The possibility of influencing extreme weather phenomena has been discussed for decades; however, it remains far from operational practice, and there is still no established framework for designing small, spatially localized perturbations that can reliably steer chaotic geophysical flows. In this study, we propose a hybrid control method, termed ensemble-Kalman-guided model predictive path integral control (EKG-MPPI), which combines ensemble Kalman control (EnKC) with model predictive path integral (MPPI) control. Within a control simulation experiment framework, an ensemble Kalman filter is first used for state estimation, after which EnKC computes a candidate perturbation by treating the control objective as a pseudo-observation. An adaptive thresholding procedure then enforces spatial sparsity, so that the EnKC perturbation identifies candidate actuator locations and their nominal amplitudes. This information is embedded into the mean and covariance of Gaussian proposal distributions for MPPI, which subsequently refines the perturbation through sampling-based optimization with nonlinear rollouts, without linearizing the dynamics or computing gradients. Numerical experiments with the Lorenz-96 model and the surface quasi-geostrophic (SQG) model demonstrate that EKG-MPPI can suppress extremes in state variables and regional wind speed more effectively than EnKC alone, while using comparable or smaller control inputs. These results highlight EKG-MPPI as a promising building block for simulation-based assessment of localized intervention strategies in geophysical flows.
During the past decade, a succession of record low sea ice events has led to the suggestion that a shift in the overall dynamics of sea ice in the Antarctic region is underway. We attempt to gain fresh insight into how persistent atmospheric states may play a role in these anomalous events, particularly their influence on Antarctic sea ice retreat in the warmer months, by studying coupled atmosphere-sea ice variability during the 2016–2017, 2021–2022, and 2023 low sea ice concentration years. We construct a reduced-order model from reanalyzed observations based on a well-developed machine learning algorithm incorporating coupling across subsystems, namely the atmosphere and sea ice. Background persistent events occurring throughout the years of interest are then extracted via non-stationary transition matrix methods to the resultant temporal sequence of states. These events are analyzed by considering the associated surface pressure, winds, temperature, and sea ice concentration. The results show that persistent patterns in the atmosphere covary with the rate and spatial patterns of sea ice growth and retreat, noting that some periods of retreat coincide with warm surface temperatures and relatively quiescent synoptic winds. Our non-stationary approach provides additional insight over and above what can be inferred from simple monthly or seasonal averages alone, particularly in capturing events across varying temporal scales.
The reliability of physics-based landslide runout models depends on the effective calibration of their parameters, which are often conceptual and cannot be physically measured. Bayesian methods offer a robust framework to incorporate uncertainties in both model and observations into the calibration process. Therefore, they are increasingly used to calibrate physics-based landslide runout models. However, the reliability of Bayesian calibration in its practical application to real-world landslide events depends on the availability and quality of observational data – from aggregated post-event measurements such as impact area to time-resolved data such as force time histories. Despite this, systematic investigation of the influence of observational data on the Bayesian calibration of landslide runout models has been limited.We propose quantifying the impact of observational data on calibration outcomes by measuring the information gained during the calibration process using an information-theoretic measure called Kullback-Leibler (KL) divergence. Building on this, we present a unified Bayesian data selection workflow to identify the most informative dataset for calibrating a given parameter. The workflow runs parallel calibration routines across available observation datasets. It then computes the information gained relative to the observations by calculating the KL divergence between prior and posterior distributions and selects the dataset that yields the highest KL divergence.We demonstrate our workflow using an elementary landslide runout model, calibrating friction parameters with a diverse set of synthetic observations to evaluate the impact of data selection on parameter calibration. Specifically, we compare and quantify the information gained from calibration routines using observations that differ in information content (runout distance versus maximum velocity), granularity (aggregated versus time series data), and temporal characteristics (resolution and length of the velocity time series). The insights from this study will optimize the use of available observations for calibration and guide the design of effective data acquisition strategies.
