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Calibration of unknown model parameters is a common task in many ocean model applications. We present an adjoint‐based optimization of an unstructured mesh shallow water model for the Baltic Sea. Spatially varying bottom friction parameter is tuned to minimize the misfit with respect to tide gauge sea surface height (SSH) observations. A key benefit of adjoint‐based optimization is that computational cost does not depend on the number of unknown variables. Adjoint models are, however, typically very laborious to implement. In this work, we leverage a domain specific language framework in which the discrete adjoint model can be obtained automatically. The adjoint model is both exactly compatible with the discrete forward model and computationally efficient. A gradient‐based quasi‐Newton method is used to minimize the misfit. Optimizing spatially‐variable parameters is typically an under‐determined problem and can lead to over‐fitting. We employ Hessian‐based regularization to penalize the spatial curvature of the friction field to overcome this problem. The SSH dynamics in the Baltic Sea are simulated for a 3‐month period. Optimization of the bottom friction parameter results in significant improvement of the model performance. The results are especially encouraging in the complex Danish Straits region, highlighting the benefit of unstructured meshes. Domain specific language frameworks enable automated model analysis and provide easy access to adjoint modeling. Our application shows that this capability can be enabled with few efforts, and the optimization procedure is robust and computationally efficient. Plain Language Summary Ocean circulation models have several unknown parameters that must be tuned for each application in order to produce physically meaningful results. The tuning process can be a very laborious and time consuming task. In this paper, we investigate an automated way to tune the model's friction at the sea bed to minimize the model's error in predicted sea surface height (SSH). The method is based on a novel way of defining the model's equations which enables solving such optimization problems automatically. The methodology is tested in the Baltic Sea. The modeled SSH is compared against observations at several tide gauges. We show that by optimizing the bottom friction, the model's capability to predict SSH improves significantly. Moreover, we show that the optimization process is robust and computationally efficient. Key Points Adjoint‐based optimization is used to optimize bottom friction coefficient in 2D water elevation model for the Baltic Sea The discrete adjoint model is automatically generated by leveraging a symbolic representation of the discrete forward model equations The optimization method is robust and results in significant improvement in the sea surface height performance at tide gauge locations

Machine learning (ML) weather models like GraphCast and NeuralGCM show forecasting promise but face fundamental limitations for data assimilation (DA) integration. This study reveals critical problems in error covariance representation and adjoint sensitivity patterns challenging their operational viability. We evaluate tangent linear and adjoint models of GraphCast and NeuralGCM by comparing perturbation responses with MPAS‐A, a well‐established numerical weather prediction model. ML models exhibit unphysical adjoint sensitivities, including persistent localized responses and excessive noise at various atmospheric levels, contrasting sharply with physically consistent MPAS‐A patterns and indicating fundamental error covariance representation issues. Implications extend across DA methodologies. Unrealistic sensitivity patterns would generate distorted error covariances in ensemble systems and unphysical analysis increments in variational approaches like 4DVar. Assimilating single observations could create spurious corrections far from observation locations, degrading forecast skill. Despite forecasting capabilities, current ML weather models require significant improvements in linearization properties before reliable operational DA integration.

We assess the ability of neural network emulators of physical parametrization schemes in numerical weather prediction models to aid in the construction of linearized models required by four‐dimensional variational (4D‐Var) data assimilation. Neural networks can be differentiated trivially, and so if a physical parametrization scheme can be accurately emulated by a neural network then its tangent‐linear and adjoint versions can be obtained with minimal effort, compared with the standard paradigms of manual or automatic differentiation of the model code. Here we apply this idea by emulating the non‐orographic gravity wave drag parametrization scheme in an atmospheric model with a neural network, and deriving its tangent‐linear and adjoint models. We demonstrate that these neural network‐derived tangent‐linear and adjoint models not only pass the standard consistency tests but also can be used successfully to do 4D‐Var data assimilation. This technique holds the promise of significantly easing maintenance of tangent‐linear and adjoint codes in weather forecasting centers, if accurate neural network emulators can be constructed. Plain Language Summary The neural network is an algorithm developed in the field of artificial intelligence that can in principle learn the relationship between any two variables, provided you give it enough real‐world data. There are countless applications for such an algorithm in the field of weather and climate simulation. The application that we focus on here is to use the neural network as a replacement for one component of a weather simulation. Essentially, you train the neural network so that it can accurately emulate the component that it replaces. For expensive components, using the neural network emulator instead of the original can provide a significant computational saving. Other studies have already demonstrated that this technique can be applied in weather simulations. What we show here, however, is that neural networks can also be used to easily and automatically calculate the slope of the line relating the two variables in question, through a slight modification of the network. This is an essential procedure for constructing the initial conditions for weather forecasts through a process known as data assimilation. Key Points Neural network emulators of physical parametrization schemes can be used to easily construct tangent‐linear and adjoint models These neural network‐based linear models are potentially much easier to maintain compared with the traditional approach We test these tangent‐linear and adjoint models in data assimilation experiments in a state‐of‐the‐art weather forecasting model

