Preprints
https://doi.org/10.48550/arXiv.2304.06058
https://doi.org/10.48550/arXiv.2304.06058
27 Jun 2023
 | 27 Jun 2023

Consistent Point Data Assimilation in Firedrake and Icepack

Reuben W. Nixon-Hill, Daniel Shapero, Colin J. Cotter, and David A. Ham

Abstract. 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.

Reuben W. Nixon-Hill, Daniel Shapero, Colin J. Cotter, and David A. Ham

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on egusphere-2023-984', Douglas Brinkerhoff, 28 Jul 2023
  • RC2: 'Comment on egusphere-2023-984', Umberto Villa, 19 Dec 2023
  • AC1: 'Comment on egusphere-2023-984', Reuben Nixon-Hill, 16 Feb 2024
  • AC2: 'Comment on egusphere-2023-984', Reuben Nixon-Hill, 16 Feb 2024
Reuben W. Nixon-Hill, Daniel Shapero, Colin J. Cotter, and David A. Ham
Reuben W. Nixon-Hill, Daniel Shapero, Colin J. Cotter, and David A. Ham

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Short summary
Scientists often use models to study complex processes, like the movement of ice sheets, and compare them to measurements for estimating hard-to-measure quantities. We highlight an approach that ensures accurate results from point data sources (such as height measurements) by evaluating the numerical solution at true point locations. This method improves accuracy, can aid communication between scientists, and is well suited for integration with specialised software that automates the processes.