In supervised data assimilation machine learning emulation, the training data contain targets produced by an existing data assimilation scheme, such as analysis increments. By contrast, data assimilation networksIn this study, we extend the investigation of such learned analysis operators beyond the preliminary experiments reported so far. First, we analyse the emergence of local patterns encoded in the operator, which accounts for the remarkable scalability of the approach to high-dimensional state spaces. Second, we assess the performance of the learned operators in stronger nonlinear regimes of the chaotic dynamics. We show that they can match the efficiency of the iterative ensemble Kalman filter, the baseline in this context, while avoiding the need for nonlinear iterative optimisation. Throughout the paper, we seek underlying reasons for the efficiency of the approach, drawing on insights from both machine learning and nonlinear data assimilation.
The risk that the climate system may contain tipping points remains a concern. Firstly, a level of global warming may be reached at which relatively small additional warming could cause major parts of the Earth system to transition to a new state. Depending on the location and specific Earth system component, this could disproportionately impact large sectors of society. Secondly, the Earth system component may exhibit hysteresis effects, and therefore, if global temperatures are subsequently lowered after triggering a jump in state, a return to earlier conditions may not occur until warming is substantially reduced. Earth System Models (ESMs) are numerical frameworks that operate at fine spatial scales. Such models are designed to estimate how all components of the climate system will evolve in response to changes in atmospheric greenhouse gas concentrations caused by human activity. Many ESM projections suggest that various parts of the climate system are capable of tipping. Yet, these models are computationally demanding and have therefore been operated only over a small range of scenarios. Very few “overshoot” simulations with ESMs exist, where climate change is reversed, resulting in limited understanding of hysteresis effects following a tipping event.Advances in nonlinear mathematics include developing dynamical systems whose equations frequently contain a bifurcation parameter. These equations can accurately replicate tipping points, jumps in state, and hysteresis as the bifurcation parameter changes. Recent progress in ESM development often introduces higher spatial resolution, enhancing process representation and increasing accuracy in predicting local changes. However, mapping the broad behaviour of the components of ESM projections onto simpler dynamical system models may also offer many advantages, including the overall characterisation of climate models and a method to rapidly extrapolate their projections to a wider range of forcing scenarios. The bifurcation parameters in such equations may represent changing forcings, such as an increasing warming level that leads to a tipping event. Progress has already been made in mapping many components of the Earth system onto large-scale variables for representation as dynamical systems. Most advances to date have focused on understanding whether tipping events can be avoided if systems possess substantial inertia, i.e. respond over long timescales, allowing climate change to temporarily exceed thresholds that might otherwise trigger major nonlinear change. However, potential hysteresis effects in the context of climate change are less well represented in equation form. Achieving such a mathematical formulation requires a dynamic system to describe a climate system component not only at the point of tipping but also for substantial periods before and after. This behaviour corresponds to a bifurcation parameter that first increases and then decreases, with the modelled behaviours differing significantly during the return phase.To support such necessary developments, we present a parameter-sparse dynamical system model that can exhibit hysteresis following a tipping occurrence, offering the potential for characterising Earth system components with this feature. We place particular emphasis on presenting in full the algebra needed to map known or modelled key attributes of a system that can tip onto the simplified dynamical system equation. We drive the equation with a time-evolving forcing representing an “overshoot” trajectory of global warming that exceeds a threshold for potential tipping. Calculations are performed over a range of system inertia values, illustrating a threshold inertia above which full tipping and hysteresis can be avoided. We use scale analysis to relate this threshold to those reported in existing climate research on behaviour near potential tipping points.In some instances, a basic equation may be too simple to capture the dominant qualitative behaviour of a large-scale environmental system. In those cases, a more complex model from any hierarchy of suggested simulations of climate components may be appropriate. However, should a system be amenable to representation by a dynamical system, we hope the framework we present offers a simple-to-use equation structure that may quantify tipping points in the climate system. Such a framework may illustrate a more complete description of behaviours both before and after tipping, and depending on the post-tipping forcing scenario, it may involve substantial hysteresis.