The traditional method for estimating weather forecast sensitivity to initial conditions uses adjoint models, which are limited to short lead times due to linearization around a control forecast. The advent of deep‐learning frameworks enables a new approach using backpropagation and gradient descent to iteratively optimize initial conditions, minimizing forecast errors. We apply this approach to the June 2021 Pacific Northwest heatwave using the GraphCast model, yielding over 90% reduction in 10‐day forecast errors over the Pacific Northwest. Similar improvements are found for Pangu‐Weather model forecasts initialized with the GraphCast‐derived optimal, suggesting that model error is an unimportant part of the perturbations. Eliminating small scales from the perturbations also yields similar forecast improvements. Extending the length of the optimization window, we find forecast improvement to about 23 days, suggesting atmospheric predictability at the upper end of recent estimates. Plain Language Summary This study examines a deep‐learning approach to understanding how small changes to initial conditions impact weather forecasts. Traditionally, a linear approach known as the adjoint method has been used to determine the sensitivity of forecasts to initial conditions. We leverage recent advancements in machine learning to find optimal initial conditions using the backpropagation method within deep‐learning frameworks. This approach iteratively searches for initial conditions that produce the best forecasts. We apply this method to GraphCast forecasts of the June 2021 Pacific Northwest extreme heatwave. We find that small changes to the initial conditions yield nearly perfect 10‐day weather forecasts of the heatwave in both the GraphCast and the Pangu‐Weather models. This research suggests that a significant increase in forecast skill may be possible from existing observations through better estimates of initial conditions. Key Points We use nonlinear gradient descent to optimize initial conditions for weather forecasting with machine learning models Application to the Pacific Northwest June 2021 heatwave reduces 10‐day forecast error by over 90 percent Forecast improvements are not sensitive to the forecast model, and derive mainly from analysis errors on synoptic and larger scales

Recent machine learning (ML)‐based weather forecasting models have improved the accuracy and efficiency of forecasts while minimizing computational resources, yet still depend on traditional data assimilation (DA) systems to generate analysis fields. Four dimensional variational data assimilation (4DVar) enhances model states, relying on the prediction model to propagate observation to the initial field. Consequently, the initial fields from traditional DA are not optimal for ML‐based models, necessitating a customized DA system. This paper introduces an ensemble 4DVar system integrated with the FuXi model (FuXi‐En4DVar), which can independently generate accurate analysis fields. It utilizes automatic differentiation to compute gradients, and demonstrates the equivalence of these gradients with those derived from adjoint models. Experimental results indicate that this system preserves the physical balance of the analysis field and exhibits flow‐dependent characteristics. These features enhance the propagation and assimilation of observation into the initial analysis field, thereby improving the accuracy of the analysis fields. Plain Language Summary Machine learning (ML)‐based weather forecasting models have made significant progress, offering fast and accurate weather predictions. However, a critical limitation of these models is their dependence on externally provided initial fields, which they are unable to generate independently. This study addresses this limitation by developing a data assimilation (DA) system with FuXi, a state‐of‐the‐art ML‐based weather forecasting model, enabling it to generate these initial fields. Experimental results confirm the rationality and effectiveness of this system. Key Points The FuXi‐En4Dvar employ automatic differentiation to compute gradients eliminating the need for tangent linear models and adjoint models Using the rapid ensemble generation capabilities of ML‐based weather forecasting model to construct the background error covariance matrix The FuXi‐En4DVar demonstrates flow‐dependent characteristics, constraining analysis increments that adhere to physical balance relationships

