Iterative Regularisation and Uncertainty Propagation in Solving Ill-Posed Problems: A GOCE-Based Regional Gravity Field Modelling

Abstract Inverse problems are fundamental in geoscience, enabling estimation of Earth’s physical properties that cannot be measured directly. These problems are typically ill-posed, meaning their solutions are highly sensitive to noise and variability in observational data, and thus require regularisation to ensure stability. Both direct and iterative regularisation methods have been extensively studied across geoscientific applications. However, accurately quantifying uncertainty—particularly for iterative regularisation—remains a major challenge, as uncertainty estimation in these methods is less well understood. This study examines the performance and uncertainty characteristics of four iterative regularisation techniques—conjugate gradient least-squares (CGLS), minimum residual 2 (MR2), algebraic reconstruction technique (ART), and the ν -method (Nu). Here, a comprehensive conceptual overview of these methods is provided, along with an evaluation of two internal uncertainty quantification approaches: (a) the error propagation using numerical Jacobians, and (b) the Monte Carlo method. Although computationally demanding, these techniques offer valuable insights into how observational errors propagate through nonlinear iterative processes. The methods are applied to the regional recovery of gravity anomalies at sea level from the Gravity field and steady-state Ocean Circulation Explorer (GOCE) 2nd-order radial gradients over Alpine and surrounding south-central European region as a case study. Results demonstrate that different iterative techniques yield distinct solutions even under identical regularisation criteria (e.g., the L-curve), and that the magnitude of propagated error alone is not a reliable indicator of solution quality. The estimated gravity anomalies and their uncertainties by the MR2 and ART are close to each other. Nu provides faster, smoother solutions with lower uncertainties, while the CGLS preserves more high-frequency geophysical signals at the cost of higher propagated errors. Error propagation with numerical Jacobians produces smaller error estimates compared to with the Monte Carlo method, which consistently reports larger uncertainties, reflecting its ability to account for both random noise and nonlinear effects. The difference between the estimated uncertainties by these methods over the study area are around 2 mGal or less.