Westerly wind bursts (WWBs) have long been known to have a major impact on the development of El Niño events. In particular, they amplify these events, with stronger events associated with a higher number and stronger WWBs. We consider here a noise-driven recharge oscillator model of ENSO. Commonly, WWBs are represented by a state-dependent Gaussian noise that naturally reproduces the amplification of warm events. However, we show that many properties of WWBs and their effects on sea surface temperature (SST) are better captured by a conditional additive and multiplicative (CAM) noise, which presents a promising alternative to represent WWBs. In addition to recovering the sporadic nature of WWBs, CAM noise leads to an asymmetry between El Niño and La Niña events without the need for deterministic nonlinearities. Furthermore, CAM noise generates SST dynamics with a higher frequency of WWBs accompanying the largest events. This suggests that extreme warm events are better modelled by CAM noise. To cover the full spectrum of warm events, we propose a conditional noise model in which the wind stress is modelled by additive Gaussian noise for sufficiently small SSTs and by additive CAM noise once the SST exceeds a certain threshold. We show that this conditional noise model captures observed bulk statistical properties of ENSO equally well as the commonly used multiplicative Gaussian red noise model, but additionally better reproduces dynamical signatures such as the increased number of WWBs preceding large El Niño events.
Operational smoke plume assimilation systems require fast, yet accurate error estimations to represent the large, case-dependent and spatio-temporally varying uncertainties. This study exploits parametric uncertainty forecasts (PUF) as a computationally efficient alternative to existing ensemble approaches, where the dynamics of error parameters (such as error standard deviations) are explicitly evolved numerically at a fraction of the cost of ensemble-based methods. It focuses on (1) the theoretical derivation of forecast dynamics tailored to near-surface air quality applications with uncertain emissions, and (2) the implementation of PM2.52.52.5
The ensemble Kalman filter (EnKF) is widely used for state estimation in chaotic dynamical systems, including atmospheric and oceanic flows. One of the fundamental questions is how many samples are required for accurate long-term performance of the EnKF. In this study, we introduce a notion of time-asymptotic filter accuracy based on the scaling of the analysis error with respect to the observation noise level. This formulation provides a qualitative distinction between convergent and divergent filtering behavior, beyond standard criteria based on time-averaged RMSE at a fixed noise level. We investigate the minimum ensemble size m*N+. Because spanning the unstable directions by a limited ensemble is essential for long-term accuracy, we propose an ensemble spin-up and downsizing strategy. Numerical experiments with the EnKF applied to the Lorenz 96 model indicate that the minimum ensemble size required for this filter accuracy satisfies
Causal inference methods present a statistical approach to the analysis and reconstruction of dynamic systems as observed in nature or in experiments. Climate tipping points are likely present in several core components of the Earth system, such as the Greenland ice sheet or the Atlantic Meridional Overturning Circulation (AMOC), and are characterized by an abrupt and irreversible degradation under sustained global temperatures above their corresponding thresholds. Causal inference methods may provide a promising way to study the interactions of climate tipping elements, which are currently highly uncertain due to limitations in model-based approaches. However, the data-driven analysis of climate tipping elements presents several challenges, e.g., with regard to nonlinearity, delayed effects and confoundedness. In this study, we quantify the accuracy of three commonly used multivariate causal inference methods with regard to these challenges and find unique advantages of each method: The Liang–Kleeman Information Flow (LKIF) is preferable in simple settings with limited data availability, the Peter–Clark Momentary Conditional Independence (PCMCI) provides the most control, e.g., to integrate expert knowledge, and the Granger Causality for State Space Models is advantageous for large datasets and delayed interactions. In general, data sampling intervals should be aligned with the interaction delays, and the inclusion of a confounder (like global temperatures) is crucial to deal with the nonlinear response to (climate) forcing. Based on these findings and given their data masking capabilities, we apply the LKIF and PCMCI methods to reanalysis data to detect tipping point interactions between the AMOC and Arctic summer sea ice, which imply a bidirectional stabilizing interaction, in agreement with physical mechanisms. Our results therefore contribute robust evidence to the study of interactions of the AMOC and the cryosphere.
