In contemporary Earth observation (EO), measurement models are implemented as end‑to‑end data‑processing codes that transform raw satellite observations into geophysical products. These codes evolve continuously as algorithms, auxiliary data, and implementations change, which makes traditional, hand‑crafted uncertainty bookkeeping brittle and difficult to maintain. Classical GUM‑style treatments, based on fixed analytical expressions and static data flows, provide important principles but do not naturally accommodate this reality of evolving software systems.
A different route is to treat the data‑processing code itself as the primary object of differentiation. Modern machine learning frameworks already provide highly optimized algorithmic differentiation (AD) backends, which can be repurposed to obtain Jacobians directly from the actual implementation whenever needed. This allows local linearizations to remain exactly consistent with the running code, rather than with a separate analytical surrogate, and makes it possible to formulate uncertainty propagation directly in terms of algorithmically differentiable programs.
On top of these AD backends, tensor‑level formulations of the law of propagation of uncertainty can be implemented that operate directly on multidimensional EO data. Instead of flattening images, spectra, and spatiotemporal cubes into vectors, one can work with Jacobian and covariance tensors whose shapes reflect the underlying spatial and temporal structure. This preserves correlations and aligns more naturally with how EO scientists think about their data, while still enabling efficient propagation of full covariance information.
A central novelty of this approach is that Jacobian and uncertainty tensors become active citizens inside the data‑processing codes, not just external annotations or post‑hoc diagnostics. Sensitivities and uncertainties can be computed, transformed, and consumed as part of the workflow itself—for example to steer parameter estimation, drive adaptive algorithms, or expose uncertainty‑aware outputs at multiple stages of a processing chain. The concept is currently being explored on concrete EO sub‑algorithms as initial testbeds, but is designed to be generic: any data‑processing pipeline that can be expressed in an AD‑enabled framework (such as JAX) can, in principle, be turned into an algorithmically differentiable measurement model with automatic, tensor‑level uncertainty propagation built in.
Abstract submitted to ESA
$\Phi$ -innovation Summit 2026 https://philab.esa.int/phinnovation/.