  When estimating quantities and fields that are difficult to measure directly,
such as the fluidity of ice, from point data sources, such as satellite
altimetry, it is important to solve a numerical inverse problem that is
formulated with Bayesian consistency. Otherwise, the resultant probability
density function for the difficult to measure quantity or field will not be
appropriately clustered around the truth. In particular, the inverse problem
should be formulated by evaluating the numerical solution at the true point
locations for direct comparison with the point data source. If the data are
first fitted to a gridded or meshed field on the computational grid or mesh,
and the inverse problem formulated by comparing the numerical solution to the
fitted field, the benefits of additional point data values below the grid
density will be lost. We demonstrate, with examples in the fields of
groundwater hydrology and glaciology, that a consistent formulation can
increase the accuracy of results and aid discourse between modellers and
observationalists.
  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). This new abstraction is well-suited to automation, including
automatic differentiation. We demonstrate this through implementation in
Firedrake, which generates highly optimised code for solving Partial
Differential Equations (PDEs) with the finite element method. Our solution
integrates with dolfin-adjoint/pyadjoint, allowing PDE-constrained optimisation
problems, such as data assimilation, to be solved through forward and adjoint
mode automatic differentiation.
