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A four‐dimensional ensemble‐variational (4DEnVar) data assimilation (DA) system was developed based on the global forecast system of the Global/Regional Assimilation and Prediction System (GRAPES‐GFS). Instead of using the adjoint technique, this system utilizes a dimension‐reduced projection (DRP) technique to minimize the cost function of the standard four‐dimensional variational (4DVar) DA. It dynamically predicts ensemble background error covariance (BEC) and realizes the explicit flow‐dependence of BEC in the variational configuration. An inflation technique based on a linear combination of analysis increments and balanced random perturbations, is utilized to overcome the problem of underestimation of BEC matrix (B‐matrix) during the assimilation cycle. To mitigate the spurious correlations in the ensemble B‐matrix caused by the insufficient ensemble members, an ensemble‐sample‐based subspace localization method is utilized. In order to evaluate the new system, single‐point observation experiments (SOEs) and observing system simulation experiments (OSSEs) were conducted with sounding and cloud‐derived wind data based on GRAPES‐GFS. The explicit flow‐dependent characteristic of the 4DEnVar system using a localized ensemble covariance was verified in the SOEs. In the OSSEs, the ensemble mean analysis of 4DEnVar outperforms the analysis of 4DVar. The deterministic forecast initialized from the 4DEnVar ensemble mean analysis has better performance in the short‐range forecasts, better (worse) performance in the early (late) period of the medium‐range forecasts in the Northern Extratropics, and opposite performance in the Southern Extratropics, and exhibits slightly worse effects in the Tropics. Moreover, the ensemble mean forecast initialized by the 4DEnVar system has higher forecast skills. Plain Language Summary Medium‐range numerical weather prediction aims to predict weather states for future 1–10 days from the current state by solving the initial value problem of a set of partial differential equations. Data assimilation (DA) is one of the key techniques to improve forecast skills, which attempts to provide an optimal estimation of the current state by combining observations and forecasts. This study developed a four‐dimensional ensemble‐variational (4DEnVar) DA system based on the global forecast system of the Global/Regional Assimilation and Prediction System (GRAPES‐GFS) applying the dimension‐reduced projection (DRP) four‐dimensional variational (4DVar) approach. Compared with the standard 4DVar, which is generally recognized as one of the most advanced DA methods, this new system has three unique features. First, it dynamically estimates background error covariance (BEC) during the assimilation cycle instead of adopting a pre‐estimated static BEC as 4DVar does. Second, it uses a pure anisotropic ensemble covariance. Third, it can avoid using adjoint models and handle nonlinear problems well. The observing system simulation experiments based on GRAPES‐GFS verify that 4DEnVar has smaller analysis errors, and better ensemble mean forecast skills than 4DVar, and comparable skills of deterministic forecast initialized from the ensemble mean analysis to 4DVar. Key Points A DRP‐4DVar based 4DEnVar data assimilation system with the flow‐dependent background error covariance was developed for global numerical weather prediction The deterministic forecast initialized from the 4DEnVar ensemble mean analysis has performance comparable to 4DVar in the Extratropics Higher quality of analyses and ensemble forecasts can be produced by the 4DEnVar system relative to the 4DVar system

Ensemble‐based convective‐scale radar data assimilation commonly suffers from an underdispersive background ensemble. This study introduces a multivariate additive‐inflation method to address such deficiency. The multivariate additive inflation (AI) approach generates coherent random perturbations drawn from a newly constructed convective‐scale static background error covariance matrix for all state variables including hydrometeors and vertical velocity. This method is compared with a previously proposed univariate AI approach, which perturbs each variable individually without cross‐variable coherency. Comparisons are performed on the analyses and forecasts of the 8 May 2003 Oklahoma City tornadic supercell. Within assimilation cycles, the multivariate approach is more efficient in increasing reflectivity spread and thus has a reduced spinup time than the univariate approach; the additional inclusion of hydrometeors and vertical velocity results in more background spread for both reflectivity and radial velocity. Significant differences among AI experiments also exist in the subsequent forecasts and are more pronounced for the forecasts initialized from the earlier assimilation cycles. The multivariate approach yields better forecasts of low‐level rotation, reflectivity distributions, and storm maintenance for most lead times. The additional inclusion of hydrometeor and vertical velocity in the multivariate method is beneficial in forecasts. Conversely, the additional inclusion of hydrometeor and vertical velocity in the univariate method poses negative impacts for the majority of forecast lead times. Plain Language Summary Data assimilation (DA) requires an accurate estimation of the background uncertainty. In ensemble‐based DA, such uncertainty is represented by the statistics of a background ensemble. However, it is difficult to construct ideal background ensembles that truly represent forecast errors. The deficient background ensemble becomes more problematic for convective‐scale radar DA when all ensemble members miss the observed storms. This study proposes a multivariate additive‐inflation approach to address such deficiency. The mitigation of ensemble deficiency is achieved by drawing spatially and physically coherent random perturbations from a recently constructed convective‐scale static background error covariance matrix and adding them to each ensemble member. This study assesses the new approach by comparing it with a previously proposed univariate approach, which perturbs each variable individually without cross‐variable coherency. Results and diagnostics from a tornadic supercell study show that the multivariate approach improves the analysis and forecast compared to the univariate approach. The multivariate approach by additionally perturbing hydrometeors and vertical velocity further improves the forecast. In contrast, the univariate approach including the hydrometeors and vertical velocity perturbations degrades the forecast. Key Points A multivariate additive‐inflation approach is introduced to address the deficient ensemble for ensemble‐based radar data assimilation The multivariate approach improves the analysis and forecast of a tornadic supercell relative to a previous univariate approach The multivariate approach by additionally perturbing hydrometeors and vertical velocity is beneficial in analysis and forecast

