3. Online PDA
With the rapid development of artificial intelligence (AI), it has become possible to move PDA from an offline framework to an online framework by using AI-based surrogate models to efficiently propagate climate states forward in time. Consequently, information from previously assimilated climate states can be carried forward to the current time, enabling temporally consistent climate reconstructions. By integrating data assimilation with AI-based surrogate models, an online PDA framework can be developed.
3.1. Integrated Hybrid EnKF
Although AI-based surrogate models are effective at predicting climate states, they generally provide less reliable estimates of background error statistics. To mitigate sampling errors and model errors in the AI-derived error covariance, information from physical climate models can be incorporated through covariance hybridization. This framework is referred to as the Integrated Hybrid EnKF (IHEnKF), which was first proposed by Lei et al. (2021). Unlike the commonly used hybrid ensemble–variational methods, the IHEnKF achieves covariance hybridization entirely within a pure ensemble framework. Its effectiveness for PDA was further demonstrated by Sun et al. (2025).
3.1.1. IHEnKF Update
Figure 3.1 Framework of the IHEnKF. The prior ensemble is advanced from the previous posterior ensemble using the AI surrogate model and then blended with an analog ensemble. The background error covariance is obtained by hybridizing the covariance estimated from the blended ensemble with the climatological error covariance.
As illustrated in Figure 3.1, blending the AI-predicted ensemble with an analog ensemble generated by a physical climate model increases the ensemble spread and helps prevent filter divergence. Furthermore, hybridizing the blended and climatological error covariances provides a more accurate estimate of uncertainty and reduces the overconfidence associated with the AI-derived covariance alone.
3.1.2. Strongly Coupled Data Assimilation
In addition, AI models can efficiently couple different components of the Earth system (e.g., the atmosphere, ocean, and land), enabling coupled data assimilation. Strongly Coupled Data Assimilation (SCDA) allows observations or proxy records from one Earth system component to directly update the states of other components through the coupled background error covariance. SCDA has been successfully applied to PDA, leading to improved reconstructions of both surface air temperature (SAT) and sea surface temperature (SST) (Lei et al. 2026).
3.2. Integrated Hybrid EnKS
Furthermore, information can be propagated forward in time, allowing future proxy observations to improve the estimation of past climate states. This is possible because PDA is fundamentally a climate reconstruction problem rather than a forecasting problem, where future observations are available for assimilation. By extending the covariance hybridization strategy to the Ensemble Kalman Smoother (EnKS) to improve the estimation of cross-time error covariances, the Integrated Hybrid EnKS (IHEnKS) provides a more suitable framework for PDA.
3.2.1. IHEnKS Update
Figure 3.2 Framework of the IHEnKS. The posterior ensemble produced by the IHEnKF at time t is further updated using future observations at time t + 1.
3.3. Implementation: IHEnKS in SCDA
3.3.1. Proxies and Simulations
Proxies are from the Past Global Changes 2k Network (PAGES2k Emile-Geay et al. 2017) with temporal coverage of the Common Era (0-2000 CE) The proxy system model (PSM), i.e., the forward operator, is constructed by a linear univariate model for each kind of proxy, based on annual means / seasonal averages for tree-based proxies and annual means for the other kinds of proxies (Tardif et al. 2019). For the \(p^{th}\) proxy, the PSM is calibrated relative to the NASA Goddard Institute for Space Studies surface temperature analysis (GISTEMP; Hansen et al. 2010), through the linear regression that gives the best fit to the proxy data. The regression residual \(\sigma _p^2\) is used as the observation error variance, i.e., the diagonal element of the diagonal matrix R
The model data is from the Community Earth System Model-Last Millennium Ensemble Project (CESM LME; Otto-Bliesner et al., 2016). Ensemble simulations of the CESM LME with full forcing, including the transient evolution of solar intensity, volcanic activity, orbital parameters, changes in land use/land cover and greenhouse gases, give a total of 13 full-forcing simulations from 850 to 2005 CE. For annual mean and seasonal averages, the monthly output is processed by removing the climatology over the entire period.
3.3.2. Surrogate Models
The traditional online DA advances the numerical model from the posteriors, and provides priors with flow-dependent uncertainties for next DA cycle. Instead of using numerical models that are computationally expensive, data-driven surrogate models are used here. Due to the different time scales of proxy records, the PSM requires different seasonal averages. Therefore, the integrate time of the surrogate model is usually the average time scale of the proxies, which is generally 12 months for PAGES2k. Based on this, the surrogate model is designed to take in a continuous 12 months of SAT and SST fields and predict for the subsequent 12 months:
where the superscripts \(f\) and \(a\) denote prior and posterior, respectively. M is a self-attention-based surrogate model (Zhou and Zhang 2023)
Figure 3.3 Archeiture of the surrogate model.
The loss function L is defined using the mean square eroor (MSE) of the 12 predictive months and the truth:
3.3.3. DA Methods
Different from ensemble priors of offline DA that are randomly sampled from a climatological simulation, ensemble priors \({\bf{x}}_{cyc,i}^f\) (\(i =1,…,N\)) of online DA are short-term forecasts based on the surrogate models. The ensemble mean \({\bf{\bar x}}_{cyc}^f = \left( {1/N} \right)\sum\limits_{i = 1}^N {{\bf{x}}_{cyc,i}^f}\) and ensemble perturbations \({\bf{x'}}_{cyc,i}^f\) (\(i =1,…,N\)) are updated separately. The ensemble is first updated using the IHEnKF and is then further refined by the IHEnKS.