Data assimilation (DA) is integrated with machine learning in order to perform entirely data‐driven online state estimation. To achieve this, recurrent neural networks (RNNs) are implemented as pretrained surrogate models to replace key components of the DA cycle in numerical weather prediction (NWP), including the conventional numerical forecast model, the forecast error covariance matrix, and the tangent linear and adjoint models. It is shown how these RNNs can be initialized using DA methods to directly update the hidden/reservoir state with observations of the target system. The results indicate that these techniques can be applied to estimate the state of a system for the repeated initialization of short‐term forecasts, even in the absence of a traditional numerical forecast model. Further, it is demonstrated how these integrated RNN‐DA methods can scale to higher dimensions by applying domain localization and parallelization, providing a path for practical applications in NWP. Plain Language Summary Weather forecast models derived from fundamental equations of physics continue to increase in detail and complexity. While this evolution leads to consistently improving daily weather forecasts, it also leads to associated increases in computational costs. In order to make a forecast at any given moment, these models must be initialized with our best guess of the current state of the atmosphere, which typically includes information from a limited set of observations as well as forecasts from the recent past. Modern methods for initializing these computer forecasts typically require running many copies of the model, either simultaneously or in sequence, to compare with observations over the recent past and ensure that our best guess estimate of the current state of the atmosphere agrees closely with those observations before making a new forecast. This repeated execution of the computer forecast model is often a time‐consuming and costly bottleneck in the initialization process. Here, it is shown that techniques from the fields of artificial intelligence and machine learning (AI/ML) can be used to produce simple surrogate models that provide sufficiently accurate approximations to replace the original costly model in the initialization phase. The resulting process is self‐contained, and does not require any further utilization of the original computer model when making daily forecasts. Key Points Recurrent neural networks (RNNs) can replace conventional forecast models, producing accurate ensemble forecast statistics and linearized dynamics Data assimilation (DA) is compatible with RNNs by applying state estimation in the hidden state space using a modified observation operator The integrated RNN‐DA methods can be scaled to higher dimensions by applying domain localization techniques

Four-dimensional variational (4D-Var) data assimilation (DA) is developed for a coupled atmosphere–ocean quasigeostrophic application. Complications arise in coupled data assimilation (CDA) systems due to the presence of multiple spatiotemporal scales. Various formulations of the background error covariance matrix ( ), using different localization strategies, are explored to evaluate their impact on 4D-Var performance in a CDA setting. 4D-Var requires access to tangent linear and adjoint models (TLM/AM) to propagate information about the misfit between the forecast and observations within an optimization window. In practice, particularly for coupled models, the TLM and adjoint are often difficult to produce, and for some models are nonexistent in analytic form. Accordingly, a statistical data-driven alternative is also employed and evaluated to determine its feasibility for a 4D-Var CDA system. Using experiments conducted with a coupled atmosphere–ocean quasigeostrophic model, it is found that ensemble generation of flow-dependent error covariance statistics can increase the accuracy of 4D-Var CDA. When observing all variables, the hybrid climatological/flow-dependent constructions outperform either independently. The use of a hybrid matrix combined with a rapid updating ensemble transform Kalman filter (RU-ETKF) using either strongly or weakly CDA resulted in lower overall RMSE. The ocean component achieved its lowest RMSE when using a fully flow-dependent matrix generated using 4D-ETKF and using weakly CDA. These results show the importance of time scales and analysis update frequencies. The use of a statistically derived TLM/AM generated from the ETKF ensemble perturbations produces results similar to cases using the analytical coupled TLM/AM in 4D-Var.