The current NCEP operational four‐dimensional ensemble‐variational data assimilation system uses a control forecast at T1534 resolution coupled with an 80 member ensemble at T574 resolution. Given an increase in computing resources, and assuming the control forecast resolution is fixed, would it be better to increase the ensemble size and keep the ensemble resolution the same, or increase the ensemble resolution and keep the ensemble size the same? To answer this question, experiments are conducted at reduced resolutions. Two sets of experiments are conducted which both use approximately four times more computational resources than the control experiment that uses a control forecast at T670 and an 80 member ensemble at T254. One increases the ensemble size to 320 but keeps the ensemble resolution at T254; and the other increases the ensemble resolution to T670 but retains an 80 ensemble size. When ensemble size increases to 320, turning off the static component of the background‐error covariance does not degrade performance. When the data assimilation parameters are tuned for optimal performance, increasing either ensemble size or ensemble resolution can improve the forecast performance. Increasing ensemble resolution is slightly, but significantly better than increasing ensemble size for these experiments, particularly when considering errors at smaller scales. Much of the benefit of increasing ensemble resolution comes about by eliminating the need for a deterministic control forecast and running all of the background forecasts at the same resolution. In this “single‐resolution” mode, the control forecast is replaced by an ensemble average, which reduces small‐scale errors significantly. Key Points The trade‐offs between ensemble size and ensemble resolution for an ensemble‐variational data assimilation system are evaluated Increasing either ensemble size or ensemble resolution can improve the forecast performance Increasing ensemble resolution is better than increasing ensemble size, particularly when considering errors at smaller scales

We propose a method of flow-dependent large-scale blending (LSB) method for limited-area model data assimilation (LAM DA). By incorporating the information from the global model (GM), LSB methods alleviate the large-scale degradation caused by limitations in the domain size and observations. Our proposed LSB method (nested EnVar) extends the static variational DA augmented by GM information (nested 3DVar) of previous studies, thus dynamically determining the relative weights of GM information based on the uncertainties in GM. The simultaneous assimilation of GM information by the nested EnVar avoids disturbing the optimal state of DA caused by independent blending. The nested EnVar is compared against the nested 3DVar and background LSB methods in the cycled assimilation experiments using a nested system of chaotic models with a single spatial dimension. We also investigate the impact of flow-dependency on the blended analysis and forecast. All LSB methods reduce the large-scale errors in LAM DA and provide better analyses and forecasts than GM downscaling. When dense and uneven observations are assimilated into the LAM domain, the nested EnVar outperforms the conventional DA and other LSB methods. By dynamically incorporating the GM uncertainty, the nested EnVar improves the analyses and their stability across scales. These results suggest that the nested EnVar is a promising alternative to traditional LSB methods in high-resolution simulations of hierarchical phenomena with high variability.