This study aims to quantify the relationship between the duration and occurrence frequency of droughts in China, particularly focusing on different drought intensities. By analyzing daily meteorological drought composite index (MCI) data from 1897 meteorological stations across China spanning from 1961 to 2020, the study reveals a significant double-logarithmic relationship between drought duration and occurrence frequency. The results show that shorter drought durations are associated with higher occurrence frequencies, while longer durations correspond to lower frequencies. This relationship is characterized by parameter kb. Spatially, the values of the parameter exhibit a gradient from northwest to southeast, with higher values in arid and semi-arid regions and lower values in humid and semi-humid regions. Notably, the parameter k
It is widely acknowledged that the joint inversion of magnetotelluric and seismological datasets enhances the quality of the crustal structure solution, even when the physical correlation between electrical resistivity and seismic velocity is weak or indirect. The structurally coupled joint inversion approach has received considerable attention over the past two decades for its ability to estimate such parameters by penalizing their cross-gradient vectors at similar spatial positions. Despite this interest, various structural couplings and different physical directions (incremental or decremental) have been partially overlooked. We hereby propose an approach for the joint inversion of magnetotelluric (MT) and Rayleigh wave dispersion (RWD) data to estimate uncorrelated parameters by integrating particle swarm optimization (PSO) and the Pareto optimality approach. We used the optimality framework of these methods to overcome the difficulties associated with traditional joint inversion algorithms and to obtain optimal solutions that account for both similar and contrasting physical sensitivities. The significant correlation between the inverted and synthetic models under both noise-free and noisy datasets, together with the consistent results obtained from comparison with a traditional derivative-based joint inversion algorithm, further strengthened our confidence in applying the proposed modeling approach to the field data from the southeastern Biga Peninsula, western Anatolia. The models inverted from the field data corroborate the efficacy of the presented method. A notable characteristic of the proposed methodology is its capacity to estimate uncorrelated physical parameters, such as electrical resistivity and seismic velocity, without the imposition of penalties. Therefore, the presented method not only offers advantages in joint inversion but also allows modelers to observe and analyze model parameters having different sensitivities that may indicate different physical directions.
rare event samplingensemble boosting: lightly perturbing simulated moderate events into more extreme ones. We formulate a new, flexible sampling strategy and characterizes a critical parameter – the
advance split time, dictating when to perturb – in a simple atmospheric turbulence model, with generalizable entropy-based criteria.
Climate science needs more efficient ways to study high-impact, low-probability extreme events, which are rare by definition and costly to simulate in large numbers. Rare event sampling (RES), including ensemble boosting, offers a novel strategy to extract more information from those occasional simulated events, by applying small perturbations to turn a moderate event into a severe one which otherwise might not come for many more simulation-years. But how severe the events can become, and their estimated probabilities, depend sensitively on the details of the perturbation. In particular, for sudden and transient events like precipitation, performance of boosting depends sensitively on the choice of advance split time (AST) of the perturbation. Heuristically, the perturbation must come early enough before the event to let the ensemble of simulations diversify, but not so early that they forget the special initial conditions that led to the extreme. In pursuit of guidelines for choosing the AST, we study the effect of AST in the task of sampling extreme fluctuations of a passive tracer in a quasigeostrophic turbulent channel flow. This model system is idealized, but captures key elements of midlatitude storm track dynamics while exposing similar algorithmic challenges. We formulate AST selection as a concrete optimization problem for statistical accuracy against a ground truth. Given that such a ground truth would not generally be available, we propose a proxy objective function to optimize in practice: thresholded entropy, which rewards ensembles with both a high mean and a large spread. We show that ensemble boosting, when given a well-chosen AST and equipped with methods to estimate probabilities, can accurately sample extremes at long return periods. We furthermore find evidence that thresholded entropy successfully identifies an optimal AST, which is roughly 1–3  ddy turnover timescales in the quasigeostrophic system. Moreover, this proxy captures the variation
Sandy shoreline–sandbar systems exhibit complex variability arising from the interplay between hydrodynamic forcing and intrinsic morphological feedbacks. Using long-term satellite-derived shoreline and sandbar observations, we applied global polynomial modeling to reconstruct low-dimensional deterministic dynamics for 4 contrasting coastal sites. The resulting autonomous models reproduce key morphodynamic features, including self-sustained shoreline oscillations, shoreline–sandbar coupling, and intermittent transitions between quasi-stable configurations. Nonlinear stability analyses reveal that these systems behave as chaotic oscillators, characterized by locally divergent yet globally bounded trajectories. Energetic episodes correspond to rapid shoreline–sandbar exchanges, whereas long low-energy states reflect stable attractor confinement. Together, these results demonstrate that sandy coasts are governed by deterministic but chaotic dynamics, in which internal coupling and self-organization control both variability and finite predictability. The proposed framework offers a physically consistent and data-driven approach to characterize and compare coastal morphodynamics within a unified nonlinear dynamical perspective.
