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When estimating quantities and fields that are difficult to measure directly,such as the fluidity of ice, from point data sources, such as satellitealtimetry, it is important to solve a numerical inverse problem that isformulated with Bayesian consistency. Otherwise, the resultant probabilitydensity function for the difficult to measure quantity or field will not beappropriately clustered around the truth. In particular, the inverse problemshould be formulated by evaluating the numerical solution at the true pointlocations for direct comparison with the point data source. If the data arefirst fitted to a gridded or meshed field on the computational grid or mesh,and the inverse problem formulated by comparing the numerical solution to thefitted field, the benefits of additional point data values below the griddensity will be lost. We demonstrate, with examples in the fields ofgroundwater hydrology and glaciology, that a consistent formulation canincrease the accuracy of results and aid discourse between modellers andobservationalists.  To do this, we bring point data into the finite element method ecosystem asdiscontinuous fields on meshes of disconnected vertices. Point evaluation canthen be formulated as a finite element interpolation operation(dual-evaluation). This new abstraction is well-suited to automation, includingautomatic differentiation. We demonstrate this through implementation inFiredrake, which generates highly optimised code for solving PartialDifferential Equations (PDEs) with the finite element method. Our solutionintegrates with dolfin-adjoint/pyadjoint, allowing PDE-constrained optimisationproblems, such as data assimilation, to be solved through forward and adjointmode automatic differentiation.

Consistent Point Data Assimilation in Firedrake and Icepack