Supplementary material to "Sea Ice Albedo Bounded Data Assimilation and Its Impact on Modeling: A Regional Approach"

Adaptive Inflation for Sea Ice DA in QCEFFThe introduction of a bounded DA scheme complicates the use of classical inflation techniques.In traditional ensemble DA, multiplicative inflation is often applied to artificially increase model spread, enhancing the ensemble's ability to capture observed variability (Anderson and Anderson, 1999).However, this approach becomes problematic when applied to bounded variables, which must remain within strict physical limits.For example, during the melt season, SIC within a grid cell can approach zero.When is near zero, inflating the spread by a factor intended for Gaussian distributions can produce ensemble members that fall below zero-an unphysical state-or expand the distribution in a way that still fails to include nearby observed values.If, after inflation, the observation still falls outside the ensemble's acceptable range (here defined as within 2 of the mean), the DA framework may reject it as inconsistent with the model prior.This mismatch highlights the challenge of using symmetric inflation schemes for variables with hard physical bounds and skewed distributions near those bounds.Observation rejections can also occur even when values are not near physical bounds.These rejections stem from insufficient ensemble spread rather than model limitations, especially during rapid changes when diverges from the true model-generated observation.This issue is especially common during the melt season, when fast transitions often associated with albedo feedback and driven by melt-ponding, refreezing, or snowfall introduce variability that bounded ensemble systems struggle to accommodate.To address these challenges, we apply a temporally varying adaptive inflation scheme available in DART, which enforces a minimum model spread (El Gharamti, 2018).This scheme models inflation factors as inverse-gamma distributed random variables.Inflation values evolve over time alongside the ensemble state, with their means and variances updated based on observational input.In our implementation within the QCEFF (Quantile Conserving Ensemble Filter Framework), the adaptive inflation diverges from its original formulation.Rather than enforcing strict bounds through priors or hard constraints-as in traditional bounded inflation schemes-the QCEFF-compatible version emphasizes physical consistency and conservation across the ensemble.This reformulation decouples inflation from rigid statistical boundaries and instead aligns it with the QCEFF's diagnostic balance principles (Anderson, 2022(Anderson, , 2023)).This adaptive inflation framework ensures stability and consistency by preserving the integrity of the initial ensemble spread introduced during the spin-up phase.We fix the minimum inflation factor at 1.0, maintaining the original ensemble spread prior to data assimilation.The upper bound is set at 50.0, although this limit is rarely approached due to the physical constraints imposed by the bounded DA scheme.To regulate how inflation evolves over time, we constrain the standard deviation of the inverse gamma distribution used to sample the inflation factor.Specifically, the distribution's standard deviation must be no smaller than 0.6, ensuring a minimum level of ensemble variability, and the distribution's width is restricted such that the inflation standard deviation cannot grow or shrink by more than a factor of 1.05 per time step.These constraints, informed by