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Mitigating the impact of atmospheric effects on optical remote sensing data is critical for monitoring intrinsic land processes and developing Analysis Ready Data (ARD). This work develops an approach to this for the NERC NCEO medium resolution ARD Landsat 8 (L8) and Sentinel 2 (S2) products, called Sensor Invariant Atmospheric Correction (SIAC). The contribution of the work 5 is to phrase and solve that problem within a probabilistic (Bayesian) framework for medium resolution multispectral sensors S2/MSI and L8/OLI and provide per-pixel uncertainty estimates traceable from assumed top-of-atmosphere (TOA) measurement uncertainty, making progress towards an important aspect of CEOS ARD target requirements. A set of observational and a priori constraints are developed in SIAC to constrain an estimate of coarse resolution (500m) Aerosol Optical Thickness (AOT) and Total Column Water Vapour (TCWV), along with associated uncertainty. This is thenused to estimate the medium resolution (10-60m) surface reflectance and uncertainty, given an assumed uncertainty of 5% in TOA reflectance. The coarse resolution a priori constraints used are: the MODIS MCD43 BRDF/Albedo product giving a constraint on 500m surface reflectance; and Copernicus Atmosphere Monitoring Service (CAMS) operational forecasts of AOT and TCWV providing estimates of atmospheric state at core 40 km spatial resolution with an associated 500m resolution spatial correlation model. The mapping in spatial scale between medium resolution observations and the coarser resolution constraints is achieved using a calibrated effective Point Spread Function for MCD43. Efficient statistical approximations (emulators) to outputs of the 6S atmospheric radiative transfer code used to estimate the state parameters and in the atmospheric correction. SIAC is demonstrated for a set of global S2 and L8 images covering AERONET and RadCalNet sites. AOT retrievals show a very high correlation to AERONET estimates (R around 0.86, RMSE of 0.07 for both sensors), although with a small bias in AOT. TCWV is accurately retrieved from both sensors (R > 0.96, RMSE < 0.32 g/cm2). Comparisons with in situ surface reflectance measurements from the RadCalNet network show that SIAC provides accurate estimates of surface reflectance across the entire spectrum, with RMSE mismatches with the reference data between 0.01 and 0.02 in units of reflectance, for both S2 and L8. For near-simultaneous S2 and L8 acquisitions, there is a very tight relationship (R > 0.95 for all common bands) between surface reflectance from both sensors, with negligible biases. Uncertainty estimates are assessed through discrepancy analysis and found to provide viable estimates for AOT and TCWV. For surface reflectance, they give conservative estimates of uncertainty, suggesting that a lower estimate of TOA reflectance uncertainty might be appropriate. [ABSTRACT FROM AUTHOR]