Seismic inversion for estimating elastic properties is a key technique for reservoir characterization after drilling. The choice of inversion method strongly influences the accuracy, efficiency, and reliability of results. Bayesian inference based on Markov Chain Monte Carlo (MCMC) algorithms provides a robust framework for incorporating data uncertainty and prior geological knowledge. In this study, we compare the performance of four inversion methods – Metropolis-Hastings (MH), Hamiltonian Monte Carlo (HMC), the Metropolis-Adjusted Langevin Algorithm (MALA), and its variant Lip-MALA – in prestack seismic inversion using both synthetic models and real data from an eastern Venezuelan hydrocarbon reservoir. Results indicate that gradient-based methods (HMC, MALA, Lip-MALA) outperform MH in velocity estimation, while density inversion remains more challenging. MH and MALA achieve shorter execution times, whereas HMC and Lip-MALA improve accuracy at higher computational cost. This analysis evaluates mean values and standard deviation (SD) estimates for P-wave velocity, S-wave velocity, and density, with quality assessed through correlation metrics, objective function behavior, seismic traces, and Root Mean Square Error (RMSE). A two-dimensional inversion with real data further demonstrates algorithms performance under complex geological conditions. The findings highlight trade-offs between accuracy and efficiency, providing practical guidelines for selecting inversion method in seismic reservoir characterization.
This study examines whether the predictability of precipitation dynamics in Serbia has been influenced by climate change. We apply Generalized Weighted Permutation Entropy (GWPE) to evaluate the temporal structure of daily precipitation series using the parameter q, which filters subsets of small (q<0) and large (q>0) fluctuations. The analysis covers data from 14 weather stations between 1961 and 2020. Entropy values for q=0q=2, corresponding to Permutation Entropy and Weighted Permutation Entropy respectively, remained stable spatially and temporally. In contrast, GWPE values for
Accurate forecasting of aftershock distributions is vital for effective post-earthquake emergency response, early warning systems, and long-term seismic hazard mitigation. This study introduces a novel nonlinear, multiscale framework for modeling the evolution of Coulomb stress following a major earthquake. The proposed approach integrates rate-and-state friction laws, a KPP-type reaction–diffusion equation, and the Banach fixed-point theorem to simulate the dynamic redistribution of stress in space and time. Central to the model are two time-dependent parameters – α(t), which governs the decay of stress memory consistent with Omori's law, and β(t), which modulates the nonlinear diffusion and reaction dynamics. Applied to the 2018 Hualien earthquake in Taiwan, the framework resolves stress changes and their gradients at depths of 6–25 km. Results indicate that stress gradients are more predictive of aftershock occurrences within the first 50 d and at depths shallower than 12 km, while stress changes play a dominant role at greater depths and later times. Validation using AUC and Molchan error metrics demonstrates the model's strong spatial forecasting capability. The framework's adaptive convergence and modular structure support real-time seismic hazard assessment and integration into PSHA workflows, offering a promising tool for aftershock modeling and disaster resilience planning.