Investigating dryland phreatic water assets requires an in‐deep understanding of depth to water table (DWT). However, current DWT methods suffer from limited accuracy and demand refinement. Therefor, this study suggests one novel integrated model cascaded by twin, assimilation, and DWT submodels. The twin submodel clones land surface model (LSM) with machine learning method (MLM) to capture LSM uncertainties, then the assimilation submodel develops an eigen‐uncertainty‐weighted four‐dimensional variational assimilation framework to optimize LSM outputs using multiple remote sensing (RS) actual evapotranspirations (AETs), thereby, the DWT submodel proposes one physical mechanism based equation with dynamical parameters constrained by optimized LSM outputs and in situ observations. Its effectiveness is evaluated through five pairs of experiments conducted in the Tarim river basin (TRB), China. Results corroborate that the RMSE, MAPE, MAE, R2 of its estimated DWTs are improved by 21.9%–36.1%, 52.9%–58.3%, 49.5%–59.7%, 2.6%–9.3%, respectively, compared to those from pure‐MLMs using original LSM outputs against 84 validation wells across 15 basins. Additionally, the DWT trend obtained well reflects the temporal variability and spatial heterogeneity of the TRB from 2000 to 2020. Owe to its LSM independence and solid physical mechanism, the integrated model ameliorates DWT estimate through a novel insight into the dynamics between phreatic water and heat by maximizing the advantages of multi‐source quality RS AETs and LSM outputs without the adjoint models and running of LSMs.

Thwaites Glacier has experienced accelerating mass loss, with rates increasing over fivefold since the 1990s. We apply transient calibration to two independent ice‐sheet models (STREAMICE and ISSM) using time‐varying velocity and surface elevation data from 2004 to 2017 to project future mass loss through 2067. We test different calibration approaches: constraining to velocities only, surface elevation change only, or both combined. Models calibrated solely to surface elevation change show the best agreement with observed volume‐above‐floatation loss rates and project the largest future mass losses, reaching 180–200 Gt/a by 2067—comparable to current Antarctic‐wide mass balance. These surface‐constrained models produce focused thinning patterns extending ∼100 km inland along Thwaites' deep trough, suggesting potential marine ice sheet instability. In contrast, velocity‐only calibrations show initially high but rapidly stabilizing loss rates. Our results demonstrate that calibration methodology critically influences century‐scale projections, with surface‐elevation‐constrained models providing the most realistic representation of observed dynamical changes.

The most general quantities of interest (called “responses”) produced by the computational model of a linear physical system can depend on both the forward and adjoint state functions that describe the respective system. This work presents the Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (4th-CASAM) for linear systems, which enables the efficient computation of the exact expressions of the 1st-, 2nd-, 3rd- and 4th-order sensitivities of a generic system response, which can depend on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward/adjoint systems. Among the best known such system responses are various Lagrangians, including the Schwinger and Roussopoulos functionals, for analyzing ratios of reaction rates, the Rayleigh quotient for analyzing eigenvalues and/or separation constants, etc., which require the simultaneous consideration of both the forward and adjoint systems when computing them and/or their sensitivities (i.e., functional derivatives) with respect to the model parameters. Evidently, such responses encompass, as particular cases, responses that may depend just on the forward or just on the adjoint state functions pertaining to the linear system under consideration. This work also compares the CPU-times needed by the 4th-CASAM versus other deterministic methods (e.g., finite-difference schemes) for computing response sensitivities These comparisons underscore the fact that the 4th-CASAM is the only practically implementable methodology for obtaining and subsequently computing the exact expressions (i.e., free of methodologically-introduced approximations) of the 1st-, 2nd, 3rd- and 4th-order sensitivities (i.e., functional derivatives) of responses to system parameters, for coupled forward/adjoint linear systems. By enabling the practical computation of any and all of the 1st-, 2nd, 3rd- and 4th-order response sensitivities to model parameters, the 4th-CASAM makes it possible to compare the relative values of the sensitivities of various order, in order to assess which sensitivities are important and which may actually be neglected, thus enabling future investigations of the convergence of the (multivariate) Taylor series expansion of the response in terms of parameter variations, as well as investigating the range of validity of other important quantities (e.g., response variances/covariance, skewness, kurtosis, etc.) that are derived from Taylor-expansion of the response as a function of the model’s parameters. The 4th-CASAM presented in this work provides the basis for significant future advances towards overcoming the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.