This study examines a hurricane prediction system that uses an ensemble Kalman filter (EnKF) to assimilate high‐resolution airborne radar observations for convection‐permitting hurricane initialization and forecasting. This system demonstrated very promising performance, especially on hurricane intensity forecasts, through experiments over all 61 applicable NOAA P‐3 airborne Doppler missions during the 2008–2010 Atlantic hurricane seasons. The mean absolute intensity forecast errors initialized with the EnKF‐analysis of the airborne Doppler observations at the 24‐ to 120‐h lead forecast times were 20–40% lower than the National Hurricane Center's official forecasts issued at similar times. This prototype system was first implemented in real‐time for Hurricane Ike (2008). It represents the first time that airborne Doppler radar observations were successfully assimilated in real‐time into a hurricane prediction model. It also represents the first time that the convection‐permitting ensemble analyses and forecasts for hurricanes were performed in real‐time. Also unprecedented was the on‐demand usage of more than 23,000 computer cluster processors simultaneously in real‐time. Key Points Systematic use of airborne Doppler radar data Ensemble based analysis and prediction for cloud‐resolving hurricane prediction Significant improvement in hurricane intensity hurricane

We present a candidate mean secular variation (SV) model for the 2025.0 - 2030.0 period. The forecasted SV is produced with a data assimilation (DA) system built around a simple frozen-flux model of the core flow and magnetic field near the core–mantle boundary (CMB). An Ensemble Kalman Filter (EnKF) and smoother (EnKS) are used to assimilate Gauss coefficients from the Kalmag field model, to estimate a core flow which is then used to predict changes in the magnetic field. This forecast methodology is tested against past 5-year periods where it is found to be effective in predicting mean SV, and is superior to an otherwise identical setup using an EnKF alone (no EnKS). The inferred core flow is examined and is seen to exhibit structures consistent with the eccentric gyre and westward drift found in traditional inversions. While this study presents an SV candidate, its secondary purpose is to explore and highlight the potential of the EnKS methodology in understanding the geodynamo. Notably, the EnKS algorithm we use requires no adjoint for the model and can be implemented into already existing EnKF-based systems. The ease of implementation and improvement provided by the EnKS make it a desirable addition to other geomagnetic data assimilation systems, particularly those built around full, 3-D numerical dynamo models, for which the production and maintenance of an adjoint can be challenging. Graphical Abstract

Identifying highly channelized hydraulic conductivity fields and contaminant source parameters remains a challenging task, primarily due to the non‐Gaussian nature and high dimensionality of the parameter space, as well as the computational burden caused by repeatedly running forward numerical models. This study proposes a novel deep learning parameterization method called AEdiffusion, which combines Diffusion Denoising Probabilistic Model (DDPM) with Variational Autoencoder (VAE) for dimensionality reduction. The method employs a generator‐refiner strategy to generate high‐dimensional aquifer properties from low‐dimensional latent representations. The inversion modeling was performed on a synthetic non‐Gaussian hydraulic conductivity field with line‐source contamination using the Iterative Local Updating Ensemble Smoother (ILUES) algorithm. The results demonstrate that the AEdiffusion‐ILUES framework can accurately identify model parameters. To reduce the computational burden, an AR‐Net‐WL (ARNW) surrogate model was introduced, resulting in an efficient inversion framework (AEdiffusion‐ILUES‐ARNW) with similar prediction accuracy and predictive uncertainty estimation as the AEdiffusion‐ILUES but at a lower computational cost. Plain Language Summary Identifying highly channelized hydraulic conductivity fields and contaminant source parameters is crucial for developing groundwater remediation strategies. However, this remains a challenging task due to the non‐Gaussian nature and high dimensionality of the parameter space, as well as the computational burden caused by repeatedly running numerical models. We propose a novel deep learning‐based inversion framework to identify hydraulic conductivity fields and contaminant sources from sparse and error‐prone observations. Key Points A novel and accurate deep learning parameterization method combining DDPM and VAE is proposed to parameterize non‐Gaussian hydraulic conductivity fields A deep autoregressive neural network is integrated into the inversion framework as a surrogate to alleviate the high computational cost of the forward numerical models The integrated approach is assessed with inverse problems for the identification of a non‐Gaussian conductivity and line contaminant source parameters

