Physics-based models capture broad spatial and temporal dynamics, but often suffer from biases and numerical approximations, while observations capture localized variability but are sparse. Integrating these complementary data modalities is important to improving the accuracy and reliability of model outputs. Meanwhile, physics-based models typically generate large outputs that are challenging to manipulate. In this paper, we propose Sig-PCA, a space-time framework that integrates summary statistics from model outputs with localized observations via a neural network (NN). By leveraging reduced-order representations from physics-based models and integrating them with observational data, our approach corrects model outputs, while allowing to work with dimensionally-reduced quantities hence with smaller NNs. This framework highlights the synergy between observational data and statistical summaries of model outputs, and effectively combines multisource data by preserving essential statistical information. We demonstrate our approach on two datasets (surface temperature and surface wind) with different statistical properties and different ratios of model to observational data. Our method corrects model outputs to align closely with the observational data, specifically enabling to correct probability distributions and space-time correlation structures.

In recent years, the convergence of data-driven machine learning models with Data Assimilation (DA) offers a promising avenue for enhancing weather forecasting. This study delves into this emerging trend, presenting our methodologies and outcomes. We harnessed the UK's local ERA5 850 hPa temperature data and refined the U-STN12 global weather forecasting model, tailoring its predictions to the UK's climate nuances. From the ASOS network, we sourced T2m data, representing ground observations across the UK. We employed the advanced kriging method with a polynomial drift term for consistent spatial resolution. Furthermore, Gaussian noise was superimposed on the ERA5 T850 data, setting the stage for ensuing multi-time step synthetic observations. Probing into the assimilation impacts, the ASOS T2m data was integrated with the ERA5 T850 dataset. Our insights reveal that while global forecast models can adapt to specific regions, incorporating atmospheric data in DA significantly bolsters model accuracy. Conversely, the direct assimilation of surface temperature data tends to mitigate this enhancement, tempering the model's predictive prowess.