in Firedrake and Icepack

We present methods and tools that significantly improve the ability to estimate quantities and fields which are difficult to directly measure, such as the fluidity of ice, using point data sources, such as satellite altimetry. These work with both sparse and dense point data with estimated quantities and fields becoming more accurate as the number of measurements are increased. Such quantities and fields are often used as inputs to mathematical models that are used to make predictions so improving their accuracy is of vital importance. We demonstrate how our methods and tools can increase the accuracy of results, ensure posterior consistency, and aid discourse between modellers and experimenters. To do this, we bring point data into the finite element method ecosystem as discontinuous fields on meshes of disconnected vertices. Point evaluation can then be formulated as a finite element interpolation operation (dual-evaluation). Our new abstractions are well-suited to automation. We demonstrate this by implementing them in Firedrake, which generates highly optimised code for solving PDEs with the finite element method. Our solution integrates with dolfin-adjoint/pyadjoint which allows PDE-constrained optimisation problems, such as data assimilation, to be solved through forward and adjoint mode automatic differentiation. We demonstrate our new functionality through examples in the fields of groundwater hydrology and glaciology.