Offline PDA *********** If you are familiar with data assimilation in traditional numerical weather prediction (NWP), it is typically implemented in an **online** cycling framework, where the prior weather estimate is obtained by advancing the weather model forward from previously assimilated states. In contrast, PDA is most commonly implemented in an **offline** framework. Rather than obtaining the prior climate estimate by integrating the climate model forward from previously assimilated climate states, the prior is sampled from a pre-existing climatological archive of climate simulations. This approach is motivated by two main factors: computational limitations and the limited predictability of climate models. .. rubric:: Computational limitations 1. Paleoclimate simulations require extremely long integrations, ranging from thousands of years for studies of the Last Millennium and the Holocene to millions of years for deep-time climate simulations (e.g., `Judd et al. 2024 `_). 2. Data assimilation requires an estimate of model uncertainty. In ensemble-based methods, this uncertainty is represented by an ensemble of climate simulations, making such long integrations computationally prohibitive. .. rubric:: Limited predictability 1. The predictability of climate models is limited by uncertainties in external boundary conditions, internal climate variability, and imperfect representations of physical processes. 2. The temporal resolution of paleoclimate proxy observations is typically much coarser than the predictability timescale of climate models. Consequently, information introduced through data assimilation cannot be propagated reliably through model integration until the next observation becomes available. Offline EnKF ============ Within the offline framework, when the prior ensemble is updated using the Ensemble Kalman Filter (EnKF), the method is referred to as the **Offline EnKF (OEnKF)** (e.g., `Hakim et al. 2016 `_; `Tardif et al. 2019 `_). OEnKF Update ------------ .. figure:: figs/OEnKF.png :width: 100% :align: center :name: OEnKF :alt: Framework of the Offline Ensemble Kalman Filter Framework of the Offline EnKF. The prior ensemble is sampled randomly from the climatological ensemble. As illustrated in :numref:`OEnKF`, the prior estimate of temperature at the current time is selected randomly from the climatological archive rather than being propagated from a previously assimilated state. Analog Offline EnKF =================== Because the prior ensemble in OEnKF is selected randomly from the climatological ensemble, both the prior states and their associated error statistics lose their flow dependence. A natural improvement is to use proxy observations to preconstrain the prior selection by identifying analog states from the climatology that are most similar to the proxy observations. These analog priors are then updated using the EnKF. We refer to this as the **Analog Offline EnKF (AOEnKF)** (`Sun et al. 2022 `_). .. _AOEnKF update: AOEnKF Update ------------- .. figure:: figs/AOEnKF.png :width: 100% :align: center :name: AOEnKF :alt: Framework of the Analog Offline EnKF Framework of the AOEnKF. The prior ensemble is selected from the climatological ensemble based on its similarity to the proxy observations. Framework of the AOEnKF. The prior ensemble is selected from the climatological ensemble based on its similarity to the proxy observations. Note that the black stars denote the same ensemble members shown in green in :numref:`OEnKF`. As illustrated in :numref:`AOEnKF`, the prior estimate of temperature at the current time is selected to be closer to the true climate state (i.e., a warmer temperature in this example). Hybrid Gain Analog Offline EnKF =============================== Although the analog ensemble provides a flow-dependent prior state and associated error statistics, its error covariance estimate is more susceptible to sampling errors because of the limited ensemble size. In contrast, the climatological ensemble is typically much larger and therefore provides a more robust estimate of the background error covariance. Moreover, different paleoclimate periods are characterized by different climatological states (e.g., a relatively stable climate during the Last Millennium versus a warming climate following the Last Glacial Maximum). Incorporating this climatological information into the background error statistics can therefore improve the quality of the covariance estimate. The **Hybrid Gain Analog Offline EnKF (HGAOEnKF)** (`Sun et al. 2024 `_) combines the error statistics estimated from the analog ensemble and the climatological ensemble through covariance hybridization. .. _HGAOEnKF update: HGAOEnKF Update --------------- .. figure:: figs/HGAOEnKF.png :width: 100% :align: center :name: HGAOEnKF :alt: Framework of the HGAOEnKF Framework of the HGAOEnKF. The prior state is selected from the analog ensemble, while the background error covariance is obtained by hybridizing the analog and climatological error covariances. As illustrated in :numref:`HGAOEnKF`, hybridizing the analog and climatological error covariances provides a more accurate estimate of uncertainty and reduces the overconfidence associated with the pure analog ensemble.