On ensemble‐based coupled data assimilation, cross‐component parameter estimation (CPE), has not been as extensively developed and applied as weakly coupled state and parameter estimation along with cross‐component state estimation. This discrepancy is partially attributed to the lack of emphasis on the instantaneous response of coupled model states with respect to parameters across different components. We define so‐called response as the instantaneous parameter sensitivity (IPS). Under the framework of sequential assimilation, the prior information heavily relies on the IPS of coupled states with different time scales. Based on the IPS analysis for an intermediate coupled model, a series of twin experiments of state and parameter estimation are conducted, in which an IPS‐inspired adaptive inflation scheme for parameter ensemble is introduced. Results show that the success of a parameter estimation strategy is closely tied to the significant IPS of the observed state to the parameter targeted for optimization, as it maintains a high signal‐to‐noise ratio in the error covariance between parameter and prior state, thereby enhancing parameter estimation. An interesting finding in the context of IPS‐based CPE is: an atmospheric parameter can be successfully estimated by assimilating observations from slow‐varying oceanic component, but not vice versa. In comparison with cross‐component state estimation, successful CPE significantly enhances the estimation accuracy of coupled states by mitigating model bias. Plain Language Summary The sequential data assimilation, exemplified by the ensemble‐based methods, focuses on adjusting model state and/or parameter instantaneously. Its underlying theoretical basis lies in the fact that random fluctuations characterize the Earth system in nature. Previous studies on ensemble‐based parameter estimation concentrated on the climatological response of state to parameter. However, this study emphasizes the instantaneous impact of parameter on state, denoted as the instantaneous parameter sensitivity (IPS), which is crucial for interpretating the feasibility of cross‐component parameter estimation (CPE) in the scope of strongly coupled data assimilation (CDA). Assimilating fast‐varying atmospheric observation has been employed to estimate slow‐varying oceanic state, but it cannot be taken for granted to optimize oceanic parameter. Conversely, assimilating oceanic observation can optimize atmospheric parameter and coupling coefficient since the ocean significantly responds to their direct modulation. The IPS serves as a tool for researchers to preliminarily assess the feasibility of CPE, representing a valuable advance particularly in the realm of strongly CDA. Key Points The instantaneous parameter sensitivity plays a crucial role in interpreting the feasibility of ensemble‐based parameter estimation The significant instantaneous response of model state to parameter maintains a high signal‐to‐noise ratio in their error covariance Atmospheric parameter and coupling coefficient related to ocean‐atmosphere interaction can be optimized by assimilating oceanic observation

A new approach is presented and evaluated for efficiently applying scale-dependent spatial localisation to ensemble background-error covariances within an ensemble-variational data assimilation system. The approach is primarily motivated by the requirements of future data assimilation systems for global numerical weather prediction that will be capable of resolving the convective scale. Such systems must estimate the global and synoptic scales at least as well as current global systems while also effectively making use of information from frequent and spatially dense observation networks to constrain convective-scale features. Scale-dependent covariance localisation allows a wider range of scales to be efficiently estimated while simultaneously assimilating all available observations. In the context of an idealised numerical experiment, it is shown that using scale-dependent localisation produces an improved ensemble-based estimate of spatially varying covariances as compared with standard spatial localisation. When applied to an ensemble of Arctic sea-ice concentration, it is demonstrated that strong spatial gradients in the relative contribution of different spatial scales in the ensemble covariances result in strong spatial variations in the overall amount of spatial localisation. This feature is qualitatively similar to what might be expected when applying an adaptive localisation approach that estimates a spatially varying localisation function from the ensemble itself. When compared with standard spatial localisation, scale-dependent localisation also results in a lower analysis error for sea-ice concentration over all spatial scales.

This study investigates the impact of assimilating Formosat-7/COSMIC-II (FS7/C2) radio occultation (RO) refractivity data on predicting the heavy rainfall event that occurred in Taiwan on August 13, 2019. This event was characterized by heavy rainfall over the coastal region of central and southwestern Taiwan. Our investigation is performed using the Weather Research and Forecasting-Local Ensemble Transform Kalman Filter. Generally, assimilating the RO data increases the amount of moisture over the northern South China Sea (SCS) and the Pearl River area in southern China. It was expected that assimilating the RO data would improve low-level moisture analysis, given that more RO data are available for the lower atmosphere compared to those from Formosat-3/COSMIC-I. However, our results show that the experiment that does not include the RO data below 3 km facilitates better rainfall prediction over Taiwan in terms of the intensity and location of heavy rainfall. This heavy rainfall event can be attributed to moisture transport from the Pearl River area, where the RO data at the altitude of 3–5 km provide effective moisture enhancement to deepen the high-moisture layer. The experiment using the local spectral width (LSW) to conduct the quality control (QC) also helps improve rainfall prediction. However, such an LSW-based QC procedure tends to reject significant amounts of RO data 3 km above the land. Based on this case study, our results show that the QC procedure brings a larger impact to rainfall prediction than counterparts that adjust the observation error variance. A sophisticated QC procedure should be developed to optimize the impact of low-level RO data. Key points Assimilating FS7/C2 RO data above 3 km improves moisture and rainfall prediction. LSW-based QC shows that RO data quality is sensitive to the land-sea distribution. QC is more crucial to optimize the low-level RO assimilation than R adjustment.