Regime dependencies and Granger causal relationships between tropical and extratropical teleconnections are inferred using Bayesian structure learning. Using ERA5 data, an examination of the differences between the learned graphical structures during particular phases of the Interdecadal Pacific Oscillation (IPO) are used to infer the role of the background state on interactions between the major climate teleconnections. These relationships present a clear regime dependency on the phase of IPO. In the positive phase, IPO autocorrelations are weak whereas Indian Ocean Dipole (IOD) and El Niño Southern Oscillation (ENSO) autocorrelations and the influence of the Madden Julian Oscillation (MJO) are indicative of an enhanced Walker circulation. In contrast, during the negative phase, IPO autocorrelations are strongest with evidence of an enhanced role for extratropical teleconnections on the tropics. Exclusion of MJO removes important tropical-extratropical influences while increasing posterior edge weights between ENSO, the IPO and IOD. Our analysis reveals the dependence of the ENSO autocorrelation on the phase of the background IPO state, and the role of the MJO as being key to link the extratropical tropospheric modes Pacific North American and North Atlantic Oscillation (PNA, NAO) and equatorial surface ocean temperatures (IOD, ENSO) and as a consequence convection.
Regional Climate Model (RCM) emulators enable rapid and computationally efficient RCM projections given Global Climate Model (GCM) inputs, complementing dynamical downscaling by approximating physical representations with statistical models. However, while existing RCM emulators perform well in deterministic emulations, they do not sample internal RCM variability and remain computationally expensive. Here, we present MESMER-RCM, a probabilistic RCM emulator designed for spatially resolved annual 2 m temperature. MESMER-RCM is a generative model that enables both data-efficient learning and interpretability. It can generate large ensembles of synthetic, yet physically plausible, RCM realizations, capturing the internal RCM variability at a fraction of the computational cost. This work offers a fast and reliable RCM emulation framework, supporting finer-scale what-if analyses of regional climate responses and informing local adaptation and mitigation strategies.
Most data-driven methods, among them Dynamic Mode Decomposition (DMD), focus on analysing and reconstructing the average behaviour of a system. However, the primary interest often lies in the anomalous behaviour, known as extreme events. This is especially the case in climate research, where extreme events have significant economic and societal costs. Therefore, we extend a DMD method to account for extreme events by adding a penalisation term. This extension allows us to not only better reconstruct the extreme events, but also extract the spatiotemporal structures related to those extreme events. DMD was originally developed by Schmid and Sesterhenn (Schmid and Sesterhenn, 2008)modes) from high-dimensional data. In its essence DMD uses most relevant modes to filter the noise and reconstruct the original signal. We ask “Is the noise really noise”! Or can we attribute some of these dynamic modes, that result from the DMD, to extreme events? We applied this new method to the climate system, well known for its high-dimensionality. As a proof of concept, we applied the method to two well-studied European heatwaves: those of 2003 and 2010. Across both cases, our extreme%–0.85 %
We examine the local stable and unstable manifolds of chaotic attractors and their associated growth rates for the quantification of (non-)hyperbolicity in low dimensional nonlinear autonomous dissipative models. This is motivated by a desire for a deeper understanding of transversality and hyperbolicity to inform key challenges to prediction in spatially extended chaotic systems in geophysical flows. In particular, we apply local measures of chaos to quantify the relationship between transversality, dimension, and hyperbolicity on the subspaces of the attractors' invariant manifolds. We consider unstable directions and growth rates determined over finite time intervals, specifically those predicated on information over the past evolution i.e., finite time backwards Lyapunov vectors, and those that include information from both the past and future i.e., finite time covariant Lyapunov vectors. Our study reveals general properties across a diverse set of chaotic attractors at short, intermediate and extended forecast horizons associated with the emergence of distinct locally evolving regions of instability.